add periodic table
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37
Makefile
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37
Makefile
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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 = $(SRC_DIR)/main.tex
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MAIN_PDF = $(OUT_DIR)/main.pdf
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# LaTeX and Biber commands
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LATEX = lualatex
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BIBER = biber
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LATEX_OPTS := -output-directory=$(OUT_DIR) -interaction=nonstopmode -shell-escape
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.PHONY: default release clean
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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}/' $(MAIN_TEX)
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-cd $(SRC_DIR) && latexmk -g
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mv $(MAIN_PDF) $(OUT_DIR)/$(shell date -I)_en_formula_collection.pdf
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german:
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sed -r -i 's/usepackage\[[^]]+\]\{babel\}/usepackage[german]{babel}/' $(MAIN_TEX)
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-cd $(SRC_DIR) && latexmk -g
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mv $(MAIN_PDF) $(OUT_DIR)/$(shell date -I)_de_formelsammlung.pdf
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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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6269
scripts/PeriodicTableJSON.json
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6269
scripts/PeriodicTableJSON.json
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28402
scripts/elements.json
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28402
scripts/elements.json
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78
scripts/periodic_table.py
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78
scripts/periodic_table.py
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"""
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Script to process the periodic table as json into latex stuff
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Source for `elements.json` is this amazing project:
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https://pse-info.de/de/data
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Copyright Matthias Quintern 2024
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"""
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import json
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import os
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import re
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outdir = "../src/ch"
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def gen_periodic_table():
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with open("elements.json") as file:
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ptab = json.load(file)
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# print(ptab["elements"][1])
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s = "% This file was created by the periodic_table.py script.\n% Do not edit manually. Any changes might get lost.\n"
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for i, el_key in enumerate(ptab):
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el = ptab[el_key]
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def get(*keys):
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"""get keys or return empty string"""
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val = el
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for key in keys:
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if not key in val: return ""
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val = val[key]
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return val
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# print(i, el)
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el_s = f"\\begin{{element}}{{{el['symbol']}}}{{{el['number']}}}{{{el['period']}}}{{{el['column']}}}"
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# description
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el_s += f"\n\t\\desc[english]{{{el['names']['en']}}}{{{get('appearance', 'en')}}}{{}}"
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el_s += f"\n\t\\desc[german]{{{el['names']['de']}}}{{English: {get('names', 'en')}\\\\{get('appearance', 'de')}}}{{}}"
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# simple properties
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for field in ["crystal_structure", "set", "magnetic_ordering"]:
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if field in el:
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el_s += f"\n\t\\property{{{field}}}{{{el[field]}}}"
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# mass
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m = get("atomic_mass", "value")
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if m:
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assert(get("atomic_mass", "unit") == "u")
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el_s += f"\n\t\\property{{{'atomic_mass'}}}{{{m}}}"
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# refractive indices
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temp = ""
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add_refractive_index = lambda idx: f"\\GT{{{idx['label']}}}: ${idx['value']}$, "
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idxs = get("optical", "refractive_index")
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print(idxs)
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if type(idxs) == list:
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for idx in idxs: add_refractive_index(idx)
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elif type(idxs) == dict: add_refractive_index(idxs)
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elif type(idxs) == float: temp += f"${idxs}$, "
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if temp:
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el_s += f"\n\t\\property{{{'refractive_index'}}}{{{temp[:-2]}}}"
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# electron configuration
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match = re.fullmatch(r"([A-Z][a-z]*)? ?(.+?)", el["electron_config"])
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if match:
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el_s += f"\n\t\\property{{{'electron_config'}}}{{"
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if match.groups()[0]:
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el_s += f"\\elRef{{{match.groups()[0]}}} "
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el_s += f"{match.groups()[1]}}}"
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el_s += "\n\\end{element}"
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print(el_s)
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s += el_s + "\n"
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# print(s)
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return s
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if __name__ == "__main__":
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ptable = gen_periodic_table()
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assert os.path.abspath(".").endswith("scripts"), "Please run from the `scripts` directory"
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with open(f"{outdir}/periodic_table.tex", "w") as file:
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file.write(ptable)
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@ -4,7 +4,7 @@ import numpy as np
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import math
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import scipy as scp
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outdir = "../img/"
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outdir = "../src/img/"
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filetype = ".pdf"
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skipasserts = False
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30
src/.latexmkrc
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30
src/.latexmkrc
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# Specify the auxiliary and output directories
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$aux_dir = '../.aux';
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$out_dir = '../out';
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# Set lualatex as the default engine
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$pdf_mode = 1; # Enable PDF generation mode
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$pdflatex = 'lualatex -interaction=nonstopmode -shell-escape'
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# Additional options for compilation
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# '-verbose',
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# '-file-line-error',
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# Quickfix-like filtering (warnings to ignore)
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# @warnings_to_filter = (
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# qr/Underfull \\hbox/,
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# qr/Overfull \\hbox/,
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# qr/LaTeX Warning: .+ float specifier changed to/,
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# qr/LaTeX hooks Warning/,
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# qr/Package siunitx Warning: Detected the "physics" package:/,
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# qr/Package hyperref Warning: Token not allowed in a PDF string/
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# );
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# # Filter function for warnings
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# sub filter_warnings {
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# my $warning = shift;
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# foreach my $filter (@warnings_to_filter) {
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# return 0 if $warning =~ $filter;
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# }
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# return 1;
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# }
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14
src/atom.tex
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src/atom.tex
@ -179,3 +179,17 @@
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]{mag_effects}
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\TODO{all}
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\\\TODO{Hunds rules}
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\Subsection[
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\eng{misc}
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\ger{Sonstiges}
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]{other}
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\begin{formula}{auger_effect}
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\desc{Auger-Meitner-Effekt}{Auger-Effect}{}
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\desc[german]{Auger-Meitner-Effekt}{Auger-Effekt}{}
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\ttxt{
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\eng{An excited electron relaxes into a lower, unoccupied energy level. The released energy causes the emission of another electron in a higher energy level (Auger-Electron)}
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\ger{Ein angeregtes Elektron fällt in ein unbesetztes, niedrigeres Energieniveau zurück. Durch die frei werdende Energie verlässt ein Elektron aus einer höheren Schale das Atom (Auger-Elektron).}
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}
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\end{formula}
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src/ch/ch.tex
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src/ch/ch.tex
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\Part[
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\eng{Chemie}
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\ger{Chemie}
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]{ch}
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\Section[
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\eng{Periodic table}
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\ger{Periodensystem}
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]{ptable}
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\drawPeriodicTable
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\Section[
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\eng{stuff}
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\ger{stuff}
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]{stuff}
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\begin{formula}{covalent_bond}
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\desc{Covalent bond}{}{}
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\desc[german]{Kolvalente Bindung}{}{}
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\ttxt{
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\eng{Bonds that involve sharing of electrons to form electron pairs between atoms.}
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\ger{Bindungen zwischen Atomen die durch geteilte Elektronen, welche Elektronenpaare bilden, gebildet werden.}
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}
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\end{formula}
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\Section[
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\eng{List of elements}
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\ger{Liste der Elemente}
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]{elements}
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\printAllElements
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1054
src/ch/periodic_table.tex
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1054
src/ch/periodic_table.tex
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85
src/cm/charge_transport.tex
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src/cm/charge_transport.tex
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\Section[
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\eng{Charge transport}
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\ger{Ladungstransport}
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]{charge_transport}
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\Subsection[
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\eng{Drude model}
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\ger{Drude-Modell}
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]{drude}
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\begin{ttext}
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\eng{Classical model describing the transport properties of electrons in materials (metals):
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The material is assumed to be an ion lattice and with freely moving electrons (electron gas). The electrons are
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accelerated by an electric field and decelerated through collisions with the lattice ions.
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The model disregards the Fermi-Dirac partition of the conducting electrons.
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}
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\ger{Ein klassisches Model zur Beschreibung der Transporteigenschaften von Elektronen in (v.a.) Metallen:
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Der Festkörper wird als Ionenkristall mit frei beweglichen Elektronen (Elektronengas).
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Die Elektronen werden durch ein Elektrisches Feld $E$ beschleunigt und durch Stöße mit den Gitterionen gebremst.
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Das Modell vernachlässigt die Fermi-Dirac Verteilung der Leitungselektronen.
