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2024-10-07
44
Makefile
44
Makefile
@ -1,44 +0,0 @@
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# Makefile for lualatex
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# Paths and filenames
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SRC_DIR = src
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OUT_DIR = out
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MAIN_TEX = main.tex # in SRC_DIR
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MAIN_PDF = main.pdf # in OUT_DIR
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# LaTeX and Biber commands
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LATEX = lualatex
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BIBER = biber
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LATEX_OPTS := -output-directory=$(OUT_DIR) -interaction=nonstopmode -shell-escape
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.PHONY: default release clean scripts
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default: english
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release: german english
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# Default target
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english:
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sed -r -i 's/usepackage\[[^]]+\]\{babel\}/usepackage[english]{babel}/' $(SRC_DIR)/$(MAIN_TEX)
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-cd $(SRC_DIR) && latexmk -lualatex -g main.tex
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mv $(OUT_DIR)/$(MAIN_PDF) $(OUT_DIR)/$(shell date -I)_en_Formulary.pdf
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german:
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sed -r -i 's/usepackage\[[^]]+\]\{babel\}/usepackage[german]{babel}/' $(SRC_DIR)/$(MAIN_TEX)
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-cd $(SRC_DIR) && latexmk -lualatex -g main.tex
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mv $(OUT_DIR)/$(MAIN_PDF) $(OUT_DIR)/$(shell date -I)_de_Formelsammlung.pdf
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SCRIPT_DIR = scripts
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PY_SCRIPTS = $(wildcard $(SCRIPT_DIR)/*.py)
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PY_SCRIPTS_REL = $(notdir $(PY_SCRIPTS))
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scripts:
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-#cd $(SCRIPT_DIR) && for file in $(find -type f -name '*.py'); do echo "Running $$file"; python3 "$$file"; done
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cd $(SCRIPT_DIR) && $(foreach script,$(PY_SCRIPTS_REL),echo "Running $(script)"; python3 $(script);)
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# Clean auxiliary and output files
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clean:
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rm -r $(OUT_DIR)
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# Phony targets
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.PHONY: all clean biber
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113
readme.md
113
readme.md
@ -1,113 +0,0 @@
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# Formulary
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This is supposed to be a compact, searchable collection of the most important stuff I learned during my physics studides,
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because it would be a shame if I forget it all!
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## Building the PDF
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### Dependencies
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Any recent **TeX Live** distribution should work. You need:
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- `LuaLaTeX` compiler
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- several packages from ICAN
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- `latexmk` to build it
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### With GNU make
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1. In the project directory (where this `readme` is), run `make german` or `make english`.
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2. Rendered document will be `out/<date>_<formulary>.pdf`
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### With Latexmk
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1. Choose the language: In `main.tex`, set the language in `\usepackage[english]{babel}` to either `english` or `german`
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2. In the `src` directory, run `latexmk -lualatex main.tex`
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3. Rendered document will be `out/main.pdf`
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### With LuaLatex
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1. Choose the language: In `main.tex`, set the language in `\usepackage[english]{babel}` to either `english` or `german`
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2. Create the `.aux` directory
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3. In the `src` directory, run
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- `lualatex -output-directory="../.aux" --interaction=nonstopmode --shell-escape "main.tex"` **3 times**
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4. Rendered document will be `.aux/main.pdf`
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# LaTeX Guideline
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Here is some info to help myself remember why I did things the way I did.
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In general, most content should be written with macros, so that the behaviour can be changed later.
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## Structure
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All translation keys and LaTeX labels should use a structured approach:
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`<key type>:<partname>:<section name>:<subsection name>:<...>:<name>`
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The `<partname>:...:<lowest section name>` will be defined as `\fqname` (fully qualified name) when using the `\Part`, `\Section`, ... commands.
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`<key type>` is:
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- formula: `f`
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- equation: `eq`
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- table: `tab`
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- figure: `fig`
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- parts, (sub)sections: `sec`
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Use `misc` as (sub(sub))section for anything that can not be categorized within its (sub)section/part.
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### Files and directories
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Separate parts in different source files named `<partname>.tex`.
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If a part should be split up in multiple source files itself, use a
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subdirectory named `<partname>` containing `<partname>.tex` and other source files for sections.
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This way, the `fqname` of a section or formula partially matches the path of the source file it is in.
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## `formula` environment
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The main way to display something is the formula environment:
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```tex
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\begin{formula}{<key>}
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\desc{English name}{English description}{$q$ is some variable, $s$ \qtyRef{some_quantity}}
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\desc[german]{Deutscher Name}{Deutsche Beschreibung}{$q$ ist eine Variable, $s$ \qtyRef{some_quantity}}
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<content>
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\end{formula}
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```
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Each formula automatically gets a `f:<section names...>:<key>` label.
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For the content, several macros are available:
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- `\eq{<equation>}` a wrapper for the `align` environment
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- `\fig[width]{<path>}`
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- `\quantity{<symbol>}{<units>}{<vector, scalar, extensive etc.>}` for physical quantites
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- `\constant{<symbol>}{ <values> }` for constants, where `<values>` may contain one or more `\val{value}{unit}` commands.
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### References
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**Use references where ever possible.**
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In equations, reference or explain every variable. Several referencing commands are available for easy referencing:
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- `\fqSecRef{<fqname of section>}` prints the translated title of the section
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- `\fqEqRef{<fqname of formula>}` prints the translated title of the formula (first argument of `\desc`)
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- `\qtyRef{<key>}` prints the translated name of the quantity
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- `\QtyRef{<key>}` prints the symbol and the translated name of the quantity
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- `\constRef{<key>}` prints the translated name of the constant
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- `\ConstRef{<key>}` prints the symbol and the translated name of the constant
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- `\elRef{<symbol>}` prints the symbol of the chemical element
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- `\ElRef{<symbol>}` prints the name of the chemical element
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## Multilanguage
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All text should be defined as a translation (`translations` package, see `util/translation.tex`).
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Use `\dt` or `\DT` or the the shorthand language commands `\eng`, `\Eng` etc. to define translations.
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These commands also be write the translations to an auxiliary file, which is read after the document begins.
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This means (on subsequent compilations) that the translation can be resolved before they are defined.
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Use the `gt` or `GT` macros to retrieve translations.
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The english translation of any key must be defined, because it will also be used as fallback.
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Lower case macros are relative to the current `fqname`, while upper case macros are absolute.
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Never make a macro that would have to be changed if a new language was added, eg dont do
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```tex
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% 1: key, 2: english version, 3: german version
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\newcommand{\mycmd}[3]{
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\dosomestuff{english}{#1}{#2}
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\dosomestuff{german}{#1}{#3}
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}
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\mycmd{key}{this is english}{das ist deutsch}
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```
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Instead, do
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```tex
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% [1]: lang, 2: key, 2: text
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\newcommand{\mycmd}[3][english]{
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\dosomestuff{#1}{#2}{#3}
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}
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\mycmd{key}{this is english}
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\mycmd[german]{key}{das ist deutsch}
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```
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@ -1,39 +0,0 @@
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import ase.io as io
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from ase.build import cut
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from ase.spacegroup import crystal
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a = 9.04
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skutterudite = crystal(('Co', 'Sb'),
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basis=[(0.25, 0.25, 0.25), (0.0, 0.335, 0.158)],
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spacegroup=204,
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cellpar=[a, a, a, 90, 90, 90])
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# Create a new atoms instance with Co at origo including all atoms on the
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# surface of the unit cell
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cosb3 = cut(skutterudite, origo=(0.25, 0.25, 0.25), extend=1.01)
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# Define the atomic bonds to show
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bondatoms = []
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symbols = cosb3.get_chemical_symbols()
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for i in range(len(cosb3)):
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for j in range(i):
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if (symbols[i] == symbols[j] == 'Co' and
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cosb3.get_distance(i, j) < 4.53):
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bondatoms.append((i, j))
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elif (symbols[i] == symbols[j] == 'Sb' and
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cosb3.get_distance(i, j) < 2.99):
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bondatoms.append((i, j))
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# Create nice-looking image using povray
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renderer = io.write('spacegroup-cosb3.pov', cosb3,
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rotation='90y',
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radii=0.4,
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povray_settings=dict(transparent=False,
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camera_type='perspective',
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canvas_width=320,
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bondlinewidth=0.07,
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bondatoms=bondatoms))
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renderer.render()
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@ -1,50 +0,0 @@
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# creates: bztable.rst
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# creates: 00.CUB.svg 01.FCC.svg 02.BCC.svg 03.TET.svg 04.BCT1.svg
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# creates: 05.BCT2.svg 06.ORC.svg 07.ORCF1.svg 08.ORCF2.svg 09.ORCF3.svg
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# creates: 10.ORCI.svg 11.ORCC.svg 12.HEX.svg 13.RHL1.svg 14.RHL2.svg
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# creates: 15.MCL.svg 16.MCLC1.svg 17.MCLC3.svg 18.MCLC5.svg 19.TRI1a.svg
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# creates: 20.TRI1b.svg 21.TRI2a.svg 22.TRI2b.svg
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# creates: 23.OBL.svg 24.RECT.svg 25.CRECT.svg 26.HEX2D.svg 27.SQR.svg
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# taken from https://wiki.fysik.dtu.dk/ase/gallery/gallery.html
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from formulary import *
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from ase.lattice import all_variants
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from ase.data import colors
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header = """\
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Brillouin zone data
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-------------------
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.. list-table::
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:widths: 10 15 45
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"""
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entry = """\
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* - {name} ({longname})
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- {bandpath}
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- .. image:: {fname}
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:width: 40 %
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"""
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with open('bztable.rst', 'w') as fd:
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print(header, file=fd)
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for i, lat in enumerate(all_variants()):
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id = f'{i:02d}.{lat.variant}'
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imagefname = f'out/{id}.svg'
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txt = entry.format(name=lat.variant,
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longname=lat.longname,
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bandpath=lat.bandpath().path,
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fname=imagefname)
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print(txt, file=fd)
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ax = lat.plot_bz()
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fig = ax.get_figure()
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fig.savefig(imagefname, bbox_inches='tight')
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fig.clear()
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@ -1,193 +0,0 @@
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#!/usr/bin env python3
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from formulary import *
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from scipy.constants import gas_constant, Avogadro, elementary_charge
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Faraday = Avogadro * elementary_charge
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# BUTLER VOLMER / TAFEL
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@np.vectorize
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def fbutler_volmer_anode(ac, z, eta, T):
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return np.exp((1-ac)*z*Faraday*eta/(gas_constant*T))
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@np.vectorize
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def fbutler_volmer_cathode(ac, z, eta, T):
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return -np.exp(-ac*z*Faraday*eta/(gas_constant*T))
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def fbutler_volmer(ac, z, eta, T):
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return fbutler_volmer_anode(ac, z, eta, T) + fbutler_volmer_cathode(ac, z, eta, T)
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def butler_volmer():
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fig, ax = plt.subplots(figsize=size_formula_fill_default)
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ax.set_xlabel("$\\eta$ [V]")
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ax.set_ylabel("$j/j_0$")
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etas = np.linspace(-0.1, 0.1, 400)
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T = 300
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z = 1.0
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# other a
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ac2, ac3 = 0.2, 0.8
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i2 = fbutler_volmer(0.2, z, etas, T)
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i3 = fbutler_volmer(0.8, z, etas, T)
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ax.plot(etas, i2, color="blue", linestyle="dashed", label=f"$\\alpha_\\text{{C}}={ac2}$")
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ax.plot(etas, i3, color="green", linestyle="dashed", label=f"$\\alpha_\\text{{C}}={ac3}$")
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# 0.5
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ac = 0.5
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irel_anode = fbutler_volmer_anode(ac, z, etas, T)
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irel_cathode = fbutler_volmer_cathode(ac, z, etas, T)
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ax.plot(etas, irel_anode, color="gray")
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ax.plot(etas, irel_cathode, color="gray")
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ax.plot(etas, irel_cathode + irel_anode, color="black", label=f"$\\alpha_\\text{{C}}=0.5$")
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ax.grid()
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ax.legend()
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ylim = 6
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ax.set_ylim(-ylim, ylim)
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return fig
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@np.vectorize
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def ftafel_anode(ac, z, eta, T):
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return 10**((1-ac)*z*Faraday*eta/(gas_constant*T*np.log(10)))
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@np.vectorize
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def ftafel_cathode(ac, z, eta, T):
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return -10**(-ac*z*Faraday*eta/(gas_constant*T*np.log(10)))
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def tafel():
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i0 = 1
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ac = 0.2
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z = 1
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T = 300
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eta_max = 0.2
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etas = np.linspace(-eta_max, eta_max, 400)
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i = np.abs(fbutler_volmer(ac, z, etas ,T))
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iright = i0 * np.abs(ftafel_cathode(ac, z, etas, T))
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ileft = i0 * ftafel_anode(ac, z, etas, T)
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fig, ax = plt.subplots(figsize=size_formula_normal_default)
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ax.set_xlabel("$\\eta$ [V]")
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ax.set_ylabel("$\\log_{10}\\left(\\frac{|j|}{j_0}\\right)$")
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# ax.set_ylabel("$\\log_{10}\\left(|j|/j_0\\right)$")
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ax.set_yscale("log")
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# ax.plot(etas, linear, label="Tafel slope")
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ax.plot(etas[etas >= 0], ileft[etas >= 0], linestyle="dashed", color="gray", label="Tafel Approximation")
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ax.plot(etas[etas <= 0], iright[etas <= 0], linestyle="dashed", color="gray")
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ax.plot(etas, i, label=f"Butler-Volmer $\\alpha_\\text{{C}}={ac:.1f}$")
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ax.legend()
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ax.grid()
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return fig
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# NYQUIST
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@np.vectorize
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def fZ_ohm(R, omega):
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return R
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@np.vectorize
|
||||
def fZ_cap(C, omega):
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return 1/(1j*omega*C)
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||||
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||||
@np.vectorize
|
||||
def fZ_ind(L, omega):
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return (1j*omega*L)
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||||
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||||
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||||
def nyquist():
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||||
fig, ax = plt.subplots(figsize=size_formula_fill_default)
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||||
split_z = lambda Z: (Z.real, -Z.imag)
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||||
ax.grid()
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||||
ax.set_xlabel("$\\text{Re}(Z)$ [\\si{\\ohm}]")
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||||
ax.set_ylabel("$-\\text{Im}(Z)$ [\\si{\\ohm}]")
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||||
# ax.scatter(*split_z(Z_series), label="series")
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||||
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||||
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||||
R1 = 20
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||||
R2 = 5
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||||
RS = 7.5
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||||
C1 = 1e-4
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||||
C2 = 1e-6
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||||
ws1 = np.power(10, np.linspace(1, 8, 1000))
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||||
|
||||
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||||
Z_ohm1 = fZ_ohm(R1, ws1)
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||||
Z_ohm2 = fZ_ohm(R2, ws1)
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||||
Z_ohmS = fZ_ohm(RS, ws1)
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||||
Z_cap1 = fZ_cap(C1, ws1)
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||||
Z_cap2 = fZ_cap(C2, ws1)
|
||||
Z_parallel1 = 1/(1/Z_ohm1 + 1/Z_cap1)
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||||
Z_parallel2 = 1/(1/Z_ohm2 + 1/Z_cap2)
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||||
Z_cell = Z_parallel1 + Z_parallel2 + Z_ohmS
|
||||
ax.scatter(*split_z(Z_parallel1), label="Parallel $C_1,R_1$")
|
||||
ax.scatter(*split_z(Z_parallel2), label="Parallel $C_2,R_2$")
|
||||
ax.scatter(*split_z(Z_cell), label="P1 + $R_3$ + P2")
|
||||
ax.scatter(*split_z(Z_cap1), label=f"$C_1=\\SI{{{C1:.0e}}}{{\\farad}}$")
|
||||
ax.scatter(*split_z(Z_ohm1), label=f"$R_1 = \\SI{{{R1}}}{{\\ohm}}$")
|
||||
|
||||
# wmax1 = 1/(R1 * C1)
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||||
# ZatWmax1 = Z_parallel1[np.argmin(ws1 - wmax1)]
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||||
# print(ws1[0], ws1[-1])
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||||
# print(wmax1, ZatWmax1)
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||||
# ax.scatter(*split_z(ZatWmax1), color="red")
|
||||
# ax.scatter(*split_z(Z_cell1), label="cell")
|
||||
# ax.scatter(*split_z(Z_ohm2), label="ohmic")
|
||||
# ax.scatter(*split_z(Z_cell2), label="cell")
|
||||
ax.axis('equal')
|
||||
ax.set_ylim(0,R1*1.1)
|
||||
ax.legend()
|
||||
return fig
|
||||
|
||||
def fZ_tlm(Rel, Rion, Rct, Cct, ws, N):
|
||||
Zion = fZ_ohm(Rion, ws)
|
||||
Zel = fZ_ohm(Rel, ws)
|
||||
Zct = 1/(1/fZ_ohm(Rct, ws) + 1/fZ_cap(Cct, ws))
|
||||
|
||||
Z = Zct
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||||
for _ in range(N):
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||||
Z = Zion + 1/(1/Zct + 1/Z)
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||||
Z += Zel
|
||||
return Z
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||||
|
||||
def nyquist_tlm():
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||||
fig, ax = plt.subplots(figsize=(width_formula, width_formula*0.5))
|
||||
split_z = lambda Z: (Z.real, -Z.imag)
|
||||
ax.grid()
|
||||
ax.set_xlabel("$\\text{Re}(Z)$ [\\si{\\ohm}]")
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||||
ax.set_ylabel("$-\\text{Im}(Z)$ [\\si{\\ohm}]")
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||||
Rct1 = 300
|
||||
Rct2 = 100
|
||||
Rion = 10
|
||||
ws = np.power(10, np.linspace(1e-6, 5, 1000))
|
||||
Z1 = fZ_tlm(0, Rion, Rct1, 1e-4, ws, 100)
|
||||
Z2 = fZ_tlm(0, Rion, Rct2, 1e-4, ws, 100)
|
||||
ax.scatter(*split_z(Z1), label=f"$R_\\text{{ct}} = \\SI{{{Rct1}}}{{\\ohm}}$", marker=".")
|
||||
ax.scatter(*split_z(Z2), label=f"$R_\\text{{ct}} = \\SI{{{Rct2}}}{{\\ohm}}$", marker=".")
|
||||
ax.axis('equal')
|
||||
# ax.set_ylim(0,R1*1.1)
|
||||
ax.legend()
|
||||
return fig
|
||||
|
||||
def fkohlrausch(L0, K, c):
|
||||
return L0 - K*np.sqrt(c)
|
||||
|
||||
def kohlrausch():
|
||||
fig, ax = plt.subplots(figsize=size_formula_small_quadratic)
|
||||
ax.grid()
|
||||
ax.set_xlabel("$c_\\text{salt}$")
|
||||
ax.set_ylabel("$\\Lambda_\\text{M}$")
|
||||
L0 = 10
|
||||
K1 = 1
|
||||
K2 = 2
|
||||
cs = np.linspace(0, 10)
|
||||
L1 = fkohlrausch(L0, K1, cs)
|
||||
L2 = fkohlrausch(L0, K2, cs)
|
||||
ax.plot(cs, L1, label=f"$K={K1}$")
|
||||
ax.plot(cs, L2, label=f"$K={K2}$")
|
||||
ax.legend()
|
||||
return fig
|
||||
|
||||
if __name__ == '__main__':
|
||||
export(butler_volmer(), "ch_butler_volmer")
|
||||
export(tafel(), "ch_tafel")
|
||||
export(nyquist(), "ch_nyquist")
|
||||
export(nyquist_tlm(), "ch_nyquist_tlm")
|
||||
export(kohlrausch(), "ch_kohlrausch")
|
||||
|
||||
|
||||
@ -1,51 +0,0 @@
|
||||
from formulary import *
|
||||
from util.aseutil import set_atom_color, get_pov_settings
|
||||
|
||||
"""
|
||||
Create crystal structures using ase and render them with povray
|
||||
|
||||
Rotation angle:
|
||||
To get the rotation angle, open the structure in the ase.visualize.view
|
||||
and use "View->Rotation" to get the desired angles
|
||||
"""
|
||||
|
||||
set_atom_color("Na", COLORSCHEME["fg-red"])
|
||||
set_atom_color("Cl", COLORSCHEME["fg-blue"])
|
||||
|
||||
set_atom_color("Zn", COLORSCHEME["fg-blue"])
|
||||
set_atom_color("S", COLORSCHEME["fg-yellow"])
|
||||
|
||||
from ase.lattice import compounds
|
||||
from ase.build import cut, bulk
|
||||
from ase import Atom, Atoms
|
||||
|
||||
|
||||
def zincblende():
|
||||
zns = compounds.Zincblende(("Zn", "S"), latticeconstant=5.0, size=(1,1,1))
|
||||
zns_cell = cut(zns, b=(0,0,1), origo=(0,0,0), extend=1.1)
|
||||
return zns_cell
|
||||
|
||||
# NaCl cut
|
||||
def nacl():
|
||||
nacl = compounds.NaCl(("Na", "Cl"), latticeconstant=5.0, size=(1,1,1))
|
||||
nacl_cell = cut(nacl, b=(0,0,1), origo=(0,0,0), extend=1.1)
|
||||
return nacl_cell
|
||||
|
||||
def wurtzite():
|
||||
compounds.L1_2
|
||||
wurtzite = bulk('SZn', 'wurtzite', a=3.129, c=5.017)
|
||||
wurtzite_cell = cut(wurtzite,
|
||||
a=[1, 0, 0],
|
||||
b=[-1, -1, 0],
|
||||
c=[0, 0, 1], extend=1.1)
|
||||
return wurtzite_cell
|
||||
|
||||
|
||||
if __name__ == "__main__":
|
||||
export_atoms(nacl(), "cm_crystal_NaCl", size_formula_half_quadratic)
|
||||
export_atoms(wurtzite(), "cm_crystal_wurtzite", size_formula_half_quadratic, rotation="70x,20y,174z")
|
||||
export_atoms(zincblende(), "cm_crystal_zincblende", size_formula_half_quadratic, rotation="-155x,70y,24z")
|
||||
# w = wurtzite()
|
||||
# from ase.visualize import view
|
||||
# view(w)
|
||||
|
||||
@ -1,105 +0,0 @@
|
||||
#!/usr/bin env python3
|
||||
from formulary import *
|
||||
from scipy.constants import Boltzmann as kB, hbar, electron_volt, elementary_charge, epsilon_0, electron_mass
|
||||
|
||||
# OPTICS
|
||||
def eps_r(omega, omega0, gamma, chi, N):
|
||||
return 1 + chi + N * elementary_charge**2 /(epsilon_0 * electron_mass) * (1/(omega0**2 - omega**2 - 1j*gamma*omega))
|
||||
|
||||
def eps_r_lim0(omega0, gamma, chi, N):
|
||||
return 1 + chi + N * elementary_charge**2 /(epsilon_0 * electron_mass * omega0**2)
|
||||
|
||||
def eps_r_lim_infty(omega0, gamma, chi, N):
|
||||
return 1 + chi
|
||||
|
||||
def dielectric_absorption():
|
||||
# values taken from adv. sc. physics ex 10/1
|
||||
fig, axs = plt.subplots(1, 2, figsize=size_formula_normal_default, sharex=True)
|
||||
omega0 = 100e12 # 100 THz
|
||||
gamma = 5e12 # 5 THz
|
||||
omegas = np.linspace(60e12, 140e12)
|
||||
chi = 5
|
||||
N = 1e25
|
||||
|
||||
eps_complex = eps_r(omegas, omega0, gamma, chi, N)
|
||||
eps_real = eps_complex.copy().real
|
||||
eps_imag = eps_complex.copy().imag
|
||||
eps_real_0 = eps_r_lim0(omega0, gamma, chi, N)
|
||||
eps_real_infty = eps_r_lim_infty(omega0, gamma, chi, N)
|
||||
|
||||
omegas_THz = omegas / 1e12
|
||||
anno_color = COLORSCHEME["fg2"]
|
||||
axs[0].set_ylabel(r"$\epsReal(\omega)$")
|
||||
axs[0].hlines([eps_real_0], omegas_THz[0], omegas_THz[omegas_THz.shape[0]//3], color=anno_color, linestyle="dotted")
|
||||
axs[0].hlines([eps_real_infty], omegas_THz[omegas_THz.shape[0]*2//3], omegas_THz[-1], color=anno_color, linestyle="dotted")
|
||||
axs[0].text(omegas_THz[-1], eps_real_infty, r"$\epsilon_\txr(\infty)$", ha="right", va="bottom", color=anno_color)
|
||||
axs[0].text(omegas_THz[0], eps_real_0, r"$\epsilon_\txr(0)$", ha="left", va="top", color=anno_color)
|
||||
axs[1].set_ylabel(r"$\epsImag(\omega)$")
|
||||
vals = [eps_real, eps_imag]
|
||||
for i in range(2):
|
||||
ax = axs[i]
|
||||
val = vals[i]
|
||||
ax.set_xlabel(r"$\omega\,[\si{\tera\hertz}]$")
|
||||
ax.vlines([omega0/1e12], 0, 100, color=anno_color, linestyle="dashed")
|
||||
ax.plot(omegas_THz, val)
|
||||
vmax = val.max()
|
||||
vmin = val.min()
|
||||
margin = (vmax - vmin) * 0.05
|
||||
ax.set_ylim(vmin - margin, vmax + margin)
|
||||
|
||||
return fig
|
||||
|
||||
# Free e-
|
||||
@np.vectorize
|
||||
def fn(omega: float, omega_p:float, eps_infty:float, tau:float):
|
||||
eps_real = eps_infty * (1-(omega_p**2 * tau**2)/(1+omega**2 * tau**2))
|
||||
eps_imag = eps_infty * omega_p**2 * tau/(omega*(1+omega**2 * tau**2))
|
||||
eps_complex = (eps_real + 1j*eps_imag)
|
||||
n_complex: complex = np.sqrt(eps_complex)
|
||||
# n_real = n_complex.real
|
||||
# n_imag = n_complex.imag
|
||||
# return n_real, n_imag
|
||||
return n_complex
|
||||
|
||||
def free_electrons_absorption():
|
||||
# values taken from adv. sc. physics ex 10/1
|
||||
fig, axs = plt.subplots(2, 2, figsize=size_formula_fill_default, sharex=True)
|
||||
omega_p = 30e12 # 30 THz
|
||||
eps_infty = 11.7
|
||||
tau = 1e-6
|
||||
omegas = omega_p * np.logspace(-1, 3, 1000, base=10, dtype=float)
|
||||
# omegas = omega_p * np.linspace(1e-1, 1e3, 1000)
|
||||
# print(omegas)
|
||||
n_complex = fn(omegas, omega_p, eps_infty, tau)
|
||||
n_real = n_complex.copy().real
|
||||
n_imag = n_complex.copy().imag
|
||||
R = np.abs((n_complex-1)/(n_complex+1))
|
||||
alpha = 2*omegas*n_imag/3e8
|
||||
for ax in axs[1,:]:
|
||||
ax.set_xlabel(r"$\omega/\omega_\text{p}$")
|
||||
ax.set_xscale("log")
|
||||
ax.set_xticks([1e-1,1e0,1e1,1e2,1e3])
|
||||
ax.set_xticklabels([0.1, 1, 10, 100, 1000])
|
||||
|
||||
omegas_rel = omegas/omega_p
|
||||
axs[0,0].plot(omegas_rel, n_real)
|
||||
axs[0,0].set_yticks([0,1,2,np.sqrt(eps_infty)])
|
||||
axs[0,0].set_yticklabels([0,1,2,r"$\epsilon_\infty$"])
|
||||
axs[0,0].set_ylim(-0.2,0.2+np.sqrt(eps_infty))
|
||||
axs[0,0].set_ylabel(r"$n^\prime(\omega)$")
|
||||
|
||||
axs[1,0].plot(omegas_rel, n_imag)
|
||||
axs[1,0].set_ylabel(r"$n^{\prime\prime}(\omega)$")
|
||||
axs[1,0].set_yscale("log")
|
||||
|
||||
axs[0,1].plot(omegas_rel, R)
|
||||
axs[0,1].set_ylabel(r"$R(\omega)$")
|
||||
|
||||
axs[1,1].plot(omegas_rel, alpha)
|
||||
axs[1,1].set_ylabel(r"$\alpha(\omega)$")
|
||||
axs[1,1].set_yscale("log")
|
||||
return fig
|
||||
|
||||
if __name__ == '__main__':
|
||||
export(free_electrons_absorption(), "cm_optics_absorption_free_electrons")
|
||||
export(dielectric_absorption(), "cm_optics_absorption_dielectric")
|
||||
@ -1,116 +0,0 @@
|
||||
#!/usr/bin env python3
|
||||
from formulary import *
|
||||
from scipy.constants import Boltzmann as kB, hbar
|
||||
|
||||
hbar = 1
|
||||
kB = 1
|
||||
|
||||
def fone_atom_basis(q, a, M, C1, C2):
|
||||
return np.sqrt(4*C1/M * (np.sin(q*a/2)**2 + C2/C1 * np.sin(q*a)**2))
|
||||
|
||||
def one_atom_basis():
|
||||
a = 1.
|
||||
C1 = 0.25
|
||||
C2 = 0
|
||||
M = 1.
|
||||
qs = np.linspace(-2*np.pi/a, 2*np.pi/a, 300)
|
||||
omega = fone_atom_basis(qs, a, M, C1, C2)
|
||||
fig, ax = plt.subplots(figsize=size_formula_normal_default)
|
||||
ax.set_xlabel(r"$q$")
|
||||
ax.set_xticks([i * np.pi/a for i in range(-2, 3)])
|
||||
ax.set_xticklabels([f"${i}\\pi/a$" if i != 0 else "0" for i in range(-2, 3)])
|
||||
ax.set_ylabel(r"$\omega$ in $\left[4C_1/M\right]$")
|
||||
yunit = np.sqrt(4*C1/M)
|
||||
ax.set_ylim(0, yunit+0.1)
|
||||
ax.set_yticks([0,yunit])
|
||||
ax.set_yticklabels(["0","1"])
|
||||
ax.plot(qs, omega)
|
||||
ax.text(-1.8*np.pi/a, 0.8, "NN\n$C_2=0$", ha='center')
|
||||
ax.text(0, 0.8, "1. BZ", ha='center')
|
||||
ax.vlines([-np.pi/a, np.pi/a], ymin=-2, ymax=2, color="black")
|
||||
ax.grid()
|
||||
return fig
|
||||
|
||||
def ftwo_atom_basis_acoustic(q, a, M1, M2, C):
|
||||
return np.sqrt(C*(1/M1+1/M2) - C * np.sqrt((1/M1+1/M2)**2 - 4/(M1*M2) * np.sin(q*a/2)**2))
|
||||
|
||||
def ftwo_atom_basis_optical(q, a, M1, M2, C):
|
||||
return np.sqrt(C*(1/M1+1/M2) + C * np.sqrt((1/M1+1/M2)**2 - 4/(M1*M2) * np.sin(q*a/2)**2))
|
||||
|
||||
def two_atom_basis():
|
||||
a = 1.
|
||||
C = 0.25
|
||||
M1 = 1.
|
||||
M2 = 0.7
|
||||
qs = np.linspace(-2*np.pi/a, 2*np.pi/a, 300)
|
||||
omega_a = ftwo_atom_basis_acoustic(qs, a, M1, M2, C)
|
||||
omega_o = ftwo_atom_basis_optical(qs, a, M1, M2, C)
|
||||
fig, ax = plt.subplots(figsize=size_formula_normal_default)
|
||||
ax.plot(qs, omega_a, label="acoustic")
|
||||
ax.plot(qs, omega_o, label="optical")
|
||||
ax.text(0, 0.8, "1. BZ", ha='center')
|
||||
ax.vlines([-np.pi/a, np.pi/a], ymin=-2, ymax=2, color="black")
|
||||
ax.set_ylim(-0.03, 1.03)
|
||||
ax.set_ylabel(r"$\omega$ in $\left[\sqrt{2C\mu^{-1}}\right]$")
|
||||
yunit = np.sqrt(2*C*(1/M1+1/M2))
|
||||
ax.set_ylim(0, yunit+0.1)
|
||||
ax.set_yticks([0,yunit])
|
||||
ax.set_yticklabels(["0","1"])
|
||||
ax.set_xlabel(r"$q$")
|
||||
ax.set_xticks([i * np.pi/a for i in range(-2, 3)])
|
||||
ax.set_xticklabels([f"${i}\\pi/a$" if i != 0 else "0" for i in range(-2, 3)])
|
||||
ax.legend()
|
||||
ax.grid()
|
||||
return fig
|
||||
|
||||
|
||||
def fcv_einstein(T, N, omegaE):
|
||||
ThetaT = hbar * omegaE / (kB * T)
|
||||
return 3 * N * kB * ThetaT**2 * np.exp(ThetaT) / (np.exp(ThetaT) - 1)**2
|
||||
|
||||
def fcv_debye_integral(x):
|
||||
print(np.exp(x), (np.exp(x) - 1)**2)
|
||||
return x**4 * np.exp(x) / ((np.exp(x) - 1)**2)
|
||||
|
||||
def heat_capacity_einstein_debye():
|
||||
Ts = np.linspace(0, 10, 500)
|
||||
omegaD = 1e1
|
||||
omegaE = 1
|
||||
# N = 10**23
|
||||
N = 1
|
||||
cvs_einstein = fcv_einstein(Ts, N, omegaE)
|
||||
cvs_debye = np.zeros(Ts.shape, dtype=float)
|
||||
integral = np.zeros(Ts.shape, dtype=float)
|
||||
# cvs_debye = [0.0 for _ in range(Ts.shape[0])] # np.zeros(Ts.shape, dtype=float)
|
||||
# integral = [0.0 for _ in range(Ts.shape[0])] # np.zeros(Ts.shape, dtype=float)
|
||||
dT = Ts[1] - Ts[0]
|
||||
dThetaT = kB*dT/(hbar*omegaD)
|
||||
for i, T in enumerate(Ts):
|
||||
if i == 0: continue
|
||||
ThetaT = kB*T/(hbar*omegaD)
|
||||
dIntegral = fcv_debye_integral(ThetaT) * dThetaT
|
||||
integral[i] = dIntegral
|
||||
# print(integral)
|
||||
integral[i] += integral[i-1]
|
||||
C_debye = 9 * N * kB * ThetaT**3 * integral[i]
|
||||
cvs_debye[i] = C_debye
|
||||
print(i, T, ThetaT, dIntegral, C_debye, integral[i])
|
||||
fig, ax = plt.subplots(1, 1, figsize=size_formula_normal_default)
|
||||
ax.set_xlabel("$T$")
|
||||
ax.set_ylabel("$c_V$")
|
||||
ax.plot(Ts, cvs_einstein, label="Einstein")
|
||||
ax.plot(Ts, cvs_debye, label="Debye")
|
||||
ax.plot(Ts, integral, label="integral")
|
||||
ax.hlines([3*N*kB], xmin=0, xmax=Ts[-1], colors=COLORSCHEME["fg1"], linestyles="dashed")
|
||||
# print(cvs_debye)
|
||||
ax.legend()
|
||||
return fig
|
||||
|
||||
|
||||
|
||||
|
||||
if __name__ == '__main__':
|
||||
export(one_atom_basis(), "cm_vib_dispersion_one_atom_basis")
|
||||
export(two_atom_basis(), "cm_vib_dispersion_two_atom_basis")
|
||||
export(heat_capacity_einstein_debye(), "cm_vib_heat_capacity_einstein_debye")
|
||||
print(kB)
|
||||
@ -1,48 +0,0 @@
|
||||
#!/usr/bin env python3
|
||||
from formulary import *
|
||||
|
||||
# recreating a plot from adv. sc physics ex7/2
|
||||
# showing different scattering mechanisms
|
||||
|
||||
def fionized(T):
|
||||
return 10**4 * T**(3/2)
|
||||
|
||||
@np.vectorize
|
||||
def fpolar_optic(T, TDebye):
|
||||
return np.exp(TDebye/T) if T < TDebye else np.nan
|
||||
|
||||
def fpiezoelectric(T):
|
||||
return 0.8 * 10**7 * T**(-1/2)
|
||||
|
||||
def facoustic(T):
|
||||
return 10**9 * T**(-3/2)
|
||||
|
||||
|
||||
def scattering():
|
||||
fig, ax = plt.subplots(1, 1, figsize=size_formula_normal_default)
|
||||
Ts = np.power(10, np.linspace(0, 3, 100))
|
||||
|
||||
TDebye = 10**3
|
||||
mu_ionized = fionized(Ts)
|
||||
mu_polar_optic = fpolar_optic(Ts, TDebye)
|
||||
mu_piezoelectric = fpiezoelectric(Ts)
|
||||
mu_acoustic = facoustic(Ts)
|
||||
|
||||
mu_sum = 1/(1/mu_ionized + 1/mu_polar_optic + 1/mu_piezoelectric + 1/mu_acoustic)
|
||||
ax.plot(Ts, mu_ionized, label="Ionized")
|
||||
ax.plot(Ts, mu_polar_optic, label="Polar-optic")
|
||||
ax.plot(Ts, mu_piezoelectric, label="Piezoelectric")
|
||||
ax.plot(Ts, mu_acoustic, label="Acoustic")
|
||||
ax.plot(Ts, mu_sum, label="Total", color="black", linestyle="dashed")
|
||||
ax.set_xscale("log")
|
||||
ax.set_yscale("log")
|
||||
ax.set_xlabel("$T$")
|
||||
ax.set_ylabel("$\\mu$")
|
||||
ax.legend()
|
||||
ax.set_ylim(1e4, 1e7)
|
||||
return fig
|
||||
|
||||
|
||||
if __name__ == '__main__':
|
||||
export(scattering(), "cm_scattering")
|
||||
|
||||
@ -1,159 +0,0 @@
|
||||
#!/usr/bin env python3
|
||||
from formulary import *
|
||||
from scipy.constants import Boltzmann as kB, hbar, electron_volt
|
||||
|
||||
|
||||
# DEVICES
|
||||
# metal sc: schottky barrier
|
||||
def schottky_barrier():
|
||||
fig, axs = plt.subplots(1, 3, figsize=(width_formula, height_default*0.6))
|
||||
WD = 5
|
||||
q = 1
|
||||
ND = 1
|
||||
eps = 11
|
||||
dx = WD/10
|
||||
xs = np.linspace(-WD/5, 6/5*WD, 300)
|
||||
rho_S = q*ND
|
||||
Q = rho_S * WD
|
||||
rho_M = -Q/dx
|
||||
@np.vectorize
|
||||
def rho_approx(x):
|
||||
if x < -dx: return 0.0
|
||||
if x < 0: return rho_M
|
||||
if x < WD: return rho_S
|
||||
return 0.0
|
||||
rhos_approx = rho_approx(xs)
|
||||
|
||||
@np.vectorize
|
||||
def E(x):
|
||||
if x < -dx: return 0.0
|
||||
if x < 0: return -rho_M/eps * (-dx-x)
|
||||
if x < WD: return -rho_S/eps * (WD-x)
|
||||
return 0.0
|
||||
Es = E(xs)
|
||||
|
||||
@np.vectorize
|
||||
def phi(x):
|
||||
# if x < -dx: return 0.0
|
||||
# if x < 0: return -rho_M/(2*eps) * (dx**2-(dx-x)**2)
|
||||
if x < 0: return 0.0
|
||||
if x < WD: return rho_S/(2*eps) * (WD**2-(WD-x)**2)
|
||||
return q*ND/(2*eps) * WD**2
|
||||
phis = phi(xs)
|
||||
|
||||
for ax in axs:
|
||||
ax.set_xlabel("$x$")
|
||||
ax.set_xticks([0,WD])
|
||||
ax.set_xticklabels(["0", r"$W_\text{D}$"])
|
||||
ax.set_yticks([0])
|
||||
ax.set_yticklabels(["0"])
|
||||
axs[0].plot(xs, rhos_approx)
|
||||
axs[0].set_ylabel(r"$\rho(x)$")
|
||||
axs[1].plot(xs, Es)
|
||||
axs[1].set_ylabel(r"$\mathcal{E}(x)$")
|
||||
|
||||
axs[2].plot(xs, phis)
|
||||
axs[2].set_ylabel(r"$\phi(x)$")
|
||||
|
||||
return fig
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
# Charge carrier density
|
||||
def fn_i(T, NV, NC, Egap):
|
||||
return np.sqrt(NV*NC) * np.exp(-Egap*electron_volt/(2*kB*T))
|
||||
|
||||
def fn_freeze(T, NV, NC, Egap):
|
||||
return np.sqrt(NV*NC) * np.exp(-0.07*Egap*electron_volt/(2*kB*T))
|
||||
|
||||
def charge_carrier_density():
|
||||
fig, ax = plt.subplots(1, 1, figsize=size_formula_normal_default)
|
||||
# Ts = np.power(10, np.linspace(-4, 0, 100))
|
||||
N = 500
|
||||
ts = np.linspace(0.01, 20, N)
|
||||
Ts = 1000/ts
|
||||
# Ts = np.linspace(2000, 50, N)
|
||||
# ts = 1000/Ts
|
||||
# ts = 1/Ts
|
||||
|
||||
NV = 1e23
|
||||
NC = 1e21
|
||||
Egap = 2.4
|
||||
|
||||
n_is = fn_i(Ts, NV, NC, Egap)
|
||||
|
||||
n_sat = np.empty(Ts.shape)
|
||||
n_sat.fill(1e15)
|
||||
|
||||
|
||||
idx1 = np.argmin(np.abs(n_is-n_sat))
|
||||
print(idx1, N)
|
||||
# TODO make quadratic simple
|
||||
n_total = blend_arrays(n_is, n_sat, idx1, idx1+10, "linear")
|
||||
|
||||
n_freeze = fn_freeze(Ts, NV, NC, Egap)
|
||||
idx2 = np.argmin(np.abs(n_freeze-n_sat))
|
||||
|
||||
print(idx1, idx2, N)
|
||||
n_total = blend_arrays(n_total, n_freeze, idx2-10, idx2+10, "quadratic_simple")
|
||||
|
||||
# ax.plot(ts, n_is, label="Intrinsic")
|
||||
# ax.plot(ts, n_sat, label="Saturation")
|
||||
# ax.plot(ts, n_freeze, label="Freeze-out")
|
||||
# ax.plot(ts, n_total, label="Total", linestyle="dashed", color="black")
|
||||
n_total2 = n_is.copy()
|
||||
n_total2[idx1:idx2] = n_sat[idx1:idx2]
|
||||
n_total2[idx2:] = n_freeze[idx2:]
|
||||
ax.plot(ts, n_is, label="Intrinsic", linestyle="dashed")
|
||||
ax.plot(ts, n_total2, label="Total", color="black")
|
||||
ax.set_yscale("log")
|
||||
ax.set_ylim(1e13, 1e17)
|
||||
|
||||
idx = int(N*0.9)
|
||||
ax.text(ts[idx], n_freeze[idx], "Freeze-out", ha="right", va="top")
|
||||
idx = int(N*0.5)
|
||||
ax.text(ts[idx], n_sat[idx], "Saturation", ha="center", va="bottom")
|
||||
idx = np.argmin(np.abs(n_is-3e16))
|
||||
ax.text(ts[idx+10], n_is[idx], "Intrinsic", ha="left", va="bottom")
|
||||
# ax.set_xlim(ts[0], ts[idx1+N//6])
|
||||
# ax.legend()
|
||||
ax.set_xlabel(r"$1000/T\,[\si{\per\kelvin}]$")
|
||||
ax.set_ylabel(r"$n\,[\si{\per\cm^3}]$")
|
||||
|
||||
|
||||
return fig
|
||||
|
||||
|
||||
def test():
|
||||
fig, ax = plt.subplots()
|
||||
N = 100
|
||||
left = np.empty(N)
|
||||
left.fill(4.0)
|
||||
left = np.linspace(5.0, 0.0, N)
|
||||
right = np.empty(N)
|
||||
right.fill(2.5)
|
||||
right = np.linspace(3.0, 2.0, N)
|
||||
ax.plot(left, label="l",linestyle="dashed")
|
||||
ax.plot(right, label="r",linestyle="dashed")
|
||||
|
||||
total_lin = blend_arrays(left, right, 40, 60, "linear")
|
||||
ax.plot(total_lin, label="lin")
|
||||
# total_tanh = blend_arrays(left, right, 40, 60, "tanh")
|
||||
# ax.plot(total_tanh)
|
||||
total_q = blend_arrays(left, right, 40, 60, "quadratic_simple")
|
||||
ax.plot(total_q, label="q")
|
||||
ax.legend()
|
||||
return fig
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
if __name__ == '__main__':
|
||||
# export(test(), "cm_sc_charge_carrier_density")
|
||||
export(schottky_barrier(), "cm_sc_devices_metal-n-sc_schottky")
|
||||
# export(charge_carrier_density(), "cm_sc_charge_carrier_density")
|
||||
|
||||
@ -1,118 +0,0 @@
|
||||
#!/usr/bin env python3
|
||||
from formulary import *
|
||||
|
||||
# Define the functions
|
||||
def psi_squared(x, xi):
|
||||
return np.tanh(x/(np.sqrt(2)*xi))**2
|
||||
|
||||
def B_z(x, B0, lam):
|
||||
return B0 * np.exp(-x/lam)
|
||||
|
||||
|
||||
def n_s_boundary():
|
||||
xs = np.linspace(0, 6, 400)
|
||||
xn = np.linspace(-1, 0, 10)
|
||||
B0 = 1.0
|
||||
fig, ax = plt.subplots(figsize=size_formula_fill_default)
|
||||
ax.axvline(x=0, color='gray', linestyle='--', linewidth=0.8)
|
||||
ax.axhline(y=1, color='gray', linestyle='--', linewidth=0.8)
|
||||
ax.axhline(y=0, color='gray', linestyle='--', linewidth=0.8)
|
||||
ax.fill_between(xn, -2, 2 , color=COLORSCHEME["bg-yellow"], alpha=0.5)
|
||||
ax.fill_between(xs, -2, 2 , color=COLORSCHEME["bg-blue"], alpha=0.5)
|
||||
ax.text(-0.5, 0.9, 'N', color=COLORSCHEME["fg-yellow"], fontsize=14, ha="center", va="center")
|
||||
ax.text(3, 0.9, 'S', color=COLORSCHEME["fg-blue"], fontsize=14, ha="center", va="center")
|
||||
ax.set_xlabel("$x$")
|
||||
ax.set_ylabel(r"$|\Psi|^2$, $B_z(x)/B_\text{ext}$")
|
||||
ax.set_ylim(-0.1, 1.1)
|
||||
ax.set_xlim(-1, 6)
|
||||
ax.grid()
|
||||
lines = []
|
||||
for i, (xi, lam, color) in enumerate([(0.5, 2, "blue"), (2, 0.5, "red")]):
|
||||
psi = psi_squared(xs, xi)
|
||||
B = B_z(xs, B0, lam)
|
||||
line, = ax.plot(xs, psi, color=color, linestyle="solid", label=f"$\\xi_\\text{{GL}}={xi}$, $\\lambda_\\text{{GL}}={lam}$")
|
||||
lines.append(line)
|
||||
ax.plot(xs, B, color=color, linestyle="dashed")
|
||||
if i == 1:
|
||||
ylam = 1/np.exp(1)
|
||||
ax.plot([0, lam], [ylam, ylam], linestyle="dashed", color=COLORSCHEME["fg2"])
|
||||
ax.text(lam/2, ylam, r'$\lambda_\text{GL}$', color=color, ha="center", va="bottom")
|
||||
yxi = psi_squared(xi, xi)
|
||||
ax.plot([0, xi], [yxi, yxi], linestyle="dotted", color=COLORSCHEME["fg2"])
|
||||
ax.text(xi/2, yxi, r'$\xi_\text{GL}$', color=color, ha="center", va="bottom")
|
||||
lines.append(mpl.lines.Line2D([], [], color="black", label=r"$\lvert\Psi\rvert^2$"))
|
||||
lines.append(mpl.lines.Line2D([], [], color="black", linestyle="dashed", label=r"$B_z(x)/B_\text{ext}$"))
|
||||
ax.legend(loc='center right', handles=lines)
|
||||
return fig
|
||||
|
||||
from mpl_toolkits.mplot3d import Axes3D
|
||||
from scipy.interpolate import griddata
|
||||
|
||||
def critical_type2():
|
||||
Jc0 = 100
|
||||
Bc2_0 = 30
|
||||
Tc = 90
|
||||
|
||||
T = np.linspace(0, Tc, 100)
|
||||
Jc_T = Jc0 * (1 - (T / Tc)**2)
|
||||
Bc2_T = Bc2_0 * (1 - (T / Tc)**2)
|
||||
B = np.linspace(0, Bc2_0, 100)
|
||||
Jc_B = Jc0 * (1 - B / Bc2_0)
|
||||
|
||||
fig = plt.figure(figsize=size_formula_normal_default)
|
||||
ax = fig.add_subplot(111, projection='3d')
|
||||
|
||||
ax.plot(T, np.zeros_like(Jc_T), Jc_T, label='$J_c(T)$', color='r')
|
||||
ax.plot(T, Bc2_T, np.zeros_like(Bc2_T), label='$B_{c2}(T)$', color='g')
|
||||
ax.plot(np.zeros_like(Jc_B), B, Jc_B, label='$J_c(B)$', color='b')
|
||||
|
||||
ax.set_xlim(0, Tc)
|
||||
ax.set_ylim(0, Bc2_0)
|
||||
ax.set_zlim(0, Jc0)
|
||||
|
||||
# surface
|
||||
# T_grid, B_grid = np.meshgrid(T, B)
|
||||
# Jc_grid = Jc0 * (1 - (T_grid / Tc)**2) * (1 - B_grid / Bc2_0)
|
||||
# surf = ax.plot_surface(T_grid, B_grid, Jc_grid, color='cyan', alpha=0.5)
|
||||
ax.set_xlabel('$T$')
|
||||
ax.set_ylabel('$B_{c2}$')
|
||||
ax.set_zlabel('$J_c$')
|
||||
# ax.legend()
|
||||
ax.grid(True)
|
||||
|
||||
ax.view_init(elev=30., azim=45)
|
||||
ax.set_box_aspect(None, zoom=0.85)
|
||||
return fig
|
||||
|
||||
|
||||
|
||||
def heat_capacity():
|
||||
fig, ax = plt.subplots(1, 1, figsize=size_formula_small_quadratic)
|
||||
|
||||
T_max = 1.7
|
||||
Cn_max = 3
|
||||
f_Cn = lambda T: T * Cn_max/T_max
|
||||
Delta_C = f_Cn(1.0) * 1.43 # BCS prediction
|
||||
CsTc = f_Cn(1.0) * (1+1.43) # BCS prediction
|
||||
# exp decay from there
|
||||
f_Cs = lambda T: np.exp(-1 / T + 1) * CsTc
|
||||
|
||||
Tns = np.linspace(0.0, T_max, 100)
|
||||
Tss = np.linspace(0.0, 1.0, 100)
|
||||
Cns = f_Cn(Tns)
|
||||
Css = f_Cs(Tss)
|
||||
ax.plot(Tns, Cns, label=r"$c_\text{n}$")
|
||||
ax.plot(Tss, Css, label=r"$c_\text{s}$")
|
||||
ax.vlines([1.0], ymin=f_Cn(1.0), ymax=(CsTc), color=COLORSCHEME["fg1"], linestyles="dashed")
|
||||
ax.text(1.05, CsTc - Delta_C/2, "$\\Delta c$", color=COLORSCHEME["fg1"])
|
||||
ax.set_xlabel(r"$T/T_\text{c}$")
|
||||
ax.set_ylabel(r"$c$ [a.u.]")
|
||||
ax.legend()
|
||||
return fig
|
||||
|
||||
|
||||
|
||||
if __name__ == "__main__":
|
||||
export(n_s_boundary(), "cm_super_n_s_boundary")
|
||||
export(critical_type2(), "cm_super_critical_type2")
|
||||
export(heat_capacity(), "cm_super_heat_capacity")
|
||||
@ -1,189 +0,0 @@
|
||||
#!/usr/bin env python3
|
||||
import os
|
||||
import matplotlib.pyplot as plt
|
||||
import numpy as np
|
||||
import math
|
||||
import scipy as scp
|
||||
|
||||
import matplotlib as mpl
|
||||
mpl.rcParams["font.family"] = "serif"
|
||||
mpl.rcParams["mathtext.fontset"] = "stix"
|
||||
mpl.rcParams["text.usetex"] = True
|
||||
mpl.rcParams['text.latex.preamble'] = f'''
|
||||
\\usepackage{{amsmath}}
|
||||
\\usepackage{{siunitx}}
|
||||
\\input{{{os.path.abspath("../src/util/math-macros.tex")}}}
|
||||
'''
|
||||
|
||||
|
||||
if __name__ == "__main__": # make relative imports work as described here: https://peps.python.org/pep-0366/#proposed-change
|
||||
if __package__ is None:
|
||||
__package__ = "formulary"
|
||||
import sys
|
||||
filepath = os.path.realpath(os.path.abspath(__file__))
|
||||
sys.path.insert(0, os.path.dirname(os.path.dirname(filepath)))
|
||||
|
||||
from util.mpl_colorscheme import set_mpl_colorscheme
|
||||
import util.colorschemes as cs
|
||||
from util.gen_tex_colorscheme import generate_latex_colorscheme
|
||||
# SET THE COLORSCHEME
|
||||
# hard white and black
|
||||
# cs.p_gruvbox["fg0-hard"] = "#000000"
|
||||
# cs.p_gruvbox["bg0-hard"] = "#ffffff"
|
||||
COLORSCHEME = cs.gruvbox_light_no_beige()
|
||||
# COLORSCHEME = cs.gruvbox_dark()
|
||||
# cs.p_tum["fg0"] = cs.p_tum["alt-blue"]
|
||||
# COLORSCHEME = cs.tum()
|
||||
# COLORSCHEME = cs.legacy()
|
||||
# COLORSCHEME = cs.stupid()
|
||||
|
||||
tex_aux_path = "../.aux/"
|
||||
tex_src_path = "../src/"
|
||||
img_out_dir = os.path.abspath(os.path.join(tex_src_path, "img"))
|
||||
filetype = ".pdf"
|
||||
skipasserts = False
|
||||
|
||||
def pt_2_inch(pt):
|
||||
return 0.0138888889 * pt
|
||||
def cm_2_inch(cm):
|
||||
return 0.3937007874 * cm
|
||||
|
||||
# A4 - margins
|
||||
width_line = cm_2_inch(21.0 - 2 * 2.0)
|
||||
# width of a formula box, the prefactor has to match \eqwidth
|
||||
width_formula = 0.69 * width_line
|
||||
# arbitrary choice
|
||||
height_default = width_line * 2 / 5
|
||||
|
||||
size_bigformula_fill_default = (width_line, height_default)
|
||||
size_bigformula_half_quadratic = (width_line*0.5, width_line*0.5)
|
||||
size_bigformula_small_quadratic = (width_line*0.33, width_line*0.33)
|
||||
size_formula_fill_default = (width_formula, height_default)
|
||||
size_formula_normal_default = (width_formula*0.8, height_default*0.8)
|
||||
size_formula_half_quadratic = (width_formula*0.5, width_formula*0.5)
|
||||
size_formula_small_quadratic = (width_formula*0.4, width_formula*0.4)
|
||||
|
||||
def assert_directory():
|
||||
if not skipasserts:
|
||||
assert os.path.abspath(".").endswith("scripts"), "Please run from the `scripts` directory"
|
||||
|
||||
def texvar(var, val, math=True):
|
||||
s = "$" if math else ""
|
||||
s += f"\\{var} = {val}"
|
||||
if math: s += "$"
|
||||
return s
|
||||
|
||||
def export(fig, name, tight_layout=True):
|
||||
assert_directory()
|
||||
filename = os.path.join(img_out_dir, name + filetype)
|
||||
if tight_layout:
|
||||
fig.tight_layout()
|
||||
fig.savefig(filename, bbox_inches="tight", pad_inches=0.0)
|
||||
|
||||
|
||||
def export_atoms(atoms, name, size, rotation="-30y,20x", get_bonds=True):
|
||||
"""Export a render of ase atoms object"""
|
||||
assert_directory()
|
||||
wd = os.getcwd()
|
||||
from util.aseutil import get_bondatoms, get_pov_settings
|
||||
from ase import io
|
||||
|
||||
tmp_dir = os.path.join(os.path.abspath(tex_aux_path), "scripts_aux")
|
||||
os.makedirs(tmp_dir, exist_ok=True)
|
||||
os.chdir(tmp_dir)
|
||||
|
||||
out_filename = f"{name}.png"
|
||||
|
||||
bondatoms = None
|
||||
if get_bonds:
|
||||
bondatoms = get_bondatoms(atoms)
|
||||
renderer = io.write(f'{name}.pov', atoms,
|
||||
rotation=rotation,# text string with rotation (default='' )
|
||||
radii=0.4, # float, or a list with one float per atom
|
||||
show_unit_cell=2, # 0, 1, or 2 to not show, show, and show all of cell
|
||||
colors=None, # List: one (r, g, b, t) tuple per atom
|
||||
povray_settings=get_pov_settings(size, COLORSCHEME, bondatoms),
|
||||
)
|
||||
renderer.render()
|
||||
os.chdir(wd)
|
||||
os.rename(os.path.join(tmp_dir, out_filename), os.path.join(img_out_dir, out_filename))
|
||||
|
||||
@np.vectorize
|
||||
def smooth_step(x: float, left_edge: float, right_edge: float):
|
||||
x = (x - left_edge) / (right_edge - left_edge)
|
||||
if x <= 0: return 0.
|
||||
elif x >= 1: return 1.
|
||||
else: return 3*(x*2) - 2*(x**3)
|
||||
|
||||
def blend_arrays(a_left, a_right, idx_left, idx_right, mode="linear"):
|
||||
"""
|
||||
Return a new array thats an overlap two arrays a_left and a_right.
|
||||
Left of idx_left = a_left
|
||||
Right of idx_right = a_right
|
||||
Between: Smooth connection of both
|
||||
"""
|
||||
assert(a_left.shape == a_right.shape)
|
||||
assert(idx_left < idx_right)
|
||||
assert(idx_left > 0)
|
||||
assert(idx_right < a_left.shape[0]-1)
|
||||
ret = np.empty(a_left.shape)
|
||||
ret[:idx_left] = a_left[:idx_left]
|
||||
ret[idx_right:] = a_right[idx_right:]
|
||||
n = idx_right - idx_left
|
||||
left = a_left[idx_left]
|
||||
right = a_right[idx_right]
|
||||
for i in range(idx_left, idx_right):
|
||||
j = i-idx_left # 0-based
|
||||
if mode == "linear":
|
||||
val = left * (n-j)/n + right * (j)/n
|
||||
# connect with a single quadratic function
|
||||
elif mode == "quadratic_simple":
|
||||
slope_left = a_left[idx_left] - a_left[idx_left-1]
|
||||
slope_right = a_right[idx_right+1] - a_right[idx_right]
|
||||
b = slope_left
|
||||
a = (slope_right-b)/(2*n)
|
||||
c = left
|
||||
# val = (left + slope_left * (j/n))*(j/n) + (right+slope_right*(1-j/n))*(1-j/n)
|
||||
val = a*(j**2) + b*j +c
|
||||
print(a,b,c)
|
||||
# connect with two quadratic functions
|
||||
# TODO: fix
|
||||
elif mode == "quadratic":
|
||||
slope_left = a_left[idx_left] - a_left[idx_left-1]
|
||||
slope_right = a_right[idx_right+1] - a_right[idx_right]
|
||||
c1 = left
|
||||
b1 = slope_left
|
||||
b2 = 2*slope_right - b1
|
||||
a1 = (b2-b1)/4
|
||||
a2 = -a1
|
||||
c2 = right - a2*n**2-b2*n
|
||||
m = 2 * (c2-c1)/(b1-b2)
|
||||
print(m, a1, b1, c1)
|
||||
print(m, a2, b2, c2)
|
||||
|
||||
if j < m:
|
||||
val = a1*(j**2) + b1*j +c1
|
||||
else:
|
||||
val = a2*(j**2) + b2*j +c2
|
||||
elif mode == "tanh":
|
||||
TANH_FULL = 2 # x value where tanh is assumed to be 1/-1
|
||||
x = (2 * j / n - 1) * TANH_FULL
|
||||
tanh_error = 1-np.tanh(TANH_FULL)
|
||||
amplitude = 0.5*(right-left)
|
||||
val = amplitude * (1+tanh_error) * np.tanh(x) + (left+amplitude)
|
||||
else: raise ValueError(f"Invalid mode: {mode}")
|
||||
ret[i] = val
|
||||
return ret
|
||||
|
||||
# run even when imported
|
||||
set_mpl_colorscheme(COLORSCHEME)
|
||||
|
||||
if __name__ == "__main__":
|
||||
assert_directory()
|
||||
s = \
|
||||
"""% This file was generated by scripts/formulary.py\n% Do not edit it directly, changes will be overwritten\n""" + generate_latex_colorscheme(COLORSCHEME)
|
||||
filename = os.path.join(tex_src_path, "util/colorscheme.tex")
|
||||
print(f"Writing tex colorscheme to {filename}")
|
||||
with open(filename, "w") as file:
|
||||
file.write(s)
|
||||
|
||||
@ -1,149 +0,0 @@
|
||||
"""
|
||||
Set the colorscheme for matplotlib plots and latex.
|
||||
|
||||
Calling this script generates util/colorscheme.tex containing xcolor definitions.
|
||||
"""
|
||||
import matplotlib as mpl
|
||||
import matplotlib.pyplot as plt
|
||||
from cycler import cycler
|
||||
|
||||
skipasserts = False
|
||||
|
||||
GRUVBOX = {
|
||||
"bg0": "#282828",
|
||||
"bg0-hard": "#1d2021",
|
||||
"bg0-soft": "#32302f",
|
||||
"bg1": "#3c3836",
|
||||
"bg2": "#504945",
|
||||
"bg3": "#665c54",
|
||||
"bg4": "#7c6f64",
|
||||
"fg0": "#fbf1c7",
|
||||
"fg0-hard": "#f9f5d7",
|
||||
"fg0-soft": "#f2e5bc",
|
||||
"fg1": "#ebdbb2",
|
||||
"fg2": "#d5c4a1",
|
||||
"fg3": "#bdae93",
|
||||
"fg4": "#a89984",
|
||||
"dark-red": "#cc241d",
|
||||
"dark-green": "#98971a",
|
||||
"dark-yellow": "#d79921",
|
||||
"dark-blue": "#458588",
|
||||
"dark-purple": "#b16286",
|
||||
"dark-aqua": "#689d6a",
|
||||
"dark-orange": "#d65d0e",
|
||||
"dark-gray": "#928374",
|
||||
"light-red": "#fb4934",
|
||||
"light-green": "#b8bb26",
|
||||
"light-yellow": "#fabd2f",
|
||||
"light-blue": "#83a598",
|
||||
"light-purple": "#d3869b",
|
||||
"light-aqua": "#8ec07c",
|
||||
"light-orange": "#f38019",
|
||||
"light-gray": "#a89984",
|
||||
"alt-red": "#9d0006",
|
||||
"alt-green": "#79740e",
|
||||
"alt-yellow": "#b57614",
|
||||
"alt-blue": "#076678",
|
||||
"alt-purple": "#8f3f71",
|
||||
"alt-aqua": "#427b58",
|
||||
"alt-orange": "#af3a03",
|
||||
"alt-gray": "#7c6f64",
|
||||
}
|
||||
|
||||
FORMULASHEET_COLORSCHEME = GRUVBOX
|
||||
|
||||
colors = ["red", "orange", "yellow", "green", "aqua", "blue", "purple", "gray"]
|
||||
|
||||
# default order for matplotlib
|
||||
color_order = ["blue", "orange", "green", "red", "purple", "yellow", "aqua", "gray"]
|
||||
|
||||
def set_mpl_colorscheme(palette: dict[str, str], variant="dark"):
|
||||
P = palette
|
||||
if variant == "dark":
|
||||
FIG_BG_COLOR = P["bg0"]
|
||||
PLT_FG_COLOR = P["fg0"]
|
||||
PLT_BG_COLOR = P["bg0"]
|
||||
PLT_GRID_COLOR = P["bg2"]
|
||||
LEGEND_FG_COLOR = PLT_FG_COLOR
|
||||
LEGEND_BG_COLOR = P["bg1"]
|
||||
LEGEND_BORDER_COLOR = P["bg2"]
|
||||
else:
|
||||
FIG_BG_COLOR = P["fg0"]
|
||||
PLT_FG_COLOR = P["bg0"]
|
||||
PLT_BG_COLOR = P["fg0"]
|
||||
PLT_GRID_COLOR = P["fg2"]
|
||||
LEGEND_FG_COLOR = PLT_FG_COLOR
|
||||
LEGEND_BG_COLOR = P["fg1"]
|
||||
LEGEND_BORDER_COLOR = P["fg2"]
|
||||
COLORS = [P[f"{variant}-{c}"] for c in color_order]
|
||||
|
||||
|
||||
color_rcParams = {
|
||||
'axes.edgecolor': PLT_FG_COLOR,
|
||||
'axes.facecolor': PLT_BG_COLOR,
|
||||
'axes.labelcolor': PLT_FG_COLOR,
|
||||
'axes.titlecolor': 'auto',
|
||||
# 'axes.prop_cycle': cycler('color', ['#1f77b4', '#ff7f0e', '#2ca02c', '#d62728', '#9467bd', '#8c564b', '#e377c2', '#7f7f7f', '#bcbd22', '#17becf']),
|
||||
'axes.prop_cycle': cycler('color', COLORS),
|
||||
# 'axes3d.xaxis.panecolor': (0.95, 0.95, 0.95, 0.5),
|
||||
# 'axes3d.yaxis.panecolor': (0.9, 0.9, 0.9, 0.5),
|
||||
# 'axes3d.zaxis.panecolor': (0.925, 0.925, 0.925, 0.5),
|
||||
# 'boxplot.boxprops.color': 'black',
|
||||
# 'boxplot.capprops.color': 'black',
|
||||
# 'boxplot.flierprops.color': 'black',
|
||||
# 'boxplot.flierprops.markeredgecolor': 'black',
|
||||
# 'boxplot.flierprops.markeredgewidth': 1.0,
|
||||
# 'boxplot.flierprops.markerfacecolor': 'none',
|
||||
# 'boxplot.meanprops.color': 'C2',
|
||||
# 'boxplot.meanprops.markeredgecolor': 'C2',
|
||||
# 'boxplot.meanprops.markerfacecolor': 'C2',
|
||||
# 'boxplot.meanprops.markersize': 6.0,
|
||||
# 'boxplot.medianprops.color': 'C1',
|
||||
# 'boxplot.whiskerprops.color': 'black',
|
||||
'figure.edgecolor': PLT_BG_COLOR,
|
||||
'figure.facecolor': PLT_BG_COLOR,
|
||||
# 'figure.figsize': [6.4, 4.8],
|
||||
# 'figure.frameon': True,
|
||||
# 'figure.labelsize': 'large',
|
||||
'grid.color': PLT_GRID_COLOR,
|
||||
# 'hatch.color': 'black',
|
||||
'legend.edgecolor': LEGEND_BORDER_COLOR,
|
||||
'legend.facecolor': LEGEND_BG_COLOR,
|
||||
'xtick.color': PLT_FG_COLOR,
|
||||
'ytick.color': PLT_FG_COLOR,
|
||||
'xtick.labelcolor': PLT_FG_COLOR,
|
||||
'ytick.labelcolor': PLT_FG_COLOR,
|
||||
# 'lines.color': 'C0',
|
||||
'text.color': PLT_FG_COLOR,
|
||||
}
|
||||
|
||||
for k, v in color_rcParams.items():
|
||||
plt.rcParams[k] = v
|
||||
|
||||
# override single char codes
|
||||
# TODO: use color name with variant from palette instead of order
|
||||
mpl.colors.get_named_colors_mapping()["b"] = COLORS[0]
|
||||
mpl.colors.get_named_colors_mapping()["o"] = COLORS[1]
|
||||
mpl.colors.get_named_colors_mapping()["g"] = COLORS[2]
|
||||
mpl.colors.get_named_colors_mapping()["r"] = COLORS[3]
|
||||
mpl.colors.get_named_colors_mapping()["m"] = COLORS[4]
|
||||
mpl.colors.get_named_colors_mapping()["y"] = COLORS[5]
|
||||
mpl.colors.get_named_colors_mapping()["c"] = COLORS[6]
|
||||
mpl.colors.get_named_colors_mapping()["k"] = P["fg0"]
|
||||
mpl.colors.get_named_colors_mapping()["w"] = P["bg0"]
|
||||
|
||||
|
||||
|
||||
def color_latex_def(name, color):
|
||||
name = "{" + name.replace("-", "_") + "}"
|
||||
color = "{" + color.strip("#") + "}"
|
||||
return f"\\definecolor{name:10}{{HTML}}{color}"
|
||||
|
||||
def generate_latex_colorscheme(palette, variant="light"):
|
||||
s = ""
|
||||
for n, c in palette.items():
|
||||
s += color_latex_def(n, c) + "\n"
|
||||
return s
|
||||
|
||||
|
||||
|
||||
File diff suppressed because it is too large
Load Diff
@ -1,55 +0,0 @@
|
||||
{
|
||||
"cells": [
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 1,
|
||||
"id": "790c45a0-a10a-411d-bfc0-bdd52e2c2492",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"import tikz as t"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 2,
|
||||
"id": "6c5d640f-c287-4e8d-a0c8-8a8d801b6fae",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"pic = t.Picture()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "9b7f8347-1619-40cd-b864-24840901e7a1",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"pic.draw(t.node("
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"kernelspec": {
|
||||
"display_name": "conda",
|
||||
"language": "python",
|
||||
"name": "conda"
|
||||
},
|
||||
"language_info": {
|
||||
"codemirror_mode": {
|
||||
"name": "ipython",
|
||||
"version": 3
|
||||
},
|
||||
"file_extension": ".py",
|
||||
"mimetype": "text/x-python",
|
||||
"name": "python",
|
||||
"nbconvert_exporter": "python",
|
||||
"pygments_lexer": "ipython3",
|
||||
"version": "3.12.3"
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 5
|
||||
}
|
||||
File diff suppressed because one or more lines are too long
File diff suppressed because it is too large
Load Diff
@ -1,79 +0,0 @@
|
||||
#!/usr/bin env python3
|
||||
"""
|
||||
Script to process the periodic table as json into latex stuff
|
||||
Source for `elements.json` is this amazing project:
|
||||
https://pse-info.de/de/data
|
||||
|
||||
Copyright Matthias Quintern 2025
|
||||
"""
|
||||
import json
|
||||
import os
|
||||
import re
|
||||
|
||||
outdir = "../src/ch"
|
||||
|
||||
|
||||
def gen_periodic_table():
|
||||
with open("other/elements.json") as file:
|
||||
ptab = json.load(file)
|
||||
# print(ptab["elements"][1])
|
||||
s = "% This file was created by the periodic_table.py script.\n% Do not edit manually. Any changes might get lost.\n"
|
||||
for i, el_key in enumerate(ptab):
|
||||
el = ptab[el_key]
|
||||
def get(*keys):
|
||||
"""get keys or return empty string"""
|
||||
val = el
|
||||
for key in keys:
|
||||
if not key in val: return ""
|
||||
val = val[key]
|
||||
return val
|
||||
# print(i, el)
|
||||
el_s = f"\\begin{{element}}{{{el['symbol']}}}{{{el['number']}}}{{{el['period']}}}{{{el['column']}}}"
|
||||
# description
|
||||
el_s += f"\n\t\\desc[english]{{{el['names']['en']}}}{{{get('appearance', 'en')}}}{{}}"
|
||||
el_s += f"\n\t\\desc[german]{{{el['names']['de']}}}{{English: {get('names', 'en')}\\\\{get('appearance', 'de')}}}{{}}"
|
||||
|
||||
# simple properties
|
||||
for field in ["crystal_structure", "set", "magnetic_ordering"]:
|
||||
if field in el:
|
||||
el_s += f"\n\t\\property{{{field}}}{{{el[field]}}}"
|
||||
# mass
|
||||
m = get("atomic_mass", "value")
|
||||
if m:
|
||||
assert(get("atomic_mass", "unit") == "u")
|
||||
el_s += f"\n\t\\property{{{'atomic_mass'}}}{{{m}}}"
|
||||
|
||||
# refractive indices
|
||||
temp = ""
|
||||
add_refractive_index = lambda idx: f"\\GT{{{idx['label']}}}: ${idx['value']}$, "
|
||||
idxs = get("optical", "refractive_index")
|
||||
# print(idxs)
|
||||
if type(idxs) == list:
|
||||
for idx in idxs: add_refractive_index(idx)
|
||||
elif type(idxs) == dict: add_refractive_index(idxs)
|
||||
elif type(idxs) == float: temp += f"${idxs}$, "
|
||||
if temp:
|
||||
el_s += f"\n\t\\property{{{'refractive_index'}}}{{{temp[:-2]}}}"
|
||||
|
||||
|
||||
# electron configuration
|
||||
match = re.fullmatch(r"([A-Z][a-z]*)? ?(.+?)", el["electron_config"])
|
||||
if match:
|
||||
el_s += f"\n\t\\property{{{'electron_config'}}}{{"
|
||||
if match.groups()[0]:
|
||||
el_s += f"\\elRef{{{match.groups()[0]}}} "
|
||||
el_s += f"{match.groups()[1]}}}"
|
||||
|
||||
el_s += "\n\\end{element}"
|
||||
# print(el_s)
|
||||
s += el_s + "\n"
|
||||
# print(s)
|
||||
return s
|
||||
|
||||
|
||||
if __name__ == "__main__":
|
||||
ptable = gen_periodic_table()
|
||||
assert os.path.abspath(".").endswith("scripts"), "Please run from the `scripts` directory"
|
||||
with open(f"{outdir}/periodic_table.tex", "w") as file:
|
||||
file.write(ptable)
|
||||
|
||||
@ -1,24 +0,0 @@
|
||||
# Scripts
|
||||
Put all scripts that generate plots or tex files here.
|
||||
You can run all files at once using `make scripts`
|
||||
|
||||
## Plots
|
||||
### `matplotlib`
|
||||
For plots with `matplotlib`:
|
||||
1. import `formulary.py`
|
||||
2. use one of the preset figsizes
|
||||
3. save the image using the `export` function in the `if __name__ == '__main__'` part
|
||||
|
||||
### `ase` - Atomic Simulation Environment
|
||||
For plots with `ase`:
|
||||
1. import `formulary.py` and `util.aseutil`
|
||||
2. Use `util.aseutil.set_atom_color` to change the color of all used atoms to one in the colorscheme
|
||||
3. export the render using the `export_atoms` function in the `if __name__ == '__main__'` part.
|
||||
Pass one of the preset figsizes as size.
|
||||
|
||||
## Colorscheme
|
||||
To ensure a uniform look of the tex source and the python plots,
|
||||
the tex and matplotlib colorschemes are both handled in `formulary.py`.
|
||||
Set the `COLORSCHEME` variable to the desired colors.
|
||||
Importing `formulary.py` will automatically apply the colors to matplotlib,
|
||||
and running it will generate `util/colorscheme.tex` for LaTeX.
|
||||
@ -1,47 +0,0 @@
|
||||
from util.colorschemes import hex_to_rgb_float
|
||||
|
||||
def set_atom_color(symbol, hexcolor):
|
||||
from ase.data import atomic_numbers
|
||||
from ase.data.colors import jmol_colors, cpk_colors
|
||||
float_color = hex_to_rgb_float(hexcolor)
|
||||
n = atomic_numbers[symbol]
|
||||
jmol_colors[n] = float_color
|
||||
cpk_colors[n] = float_color
|
||||
|
||||
|
||||
from scipy.spatial.distance import pdist, squareform
|
||||
import numpy as np
|
||||
def get_bondatoms(atoms):
|
||||
site_positions = [site.position for site in atoms]
|
||||
pair_distances = squareform(pdist(np.stack(site_positions)))
|
||||
vs = pair_distances
|
||||
bondatoms = []
|
||||
for i in range(vs.shape[0]):
|
||||
for j in range(i):
|
||||
if vs[i, j] < 3: # up to 3 angstrom distance show a bond TODO
|
||||
bondatoms.append((i, j))
|
||||
return bondatoms
|
||||
# returns to many
|
||||
# from ase.io.pov import get_bondpairs
|
||||
# bondatoms=get_bondpairs(lat, 5)
|
||||
|
||||
|
||||
TARGET_DPI = 300
|
||||
# doc: https://github.com/WMD-group/ASE-Tutorials/blob/master/povray-tools/ase_povray.py
|
||||
def get_pov_settings(size, COLORSCHEME, bondatoms=None):
|
||||
white = hex_to_rgb_float(COLORSCHEME["bg0"])
|
||||
other = hex_to_rgb_float(COLORSCHEME["fg-yellow"])
|
||||
pixels = TARGET_DPI * size[0]
|
||||
pov_settings=dict(
|
||||
transparent=True,
|
||||
display=False,
|
||||
# camera_type='orthographic',
|
||||
camera_type='perspective',
|
||||
canvas_width=pixels,
|
||||
# point_lights : [], #[(18,20,40), 'White'],[(60,20,40),'White'], # [[loc1, color1], [loc2, color2],...]
|
||||
point_lights=[[(18,20,40), white],[(60,20,40),other]], # [[loc1, color1], [loc2, color2],...]
|
||||
background=(0, 0, 0, 1.,),
|
||||
bondlinewidth=0.07,
|
||||
bondatoms=bondatoms
|
||||
)
|
||||
return pov_settings
|
||||
@ -1,208 +0,0 @@
|
||||
"""
|
||||
A colorscheme for this project needs:
|
||||
fg and bg 0-4, where 0 is used as default font / background
|
||||
fg-<color> and bg-<color> where <color> is "red", "orange", "yellow", "green", "aqua", "blue", "purple", "gray"
|
||||
"""
|
||||
|
||||
from math import floor
|
||||
|
||||
|
||||
colors = ["red", "orange", "yellow", "green", "aqua", "blue", "purple", "gray"]
|
||||
|
||||
def duplicate_letters(color: str):
|
||||
return ''.join([c+c for c in color])
|
||||
|
||||
def hex_to_rgb_int(color: str) -> list[int]:
|
||||
color = color.strip("#")
|
||||
ctuple = []
|
||||
# turn RGBA to RRGGBBAA
|
||||
if len(color) == 3 or len(color) == 4:
|
||||
color = duplicate_letters(color)
|
||||
for i in range(len(color)//2):
|
||||
ctuple.append(int(color[i*2:i*2+2], 16))
|
||||
return ctuple
|
||||
|
||||
def hex_to_rgb_float(color: str) -> list[float]:
|
||||
clist = hex_to_rgb_int(color)
|
||||
fclist = [float(c) / 255 for c in clist]
|
||||
return fclist
|
||||
|
||||
|
||||
|
||||
def brightness(color:str, percent:float):
|
||||
if color.startswith("#"):
|
||||
color = color.strip("#")
|
||||
newcolor = "#"
|
||||
else:
|
||||
newcolor = ""
|
||||
for i in range(3):
|
||||
c = float(int(color[i*2:i*2+2], 16))
|
||||
c = int(round(max(0, min(c*percent, 0xff)), 0))
|
||||
newcolor += f"{c:02x}"
|
||||
return newcolor
|
||||
|
||||
#
|
||||
# GRUVBOX
|
||||
#
|
||||
p_gruvbox = {
|
||||
"fg0": "#282828",
|
||||
"fg0-hard": "#1d2021",
|
||||
"fg0-soft": "#32302f",
|
||||
"fg1": "#3c3836",
|
||||
"fg2": "#504945",
|
||||
"fg3": "#665c54",
|
||||
"fg4": "#7c6f64",
|
||||
"bg0": "#fbf1c7",
|
||||
"bg0-hard": "#f9f5d7",
|
||||
"bg0-soft": "#f2e5bc",
|
||||
"bg1": "#ebdbb2",
|
||||
"bg2": "#d5c4a1",
|
||||
"bg3": "#bdae93",
|
||||
"bg4": "#a89984",
|
||||
"dark-red": "#cc241d",
|
||||
"dark-green": "#98971a",
|
||||
"dark-yellow": "#d79921",
|
||||
"dark-blue": "#458588",
|
||||
"dark-purple": "#b16286",
|
||||
"dark-aqua": "#689d6a",
|
||||
"dark-orange": "#d65d0e",
|
||||
"dark-gray": "#928374",
|
||||
"light-red": "#fb4934",
|
||||
"light-green": "#b8bb26",
|
||||
"light-yellow": "#fabd2f",
|
||||
"light-blue": "#83a598",
|
||||
"light-purple": "#d3869b",
|
||||
"light-aqua": "#8ec07c",
|
||||
"light-orange": "#f38019",
|
||||
"light-gray": "#a89984",
|
||||
"alt-red": "#9d0006",
|
||||
"alt-green": "#79740e",
|
||||
"alt-yellow": "#b57614",
|
||||
"alt-blue": "#076678",
|
||||
"alt-purple": "#8f3f71",
|
||||
"alt-aqua": "#427b58",
|
||||
"alt-orange": "#af3a03",
|
||||
"alt-gray": "#7c6f64",
|
||||
}
|
||||
|
||||
def gruvbox_light():
|
||||
GRUVBOX_LIGHT = { "fg0": p_gruvbox["fg0-hard"], "bg0": p_gruvbox["bg0-hard"] } \
|
||||
| {f"fg{n}": p_gruvbox[f"fg{n}"] for n in range(1,5)} \
|
||||
| {f"bg{n}": p_gruvbox[f"bg{n}"] for n in range(1,5)} \
|
||||
| {f"fg-{n}": p_gruvbox[f"alt-{n}"] for n in colors} \
|
||||
| {f"bg-{n}": p_gruvbox[f"light-{n}"] for n in colors}
|
||||
return GRUVBOX_LIGHT
|
||||
|
||||
def gruvbox_dark():
|
||||
GRUVBOX_DARK = { "fg0": p_gruvbox["bg0-hard"], "bg0": p_gruvbox["fg0-hard"] } \
|
||||
| {f"fg{n}": p_gruvbox[f"bg{n}"] for n in range(1,5)} \
|
||||
| {f"bg{n}": p_gruvbox[f"fg{n}"] for n in range(1,5)} \
|
||||
| {f"fg-{n}": p_gruvbox[f"light-{n}"] for n in colors} \
|
||||
| {f"bg-{n}": p_gruvbox[f"dark-{n}"] for n in colors}
|
||||
return GRUVBOX_DARK
|
||||
|
||||
def gruvbox_light_no_beige():
|
||||
GRUVBOX_NO_BEIGE = \
|
||||
{ f"fg{n}": brightness("#999999", n/5) for n in range(5)} \
|
||||
| { f"bg{n}": brightness("#FFFFFF", 1-n/8) for n in range(5)} \
|
||||
| {f"fg-{n}": p_gruvbox[f"alt-{n}"] for n in colors} \
|
||||
| {f"bg-{n}": p_gruvbox[f"light-{n}"] for n in colors}
|
||||
return GRUVBOX_NO_BEIGE
|
||||
|
||||
#
|
||||
# LEGACY
|
||||
#
|
||||
p_legacy = {
|
||||
"fg0": "#fcfcfc",
|
||||
"bg0": "#333333",
|
||||
"red": "#d12229",
|
||||
"green": "#007940",
|
||||
"yellow": "#ffc615",
|
||||
"blue": "#2440fe",
|
||||
"purple": "#4D1037",
|
||||
"aqua": "#008585",
|
||||
"orange": "#f68a1e",
|
||||
"gray": "#928374",
|
||||
}
|
||||
|
||||
def legacy():
|
||||
LEGACY = \
|
||||
{ f"fg{n}": brightness(p_legacy["fg0"], 1-n/8) for n in range(5)} \
|
||||
| { f"bg{n}": brightness(p_legacy["bg0"], 1+n/8) for n in range(5)} \
|
||||
| { f"bg-{n}": c for n,c in p_legacy.items() } \
|
||||
| { f"fg-{n}": brightness(c, 2.0) for n,c in p_legacy.items() }
|
||||
return LEGACY
|
||||
|
||||
#
|
||||
# TUM
|
||||
#
|
||||
p_tum = {
|
||||
"dark-blue": "#072140",
|
||||
"light-blue": "#5E94D4",
|
||||
"alt-blue": "#3070B3",
|
||||
"light-yellow": "#FED702",
|
||||
"dark-yellow": "#CBAB01",
|
||||
"alt-yellow": "#FEDE34",
|
||||
"light-orange": "#F7811E",
|
||||
"dark-orange": "#D99208",
|
||||
"alt-orange": "#F9BF4E",
|
||||
"light-purple": "#B55CA5",
|
||||
"dark-purple": "#9B468D",
|
||||
"alt-purple": "#C680BB",
|
||||
"light-red": "#EA7237",
|
||||
"dark-red": "#D95117",
|
||||
"alt-red": "#EF9067",
|
||||
"light-green": "#9FBA36",
|
||||
"dark-green": "#7D922A",
|
||||
"alt-green": "#B6CE55",
|
||||
"light-gray": "#475058",
|
||||
"dark-gray": "#20252A",
|
||||
"alt-gray": "#333A41",
|
||||
"light-aqua": "#689d6a",
|
||||
"dark-aqua": "#427b58", # taken aquas from gruvbox
|
||||
"fg0-hard": "#000000",
|
||||
"fg0": "#000000",
|
||||
"fg0-soft": "#20252A",
|
||||
"fg1": "#072140",
|
||||
"fg2": "#333A41",
|
||||
"fg3": "#475058",
|
||||
"fg4": "#6A757E",
|
||||
"bg0-hard": "#FFFFFF",
|
||||
"bg0": "#FBF9FA",
|
||||
"bg0-soft": "#EBECEF",
|
||||
"bg1": "#DDE2E6",
|
||||
"bg2": "#E3EEFA",
|
||||
"bg3": "#F0F5FA",
|
||||
}
|
||||
|
||||
def tum():
|
||||
TUM = {}
|
||||
for n,c in p_tum.items():
|
||||
n2 = n.replace("light", "bg").replace("dark", "fg")
|
||||
TUM[n2] = c
|
||||
TUM["fg-blue"] = p_tum["alt-blue"] # dark blue is too black
|
||||
return TUM
|
||||
|
||||
#
|
||||
# STUPID
|
||||
#
|
||||
p_stupid = {
|
||||
"bg0": "#0505aa",
|
||||
"fg0": "#ffffff",
|
||||
"red": "#ff0000",
|
||||
"green": "#23ff81",
|
||||
"yellow": "#ffff00",
|
||||
"blue": "#5555ff",
|
||||
"purple": "#b00b69",
|
||||
"aqua": "#00ffff",
|
||||
"orange": "#ffa500",
|
||||
"gray": "#444444",
|
||||
}
|
||||
def stupid():
|
||||
LEGACY = \
|
||||
{ f"fg{n}": brightness(p_stupid["fg0"], 1-n/8) for n in range(5)} \
|
||||
| { f"bg{n}": brightness(p_stupid["bg0"], 1+n/8) for n in range(5)} \
|
||||
| { f"bg-{n}": c for n,c in p_stupid.items() } \
|
||||
| { f"fg-{n}": brightness(c, 2.0) for n,c in p_stupid.items() }
|
||||
return LEGACY
|
||||
|
||||
@ -1,10 +0,0 @@
|
||||
def color_latex_def(name, color):
|
||||
# name = name.replace("-", "_")
|
||||
color = color.strip("#")
|
||||
return "\\definecolor{" + name + "}{HTML}{" + color + "}"
|
||||
|
||||
def generate_latex_colorscheme(palette):
|
||||
s = ""
|
||||
for n, c in palette.items():
|
||||
s += color_latex_def(n, c) + "\n"
|
||||
return s
|
||||
@ -1,84 +0,0 @@
|
||||
"""
|
||||
Set the colorscheme for matplotlib plots and latex.
|
||||
|
||||
Calling this script generates util/colorscheme.tex containing xcolor definitions.
|
||||
"""
|
||||
import matplotlib as mpl
|
||||
import matplotlib.pyplot as plt
|
||||
from cycler import cycler
|
||||
|
||||
# default order for matplotlib
|
||||
color_order = ["blue", "orange", "green", "red", "purple", "yellow", "aqua", "gray"]
|
||||
|
||||
def set_mpl_colorscheme(palette: dict[str, str]):
|
||||
P = palette
|
||||
FIG_BG_COLOR = P["bg0"]
|
||||
PLT_FG_COLOR = P["fg0"]
|
||||
PLT_BG_COLOR = P["bg0"]
|
||||
PLT_GRID_COLOR = P["bg2"]
|
||||
LEGEND_FG_COLOR = PLT_FG_COLOR
|
||||
LEGEND_BG_COLOR = P["bg1"]
|
||||
LEGEND_BORDER_COLOR = P["bg2"]
|
||||
COLORS = [P[f"fg-{c}"] for c in color_order]
|
||||
|
||||
|
||||
color_rcParams = {
|
||||
'axes.edgecolor': PLT_FG_COLOR,
|
||||
'axes.facecolor': PLT_BG_COLOR,
|
||||
'axes.labelcolor': PLT_FG_COLOR,
|
||||
'axes.titlecolor': 'auto',
|
||||
# 'axes.prop_cycle': cycler('color', ['#1f77b4', '#ff7f0e', '#2ca02c', '#d62728', '#9467bd', '#8c564b', '#e377c2', '#7f7f7f', '#bcbd22', '#17becf']),
|
||||
'axes.prop_cycle': cycler('color', COLORS),
|
||||
# 'axes3d.xaxis.panecolor': (0.95, 0.95, 0.95, 0.5),
|
||||
# 'axes3d.yaxis.panecolor': (0.9, 0.9, 0.9, 0.5),
|
||||
# 'axes3d.zaxis.panecolor': (0.925, 0.925, 0.925, 0.5),
|
||||
# 'boxplot.boxprops.color': 'black',
|
||||
# 'boxplot.capprops.color': 'black',
|
||||
# 'boxplot.flierprops.color': 'black',
|
||||
# 'boxplot.flierprops.markeredgecolor': 'black',
|
||||
# 'boxplot.flierprops.markeredgewidth': 1.0,
|
||||
# 'boxplot.flierprops.markerfacecolor': 'none',
|
||||
# 'boxplot.meanprops.color': 'C2',
|
||||
# 'boxplot.meanprops.markeredgecolor': 'C2',
|
||||
# 'boxplot.meanprops.markerfacecolor': 'C2',
|
||||
# 'boxplot.meanprops.markersize': 6.0,
|
||||
# 'boxplot.medianprops.color': 'C1',
|
||||
# 'boxplot.whiskerprops.color': 'black',
|
||||
'figure.edgecolor': PLT_BG_COLOR,
|
||||
'figure.facecolor': PLT_BG_COLOR,
|
||||
# 'figure.figsize': [6.4, 4.8],
|
||||
# 'figure.frameon': True,
|
||||
# 'figure.labelsize': 'large',
|
||||
'grid.color': PLT_GRID_COLOR,
|
||||
# 'hatch.color': 'black',
|
||||
'legend.edgecolor': LEGEND_BORDER_COLOR,
|
||||
'legend.facecolor': LEGEND_BG_COLOR,
|
||||
'xtick.color': PLT_FG_COLOR,
|
||||
'ytick.color': PLT_FG_COLOR,
|
||||
'xtick.labelcolor': PLT_FG_COLOR,
|
||||
'ytick.labelcolor': PLT_FG_COLOR,
|
||||
# 'lines.color': 'C0',
|
||||
'text.color': PLT_FG_COLOR,
|
||||
}
|
||||
|
||||
for k, v in color_rcParams.items():
|
||||
plt.rcParams[k] = v
|
||||
|
||||
# override single char codes
|
||||
# TODO: use color name with variant from palette instead of order
|
||||
mpl.colors.get_named_colors_mapping()["b"] = COLORS[0]
|
||||
mpl.colors.get_named_colors_mapping()["o"] = COLORS[1]
|
||||
mpl.colors.get_named_colors_mapping()["g"] = COLORS[2]
|
||||
mpl.colors.get_named_colors_mapping()["r"] = COLORS[3]
|
||||
mpl.colors.get_named_colors_mapping()["m"] = COLORS[4]
|
||||
mpl.colors.get_named_colors_mapping()["y"] = COLORS[5]
|
||||
mpl.colors.get_named_colors_mapping()["c"] = COLORS[6]
|
||||
mpl.colors.get_named_colors_mapping()["k"] = P["fg0"]
|
||||
mpl.colors.get_named_colors_mapping()["w"] = P["bg0"]
|
||||
mpl.colors.get_named_colors_mapping()["black"] = P["fg0"]
|
||||
for color in color_order:
|
||||
mpl.colors.get_named_colors_mapping()[color] = P[f"fg-{color}"]
|
||||
|
||||
|
||||
|
||||
|
||||
@ -1,32 +0,0 @@
|
||||
# Specify the auxiliary and output directories
|
||||
$aux_dir = '../.aux';
|
||||
$out_dir = '../out';
|
||||
|
||||
# Set lualatex as the default engine
|
||||
$pdf_mode = 1; # Enable PDF generation mode
|
||||
# $pdflatex = 'lualatex --interaction=nonstopmode --shell-escape'
|
||||
$lualatex = 'lualatex %O --interaction=nonstopmode --shell-escape %S';
|
||||
|
||||
# Additional options for compilation
|
||||
# '-verbose',
|
||||
# '-file-line-error',
|
||||
|
||||
ensure_path('TEXINPUTS', './pkg');
|
||||
# Quickfix-like filtering (warnings to ignore)
|
||||
# @warnings_to_filter = (
|
||||
# qr/Underfull \\hbox/,
|
||||
# qr/Overfull \\hbox/,
|
||||
# qr/LaTeX Warning: .+ float specifier changed to/,
|
||||
# qr/LaTeX hooks Warning/,
|
||||
# qr/Package siunitx Warning: Detected the "physics" package:/,
|
||||
# qr/Package hyperref Warning: Token not allowed in a PDF string/
|
||||
# );
|
||||
|
||||
# # Filter function for warnings
|
||||
# sub filter_warnings {
|
||||
# my $warning = shift;
|
||||
# foreach my $filter (@warnings_to_filter) {
|
||||
# return 0 if $warning =~ $filter;
|
||||
# }
|
||||
# return 1;
|
||||
# }
|
||||
125
src/analysis.tex
Normal file
125
src/analysis.tex
Normal file
@ -0,0 +1,125 @@
|
||||
\Part[
|
||||
\eng{Calculus}
|
||||
\ger{Analysis}
|
||||
]{cal}
|
||||
|
||||
\Subsection[
|
||||
\eng{Convolution}
|
||||
\ger{Faltung / Konvolution}
|
||||
]{conv}
|
||||
\begin{ttext}
|
||||
\eng{Convolution is \textbf{commutative}, \textbf{associative} and \textbf{distributive}.}
|
||||
\ger{Die Faltung ist \textbf{kommutativ}, \textbf{assoziativ} und \textbf{distributiv}}
|
||||
\end{ttext}
|
||||
\begin{formula}{def}
|
||||
\desc{Definition}{}{}
|
||||
\desc[german]{Definition}{}{}
|
||||
\eq{(f*g)(t) = f(t) * g(t) = int_{-\infty}^\infty f(\tau) g(t-\tau) \d \tau}
|
||||
\end{formula}
|
||||
\begin{formula}{notation}
|
||||
\desc{Notation}{}{}
|
||||
\desc[german]{Notation}{}{}
|
||||
\eq{
|
||||
f(t) * g(t-t_0) &= (f*g)(t-t_0) \\
|
||||
f(t-t_0) * g(t-t_0) &= (f*g)(t-2t_0)
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{commutativity}
|
||||
\desc{Commutativity}{}{}
|
||||
\desc[german]{Kommutativität}{}{}
|
||||
\eq{f * g = g * f}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{associativity}
|
||||
\desc{Associativity}{}{}
|
||||
\desc[german]{Assoziativität]}{}{}
|
||||
\eq{(f*g)*h = f*(g*h)}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{distributivity}
|
||||
\desc{Distributivity}{}{}
|
||||
\desc[german]{Distributivität}{}{}
|
||||
\eq{f * (g + h) = f*g + f*h}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{complex_conjugate}
|
||||
\desc{Complex conjugate}{}{}
|
||||
\desc[german]{Komplexe konjugation}{}{}
|
||||
\eq{(f*g)^* = f^* * g^*}
|
||||
\end{formula}
|
||||
|
||||
\Subsection[
|
||||
\eng{Fourier analysis}
|
||||
\ger{Fourieranalyse}
|
||||
]{fourier}
|
||||
\Subsubsection[
|
||||
\eng{Fourier series}
|
||||
\ger{Fourierreihe}
|
||||
]{series}
|
||||
\begin{formula}{series}
|
||||
\desc{Fourier series}{Complex representation}{$f\in \Lebesgue^2(\R,\C)$ $T$-\GT{periodic}}
|
||||
\desc[german]{Fourierreihe}{Komplexe Darstellung}{}
|
||||
\eq{f(t) = \sum_{k=-\infty}^{\infty} c_k \Exp{\frac{2\pi \I kt}{T}}}
|
||||
\end{formula}
|
||||
\Eng[real]{real}
|
||||
\Ger[real]{reellwertig}
|
||||
\begin{formula}{coefficient}
|
||||
\desc{Fourier coefficients}{Complex representation}{}
|
||||
\desc[german]{Fourierkoeffizienten}{Komplexe Darstellung}{}
|
||||
\eq{
|
||||
c_k &= \frac{1}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} f(t)\,\Exp{-\frac{2\pi \I}{T}kt}\d t \quad\text{\GT{for}}\,k\ge0\\
|
||||
c_{-k} &= \overline{c_k} \quad \text{\GT{if} $f$ \GT{real}}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{series_sincos}
|
||||
\desc{Fourier series}{Sine and cosine representation}{$f\in \Lebesgue^2(\R,\C)$ $T$-\GT{periodic}}
|
||||
\desc[german]{Fourierreihe}{Sinus und Kosinus Darstellung}{}
|
||||
\eq{f(t) = \frac{a_0}{2} + \sum_{k=1}^{\infty} \left(a_k \Cos{\frac{2\pi}{T}kt} + b_k\Sin{\frac{2\pi}{T}kt}\right)}
|
||||
\end{formula}
|
||||
\begin{formula}{coefficient}
|
||||
\desc{Fourier coefficients}{Sine and cosine representation\\If $f$ has point symmetry: $a_{k>0}=0$, if $f$ has axial symmetry: $b_k=0$}{}
|
||||
\desc[german]{Fourierkoeffizienten}{Sinus und Kosinus Darstellung\\Wenn $f$ punktsymmetrisch: $a_{k>0}=0$, wenn $f$ achsensymmetrisch: $b_k=0$}{}
|
||||
\eq{
|
||||
a_k &= \frac{2}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} f(t)\,\Cos{-\frac{2\pi}{T}kt}\d t \quad\text{\GT{for}}\,k\ge0\\
|
||||
b_k &= \frac{2}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} f(t)\,\Sin{-\frac{2\pi}{T}kt}\d t \quad\text{\GT{for}}\,k\ge1\\
|
||||
a_k &= c_k + c_{-k} \quad\text{\GT{for}}\,k\ge0\\
|
||||
b_k &= \I(c_k - c_{-k}) \quad\text{\GT{for}}\,k\ge1
|
||||
}
|
||||
\end{formula}
|
||||
\TODO{cleanup}
|
||||
|
||||
|
||||
|
||||
\Subsubsection[
|
||||
\eng{Fourier transformation}
|
||||
\ger{Fouriertransformation}
|
||||
]{trafo}
|
||||
\begin{formula}{transform}
|
||||
\desc{Fourier transform}{}{$\hat{f}:\R^n \mapsto \C$, $\forall f\in L^1(\R^n)$}
|
||||
\desc[german]{Fouriertransformierte}{}{}
|
||||
\eq{\hat{f}(k) \coloneq \frac{1}{\sqrt{2\pi}^n} \int_{\R^n} \e^{-\I kx}f(x)\d x}
|
||||
\end{formula}
|
||||
|
||||
\Eng[linear_in]{linear in}
|
||||
\Ger[linear_in]{linear in}
|
||||
\GT{for} $f\in L^1(\R^n)$:
|
||||
\begin{enumerate}[i)]
|
||||
\item $f \mapsto \hat{f}$ \GT{linear_in} $f$
|
||||
\item $g(x) = f(x-h) \qRarrow \hat{g}(k) = \e^{-\I kn}\hat{f}(k)$
|
||||
\item $g(x) = \e^{ih\cdot x}f(x) \qRarrow \hat{g}(k) = \hat{f}(k-h)$
|
||||
\item $g(\lambda) = f\left(\frac{x}{\lambda}\right) \qRarrow \hat{g}(k)\lambda^n \hat{f}(\lambda k)$
|
||||
\end{enumerate}
|
||||
|
||||
|
||||
\Section[
|
||||
\eng{List of common integrals}
|
||||
\ger{Liste nützlicher Integrale}
|
||||
]{integrals}
|
||||
|
||||
\begin{formula}{riemann_zeta}
|
||||
\desc{Riemann Zeta Function}{}{}
|
||||
\desc[german]{Riemannsche Zeta-Funktion}{}{}
|
||||
\eq{\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} = \frac{1}{(1-2^{(1-s)})\Gamma(s)} \int_0^\infty \d\eta \frac{\eta^{(s-1)}}{\e^\eta + 1}}
|
||||
\end{formula}
|
||||
|
||||
@ -1,20 +0,0 @@
|
||||
\Part{appendix}
|
||||
\desc{Appendix}{}{}
|
||||
\desc[german]{Anhang}{}{}
|
||||
\begin{formula}{world}
|
||||
\desc{World formula}{}{}
|
||||
\desc[german]{Weltformel}{}{}
|
||||
\eq{E = mc^2 +\text{AI}}
|
||||
\end{formula}
|
||||
\Input{quantities}
|
||||
\Input{constants}
|
||||
|
||||
% \listofquantities
|
||||
% \listoffigures
|
||||
% \listoftables
|
||||
\Section{elements}
|
||||
\desc{List of elements}{}{}
|
||||
\desc[german]{Liste der Elemente}{}{}
|
||||
\printAllElements
|
||||
\newpage
|
||||
|
||||
182
src/atom.tex
Normal file
182
src/atom.tex
Normal file
@ -0,0 +1,182 @@
|
||||
\def\vecr{{\vec{r}}}
|
||||
\def\abohr{a_\textrm{B}}
|
||||
|
||||
\Section[
|
||||
\eng{Hydrogen Atom}
|
||||
\ger{Wasserstoffatom}
|
||||
]{h}
|
||||
|
||||
\begin{formula}{reduced_mass}
|
||||
\desc{Reduced mass}{}{}
|
||||
\desc[german]{Reduzierte Masse}{}{}
|
||||
\eq{\mu = \frac{\masse m_\textrm{K}}{\masse + m_\textrm{K}} \explOverEq[\approx]{$\masse \ll m_\textrm{K}$} \masse}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{potential}
|
||||
\desc{Coulumb potential}{For a single electron atom}{$Z$ atomic number}
|
||||
\desc[german]{Coulumb potential}{Für ein Einelektronenatom}{$Z$ Ordnungszahl/Kernladungszahl}
|
||||
\eq{V(\vecr) = \frac{Z\,e^2}{4\pi\epsilon_0 r}}
|
||||
\end{formula}
|
||||
\begin{formula}{hamiltonian}
|
||||
\desc{Hamiltonian}{}{}
|
||||
\desc[german]{Hamiltonian}{}{}
|
||||
% \eq{V(\vecr) = \frac{Z\,e^2}{4\pi\epsilon_0 r}}
|
||||
\eq{
|
||||
\hat{H} &= -\frac{\hbar^2}{2\mu} {\Grad_\vecr}^2 - V(\vecr) \\
|
||||
&= \frac{\hat{p}_r^2}{2\mu} + \frac{\hat{L}^2}{2\mu r} + V(r)
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{wave_function}
|
||||
\desc{Wave function}{}{}
|
||||
\desc[german]{Wellenfunktion}{}{}
|
||||
\eq{\psi_{nlm}(r, \theta, \phi) = R_{nl}(r)Y_{lm}(\theta,\phi)}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{radial}
|
||||
\desc{Radial part}{}{$L_r^s(x)$ Laguerre-polynomials}
|
||||
\desc[german]{Radialanteil}{}{$L_r^s(x)$ Laguerre-Polynome}
|
||||
\eq{
|
||||
R_{nl} &= - \sqrt{\frac{(n-l-1)!(2\kappa)^3}{2n[(n+l)!]^3}} (2\kappa r)^l \e^{-\kappa r} L_{n+1}^{2l+1}(2\kappa r)
|
||||
\shortintertext{\GT{with}}
|
||||
\kappa &= \frac{\sqrt{2\mu\abs{E}}}{\hbar} = \frac{Z}{n \abohr}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{energy}
|
||||
\desc{Energy eigenvalues}{}{}
|
||||
\desc[german]{Energieeigenwerte}{}{}
|
||||
\eq{E_n &= \frac{Z^2\mu e^4}{n^2(4\pi\epsilon_0)^2 2\hbar^2} = -E_\textrm{H}\frac{Z^2}{n^2}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{rydberg_energy}
|
||||
\desc{Rydberg energy}{}{}
|
||||
\desc[german]{Rydberg-Energy}{}{}
|
||||
\eq{E_\textrm{H} = h\,c\,R_\textrm{H} = \frac{\mu e^4}{(4\pi\epsilon_0)^2 2\hbar^2}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsection[
|
||||
\eng{Corrections}
|
||||
\ger{Korrekturen}
|
||||
]{corrections}
|
||||
|
||||
\Subsubsection[
|
||||
\eng{Darwin term}
|
||||
\ger{Darwin-Term}
|
||||
]{darwin}
|
||||
\begin{ttext}[desc]
|
||||
\eng{Relativisitc correction: Because of the electrons zitterbewegung, it is not entirely localised. \TODO{fact check}}
|
||||
\ger{Relativistische Korrektur: Elektronen führen eine Zitterbewegung aus und sind nicht vollständig lokalisiert.}
|
||||
\end{ttext}
|
||||
\begin{formula}{energy_shift}
|
||||
\desc{Energy shift}{}{}
|
||||
\desc[german]{Energieverschiebung}{}{}
|
||||
\eq{\Delta E_\textrm{rel} = -E_n \frac{Z^2\alpha^2}{n} \Big(\frac{3}{4n} - \frac{1}{l+ \frac{1}{2}}\Big)}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{fine_structure_constant}
|
||||
\desc{Fine-structure constant}{Sommerfeld constant}{}
|
||||
\desc[german]{Feinstrukturkonstante}{Sommerfeldsche Feinstrukturkonstante}{}
|
||||
\eq{\alpha = \frac{e^2}{4\pi\epsilon_0\hbar c} \approx \frac{1}{137}}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection[
|
||||
\eng{Spin-orbit coupling (LS-coupling)}
|
||||
\ger{Spin-Bahn-Kopplung (LS-Kopplung)}
|
||||
]{ls_coupling}
|
||||
\begin{ttext}[desc]
|
||||
\eng{The interaction of the electron spin with the electrostatic field of the nuclei lead to energy shifts.}
|
||||
\ger{The Wechselwirkung zwischen dem Elektronenspin und dem elektrostatischen Feld des Kerns führt zu Energieverschiebungen.}
|
||||
\end{ttext}
|
||||
|
||||
\begin{formula}{energy_shift}
|
||||
\desc{Energy shift}{}{}
|
||||
\desc[german]{Energieverschiebung}{}{}
|
||||
\eq{\Delta E_\text{LS} = \frac{\mu_0 Z e^2}{8\pi \masse^2\,r^3} \braket{\vec{S} \cdot \vec{L}}}
|
||||
\end{formula}
|
||||
\begin{formula}{sl}
|
||||
\desc{\TODO{name}}{}{}
|
||||
\desc[german]{??}{}{}
|
||||
\eq{\braket{\vec{S} \cdot \vec{L}} &= \frac{1}{2} \braket{[J^2-L^2-S^2]} \nonumber \\
|
||||
&= \frac{\hbar^2}{2}[j(j+1) -l(l+1) -s(s+1)]}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection[
|
||||
\eng{Fine-structure}
|
||||
\ger{Feinstruktur}
|
||||
]{fine_structure}
|
||||
\begin{ttext}[desc]
|
||||
\eng{The fine-structure combines relativistic corrections \ref{sec:qm:h:corrections:darwin} and the spin-orbit coupling \ref{sec:qm:h:corrections:ls_coupling}.}
|
||||
\ger{Die Feinstruktur vereint relativistische Korrekturen \ref{sec:qm:h:corrections:darwin} und die Spin-Orbit-Kupplung \ref{sec:qm:h:corrections:ls_coupling}.}
|
||||
\end{ttext}
|
||||
\begin{formula}{energy_shift}
|
||||
\desc{Energy shift}{}{}
|
||||
\desc[german]{Energieverschiebung}{}{}
|
||||
\eq{\Delta E_\textrm{FS} = \frac{Z^2\alpha^2}{n}\Big(\frac{1}{j+\frac{1}{2}} - \frac{3}{4n}\Big)}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsubsection[
|
||||
\eng{Lamb-shift}
|
||||
\ger{Lamb-Shift}
|
||||
]{lamb_shift}
|
||||
\begin{ttext}[desc]
|
||||
\eng{The interaction of the electron with virtual photons emitted/absorbed by the nucleus leads to a (very small) shift in the energy level.}
|
||||
\ger{The Wechselwirkung zwischen dem Elektron und vom Kern absorbierten/emittierten virtuellen Photonen führt zu einer (sehr kleinen) Energieverschiebung.}
|
||||
\end{ttext}
|
||||
\begin{formula}{energy}
|
||||
\desc{Potential energy}{}{$\delta r$ pertubation of $r$}
|
||||
\desc[german]{Potentielle Energy}{}{$\delta r$ Schwankung von $r$}
|
||||
\eq{\braket{E_\textrm{pot}} = -\frac{Z e^2}{4\pi\epsilon_0} \Braket{\frac{1}{r+\delta r}}}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection[
|
||||
\eng{Hyperfine structure}
|
||||
\ger{Hyperfeinstruktur}
|
||||
]{hyperfine_structure}
|
||||
\begin{ttext}[desc]
|
||||
\eng{Interaction of the nucleus spin with the magnetic field created by the electron leads to energy shifts. (Lifts degeneracy) }
|
||||
\ger{Wechselwirkung von Kernspin mit dem vom Elektron erzeugten Magnetfeld spaltet Energieniveaus}
|
||||
\end{ttext}
|
||||
\begin{formula}{nuclear_spin}
|
||||
\desc{Nuclear spin}{}{}
|
||||
\desc[german]{Kernspin}{}{}
|
||||
\eq{\vec{F} &= \vec{J} + \vec{I} \\
|
||||
\abs{\vec{I}} &= \sqrt{i(i+1)}\hbar \\
|
||||
I_z &= m_i\hbar \\
|
||||
m_i &= -i, -i+1, \dots, i-1, i
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{angular_momentum}
|
||||
\desc{Combined angular momentum}{}{}
|
||||
\desc[german]{Gesamtdrehimpuls}{}{}
|
||||
\eq{\vec{F} &= \vec{J} + \vec{I} \\
|
||||
\abs{\vec{F}} &= \sqrt{f(f+1)}\hbar \\
|
||||
F_z &= m_f\hbar
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{selection_rule}
|
||||
\desc{Selection rule}{}{}
|
||||
\desc[german]{Auswahlregel}{}{}
|
||||
\eq{f &= j \pm i \\ m_f &= -f,-f+1,\dots,f-1,f}
|
||||
\end{formula}
|
||||
\begin{formula}{constant}
|
||||
\desc{Hyperfine structure constant}{}{$B_\textrm{HFS}$ hyperfine field, $\mu_\textrm{K}$ nuclear magneton, $g_i$ nuclear g-factor \ref{qm:h:lande}}
|
||||
\desc[german]{Hyperfeinstrukturkonstante}{}{$B_\textrm{HFS}$ Hyperfeinfeld, $\mu_\textrm{K}$ Kernmagneton, $g_i$ Kern-g-Faktor \ref{qm:h:lande}}
|
||||
\eq{A = \frac{g_i \mu_\textrm{K} B_\textrm{HFS}}{\sqrt{j(j+1)}}}
|
||||
\end{formula}
|
||||
\begin{formula}{energy_shift}
|
||||
\desc{Energy shift}{}{}
|
||||
\desc[german]{Energieverschiebung}{}{}
|
||||
\eq{\Delta H_\textrm{HFS} = \frac{A}{2}[f(f+1) - j(j+1) -i(i+1)]}
|
||||
\end{formula}
|
||||
\TODO{landé factor}
|
||||
|
||||
\Subsection[
|
||||
\eng{Effects in magnetic field}
|
||||
\ger{Effekte im Magnetfeld}
|
||||
]{mag_effects}
|
||||
\TODO{all}
|
||||
|
||||
\\\TODO{Hunds rules}
|
||||
55
src/calculus.tex
Normal file
55
src/calculus.tex
Normal file
@ -0,0 +1,55 @@
|
||||
\Part[
|
||||
\eng{Analysis}
|
||||
\ger{Analysis}
|
||||
]{ana}
|
||||
|
||||
\Subsection[
|
||||
\eng{Convolution}
|
||||
\ger{Faltung / Konvolution}
|
||||
]{conv}
|
||||
\begin{ttext}
|
||||
\eng{Convolution is \textbf{commutative}, \textbf{associative} and \textbf{distributive}.}
|
||||
\ger{Die Faltung ist \textbf{kommutativ}, \textbf{assoziativ} und \textbf{distributiv}}
|
||||
\end{ttext}
|
||||
\begin{formula}{def}
|
||||
\desc{Definition}{}{}
|
||||
\desc[german]{Definition}{}{}
|
||||
\eq{(f*g)(t) = f(t) * g(t) = int_{-\infty}^\infty f(\tau) g(t-\tau) \d \tau}
|
||||
\end{formula}
|
||||
\begin{formula}{notation}
|
||||
\desc{Notation}{}{}
|
||||
\desc[german]{Notation}{}{}
|
||||
\eq{
|
||||
f(t) * g(t-t_0) &= (f*g)(t-t_0) \\
|
||||
f(t-t_0) * g(t-t_0) &= (f*g)(t-2t_0)
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{commutativity}
|
||||
\desc{Commutativity}{}{}
|
||||
\desc[german]{Kommutativität}{}{}
|
||||
\eq{f * g = g * f}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{associativity}
|
||||
\desc{Associativity}{}{}
|
||||
\desc[german]{Assoziativität]}{}{}
|
||||
\eq{(f*g)*h = f*(g*h)}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{distributivity}
|
||||
\desc{Distributivity}{}{}
|
||||
\desc[german]{Distributivität}{}{}
|
||||
\eq{f * (g + h) = f*g + f*h}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{complex_conjugate}
|
||||
\desc{Complex conjugate}{}{}
|
||||
\desc[german]{Komplexe konjugation}{}{}
|
||||
\eq{(f*g)^* = f^* * g^*}
|
||||
\end{formula}
|
||||
|
||||
\Subsection[
|
||||
\eng{Fourier analysis}
|
||||
\ger{Fourieranalyse}
|
||||
]{fourier}
|
||||
|
||||
@ -1,8 +0,0 @@
|
||||
\Part{ch}
|
||||
\desc{Chemistry}{}{}
|
||||
\desc[german]{Chemie}{}{}
|
||||
|
||||
\Section{ptable}
|
||||
\desc{Periodic table}{}{}
|
||||
\desc[german]{Periodensystem}{}{}
|
||||
\drawPeriodicTable
|
||||
678
src/ch/el.tex
678
src/ch/el.tex
@ -1,678 +0,0 @@
|
||||
\Section{el}
|
||||
\desc{Electrochemistry}{}{}
|
||||
\desc[german]{Elektrochemie}{}{}
|
||||
|
||||
\begin{formula}{chemical_potential}
|
||||
\desc{Chemical potential}{of species $i$\\Energy involved when the particle number changes}{\QtyRef{free_enthalpy}, \QtyRef{amount}}
|
||||
\desc[german]{Chemisches Potential}{der Spezies $i$\\Involvierte Energie, wenn sich die Teilchenzahl ändert}{}
|
||||
\quantity{\mu}{\joule\per\mol;\joule}{is}
|
||||
\eq{
|
||||
\mu_i \equiv \pdv{G}{n_i}_{n_j\neq n_i,p,T}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{standard_chemical_potential}
|
||||
\desc{Standard chemical potential}{In equilibrium}{\QtyRef{chemical_potential}, \ConstRef{universal_gas}, \QtyRef{temperature}, \QtyRef{activity}}
|
||||
\desc[german]{Standard chemisches Potential}{}{}
|
||||
\eq{\mu_i = \mu_i^\theta + RT \Ln{a_i}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{chemical_equilibrium}
|
||||
\desc{Chemical equilibrium}{}{\QtyRef{chemical_potential}, \QtyRef{stoichiometric_coefficient}}
|
||||
\desc[german]{Chemisches Gleichgewicht}{}{}
|
||||
\eq{\sum_\text{\GT{products}} \nu_i \mu_i = \sum_\text{\GT{educts}} \nu_i \mu_i}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{activity}
|
||||
\desc{Activity}{relative activity}{\QtyRef{chemical_potential}, \fRef{::standard_chemical_potential}, \ConstRef{universal_gas}, \QtyRef{temperature}}
|
||||
\desc[german]{Aktivität}{Relative Aktivität}{}
|
||||
\quantity{a}{}{s}
|
||||
\eq{a_i = \Exp{\frac{\mu_i-\mu_i^\theta}{RT}}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{electrochemical_potential}
|
||||
\desc{Electrochemical potential}{Chemical potential with electrostatic contributions}{\QtyRef{chemical_potential}, $z$ valency (charge), \ConstRef{faraday}, \QtyRef{electric_scalar_potential} (Galvani Potential)}
|
||||
\desc[german]{Elektrochemisches Potential}{Chemisches Potential mit elektrostatischen Enegiebeiträgen}{\QtyRef{chemical_potential}, $z$ Ladungszahl, \ConstRef{faraday}, \QtyRef{electric_scalar_potential} (Galvanisches Potential)}
|
||||
\quantity{\muecp}{\joule\per\mol;\joule}{is}
|
||||
\eq{\muecp_i \equiv \mu_i + z_i F \phi}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsection{cell}
|
||||
\desc{Electrochemical cell}{}{}
|
||||
\desc[german]{Elektrochemische Zelle}{}{}
|
||||
\eng[galvanic]{galvanic}
|
||||
\ger[galvanic]{galvanisch}
|
||||
\eng[electrolytic]{electrolytic}
|
||||
\ger[electrolytic]{electrolytisch}
|
||||
|
||||
\Eng[working_electrode]{Working electrode}
|
||||
\Eng[counter_electrode]{Counter electrode}
|
||||
\Eng[reference_electrode]{Reference electrode}
|
||||
\Ger[working_electrode]{Working electrode}
|
||||
\Ger[counter_electrode]{Gegenelektrode}
|
||||
\Ger[reference_electrode]{Referenzelektrode}
|
||||
\Eng[potentiostat]{Potentiostat}
|
||||
\Ger[potentiostat]{Potentiostat}
|
||||
|
||||
\begin{formula}{schematic}
|
||||
\desc{Schematic}{}{}
|
||||
\desc[german]{Aufbau}{}{}
|
||||
\begin{tikzpicture}[scale=1.0,transform shape]
|
||||
\pgfmathsetmacro{\W}{6}
|
||||
\pgfmathsetmacro{\H}{3}
|
||||
\pgfmathsetmacro{\elW}{\W/20}
|
||||
|
||||
\pgfmathsetmacro{\REx}{1/6*\W}
|
||||
\pgfmathsetmacro{\WEx}{3/6*\W}
|
||||
\pgfmathsetmacro{\CEx}{5/6*\W}
|
||||
|
||||
\fill[bg-blue] (0,0) rectangle (\W, \H/2);
|
||||
\draw[ultra thick] (0,0) rectangle (\W,\H);
|
||||
% Electrodes
|
||||
\draw[thick, fill=bg-gray] (\REx-\elW,\H/5) rectangle (\REx+\elW,\H);
|
||||
\draw[thick, fill=bg-purple] (\WEx-\elW,\H/5) rectangle (\WEx+\elW,\H);
|
||||
\draw[thick, fill=bg-yellow] (\CEx-\elW,\H/5) rectangle (\CEx+\elW,\H);
|
||||
\node at (\REx,3*\H/5) {R};
|
||||
\node at (\WEx,3*\H/5) {W};
|
||||
\node at (\CEx,3*\H/5) {C};
|
||||
|
||||
% potentiostat
|
||||
\pgfmathsetmacro{\potH}{\H+0.5+2}
|
||||
\pgfmathsetmacro{\potM}{\H+0.5+1}
|
||||
\draw[thick] (0,\H+0.5) rectangle (\W,\potH);
|
||||
% Wires
|
||||
\draw (\REx,\H) -- (\REx,\potM) to[voltmeter,-o] (\WEx,\potM) to[european voltage source] (\WEx+1/6*\W,\potM) to[ammeter] (\CEx,\potM);
|
||||
\draw (\WEx,\H) -- (\WEx,\H+1.5);
|
||||
\draw (\CEx,\H) -- (\CEx,\H+1.5);
|
||||
|
||||
% labels
|
||||
\node[anchor=west, align=left] at (\W+0.2, 1*\H/4) {{\color{bg-gray} \blacksquare} \GT{reference_electrode}};
|
||||
\node[anchor=west, align=left] at (\W+0.2, 2*\H/4) {{\color{bg-purple}\blacksquare} \GT{working_electrode}};
|
||||
\node[anchor=west, align=left] at (\W+0.2, 3*\H/4) {{\color{bg-yellow}\blacksquare} \GT{counter_electrode}};
|
||||
\node[anchor=west, align=left] at (\W+0.2, \potM) {\GT{potentiostat}};
|
||||
\end{tikzpicture}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{cell}
|
||||
\desc{Electrochemical cell types}{}{}
|
||||
\desc[german]{Arten der Elektrochemische Zelle}{}{}
|
||||
\ttxt{
|
||||
\eng{
|
||||
\begin{itemize}
|
||||
\item Electrolytic cell: Uses electrical energy to force a chemical reaction
|
||||
\item Galvanic cell: Produces electrical energy through a chemical reaction
|
||||
\end{itemize}
|
||||
}
|
||||
\ger{
|
||||
\begin{itemize}
|
||||
\item Elektrolytische Zelle: Nutzt elektrische Energie um eine Reaktion zu erzwingen
|
||||
\item Galvanische Zelle: Produziert elektrische Energie durch eine chemische Reaktion
|
||||
\end{itemize}
|
||||
}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
% todo group together
|
||||
\begin{formula}{faradaic}
|
||||
\desc{Faradaic process}{}{}
|
||||
\desc[german]{Faradäischer Prozess}{}{}
|
||||
\ttxt{
|
||||
\eng{Charge transfers between the electrode bulk and the electrolyte.}
|
||||
\ger{Ladung wird zwischen Elektrode und dem Elektrolyten transferiert.}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{non-faradaic}
|
||||
\desc{Non-Faradaic (capacitive) process}{}{}
|
||||
\desc[german]{Nicht-Faradäischer (kapazitiver) Prozess}{}{}
|
||||
\ttxt{
|
||||
\eng{Charge is stored at the electrode-electrolyte interface.}
|
||||
\ger{Ladung lagert sich am Elektrode-Elektrolyt Interface an.}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\begin{formula}{electrode_potential}
|
||||
\desc{Electrode potential}{}{}
|
||||
\desc[german]{Elektrodenpotential}{}{}
|
||||
\quantity{E}{\volt}{s}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{standard_cell_potential}
|
||||
\desc{Standard cell potential}{}{$\Delta_\txR G^\theta$ standard \qtyRef{free_enthalpy} of reaction, $n$ number of electrons, \ConstRef{faraday}}
|
||||
\desc[german]{Standard Zellpotential}{}{$\Delta_\txR G^\theta$ Standard \qtyRef{free_enthalpy} der Reaktion, $n$ Anzahl der Elektronen, \ConstRef{faraday}}
|
||||
\eq{E^\theta_\text{rev} = \frac{-\Delta_\txR G^\theta}{nF}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\begin{formula}{nernst_equation}
|
||||
\desc{Nernst equation}{Electrode potential for a half-cell reaction}{\QtyRef{electrode_potential}, $E^\theta$ \fRef{::standard_cell_potential}, \ConstRef{universal_gas}, \QtyRef{temperature}, $z$ \qtyRef{charge_number}, \ConstRef{faraday}, \QtyRef{activity}, \QtyRef{stoichiometric_coefficient}}
|
||||
\desc[german]{Nernst-Gleichung}{Elektrodenpotential für eine Halbzellenreaktion}{}
|
||||
\eq{E = E^\theta + \frac{RT}{zF} \Ln{\frac{ \left(\prod_{i}(a_i)^{\abs{\nu_i}}\right)_\text{oxidized}}{\left(\prod_{i}(a_i)^{\abs{\nu_i}}\right)_\text{reduced}}}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{cell_efficiency}
|
||||
\desc{Thermodynamic cell efficiency}{}{$P$ \fRef{ed:el:power}}
|
||||
\desc[german]{Thermodynamische Zelleffizienz}{}{}
|
||||
\eq{
|
||||
\eta_\text{cell} &= \frac{P_\text{obtained}}{P_\text{maximum}} = \frac{E_\text{cell}}{E_\text{cell,rev}} & & \text{\gt{galvanic}} \\
|
||||
\eta_\text{cell} &= \frac{P_\text{minimum}}{P_\text{applied}} = \frac{E_\text{cell,rev}}{E_\text{cell}} & & \text{\gt{electrolytic}}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsection{ion_cond}
|
||||
\desc{Ionic conduction in electrolytes}{}{}
|
||||
\desc[german]{Ionische Leitung in Elektrolyten}{}{}
|
||||
\eng[z]{charge number}
|
||||
\ger[z]{Ladungszahl}
|
||||
\eng[of_i]{of ion $i$}
|
||||
\ger[of_i]{des Ions $i$}
|
||||
|
||||
\begin{formula}{diffusion}
|
||||
\desc{Diffusion}{caused by concentration gradients}{$z_i$ \qtyRef{charge_number} \gt{of_i}, \ConstRef{faraday}, \QtyRef{diffusion_coefficient} \gt{of_i}, \QtyRef{concentration} \gt{of_i}}
|
||||
\desc[german]{Diffusion}{durch Konzentrationsgradienten}{}
|
||||
\eq{ i_\text{diff} = \sum_i -z_i F D_i \left(\odv{c_i}{x}\right) }
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{migration}
|
||||
\desc{Migration}{caused by potential gradients}{$z_i$ \qtyRef{charge_number} \gt{of_i}, \ConstRef{faraday}, \QtyRef{concentration} \gt{of_i}, \QtyRef{mobility} \gt{of_i}, $\nabla\phi_\txs$ potential gradient in the solution}
|
||||
\desc[german]{Migration}{durch Potentialgradienten}{$z_i$ \qtyRef{charge_number} \gt{of_i}, \ConstRef{faraday}, \QtyRef{concentration} \gt{of_i}, \QtyRef{mobility} \gt{of_i}, $\nabla\phi_\txs$ Potentialgradient in der Lösung}
|
||||
\eq{ i_\text{mig} = \sum_i -z_i^2 F^2 \, c_i \, \mu_i \, \nabla\Phi_\txs }
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{convection}
|
||||
\desc{Convection}{caused by pressure gradients}{$z_i$ \qtyRef{charge_number} \gt{of_i}, \ConstRef{faraday}, \QtyRef{concentration} \gt{of_i}, $v_i^\text{flow}$ \qtyRef{velocity} \gt{of_i} in flowing electrolyte}
|
||||
\desc[german]{Convection}{durch Druckgradienten}{$z_i$ \qtyRef{charge_number} \gt{of_i}, \ConstRef{faraday}, \QtyRef{concentration} \gt{of_i}, $v_i^\text{flow}$ \qtyRef{velocity} \gt{of_i} im fliessenden Elektrolyt}
|
||||
\eq{ i_\text{conv} = \sum_i -z_i F \, c_i \, v_i^\text{flow} }
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{ionic_mobility}
|
||||
\desc{Ionic mobility}{}{$v_\pm$ steady state drift \qtyRef{velocity}, $\phi$ \qtyRef{electric_scalar_potential}, $z$ \qtyRef{charge_number}, \ConstRef{faraday}, \ConstRef{charge}, \QtyRef{viscosity}, $r_\pm$ ion radius}
|
||||
\desc[german]{Ionische Moblilität}{}{}
|
||||
\quantity{u_\pm}{\cm^2\mol\per\joule\s}{}
|
||||
\eq{u_\pm = - \frac{v_\pm}{\nabla \phi \,z_\pm F} = \frac{e}{6\pi F \eta_\text{dyn} r_\pm}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{stokes_friction}
|
||||
\desc{Stokes's law}{Frictional force exerted on spherical objects moving in a viscous fluid at low Reynolds numbers}{$r$ particle radius, \QtyRef{viscosity}, $v$ particle \qtyRef{velocity}}
|
||||
\desc[german]{Gesetz von Stokes}{Reibungskraft auf ein sphärisches Objekt in einer Flüssigkeit bei niedriger Reynolds-Zahl}{$r$ Teilchenradius, \QtyRef{viscosity}, $v$ Teilchengeschwindigkeit}
|
||||
\eq{F_\txR = 6\pi\,r \eta v}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\begin{formula}{ionic_conductivity}
|
||||
\desc{Ionic conductivity}{}{\ConstRef{faraday}, $z_i$, $c_i$, $u_i$ charge number, \qtyRef{concentration} and \qtyRef{ionic_mobility} of the positive (+) and negative (-) ions}
|
||||
\desc[german]{Ionische Leitfähigkeit}{}{\ConstRef{faraday}, $z_i$, $c_i$, $u_i$ Ladungszahl, \qtyRef{concentration} und \qtyRef{ionic_mobility} der positiv (+) und negativ geladenen Ionen}
|
||||
\quantity{\kappa}{\per\ohm\cm=\siemens\per\cm}{}
|
||||
\eq{\kappa = F^2 \left(z_+^2 \, c_+ \, u_+ + z_-^2 \, c_- \, u_-\right)}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{ionic_resistance}
|
||||
\desc{Ohmic resistance of ionic current flow}{}{$L$ \qtyRef{length}, $A$ \qtyRef{area}, \QtyRef{ionic_conductivity}}
|
||||
\desc[german]{Ohmscher Widerstand für Ionen-Strom}{}{}
|
||||
\eq{R_\Omega = \frac{L}{A\,\kappa}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{transference}
|
||||
\desc{Transference number}{Ion transport number \\Fraction of the current carried by positive / negative ions}{$i_{+/-}$ current through positive/negative charges}
|
||||
\desc[german]{Überführungszahl}{Anteil der positiv / negativ geladenen Ionen am Gesamtstrom}{$i_{+/-}$ Strom durch positive / negative Ladungn}
|
||||
\eq{t_{+/-} = \frac{i_{+/-}}{i_+ + i_-}}
|
||||
\end{formula}
|
||||
|
||||
\eng[csalt]{electrolyte \qtyRef{concentration}}
|
||||
\eng[csalt]{\qtyRef{concentration} des Elektrolyts}
|
||||
\begin{formula}{molar_conductivity}
|
||||
\desc{Molar conductivity}{}{\QtyRef{ionic_conductivity}, $c_\text{salt}$ \gt{csalt}}
|
||||
\desc[german]{Molare Leitfähigkeit}{}{\QtyRef{ionic_conductivity}, $c_\text{salt}$ \gt{salt}}
|
||||
\quantity{\Lambda_\txM}{\siemens\cm^2\per\mol=\ampere\cm^2\per\volt\mol}{ievs}
|
||||
\eq{\Lambda_\txM = \frac{\kappa}{c_\text{salt}}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{kohlrausch_law}
|
||||
\desc{Kohlrausch's law}{For strong electrolytes}{$\Lambda_\txM^0$ \qtyRef{molar_conductivity} at infinite dilution, $c_\text{salt}$ \gt{csalt}, $K$ \GT{constant}}
|
||||
\desc[german]{}{}{$\Lambda_\txM^0$ \qtyRef{molar_conductivity} bei unendlicher Verdünnung, $\text{salt}$ \gt{csalt},$K$ \GT{constant}}
|
||||
\eq{\Lambda_\txM = \Lambda_\txM^0 - K \sqrt{c_\text{salt}}}
|
||||
\fig{img/ch_kohlrausch.pdf}
|
||||
\end{formula}
|
||||
|
||||
% Electrolyte conductivity
|
||||
\begin{formula}{molality}
|
||||
\desc{Molality}{Amount per mass}{\QtyRef{amount} of the solute, \QtyRef{mass} of the solvent}
|
||||
\desc[german]{Molalität}{Stoffmenge pro Masse}{\QtyRef{amount} des gelösten Stoffs, \QtyRef{mass} des Lösungsmittels}
|
||||
\quantity{b}{\mol\per\kg}{}
|
||||
\eq{b = \frac{n}{m}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{molarity}
|
||||
\desc{Molarity}{Amount per volume\\\qtyRef{concentration}}{\QtyRef{amount} of the solute, \QtyRef{volume} of the solvent}
|
||||
\desc[german]{Molarität}{Stoffmenge pro Volumen\\\qtyRef{concentration}}{\QtyRef{amount} des gelösten Stoffs, \QtyRef{volume} des Lösungsmittels}
|
||||
\quantity{c}{\mol\per\litre}{}
|
||||
\eq{c = \frac{n}{V}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{ionic_strength}
|
||||
\desc{Ionic strength}{Measure of the electric field in a solution through solved ions}{\QtyRef{molality}, \QtyRef{molarity}, $z$ \qtyRef{charge_number}}
|
||||
\desc[german]{Ionenstärke}{Maß einer Lösung für die elektrische Feldstärke durch gelöste Ionen}{}
|
||||
\quantity{I}{\mol\per\kg;\mol\per\litre}{}
|
||||
\eq{I_b &= \frac{1}{2} \sum_i b_i z_i^2 \\ I_c &= \frac{1}{2} \sum_i c_i z_i^2}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{debye_screening_length}
|
||||
\desc{Debye screening length}{}{\ConstRef{avogadro}, \ConstRef{charge}, \QtyRef{ionic_strength}, \QtyRef{permittivity}, \ConstRef{boltzmann}, \QtyRef{temperature}}
|
||||
\desc[german]{Debye-Länge / Abschirmlänge}{}{}
|
||||
\eq{\lambda_\txD = \sqrt{\frac{\epsilon \kB T}{2\NA e^2 I_C}}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{mean_ionic_activity}
|
||||
\desc{Mean ionic activity coefficient}{Accounts for decreased reactivity because ions must divest themselves of their ion cloud before reacting}{\QtyRef{activity}, $m_i$ \qtyRef{molality}, $m_0 = \SI{1}{\mol\per\kg}$}
|
||||
\desc[german]{Mittlerer ionischer Aktivitätskoeffizient}{Berücksichtigt dass Ionen sich erst von ihrer Ionenwolke lösen müssen, bevor sie reagieren können}{}
|
||||
\quantity{\gamma}{}{s}
|
||||
\eq{\gamma_\pm = \left(\gamma_+^{\nu_+} \, \gamma_-^{\nu_-}\right)^{\frac{1}{\nu_+ + \nu_-}}}
|
||||
\eq{a_i \equiv \gamma_i \frac{m_i}{m^0}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{debye_hueckel_law}
|
||||
\desc{Debye-Hückel limiting law}{For an infinitely dilute solution}{\QtyRef{mean_ionic_activity}, $A$ solvent dependant constant, $z$ \qtyRef{charge_number}, \QtyRef{ionic_strength} in [\si{\mol\per\kg}]}
|
||||
\desc[german]{Debye-Hückel Gesetz}{Für eine unendlich verdünnte Lösung}{}
|
||||
\eq{\Ln{\gamma_{\pm}} = -A \abs{z_+ \, z_-} \sqrt{I_b}}
|
||||
\end{formula}
|
||||
|
||||
\Subsection{kin}
|
||||
\desc{Kinetics}{}{}
|
||||
\desc[german]{Kinetik}{}{}
|
||||
\begin{formula}{transfer_coefficient}
|
||||
\desc{Transfer coefficient}{}{}
|
||||
\desc[german]{Durchtrittsfaktor}{Transferkoeffizient\\Anteil des Potentials der sich auf die freie Reaktionsenthalpie des anodischen Prozesses auswirkt}{}
|
||||
\eq{
|
||||
\alpha_\txA &= \alpha \\
|
||||
\alpha_\txC &= 1-\alpha
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{overpotential}
|
||||
\desc{Overpotential}{}{}
|
||||
\desc[german]{Überspannung}{}{}
|
||||
\ttxt{
|
||||
\eng{Potential deviation from the equilibrium cell potential}
|
||||
\ger{Abweichung der Spannung von der Zellspannung im Gleichgewicht}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{activation_overpotential}
|
||||
\desc{Activation verpotential}{}{$E_\text{electrode}$ potential at which the reaction starts $E_\text{ref}$ thermodynamic potential of the reaction}
|
||||
\desc[german]{Aktivierungsüberspannung}{}{$E_\text{electrode}$ Potential bei der die Reaktion beginnt, $E_\text{ref}$ thermodynamisches Potential der Reaktion}
|
||||
\eq{\eta_\text{act} = E_\text{electrode} - E_\text{ref}}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection{mass}
|
||||
\desc{Mass transport}{}{}
|
||||
\desc[german]{Massentransport}{}{}
|
||||
\begin{formula}{concentration_overpotential}
|
||||
\desc{Concentration overpotential}{Due to concentration gradient near the electrode, the ions need to \fRef[diffuse]{ch:el:ion_cond:diffusion} to the electrode before reacting}{\ConstRef{universal_gas}, \QtyRef{temperature}, $\c_{0/\txS}$ ion concentration in the electrolyte / at the double layer, $z$ \qtyRef{charge_number}, \ConstRef{faraday}}
|
||||
\desc[german]{Konzentrationsüberspannung}{Durch einen Konzentrationsgradienten an der Elektrode müssen Ionen erst zur Elektrode \fRef[diffundieren]{ch:el:ion_cond:diffusion}, bevor sie reagieren können}{}
|
||||
\eq{
|
||||
\eta_\text{conc,anodic} &= -\frac{RT}{\alpha \,zF} \ln \left(\frac{c_\text{red}^0}{c_\text{red}^\txS}\right) \\
|
||||
\eta_\text{conc,cathodic} &= -\frac{RT}{(1-\alpha) zF} \ln \left(\frac{c_\text{ox}^0}{c_\text{ox}^\txS}\right)
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{diffusion_overpotential}
|
||||
\desc{Diffusion overpotential}{Due to mass transport limitations}{$j_\infty$ \fRef{::limiting_current}, $j_\text{meas}$ measured \qtyRef{current_density}, \ConstRef{universal_gas}, \QtyRef{temperature}, $n$ \qtyRef{charge_number}, \ConstRef{faraday}}
|
||||
\desc[german]{Diffusionsüberspannung}{Durch Limit des Massentransports}{}
|
||||
% \eq{\eta_\text{diff} = \frac{RT}{nF} \ln \left( \frac{\cfrac{c^\txs_\text{ox}}{c^0_\text{ox}}}{\cfrac{c^\txs_\text{red}}{c^0_\text{red}}} \right)}
|
||||
\eq{\eta_\text{diff} = \frac{RT}{nF} \Ln{\frac{j_\infty}{j_\infty - j_\text{meas}}}}
|
||||
\end{formula}
|
||||
|
||||
% 1: ion radius
|
||||
% 2: ion color
|
||||
% 3: ion label
|
||||
% 4: N solvents, leave empty for none
|
||||
% 5: solvent radius 6: solvent color
|
||||
% 7:position
|
||||
\newcommand{\drawIon}[7]{%
|
||||
\fill[#2] (#7) circle[radius=#1] node[fg0] {#3};
|
||||
\ifstrempty{#4}{}{
|
||||
\foreach \j in {1,...,#4} {
|
||||
\pgfmathsetmacro{\angle}{\j * 360/#4}
|
||||
\fill[#6] (#7) ++(\angle:#1 + #5) circle[radius=#5];
|
||||
}
|
||||
}
|
||||
}
|
||||
\newcommand{\drawAnion}[1]{\drawIon{\Ranion}{bg-blue}{-}{}{}{}{#1}}
|
||||
\newcommand{\drawCation}[1]{\drawIon{\Rcation}{bg-red}{+}{}{}{}{#1}}
|
||||
\newcommand{\drawAnionSolved}[1]{\drawIon{\Ranion}{bg-blue}{-}{6}{\Rsolvent}{fg-blue!50!bg2}{#1}}
|
||||
|
||||
\Eng[electrode]{Electrode}
|
||||
\Ger[electrode]{Elektrode}
|
||||
\Eng[nernst_layer]{Nernst layer}
|
||||
\Ger[nernst_layer]{Nernst-Schicht}
|
||||
\Eng[electrolyte]{Electrolyte}
|
||||
\Ger[electrolyte]{Elektrolyt}
|
||||
\Eng[c_surface]{surface \qtyRef{concentration}}
|
||||
\Eng[c_bulk]{bulk \qtyRef{concentration}}
|
||||
\Ger[c_surface]{Oberflächen-\qtyRef{concentration}}
|
||||
\Ger[c_bulk]{Bulk-\qtyRef{concentration}}
|
||||
|
||||
\begin{formula}{diffusion_layer}
|
||||
\desc{Cell layers}{}{IHP/OHP inner/outer Helmholtz-plane, $c^0$ \GT{c_bulk}, $c^\txS$ \GT{c_surface}}
|
||||
\desc[german]{Zellschichten}{}{IHP/OHP innere/äußere Helmholtzschicht, $c^0$ \GT{c_bulk}, $c^\txS$ \GT{c_surface}}
|
||||
\begin{tikzpicture}
|
||||
\tikzset{
|
||||
label/.style={color=fg1,anchor=center,rotate=90},
|
||||
}
|
||||
\pgfmathsetmacro{\Ranion}{0.15}
|
||||
\pgfmathsetmacro{\Rcation}{0.2}
|
||||
\pgfmathsetmacro{\Rsolvent}{0.06}
|
||||
|
||||
\pgfmathsetmacro{\tkW}{8} % Total width
|
||||
\pgfmathsetmacro{\tkH}{4} % Total height
|
||||
\pgfmathsetmacro{\edW}{1} % electrode width
|
||||
\pgfmathsetmacro{\hhW}{4*\Rsolvent+2*\Ranion} % helmholtz width
|
||||
\pgfmathsetmacro{\ndW}{3} % nernst diffusion with
|
||||
\pgfmathsetmacro{\eyW}{\tkW-\edW-\hhW-\ndW} % electrolyte width
|
||||
\pgfmathsetmacro{\edX}{0} % electrode width
|
||||
\pgfmathsetmacro{\hhX}{\edW} % helmholtz width
|
||||
\pgfmathsetmacro{\ndX}{\edW+\hhW} % nernst diffusion with
|
||||
\pgfmathsetmacro{\eyX}{\tkW-\eyW} % electrolyte width
|
||||
|
||||
\path[fill=bg-orange] (\edX,0) rectangle (\edX+\edW,\tkH);
|
||||
\path[fill=bg-green!90!bg0] (\hhX,0) rectangle (\hhX+\hhW,\tkH);
|
||||
\path[fill=bg-green!60!bg0] (\ndX,0) rectangle (\ndX+\ndW,\tkH);
|
||||
\path[fill=bg-green!20!bg0] (\eyX,0) rectangle (\eyX+\eyW,\tkH);
|
||||
\draw (\ndX,2) -- (\eyX,3) -- (\tkW,3);
|
||||
% axes
|
||||
\draw[->] (0,0) -- (\tkW+0.2,0) node[anchor=north] {$x$};
|
||||
\draw[->] (0,0) -- (0,\tkH+0.2) node[anchor=east] {$c$};
|
||||
\tkYTick{2}{$c^\txS$};
|
||||
\tkYTick{3}{$c^0$};
|
||||
\foreach \i in {1,...,5} {
|
||||
\drawCation{\edW-\Ranion, \tkH * \i /6}
|
||||
\drawAnionSolved{\edW+\Rcation+2*\Rsolvent, \tkH * \i /6}
|
||||
}
|
||||
\drawCation{\ndX+\ndW * 0.1, \tkH * 2/10}
|
||||
\drawCation{\ndX+\ndW * 0.15, \tkH * 4/10}
|
||||
\drawCation{\ndX+\ndW * 0.1, \tkH * 6/10}
|
||||
\drawCation{\ndX+\ndW * 0.1, \tkH * 9/10}
|
||||
\drawAnion{ \ndX+\ndW * 0.2, \tkH * 7/10}
|
||||
\drawAnion{ \ndX+\ndW * 0.4, \tkH * 4/10}
|
||||
\drawAnion{ \ndX+\ndW * 0.3, \tkH * 3/10}
|
||||
\drawAnion{ \ndX+\ndW * 0.5, \tkH * 6/10}
|
||||
\drawAnion{ \ndX+\ndW * 0.8, \tkH * 3/10}
|
||||
\drawAnion{ \ndX+\ndW * 0.3, \tkH * 1/10}
|
||||
\drawAnion{ \ndX+\ndW * 0.4, \tkH * 9/10}
|
||||
\drawAnion{ \ndX+\ndW * 0.6, \tkH * 7/10}
|
||||
\drawCation{\ndX+\ndW * 0.3, \tkH * 3/10}
|
||||
\drawCation{\ndX+\ndW * 0.6, \tkH * 8/10}
|
||||
\draw (\edX+\Rcation, 0) -- ++(0, -0.5) node[anchor=west,rotate=-45] {\GT{electrode}};
|
||||
\draw (\edX+\edW-\Rcation, 0) -- ++(0, -0.5) node[anchor=west,rotate=-45] {{IHP}};
|
||||
\draw (\hhX+\hhW/2, 0) -- ++(0, -0.5) node[anchor=west,rotate=-45] {{OHP}};
|
||||
\draw (\ndX+\ndW/2, 0) -- ++(0, -0.5) node[anchor=west,rotate=-45] {\GT{nernst_layer}};
|
||||
\draw (\eyX+\eyW/2, 0) -- ++(0, -0.5) node[anchor=west,rotate=-45] {\GT{electrolyte}};
|
||||
% TODO
|
||||
\end{tikzpicture}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\begin{formula}{diffusion_layer_thickness}
|
||||
\desc{Nerst Diffusion layer thickness}{}{$c^0$ \GT{c_bulk}, $c^\txS$ \GT{c_surface}}
|
||||
\desc[german]{Dicke der Nernstschen Diffusionsschicht}{}{}
|
||||
\eq{\delta_\txN = \frac{c^0 - c^\txS}{\odv{c}{x}_{x=0}}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{limiting_current}
|
||||
\desc{(Limiting) current density}{}{$n$ \QtyRef{charge_number}, \ConstRef{faraday}, $c^0$ \GT{c_bulk}, $D$ \qtyRef{diffusion_coefficient}, $\delta_\text{diff}$ \fRef{::diffusion_layer_thickness}}
|
||||
% \desc[german]{Limitierender Strom}{}{}
|
||||
\eq{
|
||||
\abs{j} &= nFD \frac{c^0-c^\txS}{\delta_\text{diff}}
|
||||
\shortintertext{\GT{for} $c^\txS \to 0$}
|
||||
\abs{j_\infty} &= nFD \frac{c^0}{\delta_\text{diff}}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{relation?}
|
||||
\desc{Current - concentration relation}{}{$c^0$ \GT{c_bulk}, $c^\txS$ \GT{c_surface}, $j$ \fRef{::limiting_current}}
|
||||
\desc[german]{Strom - Konzentrationsbeziehung}{}{}
|
||||
\eq{\frac{j}{j_\infty} = 1 - \frac{c^\txS}{c^0}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{kinetic_current}
|
||||
\desc{Kinetic current density}{}{$j_\text{meas}$ measured \qtyRef{current_density}, $j_\infty$ \fRef{::limiting_current}}
|
||||
\desc[german]{Kinetische Stromdichte}{}{$j_\text{meas}$ gemessene \qtyRef{current_density}, $j_\infty$ \fRef{::limiting_current}}
|
||||
\eq{j_\text{kin} = \frac{j_\text{meas} j_\infty}{j_\infty - j_\text{meas}}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{roughness_factor}
|
||||
\desc{Roughness factor}{Surface area related to electrode geometry}{}
|
||||
\eq{\rfactor}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{butler_volmer}
|
||||
\desc{Butler-Volmer equation}{Reaction kinetics near the equilibrium potentential}
|
||||
{$j$ \qtyRef{current_density}, $j_0$ exchange current density, $\eta$ \fRef{ch:el:kin:overpotential}, \QtyRef{temperature}, $z$ \qtyRef{charge_number}, \ConstRef{faraday}, \ConstRef{universal_gas}, $\alpha_{\txC/\txA}$ cathodic/anodic charge transfer coefficient, $\text{rf}$ \fRef{::roughness_factor}}
|
||||
%Current through an electrode iof a unimolecular redox reaction with both anodic and cathodic reaction occuring on the same electrode
|
||||
\desc[german]{Butler-Volmer-Gleichung}{Reaktionskinetik in der Nähe des Gleichgewichtspotentials}
|
||||
{$j$ \qtyRef{current_density}, $j_0$ Austauschstromdichte, $\eta$ \fRef{ch:el:kin:overpotential}, \QtyRef{temperature}, $z$ \qtyRef{charge_number}, \ConstRef{faraday}, \ConstRef{universal_gas}, $\alpha_{\txC/\txA}$ Ladungstransferkoeffizient an der Kathode/Anode, $\text{rf}$ \fRef{::roughness_factor}}
|
||||
\newFormulaEntry
|
||||
\begin{gather}
|
||||
j = j_0 \,\rfactor\, \left[ \Exp{\frac{(1-a_\txC) z F \eta}{RT}} - \Exp{-\frac{\alpha_\txC z F \eta}{RT}}\right]
|
||||
\intertext{\GT{with}}
|
||||
\alpha_\txA = 1 - \alpha_\txC
|
||||
\end{gather}
|
||||
\fig{img/ch_butler_volmer.pdf}
|
||||
\end{formula}
|
||||
|
||||
% \Subsubsection[
|
||||
% \eng{Tafel approximation}
|
||||
% \ger{Tafel Näherung}
|
||||
% ]{tafel}
|
||||
|
||||
% \begin{formula}{slope}
|
||||
% \desc{Tafel slope}{}{}
|
||||
% \desc[german]{Tafel Steigung}{}{}
|
||||
% \eq{}
|
||||
% \end{formula}
|
||||
|
||||
\begin{formula}{equation}
|
||||
\desc{Tafel approximation}{For slow kinetics: $\abs{\eta} > \SI{0.1}{\volt}$}{}
|
||||
\desc[german]{Tafel Näherung}{Für langsame Kinetik: $\abs{\eta} > \SI{0.1}{\volt}$}{}
|
||||
\eq{
|
||||
\Log{j} &\approx \Log{j_0} + \frac{\alpha_\txC zF \eta}{RT\ln(10)} && \eta \gg \SI{0.1}{\volt}\\
|
||||
\Log{\abs{j}} &\approx \Log{j_0} - \frac{(1-\alpha_\txC) zF \eta}{RT\ln(10)} && \eta \ll -\SI{0.1}{\volt}
|
||||
}
|
||||
\fig{img/ch_tafel.pdf}
|
||||
\end{formula}
|
||||
|
||||
|
||||
|
||||
\Subsection{tech}
|
||||
\desc{Techniques}{}{}
|
||||
\desc[german]{Techniken}{}{}
|
||||
|
||||
\Subsubsection{ref}
|
||||
\desc{Reference electrodes}{}{}
|
||||
\desc[german]{Referenzelektroden}{}{}
|
||||
\begin{ttext}
|
||||
\eng{Defined as reference for measuring half-cell potententials}
|
||||
\ger{Definiert als Referenz für Messungen von Potentialen von Halbzellen}
|
||||
\end{ttext}
|
||||
|
||||
\begin{formula}{she}
|
||||
\desc{Standard hydrogen elektrode (SHE)}{}{$p=\SI{e5}{\pascal}$, $a_{\ce{H+}}=\SI{1}{\mol\per\litre}$ (\Rightarrow $\pH=0$)}
|
||||
\desc[german]{Standardwasserstoffelektrode (SHE)}{}{}
|
||||
\ttxt{
|
||||
\eng{Potential of the reaction: \ce{2H^+ +2e^- <--> H2}}
|
||||
\ger{Potential der Reaktion: \ce{2H^+ +2e^- <--> H2}}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{rhe}
|
||||
\desc{Reversible hydrogen electrode (RHE)}{RHE Potential does not change with the pH value}{$E^0\equiv \SI{0}{\volt}$, \QtyRef{activity}, \QtyRef{pressure}, \GT{see} \fRef{ch:el:cell:nernst_equation}}
|
||||
\desc[german]{Reversible Wasserstoffelektrode (RHE)}{Potential ändert sich nicht mit dem pH-Wert}{}
|
||||
\eq{
|
||||
E_\text{RHE} &= E^0 + \frac{RT}{F} \Ln{\frac{a_{\ce{H^+}}}{p_{\ce{H2}}}}
|
||||
% \\ &= \SI{0}{\volt} - \SI{0.059}{\volt}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
|
||||
|
||||
\Subsubsection{cv}
|
||||
\desc{Cyclic voltammetry}{}{}
|
||||
\desc[german]{Zyklische Voltammetrie}{}{}
|
||||
\begin{bigformula}{duck}
|
||||
\desc{Cyclic voltammogram}{}{}
|
||||
% \desc[german]{}{}{}
|
||||
|
||||
\begin{minipage}{0.44\textwidth}
|
||||
|
||||
\begin{tikzpicture}
|
||||
\pgfmathsetmacro{\Ax}{-2.3}
|
||||
\pgfmathsetmacro{\Ay}{ 0.0}
|
||||
\pgfmathsetmacro{\Bx}{ 0.0}
|
||||
\pgfmathsetmacro{\By}{ 1.0}
|
||||
\pgfmathsetmacro{\Cx}{ 0.4}
|
||||
\pgfmathsetmacro{\Cy}{ 1.5}
|
||||
\pgfmathsetmacro{\Dx}{ 2.0}
|
||||
\pgfmathsetmacro{\Dy}{ 0.5}
|
||||
\pgfmathsetmacro{\Ex}{ 0.0}
|
||||
\pgfmathsetmacro{\Ey}{-1.5}
|
||||
\pgfmathsetmacro{\Fx}{-0.4}
|
||||
\pgfmathsetmacro{\Fy}{-2.0}
|
||||
\pgfmathsetmacro{\Gx}{-2.3}
|
||||
\pgfmathsetmacro{\Gy}{-0.3}
|
||||
\pgfmathsetmacro{\x}{3}
|
||||
\pgfmathsetmacro{\y}{3}
|
||||
\begin{axis}[ymin=-\y,ymax=\y,xmax=\x,xmin=-\x,
|
||||
% equal axis,
|
||||
minor tick num=1,
|
||||
xlabel={$E$}, xlabel style={at={(axis description cs:0.5,-0.06)}},
|
||||
ylabel={$j$}, ylabel style={at={(axis description cs:-0.06,0.5)}},
|
||||
anchor=center, at={(0,0)},
|
||||
axis equal image,clip=false,
|
||||
]
|
||||
% CV with beziers
|
||||
\draw[thick, fg-blue] (axis cs:\Ax,\Ay) coordinate (A) node[left] {A}
|
||||
..controls (axis cs:\Ax+1.8, \Ay+0.0) and (axis cs:\Bx-0.2, \By-0.4) .. (axis cs:\Bx,\By) coordinate (B) node[left] {B}
|
||||
..controls (axis cs:\Bx+0.1, \By+0.2) and (axis cs:\Cx-0.3, \Cy+0.0) .. (axis cs:\Cx,\Cy) coordinate (C) node[above] {C}
|
||||
..controls (axis cs:\Cx+0.5, \Cy+0.0) and (axis cs:\Dx-1.3, \Dy+0.1) .. (axis cs:\Dx,\Dy) coordinate (D) node[right] {D}
|
||||
..controls (axis cs:\Dx-2.0, \Dy-0.1) and (axis cs:\Ex+0.3, \Ey+0.8) .. (axis cs:\Ex,\Ey) coordinate (E) node[right] {E}
|
||||
..controls (axis cs:\Ex-0.1, \Ey-0.2) and (axis cs:\Fx+0.2, \Fy+0.0) .. (axis cs:\Fx,\Fy) coordinate (F) node[below] {F}
|
||||
..controls (axis cs:\Fx-0.2, \Fy+0.0) and (axis cs:\Gx+1.5, \Gy-0.2) .. (axis cs:\Gx,\Gy) coordinate (G) node[left] {G};
|
||||
\node[above] at (A) {\rightarrow};
|
||||
|
||||
\draw[dashed, fg2] (axis cs: \Bx,\By) -- (axis cs: \Ex, \Ey);
|
||||
|
||||
\draw[->] (axis cs:-\x-0.6, 0.4) -- (axis cs:-\x-0.6, \y) node[left=0.3cm, anchor=east, rotate=90] {Cath / Red};
|
||||
\draw[->] (axis cs:-\x-0.6,-0.4) -- (axis cs:-\x-0.6,-\y) node[left=0.3cm, anchor=west, rotate=90] {An / Ox};
|
||||
\end{axis}
|
||||
\end{tikzpicture}
|
||||
\end{minipage}
|
||||
\begin{minipage}{0.55\textwidth}
|
||||
\begin{ttext}
|
||||
\eng{\begin{itemize}
|
||||
\item {\color{fg-blue}A-D}: Diffusion layer growth \rightarrow decreased current after peak
|
||||
\item {\color{fg-blue}D}: Switching potential
|
||||
\item {\color{fg-blue}B,E}: Equal concentrations of reactants
|
||||
\item {\color{fg-blue}C,F}: Formal potential of redox pair: $E \approx \frac{E_\txC - E_\txF}{2}$
|
||||
\item {\color{fg-blue}C,F}: Peak separation for reverisble processes: $\Delta E_\text{rev} = E_\txC - E_\txF = n\,\SI{59}{\milli\volt}$
|
||||
\item Information about surface chemistry
|
||||
\item Double-layer capacity (horizontal lines): $I = C v$
|
||||
\end{itemize}}
|
||||
\end{ttext}
|
||||
\end{minipage}
|
||||
\end{bigformula}
|
||||
|
||||
\begin{formula}{charge}
|
||||
\desc{Charge}{Area under the curve}{$v$ \qtyRef{scan_rate}}
|
||||
\desc[german]{Ladung}{Fläche unter der Kurve}{}
|
||||
\eq{q = \frac{1}{v} \int_{E_1}^{E_2}j\,\d E}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{peak_current}
|
||||
\desc{Randles-Sevcik equation}{For reversible faradaic reaction.\\Peak current depends on square root of the scan rate}{$n$ \qtyRef{charge_number}, \ConstRef{faraday}, $A$ electrode surface area, $c^0$ bulk \qtyRef{concentration}, $v$ \qtyRef{scan_rate}, $D_\text{ox}$ \qtyRef{diffusion_coefficient} of oxidized analyte, \ConstRef{universal_gas}, \QtyRef{temperature}}
|
||||
\desc[german]{Randles-Sevcik Gleichung}{Für eine reversible, faradäische Reaktion\\Spitzenstrom hängt von der Wurzel der Scanrate ab}{}
|
||||
\eq{i_\text{peak} = 0.446\,nFAc^0 \sqrt{\frac{nFvD_\text{ox}}{RT}}}
|
||||
\end{formula}
|
||||
|
||||
\begin{hiddenformula}{scan_rate}
|
||||
\desc{Scan rate}{}{}
|
||||
\desc[german]{Scanrate}{}{}
|
||||
\hiddenQuantity{v}{\volt\per\s}{s}
|
||||
\end{hiddenformula}
|
||||
|
||||
|
||||
\begin{formula}{upd}
|
||||
\desc{Underpotential deposition (UPD)}{}{}
|
||||
% \desc[german]{}{}{}
|
||||
\ttxt{\eng{
|
||||
Reversible deposition of metal onto a foreign metal electrode at potentials positive of the Nernst potential.
|
||||
}\ger{
|
||||
Reversible Ablagerung von Metall auf eine Elektrode aus einem anderen Metall bei positiveren Potentialen als das Nernst-Potential.