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}
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\end{ttext}
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\begin{formula}{motion}
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\desc{Equation of motion}{}{$v$ electron speed, $\vec{v}_\text{D}$ drift velocity, $\tau$ mean free time between collisions}
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\desc[german]{Bewegungsgleichung}{}{$v$ Elektronengeschwindigkeit, $\vec{v}_\text{D}$ Driftgeschwindigkeit, $\tau$ Stoßzeit}
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\eq{\masse \odv{\vec{v}}{t} + \frac{\masse}{\tau} \vec{v}_\text{D} = -e \vec{\E}}
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\end{formula}
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\begin{formula}{scattering_time}
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\desc{Scattering time}{Momentum relaxation time}{}
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\desc[german]{Streuzeit}{}{}
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\ttxt{
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\eng{$\tau$\\ the average time between scattering events weighted by the characteristic momentum change cause by the scattering process.}
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}
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\end{formula}
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\begin{formula}{current_density}
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\desc{Current density}{Ohm's law}{$n$ charge particle density}
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\desc[german]{Stromdichte}{Ohmsches Gesetz}{$n$ Ladungsträgerdichte}
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\eq{\vec{j} = -ne\vec{v}_\text{D} = ne\mu \vec{\E}}
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\end{formula}
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\begin{formula}{conductivity}
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\desc{Drude-conductivity}{}{}
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\desc[german]{Drude-Leitfähigkeit}{}{}
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\eq{\sigma = \frac{\vec{j}}{\vec{\E}} = \frac{e^2 \tau n}{\masse} = n e \mu}
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\end{formula}
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\Subsection[
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\eng{Sommerfeld model}
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\ger{Sommerfeld-Modell}
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]{sommerfeld}
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\begin{ttext}
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\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.}
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\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.}
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\end{ttext}
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\begin{formula}{current_density}
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\desc{Current density}{}{}
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\desc[german]{Stromdichte}{}{}
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\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}}
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\end{formula}
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\TODO{The formula for the conductivity is the same as in the drude model?}
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\Subsection[
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\eng{Boltzmann-transport}
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\ger{Boltzmann-Transport}
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]{boltzmann}
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\begin{ttext}
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\eng{Semiclassical description using a probability distribution (\fqEqRef{stat:todo:fermi_dirac}) to describe the particles.}
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\ger{Semiklassische Beschreibung, benutzt eine Wahrscheinlichkeitsverteilung (\fqEqRef{stat:todo:fermi_dirac}).}
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\end{ttext}
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\begin{formula}{boltzmann_transport}
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\desc{Boltzmann Transport equation}{for charge transport}{$f$ \ref{stat:todo:fermi-dirac}}
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\desc[german]{Boltzmann-Transportgleichung}{für Ladungstransport}{}
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\eq{
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\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}}}
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}
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\end{formula}
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\Subsection[
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\eng{misc}
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\ger{misc}
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]{misc}
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\begin{formula}{tsu_esaki}
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\desc{Tsu-Esaki tunneling current}{Describes the current $I_{\txL \leftrightarrow \txR}$ through a barrier}{$\mu_i$ \qtyRef{chemical_pot} at left/right side, $U_i$ voltage on left/right side. Electrons occupy region between $U_i$ and $\mu_i$}
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\desc[german]{Tsu-Esaki Tunnelstrom}{Beschreibt den Strom $I_{\txL \leftrightarrow \txR}$ durch eine Barriere }{$\mu_i$ \qtyRef{chemical_pot} links/rechts, $U_i$ Spannung links/rechts. Elektronen besetzen Bereich zwischen $U_i$ und $\mu_i$}
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\eq{
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I_\text{T} = \frac{2e}{h} \int_{U_\txL}^\infty \left(f(E, \mu_\txL) -f(E, \mu_\txR)\right) T(E) \d E
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}
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\end{formula}
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@ -1,12 +1,12 @@
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\Part[
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\eng{Condensed matter physics}
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\ger{Festkörperphysik}
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]{cm}
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]{cm}
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\TODO{Bonds, hybridized orbitals, tight binding}
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\Section[
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\eng{Bravais lattice}
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\ger{Bravais-Gitter}
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]{bravais}
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]{bravais}
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% \begin{ttext}
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% \eng{
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@ -37,7 +37,7 @@
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\newcolumntype{Z}{>{\centering\let\newline\\\arraybackslash\hspace{0pt}}X}
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\begin{table}[H]
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\centering
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\caption{\gt{bravais_table2}}
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\expandafter\caption\expandafter{\gt{bravais_table2}}
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\label{tab:bravais2}
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\begin{adjustbox}{width=\textwidth}
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@ -85,12 +85,70 @@
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\end{tabularx}
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\end{adjustbox}
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\end{table}
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\TODO{FCC, BCC, diamond/Zincblende wurtzize cell/lattice vectors}
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\TODO{primitive unit cell: contains one lattice point}\\
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family of plane that are equivalent due to crystal symmetry
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\begin{quantity}{lattice_constant}{a}{}{s}
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\desc{Lattice constant}{Parameter (length or angle) describing the smallest unit cell}{}
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\desc[german]{Gitterkonstante}{Parameter (Länge oder Winkel) der die Einheitszelle beschreibt}{}
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\end{quantity}
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\begin{formula}{sc}
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\desc{Simple cubic (SC)}{Reciprocal: Simple cubic}{\QtyRef{lattice_constant}}
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\desc[german]{Einfach kubisch (SC)}{Reziprok: Einfach kubisch}{}
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\eq{
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\vec{a}_{1}=a \begin{pmatrix} 1\\0\\0 \end{pmatrix},\,
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\vec{a}_{2}=a \begin{pmatrix} 0\\1\\0 \end{pmatrix},\,
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\vec{a}_{3}=a \begin{pmatrix} 0\\0\\1 \end{pmatrix}
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}
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\end{formula}
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\begin{formula}{bcc}
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\desc{Body centered cubic (BCC)}{Reciprocal: \fqEqRef{cm:bravais:fcc}}{\QtyRef{lattice_constant}}
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\desc[german]{Kubisch raumzentriert (BCC)}{Reziprok: \fqEqRef{cm:bravais:fcc}}{}
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\eq{
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\vec{a}_{1}=\frac{a}{2} \begin{pmatrix} -1\\1\\1 \end{pmatrix},\,
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\vec{a}_{2}=\frac{a}{2} \begin{pmatrix} 1\\-1\\1 \end{pmatrix},\,
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\vec{a}_{3}=\frac{a}{2} \begin{pmatrix} 1\\1\\-1 \end{pmatrix}
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}
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\end{formula}
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\begin{formula}{fcc}
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\desc{Face centered cubic (FCC)}{Reciprocal: \fqEqRef{cm:bravais:bcc}}{\QtyRef{lattice_constant}}
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\desc[german]{Kubisch flächenzentriert (FCC)}{Reziprok: \fqEqRef{cm:bravais:bcc}}{}
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\eq{
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\vec{a}_{1}=\frac{a}{2} \begin{pmatrix} 0\\1\\1 \end{pmatrix},\,
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\vec{a}_{2}=\frac{a}{2} \begin{pmatrix} 1\\0\\1 \end{pmatrix},\,
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\vec{a}_{3}=\frac{a}{2} \begin{pmatrix} 1\\1\\0 \end{pmatrix}
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}
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\end{formula}
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\begin{formula}{diamond}
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\desc{Diamond lattice}{}{}
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\desc[german]{Diamantstruktur}{}{}
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\ttxt{
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\eng{\fqEqRef{cm:bravais:fcc} with basis $\begin{pmatrix} 0 & 0 & 0 \end{pmatrix}$ and $\begin{pmatrix} \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \end{pmatrix}$}
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\ger{\fqEqRef{cm:bravais:fcc} mit Basis $\begin{pmatrix} 0 & 0 & 0 \end{pmatrix}$ und $\begin{pmatrix} \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \end{pmatrix}$}
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}
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\end{formula}
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\begin{formula}{zincblende}
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\desc{Zincblende lattice}{}{}
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\desc[german]{Zinkblende-Struktur}{}{}
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\ttxt{
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\includegraphics[width=0.5\textwidth]{img/cm_zincblende.png}
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\eng{Like \fqEqRef{cm:bravais:diamond} but with different species on each basis}
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\ger{Wie \fqEqRef{cm:bravais:diamond} aber mit unterschiedlichen Spezies auf den Basen}
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}
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\end{formula}
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\begin{formula}{wurtzite}
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\desc{Wurtzite structure}{hP4}{}
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\desc[german]{Wurtzite-Struktur}{hP4}{}
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\ttxt{
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||||
\includegraphics[width=0.5\textwidth]{img/cm_wurtzite.png}
|
||||
Placeholder
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\TODO{primitive unit cell: contains one lattice point}\\
|
||||
\begin{formula}{miller}
|
||||
\desc{Miller index}{}{}
|
||||
\desc{Miller index}{}{Miller family: planes that are equivalent due to crystal symmetry}
|
||||
\desc[german]{Millersche Indizes}{}{}
|
||||
\eq{
|
||||
(hkl) & \text{\GT{plane}}\\
|
||||
@ -122,7 +180,7 @@ family of plane that are equivalent due to crystal symmetry
|
||||
\Subsection[
|
||||
\eng{Scattering processes}
|
||||
\ger{Streuprozesse}
|
||||
]{scatter}
|
||||
]{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}
|
||||
@ -169,7 +227,7 @@ family of plane that are equivalent due to crystal symmetry
|
||||
\Subsection[
|
||||
\eng{2D electron gas}
|
||||
\ger{2D Elektronengas}
|
||||
]{2deg}
|
||||
]{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 $L$.}
|
||||
@ -194,24 +252,72 @@ family of plane that are equivalent due to crystal symmetry
|
||||
\Subsection[
|
||||
\eng{1D electron gas / quantum wire}
|
||||
\ger{1D Eleltronengas / Quantendraht}
|
||||
]{1deg}
|
||||
]{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}
|
||||
\TODO{condunctance}
|
||||
|
||||
\Subsection[
|
||||
\eng{0D electron gas / quantum dot}
|
||||
\ger{0D Elektronengase / Quantenpunkt}
|
||||
]{0deg}
|
||||
]{0deg}
|
||||
|
||||
\TODO{TODO}
|
||||
|
||||
\Section[
|
||||
\eng{Band theory}
|
||||
\ger{Bändermodell}
|
||||
]{band}
|
||||
\Subsection[
|
||||
\eng{Hybrid orbitals}
|
||||
\ger{Hybridorbitale}
|
||||
]{hybrid_orbitals}
|
||||
\begin{ttext}
|
||||
\eng{Hybrid orbitals are linear combinations of other atomic orbitals.}
|
||||
\ger{Hybridorbitale werden durch Linearkombinationen von anderen atomorbitalen gebildet.}
|
||||
\end{ttext}
|
||||
|
||||
% chemmacros package
|
||||
\begin{formula}{sp3}
|
||||
\desc{sp3 Orbital}{\GT{eg} \ce{CH4}}{}
|
||||
\desc[german]{sp3 Orbital}{}{}
|
||||
\eq{
|
||||
1\text{s} + 3\text{p} = \text{sp3}
|
||||
\orbital{sp3}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{sp2}
|
||||
\desc{sp2 Orbital}{}{}
|
||||
\desc[german]{sp2 Orbital}{}{}
|
||||
\eq{
|
||||
1\text{s} + 2\text{p} = \text{sp2}
|
||||
\orbital{sp2}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{sp}
|
||||
\desc{sp Orbital}{}{}
|
||||
\desc[german]{sp Orbital}{}{}
|
||||
\eq{
|
||||
1\text{s} + 1\text{p} = \text{sp}
|
||||
\orbital{sp}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Section[
|
||||
\eng{\GT{misc}}
|
||||
\ger{\GT{misc}}
|
||||
]{misc}
|
||||
|
||||
\begin{formula}{exciton}
|
||||
\desc{Exciton}{}{}
|
||||
\desc[german]{Exziton}{}{}
|
||||
\ttxt{
|
||||
\eng{Quasi particle, excitation in condensed matter as bound electron-hole pair.}
|
||||
\ger{Quasiteilchen, Anregung im Festkörper als gebundenes Elektron-Loch-Paar}
|
||||
}
|
||||
\end{formula}
|
||||
141
src/cm/low_temp.tex
Normal file
141
src/cm/low_temp.tex
Normal file
@ -0,0 +1,141 @@
|
||||
\def\L{\text{L}}
|
||||
\def\gl{\text{GL}}
|
||||
\def\GL{Ginzburg-Landau }
|
||||
\def\Tcrit{T_\text{c}}
|
||||
\def\Bcrit{B_\text{c}}
|
||||
\def\ssc{\text{s}}
|
||||
\def\ssn{\text{n}}
|
||||
|
||||
\Section[
|
||||
\eng{Superconductivity}
|
||||
\ger{Supraleitung}
|
||||
]{sc}
|
||||
\begin{ttext}
|
||||
\eng{
|
||||
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 $\Bcrit$.
|
||||
\\\textbf{Type I}: Has a single critical magnetic field at which the superconuctor becomes a normal conductor.
|
||||
\\\textbf{Type II}: Has two critical
|
||||
}
|
||||
\ger{
|
||||
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 $\Bcrit$.
|
||||
|
||||
}
|
||||
\end{ttext}
|
||||
|
||||
\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.
|
||||
(\fqEqRef{ed:fields:mag: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.
|
||||
(\fqEqRef{ed:fields:mag:induction:lenz})
|
||||
}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{meissner_effect}
|
||||
\desc{Meißner-Ochsenfeld effect}{Perfect diamagnetism}{}
|
||||
\desc[german]{Meißner-Ochsenfeld Effekt}{Idealer Diamagnetismus}{}
|
||||
\ttxt{
|
||||
\eng{External magnetic field decays exponetially inside the superconductor below a critical temperature and a critical magnetic field.}
|
||||
\ger{Externes Magnetfeld fällt im Supraleiter exponentiell unterhalb einer kritischen Temperatur und unterhalb einer kritischen Feldstärke ab.}
|
||||
}
|
||||
\end{formula}
|
||||
\Subsection[
|
||||
\eng{London equations}
|
||||
\ger{London-Gleichungen}
|
||||
]{london}
|
||||
\begin{ttext}
|
||||
\eng{
|
||||
Quantitative description of the \fqEqRef{cm:sc:meissner_effect}.
|
||||
}
|
||||
\ger{
|
||||
Quantitative Beschreibung des \fqEqRef{cm:sc:meissner_effect}s.