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection[
|
||||
\eng{Rotating disk electrodes}
|
||||
% \ger{}
|
||||
]{rde} \abbrLink{rde}{RDE}
|
||||
\begin{formula}{viscosity}
|
||||
\desc{Dynamic viscosity}{}{}
|
||||
\desc[german]{Dynamisch Viskosität}{}{}
|
||||
\quantity{\eta,\mu}{\pascal\s=\newton\s\per\m^2=\kg\per\m\s}{}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{kinematic_viscosity}
|
||||
\desc{Kinematic viscosity}{\qtyRef{viscosity} related to density of a fluid}{\QtyRef{viscosity}, \QtyRef{density}}
|
||||
\desc[german]{Kinematische Viskosität}{\qtyRef{viscosity} im Verhältnis zur Dichte der Flüssigkeit}{}
|
||||
\quantity{\nu}{\cm^2\per\s}{}
|
||||
\eq{\nu = \frac{\eta}{\rho}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{diffusion_layer_thickness}
|
||||
\desc{Diffusion layer thickness}{}{$D$ \qtyRef{diffusion_coefficient}, $\nu$ \qtyRef{kinematic_viscosity}, \QtyRef{angular_frequency}}
|
||||
\desc[german]{Diffusionsschichtdicke}{}{}
|
||||
\eq{\delta_\text{diff}= 1.61 D{^\frac{1}{3}} \nu^{\frac{1}{6}} \omega^{-\frac{1}{2}}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{limiting_current}
|
||||
\desc{Limiting current density}{for a \abbrRef{rde}}{$n$ \QtyRef{charge_number}, \ConstRef{faraday}, $c^0$ \GT{c_bulk}, $D$ \qtyRef{diffusion_coefficient}, $\delta_\text{diff}$ \fRef{::diffusion_layer_thickness}, $\nu$ \qtyRef{kinematic_viscosity}, \QtyRef{angular_frequency}}
|
||||
% \desc[german]{Limitierender Strom}{}{}
|
||||
\eq{j_\infty = nFD \frac{c^0}{\delta_\text{diff}} = \frac{1}{1.61} nFD^{\frac{2}{3}} v^{\frac{-1}{6}} c^0 \sqrt{\omega}}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection{ac}
|
||||
\desc{AC-Impedance}{}{}
|
||||
\desc[german]{AC-Impedanz}{}{}
|
||||
\begin{formula}{nyquist}
|
||||
\desc{Nyquist diagram}{Real and imaginary parts of \qtyRef{impedance} while varying the frequency}{}
|
||||
\desc[german]{Nyquist-Diagram}{Real und Imaginaärteil der \qtyRef{impedance} während die Frequenz variiert wird}{}
|
||||
\fig{img/ch_nyquist.pdf}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{tlm}
|
||||
\desc{Transmission line model}{Model of porous electrodes as many slices}{$R_\text{ion}$ ion conduction resistance in electrode slice, $R$ / $C$ resistance / capacitance of electode slice}
|
||||
% \desc[german]{}{}{}
|
||||
\ctikzsubcircuitdef{rcpair}{in, out}{%
|
||||
coordinate(#1-in)
|
||||
(#1-in) -- ++(0, -\rcpairH)
|
||||
-- ++(\rcpairW, 0) to[R, l=$R$] ++(0,-\rcpairL) -- ++(-\rcpairW, 0)
|
||||
-- ++(0,-\rcpairH) coordinate (#1-out) ++(0,\rcpairH)
|
||||
-- ++(-\rcpairW, 0) to[C, l=$C$] ++(0,\rcpairL) -- ++(\rcpairW,0)
|
||||
(#1-out)
|
||||
}
|
||||
|
||||
\pgfmathsetmacro{\rcpairH}{0.5}
|
||||
\pgfmathsetmacro{\rcpairW}{0.5}
|
||||
\pgfmathsetmacro{\rcpairL}{1.8}
|
||||
\ctikzsubcircuitactivate{rcpair}
|
||||
\pgfmathsetmacro{\rcpairD}{3.0} % distance
|
||||
\centering
|
||||
\begin{circuitikz}[/tikz/circuitikz/bipoles/length=1cm,scale=0.7]
|
||||
\draw (0,0) to[R,l=$R_\text{electrolyte}$] ++(2,0) -- ++(1,0)
|
||||
\rcpair{rc1}{} (rc1-in) to[R,l=$R_\text{ion}$] ++(\rcpairD,0) \rcpair{rc2}{} (rc2-in) to[R,l=$R_\text{ion}$] ++(\rcpairD,0) ++(\rcpairD,0) \rcpair{rc3}{};
|
||||
\draw[dashed] (rc2-in) ++(\rcpairD,0) -- (rc3-in) (rc2-out) ++(\rcpairD,0) -- (rc3-out);
|
||||
\draw (rc1-out) -- (rc2-out) -- ++(\rcpairD,0) (rc3-out) -- ++(\rcpairD/2,0);
|
||||
\end{circuitikz}
|
||||
\fig{img/ch_nyquist_tlm.pdf}
|
||||
\end{formula}
|
||||
|
||||
106
src/ch/misc.tex
106
src/ch/misc.tex
@ -1,106 +0,0 @@
|
||||
\Section{thermo}
|
||||
\desc{Thermoelectricity}{}{}
|
||||
\desc[german]{Thermoelektrizität}{}{}
|
||||
|
||||
\begin{formula}{seebeck}
|
||||
\desc{Seebeck coefficient}{Thermopower}{$V$ voltage, \QtyRef{temperature}}
|
||||
\desc[german]{Seebeck-Koeffizient}{}{}
|
||||
\quantity{S}{\micro\volt\per\kelvin}{s}
|
||||
\eq{S = -\frac{\Delta V}{\Delta T}}
|
||||
\end{formula}
|
||||
\begin{formula}{seebeck_effect}
|
||||
\desc{Seebeck effect}{Elecromotive force across two points of a material with a temperature difference}{\QtyRef{conductivity}, $V$ local voltage, \QtyRef{seebeck}, \QtyRef{temperature}}
|
||||
\desc[german]{Seebeck-Effekt}{}{}
|
||||
\eq{\vec{j} = \sigma(-\Grad V - S \Grad T)}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{thermal_conductivity}
|
||||
\desc{Thermal conductivity}{Conduction of heat, without mass transport}{\QtyRef{heat}, \QtyRef{length}, \QtyRef{area}, \QtyRef{temperature}}
|
||||
\desc[german]{Wärmeleitfähigkeit}{Leitung von Wärme, ohne Stofftransport}{}
|
||||
\quantity{\kappa,\lambda,k}{\watt\per\m\K=\kg\m\per\s^3\kelvin}{s}
|
||||
\eq{\kappa = \frac{\dot{Q} l}{A\,\Delta T}}
|
||||
\eq{\kappa_\text{tot} = \kappa_\text{lattice} + \kappa_\text{electric}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{wiedemann-franz}
|
||||
\desc{Wiedemann-Franz law}{}{$\kappa$ Electric \qtyRef{thermal_conductivity}, $L$ in \si{\watt\ohm\per\kelvin} Lorentz number, \QtyRef{conductivity}}
|
||||
\desc[german]{Wiedemann-Franz Gesetz}{}{$\kappa$ Elektrische \qtyRef{thermal_conductivity}, $L$ in \si{\watt\ohm\per\kelvin} Lorentzzahl, \QtyRef{conductivity}}
|
||||
\eq{\kappa = L\sigma T}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{zt}
|
||||
\desc{Thermoelectric figure of merit}{Dimensionless quantity for comparing different materials}{\QtyRef{seebeck}, \QtyRef{conductivity}, $\kappa$ \qtyRef{thermal_conductivity}, \QtyRef{temperature}}
|
||||
\desc[german]{Thermoelektrische Gütezahl}{Dimensionsoser Wert zum Vergleichen von Materialien}{}
|
||||
\eq{zT = \frac{S^2\sigma}{\kappa} T}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Section{misc}
|
||||
\desc{misc}{}{}
|
||||
\desc[german]{misc}{}{}
|
||||
|
||||
% TODO: hide
|
||||
\begin{formula}{stoichiometric_coefficient}
|
||||
\desc{Stoichiometric coefficient}{}{}
|
||||
\desc[german]{Stöchiometrischer Koeffizient}{}{}
|
||||
\quantity{\nu}{}{s}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{std_condition}
|
||||
\desc{Standard temperature and pressure}{}{}
|
||||
\desc[german]{Standardbedingungen}{}{}
|
||||
\eq{
|
||||
T &= \SI{273.15}{\kelvin} = \SI{0}{\celsius} \\
|
||||
p &= \SI{100000}{\pascal} = \SI{1.000}{\bar}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{ph}
|
||||
\desc{pH definition}{}{$a_{\ce{H+}}$ hyrdrogen ion \qtyRef{activity}}
|
||||
\desc[german]{pH-Wert definition}{}{$a_{\ce{H+}}$ Wasserstoffionen-\qtyRef{activity}}
|
||||
\eq{\pH = -\log_{10}(a_{\ce{H+}})}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{ph_rt}
|
||||
\desc{pH}{At room temperature \SI{25}{\celsius}}{}
|
||||
\desc[german]{pH-Wert}{Bei Raumtemperatur \SI{25}{\celsius}}{}
|
||||
\eq{
|
||||
\pH > 7 &\quad\tGT{basic} \\
|
||||
\pH < 7 &\quad\tGT{acidic} \\
|
||||
\pH = 7 &\quad\tGT{neutral}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{grotthuss}
|
||||
\desc{Grotthuß-mechanism}{}{}
|
||||
\desc[german]{Grotthuß-Mechanismus}{}{}
|
||||
\ttxt{
|
||||
\eng{The mobility of protons in aqueous solutions is much higher than that of other ions because they can "move" by breaking and reforming covalent bonds of water molecules.}
|
||||
\ger{The Moblilität von Protononen in wässrigen Lösungen ist wesentlich größer als die anderer Ionen, da sie sich "bewegen" können indem die Wassertsoffbrückenbindungen gelöst und neu gebildet werden.}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\begin{formula}{common_chemicals}
|
||||
\desc{Common chemicals}{}{}
|
||||
\desc[german]{Häufige Chemikalien}{}{}
|
||||
\centering
|
||||
\begin{tabular}{l|c}
|
||||
\GT{name} & \GT{formula} \\ \hline\hline
|
||||
\begin{ttext}[cyanide]\eng{Cyanide}\ger{Zyanid}\end{ttext} & \ce{CN} \\ \hline
|
||||
\begin{ttext}[ammonia]\eng{Ammonia}\ger{Ammoniak}\end{ttext} & \ce{NH3} \\ \hline
|
||||
\begin{ttext}[hydrogen peroxide]\eng{Hydrogen Peroxide}\ger{Wasserstoffperoxid}\end{ttext} & \ce{H2O2} \\ \hline
|
||||
\begin{ttext}[sulfuric acid]\eng{Sulfuric Acid}\ger{Schwefelsäure}\end{ttext} & \ce{H2SO4} \\ \hline
|
||||
\begin{ttext}[ethanol]\eng{Ethanol}\ger{Ethanol}\end{ttext} & \ce{C2H5OH} \\ \hline
|
||||
\begin{ttext}[acetic acid]\eng{Acetic Acid}\ger{Essigsäure}\end{ttext} & \ce{CH3COOH} \\ \hline
|
||||
\begin{ttext}[methane]\eng{Methane}\ger{Methan}\end{ttext} & \ce{CH4} \\ \hline
|
||||
\begin{ttext}[hydrochloric acid]\eng{Hydrochloric Acid}\ger{Salzsäure}\end{ttext} & \ce{HCl} \\ \hline
|
||||
\begin{ttext}[sodium hydroxide]\eng{Sodium Hydroxide}\ger{Natriumhydroxid}\end{ttext} & \ce{NaOH} \\ \hline
|
||||
\begin{ttext}[nitric acid]\eng{Nitric Acid}\ger{Salpetersäure}\end{ttext} & \ce{HNO3} \\ \hline
|
||||
\begin{ttext}[calcium carbonate]\eng{Calcium Carbonate}\ger{Calciumcarbonat}\end{ttext} & \ce{CaCO3} \\ \hline
|
||||
\begin{ttext}[glucose]\eng{Glucose}\ger{Glukose}\end{ttext} & \ce{C6H12O6} \\ \hline
|
||||
\begin{ttext}[benzene]\eng{Benzene}\ger{Benzol}\end{ttext} & \ce{C6H6} \\ \hline
|
||||
\begin{ttext}[acetone]\eng{Acetone}\ger{Aceton}\end{ttext} & \ce{C3H6O} \\ \hline
|
||||
\begin{ttext}[ethylene]\eng{Ethylene}\ger{Ethylen}\end{ttext} & \ce{C2H4} \\ \hline
|
||||
\begin{ttext}[potassium permanganate]\eng{Potassium Permanganate}\ger{Kaliumpermanganat}\end{ttext} & \ce{KMnO4} \\ \hline
|
||||
\end{tabular}
|
||||
\end{formula}
|
||||
File diff suppressed because it is too large
Load Diff
@ -1,270 +0,0 @@
|
||||
\Section{charge_transport}
|
||||
\desc{Charge transport}{}{}
|
||||
\desc[german]{Ladungstransport}{}{}
|
||||
|
||||
\Subsection{drude}
|
||||
\desc{Drude model}{
|
||||
Classical model describing the transport properties of electrons in materials (metals):
|
||||
The material is assumed to be an ion lattice and with freely moving electrons (electron gas). The electrons are
|
||||
accelerated by an electric field and decelerated through collisions with the lattice ions.
|
||||
The model disregards the Fermi-Dirac partition of the conducting electrons.
|
||||
}{}
|
||||
\desc[german]{Drude-Modell}{
|
||||
Ein klassisches Model zur Beschreibung der Transporteigenschaften von Elektronen in (v.a.) Metallen:
|
||||
Der Festkörper wird als Ionenkristall mit frei beweglichen Elektronen (Elektronengas).
|
||||
Die Elektronen werden durch ein Elektrisches Feld $E$ beschleunigt und durch Stöße mit den Gitterionen gebremst.
|
||||
Das Modell vernachlässigt die Fermi-Dirac Verteilung der Leitungselektronen.
|
||||
}{}
|
||||
\begin{formula}{eom}
|
||||
\desc{Equation of motion}{}{$v$ electron speed, $\vec{v}_\text{D}$ drift velocity, \QtyRef{scattering_time}}
|
||||
\desc[german]{Bewegungsgleichung}{}{$v$ Elektronengeschwindigkeit, $\vec{v}_\text{D}$ Driftgeschwindigkeit, \QtyRef{scattering_time}}
|
||||
\eq{\masse \odv{\vec{v}}{t} + \frac{\masse}{\tau} \vec{v}_\text{D} = -e \vec{\E}}
|
||||
\end{formula}
|
||||
\begin{formula}{current_density}
|
||||
\desc{(Drift) Current density}{Ohm's law}{\QtyRef{charge_carrier_density}, \ConstRef{charge}, \QtyRef{drift_velocity}, \QtyRef{mobility}, \QtyRef{electric_field}}
|
||||
\desc[german]{(Drift-) Stromdichte}{Ohmsches Gesetz}{}
|
||||
\quantity{\vec{j}}{\ampere\per\m^2}{v}
|
||||
\eq{\vec{j} = -ne\vec{v}_\text{D} = ne\mu \vec{\E}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{conductivity}
|
||||
\desc{Electrical conductivity}{Both from Drude model and Sommerfeld model}{\QtyRef{current_density}, \QtyRef{electric_field}, \QtyRef{charge_carrier_density}, \ConstRef{charge}, \QtyRef{scattering_time}, \ConstRef{electron_mass}, \QtyRef{mobility}}
|
||||
\desc[german]{Elektrische Leitfähigkeit}{Aus dem Drude-Modell und dem Sommerfeld-Modell}{}
|
||||
\quantity{\sigma}{\siemens\per\m=\per\ohm\m=\ampere^2\s^3\per\kg\m^3}{t}
|
||||
\eq{\sigma = \frac{\vec{j}}{\vec{\E}} = \frac{n e^2 \tau}{\masse} = n e \mu}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{drift_velocity}
|
||||
\desc{Drift velocity}{Velocity component induced by an external force (eg. electric field)}{$v_\text{th}$ thermal velocity}
|
||||
\desc[german]{Driftgeschwindgkeit}{Geschwindigkeitskomponente durch eine externe Kraft (z.B. ein elektrisches Feld)}{$v_\text{th}$ thermische Geschwindigkeit}
|
||||
\hiddenQuantity{\vecv_\txD}{\m\per\s}{v}
|
||||
\eq{\vec{v}_\text{D} = \vec{v} - \vec{v}_\text{th}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{mean_free_path}
|
||||
\abbrLabel{MFP}
|
||||
\desc{Mean free path}{}{}
|
||||
\desc[german]{Mittlere freie Weglänge}{}{}
|
||||
\eq{\ell = \braket{v} \tau}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{mobility}
|
||||
\desc{Electrical mobility}{How quickly a particle moves through a material when moved by an electric field}{$q$ \qtyRef{charge}, $m$ \qtyRef{mass}, $\tau$ \qtyRef{scattering_time}}
|
||||
\desc[german]{Elektrische Mobilität / Beweglichkeit}{Leichtigkeit mit der sich durch ein Elektrisches Feld beeinflusstes Teilchen im Material bewegt}{}
|
||||
\quantity{\mu}{\centi\m^2\per\volt\s}{s}
|
||||
\eq{\mu = \frac{q \tau}{m}}
|
||||
\end{formula}
|
||||
|
||||
\Subsection{sommerfeld}
|
||||
\desc{Sommerfeld model}{Assumes a gas of free fermions underlying the pauli-exclusion principle. Only electrons in an energy range of $\kB T$ around the Fermi energy $\EFermi$ participate in scattering processes. The \qtyRef{conductivity} is the same as in \fRef{::drude}}{}
|
||||
\desc[german]{Sommerfeld-Modell}{Annahme eines freien Fermionengases, welches dem Pauli-Prinzip unterliegt. Nur Elektronen in einem Energiebereich von $\kB T$ um die Fermi Energe $\EFermi$ nehmen an Streuprozessen teil. Die \qtyRef{conductivity} ist die selbe wie im \fRef{::drude}}{}
|
||||
\begin{formula}{current_density}
|
||||
\desc{Electrical current density}{}{}
|
||||
\desc[german]{Elektrische Stromdichte}{}{}
|
||||
\eq{\vec{j} = -en\braket{v} = -e n \frac{\hbar}{\masse}\braket{\vec{k}} = -e \frac{1}{V} \sum_{\vec{k},\sigma} \frac{\hbar \vec{k}}{\masse}}
|
||||
\end{formula}
|
||||
|
||||
\Subsection{boltzmann}
|
||||
\desc{Boltzmann-transport}{Semiclassical description using a probability distribution (\fRef{cm:sc:fermi_dirac}) to describe the particles.}{}
|
||||
\desc[german]{Boltzmann-Transport}{Semiklassische Beschreibung, benutzt eine Wahrscheinlichkeitsverteilung (\fRef{cm:sc:fermi_dirac}).}{}
|
||||
\begin{formula}{boltzmann_transport}
|
||||
\desc{Boltzmann Transport equation}{for charge transport}{$f$ \fRef{cm:sc:fermi_dirac}}
|
||||
\desc[german]{Boltzmann-Transportgleichung}{für Ladungstransport}{}
|
||||
\eq{
|
||||
\pdv{f(\vec{r},\vec{k},t)}{t} = -\vec{v} \cdot \Grad_{\vec{r}} f - \frac{e}{\hbar}(\vec{\mathcal{E}} + \vec{v} \times \vec{B}) \cdot \Grad_{\vec{k}} f + \left(\pdv{f(\vec{r},\vec{k},t)}{t}\right)_{\text{\GT{scatter}}}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\Subsection{mag}
|
||||
\desc{Magneto-transport}{}{}
|
||||
\desc[german]{Magnetotransport}{}{}
|
||||
\begin{formula}{cyclotron_frequency}
|
||||
\desc{Cyclotron frequency}{Moving charge carriers move in cyclic orbits under applied magnetic field}{$q$ \qtyRef{charge}, \QtyRef{magnetic_flux_density}, m \qtyRef[effective]{mass}}
|
||||
\desc[german]{Zyklotronfrequenz}{Ladungstraäger bewegen sich in einem Magnetfeld auf einer Kreisbahn}{}
|
||||
\eq{w_\txc = \frac{qB}{m}}
|
||||
\end{formula}
|
||||
|
||||
\TODO{TODO}
|
||||
% \begin{formula}{cyclotron_resonance}
|
||||
% \desc{}{}{}
|
||||
% \desc[german]{}{}{}
|
||||
% \eq{}
|
||||
% \end{formula}
|
||||
|
||||
\Subsubsection{hall}
|
||||
\desc{Hall-Effect}{}{}
|
||||
\desc[german]{Hall-Effekt}{}{}
|
||||
|
||||
|
||||
\Paragraph{classic}
|
||||
\desc{Classical Hall-Effect}{Current flowing in $x$ direction in a conductor ($l \times b \times d$) with a magnetic field $B$ in $z$ direction leads to a hall voltage $U_\text{H}$ in $y$ direction.}{}
|
||||
\desc[german]{Klassischer Hall-Effekt}{Fließt in einem Leiter ($l \times b \times d$) ein Strom in $x$ Richtung, während der Leiter von einem Magnetfeld $B$ in $z$-Richtung durchdrungen, wird eine Hallspannung $U_\text{H}$ in $y$-Richtung induziert.}{}
|
||||
\begin{formula}{voltage}
|
||||
\desc{Hall voltage}{}{$n$ charge carrier density}
|
||||
\desc[german]{Hallspannung}{}{$n$ Ladungsträgerdichte}
|
||||
\eq{U_\text{H} = \frac{I B}{ne d}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{coefficient}
|
||||
\desc{Hall coefficient}{Sometimes $R_\txH$}{}
|
||||
\desc[german]{Hall-Koeffizient}{Manchmal $R_\txH$}{}
|
||||
\eq{A_\text{H} := -\frac{E_y}{j_x B_z} \explOverEq{\GT{metals}} \frac{1}{ne} = \frac{\rho_{xy}}{B_z}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{resistivity}
|
||||
\desc{Resistivity}{}{\QtyRef{momentum_relaxation_time}, \QtyRef{magnetic_flux_density}, $n$ \qtyRef{charge_carrier_density}, \ConstRef{charge}}
|
||||
\desc[german]{Spezifischer Widerstand}{}{}
|
||||
\eq{\rho_{xx} &= \frac{\masse}{ne^2\tau} \\ \rho_{xy} &= \frac{B}{ne}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Paragraph{quantum}
|
||||
\desc{Quantum hall effects}{}{}
|
||||
\desc[german]{Quantenhalleffekte}{}{}
|
||||
\begin{formula}{types}
|
||||
\desc{Types of quantum hall effects}{}{}
|
||||
\desc[german]{Arten von Quantenhalleffekten}{}{}
|
||||
\ttxt{\eng{
|
||||
\begin{itemize}
|
||||
\item \textbf{Integer} (QHE): filling factor $\nu$ is an integer
|
||||
\item \textbf{Fractional} (FQHE): filling factor $\nu$ is a fraction
|
||||
\item \textbf{Spin} (QSHE): spin currents instead of charge currents
|
||||
\item \textbf{Anomalous} (QAHE): symmetry breaking by internal effects instead of external magnetic fields
|
||||
\end{itemize}
|
||||
}\ger{
|
||||
\begin{itemize}
|
||||
\item \textbf{Integer} (QHE): Füllfaktor $\nu$ ist ganzzahlig
|
||||
\item \textbf{Fractional} (FQHE): Füllfaktor $\nu$ ist ein Bruch
|
||||
\item \textbf{Spin} (QSHE): Spin Ströme anstatt Ladungsströme
|
||||
\item \textbf{Anomalous} (QAHE): Symmetriebruch durch interne Effekte anstatt druch ein externes Magnetfeld
|
||||
\end{itemize}
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
|
||||
\begin{formula}{conductivity}
|
||||
\desc{Conductivity tensor}{}{}
|
||||
\desc[german]{Leitfähigkeitstensor}{}{}
|
||||
\eq{\sigma = \begin{pmatrix} \sigma_{xy} & \sigma_{xy} \\ \sigma_{yx} & \sigma_{yy} \end{pmatrix} }
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{resistivity_tensor}
|
||||
\desc{Resistivity tensor}{}{}
|
||||
\desc[german]{Spezifischer Widerstands-tensor}{}{}
|
||||
\eq{
|
||||
\rho = \sigma^{-1}
|
||||
% \sigma = \begin{pmatrix} \sigma_{xy} & \sigma_{xy} \\ \sigma_{yx} & \sigma_{yy} \end{pmatrix} }
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{resistivity}
|
||||
\desc{Resistivity}{}{$\nu \in \mathbb{Z}$ filing factor}
|
||||
\desc[german]{Spezifischer Hallwiderstand}{}{$\nu \in \mathbb{Z}$ Füllfaktor}
|
||||
\eq{\rho_{xy} = \frac{2\pi\hbar}{e^2} \frac{1}{\nu}}
|
||||
\end{formula}
|
||||
|
||||
% \begin{formula}{qhe}
|
||||
% \desc{Integer quantum hall effect}{}{}
|
||||
% \desc[german]{Ganzahliger Quanten-Hall-Effekt}{}{}
|
||||
% \fig{img/qhe-klitzing.jpeg}
|
||||
% \end{formula}
|
||||
|
||||
\begin{formula}{fqhe}
|
||||
\desc{Fractional quantum hall effect}{}{$\nu$ fraction of two numbers without shared divisors}
|
||||
\desc[german]{Fraktionaler Quantum-Hall-Effekt}{}{$\nu$ Bruch aus Zahlen ohne gemeinsamen Teiler}
|
||||
\eq{\nu = \frac{1}{3},\frac{2}{5},\frac{3}{7},\frac{2}{3}...}
|
||||
\end{formula}
|
||||
|
||||
\Subsection{scatter}
|
||||
\desc{Scattering processes}{Limits the \qtyRef{drift_velocity}}{}
|
||||
\desc[german]{Streuprozesse}{Begrenzt die \qtyRef{drift_velocity}}{}
|
||||
\Eng[elastic]{elastic}
|
||||
\Ger[elastic]{elastisch}
|
||||
\Eng[inelastic]{inelastic}
|
||||
\Ger[inelastic]{inelastisch}
|
||||
\begin{formula}{types}
|
||||
\desc{Types}{}{}
|
||||
\desc[german]{Arten}{}{}
|
||||
\ttxt{\eng{
|
||||
\textbf{Elastic}: constant $\abs{\veck}$ and $E$, direction of $\veck$ changes
|
||||
\\ \textbf{Inelastic}: $\abs{\veck}$, $E$ and direction of $\veck$ change
|
||||
}\ger{
|
||||
\textbf{Elastisch}: $\abs{\veck}$ und $E$ konstant, Richtung von $\veck$ ändert sich
|
||||
\\ \textbf{Inelastisch}: $\abs{\veck}$, $E$ und Richtung von $\veck$ ändern sich
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{scattering_time}
|
||||
\desc{Momentum relaxation time}{Scattering time}{}
|
||||
\desc[german]{}{Streuzeit}{}
|
||||
\quantity{\tau}{\s}{s}
|
||||
\hiddenQuantity[momentum_relaxation_time]{\tau}{\s}{s}
|
||||
\ttxt{
|
||||
\eng{The average time between scattering events weighted by the characteristic momentum change cause by the scattering process (If the momentum and momentum direction do not change, the scattering event is irrelevant for the resistance).}
|
||||
\ger{Die durschnittliche Zeit zwischen Streuprozessen, gewichtet durch die verursachte Impulsänderung (Wenn sich der Impuls und die Richtung nicht ändern ist das Streuevent irrelevant für den Widerstand)}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{matthiessen}
|
||||
\desc{Matthiessen's rule}{Approximation, only holds if the processes are independent of each other}{\QtyRef{mobility}, \QtyRef{scattering_time}}
|
||||
\desc[german]{Matthiessensche Regel}{Näherung, nur gültig wenn die einzelnen Streuprozesse von einander unabhängig sind}{}
|
||||
\eq{
|
||||
\frac{1}{\mu} &= \sum_{i = \textrm{\GT{:::scatter}}} \frac{1}{\mu_i} \\
|
||||
\frac{1}{\tau} &= \sum_{i = \textrm{\GT{:::scatter}}} \frac{1}{\tau_i}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{processes}
|
||||
\desc{Scattering processes}{}{$\theta_\txD$ \qtyRef{debye_temperature}, \QtyRef{temperature}, \QtyRef{mobility}}
|
||||
\desc[german]{Streuprozesse}{}{}
|
||||
\newFormulaEntry
|
||||
\centering
|
||||
\begin{tabular}{c|C|c}
|
||||
Process & \mu\propto T\\ \hline
|
||||
Acoustic phonons & \mu\propto T^{-3/2} & \string~ \GT{elastic}\\
|
||||
Ionized impurities & \mu\propto T^{3/2} & \GT{elastic} \\
|
||||
Piezoelectric & \mu\propto T^{-1/2} & \GT{elastic} \\
|
||||
Polar optical phonons & \mu\propto \Exp{\theta_\txD/T} \text{ for } T < \theta_\txD & \GT{inelastic}
|
||||
\end{tabular}
|
||||
\fig{img/cm_scattering.pdf}
|
||||
\TODO{Impurities at low T, phonon scattering at high T (>100K), maybe plot at slide 317 combined notes adv. sc}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{gunn_effect}
|
||||
\desc{Gunn effect}{through Intervalley scattering}{}
|
||||
\desc[german]{Gunn-Effekt}{durch Intervalley Streuung}{}
|
||||
\ttxt{\eng{
|
||||
If the carrier energies are high enough, they can scatter into a neighboring band minimum, where they have a higher \qtyRef{effective_mass} and lower \qtyRef{mobility}.
|
||||
\Rightarrow current decreases again (negative differential resistivity)
|
||||
}\ger{
|
||||
Bei ausreichend hoher Energie der Ladungsträger können diese in ein benachbartes Minimum streuen.
|
||||
Da sie dort eine höhere \qtyRef{effective_mass} und dadurch niedrigere \qty-ref{mobility} haben, verringert sich der Strom wieder (negative Resistivität)
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\Subsection{misc}
|
||||
\desc{misc}{}{}
|
||||
\desc[german]{misc}{}{}
|
||||
\begin{formula}{tsu_esaki}
|
||||
\desc{Tsu-Esaki tunneling current}{Describes the current $I_{\txL \leftrightarrow \txR}$ through a barrier}{$\mu_i$ \qtyRef{chemical_potential} at left/right side, $U_i$ voltage on left/right side. Electrons occupy region between $U_i$ and $\mu_i$}
|
||||
\desc[german]{Tsu-Esaki Tunnelstrom}{Beschreibt den Strom $I_{\txL \leftrightarrow \txR}$ durch eine Barriere }{$\mu_i$ \qtyRef{chemical_potential} links/rechts, $U_i$ Spannung links/rechts. Elektronen besetzen Bereich zwischen $U_i$ und $\mu_i$}
|
||||
\eq{
|
||||
I_\text{T} = \frac{2e}{h} \int_{U_\txL}^\infty \left(f(E, \mu_\txL) -f(E, \mu_\txR)\right) T(E) \d E
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{diffusion}
|
||||
\desc{Diffusion current}{Equilibration of concentration gradients}{\QtyRef{diffusion_coefficient}, \ConstRef{charge}, $n,p$ \qtyRef{charge_carrier_density}}
|
||||
\desc[german]{Diffunsstrom}{Ausgleich von Konzentrationsgradienten}{}
|
||||
\eq{\vec{j}_\text{diff} = -\abs{e} D_n \left(-\Grad n\right) + \abs{e} D_p \left(-\Grad p\right)}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{continuity}
|
||||
\desc{Charge continuity equation}{Electric charge can only change by the amount of electric current}{\QtyRef{charge_density}, \QtyRef{current_density}}
|
||||
\desc[german]{Kontinuitätsgleichung der Ladung}{Elektrische Ladung kann sich nur durch die Stärke des Stromes ändern}{}
|
||||
\eq{
|
||||
\pdv{\rho}{t} = - \nabla \vec{j}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
@ -1,84 +0,0 @@
|
||||
\Part{cm}
|
||||
\desc{Condensed matter physics}{}{}
|
||||
\desc[german]{Festkörperphysik}{}{}
|
||||
|
||||
\TODO{van hove singularities}
|
||||
|
||||
|
||||
\Section{bond}
|
||||
\desc{Bonds}{}{}
|
||||
\desc[german]{Bindungen}{}{}
|
||||
\begin{formula}{metallic}
|
||||
\desc{Metallic bond}{}{}
|
||||
\desc[german]{Metallbindung}{}{}
|
||||
\ttxt{\eng{
|
||||
\begin{itemize}
|
||||
\item Delocalized electrons form a cloud
|
||||
\item High \qtyRef[electrical]{conductivity} and \qtyRef[thermal]{thermal_conductivity} conductivity
|
||||
\item No internal electric field
|
||||
\end{itemize}
|
||||
}\ger{
|
||||
\begin{itemize}
|
||||
\item Elektronen delokalisiert und bilden Wolke
|
||||
\item Hohe \qtyRef[elektrische]{conductivity} und \qtyRef[thermische]{thermal_conductivity} Leitfähigkeit
|
||||
\item Kein internes elektrisches Feld
|
||||
\end{itemize}
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{covalent}
|
||||
\desc{Covalent bond}{}{}
|
||||
\desc[german]{Kolvalente Bindung}{}{}
|
||||
\ttxt{\eng{
|
||||
\begin{itemize}
|
||||
\item \fRef{cm:band:hybrid_orbitals} of shared electrons
|
||||
\item Highly directional
|
||||
\item Varying \qtyRef[electrical]{conductivity} and high \qtyRef[thermal]{thermal_conductivity} conductivity
|
||||
\end{itemize}
|
||||
}\ger{
|
||||
\begin{itemize}
|
||||
\item \fRef{cm:band:hybrid_orbitals} geteilter Elektronen
|
||||
\item Richtungsabhängige Bindung
|
||||
\item Verschiedene \qtyRef[elektrische]{conductivity} und hohe \qtyRef[thermische]{thermal_conductivity} Leitfähigkeiten
|
||||
\end{itemize}
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{ionic}
|
||||
\desc{Ionic bond}{}{}
|
||||
\desc[german]{Ionenbindung}{}{}
|
||||
\ttxt{\eng{
|
||||
\begin{itemize}
|
||||
\item Charge transfer from anion to cation
|
||||
\item Non.directional bonding
|
||||
\item Strong bond
|
||||
\item Low \qtyRef[electrical]{conductivity} and high \qtyRef[thermal]{thermal_conductivity} conductivity
|
||||
\item Always in combination with a \fRef{:::covalent}
|
||||
\end{itemize}
|
||||
}\ger{
|
||||
\begin{itemize}
|
||||
\item Ladungstransfer von Anion zu Kation
|
||||
\item Richtungsunabängig
|
||||
\item Starke Bindung
|
||||
\item Geringe \qtyRef[elektrische]{conductivity} und hohe \qtyRef[thermische]{thermal_conductivity} Leitfähigkeit
|
||||
\item Immer in Kombination mit einer \fRef[kovalenten Bindung]{:::covalent}
|
||||
\end{itemize}
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{van-der-waals}
|
||||
\desc{Van der Waals bond}{}{}
|
||||
\desc[german]{Van-der-Waals Bindung}{}{}
|
||||
\ttxt{\eng{
|
||||
\begin{itemize}
|
||||
\item Dipole-dipole interaction from local charge fluctuations
|
||||
\item Weak bond
|
||||
\end{itemize}
|
||||
}\ger{
|
||||
\begin{itemize}
|
||||
\item Dipol-Dipol Wechselwirkung durch lokale Ladungsfluktuationen
|
||||
\item Schwache Bindung
|
||||
\end{itemize}
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
@ -1,412 +0,0 @@
|
||||
\Section{crystal}
|
||||
\desc{Crystals}{}{}
|
||||
\desc[german]{Kristalle}{}{}
|
||||
|
||||
\Subsection{bravais}
|
||||
\desc{Bravais lattice}{}{}
|
||||
\desc[german]{Bravais-Gitter}{}{}
|
||||
|
||||
\Eng[lattice_system]{Lattice system}
|
||||
\Ger[lattice_system]{Gittersystem}
|
||||
\Eng[crystal_family]{Crystal system}
|
||||
\Ger[crystal_family]{Kristall-system}
|
||||
\Eng[point_group]{Point group}
|
||||
\Ger[point_group]{Punktgruppe}
|
||||
\eng[bravais_lattices]{Bravais lattices}
|
||||
\ger[bravais_lattices]{Bravais Gitter}
|
||||
|
||||
\newcommand\bvimg[1]{\begin{center}\includegraphics[width=0.1\textwidth]{img_static/bravais/#1.pdf}\end{center}}
|
||||
\renewcommand\tabularxcolumn[1]{m{#1}}
|
||||
\newcolumntype{Z}{>{\centering\let\newline\\\arraybackslash\hspace{0pt}}X}
|
||||
|
||||
\begin{bigformula}{2d}
|
||||
\desc{2D}{In 2D, there are 5 different Bravais lattices}{}
|
||||
\desc[german]{2D}{In 2D gibt es 5 verschiedene Bravais-Gitter}{}
|
||||
\begin{adjustbox}{width=\textwidth}
|
||||
\begin{tabularx}{\textwidth}{||Z|c|Z|Z||}
|
||||
\hline
|
||||
\multirow{2}{*}{\GT{lattice_system}} & \multirow{2}{*}{\GT{point_group}} & \multicolumn{2}{c||}{5 \gt{bravais_lattices}} \\ \cline{3-4}
|
||||
& & \GT{primitive} (p) & \GT{centered} (c) \\ \hline
|
||||
\GT{monoclinic} (m) & $\text{C}_\text{2}$ & \bvimg{mp} & \\ \hline
|
||||
\GT{orthorhombic} (o) & $\text{D}_\text{2}$ & \bvimg{op} & \bvimg{oc} \\ \hline
|
||||
\GT{tetragonal} (t) & $\text{D}_\text{4}$ & \bvimg{tp} & \\ \hline
|
||||
\GT{hexagonal} (h) & $\text{D}_\text{6}$ & \bvimg{hp} & \\ \hline
|
||||
\end{tabularx}
|
||||
\end{adjustbox}
|
||||
\end{bigformula}
|
||||
|
||||
|
||||
\begin{bigformula}{3d}
|
||||
\desc{3D}{In 3D, there are 14 different Bravais lattices}{}
|
||||
\desc[german]{3D}{In 3D gibt es 14 verschiedene Bravais-Gitter}{}
|
||||
% \newcolumntype{g}{>{\columncolor[]{0.8}}}
|
||||
\begin{adjustbox}{width=\textwidth}
|
||||
\begin{tabularx}{\textwidth}{||Z|Z|c|Z|Z|Z|Z||}
|
||||
\hline
|
||||
\multirow{2}{*}{\GT{crystal_family}} & \multirow{2}{*}{\GT{lattice_system}} & \multirow{2}{*}{\GT{point_group}} & \multicolumn{4}{c||}{14 \gt{bravais_lattices}} \\ \cline{4-7}
|
||||
& & & \GT{primitive} (P) & \GT{base_centered} (S) & \GT{body_centered} (I) & \GT{face_centered} (F) \\ \hline
|
||||
\multicolumn{2}{||c|}{\GT{triclinic} (a)} & $\text{C}_\text{i}$ & \bvimg{tP} & & & \\ \hline
|
||||
\multicolumn{2}{||c|}{\GT{monoclinic} (m)} & $\text{C}_\text{2h}$ & \bvimg{mP} & \bvimg{mS} & & \\ \hline
|
||||
\multicolumn{2}{||c|}{\GT{orthorhombic} (o)} & $\text{D}_\text{2h}$ & \bvimg{oP} & \bvimg{oS} & \bvimg{oI} & \bvimg{oF} \\ \hline
|
||||
\multicolumn{2}{||c|}{\GT{tetragonal} (t)} & $\text{D}_\text{4h}$ & \bvimg{tP} & & \bvimg{tI} & \\ \hline
|
||||
\multirow{2}{*}{\GT{hexagonal} (h)} & \GT{rhombohedral} & $\text{D}_\text{3d}$ & \bvimg{hR} & & & \\ \cline{2-7}
|
||||
& \GT{hexagonal} & $\text{D}_\text{6h}$ & \bvimg{hP} & & & \\ \hline
|
||||
\multicolumn{2}{||c|}{\GT{cubic} (c)} & $\text{O}_\text{h}$ & \bvimg{cP} & & \bvimg{cI} & \bvimg{cF} \\ \hline
|
||||
\end{tabularx}
|
||||
\end{adjustbox}
|
||||
\end{bigformula}
|
||||
|
||||
\begin{formula}{lattice_constant}
|
||||
\desc{Lattice constant}{Parameter (length or angle) describing the smallest unit cell}{}
|
||||
\desc[german]{Gitterkonstante}{Parameter (Länge oder Winkel) der die Einheitszelle beschreibt}{}
|
||||
\quantity{a}{}{s}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{lattice_vector}
|
||||
\desc{Lattice vector}{}{$n_i \in \Z$}
|
||||
\desc[german]{Gittervektor}{}{}
|
||||
\quantity{\vec{R}}{}{\angstrom}
|
||||
\eq{\vec{R} = n_1 \vec{a_1} + n_2 \vec{a_2} + n_3 \vec{a_3}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{primitive_unit_cell}
|
||||
\desc{Primitve unit cell}{}{}
|
||||
\desc[german]{Primitive Einheitszelle}{}{}
|
||||
\ttxt{\eng{Unit cell containing exactly one lattice point}\ger{Einheitszelle die genau einen Gitterpunkt enthält}}
|
||||
\end{formula}
|
||||
|
||||
\Eng[miller-point]{Point}
|
||||
\Ger[miller-point]{Punkt}
|
||||
\Eng[miller-direction]{Direction}
|
||||
\Ger[miller-direction]{Richtung}
|
||||
\Eng[miller-direction-family]{Family of directions}
|
||||
\Ger[miller-direction-family]{Familie von Richtungen}
|
||||
\Eng[miller-plane]{Plane}
|
||||
\Ger[miller-plane]{Ebene}
|
||||
\Eng[miller-plane-family]{Family of planes}
|
||||
\Ger[miller-plane-family]{Familie von Ebenen}
|
||||
\begin{formula}{miller}
|
||||
\desc{Miller indices}{}{
|
||||
Miller planes: $(hkl)$, $\frac{1}{h}$/$\frac{1}{k}$/$\frac{1}{l}$ give intersection with $x$/$y$/$z$ axes\\
|
||||
Miller family: planes that are equivalent due to crystal symmetry
|
||||
}
|
||||
\desc[german]{Millersche Indizes}{}{
|
||||
Miller-Ebenen: $(hkl)$, $\frac{1}{h}$/$\frac{1}{k}$/$\frac{1}{l}$ geben die Schnittpunkte mit den $x$/$y$/$z$-Achsen\\
|
||||
Miller-Familien: Ebenen, die durch Kristallsymmetrie äquivalent sind
|
||||
}
|
||||
\centering
|
||||
\newFormulaEntry
|
||||
\begin{tabularx}{\textwidth}{clcl}
|
||||
$(h,k,l)$ & \GT{miller-point} & & \\
|
||||
$hkl$ & \GT{miller-direction} & $\langle hkl \rangle$ & \GT{miller-direction-family} \\
|
||||
$(hkl)$ & \GT{miller-plane} & $\{hkl\}$ & \GT{miller-plane-family}
|
||||
\end{tabularx}
|
||||
\pgfmathsetmacro{\rectX}{2}
|
||||
\pgfmathsetmacro{\rectZ}{2}
|
||||
\newFormulaEntry
|
||||
\begin{tikzpicture}[3d view={100}{20},perspective={p={(-55,0,0)},q={(0,25,0)},r={(0,0,-30)}}]
|
||||
% <100> direction family
|
||||
\begin{scope}
|
||||
\drawRectCS{1.4*\rectX}{1.4*\rectZ}
|
||||
\setRectPoints{R1}{(0.5*\rectX,0.5*\rectX,0)}{\rectX}{\rectX}
|
||||
\setRectPoints{R2}{(0.5*\rectX,0.5*\rectX,\rectZ)}{\rectX}{\rectX}
|
||||
\drawRectBack{R1}
|
||||
\drawRectConnectionsBack{R1}{R2}
|
||||
\draw[miller dir] (0,0,0) -- ++( \rectX,0,0) node[anchor=east] {$[100]$};
|
||||
\draw[miller dir] (0,0,0) -- ++(-\rectX,0,0) node[anchor=west] {$[\bar{1}00]$};
|
||||
\draw[miller dir] (0,0,0) -- ++(0, \rectX,0) node[anchor=south] {$[010]$};
|
||||
\draw[miller dir] (0,0,0) -- ++(0,-\rectX,0) node[anchor=south] {$[0\bar{1}0]$};
|
||||
\draw[miller dir] (0,0,0) -- ++(0,0, \rectX) node[anchor=east] {$[001]$};
|
||||
\draw[miller dir] (0,0,0) -- ++(0,0,-\rectX) node[anchor=west] {$[00\bar{1}]$};
|
||||
\drawRectFront{R1}
|
||||
\drawRectBack{R2}
|
||||
\drawRectConnectionsFront{R1}{R2}
|
||||
\drawRectFront{R2}
|
||||
\node at (1.5*\rectX,1.5*\rectX, 0) {$\langle100\rangle$};
|
||||
\end{scope}
|
||||
\pgfmathsetmacro{\rectDistance}{4.5}
|
||||
% {100} plane family
|
||||
\begin{scope}[shift={(0,\rectDistance,0)}]
|
||||
\drawRectCS{1.4*\rectX}{1.4*\rectZ}
|
||||
\setRectPoints{R1}{(0.5*\rectX,0.5*\rectX,0)}{\rectX}{\rectX}
|
||||
\setRectPoints{R2}{(0.5*\rectX,0.5*\rectX,\rectZ)}{\rectX}{\rectX}
|
||||
\drawRectBack{R1}
|
||||
\drawRectConnectionsBack{R1}{R2}
|
||||
\drawRectFront{R1}
|
||||
\drawRectBack{R2}
|
||||
\drawRectConnectionsFront{R1}{R2}
|
||||
\drawRectFront{R2}
|
||||
\fill[miller plane] (R1-C) -- (R1-D) node[anchor=north,midway] {$(100)$} -- (R2-D) -- (R2-C) -- cycle;
|
||||
\fill[miller plane] (R1-A) -- (R1-D) node[anchor=west,midway] {$(010)$} -- (R2-D) -- (R2-A) -- cycle node[anchor=north east] {$(010)$};
|
||||
\fill[miller plane] (R2-A) -- (R2-B) node[midway,anchor=south] {$(001)$} -- (R2-C) -- (R2-D) -- cycle;
|
||||
\node at (1.5*\rectX,1.5*\rectX, 0) {$\{100\}$};
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
% describe how to construct miller planes
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{miller-hexagon}
|
||||
\desc{Hexagonal miller indices}{}{}
|
||||
\desc[german]{Hexagonale Millersche Indizes}{}{}
|
||||
\eq{ (hkil) && \tGT{with}\quad i = h + k }
|
||||
\centering
|
||||
\newFormulaEntry
|
||||
\begin{tikzpicture}[3d view={0}{20}]
|
||||
\pgfmathsetmacro{\hexxY}{1.5}
|
||||
\begin{scope}
|
||||
\drawHexagonCS{1}{\hexxY}
|
||||
\setHexagonPoints{H1}{(0,0,0)}{1}{1}{1}
|
||||
\setHexagonPoints{H2}{(0,0,\hexxY)}{1}{1}{1}
|
||||
\drawHexagonBack{H1}
|
||||
\drawHexagonConnectionsBack{H1}{H2}
|
||||
\drawHexagonFront{H1}
|
||||
\drawHexagonBack{H2}
|
||||
\drawHexagonConnectionsFront{H1}{H2}
|
||||
\drawHexagonFront{H2}
|
||||
\end{scope}
|
||||
\pgfmathsetmacro{\hexDistance}{3.5}
|
||||
% 1121
|
||||
\begin{scope}[shift={(\hexDistance,0,0)}]
|
||||
\drawHexagonCS{1}{\hexxY}
|
||||
\setHexagonPoints{H1}{(0,0,0)}{1}{1}{1}
|
||||
\setHexagonPoints{H2}{(0,0,\hexxY)}{1}{1}{1}
|
||||
\drawHexagonBack{H1}
|
||||
\drawHexagonConnectionsBack{H1}{H2}
|
||||
\fill[miller plane] (H1-A) -- (H2-M) -- (H1-E) -- cycle;
|
||||
\drawHexagonFront{H1}
|
||||
\drawHexagonBack{H2}
|
||||
\drawHexagonConnectionsFront{H1}{H2}
|
||||
\drawHexagonFront{H2}
|
||||
\node[anchor=north] at (xyz cylindrical cs:radius=1.5,angle=270) {$(1211)$};
|
||||
\end{scope}
|
||||
% 1010
|
||||
\begin{scope}[shift={(2*\hexDistance,0,0)}]
|
||||
\drawHexagonCS{1}{\hexxY}
|
||||
\setHexagonPoints{H1}{(0,0,0)}{1}{1}{1}
|
||||
\setHexagonPoints{H2}{(0,0,\hexxY)}{1}{1}{1}
|
||||
\drawHexagonBack{H1}
|
||||
\drawHexagonConnectionsBack{H1}{H2}
|
||||
\drawHexagonFront{H1}
|
||||
\drawHexagonBack{H2}
|
||||
\drawHexagonConnectionsFront{H1}{H2}
|
||||
\drawHexagonFront{H2}
|
||||
\fill[miller plane] (H1-F) -- (H2-F) -- (H2-E) -- (H1-E) -- cycle;
|
||||
\node[anchor=north] at (xyz cylindrical cs:radius=1.5,angle=270) {$(1010)$};
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsection{reci}
|
||||
\desc{Reciprocal lattice}{The reciprokal lattice is made up of all the wave vectors $\vec{k}$ that ressemble standing waves with the periodicity of the Bravais lattice.}{}
|
||||
\desc[german]{Reziprokes Gitter}{Das rezioproke Gitter besteht aus dem dem Satz aller Wellenvektoren $\vec{k}$, die ebene Wellen mit der Periodizität des Bravais-Gitters ergeben.}{}
|
||||
|
||||
\begin{formula}{vectors}
|
||||
\desc{Reciprocal lattice vectors}{}{$a_i$ real-space lattice vectors, $V_c$ volume of the primitive lattice cell}
|
||||
\desc[german]{Reziproke Gittervektoren}{}{$a_i$ Bravais-Gitter Vektoren, $V_c$ Volumen der primitiven Gitterzelle}
|
||||
\eq{
|
||||
\vec{b_1} &= \frac{2\pi}{V_c} \vec{a_2} \times \vec{a_3} \\
|
||||
\vec{b_2} &= \frac{2\pi}{V_c} \vec{a_3} \times \vec{a_1} \\
|
||||
\vec{b_3} &= \frac{2\pi}{V_c} \vec{a_1} \times \vec{a_2}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{reciprocal_lattice_vector}
|
||||
\desc{Reciprokal attice vector}{}{$n_i \in \Z$}
|
||||
\desc[german]{Reziproker Gittervektor}{}{}
|
||||
\quantity{\vec{G}}{}{\angstrom}
|
||||
\eq{\vec{G}_{{hkl}} = h \vec{b_1} + k \vec{b_2} + l \vec{b_3}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsection{lat}
|
||||
\desc{Lattices}{}{}
|
||||
\desc[german]{Gitter}{}{}
|
||||
|
||||
\begin{formula}{sc}
|
||||
\desc{Simple cubic (SC)}{Reciprocal: Simple cubic}{\QtyRef{lattice_constant}}
|
||||
\desc[german]{Einfach kubisch (SC)}{Reziprok: Einfach kubisch}{}
|
||||
\eq{
|
||||
\vec{a}_{1}=a \begin{pmatrix} 1\\0\\0 \end{pmatrix},\,
|
||||
\vec{a}_{2}=a \begin{pmatrix} 0\\1\\0 \end{pmatrix},\,
|
||||
\vec{a}_{3}=a \begin{pmatrix} 0\\0\\1 \end{pmatrix}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{bcc}
|
||||
\desc{Body centered cubic (BCC)}{Reciprocal: \fRef{::fcc}}{\QtyRef{lattice_constant}}
|
||||
\desc[german]{Kubisch raumzentriert (BCC)}{Reziprok: \fRef{::fcc}}{}
|
||||
\eq{
|
||||
\vec{a}_{1}=\frac{a}{2} \begin{pmatrix} -1\\1\\1 \end{pmatrix},\,
|
||||
\vec{a}_{2}=\frac{a}{2} \begin{pmatrix} 1\\-1\\1 \end{pmatrix},\,
|
||||
\vec{a}_{3}=\frac{a}{2} \begin{pmatrix} 1\\1\\-1 \end{pmatrix}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{fcc}
|
||||
\desc{Face centered cubic (FCC)}{Reciprocal: \fRef{::bcc}}{\QtyRef{lattice_constant}}
|
||||
\desc[german]{Kubisch flächenzentriert (FCC)}{Reziprok: \fRef{::bcc}}{}
|
||||
\eq{
|
||||
\vec{a}_{1}=\frac{a}{2} \begin{pmatrix} 0\\1\\1 \end{pmatrix},\,
|
||||
\vec{a}_{2}=\frac{a}{2} \begin{pmatrix} 1\\0\\1 \end{pmatrix},\,
|
||||
\vec{a}_{3}=\frac{a}{2} \begin{pmatrix} 1\\1\\0 \end{pmatrix}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{diamond}
|
||||
\desc{Diamond lattice}{}{}
|
||||
\desc[german]{Diamantstruktur}{}{}
|
||||
\ttxt{
|
||||
\eng{\fRef{:::fcc} with basis $\begin{pmatrix} 0 & 0 & 0 \end{pmatrix}$ and $\begin{pmatrix} \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \end{pmatrix}$}
|
||||
\ger{\fRef{:::fcc} mit Basis $\begin{pmatrix} 0 & 0 & 0 \end{pmatrix}$ und $\begin{pmatrix} \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \end{pmatrix}$}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{zincblende}
|
||||
\desc{Zincblende lattice}{}{}
|
||||
\desc[german]{Zinkblende-Struktur}{}{}
|
||||
\fsplit{
|
||||
\centering
|
||||
\includegraphics[width=0.9\textwidth]{img/cm_crystal_zincblende.png}
|
||||
}{
|
||||
\ttxt{
|
||||
\eng{Like \fRef{:::diamond} but with different species on each basis}
|
||||
\ger{Wie \fRef{:::diamond} aber mit unterschiedlichen Spezies auf den Basen}
|
||||
}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{rocksalt}
|
||||
\desc{Rocksalt structure}{\elRef{Na}\elRef{Cl}}{}
|
||||
\desc[german]{Kochsalz-Struktur}{}{}
|
||||
\fsplit{
|
||||
\centering
|
||||
\includegraphics[width=0.9\textwidth]{img/cm_crystal_NaCl.png}
|
||||
}{
|
||||
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{wurtzite}
|
||||
\desc{Wurtzite structure}{hP4}{}
|
||||
\desc[german]{Wurtzite-Struktur}{hP4}{}
|
||||
\fsplit{
|
||||
\centering
|
||||
\includegraphics[width=0.9\textwidth]{img/cm_crystal_wurtzite.png}
|
||||
}{
|
||||
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\Subsection{defect}
|
||||
\desc{Defects}{}{}
|
||||
\desc[german]{Defekte}{}{}
|
||||
\Subsubsection{point}
|
||||
\desc{Point defects}{}{}
|
||||
\desc[german]{Punktdefekte}{}{}
|
||||
\begin{formula}{vacancy}
|
||||
\desc{Vacancy}{}{}
|
||||
\desc[german]{Fehlstelle}{}{}
|
||||
\ttxt{\eng{
|
||||
\begin{itemize}
|
||||
\item Lattice site missing an atom
|
||||
\item Low formation energy
|
||||
\end{itemize}
|
||||
}\ger{
|
||||
\begin{itemize}
|
||||
\item Unbesetzter Gitterpunkt
|
||||
\item Geringe Formationsenergie
|
||||
\end{itemize}
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{interstitial}
|
||||
\desc{Interstitial}{}{}
|
||||
\desc[german]{}{}{}
|
||||
\ttxt{\eng{
|
||||
\begin{itemize}
|
||||
\item Extranous atom between lattice atoms
|
||||
\item High formation energy
|
||||
\end{itemize}
|
||||
}\ger{
|
||||
\begin{itemize}
|
||||
\item Zusätzliches Atom zwischen Gitteratomen
|
||||
\item Hohe Formationsenergy
|
||||
\end{itemize}
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{schottky}
|
||||
\desc{Schottky defect}{}{}
|
||||
\desc[german]{Schottky-Defekt}{}{}
|
||||
\ttxt{\eng{
|
||||
Atom type A \fRef{:::vacancy} + atom type B \fRef{:::vacancy}.
|
||||
Only in (partially) ionic materials.
|
||||
}\ger{
|
||||
\fRef{:::vacancy} von Atomsorte A und \fRef{:::vacancy} von Atomsorte B.
|
||||
Tritt nur in ionischen Materialiern auf.
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{frenkel}
|
||||
\desc{Frenkel defect}{}{}
|
||||
\desc[german]{Frenkel Defekt}{}{}
|
||||
\ttxt{\eng{
|
||||
\fRef{:::vacancy} + \fRef{:::interstitial}
|
||||
}\ger{
|
||||
\fRef{:::vacancy} + \fRef{:::interstitial}
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection{line}
|
||||
\desc{Line defects}{}{}
|
||||
\desc[german]{Liniendefekte}{}{}
|
||||
\begin{formula}{edge}
|
||||
\desc{Edge distortion}{}{}
|
||||
\desc[german]{Stufenversetzung}{}{}
|
||||
\ttxt{\eng{
|
||||
Insertion of an extra plane of atoms
|
||||
}\ger{
|
||||
Einschiebung einer zusätzliche Atomebene
|
||||
}}
|
||||
\TODO{images}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{screw}
|
||||
\desc{Screw distortion}{}{}
|
||||
\desc[german]{Schraubenversetzung}{}{}
|
||||
\ttxt{\eng{
|
||||
\TODO{TODO}
|
||||
}\ger{
|
||||
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{burgers_vector}
|
||||
\desc{Burgers vector}{Magnitude and direction of dislocation}{}
|
||||
\desc[german]{Burgers-Vektor}{Größe und Richtung einer Versetzung}{}
|
||||
\quantity{\vecb}{units}{ievs}
|
||||
\eq{
|
||||
\TODO{TODO}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection{area}
|
||||
\desc{Area defects}{}{}
|
||||
\desc[german]{Flächendefekte}{}{}
|
||||
\begin{formula}{grain_boundary}
|
||||
\desc{Grain boundary}{}{}
|
||||
\desc[german]{Korngrenze}{}{}
|
||||
\ttxt{\eng{
|
||||
Lead to
|
||||
\begin{itemize}
|
||||
\item Secondary phases
|
||||
\item Charge carrier trapping, recombination
|
||||
\item High mass diffusion constants
|
||||
\end{itemize}
|
||||
}\ger{
|
||||
Führen zu
|
||||
\begin{itemize}
|
||||
\item Sekundärphasen
|
||||
\item Separierung, Trapping und Streuung von Ladunsträgern
|
||||
\item Hohe Massendiffusionskonstante
|
||||
\end{itemize}
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
143
src/cm/egas.tex
143
src/cm/egas.tex
@ -1,143 +0,0 @@
|
||||
\Section{egas}
|
||||
\desc{Free electron gas}{Assumptions: electrons can move freely and independent of each other. \GT{see_also}: \fRef{td:id_qgas}}{}
|
||||
\desc[german]{Freies Elektronengase}{Annahmen: Elektronen bewegen sich frei und unabhänig voneinander. \GT{see_also}: \fRef{td:id_qgas}}{}
|
||||
|
||||
\TODO{merge with stat mech egas?}
|
||||
|
||||
\begin{formula}{density_of_states}
|
||||
\abbrLabel{DOS}
|
||||
\desc{Density of states (DOS)}{}{\QtyRef{volume}, $N$ number of energy levels, \QtyRef{energy}}
|
||||
\desc[german]{Zustandsdichte (DOS)}{}{\QtyRef{volume}, $N$ Anzahl der Energieniveaus, \QtyRef{energy}}
|
||||
\quantity{D,g}{\per\m^3}{s}
|
||||
\eq{D(E) = \frac{1}{V}\sum_{i=1}^{N} \delta(E-E(\vec{k_i}))}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{fermi-dirac}
|
||||
\desc{Fermi-Dirac distribution}{For electrons and holes}{}
|
||||
\desc[german]{Fermi-Dirac Verteilung}{Für Elektronen und Löcher}{}
|
||||
\eq{
|
||||
f_\txe(E) &= \frac{1}{\Exp{\frac{E-\EFermi}{\kB T}+1}}\\
|
||||
f_\txh(E) &= 1-f_\txe(E)
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{charge_carrier_density}
|
||||
\desc{Charge carrier density}{Number of charge carriers per volume}{$N$ number of charge carriers, \QtyRef{volume}, $D$ \qtyRef{density_of_states}, $f$ \fRef{::fermi-dirac}, \QtyRef{energy}, \QtyRef{fermi_energy}}
|
||||
\desc[german]{Ladungsträgerdichte}{Anzahl der Ladungsträger pro Volumen}{}
|
||||
\quantity{n}{\per\m^3}{s}
|
||||
\eq{n = \frac{N}{V} = \int_0^\infty D(E) f(E,T=0) \d E = \int_0^{\Efermi} D(E) \d E}
|
||||
\end{formula}
|
||||
|
||||
\Subsection{3deg}
|
||||
\desc{3D electron gas}{}{}
|
||||
\desc[german]{3D Elektronengas}{}{}
|
||||
\begin{formula}{dos}
|
||||
\desc{Density of states}{}{}
|
||||
\desc[german]{Zustandsdichte}{}{}
|
||||
\eq{D_\text{3D}(E) = \frac{1}{2\pi^2} \left(\frac{2m}{\hbar^2}\right)^{3/2} \sqrt{E}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{fermi_energy}
|
||||
\desc{\qtyRef{fermi_energy}}{}{$n$ \qtyRef{charge_carrier_density}, $m$ \qtyRef{mass}}
|
||||
\desc[german]{\qtyRef{fermi_energy}}{}{}
|
||||
\eq{\EFermi = \frac{\hbar^2}{2m} \left(3\pi^2n\right)^{2/3}}
|
||||
\end{formula}
|
||||
|
||||
\Subsection{2deg}
|
||||
\desc{2D electron gas / Quantum well}{Lower dimension gases can be obtained by restricting a 3D gas with infinetly high potential walls on a narrow area with the width $L$.}{}
|
||||
\desc[german]{2D Elektronengas / Quantum well}{Niederdimensionale Elektronengase erhält man, wenn ein 3D Gas durch unendlich hohe Potentialwände auf einem schmalen Bereich mit Breite $L$ eingeschränkt wird.}{}
|
||||
\begin{formula}{confinement_energy}
|
||||
\desc{Confinement energy}{Raises ground state energy}{}
|
||||
\desc[german]{Confinement Energie}{Erhöht die Grundzustandsenergie}{}
|
||||
\eq{\Delta E = \frac{\hbar^2 \pi^2}{2\meff L^2}}
|
||||
\end{formula}
|
||||
|
||||
\Eng[plain_wave]{plain wave}
|
||||
\Ger[plain_wave]{ebene Welle}
|
||||
\begin{formula}{energy}
|
||||
\desc{Energy}{}{}
|
||||
\desc[german]{Energie}{}{}
|
||||
\eq{E_{n,k_x,k_y} = \underbrace{\frac{\hbar^2 k_\parallel^2}{2\meff}}_\text{$x$-$y$: \GT{plain_wave}} + \underbrace{\frac{\hbar^2 \pi^2}{2\meff L^2} n^2}_\text{$z$}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{wavefunction}
|
||||
\desc{Wavefunction}{}{$\chi$ envelope function}
|
||||
\desc[german]{Wellenfunktion}{}{$\chi$ Einhüllende Funktion}
|
||||
\eq{\Psi(\vecr) = \e^{\I \veck_\parallel\cdot\vecr_\parallel} \underbrace{\chi(z)}_{\text{quantized motion}}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{se}
|
||||
\desc{Ben-Daniel-Duke Schrödinger equation}{}{\QtyRef{effective_mass}, $V$ \fRef{::effective_potential}, $\chi_n$ envelope functions, $E$ \qtyRef{energy}}
|
||||
\desc[german]{Ben-Daniel-Duke Schrödingergleichung}{}{}
|
||||
\eq{\left(-\frac{\hbar^2}{2} \pdv{}{z} \frac{1}{\meff_z(z)} \pdv{}{z} + V(z)\right) \chi_n(z) = E_{n,k_x,k_y} \chi_n(z)}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{effective_potential}
|
||||
\eng[elstat]{Electrostatic}
|
||||
\ger[elstat]{Electrostatisch}
|
||||
\eng[bandedge_modulation]{bandedge modulation}
|
||||
\ger[bandedge_modulation]{Bandkantenmodulierung}
|
||||
\eng[xc]{exchange correlation}
|
||||
% \ger[xc]{}
|
||||
\eng[image_charges]{image charges}
|
||||
\ger[image_charges]{Bildladungen}
|
||||
\desc{Effective Potential}{Often self-consistent solution of \fRef[SG]{::se} and \absRef{poisson_equation} required}{
|
||||
$V_\text{bm}(z)$ \gt{bandedge_modulation}
|
||||
$V_\txe(z)$ \gt{elstat}
|
||||
$V_\text{XC}(z)$ \gt{xc}
|
||||
$V_\text{img}$ \gt{image_charges}
|
||||
}
|
||||
\desc[german]{Effekives Potential}{}{}
|
||||
\eq{ V(z) = V_\text{bm}(z) - V_\txe(z) \pm V_\text{XC}(z) + V_\text{img}
|
||||
}
|
||||
\TODO{in real heterostructure, move to sc section}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{dos}
|
||||
\desc{Density of states}{}{}
|
||||
\desc[german]{Zustandsdichte}{}{}
|
||||
\eq{D_\text{2D}(E) = \frac{m}{\pi\hbar^2}}
|
||||
\end{formula}
|
||||
\TODO{plot band structure+dos? adv sc. slide 170}
|
||||
|
||||
\Subsection{1deg}
|
||||
\desc{1D electron gas / Quantum wire}{}{}
|
||||
\desc[german]{1D Eleltronengas / Quantendraht}{}{}
|
||||
|
||||
\begin{formula}{energy}
|
||||
\desc{Energy}{}{}
|
||||
\desc[german]{Energie}{}{}
|
||||
\eq{E_{nm,k_z} = \frac{\hbar^2 k_z^2}{2\meff_z} + \frac{\hbar^2 \pi^2}{2} \left(\frac{n^2}{\meff_xL_x^2} + \frac{m^2}{\meff_yL_y^2}\right)}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{dos}
|
||||
\desc{Density of states}{}{}
|
||||
\desc[german]{Zustandsdichte}{}{}
|
||||
\eq{D_\text{1D}(E) = \frac{1}{\pi\hbar} \sqrt{\frac{m}{2}} \frac{1}{\sqrt{E}}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{conductance}
|
||||
\desc{Quantized conductance}{per 1D subband}{$T(E)$ transmission probability}
|
||||
\desc[german]{Quantisierte Leitfähigkeit}{pro 1D Subband}{}
|
||||
\eq{G = \frac{2e^2}{h} T(E) = \frac{2}{R_\txK} T(E)}
|
||||
\end{formula}
|
||||
|
||||
\Subsection{0deg}
|
||||
\desc{0D electron gas / Quantum dot}{}{}
|
||||
\desc[german]{0D Elektronengase / Quantenpunkt}{}{}
|
||||
|
||||
\begin{formula}{energy}
|
||||
\desc{Energy}{}{}
|
||||
\desc[german]{Energie}{}{}
|
||||
\eq{E_{nml} = \left(\frac{\hbar^2\pi^2}{2m_\perp^*}\right) \left[ \left(\frac{n}{L_x}\right)^2 + \left(\frac{m}{L_y}\right)^2 + \left(\frac{l}{L_z}\right)^2\right]}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{dos}
|
||||
\desc{Density of states}{}{}
|
||||
\desc[german]{Zustandsdichte}{}{}
|
||||
\eq{D_\text{0D}(E) = 2\delta(E-E_C)}
|
||||
\end{formula}
|
||||
|
||||
\TODO{TODO}
|
||||
|
||||
|
||||
@ -1,28 +0,0 @@
|
||||
\Section{mat}
|
||||
\desc{Material physics}{}{}
|
||||
\desc[german]{Materialphysik}{}{}
|
||||
|
||||
|
||||
\begin{formula}{tortuosity}
|
||||
\desc{Tortuosity}{Degree of the winding of a transport path through a porous material. \\ Multiple definitions exist}{$l$ path length, $L$ distance of the end points}
|
||||
\desc[german]{Toruosität}{Grad der Gewundenheit eines Transportweges in einem porösen Material. \\ Mehrere Definitionen existieren}{$l$ Weglänge, $L$ Distanz der Endpunkte}
|
||||
\quantity{\tau}{}{}
|
||||
\eq{
|
||||
\tau &= \left(\frac{l}{L}\right)^2 \\
|
||||
\tau &= \frac{l}{L}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{stress}
|
||||
\desc{Stress}{Force per area}{\QtyRef{force}, \QtyRef{area}}
|
||||
\desc[german]{Spannung}{(Engl. "stress") Kraft pro Fläche}{}
|
||||
\quantity{\sigma}{\newton\per\m^2}{v}
|
||||
\eq{\ten{\sigma}_{ij} = \frac{F_i}{A_j}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{strain}
|
||||
\desc{Strain}{}{$\Delta x$ distance from reference position $x_0$}
|
||||
\desc[german]{Dehnung}{(Engl. "strain")}{$\Delta x$ Auslenkung aus der Referenzposition $x_0$}
|
||||
\quantity{\epsilon}{}{s}
|
||||
\eq{\epsilon = \frac{\Delta x}{x_0}}
|
||||
\end{formula}
|
||||
212
src/cm/misc.tex
212
src/cm/misc.tex
@ -1,212 +0,0 @@
|
||||
\Section{band}
|
||||
\desc{Band theory}{}{}
|
||||
\desc[german]{Bändermodell}{}{}
|
||||
|
||||
\begin{formula}{strain}
|
||||
\desc{Influence of strain}{}{}
|
||||
\desc[german]{Einfluss von Dehnung}{}{}
|
||||
\ttxt{\eng{
|
||||
\textbf{Hydrostatic strain (I)}: widens band gap, preserves symmetries \\
|
||||
\textbf{Biaxial strain (B)}: alters symmetry \Rightarrow lifts band degeneracies
|
||||
}\ger{
|
||||
\textbf{Hydrostatische Dehnung}: vergrößert die Bandlücke, erhält Symmetrien \\
|
||||
\textbf{Biaxiale Dehnung}: verändert die Symmetrie \Rightarrow hebt Entartung der Bänder auf
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\Subsection{effective_mass}
|
||||
\desc{Effective mass approximation}{Approximation leading to parabolic band dispersion. The higher the effective mass, the stronger the band is curved.}{}
|
||||
\desc[german]{Effektive Masse - Näherung}{Näherung mit parabolischer Dispersion (Bändern). Je höher die effeltive Masse, desto stärker die Krümmung des Bandes.}{}
|
||||
|
||||
\begin{formula}{effective_mass}
|
||||
\desc{Effective mass}{}{Usually stated in terms of \ConstRef{electron_mass}}
|
||||
\desc[german]{Effektive Masse}{}{Meistens als Vielfaches der \ConstRef{electron_mass} angegeben}
|
||||
\quantity{\ten{\meff}}{\kg}{t}
|
||||
\eq{\left(\frac{1}{\meff}\right)_{ij} = \frac{1}{\hbar^2} \pdv{E}{k_i,k_j}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsubsection{kp}
|
||||
\desc{k p Method}{
|
||||
\fRef[Pertubative]{qm:qm_pertubation} method for calculating the effective mass of a band.
|
||||
Treats the $\veck\cdot\vecp$ term in the Bloch Hamiltonian as second-order pertubation near an extrmum, usually at $k=0$.
|
||||
}{}
|
||||
\desc[german]{kp Methode}{
|
||||
\fRef[Störungstheoretische]{qm:qm_pertubation} Methode zur Berechnung der effektiver Massen von Bändern.
|
||||
Betrachtung des Terms $\veck\cdot\vecp$ im Bloch-Hamiltonian als Störung zweiter Ordnung um ein Extremum, meistens bei $k=0$.
|
||||
}{}
|
||||
\begin{formula}{dispersion}
|
||||
\desc{Parabolic dispersion}{}{$n$ band, $\veck$ \qtyRef{wavevector}, \QtyRef{effective_mass}}
|
||||
\desc[german]{Parabolische Dispersion}{}{}
|
||||
\eq{E_{n,k} = E_{n,0} + \frac{\hbar^2k^2}{2\meff}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{effective_mass}
|
||||
\desc{Effective mass}{for non-degenerate Bands}{$u_{n,\veck}$ \absRef{bloch_function} of band $n$ at $\veck$, $\veck$ \fRef{wavevector}, $\vecp$ \absRef{momentum_operator}}
|
||||
\desc[german]{Effektive Masse}{für nicht-entartete Bänder}{}
|
||||
\eq{
|
||||
\frac{1}{\meff} = \frac{1}{m_0} + \frac{2}{m_0^2k^2} \sum_{m\neq n} \frac{\abs{\Braket{u_{n,0}|\veck\cdot\vecp|u_{m,0}}}}{E_{n,0} - E_{m,0}}
|
||||
}
|
||||
\ttxt{\eng{
|
||||
\begin{itemize}
|
||||
\item[\Rightarrow] Energy separation of bands $n$ and $m$ determines importance of $m$ on the effective mass of $n$
|
||||
\item[\Rightarrow] Coupling between $n$ and $m$ only if $\Braket{u_{m,0}|\vecp|u_{n,p}} \neq 0$ (matrix element from group theory)
|
||||
\end{itemize}
|
||||
}\ger{
|
||||
\begin{itemize}
|
||||
\item[\Rightarrow] Energieabstand der Bänder $n$ und $m$ bestimmt den Einfluss von $m$ auf die effektive Masse von $n$
|
||||
\item[\Rightarrow] Kopplung zwischen $n$ und $m$ nur wenn $\Braket{u_{m,0}|\vecp|u_{n,p}} \neq 0$ (Matrixelement aus der Gruppentheorie)
|
||||
\end{itemize}
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{conduction_band_mass}
|
||||
\desc{Effective mass of the conduction band}{in most group IV,III-V and II-VI semiconductors}{$P^2$ matrix element, $\Egap$ \absRef{bandgap}}
|
||||
\desc[german]{Effektive Masse des Leitungsbands}{für die meisten IV, III-V und II-VI Halbleiter}{}
|
||||
\eq{\frac{m}{\meff_\txC} \approx 1 + \frac{1}{\Egap} \frac{2P^2}{m} = 1 + \frac{\SI{20}{\electronvolt}}{\Egap}}
|
||||
\end{formula}
|
||||
|
||||
\begin{bigformula}{splitting}
|
||||
\eng[binding]{binding}
|
||||
\eng[antibinding]{anti-binding}
|
||||
\ger[binding]{bindend}
|
||||
\ger[antibinding]{anti-bindend}
|
||||
|
||||
\desc{Band splitting}{}{}
|
||||
\desc[german]{Band Aufteilung}{}{}
|
||||
\fcenter{
|
||||
\input{img_static/cm/bands_schematic.tex}
|
||||
}
|
||||
\end{bigformula}
|
||||
|
||||
\Subsection{hybrid_orbitals}
|
||||
\desc{Hybrid orbitals}{Hybrid orbitals are linear combinations of other atomic orbitals.}{}
|
||||
\desc[german]{Hybridorbitale}{Hybridorbitale werden durch Linearkombinationen von anderen atomorbitalen gebildet.}{}
|
||||
|
||||
% chemmacros package
|
||||
\begin{formula}{sp}
|
||||
\desc{sp Orbital}{\GT{eg} \ce{C2H2}}{}
|
||||
\desc[german]{sp Orbital}{}{}
|
||||
\ttxt{\eng{Linear with bond angle \SI{180}{\degree}}\ger{Linear mit Bindungswinkel \SI{180}{\degree}}}
|
||||
\eq{
|
||||
1\text{s} + 1\text{p} = \text{sp}
|
||||
\orbital{sp}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{sp2}
|
||||
\desc{sp2 Orbital}{\GT{eg} \ce{C2H4}}{}
|
||||
\desc[german]{sp2 Orbital}{}{}
|
||||
\ttxt{\eng{Trigonal planar with bond angle \SI{120}{\degree}}\ger{Trigonal planar mit Bindungswinkel \SI{120}{\degree}}}
|
||||
\eq{
|
||||
1\text{s} + 2\text{p} = \text{sp2}
|
||||
\orbital{sp2}
|
||||
% \\ \ket{p} = \cos\theta \ket{p_x} + \sin\theta \ket{p_y}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{sp3}
|
||||
\desc{sp3 Orbital}{\GT{eg} \ce{CH4}}{}
|
||||
\desc[german]{sp3 Orbital}{}{}
|
||||
\ttxt{\eng{Tetrahedral with bond angle \SI{109.5}{\degree}}\ger{Tetraedisch mit Bindungswinkel \SI{109.5}{\degree}}}
|
||||
\eq{
|
||||
1\text{s} + 3\text{p} = \text{sp3}
|
||||
\orbital{sp3}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{wave_function}
|
||||
\desc{Wave function}{of a hybrid orbital}{$N$ number of involved $p$ orbitals}
|
||||
\desc[german]{Wellenfunktion}{eines Hybridorbitals}{$N$ Anzahl der beteiligten $p$ Orbitale}
|
||||
\eq{\ket{h_{1\dots N+1}} = \frac{1}{\sqrt{N+1}} \left(\ket{s} + \sqrt{N} \ket{p}\right)}
|
||||
\end{formula}
|
||||
|
||||
|
||||
|
||||
|
||||
\Section{diffusion}
|
||||
\desc{Diffusion}{}{}
|
||||
\desc[german]{Diffusion}{}{}
|
||||
\begin{formula}{diffusion_coefficient}
|
||||
\desc{Diffusion coefficient}{}{}
|
||||
\desc[german]{Diffusionskoeffizient}{}{}
|
||||
\quantity{D}{\m^2\per\s}{s}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{particle_current_density}
|
||||
\desc{Particle current density}{Number of particles through an area}{}
|
||||
\desc[german]{Teilchenstromdichte}{Anzahl der Teilchen durch eine Fläche}{}
|
||||
\quantity{J}{1\per\s^2}{s}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{einstein_relation}
|
||||
\desc{Einstein relation}{Classical}{\QtyRef{diffusion_coefficient}, \QtyRef{mobility}, \QtyRef{temperature}, $q$ \qtyRef{charge}}
|
||||
\desc[german]{Einsteinrelation}{Klassisch}{}
|
||||
\eq{D = \frac{\mu \kB T}{q}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{concentration}
|
||||
\desc{Concentration}{A quantity per volume}{}
|
||||
\desc[german]{Konzentration}{Eine Größe pro Volumen}{}
|
||||
\quantity{c}{x\per\m^3}{s}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{fick_law_1}
|
||||
\desc{Fick's first law}{Particle movement is proportional to concentration gradient}{\QtyRef{particle_current_density}, \QtyRef{diffusion_coefficient}, \QtyRef{concentration}}
|
||||
\desc[german]{Erstes Ficksches Gesetz}{Teilchenbewegung ist proportional zum Konzentrationsgradienten}{}
|
||||
\eq{J = -D\pdv{c}{x}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{fick_law_2}
|
||||
\desc{Fick's second law}{}{\QtyRef{particle_current_density}, \QtyRef{diffusion_coefficient}, \QtyRef{concentration}}
|
||||
\desc[german]{Zweites Ficksches Gesetz}{}{}
|
||||
\eq{\pdv{c}{t} = D \pdv[2]{c}{x}}
|
||||
\end{formula}
|
||||
|
||||
\Section{misc}
|
||||
% \desc{\GT{misc}}{}{}
|
||||
% \desc[german]{\GT{misc}}{}{}
|
||||
|
||||
\begin{formula}{vdw_material}
|
||||
\desc{Van-der-Waals material}{2D materials}{}
|
||||
\desc[german]{Van-der-Waals Material}{2D Materialien}{}
|
||||
\ttxt{\eng{
|
||||
Materials consiting of multiple 2D-layers held together by Van-der-Waals forces.
|
||||
}\ger{
|
||||
Aus mehreren 2D-Schichten bestehende Materialien, die durch Van-der-Waals Kräfte zusammengehalten werden.
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{work_function}
|
||||
\desc{Work function}{Lowest energy required to remove an electron into the vacuum}{}
|
||||
\desc[german]{Austrittsarbeit}{eng. "Work function"; minimale Energie um ein Elektron aus dem Festkörper zu lösen}{}
|
||||
\quantity{\Phi}{\volt}{s}
|
||||
\eq{e\Phi = \Evac - \EFermi}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{electron_affinity}
|
||||
\desc{Electron affinity}{Energy required to remove one electron from an anion with one negative charge.\\Energy difference between vacuum level and conduction band}{}
|
||||
\desc[german]{Elektronenaffinität}{Energie, die benötigt wird um ein Elektron aus einem einfach-negativ geladenen Anion zu entfernen. Entspricht der Energiedifferenz zwischen Vakuum-Niveau und dem Leitungsband}{}
|
||||
\quantity{\chi}{\volt}{s}
|
||||
\eq{e\chi = \left(\Evac - \Econd\right)}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{vacuum}
|
||||
\desc{Vacuum ranges}{}{}
|
||||
\desc[german]{Vakuumklassen}{}{}
|
||||
\ttxt{\eng{
|
||||
\begin{itemize}
|
||||
\item \textbf{Rough vacuum}: \SI{1}{\atm} - \SI{10e-2}{\milli\bar} \\ viscous flow
|
||||
\item \textbf{Process vacuum}: \SI{10e-2}{\milli\bar} - \SI{10e-4}{\milli\bar} \\ \abbrRef{mean_free_path} $\le$ chamber size
|
||||
\item \textbf{High vacuum}: \SI{10e-5}{\milli\bar} - \SI{10e-9}{\milli\bar} \\ \abbrRef{mean_free_path} $>$ chamber size, mostly residual \ce{H20} vapor
|
||||
\item \textbf{Ultra-high vacuum}: $<$ \SI{10e-9}{\milli\bar} \\ \abbrRef{mean_free_path} $\gg$ chamber size, mostly residual \ce{H2}
|
||||
\end{itemize}
|
||||
}\ger{
|
||||
\begin{itemize}
|
||||
\item \textbf{Grobvakuum}: \SI{1}{\atm} - \SI{10e-2}{\milli\bar} \\ viskoser Fluss
|
||||
\item \textbf{Prozessvakuum}: \SI{10e-2}{\milli\bar} - \SI{10e-4}{\milli\bar} \\ \abbrRef{mean_free_path} $\le$ Kammergröße
|
||||
\item \textbf{Hochvakuum}: \SI{10e-5}{\milli\bar} - \SI{10e-9}{\milli\bar} \\ \abbrRef{mean_free_path} $>$ Kammergröße, hauptsächlich \ce{H2O} Rückstände übrig
|
||||
\item \textbf{Ultrahochvakuum}: $<$ \SI{10e-9}{\milli\bar} \\ \abbrRef{mean_free_path} $\gg$ Kammergröße, hauptsächlich \ce{H2} Rückstände übrig
|
||||
\end{itemize}
|
||||
}}
|
||||
\end{formula}
|
||||
@ -1,198 +0,0 @@
|
||||
\Section{optics}
|
||||
\desc{Optics}{}{}
|
||||
\desc[german]{Optik}{}{}
|
||||
|
||||
% \Subsection{insulator}
|
||||
% \desc{Dielectrics and Insulators}{}{}
|
||||
% \desc[german]{Dielektrika und Isolatoren}{}{}
|
||||
|
||||
\begin{formula}{eom}
|
||||
\desc{Equation of motion}{Nuclei remain quasi static, electrons respond to field}{$u$ \GT{dislocation}, $\gamma = \frac{1}{\tau}$ \GT{dampening}, \QtyRef{momentum_relaxation_time}, \QtyRef{electric_field}, \ConstRef{charge}, $\omega_0$ \GT{resonance_frequency}, \ConstRef{electron_mass}}
|
||||
\desc[german]{Bewegungsgleichung}{Kerne bleiben quasi-statisch, Elektronen beeinflusst durch äußeres Feld}{}
|
||||
\eq{m_\txe \odv{u}{t^2} = -e\E - m_\txe \gamma \pdv{u}{t} - m_\txe\omega_0^2 u}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{lorentz}
|
||||
\desc{Drude-Lorentz model}{Dipoles treated as classical harmonic oscillators}{\QtyRef{electric_suseptibility}, $N$ number of oscillators (atoms), $\omega_0$ resonance frequency, Absorption has \absRef[lorentzian shape]{lorentz_distribution}}
|
||||
\desc[german]{Drude-Lorentz-Model}{Dipole werden als klassische harmonische Oszillatoren behandelt}{\QtyRef{electric_suseptibility}, $N$ Anzahl der Oszillatoren (Atome), $\omega_0$ Resonanzfrequenz, Absorption hat Form einer \absRef[Lorentz-Verteilung]{lorentz_distribution}}
|
||||
\eq{\epsilon_\txr(\omega) = 1+\chi_\txe + \frac{Ne^2}{\epsilon_0 m_\txe} \left(\frac{1}{\omega^2-\omega^2-i\gamma\omega}\right)}
|
||||
\eq{
|
||||
\complex{\epsilon}_\txr(0) &\to 1+\chi_\txe + \frac{Ne^2}{\epsilon_0 m_\txe \omega_0^2} \\
|
||||
\complex{\epsilon}_\txr(\infty) &= \epsilon_\infty = 1+\chi_\txe
|
||||
}
|
||||
\fig{img/cm_optics_absorption_dielectric.pdf}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{clausius-mosotti}
|
||||
\desc{Clausius-Mosotti relation}{for dense optical media: local field from external contribution + field from other dipoles}{$\chi_\txA$ \qtyRef[susecptibility]{susecptibility} of one atom, \QtyRef{relative_permittivity}, $N$ number of dipoles (atoms)}
|
||||
\desc[german]{Clausius-Mosotti Beziehung}{}{}
|
||||
\eq{\frac{(\epsilon_\txr - 1)}{\epsilon_\txr + 2} = \frac{N\chi_\txA}{3}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsection{metal}
|
||||
\desc{Metals and doped semiconductors}{}{}
|
||||
\desc[german]{Metalle und gedopte Halbleiter}{}{}
|
||||
|
||||
\begin{formula}{plasma_frequency}
|
||||
\desc{Plasma frequency}{For metals and doped semiconductors.}{$\epsilon_\infty$ high frequency \qtyRef[permittivity]{permittivity}, \ConstRef{vacuum_permittivity}, \QtyRef{effetive_mass}, \ConstRef{charge}, $n$ \qtyRef{charge_carrier_density}}
|
||||
\desc[german]{Plasmafrequenz}{In Metallen dotierten Halbleitern}{$\epsilon_\infty$ Hochfrequenz-\qtyRef{permittivity}, \ConstRef{vacuum_permittivity}, \QtyRef{effetive_mass}, \ConstRef{charge}, $n$ \qtyRef{charge_carrier_density}}
|
||||
\eq{\omega_\txp = \left(\frac{en^2}{\epsilon_0 \epsilon_\infty \meff}\right)^{1/2}}
|
||||
\ttxt{\eng{
|
||||
Characteristic frequency for collective motion in external field.\\
|
||||
For free charge carriers: perfect screening (reflection) of the external field for $\omega < \omega_\txp$.
|
||||
}\ger{
|
||||
Charakteristische Frequenz der kollektiven Bewegung im externen Feld.\\
|
||||
Für freie Ladungsträger: perfekte Abschirmung (Reflektion) des äußeren Feldes bei $\omega<\omega_\txp$
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{dielectric_function}
|
||||
\desc{\qtyRef{complex_dielectric_function}}{for a free electon plasma}{$\omega_\txp$ \fRef{::plasma_frequency}, $\omega_0$ \GT{resonance_frequency}, $\gamma = \frac{1}{\tau}$ \GT{dampening}, \QtyRef{momentum_relaxation_time}}
|
||||
\desc[german]{}{für ein Plasma aus freien Elektronen}{}
|
||||
\eq{
|
||||
\complex{\epsilon}_\txr(\omega) = 1 - \frac{\omega_\txp}{\omega_0^2 + i\gamma\omega} \\
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\begin{formula}{absorption}
|
||||
\desc{Free charge carrier absorption}{Exemplary values}{$\omega_\txp$ \fRef{::plasma_frequency}, \QtyRef{refraction_index_real}, \QtyRef{refraction_index_complex}, \QtyRef{absorption_coefficient}, $R$ \fRef{ed:optics:reflectivity}}
|
||||
\desc[german]{Freie Ladungsträger}{Beispielwerte}{}
|
||||
\fig{img/cm_optics_absorption_free_electrons.pdf}
|
||||
\TODO{Include equations? aus adv sc ex 10/1c}
|
||||
\end{formula}
|
||||
|
||||
\TODO{relation to AC and DC conductivity}
|
||||
|
||||
\Subsubsection{sc_interband}
|
||||
\desc{Interband transitions in semiconductors}{}{}
|
||||
% \desc[german]
|
||||
\begin{formula}{selection}
|
||||
\desc{Selection rule}{}{}
|
||||
\desc[german]{Auswahlregel}{}{}
|
||||
\ttxt{\eng{
|
||||
Parity of wave functions must be opposite for the transition to occur
|
||||
}\ger{
|
||||
Die Wellenfunktionen müssen unterschiedliche Parität haben, damit der Übergang möglich ist
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formulagroup}{transition_rate}
|
||||
\desc{Transition rate}{}{}
|
||||
\desc[german]{Übergangsrate}{}{}
|
||||
\begin{formula}{absorption}
|
||||
\desc{Absorption}{}{}
|
||||
\desc[german]{Absorption}{}{}
|
||||
\eq{W_{1\to2} = \frac{2\pi}{\hbar} \abs{M_{12}}^2 \delta(E_1-E_2+\hbar\omega)}
|
||||
\end{formula}
|
||||
\begin{formula}{emission}
|
||||
\desc{Emission}{}{}
|
||||
\desc[german]{Emission}{}{}
|
||||
\eq{W_{2\to1} = \frac{2\pi}{\hbar} \abs{M_{12}}^2 \delta(E_1-E_2-\hbar\omega)}
|
||||
\end{formula}
|
||||
\TODO{stimulated vs spontaneous}
|
||||
\TODO{matrix element stuff, kane energy}
|
||||
\begin{formula}{possible_matrix_elements}
|
||||
\desc{Matrix elements}{}{}
|
||||
\desc[german]{}{}{}
|
||||
\eq{\Braket{p_x|\hat{p}_x|s} = \Braket{p_y|\hat{p}_y|s} = \Braket{p_z|\hat{p}_z|s} \neq 0}
|
||||
\TODO{heavy holes, light holes}
|
||||
\end{formula}
|
||||
|
||||
\end{formulagroup}
|
||||
|
||||
\begin{formula}{jdos}
|
||||
\desc{Joint density of states}{}{}
|
||||
% \desc[german]{}{}{}
|
||||
\ttxt{\eng{
|
||||
Desribes the density of states of an optical transition by combining the electron states in the valence band and hole states in the conduction band.
|
||||
}\ger{
|
||||
Beschreibt die Zustandsdichte eines optischen Übergangs durch kombinieren der Elektronenzustände im Valenzband und der Lochzustände im Leitungsband.
|
||||
}}
|
||||
\eq{}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{absorption_coefficient_direct}
|
||||
\desc{\qtyRef{absorption_coefficient}}{For a direct semiconductor}{\QtyRef{angular_frequency}, \QtyRef{refraction_index_real}, \QtyRef{permittivity_complex}}
|
||||
\desc[german]{}{Für direkte Halbleiter}{}
|
||||
\eq{
|
||||
\alpha &= \frac{\omega}{\nReal c} \epsReal \\
|
||||
\left(\hbar\omega\alpha\right)^2 \propto \hbar\omega-\Egap
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{absorption_coefficient_indirect}
|
||||
\desc{\qtyRef{absorption_coefficient}}{For an indirect semiconductor}{\QtyRef{angular_frequency}, $E_\txp$ phonon energy, $E_\text{ig}$ indirect gap}
|
||||
\desc[german]{}{Für indirekte Halbleiter}{\QtyRef{angular_frequency}, $E_\txp$ Phononenergie, $E_\text{ig}$ indirekte Bandlücke}
|
||||
\eq{
|
||||
\sqrt{\hbar\omega\alpha} \propto \hbar\omega \mp E_\txp - E_\text{ig}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{exciton}
|
||||
\desc{\fRef[Exciton]{cm:sc:exciton} absorption}{}{\QtyRef{band_gap}, $E_\text{binding}$ \fRef{cm:sc:exciton:binding_energy}}
|
||||
\desc[german]{\fRef[Exciton]{cm:sc:exciton} Absorption}{}{}
|
||||
\ttxt{\eng{
|
||||
Due to binding energy, exciton absorption can happen below the \absRef[band gap energy]{band_gap} \Rightarrow Sharp absorption peak below $\Egap$.
|
||||
At high (room) temperatures, excitons are ionized by collisions with phonons.
|
||||
}\ger{
|
||||
Aufgrund der Bindungsenergie kann die Exzitonenabsorption unterhalb der \absRef[Bandlückenenergie]{band_gap} auftreten \Rightarrow scharfer Absorptionspeak unterhalb von $\Egap$.
|
||||
Bei hohen (Raum) Temperaturen werden Exzitons durch Kollisionen mit Phononen ionisiert.
|
||||
}}
|
||||
\eq{\hbar\omega = \Egap - E_\text{binding}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsubsection{quantum_well}
|
||||
\desc{Quantum wells}{}{}
|
||||
\desc[german]{Quantum Wells}{}{}
|
||||
\begin{formula}{interband}
|
||||
\desc{Interband transitions}{}{}
|
||||
\desc[german]{Interband-Übergänge}{}{}
|
||||
\ttxt{\eng{
|
||||
Selection rules:
|
||||
\begin{itemize}
|
||||
\item $\E \parallel \text{QW}$: allowed for \abbrRef{light_hole}, \abbrRef{heavy_hole}
|
||||
\item $\E \perp \text{QW}$: allowed for \abbrRef{light_hole}, forbidden for \abbrRef{heavy_hole}
|
||||
\item In a symmetric potential: only $\Delta n=0$ transitions allowed
|
||||
\end{itemize}
|
||||
}\ger{
|
||||
Auswahlregeln:
|
||||
\begin{itemize}
|
||||
\item $\E \parallel \text{QW}$: erlaubt für \abbrRef{light_hole}, \abbrRef{heavy_hole}
|
||||
\item $\E \perp \text{QW}$: erlaubt für \abbrRef{light_hole}, verboten für \abbrRef{heavy_hole}
|
||||
\item In einem symmatrischen Potential: nur Übergänge mit $\Delta n=0$ erlaubt
|
||||
\end{itemize}
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{intersubband}
|
||||
\desc{Inter-subband transitions}{\qtyrange{3}{27}{\micro\m}}{}
|
||||
\desc[german]{Inter-Subband-Übergänge}{}{}
|
||||
\ttxt{\eng{
|
||||
Selection rules:
|
||||
\begin{itemize}
|
||||
\item $\E \parallel \text{QW}$ allowed
|
||||
\item $\E \perp \text{QW}$ forbidden
|
||||
\item Parity of intial and final state must differ
|
||||
\end{itemize}
|
||||
}\ger{
|
||||
Auswahlregeln:
|
||||
\begin{itemize}
|
||||
\item $\E \parallel \text{QW}$ erlaubt
|
||||
\item $\E \perp \text{QW}$ verboten
|
||||
\item Parität von Anfangs- und Endzustand muss unterschiedlich sein
|
||||
\end{itemize}
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{exciton}
|
||||
\desc{Exciton in a quantum well}{Increased \fRef{::binding_energy} due to larger Coulomb interaction through confinment}{}
|
||||
% \desc[german]{}{}{}
|
||||
\eq{E^\text{2D}_n = \Egap + E_{\txe0} + E_{\txh0} - \frac{R^*}{\left(n-\frac{1}{2}\right)^2}}
|
||||
\end{formula}
|
||||
|
||||
\TODO{dipole approximation}
|
||||
|
||||
|
||||
@ -1,565 +0,0 @@
|
||||
\Section{sc}
|
||||
\desc{Semiconductors}{}{}
|
||||
\desc[german]{Halbleiter}{}{}
|
||||
\begin{formula}{description}
|
||||
\desc{Description}{}{$n,p$ \fRef{cm:sc:charge_carrier_density:equilibrium}}
|
||||
\desc[german]{Beschreibung}{}{}
|
||||
\ttxt{
|
||||
\eng{
|
||||
Materials with an electrical conductivity that can be modified through \fRef[doping]{::doping}.\\
|
||||
\textbf{Intrinsic}: pure, electron density determined only by thermal excitation and $n_i^2 = n_0 p_0$\\
|
||||
\textbf{Extrinsic}: doped
|
||||
}
|
||||
\ger{
|
||||
Materialien, bei denen die elektrische Leitfähigkeit durch \fRef[Dotierung]{::doping} verändert werden kann.\\
|
||||
\textbf{Intrinsisch}: Pur, Elektronendichte gegeben durch thermische Anregung und $n_i^2 = n_0 p_0$ \\
|
||||
\textbf{Extrinsisch}: dotiert
|
||||
}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formulagroup}{band_gap}
|
||||
\absLabel
|
||||
\absLabel[bandgap]
|
||||
\desc{Band gap}{Energy band gap}{}
|
||||
\desc[german]{Bandlücke}{Energielücke}{}
|
||||
\begin{formula}{definition}
|
||||
\desc{Definition}{}{}
|
||||
\desc[german]{Definition}{}{}
|
||||
\hiddenQuantity[band_gap]{\Egap}{\electronvolt}{s}
|
||||
\ttxt{\eng{
|
||||
Energy gap between highest occupied (HO) and lowest unoccupied (LU) band/orbital.
|
||||
\begin{itemize}
|
||||
\item \textbf{direct}: HO and LU at same $\veck$
|
||||
\item \textbf{indirect} HO and LU at different $\veck$
|
||||
\end{itemize}
|
||||
}\ger{
|
||||
Energielücke zwischen höchstem besetztem (HO) und niedrigsten unbesetzten (LU) Band/Orbital.
|
||||
\begin{itemize}
|
||||
\item \textbf{direkt}: HO und LU bei gleichem $\veck$
|
||||
\item \textbf{indirekt}: HO und LU bei unterschiedlichem $\veck$
|
||||
\end{itemize}
|
||||
}}
|
||||
\eq{\Egap = \Econd - \Evalence}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{temperature_dependence}
|
||||
\desc{Temperature Dependence}{}{}
|
||||
\desc[german]{Temperaturabhängigkeit}{}{}
|
||||
\ttxt{\eng{
|
||||
$T\uparrow\quad\Rightarrow \Egap\downarrow$
|
||||
\begin{itemize}
|
||||
\item Distance of atoms increases with higher temperatures \Rightarrow less wave function overlap
|
||||
\item Low temperature: less phonons avaiable for electron-phonon scattering
|
||||
\end{itemize}
|
||||
}\ger{
|
||||
$T\uparrow\quad\Rightarrow \Egap\downarrow$
|
||||
\begin{itemize}
|
||||
\item Atomabstand nimmt bei steigender Temperatur zu \Rightarrow kleiner Überlapp der Wellenfunktionen
|
||||
\item Geringe Temperatur: weniger Phonon zur Elektron-Phonon-Streuung zur Vefügung
|
||||
\end{itemize}
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{vashni}
|
||||
\desc{Vashni formula}{Empirical temperature dependence of the band gap}{}
|
||||
\desc[german]{Vashni-Gleichung}{Empirische Temperaturabhängigket der Bandlücke}{}
|
||||
\eq{\Egap(T) = \Egap(\SI{0}{\kelvin}) - \frac{\alpha T^2}{T + \beta}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{bandgaps}
|
||||
\desc{Bandgaps of common semiconductors}{}{}
|
||||
\desc[german]{Bandlücken wichtiger Halbleiter}{}{}
|
||||
\begin{tabular}{l|CCc}
|
||||
& \Egap(\SI{0}{\kelvin}) [\si{\eV}] & \Egap(\SI{300}{\kelvin}) [\si{\eV}] & \\ \hline
|
||||
\GT{diamond} & 5,48 & 5,47 & \GT{indirect} \\
|
||||
Si & 1,17 & 1,12 & \GT{indirect} \\
|
||||
Ge & 0,75 & 0,66 & \GT{indirect} \\
|
||||
GaP & 2,32 & 2,26 & \GT{indirect} \\
|
||||
GaAs & 1,52 & 1,43 & \GT{direct} \\
|
||||
InSb & 0,24 & 0,18 & \GT{direct} \\
|
||||
InP & 1,42 & 1,35 & \GT{direct} \\
|
||||
CdS & 2.58 & 2.42 & \GT{direct}
|
||||
\end{tabular}
|
||||
\end{formula}
|
||||
\end{formulagroup}
|
||||
|
||||
|
||||
\begin{formulagroup}{charge_carrier_density}
|
||||
\desc{Charge carrier density}{}{}
|
||||
\desc[german]{Ladungsträgerichte}{}{}
|
||||
\begin{formula}{general}
|
||||
\desc{Charge carrier density}{General form}{$D$ \qtyRef{dos}, $f$ \fRef{cm:egas:fermi-dirac}, \GT{see_also} \fRef{cm:egas:charge_carrier_density}}
|
||||
\desc[german]{Ladungsträgerdichte}{Allgemeine Form}{}
|
||||
\eq{
|
||||
n &= \int_{\Econd}^\infty D_\txe f_\txe(E)\d E\\
|
||||
p &= \int_{-\infty}^{\Evalence} D_\txh f_\txh(E)\d E
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{equilibrium}
|
||||
\desc{Equilibrium charge carrier densities}{\fRef{math:cal:integral:list:boltzmann_approximation}, holds when $\frac{\Econd-\EFermi}{\kB T}>3.6$ and $\frac{\EFermi-\Evalence}{\kB T} > 3.6$}{}
|
||||
\desc[german]{Ladungsträgerdichte im Equilibrium}{\fRef{math:cal:integral:list:boltzmann_approximation}, gilt wenn $\frac{\Econd-\EFermi}{\kB T}>3.6$ und $\frac{\EFermi-\Evalence}{\kB T} > 3.6$}{}
|
||||
\eq{
|
||||
n_0 &\approx N_\txC(T) \Exp{-\frac{\Econd - \EFermi}{\kB T}} \\
|
||||
p_0 &\approx N_\txV(T) \Exp{-\frac{\EFermi - \Evalence}{\kB T}}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{intrinsic}
|
||||
\desc{Intrinsic charge carrier density}{}{$N$ \fRef{:::band_edge_dos}}
|
||||
\desc[german]{Intrinsische Ladungsträgerdichte}{}{}
|
||||
\eq{
|
||||
n_\txi \approx \sqrt{n_0 p_0} = \sqrt{N_\txC(T) N_\txV(T)} \Exp{-\frac{E_\text{gap}}{2\kB T}}
|
||||
}
|
||||
\end{formula}
|
||||
\end{formulagroup}
|
||||
|
||||
\begin{formula}{band_edge_dos}
|
||||
\desc{Band edge density of states}{}{$\meff$ \qtyRef{effective_mass}, \ConstRef{boltzmann}, \QtyRef{temperature}}
|
||||
\desc[german]{Bandkanten-Zustandsdichte}{}{}
|
||||
\eq{
|
||||
N_\txC &= 2\left(\frac{\meff_\txe\kB T}{2\pi\hbar^2}\right)^{3/2} \\
|
||||
N_\txV &= 2\left(\frac{\meff_\txh\kB T}{2\pi\hbar^2}\right)^{3/2}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{mass_action}
|
||||
\desc{Mass action law}{Charge densities at thermal equilibrium, independent of doping}{$n_0/p_0$ \fRef{::charge_carrier_density:equilibrium}, $n_i/p_i$ \fRef{::charge_carrier_density:intrinsic}}
|
||||
\desc[german]{Massenwirkungsgesetz}{Ladungsträgerdichten im Equilibrium, unabhängig der Dotierung }{}
|
||||
\eq{n_0p_0 = n_i^2 = p_i^2 }
|
||||
\end{formula}
|
||||
|
||||
|
||||
|
||||
\begin{formula}{min_maj}
|
||||
\desc{Minority / Majority charge carriers}{}{}
|
||||
\desc[german]{Minoritäts- / Majoritätsladungstraäger}{}{}
|
||||
\ttxt{
|
||||
\eng{
|
||||
Majority carriers: higher number of particles ($e^-$ in n-type, $h^+$ in p-type)\\
|
||||
Minority carriers: lower number of particles ($h^+$ in n-type, $e^-$ in p-type)
|
||||
}
|
||||
\ger{
|
||||
Majoritätsladungstraäger: höhere Teilchenzahl ($e^-$ in n-Typ, $h^+$ in p-Typ)\\
|
||||
Minoritätsladungsträger: niedrigere Teilchenzahl ($h^+$ in n-Typ, $e^-$ in p-Typ)
|
||||
}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsection{dope}
|
||||
\desc{Doping}{}{}
|
||||
\desc[german]{Dotierung}{}{}
|
||||
|
||||
\begin{formula}{description}
|
||||
\desc{Description}{}{}
|
||||
\desc[german]{Beschreibung}{}{}
|
||||
\ttxt{\eng{
|
||||
Modification of charger carrier densities through defects.
|
||||
\begin{itemize}
|
||||
\item $N_\txA \gg N_\txD$ \Rightarrow p-type semiconductor
|
||||
\item $N_\txA \ll N_\txD$ \Rightarrow n-type semiconductor
|
||||
\item Else: compensated semiconductor, acceptors filled by electrons from donors:
|
||||
\end{itemize}
|
||||
}\ger{
|
||||
Modifizierung der Ladungsträgerichten durch Einbringung von Fremdatomen.
|
||||
\begin{itemize}
|
||||
\item $N_\txA \gg N_\txD$ \Rightarrow p-Typ Halbleiter
|
||||
\item $N_\txA \ll N_\txD$ \Rightarrow n-Typ Halbleiter
|
||||
\item Sonst: Kompensierter Halbleiter, Akzeptoren nehmen Elektronen der Donatoren auf
|
||||
\end{itemize}
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{charge_neutrality}
|
||||
\desc{Charge neutrality}{Fermi level must adjust so that charge neutrality is preserved}{$N_{\txD/\txA}^{+/-}$ ionized donor/acceptor density, $n,p$ \fRef{cm:sc:charge_carrier_density}}
|
||||
\desc[german]{Ladungsneutralität}{Fermi-Level muss sich so anpassen, dass Ladungsneutralität erhalten ist}{$N_{\txD/\txA}^{+/-}$ Dichte der ionisierten Donatoren/Akzeptoren , $n,p$ \fRef{cm:sc:charge_carrier_density}}
|
||||
\eq{0 = N_\txD^+ + p - N_\txA^- -n}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{ionization_ratio}
|
||||
\desc{Fraction ionized donors/acceptors}{At thermal equilibrium}{$N_{\txD/\txA}^{+/-}$ ionized donor/acceptor density, $N_{\txD/\txA}$ donor/acceptor density, $E_{\txD/\txA}$ donor/acceptor energy level, $g$ spin degeneracy}
|
||||
\desc[german]{Anteil ionisiserter Akzeptoren/Donatoren}{Im thermischen Equilibrium}{$N_{\txD/\txA}^{+/-}$ ionisierte Donor/Akzeptordichte, $N_{\txD/\txA}$ Donor/Akzeptordichte, $E_{\txD/\txA}$ Energie der Donatoren/Akzeptoren, $g$ Spindegenierung}
|
||||
\eq{
|
||||
\frac{N_\txD^+}{N_\txD} &= 1- \frac{1}{1+\frac{1}{g}\Exp{\frac{E_\txD-\Efermi}{\kB T}}} \\
|
||||
\frac{N_\txA^-}{N_\txA} &= \frac{1}{1+g\Exp{\frac{E_\txA-\Efermi}{\kB T}}}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{electron_density}
|
||||
\desc{Charge carrier density}{In a doped semiconductor}{Here: n-type (with donors), $N_\txD$ donor density, $E_\txD$ donor energy level}
|
||||
\desc[german]{Ladungsträgeridchte}{In einem dotierten Halbleiter}{Hier: n-Typ (mit Donatoren), $N_\txD$ Donatorendichte, $E_\txD$ Energieniveau der Donatoren}
|
||||
\fig{img/cm_sc_charge_carrier_density.pdf}
|
||||
\ttxt{\eng{
|
||||
\begin{itemize}
|
||||
\item Instrinsic: $\ln(n) \propto -\frac{\Econd-\Evalence}{2\kB T}$
|
||||
\item Saturation: $n = N_\txD$ all donors are ionized
|
||||
\item Freeze-out: Some donors retrap electrons, $\ln(n) \propto -\frac{\Econd-E_\txD}{2\kB T}$
|
||||
\end{itemize}
|
||||
|
||||
}\ger{
|
||||
\begin{itemize}
|
||||
\item Intrinsisch $\ln(n) \propto -\frac{\Econd-\Evalence}{2\kB T}$
|
||||
\item Sättigung $n = N_\txD$ alle Donatoren sind ionisiert
|
||||
\item Freeze-out: Donatoren binden Elektronen wieder, $\ln(n) \propto -\frac{\Econd-E_\txD}{2\kB T}$
|
||||
\end{itemize}
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{modulation}
|
||||
\desc{Modulation doping}{}{}
|
||||
% \desc[german]{}{}{}
|
||||
\ttxt{\eng{
|
||||
Free charge carriers and donors are spatially separated \Rightarrow no scattering at donors \Rightarrow very high \qtyRef[carrier mobilities]{mobility}
|
||||
}\ger{
|
||||
Freie Ladungsträger räumlich von den Dotieratomen getrennt \Rightarrow keine Streuung an Dotieratomen \Rightarrow sehr hohe \qtyRef[Mobilität der Ladungsträger]{mobility}
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsection{Recombination}
|
||||
\desc{Recombination}{}{}
|
||||
\desc[german]{Rekombination}{}{}
|
||||
|
||||
\begin{formula}{shockley-read}
|
||||
\desc{Shockley-Read-Hall recombination}{}{}
|
||||
\desc[german]{Shockley-Read-Hall Rekombination}{}{}
|
||||
\ttxt{\eng{
|
||||
Recombination via defect states in the band gap:
|
||||
Electron capture, electron emission, hole capture, hole emission
|
||||
}\ger{
|
||||
Rekombination über Defektzustände in der Bandlücke:
|
||||
Elektroneneinfang, Elektronenemission, Locherfassung, Locheremission
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{auger}
|
||||
\desc{Auger recombination}{}{}
|
||||
\desc[german]{Auger Rekombination}{}{}
|
||||
\ttxt{\eng{
|
||||
Non-radiative recombination involving three particles.
|
||||
Recombination energy is transferred to another electron or hole.
|
||||
Important at high carrier densities, high temperatures and small band gaps.
|
||||
}\ger{
|
||||
Nicht-strahlende Rekombination unter Beteiligung von drei Teilchen.
|
||||
Die Rekombinationsenergie wird auf ein anderes Elektron oder Loch übertragen.
|
||||
Wichtig bei hohen Ladungsträgerdichten, hohen Temperaturen und kleinen Bandlücken.
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{bi-molecular}
|
||||
\desc{Bi-molecular recombination}{}{}
|
||||
\desc[german]{Bimolekulare Rekombination}{}{}
|
||||
\ttxt{\eng{
|
||||
Radiative two-particle process where an electron from conduction and a hole from the valence band recombine.
|
||||
}\ger{
|
||||
Strahlender zwei-Teilchen-Prozess, bei dem ein Elektron aus dem Leitungsband und ein Loch aus dem Valenzband rekombinieren.
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\Subsection{devices}
|
||||
\desc{Devices and junctions}{}{}
|
||||
\desc[german]{Bauelemente und Kontakte}{}{}
|
||||
\Subsubsection{metal-sc}
|
||||
\desc{Metal-semiconductor junction}{Here: with n-type}{}
|
||||
\desc[german]{Metall-Halbleiter Kontakt}{Hier: mit n-Typ}{}
|
||||
\begin{formulagroup}{schottky_barrier}
|
||||
\desc{Schottky barrier}{Tunnel contact, for $\Phi_\txM > \Phi_\txS$, Rectifying \fRef{cm:sc:junctions:metal-sc}}{}
|
||||
% \desc[german]{}{}{}
|
||||
\begin{bigformula}{band_diagram}
|
||||
\desc{Band diagram}{}{}
|
||||
\desc[german]{Banddiagramm}{}{}
|
||||
\fcenter{
|
||||
\resizebox{0.49\textwidth}{!}{\input{img_static/cm/sc_junction_metal_n_sc_separate.tex}}
|
||||
\resizebox{0.49\textwidth}{!}{\input{img_static/cm/sc_junction_metal_n_sc.tex}}
|
||||
}
|
||||
\ttxt{\eng{
|
||||
Upon contact, electrons flow from the semicondctor to the metal to align the Fermi levels \Rightarrow leaves depletion region of positively charged donors as barrier
|
||||
}\ger{
|
||||
Bei Kontakt fließen Elektronen vom Halbleiter zum Metall, um die Fermi-Niveaus anzugleichen \Rightarrow es entsteht eine Verarmungszone aus positiv geladenen Donatoren als Barriere
|
||||
}}
|
||||
\end{bigformula}
|
||||
\begin{formula}{full_depletion_approx}
|
||||
\desc{Full depletion approximation}{Assume full depletion area with width $W_\txD$}{$N_\txD$ doping concentration}
|
||||
% \desc[german]{}{}{}
|
||||
\eq{\rho(x) = \left\{ \begin{array}{ll} qN_\txD & 0<x<W_\txD\\ 0 & W_\txD < x \end{array}\right. }
|
||||
\fig{img/cm_sc_devices_metal-n-sc_schottky.pdf}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{schottky-mott_rule}
|
||||
\desc{Schottky-Mott rule}{Approximation, often not valid because of Fermi level pinning through defects at the interface}{$\Phi_\txB$ barrier potential, $\Phi_\txM$ \GT{metal} \qtyRef{work_function}, $\chi_\text{sc}$ \qtyRef{electron_affinity}}
|
||||
% \desc[german]{}{}{}
|
||||
\eq{\Phi_\txB \approx \Phi_\txM - \chi_\text{sc}}
|
||||
\end{formula}
|
||||
\end{formulagroup}
|
||||
|
||||
\begin{formulagroup}{ohmic}
|
||||
\desc{Ohmic contact}{For $\Phi_\txM < \Phi_\txS$}{}
|
||||
\desc[german]{Ohmscher Kontakt}{}{}
|
||||
\begin{bigformula}{band_diagram}
|
||||
\desc{Band diagram}{}{}
|
||||
\desc[german]{Banddiagramm}{}{}
|
||||
\fcenter{
|
||||
\resizebox{0.49\textwidth}{!}{\input{img_static/cm/sc_junction_ohmic_separate.tex}}
|
||||
\resizebox{0.49\textwidth}{!}{\input{img_static/cm/sc_junction_ohmic.tex}}
|
||||
}
|
||||
\ttxt{\eng{
|
||||
Upon contact, electrons flow from the metal to the semiconductor to align the Fermi levels \Rightarrow space charge region
|
||||
}\ger{
|
||||
|
||||
}}
|
||||
|
||||
\end{bigformula}
|
||||
\TODO{Compare adv sc. ex/2 solution, there can still be a barrier}
|
||||
\end{formulagroup}
|
||||
|
||||
\Subsubsection{sc-sc}
|
||||
\desc{Semiconducto-semiconductor junction}{}{}
|
||||
\desc[german]{Halbleiter-Halbleiter Kontakt}{}{}
|
||||
|
||||
\begin{formulagroup}{pn}
|
||||
\desc{p-n junction}{}{}
|
||||
\desc[german]{p-n Übergang}{}{}
|
||||
|
||||
\begin{bigformula}{band_diagram}
|
||||
\desc{Band diagram}{}{}
|
||||
\desc[german]{Banddiagramm}{}{}
|
||||
\fcenter{
|
||||
\input{img_static/cm/sc_junction_pn.tex}
|
||||
\resizebox{0.49\textwidth}{!}{\tikzPnJunction{1/3}{0}{0}{1/3}{0}{0}{}}
|
||||
\resizebox{0.49\textwidth}{!}{\tikzPnJunction{1/2}{0.4}{-0.4}{1/2}{-0.4}{0.4}{}}
|
||||
}
|
||||
\end{bigformula}
|
||||
|
||||
\begin{formula}{no_bias}
|
||||
\desc{No bias}{Balance of \fRef[drift]{cm:charge_transport:current_density} and \fRef[diffusion]{cm:charge_transport:misc:diffusion_current} currents}{$n_{n/p}$ \fRef[equilibrium electron density]{cm:sc:charge_carrier_density:equilibrium} in the $n$/$p$ side}
|
||||
\desc[german]{Keine angelegte Spannung}{Gleichgewicht von \fRef[Drift-]{cm:charge_transport:current_density} und \fRef[Diffusions-]{cm:charge_transport:misc:diffusion_current}strömen}{$n_{0,n/p}$ \qtyRef[Elektronendichte]{charge_carrier_density} in der $n$/$p$ Seite}
|
||||
\eq{U_\text{bias}= \left(\frac{\kB T}{e}\right) \Ln{\frac{n_{0,n}}{n_{0,p}}}}
|
||||
\end{formula}
|
||||
\end{formulagroup}
|
||||
|
||||
\TODO{Forward bias: negativ an n, positiv an p}
|
||||
|
||||
\begin{formulagroup}{2deg}
|
||||
\desc{Heterointerface}{2DEG, \fRef{cm:sc:dope:modulation}}{}
|
||||
% \desc[german]{}{}{}
|
||||
\begin{formula}{schematic}
|
||||
\desc{Schematic and band diagram}{}{}
|
||||
\desc[german]{Aufbau und Banddiagramm}{}{}
|
||||
\input{img_static/cm/sc_2deg_device.tex}
|
||||
\end{formula}
|
||||
\TODO{finish picture}
|
||||
|
||||
\end{formulagroup}
|
||||
|
||||
\begin{formula}{band_alignments}
|
||||
\desc{Band alignments}{in semiconducor heterointerfaces. Band profile also depends on $k$-space position.}{}
|
||||
% \desc[german]{}{}{}
|
||||
\fcenter{
|
||||
\begin{tikzpicture}
|
||||
\pgfmathsetmacro{\LW}{1.5}
|
||||
\pgfmathsetmacro{\RW}{1.5}
|
||||
\pgfmathsetmacro{\texty}{-0.5}
|
||||
% type I
|
||||
\pgfmathsetmacro{\LGap}{2}
|
||||
\pgfmathsetmacro{\RGap}{1}
|
||||
\pgfmathsetmacro{\DeltaEV}{0.3}
|
||||
\begin{scope}
|
||||
\node at (\LW,\texty) {\GT{type}-I};
|
||||
\draw[sc band val] (0,0) -- ++(\LW,0) -- ++(0, \DeltaEV) -- ++(\RW,0);
|
||||
\draw[sc band con] (0,\LGap) -- ++(\LW,0) -- ++(0,-\LGap+\DeltaEV+\RGap) -- ++(\RW,0);
|
||||
\drawDArrow{\LW+\RW/4}{0}{\DeltaEV}{$\Delta \Evalence$}
|
||||
\drawDArrow{\LW+\RW/4}{\LGap}{\DeltaEV+\RGap}{$\Delta \Econd$}
|
||||
\end{scope}
|
||||
% type II
|
||||
\pgfmathsetmacro{\LGap}{1.2}
|
||||
\pgfmathsetmacro{\RGap}{1.2}
|
||||
\pgfmathsetmacro{\DeltaEV}{0.4}
|
||||
\begin{scope}[shift={(\LW+\RW+1,0)}]
|
||||
\node at (\LW,\texty) {\GT{type}-II};
|
||||
\draw[sc band val] (0,0) -- ++(\LW,0) -- ++(0, \DeltaEV) -- ++(\RW,0);
|
||||
\draw[sc band con] (0,\LGap) -- ++(\LW,0) -- ++(0,-\LGap+\DeltaEV+\RGap) -- ++(\RW,0);
|
||||
\drawDArrow{\LW+\RW/4}{0}{\DeltaEV}{$\Delta \Evalence$}
|
||||
\drawDArrow{\LW/4}{\LGap}{\DeltaEV+\RGap}{$\Delta \Econd$}
|
||||
\end{scope}
|
||||
% type III
|
||||
\pgfmathsetmacro{\LGap}{0.8}
|
||||
\pgfmathsetmacro{\RGap}{1.0}
|
||||
\pgfmathsetmacro{\DeltaEV}{1.0}
|
||||
\begin{scope}[shift={($2*(\LW+\RW+1,0)$)}]
|
||||
\node at (\LW,\texty) {\GT{type}-III};
|
||||
\draw[sc band val] (0,0) -- ++(\LW,0) -- ++(0, \DeltaEV) -- ++(\RW,0);
|
||||
\draw[sc band con] (0,\LGap) -- ++(\LW,0) -- ++(0,-\LGap+\DeltaEV+\RGap) -- ++(\RW,0);
|
||||
\drawDArrow{\LW+\RW/4}{0}{\DeltaEV}{$\Delta \Evalence$}
|
||||
\drawDArrow{\LW/4}{\LGap}{\DeltaEV+\RGap}{$\Delta \Econd$}
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{anderson}
|
||||
\desc{Anderson's rule}{Approximation for band offsets}{$\chi$/$\Egap$ \qtyRef{electron_affinity}/\fRef{cm:sc:band_gap} of semiconductors A and B}
|
||||
\desc[german]{Andersons Regel}{Näherung für die Bandabstände}{$\chi$/$\Egap$ \qtyRef{electron_affinity}/\fRef{cm:sc:band_gap} der Halbleiter A und B}
|
||||
\eq{
|
||||
\Delta\Econd &\approx \chi_\txA - \chi_\txB \\
|
||||
\Delta\Evalence &\approx \left(\Egap^\txA-\Egap^\txB\right) - (\chi_\txA - \chi_\txB)
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection{led}
|
||||
\desc{Led emighting diodes (LED)}{Based around forward biased $p^+n$ or $n^+p$ \fRef[junctions]{::sc-sc:pn}}{}
|
||||
\desc[german]{}{Basieren auf $p^+n$ oder $n^+p$ \fRef[Kontakten]{::sc-sc:pn} im forward bias}{}
|
||||
\begin{formula}{principle}
|
||||
\desc{Principle}{}{}
|
||||
\desc[german]{Prinzip}{}{}
|
||||
\ttxt{\eng{
|
||||
Under external bias a net diffusion current flows across the junction. Injected minority carriers recombine in the vicinity of the depletion region and generate light.
|
||||
}\ger{
|
||||
Unter äußerer Spannung fließt ein Nettodiffusionsstrom über den Übergang.
|
||||
Injizierte Minoritätsträger rekombinieren in der Nähe der Verarmungszone und erzeugen Licht.
|
||||
}}
|
||||
\end{formula}
|
||||
\begin{formula}{efficiency}
|
||||
\desc{Power conversion}{}{
|
||||
$\eta_\text{int} = \frac{\frac{P_\text{int}}{\hbar\omega}}{\frac{j}{e}}$ internal quantum efficiency,
|
||||
$\eta_\text{extraction} \approx \SI{3}{\percent}$ light extraction efficiency,
|
||||
$\eta_\text{inj} = \frac{j_n}{j_n + j_p + j_\text{NR}}$ injection efficiency (for $n^+p$ junction)
|
||||
}
|
||||
\desc[german]{Umwandlungseffizienz}{}{}
|
||||
\eq{
|
||||
\eta_\text{ext} = \frac{\frac{P_\text{ext}}{\hbar\omega}}{\frac{j}{e}} = \eta_\text{int} \eta_\text{extraction} \eta_\text{inj}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection{laser}
|
||||
\desc{Laser}{Light Amplifictation by Stimulated Emission of Radiation}{}
|
||||
\desc[german]{Laser}{}{}
|
||||
\begin{formula}{laser}
|
||||
\desc{Laser}{}{}
|
||||
\desc[german]{Laser}{}{}
|
||||
\ttxt{\eng{
|
||||
\textit{Gain medium} is energized by \textit{pumping energy} (electric current or light), light of certain wavelength is amplified in the gain medium
|
||||
Components:
|
||||
\begin{itemize}
|
||||
\item Gain medium: amplify light by stimulated emission
|
||||
\item Pump: add energy to the gain medium to keep the gain positive
|
||||
\item Positive feedback
|
||||
\item Output coupler: extract light from the oscillator cavity
|
||||
\end{itemize}
|
||||
}\ger{
|
||||
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formulagroup}{stimulated_emission}
|
||||
\desc{Stimulated emission}{}{$F$ \fRef{cm:egas:fermi-dirac}, $E$ \qtyRef{energy} of the electrons/holes}
|
||||
\desc[german]{Stimulierte Emission}{}{$F$ \fRef{cm:egas:fermi-dirac}, $E$ \qtyRef{energy} der Elektronen/Löcher}
|
||||
\begin{formula}{stimulated_emission}
|
||||
\desc{Stimulated emission}{}{}
|
||||
\desc[german]{Stimulierte Emission}{}{}
|
||||
\ttxt{\eng{
|
||||
Emitted photons are identical: phase coherent, same polarization and optical mode, same propagation direction.\\
|
||||
Requires \textit{population inversion}, where most emitters are in the excited state.
|
||||
}\ger{
|
||||
Emittierte Photonen sind identisch: phasenkohärent, gleiche Polarisation und optischer Modus, gleiche Ausbreitungsrichtung.\\
|
||||
Erfordert \textit{Besetzungsinversion}, bei der sich die meisten Emitter im angeregten Zustand befinden.
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{coefficient}
|
||||
\desc{Stimulated emission coefficient}{}{}
|
||||
\desc[german]{Koeffizient der stimulierten Emission}{}{}
|
||||
\eq{\alpha(\hbar\omega) \propto \left(1-F_\txe(E_\txe)\right) \left(1-F_\txh(E_\txh)\right)}
|
||||
\end{formula}
|
||||
\begin{formula}{gain_coefficient}
|
||||
\desc{Bernard condition}{Both quasi fermi levels must lie within the bands. If fulfilled, gain coefficient is positive}{$\Efermi$ electron/hole quasi-\qtyRef[fermi level]{fermi_energy}}
|
||||
\desc[german]{Bernard-Bedingung}{Beide quasi-Fermi Level müssen innerhalb der Bänder liegen. Verstärkungskoeffizient ist positiv wenn erfüllt}{$\Efermi$ Elektron/Loch Quasi-\qtyRef[Fermi-Niveau]{fermi_energy}}
|
||||
\eq{
|
||||
E_\txe - E_\txh = E_\text{photon} < \left(\Efermi^\txe - \Efermi^\txh\right)
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{gain_spectrum}
|
||||
\desc{Gain spectrum}{Gain is frequency dependent}{}
|
||||
% \desc[german]{}{}{}
|
||||
\fig[width=0.7\textwidth]{img_static/cm_sc_laser_gain_spectrum.png}
|
||||
\TODO{plot}
|
||||
\end{formula}
|
||||
\end{formulagroup}
|
||||
|
||||
|
||||
\Subsubsection{other}
|
||||
\desc{Other}{}{}
|
||||
\desc[german]{Andere}{}{}
|
||||
\begin{formula}{single_electron_box}
|
||||
\desc{Single electron box}{Allows discrete changes of single electrons}{$C_\txg/V_\txg$ gate \qtyRef{capacitance}/\qtyRef{voltage}, T tunnel barrier, $n\in\N_0$ number of electrons}
|
||||
\desc[german]{Ein-Elektronen-Box}{}{}
|
||||
\fcenter{
|
||||
\begin{tikzpicture}
|
||||
\draw (0,0) node[ground]{} to[resistor={$R_\txT,C_\txT$}] ++(2,0) node[circle,color=bg3,fill=fg-blue] {QD} to[capacitor=$C_\txg$] ++(2,0) node[vcc] {$V\txg$};
|
||||
\end{tikzpicture}
|
||||
\TODO{fix, use tunnel contact symbol instead of resistor}
|
||||
}
|
||||
\eq{
|
||||
E &= \frac{Q_\txT^2}{2C_\txT} + \frac{Q_\txg^2}{2C_\txg} - Q_\txg V_\txg
|
||||
&\propto \frac{e^2}{2(C_\txg+C_\txT)} \left(n-\frac{C_\txg V_\txg}{e}\right)^2
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{single_electron_transistor}
|
||||
\desc{Single electron transistor}{}{}
|
||||
% \desc[german]{}{}{}
|
||||
\TODO{circuit adv sc slide 397}
|
||||
\end{formula}
|
||||
|
||||
|
||||
|
||||
\Subsection{exciton}
|
||||
\desc{Excitons}{}{}
|
||||
\desc[german]{Exzitons}{}{}
|
||||
\begin{formula}{description}
|
||||
\desc{Description}{}{}
|
||||
\desc[german]{Beschreibung}{}{}
|
||||
\ttxt{
|
||||
\eng{
|
||||
Quasi particle, excitation in condensed matter as bound electron-hole pair.
|
||||
\\ Free (Wannier) excitons: delocalised over many lattice sites
|
||||
\\ Bound (Frenkel) excitons: localised in single unit cell
|
||||
}
|
||||
\ger{
|
||||
Quasiteilchen, Anregung im Festkörper als gebundenes Elektron-Loch-Paar
|
||||
\\ Freie (Wannier) Exzitons: delokalisiert, über mehrere Einheitszellen
|
||||
\\ Gebundene (Frenkel) Exzitons: lokalisiert in einer Einheitszelle
|
||||
}
|
||||
}
|
||||
\end{formula}
|
||||
\eng[free_X]{for free Excitons}
|
||||
\ger[free_X]{für freie Exzitons}
|
||||
\begin{formula}{rydbrg}
|
||||
\desc{Exciton Rydberg energy}{\GT{::free_X}}{$R_\txH$ \fRef{qm:h:rydberg_energy}}
|
||||
\desc[german]{}{}{}
|
||||
\eq{
|
||||
E(n) = - \left(\frac{\mu}{m_0\epsilon_r^2}\right) R_\txH \frac{1}{n^2}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{bohr_radius}
|
||||
\desc{Exciton Bohr radius}{\GT{::free_X}. \qtyrange{2}{20}{\nm}}{\QtyRef{relative_permittivity}, \ConstRef{bohr_radius}, \ConstRef{electron_mass}, $\mu$ \GT{reduced_mass}}
|
||||
\desc[german]{Exziton-Bohr Radius}{}{}
|
||||
\eq{
|
||||
r_n = \left(\frac{m_\txe\epsilon_r a_\txB}{\mu}\right) n^2
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{binding_energy}
|
||||
\desc{Binding energy}{\GT{::free_X}. \qtyrange{0.2}{8}{\meV}}{$R^* = 1\,\text{Ry} \frac{\mu}{\epsilon_\txr^2}$, $\vecK_\text{CM} = \veck_\txe - \veck_\txh$, $\mu$ \TODO{reduced mass, of what?}, $n$ exciton state}
|
||||
\desc[german]{Bindungsenergie}{}{}
|
||||
\eq{E_{n,K_\text{CM}} = \Egap - \frac{R^*}{n^2} + \frac{\hbar^2}{2 \left(\meff_\txe + \meff_\txh\right)} K^2_\text{CM}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{electric_field}
|
||||
\desc{Response to electric field}{Polarization and eventually ionisation (breaks apart)}{$a_\txX$ \fRef{::bohr_radius}}
|
||||
% \desc[german]{}{}{}
|
||||
\eq{\abs{\E_\text{ion}} \approx \frac{2R^*}{e a_\txX}}
|
||||
\end{formula}
|
||||
|
||||
\TODO{stark effect/shift, adv sc. slide 502}
|
||||
@ -1,531 +0,0 @@
|
||||
\def\txL{\text{L}}
|
||||
\def\gl{\text{GL}}
|
||||
\def\GL{Ginzburg-Landau }
|
||||
\def\Tcrit{T_\text{c}}
|
||||
\def\Bcth{B_\text{c,th}}
|
||||
|
||||
\Section{super}
|
||||
\desc{Superconductivity}{
|
||||
Materials for which the electric resistance jumps to 0 under a critical temperature $\Tcrit$.
|
||||
Below $\Tcrit$ they have perfect conductivity and perfect diamagnetism, up until a critical magnetic field $\Bcth$.
|
||||
}{}
|
||||
\desc[german]{Supraleitung}{
|
||||
Materialien, bei denen der elektrische Widerstand beim unterschreiten einer kritischen Temperatur $\Tcrit$ auf 0 springt.
|
||||
Sie verhalten sich dann wie ideale Leiter und ideale Diamagnete, bis zu einem kritischen Feld $\Bcth$.
|
||||
}{}
|
||||
|
||||
\begin{formula}{type1}
|
||||
\desc{Type-I superconductor}{}{}
|
||||
\desc[german]{Typ-I Supraleiter}{}{}
|
||||
\ttxt{\eng{
|
||||
Has a single critical magnetic field, $\Bcth$.
|
||||
\\$B < \Bcth$: \fRef{:::meissner_effect}
|
||||
\\$B > \Bcth$: Normal conductor
|
||||
\\ Very small usable current density because current only flows within the \fRef{cm:super:london:penetration_depth} of the surface.
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{type2}
|
||||
\desc{Type-II superconductor}{}{}
|
||||
\desc[german]{Typ-II Supraleiter}{}{}
|
||||
\ttxt{\eng{
|
||||
Has a two critical magnetic fields.
|
||||
\\$B < B_\text{c1}$: \fRef{:::meissner_effect}
|
||||
\\$B_\text{c1} < B < B_\text{c2}$: \fRef{:::shubnikov_phase}
|
||||
\\$B > B_\text{c2}$: Normal conductor
|
||||
\\ In \fRef{:::shubnikov_phase} larger usable current density because current flows within the \fRef{cm:super:london:penetration_depth} of the surface and the penetrating flux lines.
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{perfect_conductor}
|
||||
\desc{Perfect conductor}{}{}
|
||||
\desc[german]{Ideale Leiter}{}{}
|
||||
\ttxt{
|
||||
\eng{
|
||||
In contrast to a superconductor, perfect conductors become diamagnetic only when the external magnetic field is turned on \textbf{after} the material was cooled below the critical temperature.
|
||||
(\fRef{ed:em:induction:lenz})
|
||||
}
|
||||
\ger{
|
||||
Im Gegensatz zu einem Supraleiter werden ideale Leiter nur dann diamagnetisch, wenn das externe magnetische Feld \textbf{nach} dem Abkühlen unter die kritische Temperatur eingeschaltet wird.
|
||||
(\fRef{ed:em:induction:lenz})
|
||||
}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{meissner_effect}
|
||||
\desc{Meißner-Ochsenfeld effect}{Perfect diamagnetism}{$\chi=-1$ \qtyRef{magnetic_susceptibility}}
|
||||
\desc[german]{Meißner-Ochsenfeld Effekt}{Perfekter Diamagnetismus}{}
|
||||
\ttxt{
|
||||
\eng{External magnetic field decays exponetially inside the superconductor below a critical temperature and a critical magnetic field, path-independant.}
|
||||
\ger{Externes Magnetfeld fällt im Supraleiter exponentiell unterhalb einer kritischen Temperatur und unterhalb einer kritischen Feldstärke ab, wegunabhängig.}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{bcth}
|
||||
\desc{Thermodynamic cricitial field}{for \fRef[type I]{::type1} and \fRef[type II]{::type2}}{}
|
||||
\desc[german]{Thermodynamisches kritische Feldstärke}{für \fRef[type I]{::type1} und \Ref[type II]{::type2}}{}
|
||||
\eq{g_\txs - g_\txn = - \frac{\Bcth^2(T)}{2\mu_0}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{shubnikov_phase}
|
||||
\desc{Shubnikov phase}{in \fRef{::type2}}{}
|
||||
\desc[german]{Shubnikov-Phase}{}{}
|
||||
\ttxt{\eng{
|
||||
Mixed phase in which some magnetic flux penetrates the superconductor.
|
||||
}\ger{
|
||||
Gemischte Phase in der der Supraleiter teilweise von magnetischem Fluss durchdrungen werden kann.
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{condensation_energy}
|
||||
\desc{Condensation energy}{}{\QtyRef{free_enthalpy}, \ConstRef{magnetic_vacuum_permeability}}
|
||||
\desc[german]{Kondensationsenergie}{}{}
|
||||
\eq{
|
||||
\d G &= -S \d T + V \d p - V \vecM \cdot \d\vecB \\
|
||||
G_\text{con} &= G_\txn(B=0,T) - G_\txs(B=0,T) = \frac{V \Bcth^2(T)}{2\mu_0}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\Subsection{london}
|
||||
\desc{London Theory}{}{}
|
||||
\desc[german]{London-Theorie}{}{}
|
||||
\begin{formula}{description}
|
||||
\desc{Description}{}{}
|
||||
\desc[german]{Beschreibung}{}{}
|
||||
\ttxt{\eng{
|
||||
\begin{itemize}
|
||||
\item Phenomenological theory
|
||||
\item Quantitative description of the \fRef{cm:super:meissner_effect}.
|
||||
\item Assumies uniform charge density $n(\vecr,t) = n(t)$ (London-approximation).
|
||||
\item Does not work near $T_\txc$
|
||||
\end{itemize}
|
||||
}\ger{
|
||||
\begin{itemize}
|
||||
\item Phänomenologische Theorie
|
||||
\item Quantitative Beschreibung des \fRef{cm:super:meissner_effect}s.
|
||||
\item Annahme: uniforme Ladungsdichte $n(\vecr,t) = n(t)$ (London-Näherung)
|
||||
\item Funktioniert nicht nahe $T_\txc$
|
||||
\end{itemize}
|
||||
}}
|
||||
\end{formula}
|
||||
% \begin{formula}{coefficient}
|
||||
% \desc{London-coefficient}{}{}
|
||||
% \desc[german]{London-Koeffizient}{}{}
|
||||
% \eq{\txLambda = \frac{m_\txs}{n_\txs q_\txs^2}}
|
||||
% \end{formula}
|
||||
\Eng[of_sc_particle]{of the superconducting particle}
|
||||
\Ger[of_sc_particle]{der Supraleitenden Teilchen}
|
||||
\begin{formula}{first}
|
||||
% \vec{j} = \frac{nq\hbar}{m}\Grad S - \frac{nq^2}{m}\vec{A}
|
||||
\desc{First London Equation}{}{$\vec{j}$ \qtyRef{current_density}, $m_\txs$/$n_\txs$/$q_\txs$ \qtyRef{mass}/\qtyRef{charge_carrier_density}/\qtyRef{charge} \GT{of_sc_particle}, \QtyRef{electric_field}}
|
||||
\desc[german]{Erste London-Gleichun-}{}{}
|
||||
\eq{
|
||||
\pdv{\vec{j}_{\txs}}{t} = \frac{n_\txs q_\txs^2}{m_\txs}\vec{\E} {\color{gray}- \Order{\vec{j}_\txs^2}}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{second}
|
||||
\desc{Second London Equation}{Describes the \fRef{cm:super:meissner_effect}}{$\vec{j}$ \qtyRef{current_density}, $m_\txs$/$n_\txs$/$q_\txs$ \qtyRef{mass}/\qtyRef{charge_carrier_density}/\qtyRef{charge} \GT{of_sc_particle}, \QtyRef{magnetic_flux_density}}
|
||||
\desc[german]{Zweite London-Gleichung}{Beschreibt den \fRef{cm:super:meissner_effect}}{}
|
||||
\eq{
|
||||
\Rot \vec{j_\txs} = -\frac{n_\txs q_\txs^2}{m_\txs} \vec{B}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{penetration_depth}
|
||||
\desc{London penetration depth}{Depth at which $B$ is $1/\e$ times the value of $B_\text{ext}$}{$m_\txs$/$n_\txs$/$q_\txs$ \qtyRef{mass}/\qtyRef{charge_carrier_density}/\qtyRef{charge} \GT{of_sc_particle}}
|
||||
\desc[german]{London Eindringtiefe}{Tiefe bei der $B$ das $1/\e$-fache von $B_\text{ext}$ ist}{}
|
||||
\eq{\lambda_\txL = \sqrt{\frac{m_\txs}{\mu_0 n_\txs q_\txs^2}}}
|
||||
\end{formula}
|
||||
\begin{formula}{penetration_depth_temp}
|
||||
\desc{Temperature dependence of \fRef{::penetration_depth}}{}{}
|
||||
\desc[german]{Temperaturabhängigkeit der \fRef{::penetration_depth}}{}{}
|
||||
\eq{\lambda_\txL(T) = \lambda_\txL(0) \frac{1}{\sqrt{1- \left(\frac{T}{T_\txc}\right)^4}}}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection{macro}
|
||||
\desc{Macroscopic wavefunction}{}{}
|
||||
\desc[german]{Makroskopische Wellenfunktion}{}{}
|
||||
\begin{formula}{ansatz}
|
||||
\desc{Ansatz}{}{}
|
||||
\desc[german]{Ansatz}{}{}
|
||||
\ttxt{\eng{Alternative derivation of London equations by assuming a macroscopic wavefunction which is uniform in space}\ger{Alternative Herleitung der London-Gleichungen durch Annahme einer makroskopischen Wellenfunktion, welche nicht Ortsabhängig ist}}
|
||||
\eq{\Psi(\vecr,t) = \Psi_0(\vecr,t) \e^{\theta(\vecr,t)}}
|
||||
\end{formula}
|
||||
\begin{formula}{energy-phase_relation}
|
||||
\desc{Energy-phase relation}{}{$\theta$ \qtyRef{phase}, $m_\txs$/$n_\txs$/$q_\txs$ \qtyRef{mass}/\qtyRef{charge_carrier_density}/\qtyRef{charge} \GT{of_sc_particle}, \QtyRef{current_density}, $\phi_\text{el}$ \qtyRef{electric_scalar_potential}, \QtyRef{chemical_potential}}
|
||||
\desc[german]{Energie-Phase Beziehung}{}{}
|
||||
\eq{\hbar \pdv{\theta(\vecr,t)}{t} = - \left(\frac{m_\txs}{n_\txs^2 q_\txs^2} \vecj_\txs^2(\vecr,t) + q_\txs\phi_\text{el}(\vecr,t) + \mu(\vecr,t)\right)}
|
||||
\end{formula}
|
||||
\begin{formula}{current-phase_relation}
|
||||
\desc{Current-phase relation}{}{$\theta$ \qtyRef{phase}, $m_\txs$/$n_\txs$/$q_\txs$ \qtyRef{mass}/\qtyRef{charge_carrier_density}/\qtyRef{charge} \GT{of_sc_particle}, \QtyRef{current_density}, \QtyRef{magnetic_vector_potential}}
|
||||
\desc[german]{Strom-Phase Beziehung}{}{}
|
||||
\eq{\vecj_\txs(\vecr,t) = \frac{q_\txs^2 n_\txs(\vecr,t)}{m_\txs} \left(\frac{\hbar}{q_\txs} \Grad\theta(\vecr,t) - \vecA(\vecr,t)\right) }
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsubsection{josephson}
|
||||
\desc{Josephson Effect}{}{}
|
||||
\desc[german]{Josephson Effekt}{}{}
|
||||
\begin{formula}{1st_relation}
|
||||
\desc{1. Josephson relation}{Dissipationless supercurrent accros junction at zero applied voltage}{$\vecj_\text{C}=\frac{2e}{\hbar}E_\text{J}$ critical current, $\phi$ phase difference accross junction}
|
||||
\desc[german]{1. Josephson Gleichung}{Dissipationsloser Suprastrom durch die Kreuzung ohne angelegte Spannung}{$\vecj_\text{C}=\frac{2e}{\hbar}E_\text{J}$ kritischer Strom, $\phi$ Phasendifferenz zwischen den Supraleitern}
|
||||
\eq{\vecj_\txs(\vecr,t) = \vecj_\text{C}(\vecr,t) \sin\phi(\vecr,t)}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{2nd_relation}
|
||||
\desc{2. Josephson relation}{Superconducting phase change is proportional to applied voltage}{$\phi$ phase differnce accross junction, \ConstRef{flux_quantum}, \QtyRef{voltage}}
|
||||
\desc[german]{2. Josephson Gleichung}{Supraleitende Phasendifferenz is proportional zur angelegten Spannung}{$\phi$ Phasendifferenz, \ConstRef{flux_quantum}, \QtyRef{voltage}}
|
||||
\eq{\odv{\phi(t)}{t} = \frac{2\pi}{\Phi_0} U(t)}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{coupling_energy}
|
||||
\desc{Josephson coupling energy}{}{$A$ junction \qtyRef{area}, \ConstRef{flux_quantum}, $\vecj_\txc$ \fRef[critical current density]{::1st_relation}, $\phi$ phase differnce accross junction}
|
||||
\desc[german]{Josephson}{}{$A$ junction \qtyRef{area}, \ConstRef{flux_quantum}, $\vecj_\txc$ \fRef[kritische Stromdichte]{::1st_relation}, $\phi$ Phasendifferenz zwischen den Supraleitern}
|
||||
\eq{\frac{E_\txJ}{A} = \frac{\Phi_0 \vecj_\txc}{2\pi}(1-\cos\phi)}
|
||||
\end{formula}
|
||||
|
||||
|
||||
|
||||
|
||||
\Subsection{gl}
|
||||
\desc{\GL Theory (GLAG)}{}{}
|
||||
\desc[german]{\GL Theorie (GLAG)}{}{}
|
||||
\begin{formula}{description}
|
||||
\desc{Description}{}{}
|
||||
\desc[german]{Beschreibung}{}{}
|
||||
\ttxt{\eng{
|
||||
\begin{itemize}
|
||||
\item Phenomenological theory
|
||||
\item Improvement on the Landau-Theory of 2nd order phase transitions
|
||||
% which introduces an order parameter that is $0$ in the normal state and rises to saturation in the superconducting state.
|
||||
\item Additional complex, position-dependent order parameter is introduced $\Psi(\vecr)$
|
||||
\item Only valid close to $T_\txc$.
|
||||
\item Does not have time dependancy
|
||||
\end{itemize}
|
||||
}\ger{
|
||||
\begin{itemize}
|
||||
\item Phänomenologische Theorie
|
||||
\item Weiterentwicklung der Landau-Theorie für Phasenübergänge zweiter Ordnung,
|
||||
% in der ein Ordnungsparameter in the normalen Phase 0 ist und ein der supraleitenden Phase bis zur Sättigung ansteigt.
|
||||
\item Zusätzlicher, komplexer, ortsabhängiger Ordnungsparameter $\Psi(\vecr)$
|
||||
\item Nur nahe $T_\txc$ gültig.
|
||||
\item Beschreibt keine Zeitabhängigkeit
|
||||
\end{itemize}
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{expansion}
|
||||
\desc{Expansion}{Expansion of free enthalpy of superconducting state}{
|
||||
$g_{\txs/\txn}$ specific \qtyRef{free_enthalpy} of superconducting/normal state,
|
||||
$\Psi(\vecr) = \abs{\Psi_0(\vecr)} \e^{\I\theta(\vecr)}$ order parameter,
|
||||
$n(\vecr) = \abs{\Psi}^2$ Cooper-Pair density,
|
||||
\QtyRef{magnetic_flux_density},
|
||||
\QtyRef{magnetic_vector_potential},
|
||||
$\alpha(T) = -\bar{\alpha} \left(1-\frac{T}{T_\txc}\right)^2$,
|
||||
% $\alpha > 0$ for $T > T_\txc$ and $\alpha < 0$ for $T< T_\txc$,
|
||||
$\beta = \const > 0$
|
||||
}
|
||||
% \desc[german]{}{}{}
|
||||
\begin{multline}
|
||||
g_\txs = g_\txn + \alpha \abs{\Psi}^2 + \frac{1}{2}\beta \abs{\Psi}^4 +
|
||||
\\ \frac{1}{2\mu_0}(\vecB_\text{ext} -\vecB_\text{inside})^2 + \frac{1}{2m_\txs} \abs{ \left(-\I\hbar\Grad - q_\txs \vecA\right)\Psi}^2 + \dots
|
||||
\end{multline}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{first}
|
||||
\desc{First Ginzburg-Landau Equation}{Obtained by minimizing $g_\txs$ with respect to $\delta\Psi$ in \fRef{::expansion}}{
|
||||
$\xi_\gl$ \fRef{cm:super:gl:coherence_length},
|
||||
$\lambda_\gl$ \fRef{cm:super:gl:penetration_depth}
|
||||
}
|
||||
\desc[german]{Erste Ginzburg-Landau Gleichung}{}{}
|
||||
\eq{\alpha\Psi + \beta\abs{\Psi}^2 \Psi + \frac{1}{2m} (-i\hbar \Grad + 2e\vec{A})^2\Psi = 0}
|
||||
\end{formula}
|
||||
\begin{formula}{second}
|
||||
\desc{Second Ginzburg-Landau Equation}{Obtained by minimizing $g_\txs$ with respect to $\delta\vec{A}$ in \fRef{::expansion}}{}
|
||||
\desc[german]{Zweite Ginzburg-Landau Gleichung}{}{}
|
||||
\eq{\vec{j_\txs} = \frac{ie\hbar}{m}(\Psi^*\Grad\Psi - \Psi\Grad\Psi^*) - \frac{4e^2}{m}\abs{\Psi}^2 \vec{A}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\begin{formula}{coherence_length}
|
||||
\desc{\GL Coherence Length}{Depth in the superconductor where $\abs{\Psi}$ goes from 0 to 1}{}
|
||||
\desc[german]{\GL Kohärenzlänge}{Tiefe im Supraleiter, bei der $\abs{\Psi}$ von 0 auf 1 steigt}{}
|
||||
\eq{
|
||||
\xi_\gl &= \frac{\hbar}{\sqrt{2m \abs{\alpha}}} \\
|
||||
\xi_\gl(T) &= \xi_\gl(0) \frac{1}{\sqrt{1-\frac{T}{\Tcrit}}}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{penetration_depth}
|
||||
\desc{\GL Penetration Depth}{Field screening length\\Depth in the supercondcutor where $B_\text{ext}$ decays}{}
|
||||
\desc[german]{\GL Eindringtiefe}{Tiefe im Supraleiter, bei der $B_\text{ext}$ abfällt}{}
|
||||
\eq{
|
||||
\lambda_\gl &= \sqrt{\frac{m_\txs\beta}{\mu_0 \abs{\alpha} q_s^2}} \\
|
||||
\lambda_\gl(T) &= \lambda_\gl(0) \frac{1}{\sqrt{1-\frac{T}{\Tcrit}}}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{boundary_energy}
|
||||
\desc{Boundary energy}{Negative for \fRef{:::type2}, positive for \fRef{:::type1}}{$\Delta E_\text{B}$ energy gained by expelling the external magnetic field, $\Delta E_\text{cond}$ \fRef{:::condensation_energy}}
|
||||
\desc[german]{Grenzflächenenergie}{Negativ für \fRef{:::type2}, positiv für \fRef{:::type1}}{}
|
||||
\eq{\Delta E_\text{boundary} = \Delta E_\text{con} - \Delta E_\txB = (\xi_\gl - \lambda_\gl) \frac{B_\text{c,th}^2}{2\mu_0}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{parameter}
|
||||
\desc{Ginzburg-Landau parameter}{}{}
|
||||
\desc[german]{Ginzburg-Landau Parameter}{}{}
|
||||
\eq{\kappa \equiv \frac{\lambda_\gl}{\xi_\gl}}
|
||||
\eq{
|
||||
\kappa \le \frac{1}{\sqrt{2}} &\quad\Rightarrow\quad\text{\fRef{cm:super:type1}} \\
|
||||
\kappa \ge \frac{1}{\sqrt{2}} &\quad\Rightarrow\quad\text{\fRef{cm:super:type2}}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{ns_boundary}
|
||||
\desc{Normal-superconductor boundary}{}{}
|
||||
\desc[german]{Normal-Supraleiter Grenzfläche}{}{}
|
||||
\eq{
|
||||
\abs{\Psi(x)}^2 &= \frac{n_\txs(x)}{n_\txs(\infty)} = \tanh^2 \left(\frac{x}{\sqrt{2}\xi_\gl}\right) \\
|
||||
B_z(x) &= B_z(0) \Exp{-\frac{x}{\lambda_\gl}}
|
||||
}
|
||||
\fig{img/cm_super_n_s_boundary.pdf}
|
||||
% \TODO{plot, slide 106}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{bcth}
|
||||
\desc{Thermodynamic critical field}{}{}
|
||||
\desc[german]{Thermodynamisches kritisches Feld}{}{}
|
||||
\eq{\Bcth = \frac{\Phi_0}{2\pi \sqrt{2} \xi_\gl \lambda_\gl}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{bc1}
|
||||
\desc{Lower critical magnetic field}{Above $B_\text{c1}$, flux starts to penetrate the superconducting phase}{\ConstRef{flux_quantum}, $\lambda_\gl$ \fRef{::penetration_depth} $\kappa$ \fRef{::parameter}}
|
||||
\desc[german]{Unteres kritisches Magnetfeld}{Über $B_\text{c1}$ dringt erstmals Fluss in die supraleitende Phase ein}{}
|
||||
\eq{B_\text{c1} = \frac{\Phi_0}{4\pi\lambda_\gl^2}(\ln\kappa+0.08) = \frac{1}{\sqrt{2}\kappa}(\ln\kappa + 0.08) \Bcth}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{bc2}
|
||||
\desc{Upper critical magnetic field}{Above $B_\text{c2}$, superconducting phase is is destroyed}{\ConstRef{flux_quantum}, $\xi_\gl$ \fRef{::coherence_length}}
|
||||
\desc[german]{Oberes kritisches Magnetfeld}{Über $B_\text{c2}$ ist die supraleitende Phase zerstört}{}
|
||||
\eq{B_\text{c2} = \frac{\Phi_0}{2\pi\xi_\gl^2}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{proximity_effect}
|
||||
\desc{Proximity-Effect}{}{}
|
||||
% \desc[german]{}{}{}
|
||||
\ttxt{\eng{
|
||||
Superconductor wavefunction extends into the normal conductor or isolator
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\Subsection{micro}
|
||||
\desc{Microscopic theory}{}{}
|
||||
\desc[german]{Mikroskopische Theorie}{}{}
|
||||
\begin{formula}{isotop_effect}
|
||||
\desc{Isotope effect}{Superconducting behaviour depends on atomic mass and thereby on the lattice \Rightarrow Microscopic origin}{$\Tcrit$ critial temperature, $M$ isotope mass, $\omega_\text{ph}$}
|
||||
\desc[german]{Isotopeneffekt}{Supraleitung hängt von der Atommasse und daher von den Gittereigenschaften ab \Rightarrow Mikroskopischer Ursprung}{$\Tcrit$ kritische Temperatur, $M$ Isotopen-Masse, $\omega_\text{ph}$}
|
||||
\eq{
|
||||
\Tcrit &\propto \frac{1}{\sqrt{M}} \\
|
||||
\omega_\text{ph} &\propto \frac{1}{\sqrt{M}} \Rightarrow \Tcrit \propto \omega_\text{ph}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{cooper_pairs}
|
||||
\desc{Cooper pairs}{}{}
|
||||
\desc[german]{Cooper-Paars}{}{}
|
||||
\ttxt{
|
||||
\eng{Conduction electrons reduce their energy through an attractive interaction: One electron passing by atoms attracts the these, which creats a positive charge region behind the electron, which in turn attracts another electron. }
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection{bcs}
|
||||
\desc{BCS-Theory}{}{}
|
||||
\desc[german]{BCS-Theorie}{}{}
|
||||
\begin{formula}{description}
|
||||
\desc{Description}{}{}
|
||||
\desc[german]{Beschreibung}{}{}
|
||||
\ttxt{\eng{
|
||||
\begin{itemize}
|
||||
\item Electron pairs form bosonic quasi-particles called Cooper pairs which can condensate into the ground state
|
||||
\item The wave function spans the whole material, which makes it conduct without resistance
|
||||
\item The exchange bosons between the electrons are phonons
|
||||
\end{itemize}
|
||||
}\ger{
|
||||
\begin{itemize}
|
||||
\item Elektronenpaar bilden bosonische Quasipartikel (Cooper Paare) welche in den Grundzustand kondensieren können.
|
||||
\item Die Wellenfunktion übersoannt den gesamten Festkörper, was einen widerstandslosen Ladungstransport garantiert
|
||||
\item Die Austauschbosononen zwischen den Elektronen sind Bosonen
|
||||
\end{itemize}
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\def\BCS{{\text{BCS}}}
|
||||
\begin{formula}{hamiltonian}
|
||||
\desc{BCS Hamiltonian}{for $N$ interacting electrons}{
|
||||
$c_{\veck\sigma}$ creation/annihilation operators create/destroy at $\veck$ with spin $\sigma$ \\
|
||||
First term: non-interacting free electron gas\\
|
||||
Second term: interaction energy
|
||||
}
|
||||
\desc[german]{BCS Hamiltonian}{}{}
|
||||
\eq{
|
||||
\hat{H}_\BCS =
|
||||
\sum_{\sigma} \sum_\veck \epsilon_\veck \hat{c}_{\veck\sigma}^\dagger \hat{c}_{\veck\sigma}
|
||||
+ \sum_{\veck,\veck^\prime} V_{\veck,\veck^\prime}
|
||||
\hat{c}_{\veck\uparrow}^\dagger \hat{c}_{-\veck\downarrow}^\dagger
|
||||
\hat{c}_{-\veck^\prime\downarrow} \hat{c}_{\veck^\prime,\uparrow}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{ansatz}
|
||||
\desc{BCS ground state wave function Ansatz}{\fRef{comp:est:mean_field} approach\\Coherent fermionic state}{}
|
||||
\desc[german]{BCS Grundzustandswellenfunktion-Ansatz}{\fRef{comp:est:mean_field} Ansatz\\Kohärenter, fermionischer Zustand}{}
|
||||
\eq{\Ket{\Psi_\BCS} = \prod_{\veck=\veck_1,\dots,\veck_M} \left(u_\veck + v_\veck \hat{c}_{\veck\uparrow}^\dagger \hat{c}_{-\veck\downarrow}^\dagger\right) \ket{0} }
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{coherence_factors}
|
||||
\desc{BCS coherence factors}{}{$\abs{u_\veck}^2$/$\abs{v_\veck}^2$ probability that pair state is $(\veck\uparrow,\,-\veck\downarrow)$ is empty/occupied, $\abs{u_\veck}^2+\abs{v_\veck}^2 = 1$}
|
||||
\desc[german]{BCS Kohärenzfaktoren}{}{$\abs{u_\veck}^2$/$\abs{v_\veck}^2$ Wahrscheinlichkeit, dass Paarzustand $(\veck\uparrow,\,-\veck\downarrow)$ leer/besetzt ist, $\abs{u_\veck}^2+\abs{v_\veck}^2 = 1$}
|
||||
\eq{
|
||||
u_\veck &= \frac{1}{\sqrt{1+\abs{\alpha_\veck}^2}} \\
|
||||
v_\veck &= \frac{\alpha_\veck}{\sqrt{1+\abs{\alpha_\veck}^2}}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{potential}
|
||||
\desc{BCS potential approximation}{}{}
|
||||
\desc[german]{BCS Potentialnäherung}{}{}
|
||||
\eq{
|
||||
V_{\veck,\veck^\prime} =
|
||||
\left\{ \begin{array}{rc}
|
||||
-V_0 & k^\prime > k_\txF,\, k<k_\txF + \Delta k\\
|
||||
0 & \tGT{else}
|
||||
\end{array}\right.