|
||||
}
|
||||
|
||||
\end{ttext}
|
||||
% \begin{formula}{coefficient}
|
||||
% \desc{London-coefficient}{}{}
|
||||
% \desc[german]{London-Koeffizient}{}{}
|
||||
% \eq{\Lambda = \frac{m_\ssc}{n_\ssc q_\ssc^2}}
|
||||
% \end{formula}
|
||||
\begin{formula}{first}
|
||||
% \vec{j} = \frac{nq\hbar}{m}\Grad S - \frac{nq^2}{m}\vec{A}
|
||||
\desc{First London Equation}{}{$\vec{j}$ current density, $n_\ssc$, $m_\ssc$, $q_\ssc$ density, mass and charge of superconduticng particles}
|
||||
\desc[german]{Erste London-Gleichun-}{}{$\vec{j}$ Stromdichte, $n_\ssc$, $m_\ssc$, $q_\ssc$ Dichte, Masse und Ladung der supraleitenden Teilchen}
|
||||
\eq{
|
||||
\pdv{\vec{j}_{\ssc}}{t} = \frac{n_\ssc q_\ssc^2}{m_\ssc}\vec{E} {\color{gray}- \Order{\vec{j}_\ssc^2}}
|
||||
% \\{\color{gray} = \frac{q}{m}\Grad \left(\frac{1}{2} \TODO{FActor} \vec{j}^2\right)}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{second}
|
||||
\desc{Second London Equation}{Describes the \fqEqRef{cm:sc:meissner_effect}}{$\vec{j}$ current density, $n_\ssc$, $m_\ssc$, $q_\ssc$ density, mass and charge of superconduticng particles}
|
||||
\desc[german]{Zweite London-Gleichung}{Beschreibt den \fqEqRef{cm:sc:meissner_effect}}{$\vec{j}$ Stromdichte, $n_\ssc$, $m_\ssc$, $q_\ssc$ Dichte, Masse und Ladung der supraleitenden Teilchen}
|
||||
\eq{
|
||||
\Rot \vec{j_\ssc} = -\frac{n_\ssc q_\ssc^2}{m_\ssc} \vec{B}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{penetration_depth}
|
||||
\desc{London penetration depth}{}{}
|
||||
\desc[german]{London Eindringtiefe}{}{}
|
||||
\eq{\lambda_\L = \sqrt{\frac{m_\ssc}{\mu_0 n_\ssc q_\ssc^2}}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
|
||||
\Subsection[
|
||||
\eng{\GL Theory (GLAG)}
|
||||
\ger{\GL Theorie (GLAG)}
|
||||
]{gl}
|
||||
\begin{ttext}
|
||||
\eng{
|
||||
|
||||
}
|
||||
|
||||
\end{ttext}
|
||||
\begin{formula}{coherence_length}
|
||||
\desc{\GL Coherence Length}{}{}
|
||||
\desc[german]{\GL Kohärenzlänge}{}{}
|
||||
\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}{}{}
|
||||
\desc[german]{\GL Eindringtiefe}{}{}
|
||||
\eq{
|
||||
\lambda_\gl &= \sqrt{\frac{m_\ssc\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}{first}
|
||||
\desc{First Ginzburg-Landau Equation}{}{$\xi_\gl$ \fqEqRef{cm:sc:gl:coherence_length}, $\lambda_\gl$ \fqEqRef{cm:sc: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}{}{}
|
||||
\desc[german]{Zweite Ginzburg-Landau Gleichung}{}{}
|
||||
\eq{\vec{j_\ssc} = \frac{ie\hbar}{m}(\Psi^*\Grad\Psi - \Psi\Grad\Psi^*) - \frac{4e^2}{m}\abs{\Psi}^2 \vec{A}}
|
||||
\end{formula}
|
||||
|
||||
\TODO{proximity effect}
|
||||
|
||||
\Subsection[
|
||||
\eng{Microscopic theory}
|
||||
\ger{Mikroskopische Theorie}
|
||||
]{micro}
|
||||
|
||||
\Subsection[
|
||||
\eng{BCS-Theory}
|
||||
\ger{BCS-Theorie}
|
||||
]{BCS}
|
||||
|
||||
|
||||
58
src/cm/semiconductors.tex
Normal file
58
src/cm/semiconductors.tex
Normal file
@ -0,0 +1,58 @@
|
||||
\Section[
|
||||
\eng{Semiconductors}
|
||||
\ger{Halbleiter}
|
||||
]{semic}
|
||||
\begin{formula}{types}
|
||||
\desc{Intrinsic/extrinsic}{}{$n,p$ \fqEqRef{cm:semic:charge_density_eq}}
|
||||
\desc[german]{Intrinsisch/Extrinsisch}{}{}
|
||||
\ttxt{
|
||||
\eng{
|
||||
Intrinsic: pure, electron density determiend only by thermal excitation and $n_i^2 = n_0 p_0$\\
|
||||
Extrinsic: doped
|
||||
}
|
||||
\ger{
|
||||
Intrirnsisch: Pur, Elektronendichte gegeben durch thermische Anregung und $n_i^2 = n_0 p_0$ \\
|
||||
Extrinsisch: gedoped
|
||||
}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{charge_density_eq}
|
||||
\desc{Equilibrium charge densitites}{Holds when $\frac{\Econd-\EFermi}{\kB T}>3.6$ and $\frac{\EFermi-\Evalence}{\kB T} > 3.6$}{}
|
||||
\desc[german]{Ladungsträgerdichte im Equilibrium}{Gilt wenn $\frac{\Econd-\EFermi}{\kB T}>3.6$ und $\frac{\EFermi-\Evalence}{\kB T} > 3.6$}{}
|
||||
\eq{
|
||||
n_0 &\approx N_\text{c}(T) \Exp{-\frac{E_\text{c} - \EFermi}{\kB T}} \\
|
||||
p_0 &\approx N_\text{v}(T) \Exp{-\frac{\EFermi - E_\text{v}}{\kB T}}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{charge_density_intrinsic}
|
||||
\desc{Intrinsic charge density}{}{}
|
||||
\desc[german]{Intrinsische Ladungsträgerdichte}{}{}
|
||||
\eq{
|
||||
n_\text{i} \approx \sqrt{n_0 p_0} = \sqrt{N_\text{c}(T) N_\text{v}(T)} \Exp{-\frac{E_\text{gap}}{2\kB T}}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{mass_action}
|
||||
\desc{Mass action law}{Charge densities at thermal equilibrium, independent of doping}{}
|
||||
\desc[german]{Massenwirkungsgesetz}{Ladungsträgerdichten im Equilibrium, unabhängig der Dotierung }{}
|
||||
\eq{np = n_i^2}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\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}
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
154
src/cm/techniques.tex
Normal file
154
src/cm/techniques.tex
Normal file
@ -0,0 +1,154 @@
|
||||
\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}
|
||||
|
||||
@ -6,5 +6,31 @@
|
||||
\desc{Planck Constant}{}{}
|
||||
\desc[german]{Plancksches Wirkumsquantum}{}{}
|
||||
\val{6.62607015\cdot 10^{-34}}{\joule\s}
|
||||
\val{4.135667969\dots\cdot 10^{-15}}{\eV\s}
|
||||
\val{4.135667969\dots\xE{-15}}{\eV\s}
|
||||
\end{constant}
|
||||
|
||||
\begin{constant}{universal_gas}{R}{def}
|
||||
\desc{Universal gas constant}{Proportionality factor for ideal gases}{\ConstRef{avogadro}, \ConstRef{boltzmann}}
|
||||
\desc[german]{Universelle Gaskonstante}{Proportionalitätskonstante für ideale Gase}{}
|
||||
\val{8.31446261815324}{\joule\per\mol\kelvin}
|
||||
\val{\NA \cdot \kB}{}
|
||||
\end{constant}
|
||||
|
||||
\begin{constant}{avogadro}{\NA}{def}
|
||||
\desc{Avogadro constant}{Number of molecules per mole}{}
|
||||
\desc[german]{Avogadro-Konstante}{Anzahl der Moleküle pro mol}{}
|
||||
\val{6.02214076 \xE{23}}{1\per\mole}
|
||||
\end{constant}
|
||||
|
||||
\begin{constant}{boltzmann}{\kB}{def}
|
||||
\desc{Boltzmann constant}{Temperature-Energy conversion factor}{}
|
||||
\desc[german]{Boltzmann-Konstante}{Temperatur-Energie Umrechnungsfaktor}{}
|
||||
\val{1.380649 \xE{-23}}{\joule\per\kelvin}
|
||||
\end{constant}
|
||||
|
||||
\begin{constant}{faraday}{F}{def}
|
||||
\desc{Faraday constant}{Electric charge of one mol of single-charged ions}{\ConstRef{avogadro}, \ConstRef{boltzmann}}
|
||||
\desc[german]{Faraday-Konstante}{Elektrische Ladungs von einem Mol einfach geladener Ionen}{}
|
||||
\val{9.64853321233100184}{\coulomb\per\mol}
|
||||
\val{\NA\,e}{}
|
||||
\end{constant}
|
||||
|
||||
@ -5,33 +5,9 @@
|
||||
]{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}
|
||||
\TODO{Polarization}
|
||||
|
||||
% pure electronic stuff in el
|
||||
% pure magnetic stuff in mag
|
||||
% electromagnetic stuff in em
|
||||
|
||||
\Section[
|
||||
\eng{Electric field}
|
||||
@ -40,47 +16,104 @@
|
||||
\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}}
|
||||
\eq{\PhiE = \iint_S \vec{\E}\cdot\d\vec{S} = \frac{Q}{\varepsilon_0}}
|
||||
\end{formula}
|
||||
|
||||
\begin{quantity}{permittivity}{\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}{}
|
||||
\desc{Permittivity}{Electric polarizability of a dielectric material}{}
|
||||
\desc[german]{Permitivität}{Dielektrische Konstante\\Elektrische Polarisierbarkeit eines dielektrischen Materials}{}
|
||||
\end{quantity}
|
||||
\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}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{constant}{vacuum_permittivity}{\epsilon_0}{exp}
|
||||
\desc{Vacuum permittivity}{Electric constant}{}
|
||||
\desc[german]{Vakuum Permittivität}{Elektrische Feldkonstante}{}
|
||||
\val{8.8541878188(14)\E{-1}}{\ampere\s\per\volt\m}
|
||||
\end{constant}
|
||||
|
||||
\begin{formula}{electric_susceptibility}
|
||||
\desc{Electric susceptibility}{Describes how polarized a dielectric material becomes when an electric field is applied}{$\epsilon_\txr$ \fqEqRef{ed:el:relative_permittivity}}
|
||||
\desc[german]{Elektrische Suszeptibilität}{Beschreibt wie stark ein dielektrisches Material polarisiert wird, wenn ein elektrisches Feld angelegt wird}{}
|
||||
\eq{
|
||||
\epsilon_\txr = 1 + \chi_\txe
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{dielectric_polarization_density}
|
||||
\desc{Dielectric polarization density}{}{\ConstRef{vacuum_permittivity}, $\fqEqRef{ed:el:electric_susceptibility}$, \QtyRef{electric_field}}
|
||||
\desc[german]{Dielektrische Polarisationsdichte}{}{}
|
||||
\eq{\vec{P} = \epsilon_0 \chi_\txe \vec{\E}}
|
||||
\end{formula}
|
||||
|
||||
\Section[
|
||||
\eng{Magnetic field}
|
||||
\ger{Magnetfeld}
|
||||
]{mag}
|
||||
\begin{constant}{h_joule}{\hbar}{def}
|
||||
\desc{Planck Constant}{}{}
|
||||
\desc[german]{Plancksches Wirkumsquantum}{}{}
|
||||
\val{6.62607015\cdot 10^{-34}}{\joule\s}
|
||||
\val{4.135667969\dots\cdot 10^{-15}}{\eV\s}
|
||||
\end{constant}
|
||||
|
||||
\Eng[magnetic_flux]{Magnetix flux density}
|
||||
\Ger[magnetic_flux]{Magnetische Flussdichte}
|
||||
|
||||