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{gap_at_t0}
|
||||
\desc{BCS Gap at $T=0$}{}{\QtyRef{debye_frequency}, $V_0$ \fRef{::potential}, $D$ \qtyRef{dos}, $\gamma$ Sommerfeld constant}
|
||||
\desc[german]{BCS Lücke bei $T=0$}{}{}
|
||||
\eq{
|
||||
\Delta(T=0) &= \frac{\hbar\omega_\txD}{\Sinh{\frac{2}{V_0\.D(E_\txF)}}} \approx 2\hbar \omega_\txD\\
|
||||
\frac{\Delta(T=0)}{\kB T_\txc} &= \frac{\pi}{\e^\gamma} = 1.764
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\begin{formula}{cooper_pair_binding_energy}
|
||||
\desc{Binding energy of Cooper pairs}{}{$E_\txF$ \absRef{fermi_energy}, \QtyRef{debye_frequency}, $V_0$ retarded potential, $D$ \qtyRef{dos}}
|
||||
\desc[german]{Bindungsenergie von Cooper-Paaren}{}{}
|
||||
\eq{E \approx 2E_\txF - 2\hbar\omega_\txD \Exp{-\frac{4}{V_0 D(E_\txF)}}}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection{excite}
|
||||
\desc{Excitations and finite temperatures}{}{}
|
||||
\desc[german]{Anregungen und endliche Temperatur}{}{}
|
||||
\begin{formula}{description}
|
||||
\desc{Description}{}{}
|
||||
\desc[german]{Beschreibung}{}{}
|
||||
\ttxt{\eng{
|
||||
The ground state consists of \fRef{cm:super:micro:cooper_pairs} and the excited state of Bogoliubov quasi-particles (electron-hole pairs).
|
||||
The states are separated by an energy gap $\Delta$.
|
||||
}\ger{
|
||||
Den Grundzustand bilden \fRef{cm:super:micro:cooper_pairs} und den angeregten Zustands Bogoloiubons (Elektron-Loch Quasipartikel).
|
||||
Die Zustände sind durch eine Energielücke $\Delta$ getrennt.
|
||||
}}
|
||||
\end{formula}
|
||||
\begin{formula}{bogoliubov-valatin}
|
||||
\desc{Bogoliubov-Valatin transformation}{Diagonalization of the \fRef{cm:super:micro:bcs:hamiltonian} to derive excitation energies}{
|
||||
$\xi_\veck = \epsilon_\veck-\mu$ Energy relative to the \qtyRef{chemical_potential},
|
||||
\\ $E_\veck$ \fRef{::excitation_energy},
|
||||
\\ $\Delta$ Gap
|
||||
\\ $g_\veck$ \fRef{::pairing_amplitude},
|
||||
\\ $\alpha / \beta$ create and destroy symmetric/antisymmetric Bogoliubov quasiparticles
|
||||
}
|
||||
\desc[german]{Bogoliubov-Valatin transformation}{}{}
|
||||
\eq{
|
||||
\hat{H}_\BCS - N\mu = \sum_\veck \big[\xi_\veck - E_\veck + \Delta_\veck g_\veck^*\big] + \sum_\veck \big[E_\veck \alpha_\veck^\dagger \alpha_\veck + E_\veck \beta_{-\veck}^\dagger \beta_{-\veck}\big]
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{pairing_amplitude}
|
||||
\desc{Pairing amplitude}{}{}
|
||||
\desc[german]{Paarungsamplitude}{}{}
|
||||
\eq{g_\veck \equiv \Braket{\hat{c}_{-\veck\downarrow} \hat{c}_{\veck\uparrow}}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{excitation_energy}
|
||||
\desc{Excitation energy}{}{}
|
||||
\desc[german]{Anregungsenergie}{}{}
|
||||
\eq{E_\veck = \pm \sqrt{\xi^2_\veck + \abs{\Delta_\veck}^2}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{coherence_factors_energy}
|
||||
\desc{Energy dependance of the \fRef{:::bcs:coherence_factors}}{}{$E_\veck$ \fRef{::pairing_amplitude}, \GT{see} \fRef{:::bcs:coherence_factors}}
|
||||
\desc[german]{Energieabhängigkeit der \fRef{:::bcs:coherence_factors}}{}{}
|
||||
\eq{
|
||||
\abs{u_\veck}^2 &= \frac{1}{2} \left(1+\frac{\xi_\veck}{E_\veck}\right) \\
|
||||
\abs{v_\veck}^2 &= \frac{1}{2} \left(1-\frac{\xi_\veck}{E_\veck}\right) \\
|
||||
u_\veck^* v_\veck &= \frac{\Delta_\veck}{2E_\veck}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\begin{formula}{gap_equation}
|
||||
\desc{Self-consistend gap equation}{}{}
|
||||
\desc[german]{Selbstkonsitente Energielückengleichung}{}{}
|
||||
\eq{\Delta_\veck^* = -\sum_{\veck^\prime} V_{\veck,\veck^\prime} \frac{\Delta_{\veck^\prime}}{2E_\veck} \tanh \left(\frac{E_{\veck^\prime}}{2\kB T}\right)}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\begin{formula}{gap_t}
|
||||
\desc{Temperature dependence of the BCS gap}{}{}
|
||||
\desc[german]{Temperaturabhängigkeit der BCS-Lücke}{}{}
|
||||
\eq{\frac{\Delta(T)}{\Delta(T=0)} \approx 1.74 \sqrt{1-\frac{T}{T_\txC}}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{dos}
|
||||
\desc{Quasiparticle density of states}{}{}
|
||||
\desc[german]{Quasiteilchen Zustandsdichte}{}{}
|
||||
\eq{D_\txs(E_\veck) = D_\txn(\xi_\veck) \pdv{\xi_\veck}{E_\veck} = \left\{
|
||||
\begin{array}{ll}
|
||||
D_\txn(E_\txF) \frac{E_\veck}{\sqrt{E^2_\veck -\Delta^2}} & E_\veck > \Delta \\
|
||||
& E_\veck < \Delta
|
||||
\end{array}
|
||||
\right.}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{Bcth_temp}
|
||||
\desc{Temperature dependance of the crictial magnetic field}{Jump at $T_\txc$, then exponential decay}{}
|
||||
\desc[german]{Temperaturabhängigkeit des kritischen Magnetfelds}{Sprung bei $T_\txc$, denn exponentieller Abfall}{}
|
||||
\eq{ \Bcth(T) = \Bcth(0) \left[1- \left(\frac{T}{T_\txc}\right)^2 \right] }
|
||||
% \TODO{empirical relation, relate to BCS}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{heat_capacity}
|
||||
\desc{Heat capacity in superconductors}{}{}
|
||||
\desc[german]{Wärmekapazität in Supraleitern}{}{}
|
||||
\fsplit{
|
||||
\fig{img/cm_super_heat_capacity.pdf}
|
||||
}{
|
||||
\eq{c_\txs \propto T^{-\frac{3}{2}} \e^{\frac{\Delta(0)}{\kB T}}}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection{pinning}
|
||||
\desc{Flux pinning}{}{}
|
||||
\desc[german]{Haftung von Flusslinien}{}{}
|
||||
\begin{formula}{description}
|
||||
\desc{Description}{}{}
|
||||
\desc[german]{Beschreibung}{}{}
|
||||
\ttxt{\eng{
|
||||
If a current flows in a \fRef{cm:super:type2}s in the \fRef{cm:super:shubnikov_phase} perpendicular to the penetrating flux lines,
|
||||
the lines experience a Lorentz force. This leads to ohmic behaviour of the superconductor.
|
||||
The flux lines can be pinned to defects, in which the superconducting order parameter is reduced.
|
||||
To move the flux line out of the defect, work would have to be spent overcoming the \fRef{cm:super:micro:pinning:potential}.
|
||||
This restores the superconductivity.
|
||||
}\ger{
|
||||
Wenn ein Strom in einem \fRef{cm:super:type2}s in der \fRef{cm:super:shubnikov_phase} senkrecht zu den eindringenden Flusslinien fließt, erfahren die Linien eine Lorentzkraft.
|
||||
Dies führt zu einem ohmschen Verhalten des Supraleiters.
|
||||
Die Flusslinien können an Defekten festgehalten werden, in denen der supraleitende Ordnungsparameter reduziert ist.
|
||||
Um die Flusslinie aus dem Defekt zu bewegen, müsste Arbeit aufgewendet werden, um das \fRef{cm:super:micro:pinning:potential} zu überwinden.
|
||||
Dies stellt die Supraleitfähigkeit wieder her.
|
||||
}}
|
||||
\end{formula}
|
||||
@ -1,210 +0,0 @@
|
||||
\Section{tech}
|
||||
\desc{Techniques}{}{}
|
||||
\desc[german]{Techniken}{}{}
|
||||
|
||||
|
||||
\Subsection{meas}
|
||||
\desc{Measurement techniques}{}{}
|
||||
\desc[german]{Messtechniken}{}{}
|
||||
|
||||
\Eng[name]{Name}
|
||||
\Ger[name]{Name}
|
||||
\Eng[application]{Application}
|
||||
\Ger[application]{Anwendung}
|
||||
|
||||
\Subsubsection{raman}
|
||||
\desc{Raman spectroscopy}{}{}
|
||||
\desc[german]{Raman Spektroskopie}{}{}
|
||||
|
||||
% TODO remove fqname from minipagetable?
|
||||
|
||||
\begin{bigformula}{raman}
|
||||
\desc{Raman spectroscopy}{}{}
|
||||
\desc[german]{Raman-Spektroskopie}{}{}
|
||||
\begin{minipagetable}{raman}
|
||||
\tentry{application}{
|
||||
\eng{Vibrational modes, Crystal structure, Doping, Band Gaps, Layer thickness in \fRef{cm:misc:vdw_material}}
|
||||
\ger{Vibrationsmoden, Kristallstruktur, Dotierung, Bandlücke, Schichtdicke im \fRef{cm:misc:vdw_material}}
|
||||
}
|
||||
\tentry{how}{
|
||||
\eng{Monochromatic light (\fRef{Laser}) shines on sample, inelastic scattering because of rotation-, vibration-, phonon and spinflip-processes, plot spectrum as shift of the laser light (in \si{\per\cm})}
|
||||
\ger{Monochromatisches Licht (\fRef{Laser}) bestrahlt Probe, inelastische Streuung durch Rotations-, Schwingungs-, Phonon und Spin-Flip-Prozesse, plotte Spektrum als Verschiebung gegen das Laser Licht (in \si{\per\cm}) }
|
||||
}
|
||||
\end{minipagetable}
|
||||
\begin{minipage}{0.45\textwidth}
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
% \includegraphics[width=0.8\textwidth]{img/cm_amf.pdf}
|
||||
% \caption{\cite{Bian2021}}
|
||||
\end{figure}
|
||||
\end{minipage}
|
||||
\end{bigformula}
|
||||
|
||||
|
||||
\begin{bigformula}{pl}
|
||||
\desc{Photoluminescence spectroscopy}{}{}
|
||||
\desc[german]{Photolumeszenz-Spektroskopie}{}{}
|
||||
\begin{minipagetable}{pl}
|
||||
\tentry{application}{
|
||||
\eng{Crystal structure, Doping, Band Gaps, Layer thickness in \fRef{cm:misc:vdw_material}}
|
||||
\ger{Kristallstruktur, Dotierung, Bandlücke, Schichtdicke im \fRef{cm:misc:vdw_material}}
|
||||
}
|
||||
\tentry{how}{
|
||||
\eng{Monochromatic light (\fRef{Laser}) shines on sample, electrons are excited, relax to the conduction band minimum and finally accross the band gap under photon emission}
|
||||
\ger{Monochromatisches Licht (\fRef{Laser}) bestrahlt Probe, Elektronen werden angeregt und relaxieren in das Leitungsband-Minimum und schließlich über die Bandlücke unter Photonemission}
|
||||
}
|
||||
\end{minipagetable}
|
||||
\begin{minipage}{0.45\textwidth}
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
% \includegraphics[width=0.8\textwidth]{img_static/cm_amf.pdf}
|
||||
% \caption{\cite{Bian2021}}
|
||||
\end{figure}
|
||||
\end{minipage}
|
||||
\end{bigformula}
|
||||
|
||||
|
||||
\Subsubsection{arpes}
|
||||
\desc{ARPES}{}{}
|
||||
\desc[german]{ARPES}{}{}
|
||||
what?
|
||||
in?
|
||||
how?
|
||||
plot
|
||||
|
||||
\Subsubsection{spm}
|
||||
\desc{Scanning probe microscopy SPM}{}{}
|
||||
\desc[german]{Rastersondenmikroskopie (SPM)}{}{}
|
||||
\begin{ttext}
|
||||
\eng{Images of surfaces are taken by scanning the specimen with a physical probe.}
|
||||
\ger{Bilder der Oberfläche einer Probe werden erstellt, indem die Probe mit einer Sonde abgetastet wird.}
|
||||
\end{ttext}
|
||||
|
||||
|
||||
\begin{bigformula}{amf}
|
||||
\desc{Atomic force microscopy (AMF)}{}{}
|
||||
\desc[german]{Atomare Rasterkraftmikroskopie (AMF)}{}{}
|
||||
\begin{minipagetable}{amf}
|
||||
\tentry{application}{
|
||||
\eng{Surface stuff}
|
||||
\ger{Oberflächenzeug}
|
||||
}
|
||||
\tentry{how}{
|
||||
\eng{With needle}
|
||||
\ger{Mit Nadel}
|
||||
}
|
||||
\end{minipagetable}
|
||||
\begin{minipage}{0.45\textwidth}
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.8\textwidth]{img_static/cm_amf.pdf}
|
||||
\caption{\cite{Bian2021}}
|
||||
\end{figure}
|
||||
\end{minipage}
|
||||
\end{bigformula}
|
||||
|
||||
|
||||
|
||||
|
||||
\begin{bigformula}{stm}
|
||||
\desc{Scanning tunneling microscopy (STM)}{}{}
|
||||
\desc[german]{Rastertunnelmikroskop (STM)}{}{}
|
||||
\begin{minipagetable}{stm}
|
||||
\tentry{application}{
|
||||
\eng{Surface stuff}
|
||||
\ger{Oberflächenzeug}
|
||||
}
|
||||
\tentry{how}{
|
||||
\eng{With TUnnel}
|
||||
\ger{Mit TUnnel}
|
||||
}
|
||||
\end{minipagetable}
|
||||
\begin{minipage}{0.45\textwidth}
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.8\textwidth]{img_static/cm_stm.pdf}
|
||||
\caption{\cite{Bian2021}}
|
||||
\end{figure}
|
||||
\end{minipage}
|
||||
\end{bigformula}
|
||||
|
||||
\Subsection{fab}
|
||||
\desc{Fabrication techniques}{}{}
|
||||
\desc[german]{Herstellungsmethoden}{}{}
|
||||
|
||||
\begin{bigformula}{cvd}
|
||||
\desc{Chemical vapor deposition (CVD)}{}{}
|
||||
\desc[german]{Chemische Gasphasenabscheidung (CVD)}{}{}
|
||||
\begin{minipagetable}{cvd}
|
||||
\tentry{how}{
|
||||
\eng{
|
||||
A substrate is exposed to volatile precursors, which react and/or decompose on the heated substrate surface to produce the desired deposit.
|
||||
By-products are removed by gas flow through the chamber.
|
||||
}
|
||||
\ger{
|
||||
An der erhitzten Oberfläche eines Substrates wird aufgrund einer chemischen Reaktion mit einem Gas eine Feststoffkomponente abgeschieden.
|
||||
Nebenprodukte werden durch den Gasfluss durch die Kammer entfernt.
|
||||
}
|
||||
}
|
||||
\tentry{application}{
|
||||
\eng{
|
||||
\begin{itemize}
|
||||
\item Polysilicon \ce{Si}
|
||||
\item Silicon dioxide \ce{SiO_2}
|
||||
\item Graphene
|
||||
\item Diamond
|
||||
\end{itemize}
|
||||
}
|
||||
\ger{
|
||||
\begin{itemize}
|
||||
\item Poly-silicon \ce{Si}
|
||||
\item Siliziumdioxid \ce{SiO_2}
|
||||
\item Graphen
|
||||
\item Diamant
|
||||
\end{itemize}
|
||||
}
|
||||
}
|
||||
\end{minipagetable}
|
||||
\begin{minipage}{0.45\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{img_static/cm_cvd_english.pdf}
|
||||
\end{minipage}
|
||||
\end{bigformula}
|
||||
|
||||
|
||||
\Subsubsection{epitaxy}
|
||||
\desc{Epitaxy}{}{}
|
||||
\desc[german]{Epitaxie}{}{}
|
||||
\begin{ttext}
|
||||
\eng{A type of crystal groth in which new layers are formed with well-defined orientations with respect to the crystalline seed layer.}
|
||||
\ger{Eine Art des Kristallwachstums, bei der mindestens eine kristallographische Ordnung der wachsenden Schicht der des Substrates entspricht.}
|
||||
\end{ttext}
|
||||
|
||||
\begin{bigformula}{mbe}
|
||||
\desc{Molecular Beam Epitaxy (MBE)}{}{}
|
||||
\desc[german]{Molekularstrahlepitaxie (MBE)}{}{}
|
||||
\begin{minipagetable}{mbe}
|
||||
\tentry{how}{
|
||||
\eng{In a ultra-high vacuum, the elements are heated until they slowly sublime. The gases then condensate on the substrate surface}
|
||||
\ger{Die Elemente werden in einem Ultrahochvakuum erhitzt, bis sie langsam sublimieren. Die entstandenen Gase kondensieren dann auf der Oberfläche des Substrats}
|
||||
}
|
||||
\tentry{application}{
|
||||
\eng{
|
||||
\begin{itemize}
|
||||
\item Gallium arsenide \ce{GaAs}
|
||||
\end{itemize}
|
||||
\TODO{Link to GaAs}
|
||||
}
|
||||
\ger{
|
||||
\begin{itemize}
|
||||
\item Galliumarsenid \ce{GaAs}
|
||||
\end{itemize}
|
||||
}
|
||||
}
|
||||
\end{minipagetable}
|
||||
\begin{minipage}{0.45\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{img_static/cm_mbe_english.pdf}
|
||||
\end{minipage}
|
||||
\end{bigformula}
|
||||
|
||||
116
src/cm/vib.tex
116
src/cm/vib.tex
@ -1,116 +0,0 @@
|
||||
\Section{vib}
|
||||
\desc{Lattice vibrations}{}{}
|
||||
\desc[german]{Gitterschwingungen}{}{}
|
||||
|
||||
\begin{formula}{speed_of_sound}
|
||||
\desc{Speed of sound}{Speed with which vibrations propagate through an elastic medium}{}
|
||||
\desc[german]{Schallgeschwindigkeit}{Geschwindigkeit, mit der sich Vibrationen in einem elastischem Medium ausbreiten}{}
|
||||
\quantity{v}{\m\per\s}{s}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{dispersion_1atom_basis}
|
||||
\desc{Phonon dispersion of a lattice with a one-atom basis}{same as the dispersion of a linear chain}{$C_n$ force constants between layer $s$ and $s+n$, $M$ \qtyRef{mass} of the reference atom, $a$ \qtyRef{lattice_constant}, $q$ phonon \qtyRef{wavevector}, $u$ Ansatz for the atom displacement}
|
||||
\desc[german]{Phonondispersion eines Gitters mit zweiatomiger Basis}{gleich der Dispersion einer linearen Kette}{$C_n$ Kraftkonstanten zwischen Ebene $s$ und $s+n$, $M$ \qtyRef{mass} des Referenzatoms, $a$ \qtyRef{lattice_constant}, $q$ Phonon \qtyRef{wavevector}, $u$ Ansatz für die Atomauslenkung}
|
||||
\begin{gather}
|
||||
\omega^2 = \frac{4C_1}{M}\left[\sin^2 \left(\frac{qa}{2}\right) + \frac{C2}{C1} \sin^2(qa)\right] \\
|
||||
\intertext{\GT{with}}
|
||||
u_{s+n} = U\e^{-i \left[\omega t - q(s+n)a \right]}
|
||||
\end{gather}
|
||||
\newFormulaEntry
|
||||
\fig{img/cm_vib_dispersion_one_atom_basis.pdf}
|
||||
\end{formula}
|
||||
\begin{formula}{dispersion_2atom_basis}
|
||||
\desc{Phonon dispersion of a lattice with a two-atom basis}{}{$C$ force constant between layers, $M_i$ \qtyRef{mass} of the basis atoms, $a$ \qtyRef{lattice_constant}, $q$ phonon \qtyRef{wavevector}, $u, v$ Ansatz for the displacement of basis atom 1 and 2, respectively}
|
||||
\desc[german]{Phonondispersion eines Gitters mit einatomiger Basis}{}{$C$ Kraftkonstanten zwischen Ebene $s$ und $s+n$, $M_i$ \qtyRef{mass} der Basisatome, $a$ \qtyRef{lattice_constant}, $q$ Phonon \qtyRef{wavevector}, $u, v$ jeweils Ansatz für die Atomauslenkung des Basisatoms 1 und 2}
|
||||
\begin{gather}
|
||||
\omega^2_{\txa,\txo} = C \left(\frac{1}{M_1}+\frac{1}{M_2}\right) \mp C \sqrt{\left(\frac{1}{M_1}+\frac{1}{M_2}\right)^2 - \frac{4}{M_1M_2} \sin^2 \left(\frac{qa}{2}\right)}
|
||||
\intertext{\GT{with}}
|
||||
u_{s} = U\e^{-i \left(\omega t - qsa \right)}, \quad
|
||||
v_{s} = V\e^{-i \left(\omega t - qsa \right)}
|
||||
\end{gather}
|
||||
\newFormulaEntry
|
||||
\fig{img/cm_vib_dispersion_two_atom_basis.pdf}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{branches}
|
||||
\desc{Vibration branches}{}{}
|
||||
\desc[german]{Vibrationsmoden}{}{}
|
||||
\ttxt{\eng{
|
||||
\textbf{Acoustic}: 3 modes (1 longitudinal, 2 transversal), the two basis atoms oscillate in phase.
|
||||
\\\textbf{Optical}: 3 modes, the two basis atoms oscillate in opposition. A dipole moment is created that can couple to photons.
|
||||
}\ger{
|
||||
\textbf{Akustisch}: 3 Moden (1 longitudinal, 2 transversal), die zwei Basisatome schwingen in Phase.
|
||||
\\ \textbf{Optisch}: 3 Moden, die zwei Basisatome schwingen gegenphasig. Das dadurch entstehende Dipolmoment erlaubt die Wechselwirkung mit Photonen.
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{petit-dulong}
|
||||
\absLabel
|
||||
\desc{Petit-Dulong law}{Empirical heat capacity at high temperatures}{$C_\txm$ molar \qtyRef{heat_capacity}, \ConstRef{avogadro}, \ConstRef{boltzmann}, \ConstRef{gas}}
|
||||
\desc[german]{Petit-Dulong Gesetz}{Empirische Wärmekapazität bei hohen Temperaturen}{}
|
||||
\eq{C_\txm = 3\NA \kB = 3R \approx \SI{25}{\joule\per\mol\kelvin}}
|
||||
\end{formula}
|
||||
|
||||
\Subsection{einstein}
|
||||
\desc{Einstein model}{}{}
|
||||
\desc[german]{Einstein-Modell}{}{}
|
||||
\begin{formula}{description}
|
||||
\desc{Description}{}{}
|
||||
\desc[german]{Beschreibung}{}{}
|
||||
\ttxt{\eng{
|
||||
All lattice vibrations have the \fRef[same frequency]{:::frequency}.
|
||||
Underestimates the \fRef{:::heat_capacity} for low temperatures.
|
||||
}\ger{
|
||||
Alle Gittereigenschwingungen haben die \fRef[selbe Frequenz]{:::frequency}
|
||||
Sagt zu kleine \fRef[Wärmekapazitäten]{:::heat_capacity} für tiefe Temperaturen voraus.
|
||||
}}
|
||||
\end{formula}
|
||||
\begin{formula}{frequency}
|
||||
\desc{Einstein frequency}{}{}
|
||||
\desc[german]{Einstein-Frequenz}{}{}
|
||||
\eq{\omega_\txE}
|
||||
\end{formula}
|
||||
\begin{formula}{heat_capacity}
|
||||
\desc{\qtyRef{heat_capacity}}{according to the Einstein model}{}
|
||||
\desc[german]{}{nach dem Einstein-Modell}{}
|
||||
\eq{C_V^\txE = 3N\kB \left( \frac{\hbar\omega_\txE}{\kB T}\right)^2 \frac{\e^{\frac{\hbar\omega_\txE}{\kB T}}}{ \left(\e^{\frac{\hbar\omega_\txE}{\kB T}} - 1\right)^2}}
|
||||
\end{formula}
|
||||
|
||||
\Subsection{debye}
|
||||
\desc{Debye model}{}{}
|
||||
\desc[german]{Debye-Modell}{}{}
|
||||
\begin{formula}{description}
|
||||
\desc{Description}{}{}
|
||||
\desc[german]{Beschreibung}{}{}
|
||||
\ttxt{\eng{
|
||||
Atoms behave like coupled \fRef[quantum harmonic oscillators]{sec:qm:hosc}. The finite sample size leads to periodic boundary conditio. The finite sample size leads to periodic boundary conditions for the vibrations.
|
||||
}\ger{
|
||||
Atome verhalten sich wie gekoppelte \fRef[quantenmechanische harmonische Oszillatoren]{sec:qm:hosc}. Die endliche Ausdehnung des Körpers führt zu periodischen Randbedingungen.
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{phonon_dos}
|
||||
\desc{Phonon density of states}{}{\QtyRef{volume}, $v$ \qtyRef{speed_of_sound} of the phonon mode, $\omega$ phonon frequency}
|
||||
\desc[german]{Phononenzustandsdichte}{}{\QtyRef{volume}, $v$ \qtyRef{speed_of_sound} des Dispersionszweigs, $\omega$ Phononfrequenz}
|
||||
\eq{D(\omega) \d \omega = \frac{V}{2\pi^2} \frac{\omega^2}{v^3} \d\omega}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{frequency}
|
||||
\desc{Debye frequency}{Maximum phonon frequency}{$v$ \qtyRef{speed_of_sound}, $N/V$ atom density}
|
||||
\desc[german]{Debye-Frequenz}{Maximale Phononenfrequenz}{$v$ \qtyRef{speed_of_sound}, $N/V$ Atomdichte}
|
||||
\eq{\omega_\txD = v \left(6\pi^2 \frac{N}{V}\right)^{1/3}}
|
||||
\hiddenQuantity[debye_frequency]{\omega_\txD}{\per\s}{s}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{temperature}
|
||||
\desc{Debye temperature}{Temperature at which all possible states are occupied}{\ConstRef{planck2pi}, \QtyRef{debye_frequency}, \ConstRef{boltzmann}}
|
||||
\desc[german]{Debye-Frequenz}{Temperatur, bei der alle möglichen Zustände besetzt sind}{}
|
||||
\eq{\theta_\txD = \frac{\hbar\omega_\txD}{\kB}}
|
||||
\hiddenQuantity[debye_temperature]{\theta\txD}{\kelvin}{s}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{heat_capacity}
|
||||
\desc{\qtyRef{heat_capacity}}{according to the Debye model}{$N$ number of atoms, \ConstRef{boltzmann}, \QtyRef{debye_frequency}}
|
||||
\desc[german]{}{nach dem Debye-Modell}{$N$ Anzahl der Atome, \ConstRef{boltzmann}, \QtyRef{debye_frequency}}
|
||||
\eq{C_V^\txD = 9N\kB \left(\frac{\kB T}{\hbar \omega_\txD}\right)^3 \int_0^{\frac{\hbar\omega_\txD}{\kB T}} \d x \frac{x^4 \e^x}{(\e^x-1)^2} }
|
||||
\end{formula}
|
||||
370
src/comp/ad.tex
370
src/comp/ad.tex
@ -1,370 +0,0 @@
|
||||
\Section{ad}
|
||||
\desc{Atomic dynamics}{}{}
|
||||
% \desc[german]{}{}{}
|
||||
|
||||
\begin{formula}{hamiltonian}
|
||||
\desc{Electron Hamiltonian}{}{$\hat{T}$ \fRef{comp:est:kinetic_energy}, $\hat{V}$ \fRef{comp:est:potential_energy}, $\txe$ \GT{electrons}, $\txn$ \GT{nucleons}}
|
||||
\desc[german]{Hamiltonian der Elektronen}{}{}
|
||||
\eq{\hat{H}_\txe = \hat{T}_\txe + V_{\txe \leftrightarrow \txe} + V_{\txn \leftrightarrow \txe}}
|
||||
\end{formula}
|
||||
\begin{formula}{ansatz}
|
||||
\desc{Wave function ansatz}{}{$\psi_\text{en}^n$ eigenstate $n$ of \fRef{comp:est:hamiltonian}, $\psi_\txe^i$ eigenstate $i$ of \fRef{comp:ad:hamiltonian}, $\vecr,\vecR$ electron/nucleus positions, $\sigma$ electron spin, $c^{ni}$ coefficients}
|
||||
\desc[german]{Wellenfunktion Ansatz}{}{}
|
||||
\eq{\psi_\text{en}^n\big(\{\vecr,\sigma\},\{\vecR\}\big) = \sum_i c^{ni}\big(\{\vecR\}\big)\, \psi_\txe^i\big(\{\vecr,\sigma\},\{\vecR\}\big)}
|
||||
\end{formula}
|
||||
\begin{formula}{equation}
|
||||
\desc{Equation}{}{}
|
||||
% \desc[german]{}{}{}
|
||||
\eq{
|
||||
\label{eq:\fqname}
|
||||
\left[E_\txe^j\big(\{\vecR\}\big) + \hat{T}_\txn + V_{\txn \leftrightarrow \txn} - E^n \right]c^{nj} = -\sum_i \Lambda_{ij} c^{ni}\big(\{\vecR\}\big)
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{coupling_operator}
|
||||
\desc{Exact nonadiabtic coupling operator}{Electron-phonon couplings / electron-vibrational couplings}{$\psi^i_\txe$ electronic states, $\vecR$ nucleus position, $M$ nucleus \qtyRef{mass}}
|
||||
% \desc[german]{}{}{}
|
||||
\begin{multline}
|
||||
\Lambda_{ij} = \int \d^3r (\psi_\txe^j)^* \left(-\sum_I \frac{\hbar^2\nabla_{\vecR_I}^2}{2M_I}\right) \psi_\txe^i \\
|
||||
+ \sum_I \frac{1}{M_I} \int\d^3r \left[(\psi_\txe^j)^* (-i\hbar\nabla_{\vecR_I})\psi_\txe^i\right](-i\hbar\nabla_{\vecR_I})
|
||||
\end{multline}
|
||||
\end{formula}
|
||||
|
||||
\Subsection{bo}
|
||||
\desc{Born-Oppenheimer Approximation}{}{}
|
||||
\desc[german]{Born-Oppenheimer Näherung}{}{}
|
||||
\begin{formula}{adiabatic_approx}
|
||||
\desc{Adiabatic approximation}{Electronic configuration remains the same when atoms move (\absRef{adiabatic_theorem})}{$\Lambda_{ij}$ \fRef{comp:ad:coupling_operator}}
|
||||
\desc[german]{Adiabatische Näherung}{Elektronenkonfiguration bleibt gleich bei Bewegung der Atome gleichl (\absRef{adiabatic_theorem})}{}
|
||||
\eq{\Lambda_{ij} = 0 \quad \text{\GT{for} } i\neq j}
|
||||
\end{formula}
|
||||
\begin{formula}{approx}
|
||||
\desc{Born-Oppenheimer approximation}{Electrons are not influenced by the movement of the atoms}{\GT{see} \fRef{comp:ad:equation}, $V_{\txn \leftrightarrow \txn} = \const$ absorbed into $E_\txe^j$}
|
||||
\desc[german]{Born-Oppenheimer Näherung}{Elektronen werden nicht durch die Bewegung der Atome beeinflusst}{}
|
||||
\begin{gather}
|
||||
\Lambda_{ij} = 0
|
||||
% \shortintertext{\fRef{comp:ad:bo:equation} \Rightarrow}
|
||||
\left[E_e^i\big(\{\vecR\}\big) + \hat{T}_\txn - E^n\right]c^{ni}\big(\{\vecR\}\big) = 0
|
||||
\end{gather}
|
||||
\end{formula}
|
||||
\begin{formula}{surface}
|
||||
\desc{Born-Oppenheimer surface}{Potential energy surface (PES)\\ The nuclei follow Newtons equations of motion on the BO surface if the system is in the electronic ground state}{$E_\txe^0, \psi_\txe^0$ lowest eigenvalue/eigenstate of \fRef{comp:ad:hamiltonian}}
|
||||
\desc[german]{Born-Oppenheimer Potentialhyperfläche}{Die Nukleonen Newtons klassichen Bewegungsgleichungen auf der BO Hyperfläche wenn das System im elektronischen Grundzustand ist}{$E_\txe^0, \psi_\txe^0$ niedrigster Eigenwert/Eigenzustand vom \fRef{comp:ad:hamiltonian}}
|
||||
\begin{gather}
|
||||
V_\text{BO}\big(\{\vecR\}\big) = E_\txe^0\big(\{\vecR\}\big) \\
|
||||
M_I \ddot{\vecR}_I(t) = - \Grad_{\vecR_I} V_\text{BO}\big(\{\vecR(t)\}\big)
|
||||
\end{gather}
|
||||
\end{formula}
|
||||
\begin{formula}{ansatz}
|
||||
\desc{Ansatz for \fRef{::approx}}{Product of single electronic and single nuclear state}{}
|
||||
\desc[german]{Ansatz für \fRef{::approx}}{Produkt aus einem einzelnen elektronischen Zustand und einem Nukleus-Zustand}{}
|
||||
\eq{
|
||||
\psi_\text{BO} = c^{n0} \big(\{\vecR\}\big) \,\psi_\txe^0 \big(\{\vecr,\sigma\},\{\vecR\}\big)
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{limitations}
|
||||
\desc{Limitations}{}{$\tau$ passage of time for electrons/nuclei, $L$ characteristic length scale of atomic dynamics, $\dot{\vec{R}}$ nuclear velocity, $\Delta E$ difference between two electronic states}
|
||||
\desc[german]{Limitationen}{}{}
|
||||
\ttxt{
|
||||
\eng{
|
||||
\begin{itemize}
|
||||
\item Nuclei velocities must be small and electron energy state differences large
|
||||
\item Nuclei need spin for effects like spin-orbit coupling
|
||||
\item Nonadiabitc effects in photochemistry, proteins
|
||||
\end{itemize}
|
||||
Valid when Massey parameter $\xi \gg 1$
|
||||
}
|
||||
}
|
||||
\eq{
|
||||
\xi = \frac{\tau_\txn}{\tau_\txe} = \frac{L \Delta E}{\hbar \abs{\dot{\vecR}}}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\Subsection{opt}
|
||||
\desc{Structure optimization}{}{}
|
||||
\desc[german]{Strukturoptimierung}{}{}
|
||||
\begin{formula}{forces}
|
||||
\desc{Forces}{}{}
|
||||
\desc[german]{Kräfte}{}{}
|
||||
\eq{
|
||||
\vec{F}_I = -\Grad_{\vecR_I} E
|
||||
\explOverEq{\fRef{qm:se:hellmann_feynmann}}
|
||||
-\Braket{\psi(\vecR_I) | \left(\Grad_{\vecR_I} \hat{H}(\vecR_I)\right) | \psi(\vecR)}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{ionic_cycle}
|
||||
\desc{Ionic cycle}{\fRef{comp:est:dft:ks:scf} for geometry optimization}{}
|
||||
\desc[german]{}{}{}
|
||||
\ttxt{
|
||||
\eng{
|
||||
\begin{enumerate}
|
||||
\item Initial guess for $n(\vecr)$
|
||||
\begin{enumerate}
|
||||
\item Calculate effective potential $V_\text{eff}$
|
||||
\item Solve \fRef{comp:est:dft:ks:equation}
|
||||
\item Calculate density $n(\vecr)$
|
||||
\item Repeat b-d until self consistent
|
||||
\end{enumerate}
|
||||
\item Calculate \fRef{:::forces}
|
||||
\item If $F\neq0$, get new geometry by interpolating $R$ and restart
|
||||
\end{enumerate}
|
||||
}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{transformation}
|
||||
\desc{Transformation of atomic positions under stress}{}{$\alpha,\beta=1,2,3$ position components, $R$ position, $R(0)$ zero-strain position, $\ten{\epsilon}$ \qtyRef{strain} tensor}
|
||||
\desc[german]{Transformation der Atompositionen unter Spannung}{}{$\alpha,\beta=1,2,3$ Positionskomponenten, $R$ Position, $R(0)$ Position ohne Dehnung, $\ten{\epsilon}$ \qtyRef{strain} Tensor}
|
||||
\eq{R_\alpha(\ten{\epsilon}_{\alpha\beta}) = \sum_\beta \big(\delta_{\alpha\beta} + \ten{\epsilon}_{\alpha\beta}\big)R_\beta(0)}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{stress_tensor}
|
||||
\desc{Stress tensor}{}{$\Omega$ unit cell volume, \ten{\epsilon} \qtyRef{strain} tensor}
|
||||
\desc[german]{Spannungstensor}{}{}
|
||||
\eq{\ten{\sigma}_{\alpha,\beta} = \frac{1}{\Omega} \pdv{E_\text{total}}{\ten{\epsilon}_{\alpha\beta}}_{\ten{\epsilon}=0}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{pulay_stress}
|
||||
\desc{Pulay stress}{}{}
|
||||
\desc[german]{Pulay-Spannung}{}{}
|
||||
\eq{
|
||||
N_\text{PW} \propto E_\text{cut}^\frac{3}{2} \propto \abs{\vec{G}_\text{max}}^3
|
||||
}
|
||||
\ttxt{\eng{
|
||||
Number of plane waves $N_\text{PW}$ depends on $E_\text{cut}$.
|
||||
If $G$ changes during optimization, $N_\text{PW}$ may change, thus the basis set can change.
|
||||
This typically leads to too small volumes.
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\Subsection{latvib}
|
||||
\desc{Lattice vibrations}{}{}
|
||||
\desc[german]{Gitterschwingungen}{}{}
|
||||
\begin{formula}{force_constant_matrix}
|
||||
\desc{Force constant matrix}{}{}
|
||||
% \desc[german]{}{}{}
|
||||
\eq{\Phi_{IJ}^{\mu\nu} = \pdv{V(\{\vecR\})}{R_I^\mu,R_J^\nu}_{\{\vecR_I\}=\{\vecR_I^0\}}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{harmonic_approx}
|
||||
\desc{Harmonic approximation}{Hessian matrix, 2nd order Taylor expansion of the \fRef{comp:ad:bo:surface} around every nucleus position $\vecR_I^0$}{$\Phi_{IJ}^{\mu\nu}$ \fRef{::force_constant_matrix}, $s$ displacement}
|
||||
\desc[german]{Harmonische Näherung}{Hesse matrix, Taylor Entwicklung der \fRef{comp:ad:bo:surface} in zweiter Oddnung um Atomposition $\vecR_I^0$}{}
|
||||
\eq{ V^\text{BO}(\{\vecR_I\}) \approx V^\text{BO}(\{\vecR_I^0\}) + \frac{1}{2} \sum_{I,J}^N \sum_{\mu,\nu}^3 s_I^\mu s_J^\nu \Phi_{IJ}^{\mu\nu} }
|
||||
\end{formula}
|
||||
|
||||
% solving difficult becaus we need to calculate (3N)^2 derivatives, Hellmann-Feynman cant be applied directly
|
||||
% -> DFPT
|
||||
|
||||
% finite-difference method
|
||||
\Subsubsection{fin_diff}
|
||||
\desc{Finite difference method}{}{}
|
||||
% \desc[german]{}{}{}
|
||||
|
||||
\begin{formula}{approx}
|
||||
\desc{Approximation}{Assume forces in equilibrium structure vanish}{$\Delta s$ displacement of atom $J$}
|
||||
% \desc[german]{}{}{}
|
||||
\eq{\Phi_{IJ}^{\mu\nu} \approx \frac{\vecF_I^\mu(\vecR_1^0, \dots, \vecR_J^0+\Delta s_J^\nu,\dots, \vecR_N^0)}{\Delta s_J^\nu}}
|
||||
\end{formula}
|
||||
\begin{formula}{dynamical_matrix}
|
||||
\desc{Dynamical matrix}{Mass reduced \absRef[fourier transform]{fourier_transform} of the \fRef{comp:ad:latvib:force_constant_matrix}}{$\vec{L}$ vector from origin to unit cell $n$, $\alpha/\beta$ atom index in th unit cell, $\vecq$ \qtyRef{wavevector}, $\Phi$ \fRef{comp:ad:latvib:force_constant_matrix}, $M$ \qtyRef{mass}}
|
||||
% \desc[german]{}{}{}
|
||||
\eq{D_{\alpha\beta}^{\mu\nu} = \frac{1}{\sqrt{M_\alpha M_\beta}} \sum_{n^\prime} \Phi_{\alpha\beta}^{\mu\nu}(n-n^\prime) \e^{\I \vec{q}(\vec{L}_n - \vec{L}_{n^\prime})}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{eigenvalue_equation}
|
||||
\desc{Eigenvalue equation}{For a periodic crystal, reduces number of equations from $3N_p\times N$ to $3N_p$. Eigenvalues represent phonon band structure.}{$N_p$ number of atoms per unit cell, $\vecc$ displacement amplitudes, $\vecq$ \qtyRef{wave_vector}, $\mat{D}$ \fRef{::dynamical_matrix}}
|
||||
\desc[german]{Eigenwertgleichung}{}{}
|
||||
\eq{\omega^2 \vecc(\vecq) = \mat{D}(\vecq) \vecc(\vecq) }
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection{anharmonic}
|
||||
\desc{Anharmonic approaches}{}{}
|
||||
\desc[german]{Anharmonische Ansätze}{}{}
|
||||
|
||||
\begin{formula}{qha}
|
||||
\desc{Quasi-harmonic approximation}{}{}
|
||||
\desc[german]{}{}{}
|
||||
\ttxt{\eng{
|
||||
Include thermal expansion by assuming \fRef{comp:ad:bo:surface} is volume dependant.
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{pertubative}
|
||||
\desc{Pertubative approaches}{}{}
|
||||
% \desc[german]{Störungs}{}{}
|
||||
\ttxt{\eng{
|
||||
Expand \fRef{comp:ad:latvib:force_constant_matrix} to third order.
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
|
||||
|
||||
\Subsection{md}
|
||||
\desc{Molecular Dynamics}{}{}
|
||||
\desc[german]{Molekulardynamik}{}{} \abbrLink{md}{MD}
|
||||
\begin{formula}{desc}
|
||||
\desc{Description}{}{}
|
||||
\desc[german]{Beschreibung}{}{}
|
||||
\ttxt{\eng{
|
||||
\begin{itemize}
|
||||
\item Assumes fully classical nuclei
|
||||
\item Macroscropical observables from statistical ensembles
|
||||
\item Number of points to consider does NOT scale with system size
|
||||
\item System evolves in time (\absRef{ehrenfest_theorem})
|
||||
\item Computes time-dependant observables
|
||||
\item Does not use \fRef{comp:ad:latvib:harmonic_approx} \Rightarrow Anharmonic effects included
|
||||
\end{itemize}
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{procedure}
|
||||
\desc{MD simulation procedure}{}{}
|
||||
\desc[german]{Ablauf von MD Simulationen}{}{}
|
||||
\ttxt{\eng{
|
||||
\begin{enumerate}
|
||||
\item Initialize with optimized geometry, interaction potential, ensemble, integration scheme, temperature/pressure control
|
||||
\item Equilibrate to desired temperature/pressure (eg with statistical starting velocities)
|
||||
\item Production run, run MD long enough to calculate desired observables
|
||||
\end{enumerate}
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection{ab-initio}
|
||||
\desc{Ab-initio molecular dynamics}{}{}
|
||||
\desc[german]{Ab-initio molecular dynamics}{}{}
|
||||
\begin{formula}{bomd}
|
||||
\abbrLabel{BOMD}
|
||||
\desc{Born-Oppenheimer MD (BOMD)}{}{}
|
||||
\desc[german]{Born-Oppenheimer MD (BOMD)}{}{}
|
||||
\ttxt{\eng{
|
||||
\begin{enumerate}
|
||||
\item Calculate electronic ground state of current nucleui configuration $\{\vecR(t)\}$ with \abbrRef{ksdft}
|
||||
\item \fRef[Calculate forces]{comp:ad:opt:forces} from the \fRef{comp:ad:bo:surface}
|
||||
\item Update positions and velocities
|
||||
\end{enumerate}
|
||||
\begin{itemize}
|
||||
\gooditem "ab-inito" - no empirical information required
|
||||
\baditem Many expensive \abbrRef{dft} calculations
|
||||
\end{itemize}
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{cpmd}
|
||||
\desc{Car-Parrinello MD (CPMD)}{}{$\mu$ electron orbital mass, $\varphi_i$ \abbrRef{ksdft} eigenststate, $\lambda_{ij}$ Lagrange multiplier}
|
||||
\desc[german]{Car-Parrinello MD (CPMD)}{}{}
|
||||
\ttxt{\eng{
|
||||
Evolve electronic wave function $\varphi$ (adiabatically) along with the nuclei \Rightarrow only one full \abbrRef{ksdft}
|
||||
}}
|
||||
\begin{gather}
|
||||
M_I \odv[2]{\vecR_I}{t} = -\Grad_{\vecR_I} E[\{\varphi_i\},\{\vecR_I\}] \\
|
||||
% not using pdv because of comma in parens
|
||||
% E[\{\varphi_i\}\{\vecR_I\}] = \Braket{\psi_0|H_\text{el}^\text{KS}|\psi_0}
|
||||
\mu \odv[2]{\varphi_i(\vecr,t)}{t} = - \frac{\partial}{\partial\varphi_i^*(\vecr,t)} E[\{\varphi_i\},\{\vecR_I\}] + \sum_j \lambda_{ij} \varphi_j(\vecr,t)
|
||||
\end{gather}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection{ff}
|
||||
\desc{Force-field MD}{}{}
|
||||
\desc[german]{Force-field MD}{}{}
|
||||
|
||||
\begin{formula}{ffmd}
|
||||
\desc{Force field MD (FFMD)}{}{}
|
||||
% \desc[german]{}{}{}
|
||||
\ttxt{\eng{
|
||||
\begin{itemize}
|
||||
\item Use empirical interaction potential instead of electronic structure
|
||||
\baditem Force fields need to be fitted for specific material \Rightarrow not transferable
|
||||
\gooditem Faster than \abbrRef{bomd}
|
||||
\item Example: \absRef[Lennard-Jones]{lennard_jones}
|
||||
\end{itemize}
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
|
||||
\Subsubsection{scheme}
|
||||
\desc{Integration schemes}{Procedures for updating positions and velocities to obey the equations of motion.}{}
|
||||
\desc[german]{Integrationsmethoden}{Prozeduren zum stückweisen numerischen Lösung der Bewegungsgleichungen}{}
|
||||
|
||||
\begin{formula}{euler}
|
||||
\desc{Euler method}{First-order procedure for solving \abbrRef{ode}s with a given initial value.\\Taylor expansion of $\vecR/\vecv (t+\Delta t)$}{}
|
||||
\desc[german]{Euler-Verfahren}{Prozedur um gewöhnliche DGLs mit Anfangsbedingungen in erster Ordnung zu lösen.\\Taylor Entwicklung von $\vecR/\vecv (t+\Delta t)$}{}
|
||||
\eq{
|
||||
\vecR(t+\Delta t) &= \vecR(t) + \vecv(t) \Delta t + \Order{\Delta t^2} \\
|
||||
\vecv(t+\Delta t) &= \vecv(t) + \veca(t) \Delta t + \Order{\Delta t^2}
|
||||
}
|
||||
\ttxt{\eng{
|
||||
Cumulative error scales linearly $\Order{\Delta t}$. Not time reversible.
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{verlet}
|
||||
\desc{Verlet integration}{Preverses time reversibility, does not require velocity updates. Integration in 2nd order}{}
|
||||
\desc[german]{Verlet-Algorithmus}{Zeitumkehr-symmetrisch. Interation in zweiter Ordnung}{}
|
||||
\eq{
|
||||
\vecR(t+\Delta t) = 2\vecR(t) -\vecR(t-\Delta t) + \veca(t) \Delta t^2 + \Order{\Delta t^4}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{velocity-verlet}
|
||||
\desc{Velocity-Verlet integration}{}{}
|
||||
% \desc[german]{}{}{}
|
||||
\eq{
|
||||
\vecR(t+\Delta t) &= \vecR(t) + \vecv(t)\Delta t + \frac{1}{2} \veca(t) \Delta t^2 + \Order{\Delta t^4} \\
|
||||
\vecv(t+\Delta t) &= \vecv(t) + \frac{\veca(t) + \veca(t+\Delta t)}{2} \Delta t + \Order{\Delta t^4}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{leapfrog}
|
||||
\desc{Leapfrog}{Integration in 2nd order}{}
|
||||
\desc[german]{Leapfrog}{Integration in zweiter Ordnung}{}
|
||||
\eq{
|
||||
x_{i+1} &= x_i + v_{i+1/2} \Delta t_i \\
|
||||
v_{i+1/2} &= v_{i-1/2} + a_{i} \Delta t_i
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection{stats}
|
||||
\desc{Thermostats and barostats}{}{}
|
||||
\desc[german]{Thermostate und Barostate}{}{}
|
||||
\begin{formula}{velocity_rescaling}
|
||||
\desc{Velocity rescaling}{Thermostat, keep temperature at $T_0$ by rescaling velocities. Does not allow temperature fluctuations and thus does not obey the \absRef{c_ensemble}}{$T$ target \qtyRef{temperature}, $M$ \qtyRef{mass} of nucleon $I$, $\vecv$ \qtyRef{velocity}, $f$ number of degrees of freedom, $\lambda$ velocity scaling parameter, \ConstRef{boltzmann}}
|
||||
% \desc[german]{}{}{}
|
||||
\eq{
|
||||
\Delta T(t) &= T_0 - T(t) \\
|
||||
&= \sum_I^N \frac{M_I\,(\lambda \vecv_I(t))^2}{f\kB} - \sum_I^N \frac{M_I\,\vecv_I(t)^2}{f\kB} \\
|
||||
&= (\lambda^2 - 1) T(t)
|
||||
}
|
||||
\eq{\lambda = \sqrt{\frac{T_0}{T(t)}}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{berendsen}
|
||||
\desc{Berendsen thermostat}{Does not obey \absRef{c_ensemble} but efficiently brings system to target temperature}{}
|
||||
% \desc[german]{}{}{}
|
||||
\eq{\odv{T}{t} = \frac{T_0-T}{\tau}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{nose-hoover}
|
||||
\desc{Nosé-Hoover thermostat}{Control the temperature with by time stretching with an associated mass.\\Compliant with \absRef{c_ensemble}}{$s$ scaling factor, $Q$ associated "mass", $\mathcal{L}$ \absRef{lagrangian}, $g$ degrees of freedom}
|
||||
\desc[german]{Nosé-Hoover Thermostat}{}{}
|
||||
\begin{gather}
|
||||
\d\tilde{t} = \tilde{s}\d t \\
|
||||
\mathcal{L} = \sum_{I=1}^N \frac{1}{2} M_I \tilde{s}^2 v_i^2 - V(\tilde{\vecR}_1, \ldots, \tilde{\vecR}_I, \ldots, \tilde{\vecR}_N) + \frac{1}{2} Q \dot{\tilde{s}}^2 - g \kB T_0 \ln \tilde{s}
|
||||
\end{gather}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection{obs}
|
||||
\desc{Calculating observables}{}{}
|
||||
\desc[german]{Berechnung von Observablen}{}{}
|
||||
\begin{formula}{spectral_density}
|
||||
\desc{Spectral density}{Wiener-Khinchin theorem\\\absRef{fourier_transform} of \absRef{autocorrelation}}{$C$ \absRef{autocorrelation}}
|
||||
\desc[german]{Spektraldichte}{Wiener-Khinchin Theorem\\\absRef{fourier_transform} of \absRef{autocorrelation}}{}
|
||||
\eq{S(\omega) = \int_{-\infty}^\infty \d\tau C(\tau) \e^{-\I\omega t} }
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{vdos} \abbrLabel{VDOS}
|
||||
\desc{Vibrational density of states (VDOS)}{}{$S_{v_i}$ velocity \fRef{::spectral_density} of particle $I$}
|
||||
\desc[german]{Vibrationszustandsdicht (VDOS)}{}{}
|
||||
\eq{g(\omega) \sim \sum_{I=1}^N M_I S_{v_I}(\omega)}
|
||||
\end{formula}
|
||||
@ -1,4 +0,0 @@
|
||||
\Part{comp}
|
||||
\desc{Computational Physics}{}{}
|
||||
\desc[german]{Computergestützte Physik}{}{}
|
||||
|
||||
270
src/comp/est.tex
270
src/comp/est.tex
@ -1,270 +0,0 @@
|
||||
\Section{est}
|
||||
\desc{Electronic structure theory}{}{}
|
||||
% \desc[german]{}{}{}
|
||||
|
||||
|
||||
\begin{formula}{kinetic_energy}
|
||||
\desc{Kinetic energy}{of species $i$}{$i$ = nucleons/electrons, $N$ number of particles, $m$ \qtyRef{mass}}
|
||||
\desc[german]{Kinetische Energie}{von Spezies $i$}{$i$ = Nukleonen/Elektronen, $N$ Teilchenzahl, $m$ \qtyRef{mass}}
|
||||
\eq{\hat{T}_i &= -\sum_{n=1}^{N_i} \frac{\hbar^2}{2 m_i} \vec{\nabla}^2_n}
|
||||
\end{formula}
|
||||
\begin{formula}{potential_energy}
|
||||
\desc{Electrostatic potential}{between species $i$ and $j$}{$i,j$ = nucleons/electrons, $r$ particle position, $Z_i$ charge of species $i$, \ConstRef{charge}}
|
||||
\desc[german]{Elektrostatisches Potential}{zwischen Spezies $i$ und $j$}{}
|
||||
\eq{\hat{V}_{i \leftrightarrow j} &= -\sum_{k,l} \frac{Z_i Z_j e^2}{\abs{\vecr_k - \vecr_l}}}
|
||||
\end{formula}
|
||||
\begin{formula}{hamiltonian}
|
||||
\desc{Electronic structure Hamiltonian}{}{$\hat{T}$ \fRef{comp:est:kinetic_energy}, $\hat{V}$ \fRef{comp:est:potential_energy}, $\txe$ \GT{electrons}, $\txn$ \GT{nucleons}}
|
||||
\eq{\hat{H} &= \hat{T}_\txe + \hat{T}_\txn + V_{\txe \leftrightarrow \txe} + V_{\txn \leftrightarrow \txe} + V_{\txn \leftrightarrow \txn}}
|
||||
\end{formula}
|
||||
\begin{formula}{mean_field}
|
||||
\desc{Mean field approximation}{Replaces 2-particle operator by 1-particle operator}{Example for Coulomb interaction between many electrons}
|
||||
\desc[german]{Molekularfeldnäherung}{Ersetzt 2-Teilchen Operator durch 1-Teilchen Operator}{Beispiel für Coulomb Wechselwirkung zwischen Elektronen}
|
||||
\eq{
|
||||
\frac{1}{2}\sum_{i\neq j} \frac{e^2}{\abs{\vec{r}_i - \vec{r}_j}} \approx \sum_{i} V_\text{eff}(\vec{r}_i)
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsection{tb}
|
||||
\desc{Tight-binding}{}{}
|
||||
\desc[german]{Modell der stark gebundenen Elektronen / Tight-binding}{}{}
|
||||
\begin{formula}{assumptions}
|
||||
\desc{Assumptions}{}{}
|
||||
\desc[german]{Annahmen}{}{}
|
||||
\ttxt{
|
||||
\eng{
|
||||
\begin{itemize}
|
||||
\item Atomic wave functions are localized \Rightarrow Small overlap, interaction cutoff
|
||||
\end{itemize}
|
||||
}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{hamiltonian}
|
||||
\desc{Tight-binding Hamiltonian}{in second quantized form}{$\hat{a}_i^\dagger$, $\hat{a}_i$ \GT{creation_annihilation_ops} create/destory an electron on site $i$, $\epsilon_i$ on-site energy, $t_{i,j}$ hopping amplitude, usually $\epsilon$ and $t$ are determined from experiments or other methods}
|
||||
\desc[german]{Tight-binding Hamiltonian}{in zweiter Quantisierung}{$\hat{a}_i^\dagger$, $\hat{a}_i$ \GT{creation_annihilation_ops} erzeugen/vernichten ein Elektron auf Platz $i$, $\epsilon_i$ on-site Energie, $t_{i,j}$ hopping Amplitude, meist werden $\epsilon$ und $t$ aus experimentellen Daten oder anderen Methoden bestimmt}
|
||||
\eq{\hat{H} = \sum_i \epsilon_i \hat{a}_i^\dagger \hat{a}_i - \sum_{i,j} t_{i,j} \left(\hat{a}_i^\dagger \hat{a}_j + \hat{a}_j^\dagger \hat{a}_i\right)}
|
||||
\end{formula}
|
||||
|
||||
|
||||
|
||||
\Subsection{dft}
|
||||
\desc{Density functional theory (DFT)}{}{}
|
||||
\desc[german]{Dichtefunktionaltheorie (DFT)}{}{}
|
||||
\abbrLink{dft}{DFT}
|
||||
\Subsubsection{hf}
|
||||
\desc{Hartree-Fock}{}{}
|
||||
\desc[german]{Hartree-Fock}{}{}
|
||||
\begin{formula}{description}
|
||||
\desc{Description}{}{}
|
||||
\desc[german]{Beschreibung}{}{}
|
||||
\begin{ttext}
|
||||
\eng{
|
||||
\begin{itemize}
|
||||
\item Assumes wave functions are \fRef{qm:other:slater_det} \Rightarrow Approximation
|
||||
\item \fRef{comp:est:mean_field} theory obeying the Pauli principle
|
||||
\item Self-interaction free: Self interaction is cancelled out by the Fock-term
|
||||
\end{itemize}
|
||||
}
|
||||
\end{ttext}
|
||||
\end{formula}
|
||||
\begin{formula}{equation}
|
||||
\desc{Hartree-Fock equation}{}{
|
||||
$\varphi_\xi$ single particle wavefunction of $\xi$th orbital,
|
||||
$\hat{T}$ kinetic electron energy,
|
||||
$\hat{V}_{\text{en}}$ electron-nucleus attraction,
|
||||
$h\hat{V}_{\text{HF}}$ \fRef{comp:est:dft:hf:potential},
|
||||
$x = \vecr,\sigma$ position and spin
|
||||
}
|
||||
\desc[german]{Hartree-Fock Gleichung}{}{
|
||||
$\varphi_\xi$ ein-Teilchen Wellenfunktion des $\xi$-ten Orbitals,
|
||||
$\hat{T}$ kinetische Energie der Elektronen,
|
||||
$\hat{V}_{\text{en}}$ Electron-Kern Anziehung,
|
||||
$\hat{V}_{\text{HF}}$ \fRef{comp:est:dft:hf:potential},
|
||||
$x = \vecr,\sigma$ Position and Spin
|
||||
}
|
||||
\eq{
|
||||
\left(\hat{T} + \hat{V}_{\text{en}} + \hat{V}_{\text{HF}}^\xi\right)\varphi_\xi(x) = \epsilon_\xi \varphi_\xi(x)
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{potential}
|
||||
\desc{Hartree-Fock potential}{}{}
|
||||
\desc[german]{Hartree Fock Potential}{}{}
|
||||
\eq{
|
||||
V_{\text{HF}}^\xi(\vecr) =
|
||||
\sum_{\vartheta} \int \d x'
|
||||
\frac{e^2}{\abs{\vecr - \vecr'}}
|
||||
\left(
|
||||
\underbrace{\abs{\varphi_\xi(x')}^2}_{\text{Hartree-Term}}
|
||||
- \underbrace{\frac{\varphi_{\vartheta}^*(x') \varphi_{\xi}(x') \varphi_{\vartheta}(x)}{\varphi_\xi(x)}}_{\text{Fock-Term}}
|
||||
\right)
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{scf}
|
||||
\desc{Self-consistent field cycle}{}{}
|
||||
% \desc[german]{}{}{}
|
||||
\ttxt{
|
||||
\eng{
|
||||
\begin{enumerate}
|
||||
\item Initial guess for $\varphi$
|
||||
\item Solve SG for each particle
|
||||
\item Make new guess for $\varphi$
|
||||
\end{enumerate}
|
||||
}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection{hk}
|
||||
\desc{Hohenberg-Kohn Theorems}{}{}
|
||||
\desc[german]{Hohenberg-Kohn Theoreme}{}{}
|
||||
\begin{formula}{hk1}
|
||||
\desc{Hohenberg-Kohn theorem (HK1)}{}{}
|
||||
\desc[german]{Hohenberg-Kohn Theorem (HK1)}{}{}
|
||||
\ttxt{
|
||||
\eng{For any system of interacting electrons, the ground state electron density $n(\vecr)$ determines $\hat{V}_\text{ext}$ uniquely up to a trivial constant. }
|
||||
\ger{Die Elektronendichte des Grundzustandes $n(\vecr)$ bestimmt ein einzigartiges $\hat{V}_{\text{ext}}$ eines Systems aus interagierenden Elektronen bis auf eine Konstante.}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{hk2}
|
||||
\desc{Hohenberg-Kohn theorem (HK2)}{}{}
|
||||
\desc[german]{Hohenberg-Kohn Theorem (HK2)}{}{}
|
||||
\ttxt{
|
||||
\eng{Given the energy functional $E[n(\vecr)]$, the ground state density and energy can be obtained variationally. The density that minimizes the total energy is the exact ground state density. }
|
||||
\ger{Für ein Energiefunktional $E[n(\vecr)]$ kann die Grundzustandsdichte und Energie durch systematische Variation bestimmt werden. Die Dichte, welche die Gesamtenergie minimiert ist die exakte Grundzustandsichte. }
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{density}
|
||||
\desc{Ground state electron density}{}{}
|
||||
\desc[german]{Grundzustandselektronendichte}{}{}
|
||||
\eq{n(\vecr) = \Braket{\psi_0|\sum_{i=1}^N \delta(\vecr-\vecr_i)|\psi_0}}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection{ks}
|
||||
\desc{Kohn-Sham DFT}{}{}
|
||||
\desc[german]{Kohn-Sham DFT}{}{}
|
||||
\abbrLink{ksdft}{KS-DFT}
|
||||
\begin{formula}{map}
|
||||
\desc{Kohn-Sham map}{}{}
|
||||
\desc[german]{Kohn-Sham Map}{}{}
|
||||
\ttxt{
|
||||
\eng{Maps fully interacting system of electrons to a system of non-interacting electrons with the same ground state density $n^\prime(\vecr) = n(\vecr)$}
|
||||
}
|
||||
\eq{n(\vecr) = \sum_{i=1}^N \abs{\phi_i(\vecr)}^2}
|
||||
\end{formula}
|
||||
\begin{formula}{functional}
|
||||
\desc{Kohn-Sham functional}{}{$T_\text{KS}$ kinetic enery, $V_\text{ext}$ external potential, $E_\txH$ \fRef[Hartree term]{comp:est:dft:hf:potential}, $E_\text{XC}$ \fRef{comp:est:dft:xc:xc}}
|
||||
\desc[german]{Kohn-Sham Funktional}{}{}
|
||||
\eq{E_\text{KS}[n(\vecr)] = T_\text{KS}[n(\vecr)] + V_\text{ext}[n(\vecr)] + E_\text{H}[n(\vecr)] + E_\text{XC}[n(\vecr)] }
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{equation}
|
||||
\desc{Kohn-Sham equation}{Exact single particle \abbrRef{schroedinger_equation} (though often exact $E_\text{XC}$ is not known)\\ Solving it uses up a large portion of supercomputer resources}{$\phi_i^\text{KS}$ KS orbitals, $\int\d^3r v_\text{ext}(\vecr)n(\vecr)=V_\text{ext}[n(\vecr)]$}
|
||||
\desc[german]{Kohn-Sham Gleichung}{Exakte Einteilchen-\abbrRef{schroedinger_equation} (allerdings ist das exakte $E_\text{XC}$ oft nicht bekannt)\\ Die Lösung der Gleichung macht einen großen Teil der Supercomputer Ressourcen aus}{}
|
||||
\begin{multline}
|
||||
\biggr\{
|
||||
-\frac{\hbar^2\nabla^2}{2m}
|
||||
+ v_\text{ext}(\vecr)
|
||||
+ e^2 \int\d^3 \vecr^\prime \frac{n(\vecr^\prime)}{\abs{\vecr-\vecr^\prime}} \\
|
||||
+ \pdv{E_\txX[n(\vecr)]}{n(\vecr)}
|
||||
+ \pdv{E_\txC[n(\vecr)]}{n(\vecr)}
|
||||
\biggr\} \phi_i^\text{KS}(\vecr) =\\
|
||||
= \epsilon_i^\text{KS} \phi_i^\text{KS}(\vecr)
|
||||
\end{multline}
|
||||
\end{formula}
|
||||
\begin{formula}{scf}
|
||||
\desc{Self-consistent field cycle for Kohn-Sham}{}{}
|
||||
% \desc[german]{}{}{}
|
||||
\ttxt{
|
||||
\itemsep=\parsep
|
||||
\eng{
|
||||
\begin{enumerate}
|
||||
\item Initial guess for $n(\vecr)$
|
||||
\item Calculate effective potential $V_\text{eff}$
|
||||
\item Solve \fRef{comp:est:dft:ks:equation}
|
||||
\item Calculate density $n(\vecr)$
|
||||
\item Repeat 2-4 until self consistent
|
||||
\end{enumerate}
|
||||
}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection{xc}
|
||||
\desc{Exchange-Correlation functionals}{}{}
|
||||
\desc[german]{Exchange-Correlation Funktionale}{}{}
|
||||
\begin{formula}{xc}
|
||||
\desc{Exchange-Correlation functional}{}{}
|
||||
\desc[german]{Exchange-Correlation Funktional}{}{}
|
||||
\eq{ E_\text{XC}[n(\vecr)] = \Braket{\hat{T}} - T_\text{KS}[n(\vecr)] + \Braket{\hat{V}_\text{int}} - E_\txH[n(\vecr)] }
|
||||
\ttxt{\eng{
|
||||
Accounts for:
|
||||
\begin{itemize}
|
||||
\item Kinetic energy difference between interaction and non-interacting system
|
||||
\item Exchange energy due to Pauli principle
|
||||
\item Correlation energy due to many-body Coulomb interaction (not accounted for in mean field Hartree term $E_\txH$)
|
||||
\end{itemize}
|
||||
}}
|
||||
\end{formula}
|
||||
\begin{formula}{lda}
|
||||
\desc{Local density approximation (LDA)}{Simplest DFT functionals}{$\epsilon_\txX$ calculated exchange energy from \fRef[HEG model]{comp:qmb:models:heg}, $\epsilon_\txC$ correlation energy calculated with \fRef{comp:qmb:methods:qmonte-carlo}}
|
||||
\desc[german]{}{}{}
|
||||
\abbrLabel{LDA}
|
||||
\eq{E_\text{XC}^\text{LDA}[n(\vecr)] = \int \d^3r\,n(r) \Big[\epsilon_\txX[n(\vecr)] + \epsilon_\txC[n(\vecr)]\Big]}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{gga}
|
||||
\desc{Generalized gradient approximation (GGA)}{}{$\epsilon_\txX$ calculated exchange energy from \fRef[HEG model]{comp:qmb:models:heg}, $F_\text{XC}$ function containing exchange-correlation energy dependency on $n$ and $\Grad n$}
|
||||
\desc[german]{}{}{}
|
||||
\abbrLabel{GGA}
|
||||
\eq{E_\text{XC}^\text{GGA}[n(\vecr)] = \int \d^3r\,n(r) \epsilon_\txX[n(\vecr)]\,F_\text{XC}[n(\vecr), \Grad n(\vecr)]}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{hybrid}
|
||||
\desc{Hybrid functionals}{}{}
|
||||
\desc[german]{Hybride Funktionale}{}{$\alpha$ mixing paramter, $E_\txX$ exchange energy, $E_\txC$ correlation energy}
|
||||
\eq{\alpha E_\txX^\text{HF} + (1-\alpha) E_\txX^\text{GGA} + E_\txC^\text{GGA}}
|
||||
\ttxt{\eng{
|
||||
Include \fRef[Fock term]{comp:est:dft:hf:potential} (exact exchange) in other functional, like \abbrRef{gga}. Computationally expensive
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\begin{formula}{range-separated-hybrid}
|
||||
\desc{Range separated hyrid functionals (RSH)}{Here HSE as example}{$\alpha$ mixing paramter, $E_\txX$ exchange energy, $E_\txC$ correlation energy}
|
||||
% \desc[german]{}{}{}
|
||||
\newFormulaEntry
|
||||
\begin{gather}
|
||||
\frac{1}{r} = \frac{\erf(\omega r)}{r} + \frac{\erfc{\omega r}}{r} \\
|
||||
E_\text{XC}^\text{HSE} = \alpha E_\text{X,SR}^\text{HF}(\omega) + (1-\alpha)E_\text{X,SR}^\text{GGA}(\omega) + E_\text{X,LR}^\text{GGA}(\omega) + E_\txC^\text{GGA}
|
||||
\end{gather}
|
||||
\ttxt{\eng{
|
||||
Use \abbrRef{gga} and \fRef[Fock]{comp:est:dft:hf:potential} exchange for short ranges (SR) and only \abbrRef{GGA} for long ranges (LR).
|
||||
\abbrRef{GGA} correlation is always used. Useful when dielectric screening reduces long range interactions, saves computational cost.
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsubsection{basis}
|
||||
\desc{Basis sets}{}{}
|
||||
\desc[german]{Basis-Sets}{}{}
|
||||
\begin{formula}{plane_wave}
|
||||
\desc{Plane wave basis}{Plane wave ansatz in \fRef{comp:est:dft:ks:equation}\\Good for periodic structures, allows computation parallelization over a sample points in the \abbrRef{brillouin_zone}}{}
|
||||
\desc[german]{Ebene Wellen als Basis}{}{}
|
||||
\eq{\sum_{\vecG^\prime} \left[\frac{\hbar^2 \abs{\vecG+\veck}^2}{2m} \delta_{\vecG,\vecG^\prime} + V_\text{eff}(\vecG-\vecG^\prime)\right] c_{i,\veck,\vecG^\prime} = \epsilon_{i,\veck} c_{i,\veck,\vecG}}
|
||||
\end{formula}
|
||||
\begin{formula}{plane_wave_cutoff}
|
||||
\desc{Plane wave cutoff}{Number of plane waves included in the calculation must be finite}{}
|
||||
% \desc[german]{}{}{}
|
||||
\eq{E_\text{cutoff} = \frac{\hbar^2 \abs{\veck+\vecG}^2}{2m}}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection{pseudo}
|
||||
\desc{Pseudo-Potential method}{}{}
|
||||
\desc[german]{Pseudopotentialmethode}{}{}
|
||||
\begin{formula}{ansatz}
|
||||
\desc{Ansatz}{}{}
|
||||
\desc[german]{Ansatz}{}{}
|
||||
\ttxt{\eng{
|
||||
Core electrons are absorbed into the potential since they do not contribute much to interesting properties.
|
||||
}}
|
||||
\end{formula}
|
||||
189
src/comp/ml.tex
189
src/comp/ml.tex
@ -1,189 +0,0 @@
|
||||
\Section{ml}
|
||||
\desc{Machine-Learning}{}{}
|
||||
\desc[german]{Maschinelles Lernen}{}{}
|
||||
|
||||
\Subsection{performance}
|
||||
\desc{Performance metrics}{}{}
|
||||
\desc[german]{Metriken zur Leistungsmessung}{}{}
|
||||
|
||||
\eng[cp]{correct predictions}
|
||||
\ger[cp]{richtige Vorhersagen}
|
||||
\eng[fp]{false predictions}
|
||||
\ger[fp]{falsche Vorhersagen}
|
||||
|
||||
\eng[y]{ground truth}
|
||||
\eng[yhat]{prediction}
|
||||
\ger[y]{Wahrheit}
|
||||
\ger[yhat]{Vorhersage}
|
||||
|
||||
\eng[n_desc]{Number of data points}
|
||||
\ger[n_desc]{Anzahl der Datenpunkte}
|
||||
|
||||
\begin{formula}{accuracy}
|
||||
\desc{Accuracy}{}{}
|
||||
\desc[german]{Genauigkeit}{}{}
|
||||
\eq{a = \frac{\tGT{::cp}}{\tGT{::fp} + \tGT{::cp}}}
|
||||
\end{formula}
|
||||
\begin{formula}{mean_abs_error}
|
||||
\desc{Mean absolute error (MAE)}{}{$y$ \GT{::y}, $\hat{y}$ \GT{::yhat}, $n$ \GT{::n_desc}}
|
||||
\desc[german]{Mittlerer absoluter Fehler (MAE)}{}{}
|
||||
\eq{\text{MAE} = \frac{1}{n} \sum_{i=1}^n \abs{y_i - \hat{y}_i}}
|
||||
\end{formula}
|
||||
\begin{formula}{mean_square_error}
|
||||
\desc{Mean squared error (MSE)}{}{$y$ \GT{::y}, $\hat{y}$ \GT{::yhat}, $n$ \GT{::n_desc}}
|
||||
\desc[german]{Methode der kleinsten Quadrate (MSE)}{Quadratwurzel des mittleren quadratischen Fehlers (SME)}{}
|
||||
\eq{\text{MSE} = \frac{1}{n} \sum_{i=1}^n \left(y_i - \hat{y}_i\right)^2}
|
||||
\end{formula}
|
||||
\begin{formula}{root_mean_square_error}
|
||||
\desc{Root mean squared error (RMSE)}{}{$y$ \GT{::y}, $\hat{y}$ \GT{::yhat}, $n$ \GT{::n_desc}}
|
||||
\desc[german]{Standardfehler der Regression}{Quadratwurzel des mittleren quadratischen Fehlers (RSME)}{}
|
||||
\eq{\text{RMSE} = \sqrt{\frac{1}{n} \sum_{i=1}^n \left(y_i - \hat{y}_i\right)^2}}
|
||||
\end{formula}
|
||||
|
||||
\Subsection{reg}
|
||||
\desc{Regression}{}{}
|
||||
\desc[german]{Regression}{}{}
|
||||
\Subsubsection{linear}
|
||||
\desc{Linear Regression}{}{}
|
||||
\desc[german]{Lineare Regression}{}{}
|
||||
\begin{formula}{eq}
|
||||
\desc{Linear regression}{Fits the data under the assumption of \fRef[normally distributed errors]{math:pt:distributions:cont:normal}}{$\mat{x}\in\R^{N\times M}$ input data, $\mat{y}\in\R^{N\times L}$ output data, $\mat{b}$ bias, $\vec{\beta}$ weights, $N$ samples, $M$ features, $L$ output variables}
|
||||
\desc[german]{Lineare Regression}{Fitted Daten unter der Annahme \fRef[normalverteilter Fehler]{math:pt:distributions:cont:normal}}{}
|
||||
\eq{\mat{y} = \mat{\epsilon} + \mat{x} \cdot \vec{\beta}}
|
||||
\end{formula}
|
||||
\begin{formula}{design_matrix}
|
||||
\desc{Design matrix}{Stack column of ones to the feature vector\\Useful when $\epsilon$ is scalar}{$x_{ij}$ feature $j$ of sample $i$}
|
||||
\desc[german]{Designmatrix Ansatz}{}{}
|
||||
\eq{
|
||||
\mat{X} = \begin{pmatrix} 1 & x_{11} & \ldots & x_{1M} \\ \vdots & \vdots & \vdots & \vdots \\ 1 & x_{N1} & \ldots & x_{NM} \end{pmatrix}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{scalar_bias}
|
||||
\desc{Linear regression with scalar bias}{Using the design matrix, the scalar is absorbed into the weight vector}{$\mat{y}$ output data, $\mat{X}$ \fRef{::design_matrix}, $\vec{\beta}$ weights}
|
||||
\desc[german]{Lineare Regression mit skalarem Bias}{Durch die Designmatrix wird der Bias in den Gewichtsvektor absorbiert}{}
|
||||
\eq{\mat{y} = \mat{X} \cdot \vec{\beta}}
|
||||
\end{formula}
|
||||
\begin{formula}{normal_equation}
|
||||
\desc{Normal equation}{Solves \fRef{comp:ml:reg:linear:scalar_bias} with \fRef{comp:ml:performance:mean_square_error}}{$\mat{y}$ output data, $\mat{X}$ \fRef{::design_matrix}, $\vec{\beta}$ weights}
|
||||
\desc[german]{Normalengleichung}{Löst \fRef{comp:ml:reg:linear:scalar_bias} mit \fRef{comp:ml:performance:mean_square_error}}{}
|
||||
\eq{\vec{\beta} = \left(\mat{X}^\T \mat{X}\right)^{-1} \mat{X}^T \mat{y}}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection{kernel}
|
||||
\desc{Kernel method}{}{}
|
||||
\desc[german]{Kernelmethode}{}{}
|
||||
\begin{formula}{kernel_trick}
|
||||
\desc{Kernel trick}{}{$\vecx_i \in \R^{M_1}$ input vectors, $M_1$ dimension of data vector space, $M_2$ dimension of feature space}
|
||||
% \desc[german]{}{}{}
|
||||
\ttxt{\eng{
|
||||
Useful when transforming the input data $x$ into a much higher dimensional space ($M_2 \gg M_1$) $\Phi: \R^{M_1} \mapsto \R^{M_2},\quad \vecx \to \Phi(\vecx)$
|
||||
and only the dot product of this transformed data $\Phi(x)^\T\Phi(x)$ is required.
|
||||
Then the dot product can be replaced by a suitable kernel function $\kappa$.
|
||||
}}
|
||||
\eq{
|
||||
k(\vecx_i,\vecx_j) \equiv \Phi(\vecx_i)^{\T} \Phi(\vecx_j)
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{common_kernels}
|
||||
\desc{Common kernels}{}{}
|
||||
% \desc[german]{}{}{}
|
||||
% \eq{}
|
||||
\ttxt{\eng{
|
||||
Linear, Polynomial, Sigmoid, Laplacian, radial basis funciton (RBF)
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{radial_basis_function}
|
||||
\abbrLabel{RBF}
|
||||
\desc{Radial basis function kernel (RBF)}{RBF = Real function of which the value only depends on the distance of the input}{}
|
||||
\desc[german]{Radiale Basisfunktion-Kernel (RBF)}{RBF = Reelle Funktion, deren Wert nur vom Abstand zum Ursprung abängt}{}
|
||||
\eq{k(\vecx_i, \vecx_j) = \Exp{-\frac{\norm{\vecx_i - \vecx_j}_2^2}{\sigma}}}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection{bayes}
|
||||
\desc{Bayesian regression}{}{}
|
||||
\desc[german]{Bayes'sche Regression}{}{}
|
||||
|
||||
\begin{formula}{linear_regression}
|
||||
\desc{Bayesian linear regression}{}{}
|
||||
\desc[german]{Bayes'sche lineare Regression}{}{}
|
||||
\ttxt{\eng{
|
||||
Assume a \fRef{math:pt:bayesian:prior} distribution over the weights.
|
||||
Offers uncertainties in addition to the predictions.
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{ridge}
|
||||
\desc{Ridge regression}{Regularization method}{}
|
||||
\desc[german]{Ridge Regression}{}{}
|
||||
\ttxt{\eng{
|
||||
Applies a L2 norm penalty on the weights.
|
||||
This ensures unimportant features are less regarded and do not encode noise.
|
||||
\\Corresponds to assuming a \fRef{math:pt:bayesian:prior} \absRef{multivariate_normal_distribution} with $\vec{\mu} = 0$ and independent components ($\mat{\Sigma}$) for the weights.
|
||||
}\ger{
|
||||
Reduziert Gewichte mit der L2-Norm.
|
||||
Dadurch werden unwichtige Features nicht berücksichtigt (kleines Gewicht) und enkodieren nicht Noise.
|
||||
\\Entspricht der Annahme einer \absRef[Normalverteilung]{multivariate_normal_distribution} mit $\vec{\mu}=0$ und unanhängingen Komponenten ($\mat{Sigma}$ diagonaol) der die Gewichte als \fRef{math:pt:bayesian:prior}.
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{ridge_weights}
|
||||
\desc{Optimal weights}{for ridge regression}{$\lambda = \frac{\sigma^2}{\xi^2}$ shrinkage parameter, $\xi$ \absRef{variance} of the gaussian \fRef{math:pt:bayesian:prior}, $\sigma$ \absRef{variance} of the gaussian likelihood of the data}
|
||||
\desc[german]{Optimale Gewichte}{für Ridge Regression}{}
|
||||
\eq{\vec{\beta} = \left(\mat{X}^\T \mat{X} + \lambda \mathcal{1} \right)^{-1} \mat{X}^\T \vecy}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{lasso}
|
||||
\desc{Lasso regression}{Least absolute shrinkage and selection operator\\Regularization method}{}
|
||||
\desc[german]{Lasso Regression}{}{}
|
||||
\ttxt{\eng{
|
||||
Applies a L1 norm penalty on the weights, which means features can be disregarded entirely.
|
||||
\\Corresponds to assuming a \absRef{laplace_distribution} for the weights as \fRef{math:pt:bayesian:prior}.
|
||||
}\ger{
|
||||
Reduziert Gewichte mit der L1-Norm.
|
||||
Unwichtige Features werden reduziert und können auch ganz vernachlässigt werden und enkodieren nicht Noise.
|
||||
\\Entspricht der Annahme einer \absRef[Laplace-Verteilung]{laplace_distribution} der die Gewichte als \fRef{math:pt:bayesian:prior}.
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
|
||||
\begin{formula}{gaussion_process_regression}
|
||||
\desc{Gaussian process regression (GPR)}{}{}
|
||||
% \desc[german]{}{}{}
|
||||
\ttxt{\eng{
|
||||
Gaussian process: A distribtuion over functions that produce jointly gaussian distribution.
|
||||
Multivariate normal distribution like \fRef{:::linear_regression}, except that $\vec{\mu}$ and $\mat{\Sigma}$ are functions.
|
||||
GPR: non-parametric Bayesion regressor, does not assume fixed functional form for the underlying data, instead, the data determines the functional shape,
|
||||
with predictions governed by the covariance structure defined by the kernel (often \abbrRef{radial_basis_function}).
|
||||
|
||||
Offers uncertainties in addition to the predictions.
|
||||
\TODO{cleanup}
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\begin{formula}{soap}
|
||||
\desc{Smooth overlap of atomic atomic positions (SOAP)}{}{}
|
||||
% \desc[german]{}{}{}
|
||||
\ttxt{\eng{
|
||||
Goal: symmetric invariance, smoothness, completeness (completeness not achieved)
|
||||
\\Gaussian smeared density expanded in \abbrRef{radial_basis_function} and spherical harmonics.
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{gaussian_approximation_potential}
|
||||
\desc{Gaussian approximation potential}{Bond-order potential}{$V_\text{rep/attr}$ repulsive / attractive potential}
|
||||
% \desc[german]{}{}{}
|
||||
\ttxt{\eng{
|
||||
Models atomic interactions via a \textit{bond-order} term $b$.
|
||||
}}
|
||||
\eq{V_\text{BondOrder}(\vecR_M, \vecR_N) = V_\text{rep}(\vecR_M, \vecR_N) + b_{MNK} V_\text{attr}(\vecR_M, \vecR_N)}
|
||||
\end{formula}
|
||||
|
||||
\Subsection{gd}
|
||||
\desc{Gradient descent}{}{}
|
||||
\desc[german]{Gradientenverfahren}{}{}
|
||||
\TODO{in lecture 30 CMP}
|
||||
|
||||
@ -1,34 +0,0 @@
|
||||
\Section{qmb}
|
||||
\desc{Quantum many-body physics}{}{}
|
||||
\desc[german]{Quanten-Vielteilchenphysik}{}{}
|
||||
|
||||
\Subsection{models}
|
||||
\desc{Quantum many-body models}{}{}
|
||||
\desc[german]{Quanten-Vielteilchenmodelle}{}{}
|
||||
\begin{formula}{heg}
|
||||
\desc{Homogeneous electron gas (HEG)}{Also "Jellium"}{}
|
||||
% \desc[german]{}{}{}
|
||||
\ttxt{
|
||||
\eng{Both positive (nucleus) and negative (electron) charges are distributed uniformly.}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\Subsection{methods}
|
||||
\desc{Methods}{}{}
|
||||
\desc[german]{Methoden}{}{}
|
||||
\Subsubsection{qmonte-carlo}
|
||||
\desc{Quantum Monte-Carlo}{}{}
|
||||
\desc[german]{Quantum Monte-Carlo}{}{}
|
||||
|
||||
|
||||
\TODO{TODO}
|
||||
\Subsection{importance_sampling}
|
||||
\desc{Importance sampling}{}{}
|
||||
\desc[german]{Importance sampling / Stichprobenentnahme nach Wichtigkeit}{}{}
|
||||
\TODO{Monte Carlo}
|
||||
|
||||
\Subsection{mps}
|
||||
\desc{Matrix product states}{}{}
|
||||
\desc[german]{Matrix Produktzustände}{}{}
|
||||
|
||||
|
||||
397
src/condensed_matter.tex
Normal file
397
src/condensed_matter.tex
Normal file
@ -0,0 +1,397 @@
|
||||
\Part[
|
||||
\eng{Condensed matter physics}
|
||||
\ger{Festkörperphysik}
|
||||
]{cm}
|
||||
\Section[
|
||||
\eng{Bravais lattice}
|
||||
\ger{Bravais-Gitter}
|
||||
]{bravais}
|
||||
|
||||
% \begin{ttext}
|
||||
% \eng{
|
||||
|
||||
% }
|
||||
% \ger{
|
||||
|
||||
% }
|
||||
% \end{ttext}
|
||||
|
||||
\eng[bravais_table2]{In 2D, there are 5 different Bravais lattices}
|
||||
\ger[bravais_table2]{In 2D gibt es 5 verschiedene Bravais-Gitter}
|
||||
|
||||
\eng[bravais_table3]{In 3D, there are 14 different Bravais lattices}
|
||||
\ger[bravais_table3]{In 3D gibt es 14 verschiedene Bravais-Gitter}
|
||||
|
||||
\Eng[lattice_system]{Lattice system}
|
||||
\Ger[lattice_system]{Gittersystem}
|
||||
\Eng[crystal_family]{Crystal system}
|
||||
\Ger[crystal_family]{Kristall-system}
|
||||
\Eng[point_group]{Point group}
|
||||
\Ger[point_group]{Punktgruppe}
|
||||
\eng[bravais_lattices]{Bravais lattices}
|
||||
\ger[bravais_lattices]{Bravais Gitter}
|
||||
|
||||
\newcommand\bvimg[1]{\begin{center}\includegraphics[width=0.1\textwidth]{img/bravais/#1.pdf}\end{center}}
|
||||
\renewcommand\tabularxcolumn[1]{m{#1}}
|
||||
\newcolumntype{Z}{>{\centering\let\newline\\\arraybackslash\hspace{0pt}}X}
|
||||
\begin{table}[H]
|
||||
\centering
|
||||
\caption{\gt{bravais_table2}}
|
||||
\label{tab:bravais2}
|
||||
|
||||
\begin{adjustbox}{width=\textwidth}
|
||||
\begin{tabularx}{\textwidth}{||Z|c|Z|Z||}
|
||||
\hline
|
||||
\multirow{2}{*}{\GT{lattice_system}} & \multirow{2}{*}{\GT{point_group}} & \multicolumn{2}{c||}{5 \gt{bravais_lattices}} \\ \cline{3-4}
|
||||
& & \GT{primitive} (p) & \GT{centered} (c) \\ \hline
|
||||
\GT{monoclinic} (m) & $\text{C}_\text{2}$ & \bvimg{mp} & \\ \hline
|
||||
\GT{orthorhombic} (o) & $\text{D}_\text{2}$ & \bvimg{op} & \bvimg{oc} \\ \hline
|
||||
\GT{tetragonal} (t) & $\text{D}_\text{4}$ & \bvimg{tp} & \\ \hline
|
||||
\GT{hexagonal} (h) & $\text{D}_\text{6}$ & \bvimg{hp} & \\ \hline
|
||||
\end{tabularx}
|
||||
\end{adjustbox}
|
||||
\end{table}
|
||||
|
||||
|
||||
|
||||
\begin{table}[H]
|
||||
\centering
|
||||
\caption{\gt{bravais_table3}}
|
||||
\label{tab:bravais3}
|
||||
|
||||
% \newcolumntype{g}{>{\columncolor[]{0.8}}}
|
||||
\begin{adjustbox}{width=\textwidth}
|
||||
% \begin{tabularx}{\textwidth}{|c|}
|
||||
% asdfasdfadslfasdfaasdofiuapsdoifuapodisufpaoidsufpaoidsufpaoisdfaoisdfpaosidfupaoidsufpaoidsufpaoidsufpaoisdufpaoidsufpoaiudsfpioaspdoifuaposidufpaoisudpfoiaupsdoifupasodf \\
|
||||
% asdfasdfadslfasdfaasdofiuapsdoifuapodisufpaoidsufpaoidsufpaoisdfaoisdfpaosidfupaoidsufpaoidsufpaoidsufpaoisdufpaoidsufpoaiudsfpioaspdoifuaposidufpaoisudpfoiaupsdoifupasodf \\
|
||||
% \end{tabularx}
|
||||
% \begin{tabular}{|c|}
|
||||
% asdfasdfadslfasdfaasdofiuapsdoifuapodisufpaoidsufpaoidsufpaoisdfaoisdfpaosidfupaoidsufpaoidsufpaoidsufpaoisdufpaoidsufpoaiudsfpioaspdoifuaposidufpaoisudpfoiaupsdoifupasodf \\
|
||||
% asdfasdfadslfasdfaasdofiuapsdoifuapodisufpaoidsufpaoidsufpaoisdfaoisdfpaosidfupaoidsufpaoidsufpaoidsufpaoisdufpaoidsufpoaiudsfpioaspdoifuaposidufpaoisudpfoiaupsdoifupasodf \\
|
||||
% \end{tabular}
|
||||
% \\
|
||||
\begin{tabularx}{\textwidth}{||Z|Z|c|Z|Z|Z|Z||}
|
||||
\hline
|
||||
\multirow{2}{*}{\GT{crystal_family}} & \multirow{2}{*}{\GT{lattice_system}} & \multirow{2}{*}{\GT{point_group}} & \multicolumn{4}{c||}{14 \gt{bravais_lattices}} \\ \cline{4-7}
|
||||
& & & \GT{primitive} (P) & \GT{base_centered} (S) & \GT{body_centered} (I) & \GT{face_centered} (F) \\ \hline
|
||||
\multicolumn{2}{||c|}{\GT{triclinic} (a)} & $\text{C}_\text{i}$ & \bvimg{tP} & & & \\ \hline
|
||||
\multicolumn{2}{||c|}{\GT{monoclinic} (m)} & $\text{C}_\text{2h}$ & \bvimg{mP} & \bvimg{mS} & & \\ \hline
|
||||
\multicolumn{2}{||c|}{\GT{orthorhombic} (o)} & $\text{D}_\text{2h}$ & \bvimg{oP} & \bvimg{oS} & \bvimg{oI} & \bvimg{oF} \\ \hline
|
||||
\multicolumn{2}{||c|}{\GT{tetragonal} (t)} & $\text{D}_\text{4h}$ & \bvimg{tP} & & \bvimg{tI} & \\ \hline
|
||||
\multirow{2}{*}{\GT{hexagonal} (h)} & \GT{rhombohedral} & $\text{D}_\text{3d}$ & \bvimg{hR} & & & \\ \cline{2-7}
|
||||
& \GT{hexagonal} & $\text{D}_\text{6h}$ & \bvimg{hP} & & & \\ \hline
|
||||
\multicolumn{2}{||c|}{\GT{cubic} (c)} & $\text{O}_\text{h}$ & \bvimg{cP} & & \bvimg{cI} & \bvimg{cF} \\ \hline
|
||||
\end{tabularx}
|
||||
\end{adjustbox}
|
||||
\end{table}
|
||||
|
||||
|
||||
\Section[
|
||||
\eng{Reciprocal lattice}
|
||||
\ger{Reziprokes Gitter}
|
||||
]{reci}
|
||||
\begin{ttext}
|
||||
\eng{The reciprokal lattice is made up of all the wave vectors $\vec{k}$ that ressemble standing waves with the periodicity of the Bravais lattice.}
|
||||
\ger{Das rezioproke Gitter besteht aus dem dem Satz aller Wellenvektoren $\vec{k}$, die ebene Wellen mit der Periodizität des Bravais-Gitters ergeben.}
|
||||
\end{ttext}
|
||||
|
||||
\begin{formula}{vectors}
|
||||
\desc{Reciprocal lattice vectors}{}{$a_i$ real-space lattice vectors, $V_c$ volume of the primitive lattice cell}
|
||||
\desc[german]{Reziproke Gittervektoren}{}{$a_i$ Bravais-Gitter Vektoren, $V_c$ Volumen der primitiven Gitterzelle}
|
||||
\eq{
|
||||
\vec{b_1} &= \frac{2\pi}{V_c} \vec{a_2} \times \vec{a_3} \\
|
||||
\vec{b_2} &= \frac{2\pi}{V_c} \vec{a_3} \times \vec{a_1} \\
|
||||
\vec{b_3} &= \frac{2\pi}{V_c} \vec{a_1} \times \vec{a_2}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\Subsection[
|
||||
\eng{Scattering processes}
|
||||
\ger{Streuprozesse}
|
||||
]{scatter}
|
||||
\begin{formula}{matthiessen}
|
||||
\desc{Matthiessen's rule}{Approximation, only holds if the processes are independent of each other}{$\mu$ mobility, $\tau$ scattering time}
|
||||
\desc[german]{Matthiessensche Regel}{Näherung, nur gültig wenn die einzelnen Streuprozesse von einander unabhängig sind}{$\mu$ Moblitiät, $\tau$ Streuzeit}
|
||||
\eq{
|
||||
\frac{1}{\mu} &= \sum_{i = \textrm{\GT{\fqname}}} \frac{1}{\mu_i} \\
|
||||
\frac{1}{\tau} &= \sum_{i = \textrm{\GT{\fqname}}} \frac{1}{\tau_i}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Section[
|
||||
\eng{Free electron gas}
|
||||
\ger{Freies Elektronengase}
|
||||
]{free_e_gas}
|
||||
\begin{ttext}
|
||||
\eng{Assumptions: electrons can move freely and independent of each other.}
|
||||
\ger{Annahmen: Elektronen bewegen sich frei und unabhänig voneinander.}
|
||||
\end{ttext}
|
||||
|
||||
\begin{formula}{drift_velocity}
|
||||
\desc{Drift velocity}{Velocity component induced by an external force (eg. electric field)}{$v_\text{th}$ thermal velocity}
|
||||
\desc[german]{Driftgeschwindgkeit}{Geschwindigkeitskomponente durch eine externe Kraft (z.B. ein elektrisches Feld)}{$v_\text{th}$ thermische Geschwindigkeit}
|
||||
\eq{\vec{v}_\text{D} = \vec{v} - \vec{v}_\text{th}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{mean_free_time}
|
||||
\desc{Mean free time}{}{}
|
||||
\desc[german]{Streuzeit}{}{}
|
||||
\eq{\tau}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{mean_free_path}
|
||||
\desc{Mean free path}{}{}
|
||||
\desc[german]{Mittlere freie Weglänge}{}{}
|
||||
\eq{\ell = \braket{v} \tau}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{mobility}
|
||||
\desc{Electrical mobility}{}{$q$ charge, $m$ mass}
|
||||
\desc[german]{Beweglichkeit}{}{$q$ Ladung, $m$ Masse}
|
||||
\eq{\mu = \frac{q \tau}{m}}
|
||||
\end{formula}
|
||||
|
||||
\Subsection[
|
||||
\eng{Drude model}
|
||||
\ger{Drude-Modell}
|
||||
]{drude}
|
||||
\begin{ttext}
|
||||
\eng{Classical model describing the transport properties of electrons in materials (metals):
|
||||
The material is assumed to be an ion lattice and with freely moving electrons (electron gas). The electrons are
|
||||
accelerated by an electric field and decelerated through collisions with the lattice ions.
|
||||
The model disregards the Fermi-Dirac partition of the conducting electrons.
|
||||
}
|
||||
\ger{Ein klassisches Model zur Beschreibung der Transporteigenschaften von Elektronen in (v.a.) Metallen:
|
||||
Der Festkörper wird als Ionenkristall mit frei beweglichen Elektronen (Elektronengas).
|
||||
Die Elektronen werden durch ein Elektrisches Feld $E$ beschleunigt und durch Stöße mit den Gitterionen gebremst.
|
||||
Das Modell vernachlässigt die Fermi-Dirac Verteilung der Leitungselektronen.
|
||||
}
|
||||
\end{ttext}
|
||||
\begin{formula}{motion}
|
||||
\desc{Equation of motion}{}{$v$ electron speed, $\vec{v}_\text{D}$ drift velocity, $\tau$ mean free time between collisions}
|
||||
\desc[german]{Bewegungsgleichung}{}{$v$ Elektronengeschwindigkeit, $\vec{v}_\text{D}$ Driftgeschwindigkeit, $\tau$ Stoßzeit}
|
||||
\eq{\masse \odv{\vec{v}}{t} + \frac{\masse}{\tau} \vec{v}_\text{D} = -e \vec{E}}
|
||||
\end{formula}
|
||||
\begin{formula}{current_density}
|
||||
\desc{Current density}{Ohm's law}{$n$ charge particle density}
|
||||
\desc[german]{Stromdichte}{Ohmsches Gesetz}{$n$ Ladungsträgerdichte}
|
||||
\eq{\vec{j} = -ne\vec{v}_\text{D} = ne\mu \vec{E}}
|
||||
\end{formula}
|
||||
\begin{formula}{conductivity}
|
||||
\desc{Drude-conductivity}{}{}
|
||||
\desc[german]{Drude-Leitfähigkeit}{}{}
|
||||
\eq{\sigma = \frac{\vec{j}}{\vec{E}} = \frac{e^2 \tau n}{\masse} = n e \mu}
|
||||
\end{formula}
|
||||
|
||||
\Subsection[
|
||||
\eng{Sommerfeld model}
|
||||
\ger{Sommerfeld-Modell}
|
||||
]{sommerfeld}
|
||||
\begin{ttext}
|
||||
\eng{Assumes a gas of free fermions underlying the pauli-exclusion principle. Only electrons in an energy range of $\kB T$ around the Fermi energy $\EFermi$ participate in scattering processes.}
|
||||
\ger{Annahme eines freien Fermionengases, welches dem Pauli-Prinzip unterliegt. Nur Elektronen in einem Energiebereich von $\kB T$ um die Fermi Energe $\EFermi$ nehmen an Streuprozessen teil.}
|
||||
\end{ttext}
|
||||
\begin{formula}{current_density}
|
||||
\desc{Current density}{}{}
|
||||
\desc[german]{Stromdichte}{}{}
|
||||
\eq{\vec{j} = -en\braket{v} = -e n \frac{\hbar}{\masse}\braket{\vec{k}} = -e \frac{1}{V} \sum_{\vec{k},\sigma} \frac{\hbar \vec{k}}{\masse}}
|
||||
\end{formula}
|
||||
\TODO{The formula for the conductivity is the same as in the drude model?}
|
||||
|
||||
\Subsection[
|
||||
\eng{2D electron gas}
|
||||
\ger{2D Elektronengas}
|
||||
]{2deg}
|
||||
|
||||
\begin{ttext}
|
||||
\eng{Lower dimension gases can be obtained by restricting a 3D gas with infinetly high potential walls on a narrow area with the width $L$.}
|
||||
\ger{
|
||||
Niederdimensionale Elektronengase erhält man, wenn ein 3D Gas durch unendlich hohe Potentialwände auf einem schmalen Bereich mit Breite $L$ eingeschränkt wird.
|
||||
}
|
||||
\end{ttext}
|
||||
\begin{formula}{confinement_energy}
|
||||
\desc{Confinement energy}{Raises ground state energy}{}
|
||||
\desc[german]{Confinement Energie}{Erhöht die Grundzustandsenergie}{}
|
||||
\eq{\Delta E = \frac{\hbar^2 \pi^2}{2\masse L^2}}
|
||||
\end{formula}
|
||||
|
||||
\Eng[plain_wave]{plain wave}
|
||||
\Ger[plain_wave]{ebene Welle}
|
||||
\begin{formula}{energy}
|
||||
\desc{Energy}{}{}
|
||||
\desc[german]{Energie}{}{}
|
||||
\eq{E_n = \underbrace{\frac{\hbar^2 k_\parallel^2}{2\masse}}_\text{$x$-$y$: \GT{plain_wave}} + \underbrace{\frac{\hbar^2 \pi^2}{2\masse L^2} n^2}_\text{$z$}}
|
||||
\end{formula}
|
||||
|
||||
\Subsection[
|
||||
\eng{1D electron gas / quantum wire}
|
||||
\ger{1D Eleltronengas / Quantendraht}
|
||||
]{1deg}
|
||||
|
||||
\begin{formula}{energy}
|
||||
\desc{Energy}{}{}
|
||||
\desc[german]{Energie}{}{}
|
||||
\eq{E_n = \frac{\hbar^2 k_x^2}{2\masse} + \frac{\hbar^2 \pi^2}{2\masse L_z^2} n_1^2 + \frac{\hbar^2 \pi^2}{2\masse L_y^2} n_2^2}
|
||||
\end{formula}
|
||||
|
||||
\Subsection[
|
||||
\eng{0D electron gas / quantum dot}
|
||||
\ger{0D Elektronengase / Quantenpunkt}
|
||||
]{0deg}
|
||||
|
||||
\TODO{TODO}
|
||||
|
||||
\Section[
|
||||
\eng{Measurement techniques}
|
||||
\ger{Messtechniken}
|
||||
]{meas}
|
||||
\Subsection[
|
||||
\eng{ARPES}
|
||||
\ger{ARPES}
|
||||
]{arpes}
|
||||
what?
|
||||
in?
|
||||
how?
|
||||
plot
|
||||
|
||||
\Subsection[
|
||||
\eng{Scanning probe microscopy SPM}
|
||||
\ger{Rastersondenmikroskopie (SPM)}
|
||||
]{spm}
|
||||
\begin{ttext}
|
||||
\eng{Images of surfaces are taken by scanning the specimen with a physical probe.}
|
||||
\ger{Bilder der Oberfläche einer Probe werden erstellt, indem die Probe mit einer Sonde abgetastet wird.}
|
||||
\end{ttext}
|
||||
|
||||
\Eng[name]{Name}
|
||||
\Ger[name]{Name}
|
||||
\Eng[application]{Application}
|
||||
\Ger[application]{Anwendung}
|
||||
|
||||
|
||||
\begin{minipagetable}{amf}
|
||||
\entry{name}{
|
||||
\eng{Atomic force microscopy (AMF)}
|
||||
\ger{Atomare Rasterkraftmikroskopie (AMF)}
|
||||
}
|
||||
\entry{application}{
|
||||
\eng{Surface stuff}
|
||||
\ger{Oberflächenzeug}
|
||||
}
|
||||
\entry{how}{
|
||||
\eng{With needle}
|
||||
\ger{Mit Nadel}
|
||||
}
|
||||
\end{minipagetable}
|
||||
\begin{minipage}{0.5\textwidth}
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.8\textwidth]{img/cm_amf.pdf}
|
||||
\caption{\cite{Bian2021}}
|
||||
\end{figure}
|
||||
\end{minipage}
|
||||
|
||||
|
||||
\begin{minipagetable}{stm}
|
||||
\entry{name}{
|
||||
\eng{Scanning tunneling microscopy (STM)}
|
||||
\ger{Rastertunnelmikroskop (STM)}
|
||||
}
|
||||
\entry{application}{
|
||||
\eng{Surface stuff}
|
||||
\ger{Oberflächenzeug}
|
||||
}
|
||||
\entry{how}{
|
||||
\eng{With TUnnel}
|
||||
\ger{Mit TUnnel}
|
||||
}
|
||||
\end{minipagetable}
|
||||
\begin{minipage}{0.5\textwidth}
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.8\textwidth]{img/cm_stm.pdf}
|
||||
\caption{\cite{Bian2021}}
|
||||
\end{figure}
|
||||
\end{minipage}
|
||||
|
||||
\Section[
|
||||
\eng{Fabrication techniques}
|
||||
\ger{Herstellungsmethoden}
|
||||
]{fab}
|
||||
\begin{minipagetable}{cvd}
|
||||
\entry{name}{
|
||||
\eng{Chemical vapor deposition (CVD)}
|
||||
\ger{Chemische Gasphasenabscheidung (CVD)}
|
||||
}
|
||||
\entry{how}{
|
||||
\eng{
|
||||
A substrate is exposed to volatile precursors, which react and/or decompose on the heated substrate surface to produce the desired deposit.
|
||||
By-products are removed by gas flow through the chamber.
|
||||
}
|
||||
\ger{
|
||||
An der erhitzten Oberfläche eines Substrates wird aufgrund einer chemischen Reaktion mit einem Gas eine Feststoffkomponente abgeschieden.
|
||||
Nebenprodukte werden durch den Gasfluss durch die Kammer entfernt.
|
||||
}
|
||||
}
|
||||
\entry{application}{
|
||||
\eng{
|
||||
\begin{itemize}
|
||||
\item Polysilicon \ce{Si}
|
||||
\item Silicon dioxide \ce{SiO_2}
|
||||
\item Graphene
|
||||
\item Diamond
|
||||
\end{itemize}
|
||||
}
|
||||
\ger{
|
||||
\begin{itemize}
|
||||
\item Poly-silicon \ce{Si}
|
||||
\item Siliziumdioxid \ce{SiO_2}
|
||||
\item Graphen
|
||||
\item Diamant
|
||||
\end{itemize}
|
||||
}
|
||||
}
|
||||
\end{minipagetable}
|
||||
\begin{minipage}{0.5\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{img/cm_cvd_english.pdf}
|
||||
\end{minipage}
|
||||
|
||||
|
||||
\Subsection[
|
||||
\eng{Epitaxy}
|
||||
\ger{Epitaxie}
|
||||
]{epitaxy}
|
||||
\begin{ttext}
|
||||
\eng{A type of crystal groth in which new layers are formed with well-defined orientations with respect to the crystalline seed layer.}
|
||||
\ger{Eine Art des Kristallwachstums, bei der mindestens eine kristallographische Ordnung der wachsenden Schicht der des Substrates entspricht.}
|
||||
\end{ttext}
|
||||
|
||||
\begin{minipagetable}{mbe}
|
||||
\entry{name}{
|
||||
\eng{Molecular Beam Epitaxy (MBE)}
|
||||
\ger{Molekularstrahlepitaxie (MBE)}
|
||||
}
|
||||
\entry{how}{
|
||||
\eng{In a ultra-high vacuum, the elements are heated until they slowly sublime. The gases then condensate on the substrate surface}
|
||||
\ger{Die Elemente werden in einem Ultrahochvakuum erhitzt, bis sie langsam sublimieren. Die entstandenen Gase kondensieren dann auf der Oberfläche des Substrats}
|
||||
}
|
||||
\entry{application}{
|
||||
\eng{
|
||||
\begin{itemize}
|
||||
\item Gallium arsenide \ce{GaAs}
|
||||
\end{itemize}
|
||||
\TODO{Link to GaAs}
|
||||
}
|
||||
\ger{
|
||||
\begin{itemize}
|
||||
\item Galliumarsenid \ce{GaAs}
|
||||
\end{itemize}
|
||||
}
|
||||
}
|
||||
\end{minipagetable}
|
||||
\begin{minipage}{0.5\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{img/cm_mbe_english.pdf}
|
||||
\end{minipage}
|
||||
@ -1,71 +0,0 @@
|
||||
\Section{constants}
|
||||
\desc{Constants}{}{}
|
||||
\desc[german]{Konstanten}{}{}
|
||||
|
||||
\begin{formula}{planck}
|
||||
\desc{Planck Constant}{}{}
|
||||
\desc[german]{Plancksches Wirkumsquantum}{}{}
|
||||
\constant{h}{def}{
|
||||
\val{6.62607015\cdot 10^{-34}}{\joule\s}
|
||||
\val{4.135667969\dots\xE{-15}}{\eV\s}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{universal_gas}
|
||||
\desc{Universal gas constant}{Proportionality factor for ideal gases}{\ConstRef{avogadro}, \ConstRef{boltzmann}}
|
||||
\desc[german]{Universelle Gaskonstante}{Proportionalitätskonstante für ideale Gase}{}
|
||||
\constant{R}{def}{
|
||||
\val{8.31446261815324}{\joule\per\mol\kelvin}
|
||||
\val{\NA \cdot \kB}{}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{avogadro}
|
||||
\desc{Avogadro constant}{Number of molecules per mole}{}
|
||||
\desc[german]{Avogadro-Konstante}{Anzahl der Moleküle pro mol}{}
|
||||
\constant{\NA}{def}{
|
||||
\val{6.02214076 \xE{23}}{1\per\mole}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{boltzmann}
|
||||
\desc{Boltzmann constant}{Temperature-Energy conversion factor}{}
|
||||
\desc[german]{Boltzmann-Konstante}{Temperatur-Energie Umrechnungsfaktor}{}
|
||||
\constant{\kB}{def}{
|
||||
\val{1.380649 \xE{-23}}{\joule\per\kelvin}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{faraday}
|
||||
\desc{Faraday constant}{Electric charge of one mol of single-charged ions}{\ConstRef{avogadro}, \ConstRef{charge}}
|
||||
\desc[german]{Faraday-Konstante}{Elektrische Ladungs von einem Mol einfach geladener Ionen}{}
|
||||
\constant{F}{def}{
|
||||
\val{9.64853321233100184\xE{4}}{\coulomb\per\mol}
|
||||
\val{\NA\,e}{}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{charge}
|
||||
\desc{Unit charge}{}{}
|
||||
\desc[german]{Elementarladung}{}{}
|
||||
\constant{e}{def}{
|
||||
\val{1.602176634\xE{-19}}{\coulomb}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{flux_quantum}
|
||||
\desc{Flux quantum}{}{}
|
||||
\desc[german]{Flussquantum}{}{}
|
||||
\constant{\Phi_0}{def}{
|
||||
\val{2.067 833 848 \xE{-15}}{\weber=\volt\s=\kg\m^2\per\s^2\ampere}
|
||||
}
|
||||
\eq{\Phi_0 = \frac{h}{2e}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{atomic_mass_unit}
|
||||
\desc{Atomic mass unit}{}{}
|
||||
\desc[german]{Atomare Massneinheit}{}{}
|
||||
\constant{u}{exp}{
|
||||
\val{1.66053906892(52)\xE{-27}}{\kg}
|
||||
}
|
||||
\end{formula}
|
||||
@ -1,8 +0,0 @@
|
||||
\Part{ed}
|
||||
\desc{Electrodynamics}{}{}
|
||||
\desc[german]{Elektrodynamik}{}{}
|
||||
|
||||
|
||||
% pure electronic stuff in el
|
||||
% pure magnetic stuff in mag
|
||||
% electromagnetic stuff in em
|
||||
@ -1,81 +0,0 @@
|
||||
|
||||
\Section{el}
|
||||
\desc{Electric field}{}{}
|
||||
\desc[german]{Elektrisches Feld}{}{}
|
||||
\begin{formula}{electric_field}
|
||||
\desc{Electric field}{Surrounds charged particles}{}
|
||||
\desc[german]{Elektrisches Feld}{Umgibt geladene Teilchen}{}
|
||||
\quantity{\vec{\E}}{\volt\per\m=\kg\m\per\s^3\ampere}{v}
|
||||
\end{formula}
|
||||
|
||||
\def\Epotential{\phi}
|
||||
\begin{formula}{electric_scalar_potential}
|
||||
\desc{Electric potential}{Work required to move a unit of charge between two points}{}
|
||||
\desc[german]{Elektrisches Potential}{Benötigte Arbeit um eine Einheitsladung zwischen zwei Punkten zu bewegen}{}
|
||||
\quantity{\Epotential}{\volt=\kg\m^2\per\s^3\ampere}{s}
|
||||
\eq{\Epotential = -\int \vec{\E} \cdot\d\vecr}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{gauss_law}
|
||||
\desc{Gauss's law for electric fields}{Electric flux through a closed surface is proportional to the electric charge}{$S$ closed surface}
|
||||
\desc[german]{Gaußsches Gesetz für elektrische Felder}{Der magnetische Fluss durch eine geschlossene Fläche ist proportional zur elektrischen Ladung}{$S$ geschlossene Fläche}
|
||||
\eq{\PhiE = \iint_S \vec{\E}\cdot\d\vec{S} = \frac{Q}{\varepsilon_0}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{permittivity}
|
||||
\desc{Permittivity}{Dieletric function\\Electric polarizability of a dielectric material}{}
|
||||
\desc[german]{Permitivität}{Dielektrische Konstante / Dielektrische Funktion\\Elektrische Polarisierbarkeit eines dielektrischen Materials}{}
|
||||
\quantity{\epsilon}{\ampere\s\per\volt\m=\farad\per\m=\coulomb\per\volt\m=C^2\per\newton\m^2=\ampere^2\s^4\per\kg\m^3}{}
|
||||
\end{formula}
|
||||
\begin{formula}{relative_permittivity}
|
||||
\desc{Relative permittivity}{Dielectric constant}{\QtyRef{permittivity}, \ConstRef{vacuum_permittivity}}
|
||||
\desc[german]{Relative Permittivität}{Dielectric constant}{}
|
||||
\eq{
|
||||
\epsilon(\omega)_\txr = \frac{\epsilon(\omega)}{\epsilon_0}
|
||||
}
|
||||
\hiddenQuantity{\epsilon_\txr}{}{s}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{vacuum_permittivity}
|
||||
\desc{Vacuum permittivity}{Electric constant}{}
|
||||
\desc[german]{Vakuum Permittivität}{Elektrische Feldkonstante}{}
|
||||
\constant{\epsilon_0}{exp}{
|
||||
\val{8.8541878188(14)\xE{-1}}{\ampere\s\per\volt\m}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{electric_susceptibility}
|
||||
\desc{Electric susceptibility}{Describes how polarized a dielectric material becomes when an electric field is applied}{$\epsilon_\txr$ \fRef{ed:el:relative_permittivity}}
|
||||
\desc[german]{Elektrische Suszeptibilität}{Beschreibt wie stark ein dielektrisches Material polarisiert wird, wenn ein elektrisches Feld angelegt wird}{}
|
||||
\quantity{\chi_\txe}{}{s}
|
||||
\eq{
|
||||
\epsilon_\txr = 1 + \chi_\txe
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{dielectric_polarization_density}
|
||||
\desc{Dielectric polarization density}{}{\QtyRef{dipole_moment}, \QtyRef{volume}, \ConstRef{vacuum_permittivity}, \QtyRef{electric_susceptibility}, \QtyRef{electric_field}}
|
||||
\desc[german]{Dielektrische Polarisationsdichte}{}{}
|
||||
\quantity{\vec{P}}{\coulomb\per\m^2}{v}
|
||||
\eq{\vec{P} = \frac{\delta\vecp}{\delta V}\epsilon_0 \chi_\txe \vec{\E}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{electric_displacement_field}
|
||||
\desc{Electric displacement field}{}{\ConstRef{vacuum_permittivity}, \QtyRef{electric_field}, \QtyRef{dielectric_polarization_density}}
|
||||
\desc[german]{Elektrische Flussdichte}{Dielektrische Verschiebung}{}
|
||||
\quantity{\vec{D}}{\coulomb\per\m^2=\ampere\s\per\m^2}{v}
|
||||
\eq{\vec{D} = \epsilon_0 \vec{\E} + \vec{P}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{electric_flux}
|
||||
\desc{Electric flux}{through area $\vec{A}$}{\QtyRef{electric_displacement_field}}
|
||||
\desc[german]{Elektrischer Fluss}{durch die Fläche $\vec{A}$}{}
|
||||
\eq{\Phi_\txE = \int_A \vec{D}\cdot \d \vec{A}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{power}
|
||||
\desc{Electric power}{}{$U$ \qtyRef{electric_scalar_potential}, \QtyRef{current}}
|
||||
\desc[german]{Elektrische Leistung}{}{}
|
||||
\eq{P_\text{el} = U\,I}
|
||||
\end{formula}
|
||||
|
||||
|
||||
106
src/ed/em.tex
106
src/ed/em.tex
@ -1,106 +0,0 @@
|
||||
\Section{em}
|
||||
\desc{Electromagnetism}{}{}
|
||||
\desc[german]{Elektromagnetismus}{}{}
|
||||
|
||||
\begin{formula}{vacuum_speed_of_light}
|
||||
\desc{Speed of light}{in the vacuum}{}
|
||||
\desc[german]{Lightgeschwindigkeit}{in the vacuum}{}
|
||||
\constant{c}{exp}{
|
||||
\val{299792458}{\m\per\s}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{vacuum_relations}
|
||||
\desc{Vacuum permittivity - permeability relation}{\TODO{Does this have a name?}}{\ConstRef{vacuum_permittivity}, \ConstRef{magnetic_vacuum_permeability}, \ConstRef{vacuum_speed_of_light}}
|
||||
\desc[german]{Vakuum Permittivität - Permeabilität Beziehung}{}{}
|
||||
\eq{
|
||||
\epsilon_0 \mu_0 = \frac{1}{c^2}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{poisson_equation}
|
||||
\absLabel
|
||||
\desc{Poisson equation for electrostatics}{}{\QtyRef{charge_density}, \QtyRef{permittivity}, $\Phi$ \qtyRef{electric_scalar_potential}, $\laplace$ \absRef{laplace_operator}}
|
||||
\desc[german]{Poisson Gleichung in der Elektrostatik}{}{}
|
||||
\eq{\laplace \Phi(\vecr) = -\frac{\rho(\vecr)}{\epsilon}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{poynting}
|
||||
\desc{Poynting vector}{Directional energy flux or power flow of an electromagnetic field}{\QtyRef{electric_field}, \QtyRef{magnetic_field_intensity}}
|
||||
\desc[german]{Poynting-Vektor}{Gerichteter Energiefluss oder Leistungsfluss eines elektromgnetischen Feldes [$\si{\W\per\m^2}$]}{}
|
||||
\quantity{\vecS}{\W\per\m^2}{v}
|
||||
\eq{\vec{S} = \vec{\E} \times \vec{H}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{electric_field}
|
||||
\desc{Electric field}{}{\QtyRef{electric_field}, \QtyRef{electric_scalar_potential}, \QtyRef{magnetic_vector_potential}}
|
||||
\desc[german]{Elektrisches Feld}{}{}
|
||||
\eq{\vec{\E} = -\Grad\Epotential - \pdv{\vec{A}}{t}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{hamiltonian}
|
||||
\desc{Hamiltonian of a particle in an electromagnetic field}{In the \fRef{ed:em:maxwell:gauge:coulomb}}{\QtyRef{mass}, $\hat{p}$ \fRef{qm:se:momentum_operator}, \QtyRef{charge}, \QtyRef{magnetic_vector_potential}, \ConstRef{vacuum_speed_of_light}}
|
||||
\desc[german]{Hamiltonian eines Teilchens im elektromagnetischen Feld}{In der \fRef{ed:em:maxwell:gauge:coulomb}}{}
|
||||
\eq{
|
||||
\hat{H} = \frac{1}{2m} \left[\hat{p} \ \frac{e \vec{A}}{c}\right]^2
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsection{maxwell}
|
||||
\desc{Maxwell-Equations}{}{}
|
||||
\desc[german]{Maxwell-Gleichungen}{}{}
|
||||
\begin{formula}{vacuum}
|
||||
\desc{Vacuum}{microscopic formulation}{}
|
||||
\desc[german]{Vakuum}{Mikroskopische Formulierung}{}
|
||||
\eq{
|
||||
\Div \vec{\E} &= \frac{\rho_\text{el}}{\epsilon_0} \\
|
||||
\Div \vec{B} &= 0 \\
|
||||
\Rot \vec{\E} &= - \odv{\vec{B}}{t} \\
|
||||
\Rot \vec{B} &= \mu_0 \vec{j} + \frac{1}{c^2} \odv{\vec{\E}}{t}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{material}
|
||||
\desc{Matter}{Macroscopic formulation}{}
|
||||
\desc[german]{Materie}{Makroskopische Formulierung}{}
|
||||
\eq{
|
||||
\Div \vec{D} &= \rho_\text{el} \\
|
||||
\Div \vec{B} &= 0 \\
|
||||
\Rot \vec{\E} &= - \odv{\vec{B}}{t} \\
|
||||
\Rot \vec{H} &= \vec{j} + \odv{\vec{D}}{t}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsubsection{gauge}
|
||||
\desc{Gauges}{}{}
|
||||
\desc[german]{Eichungen}{}{}
|
||||
\begin{formula}{coulomb}
|
||||
\desc{Coulomb gauge}{}{\QtyRef{magnetic_vector_potential}}
|
||||
\desc[german]{Coulomb-Eichung}{}{}
|
||||
\eq{
|
||||
\Div \vec{A} = 0
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\Subsection{induction}
|
||||
\desc{Induction}{}{}
|
||||
\desc[german]{Induktion}{}{}
|
||||
\begin{formula}{farady_law}
|
||||
\desc{Faraday's law of induction}{}{}
|
||||
\desc[german]{Faradaysche Induktionsgesetz}{}{}
|
||||
\eq{U_\text{ind} = -\odv{}{t} \PhiB = - \odv{}{t} \iint_A\vec{B} \cdot \d\vec{A}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{lenz}
|
||||
\desc{Lenz's law}{}{}
|
||||
\desc[german]{Lenzsche Regel}{}{}
|
||||
\ttxt{
|
||||
\eng{
|
||||
Change of magnetic flux through a conductor induces a current that counters that change of magnetic flux.
|
||||
}
|
||||
\ger{
|
||||
Die Änderung des magnetischen Flußes durch einen Leiter induziert einen Strom der der Änderung entgegenwirkt.
|
||||
}
|
||||
}
|
||||
\end{formula}
|
||||
123
src/ed/mag.tex
123
src/ed/mag.tex
@ -1,123 +0,0 @@
|
||||
\Section{mag}
|
||||
\desc{Magnetic field}{}{}
|
||||
\desc[german]{Magnetfeld}{}{}
|
||||
|
||||
|
||||
\begin{formula}{magnetic_flux}
|
||||
\desc{Magnetic flux}{}{$\vec{A}$ \GT{area}}
|
||||
\desc[german]{Magnetischer Fluss}{}{}
|
||||
\quantity{\PhiB}{\weber=\volt\per\s=\kg\m^2\per\s^2\A}{scalar}
|
||||
\eq{\PhiB = \iint_A \vec{B}\cdot\d\vec{A}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{magnetic_flux_density}
|
||||
\desc{Magnetic flux density}{Defined by \fRef{ed:mag:lorentz}}{$\vec{H}$ \qtyRef{magnetic_field_intensity}, $\vec{M}$ \qtyRef{magnetization}, \ConstRef{magnetic_vacuum_permeability}}
|
||||
\desc[german]{Magnetische Flussdichte}{Definiert über \fRef{ed:mag:lorentz}}{}
|
||||
\quantity{\vec{B}}{\tesla=\volt\s\per\m^2=\newton\per\ampere\m=\kg\per\ampere\s^2}{}
|
||||
\eq{\vec{B} = \mu_0 (\vec{H}+\vec{M})}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{magnetic_vector_potential}
|
||||
\desc{Magnetic vector potential}{}{}
|
||||
\desc[german]{Magnetisches Vektorpotential}{}{}
|
||||
\quantity{\vec{A}}{\tesla\m=\volt\s\per\m=\kg\m\per\s^2\ampere}{ievs}
|
||||
\eq{\Rot\vec{A}(\vecr) = \vec{B}(\vecr)}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{magnetic_field_intensity}
|
||||
\desc{Magnetic field intensity}{}{}
|
||||
\desc[german]{Magnetische Feldstärke}{}{}
|
||||
\quantity{\vec{H}}{\ampere\per\m}{vector}
|
||||
\eq{
|
||||
\vec{H} \equiv \frac{1}{\mu_0}\vec{B} - \vec{M}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{lorentz}
|
||||
\desc{Lorentz force law}{Force on charged particle}{}
|
||||
\desc[german]{Lorentzkraft}{Kraft auf geladenes Teilchen}{}
|
||||
\eq{
|
||||
\vec{F} = q \vec{\E} + q \vec{v}\times\vec{B}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{magnetic_permeability}
|
||||
\desc{Magnetic permeability}{}{$B$ \qtyRef{magnetic_flux_density}, $H$ \qtyRef{magnetic_field_intensity}}
|
||||
\desc[german]{Magnetisch Permeabilität}{}{}
|
||||
\quantity{\mu}{\henry\per\m=\volt\s\per\ampere\m}{scalar}
|
||||
\eq{\mu=\frac{B}{H}}
|
||||
\end{formula}
|
||||
\begin{formula}{magnetic_vacuum_permeability}
|
||||
\desc{Magnetic vauum permeability}{}{}
|
||||
\desc[german]{Magnetische Vakuumpermeabilität}{}{}
|
||||
\constant{\mu_0}{exp}{
|
||||
\val{1.25663706127(20)}{\henry\per\m=\newton\per\ampere^2}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{relative_permeability}
|
||||
\desc{Relative permeability}{}{}
|
||||
\desc[german]{Realtive Permeabilität}{}{}
|
||||
\eq{
|
||||
\mu_\txr = \frac{\mu}{\mu_0}
|
||||
}
|
||||
\hiddenQuantity{\mu_\txr}{ }{}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{gauss_law}
|
||||
\desc{Gauss's law for magnetism}{Magnetic flux through a closed surface is $0$ \Rightarrow there are no magnetic monopoles}{$S$ closed surface}
|
||||
\desc[german]{Gaußsches Gesetz für Magnetismus}{Der magnetische Fluss durch eine geschlossene Fläche ist $0$ \Rightarrow es gibt keine magnetischen Monopole}{$S$ geschlossene Fläche}
|
||||
\eq{\PhiB = \iint_S \vec{B}\cdot\d\vec{S} = 0}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{magnetization}
|
||||
\desc{Magnetization}{Vector field describing the density of magnetic dipoles}{}
|
||||
\desc[german]{Magnetisierung}{Vektorfeld, welches die Dichte von magnetischen Dipolen beschreibt.}{}
|
||||
\quantity{\vec{M}}{\ampere\per\m}{vector}
|
||||
\eq{\vec{M} = \odv{\vec{m}}{V} = \chi_\txm \cdot \vec{H}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{magnetic_moment}
|
||||
\desc{Magnetic moment}{Strength and direction of a magnetic dipole}{}
|
||||
\desc[german]{Magnetisches Moment}{Stärke und Richtung eines magnetischen Dipols}{}
|
||||
\quantity{\vec{m}}{\ampere\m^2}{vector}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{angular_torque}
|
||||
\desc{Torque}{}{$m$ \qtyRef{magnetic_moment}}
|
||||
\desc[german]{Drehmoment}{}{}
|
||||
\eq{\vec{\tau} = \vec{m} \times \vec{B}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{magnetic_susceptibility}
|
||||
\desc{Susceptibility}{}{$\mu_\txr$ \fRef{ed:mag:relative_permeability}}
|
||||
\desc[german]{Suszeptibilität}{}{}
|
||||
\eq{\chi_\txm = \pdv{M}{B} = \mu_\txr - 1}
|
||||
\hiddenQuantity{\chi}{}{}
|
||||
\end{formula}
|
||||
|
||||
|
||||
|
||||
|
||||
\Subsection{materials}
|
||||
\desc{Magnetic materials}{}{}
|
||||
\desc[german]{Magnetische Materialien}{}{}
|
||||
\begin{formula}{paramagnetism}
|
||||
\desc{Paramagnetism}{Magnetic field strengthend in the material}{$\mu$ \fRef{ed:mag:magnetic_permeability}, $\chi_\txm$ \fRef{ed:mag:magnetic_susceptibility}}
|
||||
\desc[german]{Paramagnetismus}{Magnetisches Feld wird im Material verstärkt}{}
|
||||
\eq{\mu_\txr &> 1 \\ \chi_\txm &> 0}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{diamagnetism}
|
||||
\desc{Diamagnetism}{Magnetic field expelled from material}{$\mu$ \fRef{ed:mag:magnetic_permeability}, $\chi_\txm$ \fRef{ed:mag:magnetic_susceptibility}}
|
||||
\desc[german]{Diamagnetismus}{Magnetisches Feld wird aus dem Material gedrängt}{}
|
||||
\eq{0 < \mu_\txr < 1 \\ -1 < \chi_\txm < 0}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{ferromagnetism}
|
||||
\desc{Ferromagnetism}{Magnetic moments align to external magnetic field and stay aligned when the field is turned off (Remanescence)}{$\mu$ \fRef{ed:mag:magnetic_permeability}, $\chi_\txm$ \fRef{ed:mag:magnetic_susceptibility}}
|
||||
\desc[german]{Ferromagnetismus}{Magnetische Momente werden am äußeren Feld ausgerichtet und behalten diese ausrichtung auch wenn das Feld abgeschaltet wird (Remanenz)}{}
|
||||
\eq{
|
||||
\mu_\txr \gg 1
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
@ -1,95 +0,0 @@
|
||||
\Section{dipole}
|
||||
\desc{Electrical dipoles}{Represents two charges $q$ and $-q$ with fixed distance $l$}{}
|
||||
\desc[german]{Elektrische Dipole}{Stellt starre räumliche Trennung zweier Ladungen $q$ und $-q$ dar}{}
|
||||
|
||||
\begin{formulagroup}{moment}
|
||||
\desc{Dipole moment}{}{}
|
||||
\desc[german]{Dipolmoment}{}{}
|
||||
|
||||
\begin{formula}{dipole_moment}
|
||||
\desc{Dipole moment}{}{$q$ \qtyRef{charge}, $l$ distance between charges}
|
||||
\desc[german]{Dipolmoment}{}{}
|
||||
\quantity[dipole_moment]{\vecp}{\coulomb\meter}{v}
|
||||
\eq{\vecp &= ql\vece_l}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{continous}
|
||||
\desc{Continuous charge density}{}{\QtyRef{volume}, \QtyRef{charge_density}}
|
||||
\desc[german]{Kontinuierliche Ladungsdichte}{}{}
|
||||
\eq{\vecp = \int_V \rho(\vecr) \cdot \vecr \d^3r}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{discrete}
|
||||
\desc{Discrete charge density}{}{$N$ number of charges, $q_i,\vecr_i$ \qtyRef{charge} and \qtyRef{position} of charge $i$}
|
||||
\desc[german]{Diskrete Ladungsverteilung}{}{$N$ Anzahl Ladungen, $q_i,\vecr_i$ \qtyRef{charge} und \qtyRef{position} von Ladung $i$}
|
||||
\eq{\vecp = \sum_{i=1}^{N} \vecp_i = \sum_{i=1}^{N} q_i \vecr_i}
|
||||
\end{formula}
|
||||
\end{formulagroup}
|
||||
|
||||
\begin{formula}{poynting}
|
||||
\desc{Dipole radiation Poynting vector}{}{}
|
||||
\desc[german]{Dipolsrahlung Poynting-Vektor}{}{}
|
||||
\eq{\vec{S} = \left(\frac{\mu_0 p_0^2 \omega^4}{32\pi^2 c}\right)\frac{\sin^2\theta}{r^2} \vec{r}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{power}
|
||||
\desc{Time-average power}{}{}
|
||||
\desc[german]{Zeitlich mittlere Leistung}{}{}
|
||||
\eq{P = \frac{\mu_0\omega^4 p_0^2}{12\pi c}}
|
||||
\end{formula}
|
||||
|
||||
\Section{misc}
|
||||
\desc{misc}{}{}
|
||||
\desc[german]{misc}{}{}
|
||||
\begin{formula}{impedance_r}
|
||||
\desc{Impedance of an ohmic resistor}{}{\QtyRef{resistance}}
|
||||
\desc[german]{Impedanz eines Ohmschen Widerstands}{}{}
|
||||
\eq{Z_{R} = R}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{impedance_c}
|
||||
\desc{Impedance of a capacitor}{}{\QtyRef{capacitance}, \QtyRef{angular_velocity}}
|
||||
\desc[german]{Impedanz eines Kondensators}{}{}
|
||||
\eq{Z_{C} = \frac{1}{\I\omega C}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{impedance_l}
|
||||
\desc{Impedance of an inductor}{}{\QtyRef{inductance}, \QtyRef{angular_velocity}}
|
||||
\desc[german]{Impedanz eines Induktors}{}{}
|
||||
\eq{Z_{L} = \I\omega L}
|
||||
\end{formula}
|
||||
|
||||
\TODO{impedance addition for parallel / linear}
|
||||
|
||||
\begin{formula}{screened_coulomb}
|
||||
\desc{Screened coulomb potential}{}{$l_\txD$ screening length}
|
||||
\desc[german]{Abgeschirmtes Coulombpotential}{c}{}
|
||||
\eq{\Phi(r) = - \frac{q_1q_2}{4\pi\epsilon} \frac{1}{r} \Exp{-\frac{r}{l_\txD}}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{thomas-fermi_screening_lengthi}
|
||||
\desc{Length where $\Phi=\frac{\Phi(0)}{e}$}{}{}
|
||||
\desc[german]{}{}{}
|
||||
\eq{l_\txD^2 = 4\pi \frac{e^2}{\epsilon} D(E_\txF)}
|
||||
\end{formula}
|
||||
|
||||
\Subsection{capacitor}
|
||||
\desc{Capacitor}{}
|
||||
\desc[german]{Kondensator}
|
||||
|
||||
\begin{formula}{capacitance}
|
||||
\desc{Parallel plate capacitor}{}{\ConstRef{vacuum_permittivity}, \QtyRef{relative_permittivity}, \QtyRef{area}, $d$ \qtyRef{length}}
|
||||
\desc[german]{Plattenkondensator}{}{}
|
||||
\eq{C = \epsilon_0 \epsilon_\txr \frac{A}{d}}
|
||||
\end{formula}
|
||||
\TODO{more shapes: E-Field, capacity,... maybe with drawings}
|
||||
|
||||
\begin{formula}{energy}
|
||||
\desc{Electrostatic energy}{}{}
|
||||
\desc[german]{Elektrostatische Energie}{}{}
|
||||
\eq{E = \frac{Q^2}{2C}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
|
||||
|
||||
@ -1,124 +0,0 @@
|
||||
\Section{optics}
|
||||
\desc{Optics}{Properties of light and its interactions with matter}{}
|
||||
\desc[german]{Optik}{Ausbreitung von Licht und die Interaktion mit Materie}{}
|
||||
|
||||
\TODO{adv. sc slide 427 classification}
|
||||
|
||||
|
||||
\begin{formulagroup}{refraction_index}
|
||||
\desc{Refraction index}{Macroscopic}{\QtyRef{relative_permittivity}, \QtyRef{relative_permeability}, $c_0$ \constRef{vacuum_speed_of_light}, $c_\txM$ \qtyRef{phase_velocity}}
|
||||
\desc[german]{Brechungsindex}{Macroscopisch}{}
|
||||
\begin{formula}{definition}
|
||||
\desc{Refraction index}{}{}
|
||||
\desc[german]{Brechungsindex}{}{}
|
||||
\quantity{\complex{n}}{}{s}
|
||||
\eq{
|
||||
\complex{n} = \nReal + i\nImag
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{real}
|
||||
\desc{Real part of the refraction index}{}{}
|
||||
\desc[german]{Reller Teil des Brechungsindex}{}{}
|
||||
\quantity[refraction_index_real]{\nReal}{}{s}
|
||||
\eq{
|
||||
\nReal = \sqrt{\epsilon_\txr \mu_\txr}
|
||||
}
|
||||
\eq{
|
||||
\nReal = \frac{c_0}{c_\txM}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{complex}
|
||||
\desc{Extinction coefficient}{Complex part of the refraction index. Describes absorption in a medium}{\GT{sometimes} $\kappa$}
|
||||
\desc[german]{Auslöschungskoeffizient}{Komplexer Teil des Brechungsindex. Beschreibt Absorption im Medium}{}
|
||||
\quantity[refraction_index_complex]{\nImag}{}{s}
|
||||
\end{formula}
|
||||
\end{formulagroup}
|
||||
|
||||
\begin{formula}{reflectivity}
|
||||
\desc{Reflectivity}{}{\QtyRef{refraction_index}}
|
||||
\desc[german]{Reflektion}{}{}
|
||||
\eq{
|
||||
R = \abs{\frac{\complex{n}-1}{\complex{n}+1}}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{snell}
|
||||
\desc{Snell's law}{}{$\nReal_i$ \qtyRef{refraction_index_real}, $\theta_i$ incidence angle (normal to the surface)}
|
||||
\desc[german]{Snelliussches Brechungsgesetz}{}{$n_i$ \qtyRef{refraction_index}, $\theta_i$ Einfallswinkel (normal zur Fläche)}
|
||||
\eq{\nReal_1 \sin\theta_1 = \nReal_2\sin\theta_2}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{group_velocity}
|
||||
\desc{Group velocity}{Velocity with which the envelope of a wave propagates through space}{\QtyRef{angular_frequency}, \QtyRef{angular_wavenumber}}
|
||||
\desc[german]{Gruppengeschwindigkeit}{Geschwindigkeit, mit sich die Einhülende einer Welle ausbreitet}{}
|
||||
\eq{
|
||||
v_\txg \equiv \pdv{\omega}{k}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{phase_velocity}
|
||||
\desc{Phase velocity}{Velocity with which a wave propagates through a medium}{\QtyRef{angular_frequency}, \QtyRef{angular_wavenumber}, \QtyRef{wavelength}, \QtyRef{time_period}}
|
||||
\desc[german]{Phasengeschwindigkeit}{Geschwindigkeit, mit der sich eine Welle im Medium ausbreitet}{}
|
||||
\hiddenQuantity{v_\txp}{\m\per\s}{}
|
||||
\eq{
|
||||
v_\txp = \frac{\omega}{k} = \frac{\lambda}{T}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{absorption_coefficient}
|
||||
\desc{Absorption coefficient}{Intensity reduction while traversing a medium, not necessarily by energy transfer to the medium}{\QtyRef{refraction_index_complex}, \ConstRef{vacuum_speed_of_light}, \QtyRef{angular_frequency}}
|
||||
\desc[german]{Absoprtionskoeffizient}{Intensitätsverringerung beim Druchgang eines Mediums, nicht zwingend durch Energieabgabe an Medium}{}
|
||||
\quantity{\alpha}{\per\cm}{s}
|
||||
\eq{
|
||||
\alpha &= 2\nImag \frac{\omega}{c}
|
||||
}
|
||||
\TODO{Is this equation really true in general?}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\begin{formula}{intensity}
|
||||
\desc{Electromagnetic radiation intensity}{Surface power density}{$S$ \fRef{ed:em:poynting}}
|
||||
\desc[german]{Elektromagnetische Strahlungsintensität}{Flächenleistungsdichte}{}
|
||||
\quantity{I}{\watt\per\m^2=\k\per\s^3}{s}
|
||||
\eq{I = \abs{\braket{S}_t}}
|
||||
\end{formula}
|
||||
|
||||
% \begin{formula}{lambert_beer_law}
|
||||
% \desc{Beer-Lambert law}{Intensity in an absorbing medium}{$E_\lambda$ extinction, \QtyRef{absorption_coefficient}, \QtyRef{concentration}, $d$ Thickness of the medium}
|
||||
% \desc[german]{Lambert-beersches Gesetz}{Intensität in einem absorbierenden Medium}{$E_\lambda$ Extinktion, \QtyRef{refraction_index_complex}, \QtyRef{concentration}, $d$ Dicke des Mediums}
|
||||
% \eq{
|
||||
% E_\lambda = \log_{10} \frac{I_0}{I} = \kappa c d \\
|
||||
% }
|
||||
% \end{formula}
|
||||
\begin{formula}{lambert_beer_law}
|
||||
\desc{Beer-Lambert law}{Intensity in an absorbing medium}{\QtyRef{intensity}, \QtyRef{absorption_coefficient}, $z$ penetration depth}
|
||||
\desc[german]{Lambert-beersches Gesetz}{Intensität in einem absorbierenden Medium}{\QtyRef{intensity}, \QtyRef{absorption_coefficient}, $z$ Eindringtiefe}
|
||||
\eq{
|
||||
\d I = -I_0 \alpha(\omega) \d z\\
|
||||
I(z) = I_0 \e^{-\alpha(\omega) z}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formulagroup}{permittivity_complex}
|
||||
\desc{Complex relative \qtyRef[permittivity]{permittivity}}{Complex dielectric function\\Microscopic, response of a single atom to an EM wave}{\QtyRef{refraction_index_real}, \QtyRef{refraction_index_complex}}
|
||||
\desc[german]{Komplexe relative \qtyRef{permittivity}}{Komplexe dielektrische Funktion\\Mikroskopisch, Verhalten eines Atoms gegen eine EM-Welle}{}
|
||||
\begin{formula}{permittivity_complex}
|
||||
\desc{Complex relative permittivity}{}{}
|
||||
\desc[german]{Komplexe relative Permittivität}{}{}
|
||||
\eq{\epsilon_\txr &= \epsReal + i\epsImag}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{real}
|
||||
\desc{Real part}{}{}
|
||||
\desc[german]{Realteil}{}{}
|
||||
\eq{\epsReal &= {\nReal}^2 - {\nImag}^2}
|
||||
\hiddenQuantity[permittivity_real]{\epsReal}{}{}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{complex}
|
||||
\desc{Complex part}{}{}
|
||||
\desc[german]{Komplexer Teil}{}{}
|
||||
\eq{\epsImag &= 2\nReal \nImag}
|
||||
\hiddenQuantity[permittivity_complex]{\epsImag}{}{}
|
||||
\end{formula}
|
||||
\end{formulagroup}
|
||||
222
src/electrodynamics.tex
Normal file
222
src/electrodynamics.tex
Normal file
@ -0,0 +1,222 @@
|
||||
\def\PhiB{\Phi_\text{B}}
|
||||
\def\PhiE{\Phi_\text{E}}
|
||||
|
||||
\Part[
|
||||
\eng{Electrodynamics}
|
||||
\ger{Elektrodynamik}
|
||||
]{ed}
|
||||
|
||||
|
||||
\Section[
|
||||
\eng{Maxwell-Equations}
|
||||
\ger{Maxwell-Gleichungen}
|
||||
]{Maxwell}
|
||||
\begin{formula}{vacuum}
|
||||
\desc{Vacuum}{microscopic formulation}{}
|
||||
\desc[german]{Vakuum}{Mikroskopische Formulierung}{}
|
||||
\eq{
|
||||
\Div \vec{E} &= \frac{\rho_\text{el}}{\epsilon_0} \\
|
||||
\Div \vec{B} &= 0 \\
|
||||
\Rot \vec{E} &= - \odv{\vec{B}}{t} \\
|
||||
\Rot \vec{B} &= \mu_0 \vec{j} + \frac{1}{c^2} \odv{\vec{E}}{t}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{material}
|
||||
\desc{Matter}{Macroscopic formulation}{}
|
||||
\desc[german]{Materie}{Makroskopische Formulierung}{}
|
||||
\eq{
|
||||
\Div \vec{D} &= \rho_\text{el} \\
|
||||
\Div \vec{B} &= 0 \\
|
||||
\Rot \vec{E} &= - \odv{\vec{B}}{t} \\
|
||||
\Rot \vec{H} &= \vec{j} + \odv{\vec{D}}{t}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\Section[
|
||||
\eng{Fields}
|
||||
\ger{Felder}
|
||||
]{fields}
|
||||
|
||||
\Subsection[
|
||||
\eng{Electric field}
|
||||
\ger{Elektrisches Feld}
|
||||
]{mag}
|
||||
\begin{formula}{gauss_law}
|
||||
\desc{Gauss's law for electric fields}{Electric flux through a closed surface is proportional to the electric charge}{$S$ closed surface}
|
||||
\desc[german]{Gaußsches Gesetz für elektrische Felder}{Der magnetische Fluss durch eine geschlossene Fläche ist proportional zur elektrischen Ladung}{$S$ geschlossene Fläche}
|
||||
\eq{\PhiE = \iint_S \vec{E}\cdot\d\vec{S} = \frac{Q}{\varepsilon_0}}
|
||||
\end{formula}
|
||||
|
||||
\Subsection[
|
||||
\eng{Magnetic field}
|
||||
\ger{Magnetfeld}
|
||||
]{mag}
|
||||
|
||||
\Eng[magnetic_flux]{Magnetix flux density}
|
||||
\Ger[magnetic_flux]{Magnetische Flussdichte}
|
||||
|
||||
% \begin{quantity}{mag_flux}{\Phi}{\Wb}{\kg\m^2\per\s^2\A^1}{scalar}
|
||||
% \sign{}
|
||||
% \desc{Magnetic flux density}{}
|
||||
% \desc[german]{Magnetische Feldstärke}{}
|
||||
% \end{quantity}
|
||||
|
||||
\begin{formula}{magnetic_flux}
|
||||
\desc{Magnetic flux}{}{}
|
||||
\desc[german]{Magnetischer Fluss}{}{}
|
||||
\eq{\PhiB = \iint_A \vec{B}\cdot\d\vec{A}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{gauss_law}
|
||||
\desc{Gauss's law for magnetism}{Magnetic flux through a closed surface is $0$ \Rightarrow there are no magnetic monopoles}{$S$ closed surface}
|
||||
\desc[german]{Gaußsches Gesetz für Magnetismus}{Der magnetische Fluss durch eine geschlossene Fläche ist $0$ \Rightarrow es gibt keine magnetischen Monopole}{$S$ geschlossene Fläche}
|
||||
\eq{\PhiB = \iint_S \vec{B}\cdot\d\vec{S} = 0}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{name}
|
||||
\desc{}{}{}
|
||||
\desc[german]{}{}{}
|
||||
\eq{}
|
||||
\end{formula}
|
||||
\begin{formula}{magnetization}
|
||||
\desc{Magnetization}{}{$m$ mag. moment, $V$ volume}
|
||||
\desc[german]{Magnetisierung}{}{$m$ mag. Moment, $V$ Volumen}
|
||||
\eq{\vec{M} = \odv{\vec{m}}{V} = \chi_\text{m} \cdot \vec{H}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{angular_torque}
|
||||
\desc{Torque}{}{$m$ mag. moment}
|
||||
\desc[german]{Drehmoment}{}{$m$ mag. Moment}
|
||||
\eq{\vec{\tau} = \vec{m} \times \vec{B}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{suceptibility}
|
||||
\desc{Susceptibility}{}{}
|
||||
\desc[german]{Suszeptibilität}{}{}
|
||||
\eq{\chi_\text{m} = \pdv{M}{B} = \frac{\mu}{\mu_0} - 1 }
|
||||
\end{formula}
|
||||
|
||||
|
||||
\begin{formula}{poynting}
|
||||
\desc{Poynting vector}{Directional energy flux or power flow of an electromagnetic field [$\si{\W\per\m^2}$]}{}
|
||||
\desc[german]{Poynting-Vektor}{Gerichteter Energiefluss oder Leistungsfluss eines elektromgnetischen Feldes [$\si{\W\per\m^2}$]}{}
|
||||
\eq{\vec{S} = \vec{E} \times \vec{H}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
|
||||
\Subsection[
|
||||
\eng{Induction}
|
||||
\ger{Unduktion}
|
||||
]{induction}
|
||||
\begin{formula}{farady_law}
|
||||
\desc{Faraday's law of induction}{}{}
|
||||
\desc[german]{Faradaysche Induktionsgesetz}{}{}
|
||||
\eq{U_\text{ind} = -\odv{}{t} \PhiB = - \odv{}{t} \iint_A\vec{B} \cdot \d\vec{A}}
|
||||
\end{formula}
|
||||
|
||||
\Section[
|
||||
\eng{Hall-Effect}
|
||||
\ger{Hall-Effekt}
|
||||
]{hall}
|
||||
|
||||
\begin{formula}{cyclotron}
|
||||
\desc{Cyclontron frequency}{}{}
|
||||
\desc[german]{Zyklotronfrequenz}{}{}
|
||||
\eq{\omega_\text{c} = \frac{e B}{\masse}}
|
||||
\end{formula}
|
||||
\TODO{Move}
|
||||
|
||||
|
||||