\begin{quantity}{magnetic_flux}{\PhiB}{\weber=\volt\per\s=\kg\m^2\per\s^2\A}{scalar}
|
||||
\desc{Magnetic flux}{}
|
||||
\desc[german]{Magnetischer Fluss}{}
|
||||
\desc{Magnetic flux}{Test desc}{Test def}
|
||||
\desc[german]{Magnetischer Fluss}{Test desc}{Test def}
|
||||
\end{quantity}
|
||||
|
||||
\begin{quantity}{magnetic_flux_density}{\vec{B}}{\tesla=\volt\s\per\m^2=\newton\per\ampere\m=\kg\per\ampere\s^2}{}
|
||||
\desc{Magnetic flux density}{}
|
||||
\desc[german]{Magnetische Flussdichte}{}
|
||||
\desc{Magnetic flux density}{}{}
|
||||
\desc[german]{Magnetische Flussdichte}{}{}
|
||||
\end{quantity}
|
||||
\begin{formula}{magnetic_flux_density}
|
||||
\desc{\qtyRef{magnetic_flux_density}}{}{$\vec{H}$ \qtyRef{magnetic_field_density}, $\vec{M}$ \qtyRef{magnetization}, $\mu_0$ \constRef{vacuum_permeability}}
|
||||
\desc[german]{}{}{}
|
||||
\desc{\qtyRef{magnetic_flux_density}}{Defined by \fqEqRef{ed:mag:lorentz}}{$\vec{H}$ \qtyRef{magnetic_field_intensity}, $\vec{M}$ \qtyRef{magnetization}, \ConstRef{magnetic_vacuum_permeability}}
|
||||
\desc[german]{}{Definiert über \fqEqRef{ed:mag:lorentz}}{}
|
||||
\eq{\vec{B} = \mu_0 (\vec{H}+\vec{M})}
|
||||
\end{formula}
|
||||
|
||||
\begin{quantity}{magnetic_permeability}{\mu}{\henry\per\m=\volt\s\per\ampere\m}{scalar}
|
||||
\desc{Magnetic permeability}{}
|
||||
\desc[german]{Magnetisch Permeabilität}{}
|
||||
\begin{quantity}{magnetic_field_intensity}{\vec{H}}{\ampere\per\m}{vector}
|
||||
\desc{Magnetic field intensity}{}{}
|
||||
\desc[german]{Magnetische Feldstärke}{}{}
|
||||
\end{quantity}
|
||||
\begin{constant}{vacuum_permeability}{\mu_0}{exp}
|
||||
\begin{formula}{magnetic_field_intensity}
|
||||
\desc{\qtyRef{magnetic_field_intensity}}{}{}
|
||||
\desc[german]{}{}{}
|
||||
\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{quantity}{magnetic_permeability}{\mu}{\henry\per\m=\volt\s\per\ampere\m}{scalar}
|
||||
\desc{Magnetic permeability}{}{}
|
||||
\desc[german]{Magnetisch Permeabilität}{}{}
|
||||
\end{quantity}
|
||||
\begin{formula}{magnetic_permeability}
|
||||
\desc{\qtyRef{magnetic_permeability}}{}{$B$ \qtyRef{magnetic_flux_density}, $H$ \qtyRef{magnetic_field_intensity}}
|
||||
\desc[german]{}{}{}
|
||||
\eq{\mu=\frac{B}{H}}
|
||||
\end{formula}
|
||||
\begin{constant}{magnetic_vacuum_permeability}{\mu_0}{exp}
|
||||
\desc{Magnetic vauum permeability}{}{}
|
||||
\desc[german]{Magnetische Vakuumpermeabilität}{}{}
|
||||
\val{1.25663706127(20)}{\henry\per\m=\newton\per\ampere^2}
|
||||
\end{constant}
|
||||
\begin{formula}{relative_permeability}
|
||||
\desc{Relative permeability}{}{}
|
||||
\desc[german]{Realtive Permeabilität}{}{}
|
||||
\eq{
|
||||
\mu_\txr = \frac{\mu}{\mu_0}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{magnetic_flux}
|
||||
\desc{Magnetic flux}{}{$\vec{A}$ \GT{area}}
|
||||
@ -95,44 +128,114 @@
|
||||
\end{formula}
|
||||
|
||||
\begin{quantity}{magnetization}{\vec{M}}{\ampere\per\m}{vector}
|
||||
\desc{Magnetization}{Vector field describing the density of magnetic dipoles}
|
||||
\desc[german]{Magnetisierung}{Vektorfeld, welches die Dichte von magnetischen Dipolen beschreibt.}
|
||||
\desc{Magnetization}{Vector field describing the density of magnetic dipoles}{}
|
||||
\desc[german]{Magnetisierung}{Vektorfeld, welches die Dichte von magnetischen Dipolen beschreibt.}{}
|
||||
\end{quantity}
|
||||
\begin{quantity}{magnetic_moment}{\vec{m}}{\ampere\m^2}{vector}
|
||||
\desc{Magnetic moment}{Strength and direction of a magnetic dipole}
|
||||
\desc[german]{Magnetisches Moment}{Stärke und Richtung eines magnetischen Dipols}
|
||||
\end{quantity}
|
||||
|
||||
\begin{formula}{magnetization}
|
||||
\desc{\qtyRef{magnetization}}{}{$m$ \qtyRef{magnetic_moment}, $V$ \qtyRef{volume}}
|
||||
\desc[german]{}{}{}
|
||||
\eq{\vec{M} = \odv{\vec{m}}{V} = \chi_\text{m} \cdot \vec{H}}
|
||||
\eq{\vec{M} = \odv{\vec{m}}{V} = \chi_\txm \cdot \vec{H}}
|
||||
\end{formula}
|
||||
|
||||
\begin{quantity}{magnetic_moment}{\vec{m}}{\ampere\m^2}{vector}
|
||||
\desc{Magnetic moment}{Strength and direction of a magnetic dipole}{}
|
||||
\desc[german]{Magnetisches Moment}{Stärke und Richtung eines magnetischen Dipols}{}
|
||||
\end{quantity}
|
||||
|
||||
\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}{suceptibility}
|
||||
\desc{Susceptibility}{}{}
|
||||
\begin{formula}{magnetic_susceptibility}
|
||||
\desc{Susceptibility}{}{$\mu_\txr$ \fqEqRef{ed:mag:relative_permeability}}
|
||||
\desc[german]{Suszeptibilität}{}{}
|
||||
\eq{\chi_\text{m} = \pdv{M}{B} = \frac{\mu}{\mu_0} - 1 }
|
||||
\eq{\chi_\txm = \pdv{M}{B} = \mu_\txr - 1}
|
||||
\end{formula}
|
||||
|
||||
|
||||
|
||||
|
||||
\Subsection[
|
||||
\eng{Magnetic materials}
|
||||
\ger{Magnetische Materialien}
|
||||
]{materials}
|
||||
\begin{formula}{paramagnetism}
|
||||
\desc{Paramagnetism}{Magnetic field strengthend in the material}{$\mu$ \fqEqRef{ed:mag:magnetic_permeability}, $\chi_\txm$ \fqEqRef{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}{$\mu$ \fqEqRef{ed:mag:magnetic_permeability}, $\chi_\txm$ \fqEqRef{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)}{$\mu$ \fqEqRef{ed:mag:magnetic_permeability}, $\chi_\txm$ \fqEqRef{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}
|
||||
|
||||
\Section[
|
||||
\eng{Electromagnetism}
|
||||
\ger{Elektromagnetismus}
|
||||
]{em}
|
||||
\begin{constant}{speed_of_light}{c}{exp}
|
||||
\desc{Speed of light}{in the vacuum}{}
|
||||
\desc[german]{Lightgeschwindigkeit}{in the vacuum}{}
|
||||
\val{299792458}{\m\per\s}
|
||||
\end{constant}
|
||||
\begin{formula}{vacuum_relations}
|
||||
\desc{Vacuum permittivity - permeability relation}{\TODO{Does this have a name?}}{\ConstRef{vacuum_permittivity}, \ConstRef{magnetic_vacuum_permeability}, \ConstRef{speed_of_light}}
|
||||
\desc[german]{Vakuum Permittivität - Permeabilität Beziehung}{}{}
|
||||
\eq{
|
||||
\epsilon_0 \mu_0 = \frac{1}{c^2}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{poisson_equation}
|
||||
\desc{Poisson equation for electrostatics}{}{\QtyRef{charge_density}, \QtyRef{permittivity}, $\phi$}
|
||||
\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 [$\si{\W\per\m^2}$]}{}
|
||||
\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}
|
||||
|
||||
\begin{formula}{magnetic_permeability}
|
||||
\desc{\qtyRef{magnetic_permeability}}{}{$B$ \qtyRef{magnetic_flux_density}, $H$ \qtyRef{magnetic_field_intensity}}
|
||||
\desc[german]{}{}{}
|
||||
\eq{\mu=\frac{B}{H}}
|
||||
\end{formula}
|
||||
\Subsection[
|
||||
\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}
|
||||
\TODO{Polarization}
|
||||
|
||||
\Subsection[
|
||||
\eng{Induction}
|
||||
@ -157,15 +260,7 @@
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\Subsection[
|
||||
\eng{Magnetic materials}
|
||||
\ger{Magnetische Materialien}
|
||||
]{materials}
|
||||
\begin{formula}{paramagnetism}
|
||||
\desc{Paramagnetism}{}{$\mu$ \fqEqRef{ed:mag:permeability}, $\chi$ \fqEqRef{ed:mag:susecptibility}}
|
||||
\desc[german]{Paramagnetismus}{}{}
|
||||
\eq{\mu &> 1 \\ \chi > 0}
|
||||
\end{formula}
|
||||
|
||||
|
||||
|
||||
\Section[
|
||||
|
||||
BIN
src/img/cm_wurtzite.png
Normal file
BIN
src/img/cm_wurtzite.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 89 KiB |
BIN
src/img/cm_zincblende.png
Normal file
BIN
src/img/cm_zincblende.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 268 KiB |
92
src/main.tex
92
src/main.tex
@ -40,6 +40,7 @@
|
||||
% SCIENCE PACKAGES
|
||||
\usepackage{mathtools}
|
||||
\usepackage{MnSymbol} % for >>> \ggg sign
|
||||
\usepackage{chemmacros} % for orbitals
|
||||
% \usepackage{esdiff} % derivatives
|
||||
% esdiff breaks when taking \dot{q} has argument
|
||||
\usepackage{derivative}
|
||||
@ -108,31 +109,71 @@
|
||||
\label{sec:\fqname}
|
||||
}
|
||||
|
||||
% Make the translation of #1 a reference to a equation
|
||||
% 1: key
|
||||
% REFERENCES
|
||||
% All xyzRef commands link to the key using the translated name
|
||||
% Uppercase (XyzRef) commands have different link texts, but the same link target
|
||||
% 1: key/fully qualified name (without qty/eq/sec/const/el... prefix)
|
||||
% Equations/Formulas
|
||||
% <name>
|
||||
\newrobustcmd{\fqEqRef}[1]{%
|
||||
% \edef\fqeqrefname{\GT{#1}}
|
||||
% \hyperref[eq:#1]{\fqeqrefname}
|
||||
\hyperref[eq:#1]{\GT{#1}}%
|
||||
}
|
||||
\newrobustcmd{\qtyRef}[1]{%
|
||||
\hyperref[qty:#1]{\GT{#1}}%
|
||||
}
|
||||
\newrobustcmd{\constRef}[1]{%
|
||||
\hyperref[const:#1]{\GT{#1}}%
|
||||
}
|
||||
% Make the translation of #1 a reference to a section
|
||||
% 1: key
|
||||
% Section
|
||||
% <name>
|
||||
\newrobustcmd{\fqSecRef}[1]{%
|
||||
\hyperref[sec:#1]{\GT{#1}}%
|
||||
}
|
||||
% Quantities
|
||||
% <symbol>
|
||||
\newrobustcmd{\qtyRef}[1]{%
|
||||
\hyperref[qty:#1]{\GT{qty:#1}}%
|
||||
}
|
||||
% <symbol> <name>
|
||||
\newrobustcmd{\QtyRef}[1]{%
|
||||
${\luavar{quantities["#1"]["symbol"]}}$ \hyperref[qty:#1]{\GT{qty:#1}}%
|
||||
}
|
||||
% Constants
|
||||
% <name>
|
||||
\newrobustcmd{\constRef}[1]{%
|
||||
\hyperref[const:#1]{\GT{const:#1}}%
|
||||
}
|
||||
% <symbol> <name>
|
||||
\newrobustcmd{\ConstRef}[1]{%
|
||||
$\luavar{constants["#1"]["symbol"]}$ \hyperref[const:#1]{\GT{const:#1}}%
|
||||
}
|
||||
% Element from periodic table
|
||||
% <symbol>
|
||||
\newrobustcmd{\elRef}[1]{%
|
||||
\hyperref[el:#1]{{\color{dark0_hard}#1}}%
|
||||
}
|
||||
% <name>
|
||||
\newrobustcmd{\ElRef}[1]{%
|
||||
\hyperref[el:#1]{\GT{el:#1}}%
|
||||
}
|
||||
|
||||
% \usepackage{xstring}
|
||||
|
||||
% LUA sutff
|
||||
\newcommand\luavar[1]{\directlua{tex.sprint(#1)}}
|
||||
% Write directlua command to aux and run it as well
|
||||
\newcommand\directLuaAux[1]{
|
||||
\immediate\write\luaauxfile{\noexpand\directlua{\detokenize{#1}}}
|
||||
\directlua{#1}
|
||||
}
|
||||
\newwrite\luaauxfile
|
||||
\immediate\openout\luaauxfile=\jobname.lua.aux
|
||||
\immediate\write\luaauxfile{\noexpand\def\noexpand\luaAuxLoaded{lua aux loaded}}%
|
||||
\AtEndDocument{\immediate\closeout\luaauxfile}
|
||||
\IfFileExists{\jobname.lua.aux}{%
|
||||
\input{\jobname.lua.aux}
|
||||
}{}
|
||||
|
||||
\input{circuit.tex}
|
||||
\input{util/macros.tex}
|
||||
\input{util/environments.tex} % requires util/translation.tex to be loaded first
|
||||
\input{util/environments.tex} % requires util/translation.tex to be loaded first
|
||||
\input{util/periodic_table.tex} % requires util/translation.tex to be loaded first
|
||||
|
||||
\def\inputOnlyFile{\relax}
|
||||
\newcommand\Input[1]{
|
||||
@ -159,9 +200,11 @@
|
||||
|
||||
\newwrite\translationsaux
|
||||
\immediate\openout\translationsaux=\jobname.translations.aux
|
||||
\immediate\write\translationsaux{\noexpand\def\noexpand\MYVAR{AUSM AUX}}%
|
||||
\immediate\write\translationsaux{\noexpand\def\noexpand\translationsAuxLoaded{translations aux loaded}}%
|
||||
\AtEndDocument{\immediate\closeout\translationsaux}
|
||||
|
||||
\makeatletter\let\percentchar\@percentchar\makeatother
|
||||
|
||||
\maketitle
|
||||
\tableofcontents
|
||||
\newpage
|
||||
@ -169,10 +212,14 @@
|
||||
|
||||
\input{util/translations.tex}
|
||||
|
||||
% \include{maths/linalg}
|
||||
% \include{maths/geometry}
|
||||
\input{maths/analysis.tex}
|
||||
% \include{maths/probability_theory}
|
||||
\Part[
|
||||
\eng{Mathematics}
|
||||
\ger{Mathematik}
|
||||
]{math}
|
||||
% \include{math/linalg}
|
||||
% \include{math/geometry}
|
||||
% \input{math/calculus.tex}
|
||||
% \include{math/probability_theory}
|
||||
|
||||
\include{mechanics}
|
||||
|
||||
@ -183,7 +230,7 @@
|
||||
% \include{quantum_mechanics}
|
||||
% \include{atom}
|
||||
|
||||
\include{condensed_matter}
|
||||
\input{cm/cm.tex}
|
||||
\input{cm/charge_transport.tex}
|
||||
\input{cm/low_temp.tex}
|
||||
\input{cm/semiconductors.tex}
|
||||
@ -198,6 +245,9 @@
|
||||
\include{quantities}
|
||||
\include{constants}
|
||||
|
||||
\input{ch/periodic_table.tex} % only definitions
|
||||
\input{ch/ch.tex}
|
||||
|
||||
\newpage
|
||||
% \DT[english]{ttest}{TESTT EN}
|
||||
% \DT[german]{ttest}{TESTT DE}
|
||||
@ -235,6 +285,7 @@ Is defined? = \expandafter\IfTranslationExists\expandafter{\ttest:name}{yes}{no}
|
||||
Link to quantity which is defined after the reference: \qtyRef{test}\\
|
||||
\DT[qty:test]{english}{If you read this, then the translation for qty:test was expandend!}
|
||||
Link to defined quantity: \qtyRef{mass}
|
||||
\\ Link to element with name: \ElRef{H}
|
||||
\begin{equation}
|
||||
\label{qty:test}
|
||||
E = mc^2
|
||||
@ -246,9 +297,9 @@ Link to defined quantity: \qtyRef{mass}
|
||||
\gt{relative_undefined_translation_with_underscors}\\
|
||||
\GT{absolute_undefined_translation_with_&ersand}
|
||||
|
||||
|
||||
|
||||
|
||||
\paragraph{Aux files}
|
||||
\noindent Lua Aux loaded? \luaAuxLoaded\\
|
||||
Translations Aux loaded? \translationsAuxLoaded\\
|
||||
|
||||
|
||||
\newpage
|
||||
@ -261,4 +312,5 @@ Link to defined quantity: \qtyRef{mass}
|
||||
\listoftables
|
||||
% \bibliographystyle{plain}
|
||||
% \bibliography{ref}
|
||||
|
||||
\end{document}
|
||||
|
||||
@ -1,7 +1,7 @@
|
||||
\Part[
|
||||
\Section[
|
||||
\eng{Calculus}
|
||||
\ger{Analysis}
|
||||
]{cal}
|
||||
]{cal}
|
||||
|
||||
% \begin{formula}{shark}
|
||||
% \desc{Shark-midnight formula}{\emoji{shark}-s}{}
|
||||
@ -14,7 +14,7 @@
|
||||
\Subsection[
|
||||
\eng{Convolution}
|
||||
\ger{Faltung / Konvolution}
|
||||
]{conv}
|
||||
]{conv}
|
||||
\begin{ttext}
|
||||
\eng{Convolution is \textbf{commutative}, \textbf{associative} and \textbf{distributive}.}
|
||||
\ger{Die Faltung ist \textbf{kommutativ}, \textbf{assoziativ} und \textbf{distributiv}}
|
||||
@ -59,11 +59,11 @@
|
||||
\Subsection[
|
||||
\eng{Fourier analysis}
|
||||
\ger{Fourieranalyse}
|
||||
]{fourier}
|
||||
]{fourier}
|
||||
\Subsubsection[
|
||||
\eng{Fourier series}
|
||||
\ger{Fourierreihe}
|
||||
]{series}
|
||||
]{series}
|
||||
\begin{formula}{series}
|
||||
\desc{Fourier series}{Complex representation}{$f\in \Lebesgue^2(\R,\C)$ $T$-\GT{periodic}}
|
||||
\desc[german]{Fourierreihe}{Komplexe Darstellung}{}
|
||||
@ -102,7 +102,7 @@
|
||||
\Subsubsection[
|
||||
\eng{Fourier transformation}
|
||||
\ger{Fouriertransformation}
|
||||
]{trafo}
|
||||
]{trafo}
|
||||
\begin{formula}{transform}
|
||||
\desc{Fourier transform}{}{$\hat{f}:\R^n \mapsto \C$, $\forall f\in L^1(\R^n)$}
|
||||
\desc[german]{Fouriertransformierte}{}{}
|
||||
@ -148,58 +148,58 @@
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Section[
|
||||
\eng{Logarithm}
|
||||
\ger{Logarithmus}
|
||||
]{log}
|
||||
\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}{intergral}
|
||||
\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[
|
||||
\eng{Logarithm}
|
||||
\ger{Logarithmus}
|
||||
]{log}
|
||||
\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}{intergral}
|
||||
\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}
|
||||
|
||||
\Section[
|
||||
\eng{List of common integrals}
|
||||
\ger{Liste nützlicher Integrale}
|
||||
\Subection[
|
||||
\eng{List of common integrals}
|
||||
\ger{Liste nützlicher Integrale}
|
||||
]{integrals}
|
||||
|
||||
% Put links to other integrals here
|
||||
\fqEqRef{cal:log:integral}
|
||||
% Put links to other integrals here
|
||||
\fqEqRef{cal:log:integral}
|
||||
|
||||
\begin{formula}{spherical-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}{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}{spherical-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}{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}{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}
|
||||
|
||||
|
||||
|
||||
@ -1,50 +1,50 @@
|
||||
\Part[
|
||||
\Section[
|
||||
\eng{Geometry}
|
||||
\ger{Geometrie}
|
||||
]{geo}
|
||||
|
||||
\Section[
|
||||
\Subsection[
|
||||
\eng{Trigonometry}
|
||||
\ger{Trigonometrie}
|
||||
]{trig}
|
||||
]{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}{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}{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}{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_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}
|
||||
\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[
|
||||
\eng{Various theorems}
|
||||
\ger{Verschiedene Theoreme}
|
||||
]{theorems}
|
||||
\Subsection[
|
||||
\eng{Various theorems}
|
||||
\ger{Verschiedene Theoreme}
|
||||
]{theorems}
|
||||
\begin{formula}{sum}
|
||||
\desc{}{}{}
|
||||
\desc[german]{}{}{}
|
||||
@ -78,10 +78,10 @@
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsection[
|
||||
\Subsubsection[
|
||||
\eng{Table of values}
|
||||
\ger{Wertetabelle}
|
||||
]{value_table}
|
||||
]{value_table}
|
||||
\begingroup
|
||||
\setlength{\tabcolsep}{0.9em} % horizontal
|
||||
\renewcommand{\arraystretch}{2} % vertical
|
||||
@ -1,11 +1,9 @@
|
||||
\def\id{\mathbb{1}}
|
||||
|
||||
\Part[
|
||||
\Section[
|
||||
\eng{Linear algebra}
|
||||
\ger{Lineare Algebra}
|
||||
]{linalg}
|
||||
|
||||
\Section[
|
||||
\Subsection[
|
||||
\eng{Determinant}
|
||||
\ger{Determinante}
|
||||
]{determinant}
|
||||
@ -43,7 +41,7 @@
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Section[
|
||||
\Subsection[
|
||||
|
||||
]{zeug}
|
||||
|
||||
@ -95,7 +93,7 @@
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Section[
|
||||
\Subection[
|
||||
\eng{Eigenvalues}
|
||||
\ger{Eigenwerte}
|
||||
]{eigen}
|
||||
241
src/math/probability_theory.tex
Normal file
241
src/math/probability_theory.tex
Normal file
@ -0,0 +1,241 @@
|
||||
\Section[
|
||||
\eng{Probability theory}
|
||||
\ger{Wahrscheinlichkeitstheorie}
|
||||
]{pt}
|
||||
|
||||
\begin{formula}{mean}
|
||||
\desc{Mean}{Expectation value}{}
|
||||
\desc[german]{Mittelwert}{Erwartungswert}{}
|
||||
\eq{\braket{x} = \int w(x)\, x\, \d x}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{variance}
|
||||
\desc{Variance}{Square of the \fqEqRef{math:pt:std-deviation}}{}
|
||||
\desc[german]{Varianz}{Quadrat der\fqEqRef{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}
|