\Subsection[
|
||||
\eng{Classical Hall-Effect}
|
||||
\ger{Klassischer Hall-Effekt}
|
||||
]{classic}
|
||||
\begin{ttext}
|
||||
\eng{Current flowing in $x$ direction in a conductor ($l \times b \times d$) with a magnetic field $B$ in $z$ direction leads to a hall voltage $U_\text{H}$ in $y$ direction.}
|
||||
\ger{Fließt in einem Leiter ($l \times b \times d$) ein Strom in $x$ Richtung, während der Leiter von einem Magnetfeld $B$ in $z$-Richtung durchdrungen, wird eine Hallspannung $U_\text{H}$ in $y$-Richtung induziert.}
|
||||
\end{ttext}
|
||||
\begin{formula}{voltage}
|
||||
\desc{Hall voltage}{}{$n$ charge carrier density}
|
||||
\desc[german]{Hallspannung}{}{$n$ Ladungsträgerdichte}
|
||||
\eq{U_\text{H} = \frac{I B}{ne d}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{coefficient}
|
||||
\desc{Hall coefficient}{}{}
|
||||
\desc[german]{Hall-Koeffizient}{}{}
|
||||
\eq{R_\text{H} = -\frac{Eg}{j_x Bg} = \frac{1}{ne} = \frac{\rho_{xy}}{B_z}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{resistivity}
|
||||
\desc{Resistivity}{}{}
|
||||
\desc[german]{Spezifischer Widerstand}{}{}
|
||||
\eq{\rho_{xx} &= \frac{\masse}{ne^2\tau} \\ \rho_{xy} &= \frac{B}{ne}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsection[
|
||||
\eng{Integer quantum hall effect}
|
||||
\ger{Ganzahliger Quantenhalleffekt}
|
||||
]{quantum}
|
||||
|
||||
\begin{formula}{conductivity}
|
||||
\desc{Conductivity tensor}{}{}
|
||||
\desc[german]{Leitfähigkeitstensor}{}{}
|
||||
\eq{\sigma = \begin{pmatrix} \sigma_{xy} & \sigma_{xy} \\ \sigma_{yx} & \sigma_{yy} \end{pmatrix} }
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{resistivity}
|
||||
\desc{Resistivity tensor}{}{}
|
||||
\desc[german]{Spezifischer Widerstands-tensor}{}{}
|
||||
\eq{
|
||||
\rho = \sigma^{-1}
|
||||
% \sigma = \begin{pmatrix} \sigma_{xy} & \sigma_{xy} \\ \sigma_{yx} & \sigma_{yy} \end{pmatrix} }
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{resistivity}
|
||||
\desc{Resistivity}{}{$\nu \in \mathbb{Z}$}
|
||||
\desc[german]{Spezifischer Hallwiderstand}{}{$\nu \in \mathbb{Z}$}
|
||||
\eq{\rho_{xy} = \frac{2\pi\hbar}{e^2} \frac{1}{\nu}}
|
||||
\end{formula}
|
||||
|
||||
% \begin{formula}{qhe}
|
||||
% \desc{Integer quantum hall effect}{}{}
|
||||
% \desc[german]{Ganzahliger Quanten-Hall-Effekt}{}{}
|
||||
% \fig{img/qhe-klitzing.jpeg}
|
||||
% \end{formula}
|
||||
|
||||
|
||||
\TODO{sort}
|
||||
\begin{formula}{impedance_c}
|
||||
\desc{Impedance of a capacitor}{}{}
|
||||
\desc[german]{Impedanz eines Kondesnators}{}{}
|
||||
\eq{Z_{C} = \frac{1}{i\omega C}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{impedance_l}
|
||||
\desc{Impedance of an inductor}{}{}
|
||||
\desc[german]{Impedanz eines Induktors}{}{}
|
||||
\eq{Z_{L} = i\omega L}
|
||||
\end{formula}
|
||||
|
||||
\TODO{impedance addition for parallel / linear}
|
||||
|
||||
\Section[
|
||||
\eng{Dipole-stuff}
|
||||
\ger{Dipol-zeug}
|
||||
]{dipole}
|
||||
|
||||
\begin{formula}{poynting}
|
||||
\desc{Dipole radiation Poynting vector}{}{}
|
||||
\desc[german]{Dipolsrahlung Poynting-Vektor}{}{}
|
||||
\eq{\vec{S} = \left(\frac{\mu_0 p_0^2 \omega^4}{32\pi^2 c}\right)\frac{\sin^2\theta}{r^2} \vec{r}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{power}
|
||||
\desc{Time-average power}{}{}
|
||||
\desc[german]{Zeitlich mittlere Leistung}{}{}
|
||||
\eq{P = \frac{\mu_0\omega^4 p_0^2}{12\pi c}}
|
||||
\end{formula}
|
||||
@ -1,51 +1,53 @@
|
||||
\Section{geo}
|
||||
\desc{Geometry}{}{}
|
||||
\desc[german]{Geometrie}{}{}
|
||||
\Part[
|
||||
\eng{Geometry}
|
||||
\ger{Geometrie}
|
||||
]{geo}
|
||||
|
||||
\Section[
|
||||
\eng{Trigonometry}
|
||||
\ger{Trigonometrie}
|
||||
]{trig}
|
||||
|
||||
\begin{formula}{exponential_function}
|
||||
\desc{Exponential function}{}{}
|
||||
\desc[german]{Exponentialfunktion}{}{}
|
||||
\eq{\exp(x) = \sum_{n=0}^{\infty} \frac{x^n}{n!}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{sine}
|
||||
\desc{Sine}{}{}
|
||||
\desc[german]{Sinus}{}{}
|
||||
\eq{\sin(x) &= \sum_{n=0}^{\infty} (-1)^{n} \frac{x^{(2n+1)}}{(2n+1)!} \\
|
||||
&= \frac{e^{ix}-e^{-ix}}{2i}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{cosine}
|
||||
\desc{Cosine}{}{}
|
||||
\desc[german]{Kosinus}{}{}
|
||||
\eq{\cos(x) &= \sum_{n=0}^{\infty} (-1)^{n} \frac{x^{(2n)}}{(2n)!} \\
|
||||
&= \frac{e^{ix}+e^{-ix}}{2}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsection{trig}
|
||||
\desc{Trigonometry}{}{}
|
||||
\desc[german]{Trigonometrie}{}{}
|
||||
\begin{formula}{hyperbolic_sine}
|
||||
\desc{Hyperbolic sine}{}{}
|
||||
\desc[german]{Sinus hyperbolicus}{}{}
|
||||
\eq{\sinh(x) &= -i\sin{ix} \\ &= \frac{e^{x}-e^{-x}}{2}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{exponential_function}
|
||||
\desc{Exponential function}{}{}
|
||||
\desc[german]{Exponentialfunktion}{}{}
|
||||
\eq{\exp(x) = \sum_{n=0}^{\infty} \frac{x^n}{n!}}
|
||||
\end{formula}
|
||||
\begin{formula}{hyperbolic_cosine}
|
||||
\desc{Hyperbolic cosine}{}{}
|
||||
\desc[german]{Kosinus hyperbolicus}{}{}
|
||||
\eq{\cosh(x) &= \cos{ix} \\ &= \frac{e^{x}+e^{-x}}{2}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{sine}
|
||||
\desc{Sine}{}{}
|
||||
\desc[german]{Sinus}{}{}
|
||||
\eq{\sin(x) &= \sum_{n=0}^{\infty} (-1)^{n} \frac{x^{(2n+1)}}{(2n+1)!} \\
|
||||
&= \frac{e^{ix}-e^{-ix}}{2i}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{cosine}
|
||||
\desc{Cosine}{}{}
|
||||
\desc[german]{Kosinus}{}{}
|
||||
\eq{\cos(x) &= \sum_{n=0}^{\infty} (-1)^{n} \frac{x^{(2n)}}{(2n)!} \\
|
||||
&= \frac{e^{ix}+e^{-ix}}{2}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\begin{formula}{hyperbolic_sine}
|
||||
\desc{Hyperbolic sine}{}{}
|
||||
\desc[german]{Sinus hyperbolicus}{}{}
|
||||
\eq{\sinh(x) &= -i\sin{ix} \\ &= \frac{e^{x}-e^{-x}}{2}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{hyperbolic_cosine}
|
||||
\desc{Hyperbolic cosine}{}{}
|
||||
\desc[german]{Kosinus hyperbolicus}{}{}
|
||||
\eq{\cosh(x) &= \cos{ix} \\ &= \frac{e^{x}+e^{-x}}{2}}
|
||||
\end{formula}
|
||||
|
||||
\Subsection{theorems}
|
||||
\desc{Various theorems}{}{}
|
||||
\desc[german]{Verschiedene Theoreme}{}{}
|
||||
\Subsection[
|
||||
\eng{Various theorems}
|
||||
\ger{Verschiedene Theoreme}
|
||||
]{theorems}
|
||||
\begin{formula}{sum}
|
||||
\desc{Hypthenuse in the unit circle}{}{}
|
||||
\desc[german]{Hypothenuse im Einheitskreis}{}{}
|
||||
\desc{}{}{}
|
||||
\desc[german]{}{}{}
|
||||
\eq{1 &= \sin^2 x + \cos^2 x}
|
||||
\end{formula}
|
||||
|
||||
@ -69,16 +71,17 @@
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{other}
|
||||
\desc{Other}{}{$\tan\theta = b$}
|
||||
\desc[german]{Sonstige}{}{$\tan\theta = b$}
|
||||
\begin{formula}{name}
|
||||
\desc{}{}{$\tan\theta = b$}
|
||||
\desc[german]{}{}{$\tan\theta = b$}
|
||||
\eq{\cos x + b\sin x = \sqrt{1 + b^2}\cos(x-\theta)}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsubsection{value_table}
|
||||
\desc{Table of values}{}{}
|
||||
\desc[german]{Wertetabelle}{}{}
|
||||
\Subsection[
|
||||
\eng{Table of values}
|
||||
\ger{Wertetabelle}
|
||||
]{value_table}
|
||||
\begingroup
|
||||
\setlength{\tabcolsep}{0.9em} % horizontal
|
||||
\renewcommand{\arraystretch}{2} % vertical
|
||||
120
src/img-dev.tex
120
src/img-dev.tex
@ -1,120 +0,0 @@
|
||||
\documentclass{standalone}
|
||||
\usepackage{xcolor}
|
||||
\usepackage{titlesec}
|
||||
\usepackage{hyperref}
|
||||
\usepackage{amsmath}
|
||||
\input{util/math-macros.tex}
|
||||
\input{util/colorscheme.tex}
|
||||
\input{util/colors.tex} % after colorscheme
|
||||
\newcommand\gt[1]{#1}
|
||||
\newcommand\GT[1]{#1}
|
||||
|
||||
% GRAPHICS
|
||||
\usepackage{pgfplots}
|
||||
\pgfplotsset{compat=1.18}
|
||||
\usepackage{tikz} % drawings
|
||||
\usetikzlibrary{decorations.pathmorphing}
|
||||
\usetikzlibrary{decorations.pathreplacing} % braces
|
||||
\usetikzlibrary{calc}
|
||||
\usetikzlibrary{3d}
|
||||
\usetikzlibrary{perspective} % 3d view
|
||||
\usetikzlibrary{patterns}
|
||||
\usetikzlibrary{patterns}
|
||||
\input{util/tikz_macros}
|
||||
% speed up compilation by externalizing figures
|
||||
% \usetikzlibrary{external}
|
||||
% \tikzexternalize[prefix=tikz_figures]
|
||||
% \tikzexternalize
|
||||
\usepackage{circuitikz} % electrical circuits with tikz
|
||||
|
||||
|
||||
\begin{document}
|
||||
\begin{tikzpicture}
|
||||
\pgfmathsetmacro{\lvlW}{1} % width
|
||||
\pgfmathsetmacro{\lvlDst}{\lvlW*0.1} % line distance
|
||||
\pgfmathsetmacro{\atmx}{0}
|
||||
\pgfmathsetmacro{\molx}{3}
|
||||
\pgfmathsetmacro{\cstx}{7}
|
||||
\pgfmathsetmacro{\asy}{0}
|
||||
\pgfmathsetmacro{\apy}{\asy+1.5}
|
||||
\pgfmathsetmacro{\mss}{2.2} % s splitting
|
||||
\pgfmathsetmacro{\mps}{2.2} % p splitting
|
||||
\pgfmathsetmacro{\msby}{\asy-0.5*\mss} % molecule s bonding y
|
||||
\pgfmathsetmacro{\msay}{\asy+0.5*\mss} % molecule s antibonding y
|
||||
\pgfmathsetmacro{\mpby}{\apy-0.5*\mps}
|
||||
\pgfmathsetmacro{\mpay}{\apy+0.5*\mps}
|
||||
\pgfmathsetmacro{\textY}{\msby-1}
|
||||
|
||||
\tikzset{
|
||||
atom/.style={fill=fg1,circle,minimum size=0.2cm,inner sep=0},
|
||||
}
|
||||
% 1: name
|
||||
% 2: center pos
|
||||
% 3: n lines
|
||||
% 4: n atoms
|
||||
\newcommand\drawLevel[4]{
|
||||
% atoms
|
||||
\foreach \i in {1,...,#3} {
|
||||
\pgfmathsetmacro{\yy}{-\lvlDst*(#3+1)/2 + \i*\lvlDst }
|
||||
% \pgfmathsetmacro{\yy}{0}
|
||||
\draw ($#2 - (\lvlW/2,0) + (0,\yy)$) -- ($#2 + (\lvlW/2,0) + (0,\yy)$);
|
||||
}
|
||||
\path ($#2 - (\lvlW/2,0)$) coordinate (#1 left);
|
||||
\path ($#2 + (\lvlW/2,0)$) coordinate (#1 right);
|
||||
% \draw[color=red] ($#2 - (\lvlW/2,0)$) -- ($#2 + (\lvlW/2,0)$);
|
||||
% atoms
|
||||
\foreach \i in {1,...,#4} {
|
||||
\ifnum #4=0
|
||||
\else
|
||||
\pgfmathsetmacro{\xx}{-\lvlW/2+\i*\lvlW/(#4+1)}
|
||||
% \pgfmathsetmacro{\yy}{0}
|
||||
\node[atom] at ($#2 + (\xx,0)$) {};
|
||||
\fi
|
||||
}
|
||||
}
|
||||
|
||||
% 1:name
|
||||
% 2: center pos
|
||||
% 3: height
|
||||
% 4: fill options
|
||||
\newcommand\drawLevelFill[4]{
|
||||
\path ($#2 - (\lvlW/2,0)$) coordinate (#1 left);
|
||||
\path ($#2 + (\lvlW/2,0)$) coordinate (#1 right);
|
||||
\draw[#4] ($(#1 left) + (0, #3/2)$) rectangle ($(#1 right) - (0, #3/2)$);
|
||||
}
|
||||
|
||||
% atom
|
||||
\drawLevel{as}{(\atmx,\asy)}{2}{2} \node[anchor=east] at (as left) {$s$};
|
||||
\drawLevel{ap}{(\atmx,\apy)}{6}{3} \node[anchor=east] at (ap left) {$p$};
|
||||
\node at (\atmx,\textY) {\GT{atom}};
|
||||
|
||||
% molecule
|
||||
\drawLevel{msb}{(\molx,\msby)}{1}{1} \node[anchor=west] at (msb right) {$s$ \gt{binding}};
|
||||
\drawLevel{msa}{(\molx,\msay)}{1}{0} \node[anchor=west] at (msa right) {$s$ \gt{antibinding}};
|
||||
\drawLevel{mpb}{(\molx,\mpby)}{3}{3} \node[anchor=west] at (mpb right) {$p$ \gt{binding}};
|
||||
\drawLevel{mpa}{(\molx,\mpay)}{3}{0} \node[anchor=west] at (mpa right) {$p$ \gt{antibinding}};
|
||||
\node at (\molx,\textY) {\GT{molecule}};
|
||||
|
||||
\draw[dashed] (as right) -- (msb left);
|
||||
\draw[dashed] (as right) -- (msa left);
|
||||
\draw[dashed] (ap right) -- (mpb left);
|
||||
\draw[dashed] (ap right) -- (mpa left);
|
||||
|
||||
\node at (\cstx,\textY) {\GT{crystal}};
|
||||
\drawLevelFill{cv1}{(\cstx,\msby)}{0.3}{sc occupied,draw}
|
||||
\drawLevelFill{cv2}{(\cstx,\mpby)}{0.5}{sc occupied,draw} \node[anchor=west] at (cv2 right) {\gt{valence band}};
|
||||
\drawLevelFill{cc1}{(\cstx,\msay)}{0.3}{} \node[anchor=west] at (cc1 right) {\gt{conduction band}};
|
||||
\drawLevelFill{cc2}{(\cstx,\mpay)}{0.5}{}
|
||||
|
||||
% 1: x1, 2: x2, 3: y
|
||||
\newcommand\midwayArrow[3]{
|
||||
\pgfmathsetmacro{\xxmid}{#1+(#2-#1)/2}
|
||||
\draw[->] (\xxmid-0.5,#3) -- (\xxmid+0.5,#3);
|
||||
}
|
||||
\midwayArrow{\atmx}{\molx}{\textY}
|
||||
\midwayArrow{\molx}{\cstx}{\textY}
|
||||
|
||||
\end{tikzpicture}
|
||||
|
||||
|
||||
\end{document}
|
||||
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|
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|
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@ -1,87 +0,0 @@
|
||||
\begin{tikzpicture}
|
||||
\pgfmathsetmacro{\lvlW}{1} % width
|
||||
\pgfmathsetmacro{\lvlDst}{\lvlW*0.1} % line distance
|
||||
\pgfmathsetmacro{\atmx}{0}
|
||||
\pgfmathsetmacro{\molx}{3}
|
||||
\pgfmathsetmacro{\cstx}{7}
|
||||
\pgfmathsetmacro{\asy}{0}
|
||||
\pgfmathsetmacro{\apy}{\asy+1.5}
|
||||
\pgfmathsetmacro{\mss}{2.2} % s splitting
|
||||
\pgfmathsetmacro{\mps}{2.2} % p splitting
|
||||
\pgfmathsetmacro{\msby}{\asy-0.5*\mss} % molecule s bonding y
|
||||
\pgfmathsetmacro{\msay}{\asy+0.5*\mss} % molecule s antibonding y
|
||||
\pgfmathsetmacro{\mpby}{\apy-0.5*\mps}
|
||||
\pgfmathsetmacro{\mpay}{\apy+0.5*\mps}
|
||||
\pgfmathsetmacro{\textY}{\msby-1}
|
||||
|
||||
\tikzset{
|
||||
atom/.style={fill=fg1,circle,minimum size=0.2cm,inner sep=0},
|
||||
}
|
||||
% 1: name
|
||||
% 2: center pos
|
||||
% 3: n lines
|
||||
% 4: n atoms
|
||||
\newcommand\drawLevel[4]{
|
||||
% atoms
|
||||
\foreach \i in {1,...,#3} {
|
||||
\pgfmathsetmacro{\yy}{-\lvlDst*(#3+1)/2 + \i*\lvlDst }
|
||||
% \pgfmathsetmacro{\yy}{0}
|
||||
\draw ($#2 - (\lvlW/2,0) + (0,\yy)$) -- ($#2 + (\lvlW/2,0) + (0,\yy)$);
|
||||
}
|
||||
\path ($#2 - (\lvlW/2,0)$) coordinate (#1 left);
|
||||
\path ($#2 + (\lvlW/2,0)$) coordinate (#1 right);
|
||||
% \draw[color=red] ($#2 - (\lvlW/2,0)$) -- ($#2 + (\lvlW/2,0)$);
|
||||
% atoms
|
||||
\foreach \i in {1,...,#4} {
|
||||
\ifnum #4=0
|
||||
\else
|
||||
\pgfmathsetmacro{\xx}{-\lvlW/2+\i*\lvlW/(#4+1)}
|
||||
% \pgfmathsetmacro{\yy}{0}
|
||||
\node[atom] at ($#2 + (\xx,0)$) {};
|
||||
\fi
|
||||
}
|
||||
}
|
||||
|
||||
% 1:name
|
||||
% 2: center pos
|
||||
% 3: height
|
||||
% 4: fill options
|
||||
\newcommand\drawLevelFill[4]{
|
||||
\path ($#2 - (\lvlW/2,0)$) coordinate (#1 left);
|
||||
\path ($#2 + (\lvlW/2,0)$) coordinate (#1 right);
|
||||
\draw[#4] ($(#1 left) + (0, #3/2)$) rectangle ($(#1 right) - (0, #3/2)$);
|
||||
}
|
||||
|
||||
% atom
|
||||
\drawLevel{as}{(\atmx,\asy)}{2}{2} \node[anchor=east] at (as left) {$s$};
|
||||
\drawLevel{ap}{(\atmx,\apy)}{6}{3} \node[anchor=east] at (ap left) {$p$};
|
||||
\node at (\atmx,\textY) {\GT{atom}};
|
||||
|
||||
% molecule
|
||||
\drawLevel{msb}{(\molx,\msby)}{1}{1} \node[anchor=west] at (msb right) {$s$ \gt{binding}};
|
||||
\drawLevel{msa}{(\molx,\msay)}{1}{0} \node[anchor=west] at (msa right) {$s$ \gt{antibinding}};
|
||||
\drawLevel{mpb}{(\molx,\mpby)}{3}{3} \node[anchor=west] at (mpb right) {$p$ \gt{binding}};
|
||||
\drawLevel{mpa}{(\molx,\mpay)}{3}{0} \node[anchor=west] at (mpa right) {$p$ \gt{antibinding}};
|
||||
\node at (\molx,\textY) {\GT{molecule}};
|
||||
|
||||
\draw[dashed] (as right) -- (msb left);
|
||||
\draw[dashed] (as right) -- (msa left);
|
||||
\draw[dashed] (ap right) -- (mpb left);
|
||||
\draw[dashed] (ap right) -- (mpa left);
|
||||
|
||||
\node at (\cstx,\textY) {\GT{crystal}};
|
||||
\drawLevelFill{cv1}{(\cstx,\msby)}{0.3}{sc occupied,draw}
|
||||
\drawLevelFill{cv2}{(\cstx,\mpby)}{0.5}{sc occupied,draw} \node[anchor=west] at (cv2 right) {\GT{valence band}};
|
||||
\drawLevelFill{cc1}{(\cstx,\msay)}{0.3}{} \node[anchor=west] at (cc1 right) {\GT{conduction band}};
|
||||
\drawLevelFill{cc2}{(\cstx,\mpay)}{0.5}{}
|
||||
|
||||
% 1: x1, 2: x2, 3: y
|
||||
\newcommand\midwayArrow[3]{
|
||||
\pgfmathsetmacro{\xxmid}{#1+(#2-#1)/2}
|
||||
\draw[->] (\xxmid-0.5,#3) -- (\xxmid+0.5,#3);
|
||||
}
|
||||
\midwayArrow{\atmx}{\molx}{\textY}
|
||||
\midwayArrow{\molx}{\cstx}{\textY}
|
||||
|
||||
\end{tikzpicture}
|
||||
|
||||
@ -1,32 +0,0 @@
|
||||
\pgfmathsetmacro{\EgapH}{1.4}
|
||||
\pgfmathsetmacro{\EfermiY}{0}
|
||||
\pgfmathsetmacro{\EcondY}{\EfermiY+0.5*\EgapH}
|
||||
\pgfmathsetmacro{\EvalY}{\EfermiY-0.5*\EgapH}
|
||||
\pgfmathsetmacro{\DegDepth}{0.5}
|
||||
\pgfmathsetmacro{\GaAsW}{3}
|
||||
\pgfmathsetmacro{\SpacerW}{0.4}
|
||||
\pgfmathsetmacro{\AlGaAsW}{3}
|
||||
% calc
|
||||
\pgfmathsetmacro{\bendLW}{\SpacerW+\AlGaAsW}
|
||||
\pgfmathsetmacro{\bendLH}{0.8}
|
||||
\newcommand\lineBendL{
|
||||
..controls ++(-0.25*\bendLW,-1.7*\bendLH) and ++(0.5*\bendLW,-1.7*\bendLH) .. ++(-\bendLW,\bendLH)
|
||||
}
|
||||
\pgfmathsetmacro{\bendRW}{\GaAsW-\DegDepth}
|
||||
\pgfmathsetmacro{\bendRH}{0.5*\EgapH}
|
||||
\pgfmathsetmacro{\DegBottomY}{\EfermiY-\DegDepth}
|
||||
\newcommand\lineBendR{
|
||||
..controls ++(\DegDepth,\DegDepth) and ++(-0.25*\bendRW,0) .. ++(\bendRW,\bendRH)
|
||||
}
|
||||
\pgfmathsetmacro{\rightX}{0+\DegDepth+\bendRW}
|
||||
\pgfmathsetmacro{\leftX}{0-\bendLW}
|
||||
\begin{tikzpicture}
|
||||
\fill[color=bg-blue] (0,\DegBottomY) coordinate(2deg bot) -- ++(0,\DegDepth) -- ++(\DegDepth,0) -- cycle;
|
||||
\draw[sc band con] (2deg bot) -- ++(\DegDepth,\DegDepth) \lineBendR node[anchor=west] {$\Econd$};
|
||||
\draw[sc band con] (2deg bot) -- ++(0,2) \lineBendL;
|
||||
\draw[sc band val] ($(2deg bot)-(0,\EgapH)$) -- ++(0,-1) \lineBendL;
|
||||
\draw[sc band val] ($(2deg bot)-(0,\EgapH)$) -- ++(\DegDepth,\DegDepth) \lineBendR node[anchor=west] {$\Evalence$};
|
||||
\draw[sc fermi level] (\leftX, \EfermiY) -- (\rightX, \EfermiY) node[anchor=west] {$\Efermi$};
|
||||
% \draw[thick] (0,-\EgapH) coordinate (EV) node[anchor=west] {$\Evalence$} \lineBendR -- ++(-\bendRW,-\bendRH) coordinate (2deg v) -- ++(0,-0.5) \lineBendL;
|
||||
% \draw[thick] (EV) \lineBendL coordinate (middletop) -- ++(0,0.5) -- ++(\bendRW,\bendRH) \lineBendR coordinate (EV) node[anchor=west] {$\Evalence$};
|
||||
\end{tikzpicture}
|
||||
@ -1,56 +0,0 @@
|
||||
\begin{tikzpicture}[scale=0.9]
|
||||
|
||||
\pgfmathsetmacro{\tkW}{8} % Total width
|
||||
\pgfmathsetmacro{\tkH}{5} % Total height
|
||||
% left
|
||||
\pgfmathsetmacro{\tkLx}{0} % Start
|
||||
\pgfmathsetmacro{\tkLW}{2} % Right width
|
||||
\pgfmathsetmacro{\tkLyshift}{0.0} % y-shift
|
||||
\pgfmathsetmacro{\tkLBendH}{0} % Band bending height
|
||||
\pgfmathsetmacro{\tkLBendW}{0} % Band bending width
|
||||
\pgfmathsetmacro{\tkLEV}{4.0+\tkLyshift}% Vacuum energy
|
||||
\pgfmathsetmacro{\tkLEf}{1.5+\tkLyshift}% Fermi level energy
|
||||
% right
|
||||
\pgfmathsetmacro{\tkRx}{\tkLW} % Left start
|
||||
\pgfmathsetmacro{\tkRW}{\tkW-\tkRx} % Left width
|
||||
\pgfmathsetmacro{\tkRyshift}{-0.5} % y-shift
|
||||
\pgfmathsetmacro{\tkRBendH}{0.5} % Band bending height
|
||||
\pgfmathsetmacro{\tkRBendW}{\tkRW/4} % Band bending width
|
||||
\pgfmathsetmacro{\tkREv}{0.7+\tkRyshift}% Valence band energy
|
||||
\pgfmathsetmacro{\tkREc}{2.4+\tkRyshift}% Conduction band energy
|
||||
\pgfmathsetmacro{\tkREV}{4.0+\tkRyshift}% Vacuum energy
|
||||
\pgfmathsetmacro{\tkREf}{2.0+\tkRyshift}% Fermi level energy
|
||||
|
||||
% materials
|
||||
\draw[sc metal] (0,0) rectangle (\tkLW,\tkH);
|
||||
\node at (\tkLW/2,\tkH-0.2) {\GT{metal}};
|
||||
\path[sc n type] (\tkRx,0) rectangle (\tkW,\tkH);
|
||||
\node at (\tkRx+\tkRW/2,\tkH-0.2) {\GT{n-type}};
|
||||
\path[sc separate] (\tkLW,0) -- (\tkLW,\tkH);
|
||||
|
||||
% axes
|
||||
\draw[->] (0,0) -- (\tkW+0.2,0) node[anchor=north] {$x$};
|
||||
\draw[->] (0,0) -- (0,\tkH+0.2) node[anchor=east] {$E$};
|
||||
\tkXTick{\tkRx}{$0$}
|
||||
\tkXTick{\tkRx+\tkRBendW}{$W_\txD$}
|
||||
|
||||
% right bands
|
||||
\path[sc occupied] (\tkRx, 0) -- \rightBandUp{}{\tkREv} -- (\tkW, 0) -- cycle;
|
||||
\draw[sc band con] \rightBandUp{$\Econd$}{\tkREc};
|
||||
\draw[sc band val] \rightBandUp{$\Evalence$}{\tkREv};
|
||||
\draw[sc band vac] (0,\tkLEV) -- \rightBandUp{$\Evac$}{\tkREV};
|
||||
\draw[sc fermi level] \rightBand{$\Efermi$}{\tkREf};
|
||||
% left bands
|
||||
\path[sc occupied] (0,0) rectangle (\tkLW,\tkLEf);
|
||||
\draw[sc fermi level] \leftBand{$\Efermi$}{\tkLEf};
|
||||
|
||||
% work functions
|
||||
\drawDArrow{\tkLW/2}{\tkLEf}{\tkLEV}{$e\Phi_\txM$}
|
||||
\drawDArrow{\tkRx+\tkRW*3/4}{\tkREf}{\tkREV}{$e\Phi_\txS$}
|
||||
\drawDArrow{\tkRx+\tkRW*2/4}{\tkREc}{\tkREV}{$e\chi$}
|
||||
% barrier height
|
||||
\drawDArrow{\tkRx+\tkRBendW}{\tkREc}{\tkREc+\tkRBendH}{$eU_\text{Bias}$}
|
||||
\drawDArrow{\tkRx}{\tkREf}{\tkREc+\tkRBendH}{$e\Phi_\txB$}
|
||||
|
||||
\end{tikzpicture}
|
||||
|
||||
@ -1,49 +0,0 @@
|
||||
\begin{tikzpicture}[scale=0.9]
|
||||
|
||||
\pgfmathsetmacro{\tkW}{8} % Total width
|
||||
\pgfmathsetmacro{\tkH}{5} % Total height
|
||||
% left
|
||||
\pgfmathsetmacro{\tkLx}{0} % Start
|
||||
\pgfmathsetmacro{\tkLW}{2} % Right width
|
||||
\pgfmathsetmacro{\tkLyshift}{0.0} % y-shift
|
||||
\pgfmathsetmacro{\tkLBendH}{0} % Band bending height
|
||||
\pgfmathsetmacro{\tkLBendW}{0} % Band bending width
|
||||
\pgfmathsetmacro{\tkLEV}{4.0+\tkLyshift}% Vacuum energy
|
||||
\pgfmathsetmacro{\tkLEf}{1.5+\tkLyshift}% Fermi level energy
|
||||
% right
|
||||
\pgfmathsetmacro{\tkRx}{4} % Left start
|
||||
\pgfmathsetmacro{\tkRW}{\tkW-\tkRx} % Left width
|
||||
\pgfmathsetmacro{\tkRyshift}{0} % y-shift
|
||||
\pgfmathsetmacro{\tkRBendH}{0.5} % Band bending height
|
||||
\pgfmathsetmacro{\tkRBendW}{\tkRW/4} % Band bending width
|
||||
\pgfmathsetmacro{\tkREv}{0.7+\tkRyshift}% Valence band energy
|
||||
\pgfmathsetmacro{\tkREc}{2.4+\tkRyshift}% Conduction band energy
|
||||
\pgfmathsetmacro{\tkREV}{4.0+\tkRyshift}% Vacuum energy
|
||||
\pgfmathsetmacro{\tkREf}{2.0+\tkRyshift}% Fermi level energy
|
||||
|
||||
% materials
|
||||
\draw[sc metal] (0,0) rectangle (\tkLW,\tkH);
|
||||
\node at (\tkLW/2,\tkH-0.2) {\GT{metal}};
|
||||
\path[sc n type] (\tkRx,0) rectangle (\tkW,\tkH);
|
||||
\node at (\tkRx+\tkRW/2,\tkH-0.2) {\GT{n-type}};
|
||||
|
||||
% axes
|
||||
\draw[->] (0,0) -- (\tkW+0.2,0) node[anchor=north] {$x$};
|
||||
\draw[->] (0,0) -- (0,\tkH+0.2) node[anchor=east] {$E$};
|
||||
|
||||
% right bands
|
||||
\path[sc occupied] (\tkRx, 0) -- \rightBand{}{\tkREv} -- (\tkW, 0) -- cycle;
|
||||
\draw[sc band con] \rightBand{$\Econd$}{\tkREc};
|
||||
\draw[sc band val] \rightBand{$\Evalence$}{\tkREv};
|
||||
\draw[sc band vac] (0,\tkLEV) -- \rightBand{$\Evac$}{\tkREV};
|
||||
\draw[sc fermi level] \rightBand{$\Efermi$}{\tkREf};
|
||||
% left bands
|
||||
\path[sc occupied] (0,0) rectangle (\tkLW,\tkLEf);
|
||||
\draw[sc fermi level] \leftBand{$\Efermi$}{\tkLEf};
|
||||
|
||||
% work functions
|
||||
\drawDArrow{\tkLW/2}{\tkLEf}{\tkLEV}{$e\Phi_\txM$}
|
||||
\drawDArrow{\tkRx+\tkRW*2/3}{\tkREf}{\tkREV}{$e\Phi_\txS$}
|
||||
\drawDArrow{\tkRx+\tkRW*1/3}{\tkREc}{\tkREV}{$e\chi$}
|
||||
|
||||
\end{tikzpicture}
|
||||
@ -1,50 +0,0 @@
|
||||
\begin{tikzpicture}[scale=1]
|
||||
|
||||
\pgfmathsetmacro{\tkW}{8} % Total width
|
||||
\pgfmathsetmacro{\tkH}{5} % Total height
|
||||
% left
|
||||
\pgfmathsetmacro{\tkLx}{0} % Start
|
||||
\pgfmathsetmacro{\tkLW}{2} % Right width
|
||||
\pgfmathsetmacro{\tkLyshift}{-0.5} % y-shift
|
||||
\pgfmathsetmacro{\tkLBendH}{0} % Band bending height
|
||||
\pgfmathsetmacro{\tkLBendW}{0} % Band bending width
|
||||
\pgfmathsetmacro{\tkLEV}{4.0+\tkLyshift}% Vacuum energy
|
||||
\pgfmathsetmacro{\tkLEf}{2.5+\tkLyshift}% Fermi level energy
|
||||
% right
|
||||
\pgfmathsetmacro{\tkRx}{\tkLW} % Left start
|
||||
\pgfmathsetmacro{\tkRW}{\tkW-\tkRx} % Left width
|
||||
\pgfmathsetmacro{\tkRyshift}{0} % y-shift
|
||||
\pgfmathsetmacro{\tkRBendH}{-0.5} % Band bending height
|
||||
\pgfmathsetmacro{\tkRBendW}{\tkRW/4} % Band bending width
|
||||
\pgfmathsetmacro{\tkREv}{0.7+\tkRyshift}% Valence band energy
|
||||
\pgfmathsetmacro{\tkREc}{2.5+\tkRyshift}% Conduction band energy
|
||||
\pgfmathsetmacro{\tkREV}{4.0+\tkRyshift}% Vacuum energy
|
||||
\pgfmathsetmacro{\tkREf}{2.0+\tkRyshift}% Fermi level energy
|
||||
|
||||
% materials
|
||||
\draw[sc metal] (0,0) rectangle (\tkLW,\tkH);
|
||||
\node at (\tkLW/2,\tkH-0.2) {\GT{metal}};
|
||||
\path[sc n type] (\tkRx,0) rectangle (\tkW,\tkH);
|
||||
\node at (\tkRx+\tkRW/2,\tkH-0.2) {\GT{n-type}};
|
||||
\path[sc separate] (\tkRx,0) -- (\tkRx,\tkH);
|
||||
|
||||
\drawAxes
|
||||
|
||||
% right bands
|
||||
\path[sc occupied] (\tkRx, 0) -- \rightBandAuto{}{\tkREv} -- (\tkW, 0) -- cycle;
|
||||
\draw[sc band con] \rightBandAuto{$\Econd$}{\tkREc};
|
||||
\draw[sc band val] \rightBandAuto{$\Evalence$}{\tkREv};
|
||||
\draw[sc band vac] (0,\tkLEV) -- \rightBandAuto{$\Evac$}{\tkREV};
|
||||
\draw[sc fermi level] \rightBand{$\Efermi$}{\tkREf};
|
||||
% left bands
|
||||
\path[sc occupied] (0,0) rectangle (\tkLW,\tkLEf);
|
||||
\draw[sc fermi level] \leftBand{$\Efermi$}{\tkLEf};
|
||||
|
||||
% work functions
|
||||
\drawDArrow{\tkLW/2}{\tkLEf}{\tkLEV}{$e\Phi_\txM$}
|
||||
\drawDArrow{\tkRx+\tkRW*3/4}{\tkREf}{\tkREV}{$e\Phi_\txS$}
|
||||
\drawDArrow{\tkRx+\tkRW*2/4}{\tkREc}{\tkREV}{$e\chi$}
|
||||
% barrier height
|
||||
\drawDArrow{\tkRx+\tkRBendW}{\tkREc}{\tkREc+\tkRBendH}{$eU_\text{Bias}$}
|
||||
|
||||
\end{tikzpicture}
|
||||
@ -1,48 +0,0 @@
|
||||
\begin{tikzpicture}[scale=1]
|
||||
|
||||
\pgfmathsetmacro{\tkW}{8} % Total width
|
||||
\pgfmathsetmacro{\tkH}{5} % Total height
|
||||
% left
|
||||
\pgfmathsetmacro{\tkLx}{0} % Start
|
||||
\pgfmathsetmacro{\tkLW}{2} % Right width
|
||||
\pgfmathsetmacro{\tkLyshift}{0.0} % y-shift
|
||||
\pgfmathsetmacro{\tkLBendH}{0} % Band bending height
|
||||
\pgfmathsetmacro{\tkLBendW}{0} % Band bending width
|
||||
\pgfmathsetmacro{\tkLEV}{4.0+\tkLyshift}% Vacuum energy
|
||||
\pgfmathsetmacro{\tkLEf}{2.5+\tkLyshift}% Fermi level energy
|
||||
% right
|
||||
\pgfmathsetmacro{\tkRx}{4} % Left start
|
||||
\pgfmathsetmacro{\tkRW}{\tkW-\tkRx} % Left width
|
||||
\pgfmathsetmacro{\tkRyshift}{0} % y-shift
|
||||
\pgfmathsetmacro{\tkRBendH}{0.5} % Band bending height
|
||||
\pgfmathsetmacro{\tkRBendW}{\tkRW/4} % Band bending width
|
||||
\pgfmathsetmacro{\tkREv}{0.7+\tkRyshift}% Valence band energy
|
||||
\pgfmathsetmacro{\tkREc}{2.5+\tkRyshift}% Conduction band energy
|
||||
\pgfmathsetmacro{\tkREV}{4.0+\tkRyshift}% Vacuum energy
|
||||
\pgfmathsetmacro{\tkREf}{2.0+\tkRyshift}% Fermi level energy
|
||||
|
||||
% materials
|
||||
\draw[sc metal] (0,0) rectangle (\tkLW,\tkH);
|
||||
\node at (\tkLW/2,\tkH-0.2) {\GT{metal}};
|
||||
\path[sc n type] (\tkRx,0) rectangle (\tkW,\tkH);
|
||||
\node at (\tkRx+\tkRW/2,\tkH-0.2) {\GT{n-type}};
|
||||
|
||||
\drawAxes
|
||||
|
||||
% right bands
|
||||
\path[sc occupied] (\tkRx, 0) -- \rightBand{}{\tkREv} -- (\tkW, 0) -- cycle;
|
||||
\draw[sc band con] \rightBand{$\Econd$}{\tkREc};
|
||||
\draw[sc band val] \rightBand{$\Evalence$}{\tkREv};
|
||||
\draw[sc band vac] (0,\tkLEV) -- \rightBand{$\Evac$}{\tkREV};
|
||||
\draw[sc fermi level] \rightBand{$\Efermi$}{\tkREf};
|
||||
% left bands
|
||||
\path[sc occupied] (0,0) rectangle (\tkLW,\tkLEf);
|
||||
\draw[sc fermi level] \leftBand{$\Efermi$}{\tkLEf};
|
||||
|
||||
% work functions
|
||||
\drawDArrow{\tkLW/2}{\tkLEf}{\tkLEV}{$e\Phi_\txM$}
|
||||
\drawDArrow{\tkRx+\tkRW*2/3}{\tkREf}{\tkREV}{$e\Phi_\txS$}
|
||||
\drawDArrow{\tkRx+\tkRW*1/3}{\tkREc}{\tkREV}{$e\chi$}
|
||||
|
||||
\end{tikzpicture}
|
||||
|
||||
@ -1,65 +0,0 @@
|
||||
\newcommand\tikzPnJunction[7]{
|
||||
\begin{tikzpicture}[scale=1.0]
|
||||
|
||||
\pgfmathsetmacro{\tkW}{8} % Total width
|
||||
\pgfmathsetmacro{\tkH}{5} % Total height
|
||||
% left
|
||||
\pgfmathsetmacro{\tkLx}{0} % Start
|
||||
\pgfmathsetmacro{\tkLW}{\tkW*#1} % Width
|
||||
\pgfmathsetmacro{\tkLyshift}{#2} % y-shift
|
||||
\pgfmathsetmacro{\tkLBendH}{#3} % Band bending height
|
||||
\pgfmathsetmacro{\tkLBendW}{\tkLW/4} % Band bending width
|
||||
\pgfmathsetmacro{\tkLEv}{0.7+\tkLyshift}% Valence band energy
|
||||
\pgfmathsetmacro{\tkLEc}{2.3+\tkLyshift}% Conduction band energy
|
||||
\pgfmathsetmacro{\tkLEV}{4.0+\tkLyshift}% Vacuum energy
|
||||
\pgfmathsetmacro{\tkLEf}{1.1+\tkLyshift}% Fermi level energy
|
||||
% right
|
||||
\pgfmathsetmacro{\tkRx}{\tkW*(1-#4)} % Start
|
||||
\pgfmathsetmacro{\tkRW}{\tkW*#4} % Width
|
||||
\pgfmathsetmacro{\tkRyshift}{#5} % y-shift
|
||||
\pgfmathsetmacro{\tkRBendH}{#6} % Band bending height
|
||||
\pgfmathsetmacro{\tkRBendW}{\tkRW/4} % Band bending width
|
||||
\pgfmathsetmacro{\tkREv}{0.7+\tkRyshift}% Valence band energy
|
||||
\pgfmathsetmacro{\tkREc}{2.3+\tkRyshift}% Conduction band energy
|
||||
\pgfmathsetmacro{\tkREV}{4.0+\tkRyshift}% Vacuum energy
|
||||
\pgfmathsetmacro{\tkREf}{1.9+\tkRyshift}% Fermi level energy
|
||||
|
||||
% materials
|
||||
\draw[sc p type] (0,0) rectangle (\tkLW,\tkH);
|
||||
\node at (\tkLW/2,\tkH-0.2) {\GT{p-type}};
|
||||
\path[sc separate] (\tkRx,0) -- (\tkRx,\tkH);
|
||||
\path[sc n type] (\tkRx,0) rectangle (\tkW,\tkH);
|
||||
\node at (\tkRx+\tkRW/2,\tkH-0.2) {\GT{n-type}};
|
||||
\path[sc separate] (\tkLW,0) -- (\tkLW,\tkH);
|
||||
|
||||
\drawAxes
|
||||
|
||||
% right bands
|
||||
\path[sc occupied] (\tkRx, 0) -- \rightBandAuto{}{\tkREv} -- (\tkW, 0) -- cycle;
|
||||
\draw[sc band con] \rightBandAuto{$\Econd$}{\tkREc};
|
||||
\draw[sc band val] \rightBandAuto{$\Evalence$}{\tkREv};
|
||||
\draw[sc band vac] \rightBandAuto{$\Evac$}{\tkREV};
|
||||
\draw[sc fermi level] \rightBand{$\Efermi$}{\tkREf};
|
||||
% left bands
|
||||
\path[sc occupied] (\tkLx, 0) -- \leftBandAuto{}{\tkLEv} -- (\tkLW, 0) -- cycle;
|
||||
\draw[sc band con] \leftBandAuto{$\Econd$}{\tkLEc};
|
||||
\draw[sc band val] \leftBandAuto{$\Evalence$}{\tkLEv};
|
||||
\draw[sc band vac] \leftBandAuto{$\Evac$}{\tkLEV};
|
||||
\draw[sc fermi level] \leftBand{$\Efermi$}{\tkLEf};
|
||||
|
||||
% work functions
|
||||
|
||||
\drawDArrow{\tkRx+\tkRW*2/3}{\tkREf}{\tkREV}{$e\Phi_\txn$}
|
||||
\drawDArrow{\tkRx+\tkRW*1/3}{\tkREc}{\tkREV}{$e\chi_\txn$}
|
||||
\drawDArrow{\tkLx+\tkLW*2/3}{\tkLEf}{\tkLEV}{$e\Phi_\txp$}
|
||||
\drawDArrow{\tkLx+\tkLW*1/3}{\tkLEc}{\tkLEV}{$e\chi_\txp$}
|
||||
% barrier height
|
||||
% \drawDArrow{\tkRx+\tkRBendW}{\tkREc}{\tkREc+\tkRBendH}{$eU_\text{Bias}$}
|
||||
% \drawDArrow{\tkRx}{\tkREf}{\tkREc+\tkRBendH}{$e\Phi_\txB$}
|
||||
#7
|
||||
|
||||
\end{tikzpicture}
|
||||
}
|
||||
% \tikzPnJunction{1/3}{0}{0}{1/3}{0}{0}{}
|
||||
% \tikzPnJunction{1/2}{0.4}{-0.4}{1/2}{-0.4}{0.4}{}
|
||||
|
||||
Binary file not shown.
|
Before Width: | Height: | Size: 89 KiB |
Binary file not shown.
|
Before Width: | Height: | Size: 268 KiB |
@ -1,63 +1,14 @@
|
||||
\Section{linalg}
|
||||
\desc{Linear algebra}{}{}
|
||||
\desc[german]{Lineare Algebra}{}{}
|
||||
\def\id{\mathbb{1}}
|
||||
|
||||
\Part[
|
||||
\eng{Linear algebra}
|
||||
\ger{Lineare Algebra}
|
||||
]{linalg}
|
||||
|
||||
\Subsection{matrix}
|
||||
\desc{Matrix basics}{}{}
|
||||
\desc[german]{Matrizen Basics}{}{}
|
||||
|
||||
\begin{formula}{matrix_matrix_product}
|
||||
\desc{Matrix-matrix product as sum}{}{}
|
||||
\desc[german]{Matrix-Matrix Produkt als Summe}{}{}
|
||||
\eq{C_{ij} = \sum_{k} A_{ik} B_{kj}}
|
||||
\end{formula}
|
||||
\begin{formula}{matrix_vector_product}
|
||||
\desc{Matrix-vector product as sum}{}{}
|
||||
\desc[german]{Matrix-Vektor Produkt als Summe}{}{}
|
||||
\eq{\vec{c}_{i} = \sum_{j} A_{ij} \vec{b}_{j}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{symmetric}
|
||||
\desc{Symmetric matrix}{}{$A$ $n\times n$ \GT{matrix}}
|
||||
\desc[german]{Symmetrische matrix}{}{}
|
||||
\eq{A^\T = A}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{unitary}
|
||||
\desc{Unitary matrix}{}{}
|
||||
\desc[german]{Unitäre Matrix}{}{}
|
||||
\eq{U ^\dagger U = \id}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection{transposed}
|
||||
\desc{Transposed matrix}{}{}
|
||||
\desc[german]{Transponierte Matrix}{}{}
|
||||
\begin{formula}{sum}
|
||||
\desc{Sum}{}{}
|
||||
\desc[german]{Summe}{}{}
|
||||
\eq{(A+B)^\T = A^\T + B^\T}
|
||||
\end{formula}
|
||||
\begin{formula}{product}
|
||||
\desc{Product}{}{}
|
||||
\desc[german]{Produkt}{}{}
|
||||
\eq{(AB)^\T = B^\T A^\T}
|
||||
\end{formula}
|
||||
\begin{formula}{inverse}
|
||||
\desc{Inverse}{}{}
|
||||
\desc[german]{Inverse}{}{}
|
||||
\eq{(A^{-1})^\T = (A^\T)^{-1}}
|
||||
\end{formula}
|
||||
\begin{formula}{exponential}
|
||||
\desc{Exponential}{}{}
|
||||
\desc[german]{Exponential}{}{}
|
||||
\eq{\exp(A^\T) = (\exp A)^\T \\ \ln(A^\T)=(\ln A)^\T}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsection{determinant}
|
||||
\desc{Determinant}{}{}
|
||||
\desc[german]{Determinante}{}{}
|
||||
\Section[
|
||||
\eng{Determinant}
|
||||
\ger{Determinante}
|
||||
]{determinant}
|
||||
\begin{formula}{2x2}
|
||||
\desc{2x2 matrix}{}{}
|
||||
\desc[german]{2x2 Matrix}{}{}
|
||||
@ -92,24 +43,9 @@
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsection{misc}
|
||||
\desc{Misc}{}{}
|
||||
\desc[german]{Misc}{}{}
|
||||
|
||||
\begin{formula}{normal_equation}
|
||||
\desc{Normal equation}{Solves a linear regression problem}{\mat{\theta} hypothesis / weight matrix, \mat{X} design matrix, \vec{y} output vector}
|
||||
% \desc[german]{}{}{}
|
||||
\eq{
|
||||
\mat{\theta} = (\mat{X}^\T \mat{X})^{-1} \mat{X}^\T \vec{y}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{woodbury_matrix_identity}
|
||||
\desc{Woodbury matrix identity}{Inverse of a rank-$k$ correction}{$\matA\,n\times n$, $\matU\,n\times k$, $\matC\,k\times k$, $\matV \, k\times n$}
|
||||
\desc[german]{Woodbury-Matrix-Identität}{Inverse einer Rang-$k$-Korrektur}{}
|
||||
\eq{(\matA + \matU + \matC + \matV){-1} = \matA^{-1}-\matA^{-1} \matU(\matC^{-1} + \matV \matA^{-1} \matU)^{-1} \matV \matA^{-1}}
|
||||
\end{formula}
|
||||
\Section[
|
||||
|
||||
]{zeug}
|
||||
|
||||
\begin{formula}{inverse_2x2}
|
||||
\desc{Inverse $2\times 2$ matrix}{}{}
|
||||
@ -120,6 +56,12 @@
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{unitary}
|
||||
\desc{Unitary matrix}{}{}
|
||||
\desc[german]{Unitäre Matrix}{}{}
|
||||
\eq{U ^\dagger U = \id}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{svd}
|
||||
\desc{Singular value decomposition}{Factorization of complex matrices through rotating \rightarrow rescaling \rightarrow rotation.}{$A$: $m\times n$ matrix, $U$: $m\times m$ unitary matrix, $\Lambda$: $m\times n$ rectangular diagonal matrix with non-negative numbers on the diagonal, $V$: $n\times n$ unitary matrix}
|
||||
\desc[german]{Singulärwertzerlegung}{Faktorisierung einer reellen oder komplexen Matrix durch Rotation \rightarrow Skalierung \rightarrow Rotation.}{}
|
||||
@ -153,9 +95,10 @@
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsection{eigen}
|
||||
\desc{Eigenvalues}{}{}
|
||||
\desc[german]{Eigenwerte}{}{}
|
||||
\Section[
|
||||
\eng{Eigenvalues}
|
||||
\ger{Eigenwerte}
|
||||
]{eigen}
|
||||
\begin{formula}{values}
|
||||
\desc{Eigenvalue equation}{}{$\lambda$ eigenvalue, $v$ eigenvector}
|
||||
\desc[german]{Eigenwert-Gleichung}{}{$\lambda$ Eigenwert, $v$ Eigenvektor}
|
||||
255
src/main.tex
255
src/main.tex
@ -1,126 +1,142 @@
|
||||
%! TeX program = lualatex
|
||||
% (for vimtex)
|
||||
\documentclass[11pt, a4paper]{article}
|
||||
% SET LANGUAGE HERE
|
||||
% \usepackage[utf8]{inputenc}
|
||||
\usepackage[english]{babel}
|
||||
\usepackage[left=1.6cm,right=1.6cm,top=2cm,bottom=2cm]{geometry}
|
||||
\usepackage[left=2cm,right=2cm,top=2cm,bottom=2cm]{geometry}
|
||||
% ENVIRONMENTS etc
|
||||
\usepackage{adjustbox}
|
||||
\usepackage{colortbl} % color table
|
||||
\usepackage{tabularx} % bravais table
|
||||
\usepackage{array} % more array options
|
||||
\newcolumntype{C}{>{$}c<{$}} % math-mode version of "c" column type
|
||||
\usepackage{multirow} % for superconducting qubit table
|
||||
\usepackage{hhline} % for superconducting qubit table
|
||||
\usepackage{colortbl} % color table
|
||||
\usepackage{tabularx} % bravais table
|
||||
\usepackage{multirow} % for superconducting qubit table
|
||||
\usepackage{hhline} % for superconducting qubit table
|
||||
% TOOLING
|
||||
\usepackage{graphicx}
|
||||
\usepackage{etoolbox}
|
||||
% \usepackage{luacode}
|
||||
\usepackage{expl3} % switch case and other stuff
|
||||
\usepackage{luacode}
|
||||
\usepackage{expl3} % switch case and other stuff
|
||||
\usepackage{substr}
|
||||
\usepackage{xcolor}
|
||||
% FORMATING
|
||||
\usepackage{float} % float barrier
|
||||
\usepackage{subcaption} % subfigures
|
||||
\usepackage[hidelinks]{hyperref} % hyperrefs for \fRef, \qtyRef, etc
|
||||
\usepackage[shortlabels]{enumitem} % easily change enum symbols to i), a. etc
|
||||
\setlist{noitemsep} % no vertical space between items
|
||||
\setlist[1]{labelindent=\parindent} % < Usually a good idea
|
||||
\setlist[itemize]{leftmargin=*}
|
||||
% \setlist[enumerate]{labelsep=*, leftmargin=1.5pc} % horizontal indent of items
|
||||
\usepackage{float}
|
||||
\usepackage[hidelinks]{hyperref}
|
||||
\usepackage{subcaption}
|
||||
|
||||
\usepackage[shortlabels]{enumitem} % easily change enum symbols to i), a. etc
|
||||
\hypersetup{colorlinks = true, % Colours links instead of ugly boxes
|
||||
urlcolor = blue, % Colour for external hyperlinks
|
||||
linkcolor = cyan, % Colour of internal links
|
||||
citecolor = red % Colour of citations
|
||||
}
|
||||
\usepackage{translations}
|
||||
\input{util/translation.tex}
|
||||
|
||||
\usepackage{sectsty}
|
||||
\usepackage{titlesec}
|
||||
\input{util/colorscheme.tex}
|
||||
|
||||
\usepackage{titlesec} % colored titles
|
||||
% \usepackage{sectsty}
|
||||
% GRAPHICS
|
||||
\usepackage{pgfplots}
|
||||
\pgfplotsset{compat=1.18}
|
||||
\usepackage{tikz} % drawings
|
||||
\usepackage{tikz} % drawings
|
||||
\usetikzlibrary{decorations.pathmorphing}
|
||||
\usetikzlibrary{decorations.pathreplacing} % braces
|
||||
\usetikzlibrary{calc}
|
||||
\usetikzlibrary{3d}
|
||||
\usetikzlibrary{perspective} % 3d view
|
||||
\usetikzlibrary{patterns}
|
||||
\usetikzlibrary{patterns}
|
||||
\input{util/tikz_macros}
|
||||
% speed up compilation by externalizing figures
|
||||
% \usetikzlibrary{external}
|
||||
% \tikzexternalize[prefix=tikz_figures]
|
||||
% \tikzexternalize
|
||||
\usepackage{circuitikz} % electrical circuits with tikz
|
||||
\input{circuit.tex} % custom circuits TODO: move to util
|
||||
\usepackage{circuitikz}
|
||||
|
||||
% SCIENCE PACKAGES
|
||||
\usepackage{mathtools}
|
||||
\usepackage{unicode-math} % all sorts of math symbols, requires lualatex
|
||||
% \setmathfont{STIX Math Two}
|
||||
% \setmathfont{Latin Modern Math}
|
||||
% \setmathfont{Fira Math}
|
||||
% set display math skips
|
||||
\AtBeginDocument{
|
||||
\abovedisplayskip=0pt
|
||||
\abovedisplayshortskip=0pt
|
||||
\belowdisplayskip=0pt
|
||||
\belowdisplayshortskip=0pt
|
||||
}
|
||||
% \usepackage{MnSymbol} % for >>> \ggg sign
|
||||
\usepackage[version=4,arrows=pgf-filled]{mhchem}
|
||||
% \usepackage{upgreek} % upright greek letters for chemmacros -> unicode-math
|
||||
\usepackage{chemmacros} % for orbitals images
|
||||
\usepackage{MnSymbol} % for >>> \ggg sign
|
||||
% \usepackage{esdiff} % derivatives
|
||||
% esdiff breaks when taking \dot{q} has argument
|
||||
\usepackage{derivative} % \odv, \pdv
|
||||
% \usepackage{bbold} % \mathbb font -> unicode-math
|
||||
\usepackage{braket} % <bra|ket>
|
||||
\usepackage{siunitx} % \si \SI units
|
||||
\usepackage{derivative}
|
||||
\usepackage[version=4,arrows=pgf-filled]{mhchem}
|
||||
\usepackage{bbold} % \mathbb font
|
||||
\usepackage{braket}
|
||||
\usepackage{siunitx}
|
||||
\sisetup{output-decimal-marker = {,}}
|
||||
\sisetup{separate-uncertainty}
|
||||
\sisetup{per-mode = power}
|
||||
\sisetup{exponent-product=\ensuremath{\cdot}}
|
||||
|
||||
\usepackage{emoji}
|
||||
|
||||
% DEBUG
|
||||
% \usepackage{lua-visual-debug}
|
||||
% DUMB STUFF
|
||||
% \newcommand\temoji[1]{\text{\emoji{#1}}}
|
||||
% \def\sigma{\temoji{shark}}
|
||||
% \def\lambda{\temoji{sheep}}
|
||||
% \def\psi{\temoji{pickup-truck}}
|
||||
% \def\pi{\temoji{birthday-cake}}
|
||||
% % \def\Pi{\temoji{hospital}}
|
||||
% % \def\rho{\temoji{rhino}}
|
||||
% \def\nu{\temoji{unicorn}}
|
||||
% \def\mu{\temoji{mouse}}
|
||||
|
||||
\newcommand{\TODO}[1]{{\color{fg-red}TODO:#1}} % debug mode
|
||||
\renewcommand{\TODO}[1]{} % release mode
|
||||
\newcommand{\TODO}[1]{{\color{bright_red}TODO:#1}}
|
||||
\newcommand{\ts}{\textsuperscript}
|
||||
|
||||
\usepackage{mqlua}
|
||||
\usepackage{mqfqname}
|
||||
\usepackage{mqsections}
|
||||
\usepackage{mqref}
|
||||
\input{util/macros.tex} % requires mqfqname
|
||||
\input{util/math-macros.tex}
|
||||
% TRANSLATION
|
||||
% \usepackage{translations}
|
||||
\usepackage{mqtranslation}
|
||||
\input{util/colorscheme.tex}
|
||||
\input{util/colors.tex} % after colorscheme
|
||||
% put an explanation above an equal sign
|
||||
% [1]: equality sign (or anything else)
|
||||
% 2: text (not in math mode!)
|
||||
\newcommand{\explUnderEq}[2][=]{%
|
||||
\underset{\substack{\uparrow\\\mathrlap{\text{\hspace{-1em}#2}}}}{#1}}
|
||||
\newcommand{\explOverEq}[2][=]{%
|
||||
\overset{\substack{\mathrlap{\text{\hspace{-1em}#2}}\\\downarrow}}{#1}}
|
||||
|
||||
|
||||
\usepackage{mqconstant}
|
||||
\usepackage{mqquantity}
|
||||
\usepackage{mqformula}
|
||||
\usepackage{mqperiodictable}
|
||||
|
||||
\input{util/environments.tex} % requires util/translation.tex to be loaded first
|
||||
|
||||
% "automate" sectioning
|
||||
% start <section>, get heading from translation, set label
|
||||
% fqname is the fully qualified name: the keys of all previous sections joined with a ':'
|
||||
% [1]: code to run after setting \fqname, but before the \part, \section etc
|
||||
% 2: key
|
||||
\newcommand{\Part}[2][desc]{
|
||||
\newpage
|
||||
\def\partname{#2}
|
||||
\def\sectionname{}
|
||||
\def\subsectionname{}
|
||||
\def\subsubsectionname{}
|
||||
\edef\fqname{\partname}
|
||||
#1
|
||||
\part{\GT{\fqname}}
|
||||
\label{sec:\fqname}
|
||||
}
|
||||
\newcommand{\Section}[2][]{
|
||||
\def\sectionname{#2}
|
||||
\def\subsectionname{}
|
||||
\def\subsubsectionname{}
|
||||
\edef\fqname{\partname:\sectionname}
|
||||
#1
|
||||
\section{\GT{\fqname}}
|
||||
\label{sec:\fqname}
|
||||
}
|
||||
% \newcommand{\Subsection}[1]{\Subsection{#1}{}}
|
||||
\newcommand{\Subsection}[2][]{
|
||||
\def\subsectionname{#2}
|
||||
\def\subsubsectionname{}
|
||||
\edef\fqname{\partname:\sectionname:\subsectionname}
|
||||
#1
|
||||
\subsection{\GT{\fqname}}
|
||||
\label{sec:\fqname}
|
||||
}
|
||||
\newcommand{\Subsubsection}[2][]{
|
||||
\def\subsubsectionname{#2}
|
||||
\edef\fqname{\partname:\sectionname:\subsectionname:\subsubsectionname}
|
||||
#1
|
||||
\subsubsection{\GT{\fqname}}
|
||||
\label{sec:\fqname}
|
||||
}
|
||||
|
||||
% Make the translation of #1 a reference to a equation
|
||||
% 1: key
|
||||
\newcommand{\fqEqRef}[1]{
|
||||
\hyperref[eq:#1]{\GT{#1}}
|
||||
}
|
||||
% Make the translation of #1 a reference to a section
|
||||
% 1: key
|
||||
\newcommand{\fqSecRef}[1]{
|
||||
\hyperref[sec:#1]{\GT{#1}}
|
||||
}
|
||||
|
||||
% \usepackage{xstring}
|
||||
|
||||
|
||||
\input{circuit.tex}
|
||||
|
||||
% some translations
|
||||
|
||||
\title{Formelsammlung}
|
||||
\author{Matthias Quintern}
|
||||
\date{\today}
|
||||
\begin{document}
|
||||
|
||||
\makeatletter\let\percentchar\@percentchar\makeatother
|
||||
\input{util/macros.tex}
|
||||
\input{util/environments.tex}
|
||||
|
||||
\begin{document}
|
||||
|
||||
\maketitle
|
||||
\tableofcontents
|
||||
@ -129,61 +145,32 @@
|
||||
|
||||
\input{util/translations.tex}
|
||||
|
||||
% \InputOnly{cm}
|
||||
\InputOnly{test}
|
||||
\input{linalg.tex}
|
||||
|
||||
\Input{math/math}
|
||||
\Input{math/linalg}
|
||||
\Input{math/geometry}
|
||||
\Input{math/calculus}
|
||||
\Input{math/probability_theory}
|
||||
\input{geometry.tex}
|
||||
|
||||
\Input{mechanics}
|
||||
\Input{statistical_mechanics}
|
||||
\input{analysis.tex}
|
||||
|
||||
\Input{ed/ed}
|
||||
\Input{ed/el}
|
||||
\Input{ed/mag}
|
||||
\Input{ed/em}
|
||||
\Input{ed/optics}
|
||||
\Input{ed/misc}
|
||||
\input{probability_theory.tex}
|
||||
|
||||
\Input{qm/qm}
|
||||
\Input{qm/atom}
|
||||
\input{mechanics.tex}
|
||||
|
||||
\Input{cm/cm}
|
||||
\Input{cm/crystal}
|
||||
\Input{cm/vib}
|
||||
\Input{cm/egas}
|
||||
\Input{cm/charge_transport}
|
||||
\Input{cm/semiconductors}
|
||||
\Input{cm/optics}
|
||||
\Input{cm/misc}
|
||||
\Input{cm/techniques}
|
||||
\Input{cm/topo}
|
||||
\Input{cm/superconductivity}
|
||||
\Input{cm/mat}
|
||||
\input{statistical_mechanics.tex}
|
||||
|
||||
\Input{particle}
|
||||
\input{electrodynamics.tex}
|
||||
|
||||
\input{quantum_mechanics.tex}
|
||||
\input{atom.tex}
|
||||
|
||||
\Input{quantum_computing}
|
||||
\input{condensed_matter.tex}
|
||||
|
||||
\Input{comp/comp}
|
||||
\Input{comp/qmb}
|
||||
\Input{comp/est}
|
||||
\Input{comp/ad}
|
||||
\Input{comp/ml}
|
||||
% \input{topo.tex}
|
||||
|
||||
\Input{ch/periodic_table} % only definitions
|
||||
\Input{ch/ch}
|
||||
\Input{ch/el}
|
||||
\Input{ch/misc}
|
||||
% \input{quantum_computing.tex}
|
||||
|
||||
\Input{appendix}
|
||||
|
||||
\Input{test}
|
||||
% \input{many-body-simulations.tex}
|
||||
|
||||
%\newpage
|
||||
% \bibliographystyle{plain}
|
||||
% \bibliography{ref}
|
||||
|
||||
\end{document}
|
||||
|
||||
10
src/many-body-simulations.tex
Normal file
10
src/many-body-simulations.tex
Normal file
@ -0,0 +1,10 @@
|
||||
\Part[
|
||||
\eng{Many-body simulations}
|
||||
\ger{Vielteilchen Simulationen}
|
||||
]{mbsim}
|
||||
|
||||
\Section[
|
||||
\eng{Importance sampling}
|
||||
\ger{Importance sampling / Stichprobenentnahme nach Wichtigkeit}
|
||||
]{importance_sampling}
|
||||
|
||||
@ -1,337 +0,0 @@
|
||||
\Section{cal}
|
||||
\desc{Calculus}{}{}
|
||||
\desc[german]{Analysis}{}{}
|
||||
|
||||
|
||||
% \begin{formula}{shark}
|
||||
% \desc{Shark-midnight formula}{\emoji{shark}-s}{}
|
||||
% \desc[german]{Shark-Mitternachtformel}{}{}
|
||||
% \eq{
|
||||
% \temoji{seal}_{1,2} = \frac{-\temoji{shark}\pm \sqrt{\temoji{shark}^2-4\temoji{octopus}\temoji{tropical-fish}}}{2\temoji{octopus}}
|
||||
% }
|
||||
% \end{formula}
|
||||
|
||||
|
||||
\Subsection{fourier}
|
||||
\desc{Fourier analysis}{}{}
|
||||
\desc[german]{Fourieranalyse}{}{}
|
||||
\Subsubsection{series}
|
||||
\desc{Fourier series}{}{}
|
||||
\desc[german]{Fourierreihe}{}{}
|
||||
\begin{formula}{series} \absLabel[fourier_series]
|
||||
\desc{Fourier series}{Complex representation}{$f\in \Lebesgue^2(\R,\C)$ $T$-\GT{periodic}}
|
||||
\desc[german]{Fourierreihe}{Komplexe Darstellung}{}
|
||||
\eq{f(t) = \sum_{k=-\infty}^{\infty} c_k \Exp{\frac{2\pi \I kt}{T}}}
|
||||
\end{formula}
|
||||
\Eng[real]{real}
|
||||
\Ger[real]{reellwertig}
|
||||
\begin{formula}{coefficient-complex}
|
||||
\desc{Fourier coefficients}{Complex representation}{}
|
||||
\desc[german]{Fourierkoeffizienten}{Komplexe Darstellung}{}
|
||||
\eq{
|
||||
c_k &= \frac{1}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} f(t)\,\Exp{-\frac{2\pi \I}{T}kt}\d t \quad\text{\GT{for}}\,k\ge0\\
|
||||
c_{-k} &= \overline{c_k} \quad \text{\GT{if} $f$ \GT{real}}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{series_sincos}
|
||||
\desc{Fourier series}{Sine and cosine representation}{$f\in \Lebesgue^2(\R,\C)$ $T$-\GT{periodic}}
|
||||
\desc[german]{Fourierreihe}{Sinus und Kosinus Darstellung}{}
|
||||
\eq{f(t) = \frac{a_0}{2} + \sum_{k=1}^{\infty} \left(a_k \Cos{\frac{2\pi}{T}kt} + b_k\Sin{\frac{2\pi}{T}kt}\right)}
|
||||
\end{formula}
|
||||
\begin{formula}{coefficient}
|
||||
\desc{Fourier coefficients}{Sine and cosine representation\\If $f$ has point symmetry: $a_{k>0}=0$, if $f$ has axial symmetry: $b_k=0$}{}
|
||||
\desc[german]{Fourierkoeffizienten}{Sinus und Kosinus Darstellung\\Wenn $f$ punktsymmetrisch: $a_{k>0}=0$, wenn $f$ achsensymmetrisch: $b_k=0$}{}
|
||||
\eq{
|
||||
a_k &= \frac{2}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} f(t)\,\Cos{-\frac{2\pi}{T}kt}\d t \quad\text{\GT{for}}\,k\ge0\\
|
||||
b_k &= \frac{2}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} f(t)\,\Sin{-\frac{2\pi}{T}kt}\d t \quad\text{\GT{for}}\,k\ge1\\
|
||||
a_k &= c_k + c_{-k} \quad\text{\GT{for}}\,k\ge0\\
|
||||
b_k &= \I(c_k - c_{-k}) \quad\text{\GT{for}}\,k\ge1
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsubsection{trafo}
|
||||
\desc{Fourier transformation}{}{}
|
||||
\desc[german]{Fouriertransformation}{}{}
|
||||
\begin{formula}{transform} \absLabel[fourier_transform]
|
||||
\desc{Fourier transform}{}{$\hat{f}:\R^n \mapsto \C$, $\forall f\in L^1(\R^n)$}
|
||||
\desc[german]{Fouriertransformierte}{}{}
|
||||
\eq{\hat{f}(k) \coloneq \frac{1}{\sqrt{2\pi}^n} \int_{\R^n} \e^{-\I kx}f(x)\d x}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{properties}
|
||||
\desc{Properties}{}{\GT{for} $f\in L^1(\R^n)$}
|
||||
\desc[german]{Eigenschaften}{}{}
|
||||
\Eng[linear_in]{linear in}
|
||||
\Ger[linear_in]{linear in}
|
||||
\begin{enumerate}[i)]
|
||||
\item $f \mapsto \hat{f}$ \GT{linear_in} $f$
|
||||
\item $g(x) = f(x-h) \qRarrow \hat{g}(k) = \e^{-\I kn}\hat{f}(k)$
|
||||
\item $g(x) = \e^{ih\cdot x}f(x) \qRarrow \hat{g}(k) = \hat{f}(k-h)$
|
||||
\item $g(\lambda) = f\left(\frac{x}{\lambda}\right) \qRarrow \hat{g}(k)\lambda^n \hat{f}(\lambda k)$
|
||||
\end{enumerate}
|
||||
\end{formula}
|
||||
|
||||
|
||||
|
||||
\Subsubsection{conv}
|
||||
\desc{Convolution}{Convolution is \textbf{commutative}, \textbf{associative} and \textbf{distributive}.}{}
|
||||
\desc[german]{Faltung / Konvolution}{Die Faltung ist \textbf{kommutativ}, \textbf{assoziativ} und \textbf{distributiv}}{}
|
||||
\begin{formula}{def}
|
||||
\desc{Definition}{}{}
|
||||
\desc[german]{Definition}{}{}
|
||||
\eq{(f*g)(t) = f(t) * g(t) = \int_{-\infty}^\infty f(\tau) g(t-\tau) \d \tau}
|
||||
\end{formula}
|
||||
\begin{formula}{notation}
|
||||
\desc{Notation}{}{}
|
||||
\desc[german]{Notation}{}{}
|
||||
\eq{
|
||||
f(t) * g(t-t_0) &= (f*g)(t-t_0) \\
|
||||
f(t-t_0) * g(t-t_0) &= (f*g)(t-2t_0)
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{commutativity}
|
||||
\desc{Commutativity}{}{}
|
||||
\desc[german]{Kommutativität}{}{}
|
||||
\eq{f * g = g * f}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{associativity}
|
||||
\desc{Associativity}{}{}
|
||||
\desc[german]{Assoziativität]}{}{}
|
||||
\eq{(f*g)*h = f*(g*h)}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{distributivity}
|
||||
\desc{Distributivity}{}{}
|
||||
\desc[german]{Distributivität}{}{}
|
||||
\eq{f * (g + h) = f*g + f*h}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{complex_conjugate}
|
||||
\desc{Complex conjugate}{}{}
|
||||
\desc[german]{Komplexe konjugation}{}{}
|
||||
\eq{(f*g)^* = f^* * g^*}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsection{misc}
|
||||
\desc{Misc}{}{}
|
||||
\desc[german]{Verschiedenes}{}{}
|
||||
|
||||
\begin{formula}{stirling-approx}
|
||||
\desc{Stirling approximation}{}{}
|
||||
\desc[german]{Stirlingformel}{}{}
|
||||
\eq{\ln (N!) \approx N \ln(N) - N + \Order(\ln(N))}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{error-function}
|
||||
\desc{Error function}{$\erf: \C \to \C$ and complementary error function $\erfc$}{}
|
||||
\desc[german]{Fehlerfunktion}{$\erf: \C \to \C$ und komplementäre Fehlerfunktion $\erfc$}{}
|
||||
\eq{
|
||||
\erf(x) &= \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \d t \\
|
||||
\erfc(x) &= 1 - \erf(x)\\
|
||||
&= \frac{2}{\sqrt{\pi}} \int_x^\infty e^{-t^2} \d t
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{delta_of_function}
|
||||
\desc{Dirac-Delta of a function}{}{$f(x_i) = 0$}
|
||||
\desc[german]{Dirac-Delta einer Funktion}{}{}
|
||||
\eq{\delta(f(x)) = \sum_i \frac{\delta(x-x_i)}{\abs{f^\prime(x_i)}}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{geometric_series}
|
||||
\desc{Geometric series}{}{$\abs{q}<1$}
|
||||
\desc[german]{Geometrische Reihe}{}{}
|
||||
\eq{\sum_{k=0}^{\infty}q^k = \frac{1}{1-q}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsection{log}
|
||||
\desc{Logarithm}{}{}
|
||||
\desc[german]{Logarithmus}{}{}
|
||||
\begin{formula}{identities}
|
||||
\desc{Logarithm identities}{}{}
|
||||
\desc[german]{Logarithmus Identitäten}{Logarithmus Rechenregeln}{}
|
||||
\eq{
|
||||
\log(xy) &= \log(x) + \log(y) \\
|
||||
\log \left(\frac{x}{y}\right) &= \log(x) - \log(y) \\
|
||||
\log \left(x^d\right) &= d\log(x) \\
|
||||
\log \left(\sqrt[y]{x}\right) &= \frac{\log(x)}{y} \\
|
||||
x^{\log(y)} &= y^{\log(x)}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{integral}
|
||||
\desc{Integral of natural logarithm}{}{}
|
||||
\desc[german]{Integral des natürluchen Logarithmus}{}{}
|
||||
\eq{
|
||||
\int \ln(x) \d x &= x \left(\ln(x) -1\right) \\
|
||||
\int \ln(ax + b) \d x &= \frac{ax+b}{a} \left(\ln(ax + b) -1\right)
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\Subsection{vec}
|
||||
\desc{Vector calculus}{}{}
|
||||
\desc[german]{Vektor Analysis}{}{}
|
||||
\begin{formula}{laplace}
|
||||
\absLabel[laplace_operator]
|
||||
\desc{Laplace operator}{}{}
|
||||
\desc[german]{Laplace-Operator}{}{}
|
||||
\eq{\laplace = \Grad^2 = \pdv[2]{}{x} + \pdv[2]{}{y} + \pdv[2]{}{z}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{double_cross_product}
|
||||
\desc{Double cross product}{}{}
|
||||
\desc[german]{Zweifaches Kreuzproduct}{}{}
|
||||
\eq{
|
||||
\veca \times (\vecb \times \vecc) &= \vecb(\veca\cdot\vecc) - \vecc(\veca\cdot\vecb) \\
|
||||
(\veca \times \vecb) \times \vecc &= \vecb(\vecc\cdot\veca) - \veca(\vecb\cdot\vecc)
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsubsection{sphere}
|
||||
\desc{Spherical symmetry}{}{}
|
||||
\desc[german]{Kugelsymmetrie}{}{}
|
||||
\begin{formula}{coordinates}
|
||||
\desc{Spherical coordinates}{}{}
|
||||
\desc[german]{Kugelkoordinaten}{}{}
|
||||
\eq{
|
||||
x &= r \sin\phi,\cos\theta \\
|
||||
y &= r \cos\phi,\cos\theta \\
|
||||
z &= r \sin\theta
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{laplace}
|
||||
\desc{Laplace operator}{}{}
|
||||
\desc[german]{Laplace-Operator}{}{}
|
||||
\eq{\Grad^2 = \laplace = \frac{1}{r^2} \pdv{}{r} \left(r^2 \pdv{}{r}\right)}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{p-norm}
|
||||
\desc{$p$-norm}{}{}
|
||||
\desc[german]{$p$-Norm}{}{}
|
||||
\eq{\norm{\vecx}_p \equiv \left(\sum_{i=1}^{n} \abs{x_i}^p\right)^\frac{1}{p}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsection{integral}
|
||||
\desc{Integrals}{}{}
|
||||
\desc[german]{Integralrechnung}{}{}
|
||||
\begin{formula}{partial}
|
||||
\desc{Partial integration}{}{}
|
||||
\desc[german]{Partielle integration}{}{}
|
||||
\eq{
|
||||
\int_a^b f^\prime(x)\cdot g(x) \d x= \left[f(x)\cdot g(x)\right]_a^b - \int_a^b f(x)\cdot g^\prime(x) \d x
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{substitution}
|
||||
\desc{Integration by substitution}{}{}
|
||||
\desc[german]{Integration durch Substitution}{}{}
|
||||
\eq{
|
||||
\int_a^b f(g(x))\,g^\prime(x) \d x = \int_{g(a)}^{g(b)} f(z) \d z
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{gauss}
|
||||
\desc{Gauss's theorem / Divergence theorem}{Divergence in a volume equals the flux through the surface}{$A = \partial V$}
|
||||
\desc[german]{Satz von Gauss}{Divergenz in einem Volumen ist gleich dem Fluss durch die Oberfläche}{}
|
||||
\eq{
|
||||
\iiint_V \Div{\vec{F}} \d V = \oiint_A \vec{F} \cdot \d\vec{A}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{stokes}
|
||||
\desc{Stokes's theorem}{}{$S = \partial A$}
|
||||
\desc[german]{Klassischer Satz von Stokes}{}{}
|
||||
\eq{\int_A (\Rot{\vec{F}}) \cdot \d\vec{S} = \oint_{S} \vec{F} \cdot \d \vec{r}}
|
||||
\end{formula}
|
||||
\Subsubsection{list}
|
||||
\desc{List of common integrals}{}{}
|
||||
\desc[german]{Liste nützlicher Integrale}{}{}
|
||||
% Put links to other integrals here
|
||||
\fRef{math:cal:log:integral}
|
||||
|
||||
\begin{formula}{arcfunctions}
|
||||
\desc{Arcsine, arccosine, arctangent}{}{}
|
||||
\desc[german]{Arkussinus, Arkuskosinus, Arkustangens}{}{}
|
||||
\eq{
|
||||
\int \frac{1}{\sqrt{1-x^2}} \d x = \arcsin x \\
|
||||
\int -\frac{1}{\sqrt{1-x^2}} \d x = \arccos x \\
|
||||
\int \frac{1}{x^2+1} \d x = \arctan x
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{archyperbolicfunctions}
|
||||
\desc{Arcsinh, arccosh, arctanh}{}{}
|
||||
% \desc[german]{Arkussinus, Arkuskosinus, Arkustangens}{}{}
|
||||
\eq{
|
||||
\int \frac{1}{\sqrt{x^2+1}} \d x &= \arsinh x \\
|
||||
\int \frac{1}{\sqrt{x^2-1}} \d x &= \arcosh x \quad\eqnote{\GT{for} $(x > 1)$}\\
|
||||
\int \frac{1}{1-x^2} \d x &= \artanh x \quad\eqnote{\GT{for} $(\abs{x} < 1)$}\\
|
||||
\int \frac{1}{1-x^2} \d x &= \arcoth x \quad\eqnote{\GT{for} $(\abs{x} > 1)$}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{spheical-coordinates-int}
|
||||
\desc{Integration in spherical coordinates}{}{}
|
||||
\desc[german]{Integration in Kugelkoordinaten}{}{}
|
||||
\eq{\iiint\d x \d y \d z= \int_0^{\infty} \!\! \int_0^{2\pi} \!\! \int_0^\pi \d r \d\phi\d\theta \, r^2\sin\theta}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{riemann_zeta}
|
||||
\desc{Riemann Zeta Function}{}{}
|
||||
\desc[german]{Riemannsche Zeta-Funktion}{}{}
|
||||
\eq{\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} = \frac{1}{(1-2^{(1-s)})\Gamma(s)} \int_0^\infty \d\eta \frac{\eta^{(s-1)}}{\e^\eta + 1}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{gamma_function}
|
||||
\desc{Gamma function}{}{}
|
||||
\desc[german]{Gamma-Funktion}{}{}
|
||||
\eq{
|
||||
\Gamma(n) &= (n-1)! \\
|
||||
\Gamma(z) &= \int_0^\infty t^{z-1} \e^{-t} \d t \\
|
||||
\Gamma(z+1) &= z\Gamma(z)
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{upper_incomplete_gamma_function}
|
||||
\desc{Upper incomplete gamma function}{}{}
|
||||
\desc[german]{Unvollständige Gamma-Funktion der unteren Grenze}{}{}
|
||||
\eq{\Gamma(s,x) = \int_x-^\infty t^{s-1}\e^{-t} \d t}
|
||||
\end{formula}
|
||||
\begin{formula}{lower_incomplete_gamma_function}
|
||||
\desc{Lower incomplete gamma function}{}{}
|
||||
\desc[german]{Unvollständige Gamma-Funktion der oberen Grenze}{}{}
|
||||
\eq{\gamma(s,x) = \int_0^x t^{s-1}\e^{-t} \d t}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{beta_function}
|
||||
\desc{Beta function}{Complete beta function}{}
|
||||
\desc[german]{Beta-Funktion}{}{}
|
||||
\eq{
|
||||
\txB(z_1,z_2) &= \int_0^1 t^{z_1-1} (1-t)^{z_2-1} \d t \\
|
||||
\txB(z_1, z_2) &= \frac{\Gamma(z_1) \Gamma(z_2)}{\Gamma(z_1+z_2)}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{incomplete_beta_function}
|
||||
\desc{Incomplete beta function}{Complete beta function}{}
|
||||
\desc[german]{Unvollständige Beta-Funktion}{}{}
|
||||
\eq{\txB(x; z_1,z_2) = \int_0^x t^{z_1-1} (1-t)^{z_2-1} \d t}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{fermi_dirac}
|
||||
\desc{Fermi-Dirac integral}{}{$\Gamma$ \fRef{::gamma_function}}
|
||||
\desc[german]{Fermi-Dirac-Integral}{}{}
|
||||
\eq{F_j(x)= \frac{1}{\Gamma(j+1)} \int_0^\infty \frac{t^j}{\Exp{t-x}+1}\d t}
|
||||
\end{formula}
|
||||
\begin{formula}{boltzmann_approximation}
|
||||
\desc{Boltzmann approximation}{$-x\gg1$}{$F$ \fRef{::fermi_dirac_integral}}
|
||||
\desc[german]{Boltzmann-Näherung}{}{}
|
||||
\eq{F_{1/2}(x) \approx \Exp{x}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\TODO{differential equation solutions}
|
||||
@ -1,5 +0,0 @@
|
||||
\Part{math}
|
||||
\desc{Mathematics}{}{}
|
||||
\desc[german]{Mathematik}{}{}
|
||||
|
||||
|
||||
@ -1,383 +0,0 @@
|
||||
\Section{pt}
|
||||
\desc{Probability theory}{}{}
|
||||
\desc[german]{Wahrscheinlichkeitstheorie}{}{}
|
||||
|
||||
|
||||
\begin{formula}{mean}
|
||||
\absLabel
|
||||
\desc{Mean}{Expectation value}{}
|
||||
\desc[german]{Mittelwert}{Erwartungswert}{}
|
||||
\eq{\braket{x} = \int w(x)\, x\, \d x}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\begin{formula}{variance}
|
||||
\absLabel
|
||||
\desc{Variance}{Square of the \fRef{math:pt:std-deviation}}{}
|
||||
\desc[german]{Varianz}{Quadrat der\fRef{math:pt:std-deviation}}{}
|
||||
\eq{\sigma^2 = (\Delta \hat{x})^2 = \Braket{\hat{x}^2} - \braket{\hat{x}}^2 = \braket{(x - \braket{x})^2}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{covariance}
|
||||
\absLabel
|
||||
\desc{Covariance}{}{}
|
||||
\desc[german]{Kovarianz}{}{}
|
||||
\eq{\cov(x,y) = \sigma(x,y) = \sigma_{XY} = \Braket{(x-\braket{x})\,(y-\braket{y})}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{std-deviation}
|
||||
\absLabel
|
||||
\desc{Standard deviation}{}{}
|
||||
\desc[german]{Standardabweichung}{}{}
|
||||
\eq{\sigma = \sqrt{\sigma^2} = \sqrt{(\Delta x)^2}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{median}
|
||||
\desc{Median}{Value separating lower half from top half}{$x$ dataset with $n$ elements}
|
||||
\desc[german]{Median}{Teilt die untere von der oberen Hälfte}{$x$ Reihe mit $n$ Elementen}
|
||||
\eq{
|
||||
\textrm{med}(x) = \left\{ \begin{array}{ll} x_{(n+1)/2} & \text{$n$ \GT{odd}} \\ \frac{x_{(n/2)}+x_{((n/2)+1)}}{2} & \text{$n$ \GT{even}} \end{array} \right.