||||
\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}
|
||||
\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}
|
||||
\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}
|
||||
\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}{autocorrelation}
|
||||
\desc{Autocorrelation}{Correlation of $f$ to itself at an earlier point in time, $C$ is a covariance function}{}
|
||||
\desc[german]{Autokorrelation}{Korrelation vonn $f$ zu sich selbst zu einem früheren Zeitpunkt. $C$ ist auch die Kovarianzfunktion}{}
|
||||
\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}
|
||||
|
||||
\Subsection[
|
||||
\eng{Distributions}
|
||||
\ger{Verteilungen}
|
||||
]{distributions}
|
||||
\Subsubsection[
|
||||
\eng{Gauß/Normal distribution}
|
||||
\ger{Gauß/Normal-Verteilung}
|
||||
]{normal}
|
||||
\begin{minipage}{\distleftwidth}
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{img/distribution_gauss.pdf}
|
||||
\end{figure}
|
||||
\end{minipage}
|
||||
\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}
|
||||
|
||||
\begin{formula}{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}
|
||||
|
||||
\Subsubsection[
|
||||
\eng{Cauchys / Lorentz distribution}
|
||||
\ger{Cauchy / Lorentz-Verteilung}
|
||||
]{cauchy}
|
||||
\begin{minipage}{\distleftwidth}
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{img/distribution_cauchy.pdf}
|
||||
\end{figure}
|
||||
\end{minipage}
|
||||
\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}
|
||||
|
||||
\begin{ttext}
|
||||
\eng{Also known as \textbf{Cauchy-Lorentz distribution}, \textbf{Lorentz(ian) function}, \textbf{Breit-Wigner distribution}.}
|
||||
\ger{Auch bekannt als \textbf{Cauchy-Lorentz Verteilung}, \textbf{Lorentz Funktion}, \textbf{Breit-Wigner Verteilung}.}
|
||||
\end{ttext}
|
||||
|
||||
|
||||
\Subsubsection[
|
||||
\eng{Binomial distribution}
|
||||
\ger{Binomialverteilung}
|
||||
]{binomial}
|
||||
\begin{ttext}
|
||||
\eng{For the number of trials going to infinity ($n\to\infty$), the binomial distribution converges to the \hyperref[sec:pb:distributions::poisson]{poisson distribution}}
|
||||
\ger{Geht die Zahl der Versuche gegen unendlich ($n\to\infty$), konvergiert die Binomualverteilung gegen die \hyperref[sec:pb:distributions::poisson]{Poissonverteilung}}
|
||||
\end{ttext}
|
||||
\begin{minipage}{\distleftwidth}
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{img/distribution_binomial.pdf}
|
||||
\end{figure}
|
||||
\end{minipage}
|
||||
\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}
|
||||
\Subsubsection[
|
||||
\eng{Poisson distribution}
|
||||
\ger{Poissonverteilung}
|
||||
]{poisson}
|
||||
\begin{minipage}{\distleftwidth}
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{img/distribution_poisson.pdf}
|
||||
\end{figure}
|
||||
\end{minipage}
|
||||
\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}
|
||||
|
||||
|
||||
\Subsubsection[
|
||||
\eng{Maxwell-Boltzmann distribution}
|
||||
\ger{Maxwell-Boltzmann Verteilung}
|
||||
]{maxwell-boltzmann}
|
||||
\begin{minipage}{\distleftwidth}
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{img/distribution_maxwell-boltzmann.pdf}
|
||||
\end{figure}
|
||||
\end{minipage}
|
||||
\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}
|
||||
|
||||
|
||||
|
||||
% \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[
|
||||
\eng{Central limit theorem}
|
||||
\ger{Zentraler Grenzwertsatz}
|
||||
]{cls}
|
||||
\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[
|
||||
\eng{Propagation of uncertainty / error}
|
||||
\ger{Fehlerfortpflanzung}
|
||||
]{error}
|
||||
\begin{formula}{generalised}
|
||||
\desc{Generalized error propagation}{}{$V$ \fqEqRef{math:pt:covariance} matrix, $J$ \fqEqRef{math:cal:jacobi-matrix}}
|
||||
\desc[german]{Generalisiertes Fehlerfortpflanzungsgesetz}{$V$ \fqEqRef{math:pt:covariance} Matrix, $J$ \fqEqRef{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}{$\sigma$ \fqEqRef{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}{}{$w_i$ \fqEqRef{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}{}{$w_i$ \fqEqRef{math:pt:error:weight}}
|
||||
\desc[german]{Varianz des gewichteten Mittelwertes}{}{}
|
||||
\eq{\sigma^2_{\overline{x}} = \frac{1}{\sum_i w_i}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
@ -1,241 +0,0 @@
|
||||
\Part[
|
||||
\eng{Probability theory}
|
||||
\ger{Wahrscheinlichkeitstheorie}
|
||||
]{pt}
|
||||
|
||||
\begin{formula}{mean}
|
||||
\desc{Mean}{Expectation value}{}
|
||||
\desc[german]{Mittelwert}{Erwartungswert}{}
|
||||
\eq{\braket{x} = \int w(x)\, x\, \d x}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{variance}
|
||||
\desc{Variance}{Square of the \fqEqRef{pt:std-deviation}}{}
|
||||
\desc[german]{Varianz}{Quadrat der\fqEqRef{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}
|
||||
\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}
|
||||
\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}
|
||||
\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}
|
||||
\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}{autocorrelation}
|
||||
\desc{Autocorrelation}{Correlation of $f$ to itself at an earlier point in time, $C$ is a covariance function}{}
|
||||
\desc[german]{Autokorrelation}{Korrelation vonn $f$ zu sich selbst zu einem früheren Zeitpunkt. $C$ ist auch die Kovarianzfunktion}{}
|
||||
\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}
|
||||
|
||||
\Section[
|
||||
\eng{Distributions}
|
||||
\ger{Verteilungen}
|
||||
]{distributions}
|
||||
\Subsubsection[
|
||||
\eng{Gauß/Normal distribution}
|
||||
\ger{Gauß/Normal-Verteilung}
|
||||
]{normal}
|
||||
\begin{minipage}{\distleftwidth}
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{img/distribution_gauss.pdf}
|
||||
\end{figure}
|
||||
\end{minipage}
|
||||
\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}
|
||||
|
||||
\begin{formula}{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}
|
||||
|
||||
\Subsubsection[
|
||||
\eng{Cauchys / Lorentz distribution}
|
||||
\ger{Cauchy / Lorentz-Verteilung}
|
||||
]{cauchy}
|
||||
\begin{minipage}{\distleftwidth}
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{img/distribution_cauchy.pdf}
|
||||
\end{figure}
|
||||
\end{minipage}
|
||||
\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}
|
||||
|
||||
\begin{ttext}
|
||||
\eng{Also known as \textbf{Cauchy-Lorentz distribution}, \textbf{Lorentz(ian) function}, \textbf{Breit-Wigner distribution}.}
|
||||
\ger{Auch bekannt als \textbf{Cauchy-Lorentz Verteilung}, \textbf{Lorentz Funktion}, \textbf{Breit-Wigner Verteilung}.}
|
||||
\end{ttext}
|
||||
|
||||
|
||||
\Subsubsection[
|
||||
\eng{Binomial distribution}
|
||||
\ger{Binomialverteilung}
|
||||
]{binomial}
|
||||
\begin{ttext}
|
||||
\eng{For the number of trials going to infinity ($n\to\infty$), the binomial distribution converges to the \hyperref[sec:pb:distributions::poisson]{poisson distribution}}
|
||||
\ger{Geht die Zahl der Versuche gegen unendlich ($n\to\infty$), konvergiert die Binomualverteilung gegen die \hyperref[sec:pb:distributions::poisson]{Poissonverteilung}}
|
||||
\end{ttext}
|
||||
\begin{minipage}{\distleftwidth}
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{img/distribution_binomial.pdf}
|
||||
\end{figure}
|
||||
\end{minipage}
|
||||
\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}
|
||||
\Subsubsection[
|
||||
\eng{Poisson distribution}
|
||||
\ger{Poissonverteilung}
|
||||
]{poisson}
|
||||
\begin{minipage}{\distleftwidth}
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{img/distribution_poisson.pdf}
|
||||
\end{figure}
|
||||
\end{minipage}
|
||||
\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}
|
||||
|
||||
|
||||
\Subsubsection[
|
||||
\eng{Maxwell-Boltzmann distribution}
|
||||
\ger{Maxwell-Boltzmann Verteilung}
|
||||
]{maxwell-boltzmann}
|
||||
\begin{minipage}{\distleftwidth}
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{img/distribution_maxwell-boltzmann.pdf}
|
||||
\end{figure}
|
||||
\end{minipage}
|
||||
\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}
|
||||
|
||||
|
||||
|
||||
% \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[
|
||||
\eng{Central limit theorem}
|
||||
\ger{Zentraler Grenzwertsatz}
|
||||
]{cls}
|
||||
\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}
|
||||
|
||||
\Section[
|
||||
\eng{Propagation of uncertainty / error}
|
||||
\ger{Fehlerfortpflanzung}
|
||||
]{error}
|
||||
\begin{formula}{generalised}
|
||||
\desc{Generalized error propagation}{}{$V$ \fqEqRef{pt:covariance} matrix, $J$ \fqEqRef{ana:jacobi-matrix}}
|
||||
\desc[german]{Generalisiertes Fehlerfortpflanzungsgesetz}{$V$ \fqEqRef{pt:covariance} Matrix, $J$ \fqEqRef{ana: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}{$\sigma$ \fqEqRef{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}{}{$w_i$ \fqEqRef{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}{}{$w_i$ \fqEqRef{pt:error:weight}}
|
||||
\desc[german]{Varianz des gewichteten Mittelwertes}{}{}
|
||||
\eq{\sigma^2_{\overline{x}} = \frac{1}{\sum_i w_i}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
@ -6,72 +6,72 @@
|
||||
\paragraph{\GT{si_base_units}}
|
||||
|
||||
\begin{quantity}{time}{t}{\second}{}
|
||||
\desc{Time}{}
|
||||
\desc[german]{Zeit}{}
|
||||
\desc{Time}{}{}
|
||||
\desc[german]{Zeit}{}{}
|
||||
\end{quantity}
|
||||
|
||||
\begin{quantity}{Length}{l}{\m}{e}
|
||||
\desc{Length}{}
|
||||
\desc[german]{Länge}{}
|
||||
\desc{Length}{}{}
|
||||
\desc[german]{Länge}{}{}
|
||||
\end{quantity}
|
||||
|
||||
\begin{quantity}{mass}{m}{\kg}{es}
|
||||
\desc{Mass}{}
|
||||
\desc[german]{Masse}{}
|
||||
\desc{Mass}{}{}
|
||||
\desc[german]{Masse}{}{}
|
||||
\end{quantity}
|
||||
|
||||
\begin{quantity}{temperature}{T}{\kelvin}{is}
|
||||
\desc{Temperature}{}
|
||||
\desc[german]{Temperatur}{}
|
||||
\desc{Temperature}{}{}
|
||||
\desc[german]{Temperatur}{}{}
|
||||
\end{quantity}
|
||||
|
||||
\begin{quantity}{current}{I}{\ampere}{es}
|
||||
\desc{Electric current}{}
|
||||
\desc[german]{Elektrischer Strom}{}
|
||||
\desc{Electric current}{}{}
|
||||
\desc[german]{Elektrischer Strom}{}{}
|
||||
\end{quantity}
|
||||
|
||||
\begin{quantity}{amount}{n}{\mol}{es}
|
||||
\desc{Amount of substance}{}
|
||||
\desc[german]{Stoffmenge}{}
|
||||
\desc{Amount of substance}{}{}
|
||||
\desc[german]{Stoffmenge}{}{}
|
||||
\end{quantity}
|
||||
|
||||
\begin{quantity}{luminous_intensity}{I_\text{V}}{\candela}{s}
|
||||
\desc{Luminous intensity}{}
|
||||
\desc[german]{Lichtstärke}{}
|
||||
\desc{Luminous intensity}{}{}
|
||||
\desc[german]{Lichtstärke}{}{}
|
||||
\end{quantity}
|
||||
|
||||
\paragraph{\GT{other}}
|
||||
\begin{quantity}{volume}{V}{\m^d}{}
|
||||
\desc{Volume}{$d$ dimensional Volume}
|
||||
\desc[german]{Volumen}{$d$ dimensionales Volumen}
|
||||
\desc{Volume}{$d$ dimensional Volume}{}
|
||||
\desc[german]{Volumen}{$d$ dimensionales Volumen}{}
|
||||
\end{quantity}
|
||||
|
||||
\begin{quantity}{force}{\vec{F}}{\newton=\kg\m\per\second^2}{ev}
|
||||
\desc{Force}{}
|
||||
\desc[german]{Kraft}{}
|
||||
\desc{Force}{}{}
|
||||
\desc[german]{Kraft}{}{}
|
||||
\end{quantity}
|
||||
|
||||
\begin{quantity}{spring_constant}{k}{\newton\per\m=\kg\per\second^2}{s}
|
||||
\desc{Spring constant}{}
|
||||
\desc[german]{Federkonstante}{}
|
||||
\desc{Spring constant}{}{}
|
||||
\desc[german]{Federkonstante}{}{}
|
||||
\end{quantity}
|
||||
|
||||
\begin{quantity}{velocity}{\vec{v}}{\m\per\s}{v}
|
||||
\desc{Velocity}{}
|
||||
\desc[german]{Geschwindigkeit}{}
|
||||
\desc{Velocity}{}{}
|
||||
\desc[german]{Geschwindigkeit}{}{}
|
||||
\end{quantity}
|
||||
|
||||
\begin{quantity}{torque}{\tau}{\newton\m=\kg\m^2\per\s^2}{v}
|
||||
\desc{Torque}{}
|
||||
\desc[german]{Drehmoment}{}
|
||||
\desc{Torque}{}{}
|
||||
\desc[german]{Drehmoment}{}{}
|
||||
\end{quantity}
|
||||
|
||||
\begin{quantity}{heat_capacity}{C}{\joule\per\kelvin}{}
|
||||
\desc{Heat capacity}{}
|
||||
\desc[german]{Wärmekapazität}{}
|
||||
\desc{Heat capacity}{}{}
|
||||
\desc[german]{Wärmekapazität}{}{}
|
||||
\end{quantity}
|
||||
|
||||
\begin{quantity}{charge}{q}{\coulomb=\ampere\s}{}
|
||||
\desc{Charge}{}
|
||||
\desc[german]{Ladung}{}
|
||||
\desc{Charge}{}{}
|
||||
\desc[german]{Ladung}{}{}
|
||||
\end{quantity}
|
||||
|
||||
@ -1,8 +1,11 @@
|
||||
\Part{Topo}
|
||||
\Part[
|
||||
\eng{Topological Materials}
|
||||
\ger{Topologische Materialien}
|
||||
]{topo}
|
||||
\Section[
|
||||
\eng{Berry phase / Geometric phase}
|
||||
\ger{Berry-Phase / Geometrische Phase}
|
||||
]{berry_phase}
|
||||
]{berry_phase}
|
||||
|
||||
\begin{ttext}[desc]
|
||||
\eng{
|
||||
|
||||
@ -25,7 +25,7 @@
|
||||
% [1]: minipage width
|
||||
% 2: fqname of name
|
||||
% 3: fqname of a translation that holds the explanation
|
||||
\newcommand{\NameWithExplanation}[3][\descwidth]{
|
||||
\newcommand{\NameWithDescription}[3][\descwidth]{
|
||||
\begin{minipage}{#1}
|
||||
\IfTranslationExists{#2}{
|
||||
\raggedright
|
||||
@ -45,6 +45,7 @@
|
||||
\begin{minipage}{#1}
|
||||
% \vspace{-\baselineskip} % remove the space that comes from starting a new paragraph
|
||||
#2
|
||||
\smartnewline
|
||||
\noindent\IfTranslationExists{#3}{
|
||||
\begingroup
|
||||
\color{dark1}
|
||||
@ -65,7 +66,7 @@
|
||||
\par\noindent\ignorespaces
|
||||
% \textcolor{gray}{\hrule}
|
||||
\vspace{0.5\baselineskip}
|
||||
\NameWithExplanation[\descwidth]{#1}{#1_desc}
|
||||
\NameWithDescription[\descwidth]{#1}{#1_desc}
|
||||
\hfill
|
||||
\ContentBoxWithExplanation[\eqwidth]{#2}{#1_defs}
|
||||
\textcolor{dark3}{\hrule}
|
||||
@ -107,7 +108,6 @@
|
||||
}
|
||||
}
|
||||
|
||||
\newcommand\luaexpr[1]{\directlua{tex.sprint(#1)}}
|
||||
% [1]: width
|
||||
% 2: fqname
|
||||
% 3: file path
|
||||
@ -117,7 +117,7 @@
|
||||
% \textcolor{gray}{\hrule}
|
||||
\vspace{0.5\baselineskip}
|
||||
\begin{minipage}{#1\textwidth}
|
||||
\NameWithExplanation[\textwidth]{\fqname:#2}{#2_desc}
|
||||
\NameWithDescription[\textwidth]{\fqname:#2}{#2_desc}
|
||||
% TODO: why is this ignored
|
||||
\vspace{1.0cm}
|
||||
% TODO: fix box is too large without 0.9
|
||||
@ -129,7 +129,7 @@
|
||||
}{#2_defs}
|
||||
\end{minipage}
|
||||
\hfill
|
||||
\begin{minipage}{\luaexpr{1.0-#1}\textwidth}
|
||||
\begin{minipage}{\luavar{1.0-#1}\textwidth}
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{#3}
|
||||
@ -138,7 +138,6 @@
|
||||
\end{minipage}
|
||||
\textcolor{dark3}{\hrule}
|
||||
\vspace{0.5\baselineskip}
|
||||
|
||||
}
|
||||
|
||||
% 1: key
|
||||
@ -255,7 +254,42 @@
|
||||
% % \addquantity{\expandafter\gt\expandafter{\qtyname}}%
|
||||
% % \noindent\textbf{My Environment \themyenv: #1}\par%
|
||||
% }
|
||||
\directLuaAux{
|
||||
if constants == nil then
|
||||
constants = {}
|
||||
end
|
||||
}
|
||||
\newcommand\printConstant[1]{
|
||||
\edef\constName{const:#1}
|
||||
\NameLeftContentRight{\constName}{
|
||||
\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
|
||||
}
|
||||
% label it only once
|
||||
\directlua{
|
||||
if constants["#1"]["labeled"] == nil then
|
||||
constants["#1"]["labeled"] = true
|
||||
tex.print("\\label{const:#1}")
|
||||
end
|
||||
}
|
||||
\endgroup
|
||||
}
|
||||
}
|
||||
|
||||
\newcounter{constant}
|
||||
% 1: key - must expand to a valid lua string!
|
||||
% 2: symbol
|
||||
% 3: either exp or def; experimentally or defined constant
|
||||
\newenvironment{constant}[3]{
|
||||
% [1]: language
|
||||
% 2: name
|
||||
@ -267,89 +301,76 @@
|
||||
\ifblank{##3}{}{\DT[const:#1_desc]{##1}{##3}}
|
||||
\ifblank{##4}{}{\DT[const:#1_defs]{##1}{##4}}
|
||||
}
|
||||
\directlua{
|
||||
constVals = {}
|
||||
constUnits = {}
|
||||
\directLuaAux{
|
||||
constants["#1"] = {};
|
||||
constants["#1"]["symbol"] = [[\detokenize{#2}]];
|
||||
constants["#1"]["exp_or_def"] = [[\detokenize{#3}]];
|
||||
constants["#1"]["values"] = {} -- array of {value, unit};
|
||||
}
|
||||
% 1: equation for align environment
|
||||
% 1: value
|
||||
% 2: unit
|
||||
\newcommand{\val}[2]{
|
||||
\directlua{
|
||||
%--table.insert(constVals, "LOL")
|
||||
table.insert(constVals, [[##1]])
|
||||
table.insert(constUnits, [[##2]])
|
||||
%--table.insert(constUnits, "\luaescapestring{##2}")
|
||||
\directLuaAux{
|
||||
table.insert(constants["#1"]["values"], { value = [[\detokenize{##1}]], unit = [[\detokenize{##2}]] })
|
||||
}
|
||||
\def\constValue{##1}
|
||||
\def\constUnit{##2}
|
||||
}
|
||||
\edef\constName{const:#1}
|
||||
\edef\constDesc{const:#1_desc}
|
||||
\def\constSymbol{#2}
|
||||
\edef\constExpOrDef{\GT{const:#3}}
|
||||
\edef\lastConstName{#1}
|
||||
}{
|
||||
\NameLeftContentRight{\constName}{
|
||||
\begingroup % for label
|
||||
Symbol: $\constSymbol$
|
||||
\IfTranslationExists{\constDesc}{
|
||||
\\Description: \GT{\constDesc}
|
||||
}{}
|
||||
% TODO manage multiple values
|
||||
% \\Value: $\constValue\,\si{\constUnit}$
|
||||
\\\constExpOrDef:
|
||||
\directlua{
|
||||
%--tex.sprint("Hier steht Luatext" .. ":", #constVals)
|
||||
for i, v in ipairs(constVals) do
|
||||
tex.sprint("\\\\\\hspace*{1cm}${", constVals[i], "}\\,\\si{", constUnits[i], "}$")
|
||||
%--tex.sprint("VALUE ", i, v)
|
||||
end
|
||||
}
|
||||
\label{\constName}
|
||||
\endgroup
|
||||
}
|
||||
\expandafter\printConstant{\lastConstName}
|
||||
\ignorespacesafterend
|
||||
}
|
||||
|
||||
|
||||
|
||||
% 1: key
|
||||
\directLuaAux{
|
||||
if quantities == nil then
|
||||
quantities = {}
|
||||
end
|
||||
}
|
||||
\newcommand\printQuantity[1]{
|
||||
\edef\qtyName{qty:#1}
|
||||
\NameLeftContentRight{\qtyName}{
|
||||
\begingroup % for label
|
||||
Symbol: $\luavar{quantities["#1"]["symbol"]}$
|
||||
\\Unit: $\directlua{split_and_print_units(quantities["#1"]["units"])}$
|
||||
% label it only once
|
||||
\directlua{
|
||||
if quantities["#1"]["labeled"] == nil then
|
||||
quantities["#1"]["labeled"] = true
|
||||
tex.print("\\label{qty:#1}")
|
||||
end
|
||||
}
|
||||
\endgroup
|
||||
}
|
||||
|
||||
}
|
||||
% 1: key - must expand to a valid lua string!