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{pdf}
|
||||
\abbrLabel{PDF}
|
||||
\desc{Probability density function}{Random variable has density $f$. The integral gives the probability of $X$ taking a value $x\in[a,b]$.}{$f$ normalized: $\int_{-\infty}^\infty f(x) \d x= 1$}
|
||||
\desc[german]{Wahrscheinlichkeitsdichtefunktion}{Zufallsvariable hat Dichte $f$. Das Integral gibt Wahrscheinlichkeit an, dass $X$ einen Wert $x\in[a,b]$ annimmt}{$f$ normalisiert $\int_{-\infty}^\infty f(x) \d x= 1$}
|
||||
\eq{P([a,b]) := \int_a^b f(x) \d x}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{cdf}
|
||||
\abbrLabel{CDF}
|
||||
\desc{Cumulative distribution function}{}{$f$ probability density function}
|
||||
\desc[german]{Kumulative Verteilungsfunktion}{}{$f$ Wahrscheinlichkeitsdichtefunktion}
|
||||
\eq{F(x) = \int_{-\infty}^x f(t) \d t}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{pmf}
|
||||
\abbrLabel{PMF}
|
||||
\desc{Probability mass function}{Probability $p$ that \textbf{discrete} random variable $X$ has exact value $x$}{$P$ probability measure}
|
||||
\desc[german]{Wahrscheinlichkeitsfunktion / Zählfunktion}{Wahrscheinlichkeit $p$ dass eine \textbf{diskrete} Zufallsvariable $X$ einen exakten Wert $x$ annimmt}{}
|
||||
\eq{p_X(x) = P(X = x)}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{autocorrelation} \absLabel
|
||||
\desc{Autocorrelation}{Correlation of $f$ to itself at an earlier point in time, $C$ is a covariance function}{$\tau$ lag-time}
|
||||
\desc[german]{Autokorrelation}{Korrelation vonn $f$ zu sich selbst zu einem früheren Zeitpunkt. $C$ ist auch die Kovarianzfunktion}{$\tau$ Zeitverschiebung}
|
||||
\eq{C_A(\tau) &= \lim_{T \to \infty} \frac{1}{2T}\int_{-T}^{T} f(t+\tau) f(t) \d t) \\ &= \braket{f(t+\tau)\cdot f(t)}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{binomial_coefficient}
|
||||
\desc{Binomial coefficient}{Number of possibilitites of choosing $k$ objects out of $n$ objects\\}{}
|
||||
\desc[german]{Binomialkoeffizient}{Anzahl der Möglichkeiten, $k$ aus $n$ zu wählen\\ "$n$ über $k$"}{}
|
||||
\eq{\binom{n}{k} = \frac{n!}{k!(n-k)!}}
|
||||
\end{formula}
|
||||
|
||||
\Subsection{distributions}
|
||||
\desc{Distributions}{}{}
|
||||
\desc[german]{Verteilungen}{}{}
|
||||
\Subsubsection{cont}
|
||||
\desc{Continuous probability distributions}{}{}
|
||||
\desc[german]{Kontinuierliche Wahrscheinlichkeitsverteilungen}{}{}
|
||||
\begin{bigformula}{normal}
|
||||
\absLabel[normal_distribution]
|
||||
\desc{Gauß/Normal distribution}{}{}
|
||||
\desc[german]{Gauß/Normal-Verteilung}{}{}
|
||||
\fsplit[\distleftwidth]{
|
||||
\centering
|
||||
\includegraphics{img/distribution_gauss.pdf}
|
||||
}{
|
||||
\begin{distribution}
|
||||
\disteq{parameters}{\mu \in \R,\quad \sigma^2 \in \R}
|
||||
\disteq{support}{x \in \R}
|
||||
\disteq{pdf}{\frac{1}{\sqrt{2\pi\sigma^2}}\exp \left(-\frac{(x-\mu)^2}{2\sigma^2}\right)}
|
||||
\disteq{cdf}{\frac{1}{2}\left[1 + \erf \left(\frac{x-\mu}{\sqrt{2}\sigma}\right)\right]}
|
||||
\disteq{mean}{\mu}
|
||||
\disteq{median}{\mu}
|
||||
\disteq{variance}{\sigma^2}
|
||||
\end{distribution}
|
||||
}
|
||||
\end{bigformula}
|
||||
|
||||
\begin{formula}{standard_normal}
|
||||
\absLabel[standard_normal_distribution]
|
||||
\desc{Density function of the standard normal distribution}{$\mu = 0$, $\sigma = 1$}{}
|
||||
\desc[german]{Dichtefunktion der Standard-Normalverteilung}{$\mu = 0$, $\sigma = 1$}{}
|
||||
\eq{\varphi(x) = \frac{1}{\sqrt{2\pi}} \e^{-\frac{1}{2}x^2}}
|
||||
\end{formula}
|
||||
|
||||
\begin{bigformula}{multivariate_normal}
|
||||
\absLabel[multivariate_normal_distribution]
|
||||
\desc{Multivariate normal distribution}{Multivariate Gaussian distribution}{$\vec{\mu}$ \absRef{mean}, $\mat{\Sigma}$ \absRef{covariance}}
|
||||
\desc[german]{Mehrdimensionale Normalverteilung}{Multivariate Normalverteilung}{}
|
||||
\fsplit[0.3]{
|
||||
\TODO{k-variate normal plot}
|
||||
}{
|
||||
\begin{distribution}
|
||||
\disteq{parameters}{\vec{\mu} \in \R^k,+\quad \mat{\Sigma} \in \R^{k\times k}}
|
||||
\disteq{support}{\vec{x} \in \vec{\mu} + \text{span}(\mat{\Sigma})}
|
||||
\disteq{pdf}{\mathcal{N}(\vec{\mu}, \mat{\Sigma}) = \frac{1}{(2\pi)^{k/2}} \frac{1}{\sqrt{\det{\Sigma}}} \Exp{-\frac{1}{2} \left(\vecx-\vec{\mu}\right)^\T \mat{\Sigma}^{-1} \left(\vecx-\vec{\mu}\right)}}
|
||||
\disteq{mean}{\vec{\mu}}
|
||||
\disteq{variance}{\mat{\Sigma}}
|
||||
\end{distribution}
|
||||
}
|
||||
\end{bigformula}
|
||||
|
||||
\begin{bigformula}{laplace}
|
||||
\absLabel[laplace_distribution]
|
||||
\desc{Laplace-distribution}{Double exponential distribution}{}
|
||||
\desc[german]{Laplace-Verteilung}{Doppelexponentialverteilung}{}
|
||||
\fsplit[\distleftwidth]{
|
||||
\centering
|
||||
\includegraphics{img/distribution_laplace.pdf}
|
||||
}{
|
||||
\begin{distribution}
|
||||
\disteq{parameters}{\mu \in \R,\quad b > 0 \in \R}
|
||||
\disteq{support}{x \in \R}
|
||||
\disteq{pdf}{\frac{1}{\sqrt{2b}}\Exp{-\frac{\abs{x-\mu}}{b}}}
|
||||
% \disteq{cdf}{\frac{1}{2}\left[1 + \erf \left(\frac{x-\mu}{\sqrt{2}\sigma}\right)\right]}
|
||||
\disteq{mean}{\mu}
|
||||
\disteq{median}{\mu}
|
||||
\disteq{variance}{2b^2}
|
||||
\end{distribution}
|
||||
}
|
||||
\end{bigformula}
|
||||
|
||||
\begin{bigformula}{cauchy}
|
||||
\absLabel[lorentz_distribution]
|
||||
\desc{Cauchys / Lorentz distribution}{Also known as Cauchy-Lorentz distribution, Lorentz(ian) function, Breit-Wigner distribution.}{}
|
||||
\desc[german]{Cauchy / Lorentz-Verteilung}{Auch bekannt als Cauchy-Lorentz Verteilung, Lorentz Funktion, Breit-Wigner Verteilung.}{}
|
||||
\fsplit[\distleftwidth]{
|
||||
\centering
|
||||
\includegraphics{img/distribution_cauchy.pdf}
|
||||
}{
|
||||
\begin{distribution}
|
||||
\disteq{parameters}{x_0 \in \R,\quad \gamma \in \R}
|
||||
\disteq{support}{x \in \R}
|
||||
\disteq{pdf}{\frac{1}{\pi\gamma\left[1+\left(\frac{x-x_0}{\gamma}\right)^2\right]}}
|
||||
\disteq{cdf}{\frac{1}{\pi}\arctan\left(\frac{x-x_0}{\gamma}\right) + \frac{1}{2}}
|
||||
\disteq{mean}{\text{\GT{undefined}}}
|
||||
\disteq{median}{x_0}
|
||||
\disteq{variance}{\text{\GT{undefined}}}
|
||||
\end{distribution}
|
||||
}
|
||||
\end{bigformula}
|
||||
|
||||
\begin{bigformula}{maxwell-boltzmann}
|
||||
\absLabel[maxwell-boltzmann_distribution]
|
||||
\desc{Maxwell-Boltzmann distribution}{}{}
|
||||
\desc[german]{Maxwell-Boltzmann Verteilung}{}{}
|
||||
\fsplit[\distleftwidth]{
|
||||
\centering
|
||||
\includegraphics{img/distribution_maxwell-boltzmann.pdf}
|
||||
}{
|
||||
\begin{distribution}
|
||||
\disteq{parameters}{a > 0}
|
||||
\disteq{support}{x \in (0, \infty)}
|
||||
\disteq{pdf}{\sqrt{\frac{2}{\pi}} \frac{x^2}{a^3} \exp\left(-\frac{x^2}{2a^2}\right)}
|
||||
\disteq{cdf}{\erf \left(\frac{x}{\sqrt{2} a}\right) - \sqrt{\frac{2}{\pi}} \frac{x}{a} \exp\left({\frac{-x^2}{2a^2}}\right)}
|
||||
\disteq{mean}{2a \frac{2}{\pi}}
|
||||
% \disteq{median}{}
|
||||
\disteq{variance}{\frac{a^2(3\pi-8)}{\pi}}
|
||||
\end{distribution}
|
||||
}
|
||||
\end{bigformula}
|
||||
|
||||
|
||||
\begin{bigformula}{gamma}
|
||||
\absLabel[gamma_distribution]
|
||||
\desc{Gamma Distribution}{with $\lambda$ parameter}{$\Gamma$ \fRef{math:cal:integral:list:gamma_function}, $\gamma$ \fRef{math:cal:integral:list:lower_incomplete_gamma_function}}
|
||||
\desc[german]{Gamma Verteilung}{mit $\lambda$ Parameter}{}
|
||||
\fsplit[\distleftwidth]{
|
||||
\centering
|
||||
\includegraphics{img/distribution_gamma.pdf}
|
||||
}{
|
||||
\begin{distribution}
|
||||
\disteq{parameters}{\alpha > 0, \lambda > 0}
|
||||
\disteq{support}{x\in(0,1)}
|
||||
\disteq{pdf}{\frac{\lambda^\alpha}{\Gamma(\alpha) x^{\alpha-1} \e^{-\lambda x}}}
|
||||
\disteq{cdf}{\frac{1}{\Gamma(\alpha) \gamma(\alpha, \lambda x)}}
|
||||
\disteq{mean}{\frac{\alpha}{\lambda}}
|
||||
\disteq{variance}{\frac{\alpha}{\lambda^2}}
|
||||
\end{distribution}
|
||||
}
|
||||
\end{bigformula}
|
||||
|
||||
\begin{bigformula}{beta}
|
||||
\absLabel[beta_distribution]
|
||||
\desc{Beta Distribution}{}{$\txB$ \fRef{math:cal:integral:list:beta_function} / \fRef{math:cal:integral:list:incomplete_beta_function}}
|
||||
\desc[german]{Beta Verteilung}{}{}
|
||||
\fsplit[\distleftwidth]{
|
||||
\centering
|
||||
\includegraphics{img/distribution_beta.pdf}
|
||||
}{
|
||||
\begin{distribution}
|
||||
\disteq{parameters}{\alpha \in \R, \beta \in \R}
|
||||
\disteq{support}{x\in[0,1]}
|
||||
\disteq{pdf}{\frac{x^{\alpha-1} (1-x)^{\beta-1}}{\txB(\alpha,\beta)}}
|
||||
\disteq{cdf}{\frac{\txB(x;\alpha,\beta)}{\txB(\alpha,\beta)}}
|
||||
\disteq{mean}{\frac{\alpha}{\alpha+\beta}}
|
||||
% \disteq{median}{\frac{}{}} % pretty complicated, probably not needed
|
||||
\disteq{variance}{\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}}
|
||||
\end{distribution}
|
||||
}
|
||||
\end{bigformula}
|
||||
|
||||
|
||||
\Subsubsection{discrete}
|
||||
\desc{Discrete probability distributions}{}{}
|
||||
\desc[german]{Diskrete Wahrscheinlichkeitsverteilungen}{}{}
|
||||
\begin{bigformula}{binomial}
|
||||
\absLabel[binomial_distribution]
|
||||
\desc{Binomial distribution}{}{}
|
||||
\desc[german]{Binomialverteilung}{}{}
|
||||
\begin{ttext}
|
||||
\eng{For the number of trials going to infinity ($n\to\infty$), the binomial distribution converges to the \absRef[poisson distribution]{poisson_distribution}}
|
||||
\ger{Geht die Zahl der Versuche gegen unendlich ($n\to\infty$), konvergiert die Binomualverteilung gegen die \absRef[Poissonverteilung]{poisson_distribution}}
|
||||
\end{ttext}\\
|
||||
\fsplit[\distleftwidth]{
|
||||
\centering
|
||||
\includegraphics{img/distribution_binomial.pdf}
|
||||
}{
|
||||
\begin{distribution}
|
||||
\disteq{parameters}{n \in \Z, \quad p \in [0,1],\quad q = 1 - p}
|
||||
\disteq{support}{k \in \{0,\,1,\,\dots,\,n\}}
|
||||
\disteq{pmf}{\binom{n}{k} p^k q^{n-k}}
|
||||
% \disteq{cdf}{\text{regularized incomplete beta function}}
|
||||
\disteq{mean}{np}
|
||||
\disteq{median}{\floor{np} \text{ or } \ceil{np}}
|
||||
\disteq{variance}{npq = np(1-p)}
|
||||
\end{distribution}
|
||||
}
|
||||
\end{bigformula}
|
||||
|
||||
\begin{bigformula}{poisson}
|
||||
\absLabel[poisson_distribution]
|
||||
\desc{Poisson distribution}{}{}
|
||||
\desc[german]{Poissonverteilung}{}{}
|
||||
\fsplit[\distleftwidth]{
|
||||
\centering
|
||||
\includegraphics{img/distribution_poisson.pdf}
|
||||
}{
|
||||
\begin{distribution}
|
||||
\disteq{parameters}{\lambda \in (0,\infty)}
|
||||
\disteq{support}{k \in \N}
|
||||
\disteq{pmf}{\frac{\lambda^k \e^{-\lambda}}{k!}}
|
||||
\disteq{cdf}{\e^{-\lambda} \sum_{j=0}^{\floor{k}} \frac{\lambda^j}{j!}}
|
||||
\disteq{mean}{\lambda}
|
||||
\disteq{median}{\approx\floor*{\lambda + \frac{1}{3} - \frac{1}{50\lambda}}}
|
||||
\disteq{variance}{\lambda}
|
||||
\end{distribution}
|
||||
}
|
||||
\end{bigformula}
|
||||
|
||||
|
||||
|
||||
% TEMPLATE
|
||||
% \begin{distribution}{maxwell-boltzmann}
|
||||
% \distdesc{Maxwell-Boltzmann distribution}{}
|
||||
% \distdesc[german]{Maxwell-Boltzmann Verteilung}{}
|
||||
% \disteq{parameters}{}
|
||||
% \disteq{pdf}{}
|
||||
% \disteq{cdf}{}
|
||||
% \disteq{mean}{}
|
||||
% \disteq{median}{}
|
||||
% \disteq{variance}{}
|
||||
% \end{distribution}
|
||||
|
||||
|
||||
\Subsection{cls}
|
||||
\desc{Central limit theorem}{}{}
|
||||
\desc[german]{Zentraler Grenzwertsatz}{}{}
|
||||
\begin{ttext}
|
||||
\eng{
|
||||
Suppose $X_1, X_2, \dots$ is a sequence of independent and identically distributed random variables with $\braket{X_i} = \mu$ and $(\Delta X_i)^2 = \sigma^2 < \infty$.
|
||||
As $N$ approaches infinity, the random variables $\sqrt{N}(\bar{X}_N - \mu)$ converge to a normal distribution $\mathcal{N}(0, \sigma^2)$.
|
||||
\\ That means that the variance scales with $\frac{1}{\sqrt{N}}$ and statements become accurate for large $N$.
|
||||
}
|
||||
\ger{
|
||||
Sei $X_1, X_2, \dots$ eine Reihe unabhängiger und gleichverteilter Zufallsvariablen mit $\braket{X_i} = \mu$ und $(\Delta X_i)^2 = \sigma^2 < \infty$.
|
||||
Für $N$ gegen unendlich konvergieren die Zufallsvariablen $\sqrt{N}(\bar{X}_N - \mu)$ zu einer Normalverteilung $\mathcal{N}(0, \sigma^2)$.
|
||||
\\ Das bedeutet, dass die Schwankung mit $\frac{1}{\sqrt{N}}$ wächst und Aussagen für große $N$ scharf werden.
|
||||
}
|
||||
\end{ttext}
|
||||
|
||||
\Subsection{error}
|
||||
\desc{Propagation of uncertainty / error}{}{}
|
||||
\desc[german]{Fehlerfortpflanzung}{}{}
|
||||
\begin{formula}{generalised}
|
||||
\desc{Generalized error propagation}{}{$V$ \fRef{math:pt:covariance} matrix, $J$ \fRef{math:cal:jacobi-matrix}}
|
||||
\desc[german]{Generalisiertes Fehlerfortpflanzungsgesetz}{$V$ \fRef{math:pt:covariance} Matrix, $J$ \fRef{cal:jacobi-matrix}}{}
|
||||
\eq{V_y = J(x) \cdot V_x \cdot J^{\T} (x)}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{uncorrelated}
|
||||
\desc{Propagation of uncorrelated errors}{Linear approximation}{}
|
||||
\desc[german]{Fortpflanzung unabhängiger fehlerbehaftete Größen}{Lineare Näherung}{}
|
||||
\eq{u_y = \sqrt{ \sum_{i} \left(\pdv{y}{x_i}\cdot u_i\right)^2}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{weight}
|
||||
\desc{Weight}{Variance is a possible choice for a weight}{$\sigma$ \fRef{math:pt:variance}}
|
||||
\desc[german]{Gewicht}{Varianz ist eine mögliche Wahl für ein Gewicht}{}
|
||||
\eq{w_i = \frac{1}{\sigma_i^2}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{weighted-mean}
|
||||
\desc{Weighted mean}{}{$w_i$ \fRef{math:pt:error:weight}}
|
||||
\desc[german]{Gewichteter Mittelwert}{}{}
|
||||
\eq{\overline{x} = \frac{\sum_{i} (x_i w_i)}{\sum_i w_i}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{weighted-mean-error}
|
||||
\desc{Variance of weighted mean}{}{$w_i$ \fRef{math:pt:error:weight}}
|
||||
\desc[german]{Varianz des gewichteten Mittelwertes}{}{}
|
||||
\eq{\sigma^2_{\overline{x}} = \frac{1}{\sum_i w_i}}
|
||||
\end{formula}
|
||||
|
||||
\Subsection{mle}
|
||||
\desc{Maximum likelihood estimation}{}{}
|
||||
\desc[german]{Maximum likelihood Methode}{}{}
|
||||
\begin{formula}{likelihood}
|
||||
\desc{Likelihood function}{Likelihood of observing $x$ when parameter is $\theta$\\in general not normalized!}{$\rho$ \fRef{math:pt:pdf} $x\mapsto \rho(x|\theta)$ depending on parameter $\theta$, $\Theta$ parameter space}
|
||||
\desc[german]{Likelihood Funktion}{"Plausibilität" $x$ zu messen, wenn der Parameter $\theta$ ist\\nicht normalisiert!}{$\rho$ \fRef{math:pt:pdf} $x\mapsto \rho(x|\theta)$ hängt ab von Parameter $\theta$, $\Theta$ Parameterraum}
|
||||
\eq{L:\Theta \rightarrow [0,1], \quad \theta \mapsto \rho(x|\theta)}
|
||||
\end{formula}
|
||||
\begin{formula}{likelihood_independant}
|
||||
\desc{Likelihood function}{for independent and identically distributed random variables}{$x_i$ $n$ random variables, $\rho$ \fRef{math:pt:pdf} $x\mapsto f(x|\theta)$ depending on parameter $\theta$}
|
||||
\desc[german]{Likelihood function}{für unabhängig und identisch verteilte Zufallsvariablen}{$x_i$ $n$ Zufallsvariablen$\rho$ \fRef{math:pt:pdf} $x\mapsto f(x|\theta)$ hängt ab von Parameter $\theta$}
|
||||
\eq{L(\theta) = \prod_{i=1}^n f(x_i;\theta)}
|
||||
\end{formula}
|
||||
\begin{formula}{maximum_likelihood_estimate}
|
||||
\desc{Maximum likelihood estimate (MLE)}{Paramater for which outcome is most likely}{$L$ \fRef{math:pt:mle:likelihood}, $\theta$ parameter of a \fRef{math:pt:pdf}}
|
||||
\desc[german]{Maximum likelihood-Schätzung (MLE)}{Paramater, für den das Ergebnis am Wahrscheinlichsten ist}{$L$ \fRef{math:pt:mle:likelihood}, $\theta$ Parameter einer \fRef{math:pt:pdf}}
|
||||
\eq{\theta_\text{ML} &= \argmax_\theta L(\theta)\\ &= \argmax_\theta \log \big(L(\theta)\big)}
|
||||
\end{formula}
|
||||
|
||||
\Subsection{bayesian}
|
||||
\desc{Bayesian probability theory}{}{}
|
||||
\desc[german]{Bayessche Wahrscheinlichkeitstheorie}{}{}
|
||||
\begin{formula}{prior}
|
||||
\desc{Prior distribution}{Expected distribution before conducting the experiment}{$\theta$ parameter}
|
||||
\desc[german]{Prior Verteilung}{}{}
|
||||
\eq{p(\theta)}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{evidence}
|
||||
\desc{Evidence}{}{$p(\mathcal{D}|\theta)$ \fRef{math:pt:mle:likelihood}, $p(\theta)$ \fRef{math:pt:bayesian:prior}, $\mathcal{D}$ data set}
|
||||
% \desc[german]{}{}{}
|
||||
\eq{p(\mathcal{D}) = \int\d\theta \,p(\mathcal{D}|\theta)\,p(\theta)}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{theorem}
|
||||
\desc{Bayes' theorem}{}{$p(\theta|\mathcal{D})$ posterior distribution, $p(\mathcal{D}|\theta)$ \fRef{math:pt:mle:likelihood}, $p(\theta)$ \fRef{math:pt:bayesian:prior}, $p(\mathcal{D})$ \fRef{math:pt:bayesian:evidence}, $\mathcal{D}$ data set}
|
||||
\desc[german]{Satz von Bayes}{}{}
|
||||
\eq{p(\theta|\mathcal{D}) = \frac{p(\mathcal{D}|\theta)\,p(\theta)}{p(\mathcal{D})}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{map}
|
||||
\desc{Maximum a posterior estimation (MAP)}{}{}
|
||||
% \desc[german]{}{}{}
|
||||
\eq{\theta_\text{MAP} = \argmax_\theta p(\theta|\mathcal{D}) = \argmax_\theta p(\mathcal{D}|\theta)\,p(\theta)}
|
||||
\end{formula}
|
||||
|
||||
@ -1,84 +1,32 @@
|
||||
\Part{mech}
|
||||
\desc{Mechanics}{}{}
|
||||
\desc[german]{Mechanik}{}{}
|
||||
|
||||
|
||||
\Section{newton}
|
||||
\desc{Newton}{}{}
|
||||
\desc[german]{Newton}{}{}
|
||||
\begin{formula}{newton_laws}
|
||||
\desc{Newton's laws}{}{}
|
||||
\desc[german]{Newtonsche Gesetze}{}{}
|
||||
\ttxt{
|
||||
\eng{
|
||||
\begin{enumerate}
|
||||
\item A body remains at rest, or in motion at a constant speed in a straight line, except insofar as it is acted upon by a force
|
||||
\item $$\vec{F} = m \cdot \vec{a}$$
|
||||
\item If two bodies exert forces on each other, these force have the same magnitude but opposite directions
|
||||
$$\vec{F}_{\txA\rightarrow\txB} = -\vec{F}_{\txB\rightarrow\txA}$$
|
||||
\end{enumerate}
|
||||
|
||||
}
|
||||
\ger{
|
||||
\begin{enumerate}
|
||||
\item Ein kräftefreier Körper bleibt in Ruhe oder bewegt sich geradlinig mit konstanter Geschwindigkeit
|
||||
\item $$\vec{F} = m \cdot \vec{a}$$
|
||||
\item Eine Kraft von Körper A auf Körper B geht immer mit einer gleich große, aber entgegen gerichteten Kraft von Körper B auf Körper A einher:
|
||||
$$\vec{F}_{\txA\rightarrow\txB} = -\vec{F}_{\txB\rightarrow\txA}$$
|
||||
\end{enumerate}
|
||||
}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\Section{misc}
|
||||
\desc{Misc}{}{}
|
||||
\desc[german]{Verschiedenes}{}{}
|
||||
\begin{formula}{hook}
|
||||
\desc{Hooke's law}{}{$F$ \qtyRef{force}, $D$ \qtyRef{spring_constant}, $\Delta l$ spring length}
|
||||
\desc[german]{Hookesches Gesetz}{}{$F$ \qtyRef{force}, $D$ \qtyRef{spring_constant}, $\Delta l$ Federlänge}
|
||||
\eq{
|
||||
F = D\Delta l
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{centripetal_force}
|
||||
\desc{Centripetal force}{Force that must act to keep a mass on an arc trajectory}{}
|
||||
\desc[german]{Zentripetalkraft}{Kraft die auf einen Körper wirken muss, damit er sich auf einer gegrümmten Bahn bewegt}{}
|
||||
\eq{\vecF_\txc = m v^2 (-\vece_r) = m \vec{\omega}\times\vecv = -m\omega^2\vecr}
|
||||
\end{formula}
|
||||
\Part[
|
||||
\eng{Mechanics}
|
||||
\ger{Mechanik}
|
||||
]{mech}
|
||||
|
||||
\def\lagrange{\mathcal{L}}
|
||||
\Section{lagrange}
|
||||
\desc{Lagrange formalism}{}{}
|
||||
\desc[german]{Lagrange Formalismus}{}{}
|
||||
\begin{formula}{description}
|
||||
\desc{Description}{}{}
|
||||
\desc[german]{Beschreibung}{}{}
|
||||
\ttxt{
|
||||
\eng{The Lagrange formalism is often the most simple approach the get the equations of motion,
|
||||
because with suitable generalied coordinates obtaining the Lagrange function is often relatively easy.
|
||||
}
|
||||
\ger{Der Lagrange-Formalsismus ist oft der einfachste Weg die Bewegungsgleichungen zu erhalten,
|
||||
da das Aufstellen der Lagrange-Funktion mit geeigneten generalisierten Koordinaten oft relativ einfach ist.
|
||||
}
|
||||
\Section[
|
||||
\eng{Lagrange formalism}
|
||||
\ger{Lagrange Formalismus}
|
||||
]{lagrange}
|
||||
\begin{ttext}[desc]
|
||||
\eng{The Lagrange formalism is often the most simple approach the get the equations of motion,
|
||||
because with suitable generalied coordinates obtaining the Lagrange function is often relatively easy.
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{generalized_coordinates}
|
||||
\desc{Generalized coordinates}{}{}
|
||||
\desc[german]{Generalisierte Koordinaten}{}{}
|
||||
\absLabel
|
||||
\begin{ttext}[generalized_coords]
|
||||
\eng{
|
||||
The generalized coordinates are choosen so that the cronstraints are automatically fullfilled.
|
||||
For example, the generalized coordinate for a 2D pendelum is $q=\varphi$, with $\vec{x} = \begin{pmatrix} \cos\varphi \\ \sin\varphi \end{pmatrix}$.
|
||||
}
|
||||
\ger{
|
||||
Die generalisierten Koordinaten werden so gewählt, dass die Zwangsbedingungen automatisch erfüllt sind.
|
||||
Zum Beispiel findet man für ein 2D Pendel die generalisierte Koordinate $q=\varphi$, mit $\vec{x} = \begin{pmatrix} \cos\varphi \\ \sin\varphi \end{pmatrix}$.
|
||||
}
|
||||
\end{ttext}
|
||||
\end{formula}
|
||||
\begin{formula}{lagrangian} \absLabel
|
||||
\ger{Der Lagrange-Formalsismus ist oft der einfachste Weg die Bewegungsgleichungen zu erhalten,
|
||||
da das Aufstellen der Lagrange-Funktion mit geeigneten generalisierten Koordinaten oft relativ einfach ist.
|
||||
}
|
||||
\end{ttext}
|
||||
\begin{ttext}[generalized_coords]
|
||||
\eng{
|
||||
The generalized coordinates are choosen so that the cronstraints are automatically fullfilled.
|
||||
For example, the generalized coordinate for a 2D pendelum is $q=\varphi$, with $\vec{x} = \begin{pmatrix} \cos\varphi \\ \sin\varphi \end{pmatrix}$.
|
||||
}
|
||||
\ger{
|
||||
Die generalisierten Koordinaten werden so gewählt, dass die Zwangsbedingungen automatisch erfüllt sind.
|
||||
Zum Beispiel findet man für ein 2D Pendel die generalisierte Koordinate $q=\varphi$, mit $\vec{x} = \begin{pmatrix} \cos\varphi \\ \sin\varphi \end{pmatrix}$.
|
||||
}
|
||||
\end{ttext}
|
||||
\begin{formula}{lagrangian}
|
||||
\desc{Lagrange function}{}{$T$ kinetic energy, $V$ potential energy }
|
||||
\desc[german]{Lagrange-Funktion}{}{$T$ kinetische Energie, $V$ potentielle Energie}
|
||||
\eq{\lagrange = T - V}
|
||||
|
||||
114
src/particle.tex
114
src/particle.tex
@ -1,114 +0,0 @@
|
||||
\Part{particle}
|
||||
\desc{Particle physics}{}{}
|
||||
\desc[german]{Teilchenphysik}{}{}
|
||||
|
||||
|
||||
\begin{formula}{electron_mass}
|
||||
\desc{Electron mass}{}{}
|
||||
\desc[german]{Elektronenmasse}{}{}
|
||||
\constant{m_\txe}{exp}{
|
||||
\val{9.1093837139(28) \xE{-31}}{\kg}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{spin}
|
||||
\desc{Spin}{}{}
|
||||
\desc[german]{Spin}{}{}
|
||||
\quantity{\sigma}{}{v}
|
||||
\end{formula}
|
||||
|
||||
\begin{bigformula}{standard_model}
|
||||
\desc{Standard model}{}{}
|
||||
\desc[german]{Standartmodell}{}{}
|
||||
\centering
|
||||
|
||||
\tikzset{%
|
||||
label/.style = { black, midway, align=center },
|
||||
toplabel/.style = { label, above=.5em, anchor=south },
|
||||
leftlabel/.style = { midway, left=.5em, anchor=east },
|
||||
bottomlabel/.style = { label, below=.5em, anchor=north },
|
||||
force/.style = { rotate=-90,scale=0.4 },
|
||||
round/.style = { rounded corners=2mm },
|
||||
legend/.style = { anchor=east },
|
||||
nosep/.style = { inner sep=0pt },
|
||||
generation/.style = { anchor=base },
|
||||
brace/.style = { decoration={brace,mirror},decorate }
|
||||
}
|
||||
|
||||
% [1]: color
|
||||
% 2: symbol
|
||||
% 3: name
|
||||
% 4: mass
|
||||
% 5: spin
|
||||
% 6: charge
|
||||
% 7: colors
|
||||
\newcommand\drawParticle[7][white]{%
|
||||
\begin{tikzpicture}[x=2.2cm, y=2.2cm]
|
||||
% \path[fill=#1,blur shadow={shadow blur steps=5}] (0,0) -- (1,0) -- (1,1) -- (0,1) -- cycle;
|
||||
% \path[fill=#1,stroke=black,blur shadow={shadow blur steps=5},rounded corners] (0,0) rectangle (1,1);
|
||||
\path[fill=#1!20!bg0,draw=#1,thick] (0.02,0.02) rectangle (0.98,0.98);
|
||||
\node at(0.92, 0.50) [nosep,anchor=east]{\Large #2};
|
||||
% \node at(0.95, 0.15) [nosep,anchor=south east]{\footnotesize #3};
|
||||
\node at(0.05, 0.15) [nosep,anchor=south west]{\footnotesize #3};
|
||||
% \ifstrempty{#2}{}{\node at(0) [nosep,anchor=west,scale=1.5] {#2};}
|
||||
% \ifstrempty{#3}{}{\node at(0.1,-0.85) [nosep,anchor=west,scale=0.3] {#3};}
|
||||
\ifstrempty{#4}{}{\node at(0.05,0.85) [nosep,anchor=west] {\footnotesize #4};}
|
||||
\ifstrempty{#5}{}{\node at(0.05,0.70) [nosep,anchor=west] {\footnotesize #5};}
|
||||
\ifstrempty{#6}{}{\node at(0.05,0.55) [nosep,anchor=west] {\footnotesize #6};}
|
||||
% \ifstrempty{#7}{}{\node at(0.05,0.40) [nosep,anchor=west] {\footnotesize #7};}
|
||||
\end{tikzpicture}
|
||||
}
|
||||
\def\colorLepton{bg-aqua}
|
||||
\def\colorQuarks{bg-purple}
|
||||
\def\colorGauBos{bg-red}
|
||||
\def\colorScaBos{bg-yellow}
|
||||
\eng[quarks]{Quarks}
|
||||
\ger[quarks]{Quarks}
|
||||
\eng[leptons]{Leptons}
|
||||
\ger[leptons]{Leptonen}
|
||||
\eng[fermions]{Fermions}
|
||||
\ger[fermions]{Fermionen}
|
||||
\eng[bosons]{Bosons}
|
||||
\ger[bosons]{Bosonen}
|
||||
|
||||
\begin{tikzpicture}[x=2.2cm, y=2.2cm]
|
||||
\node at(0, 0) {\drawParticle[\colorQuarks]{$u$} {up} {$2.3$ MeV}{1/2}{$2/3$}{R/G/B}};
|
||||
\node at(0,-1) {\drawParticle[\colorQuarks]{$d$} {down} {$4.8$ MeV}{1/2}{$-1/3$}{R/G/B}};
|
||||
\node at(0,-2) {\drawParticle[\colorLepton]{$e$} {electron} {$511$ keV}{1/2}{$-1$}{}};
|
||||
\node at(0,-3) {\drawParticle[\colorLepton]{$\nu_e$} {$e$ neutrino} {$<2.2$ eV}{1/2}{0}{}};
|
||||
\node at(1, 0) {\drawParticle[\colorQuarks]{$c$} {charm} {$1.275$ GeV}{1/2}{$2/3$}{R/G/B}};
|
||||
\node at(1,-1) {\drawParticle[\colorQuarks]{$s$} {strange} {$95$ MeV}{1/2}{$-1/3$}{R/G/B}};
|
||||
\node at(1,-2) {\drawParticle[\colorLepton]{$\mu$} {muon} {$105.7$ MeV}{1/2}{$-1$}{}};
|
||||
\node at(1,-3) {\drawParticle[\colorLepton]{$\nu_\mu$} {$\mu$ neutrino}{$<170$ keV}{1/2}{0}{}};
|
||||
\node at(2, 0) {\drawParticle[\colorQuarks]{$t$} {top} {$173.2$ GeV}{1/2}{$2/3$}{R/G/B}};
|
||||
\node at(2,-1) {\drawParticle[\colorQuarks]{$b$} {bottom} {$4.18$ GeV}{1/2}{$-1/3$}{R/G/B}};
|
||||
\node at(2,-2) {\drawParticle[\colorLepton]{$\tau$} {tau} {$1.777$ GeV}{1/2}{$-1$}{}};
|
||||
\node at(2,-3) {\drawParticle[\colorLepton]{$\nu_\tau$} {$\tau$ neutrino} {$<15.5$ MeV}{1/2}{0}{}};
|
||||
\node at(3, 0) {\drawParticle[\colorGauBos]{$g$} {gluon} {0}{1}{0}{color}};
|
||||
\node at(3,-1) {\drawParticle[\colorGauBos]{$\gamma$} {photon} {0}{1}{0}{}};
|
||||
\node at(3,-2) {\drawParticle[\colorGauBos]{$Z$} {} {$91.2$ GeV}{1}{0}{}};
|
||||
\node at(3,-3) {\drawParticle[\colorGauBos]{$W_\pm$} {} {$80.4$ GeV}{1}{$\pm1$}{}};
|
||||
\node at(4,0) {\drawParticle[\colorScaBos]{$H$} {Higgs} {$125.1$ GeV}{0}{0}{}};
|
||||
|
||||
\draw [->] (-0.7, 0.35) node [legend] {\qtyRef{mass}} -- (-0.5, 0.35);
|
||||
\draw [->] (-0.7, 0.20) node [legend] {\qtyRef{spin}} -- (-0.5, 0.20);
|
||||
\draw [->] (-0.7, 0.05) node [legend] {\qtyRef{charge}} -- (-0.5, 0.05);
|
||||
\draw [->] (-0.7,-0.10) node [legend] {\GT{colors}} -- (-0.5,-0.10);
|
||||
|
||||
\draw [brace,draw=\colorQuarks] (-0.55, 0.5) -- (-0.55,-1.5) node[leftlabel,color=\colorQuarks] {\gt{quarks}};
|
||||
\draw [brace,draw=\colorLepton] (-0.55,-1.5) -- (-0.55,-3.5) node[leftlabel,color=\colorLepton] {\gt{leptons}};
|
||||
\draw [brace] (-0.5,-3.55) -- ( 2.5,-3.55) node[bottomlabel] {\gt{fermions}};
|
||||
\draw [brace] ( 2.5,-3.55) -- ( 4.5,-3.55) node[bottomlabel] {\gt{bosons}};
|
||||
|
||||
|
||||
\draw [brace] (0.5,0.55) -- (-0.5,0.55) node[toplabel] {\small standard matter};
|
||||
\draw [brace] (2.5,0.55) -- ( 0.5,0.55) node[toplabel] {\small unstable matter};
|
||||
\draw [brace] (4.5,0.55) -- ( 2.5,0.55) node[toplabel] {\small force carriers};
|
||||
|
||||
\node at (0,0.85) [generation] {\small I};
|
||||
\node at (1,0.85) [generation] {\small II};
|
||||
\node at (2,0.85) [generation] {\small III};
|
||||
\node at (1,1.05) [generation] {\small generation};
|
||||
\end{tikzpicture}
|
||||
|
||||
\end{bigformula}
|
||||
@ -1,72 +0,0 @@
|
||||
\ProvidesPackage{mqconstant}
|
||||
\RequirePackage{mqlua}
|
||||
\RequirePackage{etoolbox}
|
||||
|
||||
\begin{luacode}
|
||||
constants = {}
|
||||
function constantAdd(key, symbol, exp_or_def, fqname)
|
||||
constants[key] = {
|
||||
["symbol"] = symbol,
|
||||
["units"] = units,
|
||||
["exp_or_def"] = exp_or_def,
|
||||
["values"] = {} -- array of {value, unit}
|
||||
}
|
||||
if fqname == "" then
|
||||
constants[key]["fqname"] = fqnameGet()
|
||||
else
|
||||
constants[key]["fqname"] = fqname
|
||||
end
|
||||
end
|
||||
function constantAddValue(key, value, unit)
|
||||
table.insert(constants[key]["values"], { value = value, unit = unit })
|
||||
end
|
||||
function constantGetSymbol(key)
|
||||
local const = constants[key]
|
||||
if const == nil then return "???" end
|
||||
local symbol = const["symbol"]
|
||||
if symbol == nil then return "???" end
|
||||
return symbol
|
||||
end
|
||||
function constantGetFqname(key)
|
||||
local const = constants[key]
|
||||
if const == nil then return "const:"..key end
|
||||
local fqname_ = const["fqname"]
|
||||
if fqname_ == nil then return "const:"..key end
|
||||
return fqname_
|
||||
end
|
||||
\end{luacode}
|
||||
|
||||
% [1]: label to point to
|
||||
% 2: key
|
||||
% 3: symbol
|
||||
% 4: either exp or def; experimentally or defined constant
|
||||
\newcommand{\constant@new}[4][]{%
|
||||
\directLuaAuxExpand{constantAdd(\luastring{#2}, \luastringN{#3}, \luastringN{#4}, \luastring{#1})}%
|
||||
}
|
||||
|
||||
% 1: key
|
||||
% 2: value
|
||||
% 3: units
|
||||
\newcommand{\constant@addValue}[3]{%
|
||||
\directlua{constantAddValue(\luastring{#1}, \luastringN{#2}, \luastringN{#3})}%
|
||||
}
|
||||
% 1: key
|
||||
\newcommand{\constant@getSymbol}[1]{\luavar{constantGetSymbol(\luastring{#1})}}
|
||||
|
||||
% 1: key
|
||||
\newcommand\constant@print[1]{
|
||||
\begingroup % for label
|
||||
Symbol: $\luavar{constants["#1"]["symbol"]}$
|
||||
% \\Unit: $\directlua{split_and_print_units(constants["#1"]["units"])}$
|
||||
\directlua{
|
||||
tex.print("\\\\\\GT{const:"..constants["#1"]["exp_or_def"].."}")
|
||||
}
|
||||
\directlua{
|
||||
%--tex.sprint("Hier steht Luatext" .. ":", #constVals)
|
||||
for i, pair in ipairs(constants["#1"]["values"]) do
|
||||
tex.sprint("\\\\\\hspace*{1cm}${", pair["value"], "}\\,\\si{", pair["unit"], "}$")
|
||||
%--tex.sprint("VALUE ", i, v)
|
||||
end
|
||||
}
|
||||
\endgroup
|
||||
}
|
||||
@ -1,310 +0,0 @@
|
||||
\ProvidesPackage{mqformula}
|
||||
|
||||
\RequirePackage{mqfqname}
|
||||
\RequirePackage{mqsections}
|
||||
\RequirePackage{mqconstant}
|
||||
\RequirePackage{mqquantity}
|
||||
\RequirePackage{framed} % for leftbar
|
||||
|
||||
\newlength\mqformula@formulaBoxWidth
|
||||
\setlength\mqformula@formulaBoxWidth{\the\dimexpr\textwidth * 68/100\relax}
|
||||
\newlength\mqformula@formulaBoxLeftMargin
|
||||
\setlength\mqformula@formulaBoxLeftMargin{1cm}
|
||||
\newlength\mqformula@ruleWidth
|
||||
\setlength\mqformula@ruleWidth{0.5pt}
|
||||
\newlength\mqformula@ruleHMargin
|
||||
\setlength\mqformula@ruleHMargin{0.8pt}
|
||||
\newlength\mqformula@groupBarWidth
|
||||
\setlength\mqformula@groupBarWidth{\mqformula@ruleWidth+\mqformula@ruleHMargin}
|
||||
\newlength\mqformala@descriptionWidth
|
||||
|
||||
% The box should have a constant width, so the description width has to adapt
|
||||
% For example, it is smaller in a formulagroup because of the left bar
|
||||
\def\descwidth{\the\dimexpr\linewidth-\mqformula@formulaBoxWidth - \mqformula@formulaBoxLeftMargin\relax}
|
||||
% \def\descwidth{0.2\textwidth}
|
||||
|
||||
% STYLING
|
||||
% apply some transformation on the formula name/description/definitions, eg bold, color...
|
||||
\newcommand\mqformula@nameTransform[1]{#1}
|
||||
% \newcommand\mqformula@nameTransform[1]{\textbf{#1}}
|
||||
% \newcommand\mqformula@nameTransform[1]{{\color{fg-blue}#1}}
|
||||
\newcommand\mqformula@groupNameTransform[1]{\textbf{#1}}
|
||||
% \newcommand\mqformula@groupNameTransform[1]{{\color{fg-blue}\textbf{#1}}}
|
||||
|
||||
\newcommand\mqformula@descTransform[1]{{\color{fg2}#1}}
|
||||
\newcommand\mqformula@defsTransform[1]{{\color{fg2}#1}}
|
||||
|
||||
%
|
||||
% FORMULA ENVIRONMENT
|
||||
% The following commands are meant to be used with the formula environment
|
||||
%
|
||||
|
||||
% Name in black and below description in gray
|
||||
% [1]: minipage width
|
||||
% 2: fqname of name
|
||||
% 3: fqname of a translation that holds the explanation
|
||||
\newcommand{\NameWithDescription}[3][\descwidth]{%
|
||||
\makeatletter%
|
||||
\begin{minipage}{#1}
|
||||
% {\color{red}\hrule}
|
||||
\raggedright\mqformula@nameTransform{\GT{#2}}%
|
||||
\IfTranslationExists{#3}{%
|
||||
\\ \mqformula@defsTransform{\GT{#3}}%
|
||||
}{}
|
||||
\end{minipage}%
|
||||
\makeatother%
|
||||
}
|
||||
|
||||
|
||||
% TODO: rename
|
||||
\newsavebox{\contentBoxBox}
|
||||
% [1]: minipage width
|
||||
% 2: fqname of a translation that holds the explanation
|
||||
\newenvironment{ContentBoxWithExplanation}[2][\the\mqformula@formulaBoxWidth]{%
|
||||
\def\ContentFqName{#2}%
|
||||
\begin{lrbox}{\contentBoxBox}%
|
||||
\begin{minipage}{#1}
|
||||
}{
|
||||
\makeatletter%
|
||||
\IfTranslationExists{\ContentFqName}{%
|
||||
\smartnewline\noindent\raggedright%
|
||||
\mqformula@descTransform{\GT{\ContentFqName}}%
|
||||
}{}%
|
||||
\makeatother%
|
||||
\end{minipage}
|
||||
\end{lrbox}%
|
||||
\fbox{\usebox{\contentBoxBox}}%
|
||||
}
|
||||
|
||||
% defines commands that may be used in formulas and formulagroups
|
||||
% None of these commands may output anything
|
||||
% 1: key
|
||||
\newenvironment{mqformulaInvisibleCommands}[1]{
|
||||
% makes this formula referencable with \abbrRef{<name>}
|
||||
% [1]: label to use, defaults to formula key
|
||||
% 2: Abbreviation to use for references
|
||||
\forceNewCommand{\abbrLabel}[2][#1]{
|
||||
\abbrLink[\fqname]{##1}{##2}
|
||||
}
|
||||
% makes this formula referencable with \absRef{<name>}
|
||||
% [1]: <name> to use, defaults to formula key
|
||||
\forceNewCommand{\absLabel}[1][#1]{
|
||||
\absLink[\fqname]{\fqname}{##1}
|
||||
}
|
||||
|
||||
% [1]: key, defaults to formula key
|
||||
% 2: symbol
|
||||
% 3: units
|
||||
% 4: comment key to translation
|
||||
\forceNewCommand{\hiddenQuantity}[4][#1]{%
|
||||
\quantity@new[\fqname]{##1}{##2}{##3}{##4}
|
||||
}
|
||||
}{}
|
||||
|
||||
% Class defining commands shared by all formula environments
|
||||
% 1: key
|
||||
\newenvironment{mqformulaCommands}[1]{
|
||||
\begin{mqformulaInvisibleCommands}{#1}
|
||||
\directlua{n_formulaEntries = 0}
|
||||
|
||||
\newcommand{\newFormulaEntry}{
|
||||
\directlua{
|
||||
if n_formulaEntries > 0 then
|
||||
tex.print("\\vspace{0.3\\baselineskip}\\hrule\\vspace{0.3\\baselineskip}")
|
||||
end
|
||||
n_formulaEntries = n_formulaEntries + 1
|
||||
}
|
||||
% \par\noindent\ignorespaces
|
||||
}
|
||||
% 1: equation for align environment
|
||||
\newcommand{\eq}[1]{
|
||||
\newFormulaEntry
|
||||
\begin{align}
|
||||
% \label{eq:\fqname:#1}
|
||||
##1
|
||||
\end{align}
|
||||
}
|
||||
% 1: equation for flalign environment
|
||||
\newcommand{\eqFLAlign}[2]{
|
||||
\newFormulaEntry
|
||||
\begin{alignat}{##1}%
|
||||
% dont place label when one is provided
|
||||
% \IfSubStringInString{label}\unexpanded{#3}{}{
|
||||
% \label{eq:#1}
|
||||
% }
|
||||
##2%
|
||||
\end{alignat}
|
||||
}
|
||||
\newcommand{\fig}[2][]{
|
||||
\newFormulaEntry
|
||||
\centering
|
||||
\includegraphics[##1]{##2}
|
||||
}
|
||||
% 1: content for the ttext environment
|
||||
\newcommand{\ttxt}[2][text]{
|
||||
\newFormulaEntry
|
||||
\begin{ttext}[##1]
|
||||
##2
|
||||
\end{ttext}
|
||||
}
|
||||
% [1]: key, defaults to formula key
|
||||
% 2: symbol
|
||||
% 3: units
|
||||
% 4: comment key to translation
|
||||
\newcommand{\quantity}[4][#1]{%
|
||||
\quantity@new[\fqname]{##1}{##2}{##3}{##4}
|
||||
\newFormulaEntry
|
||||
\quantity@print{##1}
|
||||
}
|
||||
|
||||
% must be used only in third argument of "constant" command
|
||||
% 1: value
|
||||
% 2: unit
|
||||
\newcommand{\val}[2]{
|
||||
\constant@addValue{#1}{##1}{##2}
|
||||
}
|
||||
% 1: symbol
|
||||
% 2: either exp or def; experimentally or defined constant
|
||||
% 3: one or more \val{value}{unit} commands
|
||||
\newcommand{\constant}[4][#1]{
|
||||
\constant@new[\fqname]{##1}{##2}{##3}
|
||||
\begingroup
|
||||
##4
|
||||
\endgroup
|
||||
\newFormulaEntry
|
||||
\constant@print{##1}
|
||||
}
|
||||
|
||||
\newcommand{\fsplit}[3][0.5]{
|
||||
\newFormulaEntry
|
||||
\begingroup
|
||||
\renewcommand{\newFormulaEntry}{}
|
||||
\begin{minipage}{##1\linewidth}
|
||||
##2
|
||||
\end{minipage}
|
||||
\begin{minipage}{\luavar{0.99-##1}\linewidth}
|
||||
##3
|
||||
\end{minipage}
|
||||
\endgroup
|
||||
}
|
||||
|
||||
\newcommand{\fcenter}[1]{%
|
||||
\newFormulaEntry \centering %
|
||||
##1%
|
||||
}
|
||||
}{
|
||||
\end{mqformulaInvisibleCommands}
|
||||
}
|
||||
|
||||
\newenvironment{formula}[1]{
|
||||
\begin{sectionEntry}{#1}
|
||||
\begin{mqformulaCommands}{#1}
|
||||
\par\noindent\ignorespaces
|
||||
% {\color{red}\hrule}
|
||||
\NameWithDescription[\descwidth]{\fqname}{\fqname:desc}
|
||||
\hfill
|
||||
\begin{ContentBoxWithExplanation}{\fqname:defs}
|
||||
}{
|
||||
\end{ContentBoxWithExplanation}
|
||||
\end{mqformulaCommands}
|
||||
\end{sectionEntry}
|
||||
}
|
||||
|
||||
|
||||
% BIG FORMULA
|
||||
\newenvironment{bigformula}[1]{
|
||||
\begin{sectionEntry}{#1}
|
||||
\begin{mqformulaCommands}{#1}
|
||||
\makeatletter%
|
||||
\par\noindent
|
||||
\begin{minipage}{\textwidth} % using a minipage to now allow line breaks within the bigformula
|
||||
\par\noindent\ignorespaces
|
||||
% name
|
||||
\raggedright\mqformula@nameTransform{\GT{\fqname}}\ignorespaces%
|
||||
% description
|
||||
\IfTranslationExists{\fqname:desc}{\ignorespaces%
|
||||
: \mqformula@descTransform{\GT{\fqname:desc}}
|
||||
}{}
|
||||
\hfill
|
||||
\par
|
||||
}{
|
||||
\IfTranslationExists{\fqname:defs}{%
|
||||
\smartnewline
|
||||
\noindent
|
||||
\mqformula@defsTransform{\GT{\fqname:defs}}
|
||||
}{}
|
||||
\end{minipage}
|
||||
\makeatother%
|
||||
\end{mqformulaCommands}
|
||||
\end{sectionEntry}
|
||||
}
|
||||
|
||||
% GROUP
|
||||
\newenvironment{formulagroup}[1]{
|
||||
\begin{mqformulaInvisibleCommands}{#1}
|
||||
\begin{sectionEntry}{#1}
|
||||
% set nEntry to zero to prevent additional separations within the group
|
||||
\def\oldNEntry{\value{nEntry}}
|
||||
\setcounter{nEntry}{0}
|
||||
|
||||
% adapted from framed - leftbar
|
||||
\def\FrameCommand{{\color{bg4}\vrule width \mqformula@ruleWidth} \hspace{\mqformula@ruleHMargin}}%
|
||||
\MakeFramed {\advance\hsize-\width \FrameRestore}%
|
||||
|
||||
\makeatletter%
|
||||
\par\noindent
|
||||
\begin{minipage}{\textwidth-\mqformula@groupBarWidth}
|
||||
\mqfqname@label
|
||||
\par\noindent\ignorespaces
|
||||
% \textcolor{gray}{\hrule}
|
||||
% \vspace{0.5\baselineskip}
|
||||
% name
|
||||
\raggedright\mqformula@groupNameTransform{\GT{\fqname}}\ignorespaces%
|
||||
\IfTranslationExists{\fqname:desc}{%
|
||||
: \mqformula@defsTransform{\GT{\fqname:desc}}
|
||||
}{}%
|
||||
\hfill
|
||||
\par
|
||||
}{
|
||||
\vspace{0.5\baselineskip}
|
||||
\IfTranslationExists{\fqname:defs}{%
|
||||
\smartnewline\noindent\raggedright%
|
||||
\mqformula@defsTransform{\GT{\fqname:defs}}%
|
||||
}{}
|
||||
\end{minipage}
|
||||
\makeatother%
|
||||
\endMakeFramed
|
||||
\setcounter{nEntry}{\oldNEntry}
|
||||
\end{sectionEntry}
|
||||
\end{mqformulaInvisibleCommands}
|
||||
}
|
||||
|
||||
\newenvironment{hiddenformula}[1]{
|
||||
\begin{mqformulaCommands}{#1}
|
||||
\mqfqname@enter{#1}
|
||||
\renewcommand{\eq}[1]{}
|
||||
\renewcommand{\eqFLAlign}[2]{}
|
||||
\renewcommand{\fig}[2][]{}
|
||||
\renewcommand{\ttxt}[2][#1:desc]{}
|
||||
% [1]: key
|
||||
% 2: symbol
|
||||
% 3: units
|
||||
% 4: comment key to translation
|
||||
\renewcommand{\quantity}[4][[#1]]{%
|
||||
\quantity@new[\fqname]{##1}{##2}{##3}{##4}
|
||||
}
|
||||
% [1]: key
|
||||
% 2: symbol
|
||||
% 3: either exp or def; experimentally or defined constant
|
||||
% 4: one or more \val{value}{unit} commands
|
||||
\renewcommand{\constant}[4][#1]{
|
||||
\constant@new[\fqname]{##1}{##2}{##3}
|
||||
\begingroup
|
||||
##4
|
||||
\endgroup
|
||||
}
|
||||
}{
|
||||
\mqfqname@leave
|
||||
\end{mqformulaCommands}
|
||||
}
|
||||
|
||||
@ -1,100 +0,0 @@
|
||||
\ProvidesPackage{mqfqname}
|
||||
\edef\fqname{NULL}
|
||||
\RequirePackage{mqlua}
|
||||
\RequirePackage{etoolbox}
|
||||
|
||||
% make newcommand work on already defined commands
|
||||
\def\forceNewCommand#1{\let#1\undefined\newcommand#1}
|
||||
|
||||
\begin{luacode}
|
||||
sections = sections or {}
|
||||
|
||||
function fqnameEnter(name)
|
||||
table.insert(sections, name)
|
||||
-- table.sort(sections)
|
||||
end
|
||||
|
||||
function fqnameLeave()
|
||||
if table.getn(sections) > 0 then
|
||||
table.remove(sections)
|
||||
end
|
||||
end
|
||||
|
||||
function fqnameGet()
|
||||
return table.concat(sections, ":")
|
||||
end
|
||||
|
||||
function fqnameLeaveOnlyFirstN(n)
|
||||
if n >= 0 then
|
||||
while table.getn(sections) > n do
|
||||
table.remove(sections)
|
||||
end
|
||||
end
|
||||
end
|
||||
\end{luacode}
|
||||
|
||||
\begin{luacode}
|
||||
function fqnameGetDepth()
|
||||
return table.getn(sections)
|
||||
end
|
||||
|
||||
function fqnameGetN(N)
|
||||
if N == nil or table.getn(sections) < N then
|
||||
luatexbase.module_warning('fqnameGetN', 'N = ' .. N .. ' is larger then the table length')
|
||||
return "?!?"
|
||||
end
|
||||
s = sections[1]
|
||||
for i = 2, N do
|
||||
s = s .. ":" .. sections[i]
|
||||
end
|
||||
return s
|
||||
end
|
||||
\end{luacode}
|
||||
|
||||
|
||||
% Allow using :<key>, ::<key> and so on
|
||||
% where : points to current fqname, :: to the upper one and so on
|
||||
\begin{luacode*}
|
||||
function translateRelativeFqname(target)
|
||||
local relN = 0
|
||||
|
||||
local relTarget = ""
|
||||
-- warning('translateRelativeFqname', '(target=' .. target .. ') ');
|
||||
for i = 1, #target do
|
||||
local c = target:sub(i,i)
|
||||
if c == ":" then
|
||||
relN = relN + 1
|
||||
else
|
||||
relTarget = target:sub(i,#target)
|
||||
break
|
||||
end
|
||||
end
|
||||
if relN == 0 then
|
||||
return target
|
||||
end
|
||||
|
||||
local N = fqnameGetDepth()
|
||||
local newtarget = fqnameGetN(N - relN + 1) .. ":" .. relTarget
|
||||
-- warning('translateRelativeFqname', '(relN=' .. relN .. ') ' .. newtarget);
|
||||
return newtarget
|
||||
end
|
||||
\end{luacode*}
|
||||
|
||||
\newcommand{\mqfqname@update}{%
|
||||
\edef\fqname{\luavar{fqnameGet()}} %
|
||||
}
|
||||
\newcommand{\mqfqname@enter}[1]{%
|
||||
\directlua{fqnameEnter("\luaescapestring{#1}")}%
|
||||
\mqfqname@update
|
||||
}
|
||||
\newcommand{\mqfqname@leave}{%
|
||||
\directlua{fqnameLeave()}%
|
||||
\mqfqname@update
|
||||
}
|
||||
|
||||
\newcommand{\mqfqname@leaveOnlyFirstN}[1]{%
|
||||
\directlua{fqnameLeaveOnlyFirstN(#1)}%
|
||||
}
|
||||
|
||||
|
||||
|
||||
@ -1,93 +0,0 @@
|
||||
\ProvidesPackage{mqlua}
|
||||
|
||||
\RequirePackage{luacode}
|
||||
\LuaCodeDebugOn
|
||||
|
||||
\newcommand\luavar[1]{\directlua{tex.sprint(#1)}}
|
||||
|
||||
\begin{luacode*}
|
||||
function warning(fname, message)
|
||||
-- Get the current file name and line number
|
||||
-- local info = debug.getinfo(2, "Sl")
|
||||
-- local file_name = info.source
|
||||
-- local line_number = info.currentline
|
||||
-- tex.error(string.format("Warning %s at %s:%d", message, file_name, line_number))
|
||||
if message == nil then
|
||||
texio.write("\nWARNING: " .. fname .. "\n")
|
||||
else
|
||||
texio.write("\nWARNING: in " .. fname .. ":" .. message .. "\n")
|
||||
end
|
||||
end
|
||||
|
||||
OUTDIR = os.getenv("TEXMF_OUTPUT_DIRECTORY") or "."
|
||||
|
||||
function fileExists(file)
|
||||
local f = io.open(file, "rb")
|
||||
if f then f:close() end
|
||||
return f ~= nil
|
||||
end
|
||||
warning("TEST")
|
||||
\end{luacode*}
|
||||
|
||||
% units: siunitx units arguments, possibly chained by '='
|
||||
% returns: 1\si{unit1} = 1\si{unit2} = ...
|
||||
\begin{luacode*}
|
||||
function split_and_print_units(units)
|
||||
if units == nil or units == "" then
|
||||
tex.sprint("1")
|
||||
return
|
||||
end
|
||||
|
||||
local parts = {}
|
||||
for part in string.gmatch(units, "[^=]+") do
|
||||
table.insert(parts, part)
|
||||
end
|
||||
local result = ""
|
||||
for i, unit in ipairs(parts) do
|
||||
if i > 1 then result = result .. " = " end
|
||||
result = result .. "\\SI{1}{" .. unit .. "}"
|
||||
end
|
||||
tex.print(result)
|
||||
end
|
||||
\end{luacode*}
|
||||
|
||||
% STRING UTILITY
|
||||
\begin{luacode*}
|
||||
function string.startswith(s, start)
|
||||
return string.sub(s,1,string.len(start)) == start
|
||||
end
|
||||
|
||||
function string.sanitize(s)
|
||||
-- Use gsub to replace the specified characters with an empty string
|
||||
local result = s:gsub("[_^&]", " ")
|
||||
return result
|
||||
end
|
||||
\end{luacode*}
|
||||
% Write directlua command to aux and run it as well
|
||||
% THESE CAN ONLY BE RUN BETWEEN \begin{document} and \end{document}
|
||||
% This one expands the argument in the aux file:
|
||||
\newcommand\directLuaAuxExpand[1]{
|
||||
\immediate\write\luaAuxFile{\noexpand\directlua{#1}}
|
||||
\directlua{#1}
|
||||
}
|
||||
% This one does not:
|
||||
\newcommand\directLuaAux[1]{
|
||||
\immediate\write\luaAuxFile{\noexpand\directlua{\detokenize{#1}}}
|
||||
\directlua{#1}
|
||||
}
|
||||
|
||||
% read
|
||||
\AtBeginDocument{
|
||||
\IfFileExists{\jobname.lua.aux}{%
|
||||
\input{\jobname.lua.aux}%
|
||||
}{%
|
||||
% \@latex@warning@no@line{"Lua aux not loaded!"}
|
||||
}
|
||||
% write
|
||||
\newwrite\luaAuxFile
|
||||
\immediate\openout\luaAuxFile=\jobname.lua.aux
|
||||
\immediate\write\luaAuxFile{\noexpand\def\noexpand\luaAuxLoaded{True}}%
|
||||
}
|
||||
\def\luaAuxLoaded{False}
|
||||
|
||||
\AtEndDocument{\immediate\closeout\luaAuxFile}
|
||||
@ -1,175 +0,0 @@
|
||||
\ProvidesPackage{mqperiodictable}
|
||||
\RequirePackage{mqtranslation}
|
||||
\RequirePackage{mqlua}
|
||||
% Store info about elements in a lua table
|
||||
% Print as list or as periodic table
|
||||
% The data is taken from https://pse-info.de/de/data as json and parsed by the scripts/periodic_table.py
|
||||
|
||||
\begin{luacode}
|
||||
elements = {}
|
||||
elementsOrder = {}
|
||||
|
||||
function elementAdd(symbol, nr, period, column)
|
||||
--elementsOrder[nr] = symbol
|
||||
table.insert(elementsOrder, symbol)
|
||||
elements[symbol] = {
|
||||
symbol = symbol,
|
||||
atomic_number = nr,
|
||||
period = period,
|
||||
column = column,
|
||||
properties = {}
|
||||
}
|
||||
end
|
||||
|
||||
function elementAddProperty(symbol, key, value)
|
||||
if elements[symbol] and elements[symbol].properties then
|
||||
elements[symbol].properties[key] = value
|
||||
end
|
||||
end
|
||||
\end{luacode}
|
||||
|
||||
% 1: symbol
|
||||
% 2: nr
|
||||
% 3: period
|
||||
% 4: column
|
||||
\newenvironment{element}[4]{
|
||||
% force the fqname to el
|
||||
\directlua{
|
||||
old_sections = sections
|
||||
sections = {}
|
||||
table.insert(sections, "el")
|
||||
}
|
||||
\mqfqname@update
|
||||
\directLuaAuxExpand{
|
||||
elementAdd(\luastring{#1}, \luastring{#2}, \luastring{#3}, \luastring{#4})
|
||||
}
|
||||
% 1: key
|
||||
% 2: value
|
||||
\newcommand{\property}[2]{
|
||||
\directlua{
|
||||
elementAddProperty(\luastring{#1}, \luastringN{##1}, \luastringN{##2})
|
||||
}
|
||||
}
|
||||
\edef\lastElementName{#1}
|
||||
}{
|
||||
\ignorespacesafterend
|
||||
% restore fqname
|
||||
\directlua{
|
||||
sections = old_sections
|
||||
}
|
||||
\mqfqname@update
|
||||
}
|
||||
|
||||
% LIST
|
||||
\newcommand\printElement[1]{
|
||||
\edef\elementName{el:#1}
|
||||
\par\noindent\ignorespaces
|
||||
\vspace{0.5\baselineskip}
|
||||
\begingroup
|
||||
\directlua{
|
||||
if elements["#1"]["labeled"] == nil then
|
||||
elements["#1"]["labeled"] = true
|
||||
tex.print("\\phantomsection\\label{el:#1}")
|
||||
end
|
||||
}
|
||||
\NameWithDescription[\descwidth]{\elementName}{\elementName_desc}
|
||||
\hfill
|
||||
\begin{ContentBoxWithExplanation}{\elementName_defs}
|
||||
\directlua{
|
||||
tex.sprint("Symbol: \\ce{"..elements["#1"]["symbol"].."}")
|
||||
tex.sprint("\\\\Number: "..elements["#1"]["atomic_number"])
|
||||
for key, value in pairs(elements["#1"]["properties"]) do
|
||||
tex.sprint("\\\\\\hspace*{1cm}{\\GT{"..key.."}: "..value.."}")
|
||||
end
|
||||
}
|
||||
\end{ContentBoxWithExplanation}
|
||||
\endgroup
|
||||
\textcolor{fg3}{\hrule}
|
||||
\vspace{0.5\baselineskip}
|
||||
\ignorespacesafterend
|
||||
}
|
||||
|
||||
\newcommand{\printAllElements}{
|
||||
\directlua{
|
||||
%-- tex.sprint("\\printElement{"..val.."}")
|
||||
for key, val in ipairs(elementsOrder) do
|
||||
%-- tex.sprint(key, val);
|
||||
tex.print("\\printElement{"..val.."}")
|
||||
end
|
||||
}
|
||||
}
|
||||
|
||||
% PERIODIC TABLE
|
||||
\directlua{
|
||||
category2color = {
|
||||
metal = "bg-blue!50!bg0",
|
||||
metalloid = "fg-orange!50!bg0",
|
||||
transitionmetal = "fg-blue!50!bg0",
|
||||
lanthanoide = "bg-orange!50!bg0",
|
||||
alkalimetal = "fg-red!50!bg0",
|
||||
alkalineearthmetal = "fg-purple!50!bg0",
|
||||
nonmetal = "fg-aqua!50!bg0",
|
||||
halogen = "fg-yellow!50!bg0",
|
||||
noblegas = "bg-purple!50!bg0"
|
||||
}
|
||||
}
|
||||
\directlua{
|
||||
function getColor(cat)
|
||||
local color = category2color[cat]
|
||||
if color == nil then
|
||||
return "bg3"
|
||||
else
|
||||
return color
|
||||
end
|
||||
end
|
||||
}
|
||||
\newcommand{\drawPeriodicTable}{
|
||||
\def\ptableUnit{0.90cm}
|
||||
\begin{tikzpicture}[
|
||||
element/.style={anchor=north west, draw, minimum width=\ptableUnit, minimum height=\ptableUnit, align=center},
|
||||
element_annotation/.style={anchor=north west, font=\tiny, inner sep=1pt},
|
||||
x=\ptableUnit,
|
||||
y=\ptableUnit
|
||||
]
|
||||
\directlua{
|
||||
for k, v in pairs(elements) do
|
||||
local column = tonumber(v.column)
|
||||
local period = tonumber(v.period)
|
||||
if 5 < period and 4 <= column and column <= 17 then
|
||||
period = period + 3
|
||||
elseif column > 17 then
|
||||
column = column - 14
|
||||
end
|
||||
tex.print("\\node[element,fill=".. getColor(v.properties.set) .."] at (".. column ..", -".. period ..") {\\elRef{".. v.symbol .."}};")
|
||||
tex.print("\\node[element_annotation] at (".. column ..", -".. period ..") {".. v.atomic_number .."};")
|
||||
if v.properties.atomic_mass \string~= nil then
|
||||
tex.print("\\node[element_annotation,anchor=south west] at (".. column ..", -".. period+1 ..") {".. string.format("\percentchar .3f", v.properties.atomic_mass) .."};")
|
||||
end
|
||||
end
|
||||
}
|
||||
\draw[ultra thick,fg-purple] (4,-6) -- (4,-11);
|
||||
% color legend for categories
|
||||
\directlua{
|
||||
local x0 = 4
|
||||
local y0 = -1
|
||||
local x = 0
|
||||
local y = 0
|
||||
local ystep = 0.4
|
||||
for set, color in pairs(category2color) do
|
||||
%-- tex.print("\\draw[fill=".. color .."] ("..x0+x..","..y0+y..") rectangle ("..x0+x..","..y0+y..")()")
|
||||
tex.print("\\node[anchor=west, align=left] at ("..x0+x..","..y0-y..") {{\\color{".. color .."}\\blacksquare} \\GT{".. set .."}};")
|
||||
y = y + 1*ystep
|
||||
if y > 4*ystep then
|
||||
y = 0
|
||||
x = x+4
|
||||
end
|
||||
end
|
||||
}
|
||||
% period numbers
|
||||
\directlua{
|
||||
for i = 1, 7 do
|
||||
tex.print("\\node[anchor=east,align=right] at (1,".. -i-0.5 ..") {".. i .."};")
|
||||
end
|
||||
}
|
||||
\end{tikzpicture}
|
||||
}
|
||||
@ -1,59 +0,0 @@
|
||||
\ProvidesPackage{mqquantity}
|
||||
\RequirePackage{mqlua}
|
||||
\RequirePackage{mqfqname}
|
||||
\RequirePackage{etoolbox}
|
||||
|
||||
% TODO: MAYBE:
|
||||
% store the fqname where the quantity is defined
|
||||
% In qtyRef then use the stored label to reference it, instead of linking to qty:<name>
|
||||
% Use the mqlua hyperref function
|
||||
|
||||
\begin{luacode}
|
||||
quantities = {}
|
||||
function quantityAdd(key, symbol, units, comment, fqname)
|
||||
quantities[key] = {
|
||||
["symbol"] = symbol,
|
||||
["units"] = units,
|
||||
["comment"] = comment
|
||||
}
|
||||
if fqname == "" then
|
||||
quantities[key]["fqname"] = fqnameGet()
|
||||
else
|
||||
quantities[key]["fqname"] = fqname
|
||||
end
|
||||
end
|
||||
function quantityGetSymbol(key)
|
||||
local qty = quantities[key]
|
||||
if qty == nil then return "???" end
|
||||
local symbol = qty["symbol"]
|
||||
if symbol == nil then return "???" end
|
||||
return symbol
|
||||
end
|
||||
function quantityGetFqname(key)
|
||||
local qty = quantities[key]
|
||||
if qty == nil then return "qty:"..key end
|
||||
local fqname_ = qty["fqname"]
|
||||
if fqname_ == nil then return "qty:"..key end
|
||||
return fqname_
|
||||
end
|
||||
\end{luacode}
|
||||
|
||||
% [1]: label to point to, if not given use current fqname
|
||||
% 2: key - must expand to a valid lua string!
|
||||
% 3: symbol
|
||||
% 4: units
|
||||
% 5: comment key to translation
|
||||
\newcommand{\quantity@new}[5][]{%
|
||||
\directLuaAuxExpand{quantityAdd(\luastring{#2}, \luastringN{#3}, \luastringN{#4}, \luastringN{#5}, \luastring{#1})}
|
||||
}
|
||||
|
||||
% 1: key
|
||||
\newcommand{\quantity@getSymbol}[1]{\luavar{quantityGetSymbol(\luastring{#1})}}
|
||||
|
||||
% 1: key
|
||||
\newcommand\quantity@print[1]{
|
||||
\begingroup % for label
|
||||
Symbol: $\luavar{quantityGetSymbol(\luastring{#1})}$
|
||||
\hfill Unit: $\directlua{split_and_print_units(quantities[\luastring{#1}]["units"])}$ %
|
||||
\endgroup%
|
||||
}
|
||||
Some files were not shown because too many files have changed in this diff Show More
Loading…
x
Reference in New Issue
Block a user