|
||||
% 2: symbol
|
||||
% 3: units
|
||||
% 4: comment key to translation
|
||||
\newenvironment{quantity}[4]{
|
||||
% language, name, description
|
||||
\newcommand{\desc}[3][english]{
|
||||
% language, name, description, definitions
|
||||
\newcommand{\desc}[4][english]{
|
||||
\ifblank{##2}{}{\DT[qty:#1]{##1}{##2}}
|
||||
\ifblank{##3}{}{\DT[qty:#1_desc]{##1}{##3}}
|
||||
\ifblank{##4}{}{\DT[qty:#1_defs]{##1}{##4}}
|
||||
}
|
||||
% TODO put these in long term key - value storage for generating a full table and global referenes \qtyRef
|
||||
% for references, there needs to be a label somwhere
|
||||
\edef\qtyname{qty:#1}
|
||||
\edef\qtydesc{qty:#1_desc}
|
||||
\def\qtysymbol{#2}
|
||||
\def\qtyunits{#3}
|
||||
\edef\qtycomment{#4}
|
||||
% Unit: $\directlua{split_and_print_units([[\m\per\kg]])}$
|
||||
\directLuaAux{
|
||||
quantities["#1"] = {}
|
||||
quantities["#1"]["symbol"] = [[\detokenize{#2}]]
|
||||
quantities["#1"]["units"] = [[\detokenize{#3}]]
|
||||
quantities["#1"]["comment"] = [[\detokenize{#4}]]
|
||||
}
|
||||
\def\lastQtyName{#1}
|
||||
}
|
||||
{
|
||||
\NameLeftContentRight{\qtyname}{
|
||||
\begingroup
|
||||
Symbol: $\qtysymbol$
|
||||
\IfTranslationExists{\qtydesc}{
|
||||
\\Description: \GT{\qtydesc}
|
||||
}{}
|
||||
\\Unit: $\directlua{split_and_print_units([[\qtyunits]])}$
|
||||
\expandafter\IfTranslationExists\expandafter\qtycomment{
|
||||
\\Comment: \GT\qtycomment
|
||||
}{}%{\\No comment \color{gray}}
|
||||
\label{\qtyname}
|
||||
\endgroup
|
||||
}
|
||||
\expandafter\printQuantity{\lastQtyName}
|
||||
\ignorespacesafterend
|
||||
|
||||
% for TOC
|
||||
\refstepcounter{quantity}%
|
||||
% \addquantity{\expandafter\gt\expandafter{\qtyname}}%
|
||||
% \noindent\textbf{My Environment \themyenv: #1}\par%
|
||||
}
|
||||
\newcounter{quantity}
|
||||
\newcommand{\listofquantities}{%
|
||||
|
||||
@ -1,3 +1,4 @@
|
||||
\newcommand\smartnewline[1]{\ifhmode\\\fi} % newline only if there in horizontal mode
|
||||
\def\gooditem{\item[{$\color{neutral_red}\bullet$}]}
|
||||
\def\baditem{\item[{$\color{neutral_green}\bullet$}]}
|
||||
|
||||
@ -10,12 +11,15 @@
|
||||
\overset{\substack{\mathrlap{\text{\hspace{-1em}#2}}\\\downarrow}}{#1}}
|
||||
|
||||
% COMMON SYMBOLS WITH SUPER/SUBSCRIPTS, VECTOR ARROWS ETC.
|
||||
% \def\laplace{\Delta} % Laplace operator
|
||||
\def\laplace{\bigtriangleup} % Laplace operator
|
||||
\def\Grad{\vec{\nabla}}
|
||||
\def\Div{\vec{\nabla} \cdot}
|
||||
\def\Rot{\vec{\nabla} \times}
|
||||
\def\vecr{\vec{r}}
|
||||
\def\vecx{\vec{x}}
|
||||
\def\kB{k_\text{B}}
|
||||
\def\kB{k_\text{B}} % boltzmann
|
||||
\def\NA{N_\text{A}} % avogadro
|
||||
\def\EFermi{E_\text{F}}
|
||||
\def\Evalence{E_\text{v}}
|
||||
\def\Econd{E_\text{c}}
|
||||
@ -33,8 +37,9 @@
|
||||
\def\C{\mathbb{C}}
|
||||
\def\Z{\mathbb{Z}}
|
||||
\def\N{\mathbb{N}}
|
||||
\def\id{\mathbb{1}}
|
||||
% caligraphic
|
||||
\def\calE{\mathcal{E}}
|
||||
\def\E{\mathcal{E}} % electric field
|
||||
% upright
|
||||
\def\txA{\text{A}}
|
||||
\def\txB{\text{B}}
|
||||
@ -114,6 +119,8 @@
|
||||
% diff, for integrals and stuff
|
||||
% \DeclareMathOperator{\dd}{d}
|
||||
\renewcommand*\d{\mathop{}\!\mathrm{d}}
|
||||
% times 10^{x}
|
||||
\newcommand\xE[1]{\cdot 10^{#1}}
|
||||
% functions with paranthesis
|
||||
\newcommand\CmdWithParenthesis[2]{
|
||||
#1\left(#2\right)
|
||||
|
||||
162
src/util/periodic_table.tex
Normal file
162
src/util/periodic_table.tex
Normal file
@ -0,0 +1,162 @@
|
||||
% 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
|
||||
|
||||
% INFO
|
||||
\directLuaAux{
|
||||
if elements == nil then
|
||||
elements = {} %-- Symbol: {symbol, atomic_number, properties, ... }
|
||||
elementsOrder = {} %-- Number: Symbol
|
||||
end
|
||||
}
|
||||
|
||||
% 1: symbol
|
||||
% 2: nr
|
||||
% 3: period
|
||||
% 4: column
|
||||
\newenvironment{element}[4]{
|
||||
% [1]: language
|
||||
% 2: name
|
||||
% 3: description
|
||||
% 4: definitions/links
|
||||
\newcommand{\desc}[4][english]{
|
||||
% language, name, description, definitions
|
||||
\ifblank{##2}{}{\DT[el:#1]{##1}{##2}}
|
||||
\ifblank{##3}{}{\DT[el:#1_desc]{##1}{##3}}
|
||||
\ifblank{##4}{}{\DT[el:#1_defs]{##1}{##4}}
|
||||
}
|
||||
\directLuaAux{
|
||||
elementsOrder[#2] = "#1";
|
||||
elements["#1"] = {};
|
||||
elements["#1"]["symbol"] = [[\detokenize{#1}]];
|
||||
elements["#1"]["atomic_number"] = [[\detokenize{#2}]];
|
||||
elements["#1"]["period"] = [[\detokenize{#3}]];
|
||||
elements["#1"]["column"] = [[\detokenize{#4}]];
|
||||
elements["#1"]["properties"] = {};
|
||||
}
|
||||
% 1: key
|
||||
% 2: value
|
||||
\newcommand{\property}[2]{
|
||||
\directlua{ %-- writing to aux is only needed for references for now
|
||||
elements["#1"]["properties"]["##1"] = "\luaescapestring{\detokenize{##2}}" %-- cant use [[ ]] because electron_config ends with ]
|
||||
}
|
||||
}
|
||||
\edef\lastElementName{#1}
|
||||
}{
|
||||
% \expandafter\printElement{\lastElementName}
|
||||
\ignorespacesafterend
|
||||
}
|
||||
|
||||
% LIST
|
||||
\newcommand\printElement[1]{
|
||||
\edef\elementName{el:#1}
|
||||
\NameLeftContentRight{\elementName}{
|
||||
\begingroup % for label
|
||||
\directlua{
|
||||
tex.sprint("Symbol: \\ce{"..elements["#1"]["symbol"].."}")
|
||||
tex.sprint("\\\\Number: "..elements["#1"]["atomic_number"])
|
||||
}
|
||||
\directlua{
|
||||
%--tex.sprint("Hier steht Luatext" .. ":", #elementVals)
|
||||
for key, value in pairs(elements["#1"]["properties"]) do
|
||||
tex.sprint("\\\\\\hspace*{1cm}{\\GT{", key, "}: ", value, "}")
|
||||
%--tex.sprint("VALUE ", i, v)
|
||||
end
|
||||
}
|
||||
% label it only once
|
||||
\directlua{
|
||||
if elements["#1"]["labeled"] == nil then
|
||||
elements["#1"]["labeled"] = true
|
||||
tex.print("\\label{el:#1}")
|
||||
end
|
||||
}
|
||||
\endgroup
|
||||
}
|
||||
}
|
||||
\newcommand{\printAllElements}{
|
||||
\directlua{
|
||||
%-- tex.sprint("\\printElement{"..val.."}")
|
||||
for key, val in ipairs(elementsOrder) do
|
||||
%-- tex.sprint(key, val);
|
||||
tex.sprint("\\printElement{"..val.."}")
|
||||
end
|
||||
}
|
||||
}
|
||||
|
||||
% PERIODIC TABLE
|
||||
\directlua{
|
||||
category2color = {
|
||||
metal = "neutral_blue",
|
||||
metalloid = "bright_orange",
|
||||
transitionmetal = "bright_blue",
|
||||
lanthanoide = "neutral_orange",
|
||||
alkalimetal = "bright_red",
|
||||
alkalineearthmetal = "bright_purple",
|
||||
nonmetal = "bright_aqua",
|
||||
halogen = "bright_yellow",
|
||||
noblegas = "neutral_purple"
|
||||
}
|
||||
}
|
||||
\directlua{
|
||||
function getColor(cat)
|
||||
local color = category2color[cat]
|
||||
if color == nil then
|
||||
return "light3"
|
||||
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,faded_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}
|
||||
|
||||
|
||||
|
||||
}
|
||||
@ -21,15 +21,16 @@
|
||||
\def\tempiftranslation{\IfTranslation{english}}%
|
||||
\expandafter\tempiftranslation\expandafter{#1}%
|
||||
}
|
||||
\newcommand{\iftranslation}[3]{%
|
||||
\IfTranslationExists{\fqname:#1}{#2}{#3}%
|
||||
\newcommand{\iftranslation}[1]{%
|
||||
\IfTranslation{english}{\fqname:#1}
|
||||
% \expandafter\IfTranslationExists\expandafter{\fqname:#1}
|
||||
}
|
||||
|
||||
\newcommand{\gt}[1]{%
|
||||
\iftranslation{#1}{%
|
||||
\expandafter\GetTranslation\expandafter{\fqname:#1}%
|
||||
}{%
|
||||
\fqname:\detokenize{#1}%
|
||||
\detokenize{\fqname}:\detokenize{#1}%
|
||||
}%
|
||||
}
|
||||
\newrobustcmd{\GT}[1]{%\expandafter\GetTranslation\expandafter{#1}}
|
||||
|
||||
@ -46,3 +46,17 @@
|
||||
\Eng[const:def]{Defined value}
|
||||
\Ger[const:def]{Definierter Wert}
|
||||
|
||||
% PERIODIC TABLE
|
||||
\Eng[symbol]{Symbol}
|
||||
\Ger[symbol]{Symbol}
|
||||
|
||||
\Eng[atomic_number]{Number}
|
||||
\Ger[atomic_number]{Ordnungszahl}
|
||||
|
||||
\Eng[electron_config]{Electronic configuration}
|
||||
\Ger[electron_config]{Elektronenkonfiguration}
|
||||
|
||||
\Eng[crystal_structure]{Crystal structure}
|
||||
\Ger[crystal_structure]{Kristallstruktur}
|
||||
|
||||
|
||||
|
||||
Loading…
x
Reference in New Issue
Block a user