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61
scripts/ch_elchem.py
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@ -1,47 +1,80 @@
#!/usr/bin env python3
from formulasheet import *
from formulary import *
from scipy.constants import gas_constant, Avogadro, elementary_charge
Faraday = Avogadro * elementary_charge
@np.vectorize
def fbutler_volmer_left(ac, z, eta, T):
def fbutler_volmer_anode(ac, z, eta, T):
return np.exp((1-ac)*z*Faraday*eta/(gas_constant*T))
@np.vectorize
def fbutler_volmer_right(ac, z, eta, T):
def fbutler_volmer_cathode(ac, z, eta, T):
return -np.exp(-ac*z*Faraday*eta/(gas_constant*T))
def fbutler_volmer(ac, z, eta, T):
return fbutler_volmer_left(ac, z, eta, T) + fbutler_volmer_right(ac, z, eta, T)
return fbutler_volmer_anode(ac, z, eta, T) + fbutler_volmer_cathode(ac, z, eta, T)
def butler_volmer():
fig, ax = plt.subplots(figsize=size_half_third)
ax.set_xlabel("$\\eta$")
ax.set_ylabel("$i/i_0$")
ax.set_xlabel("$\\eta$ [V]")
ax.set_ylabel("$j/j_0$")
etas = np.linspace(-0.1, 0.1, 400)
T = 300
z = 1.0
# other a
alpha2, alpha3 = 0.2, 0.8
ac2, ac3 = 0.2, 0.8
i2 = fbutler_volmer(0.2, z, etas, T)
i3 = fbutler_volmer(0.8, z, etas, T)
ax.plot(etas, i2, color="blue", linestyle="dashed", label=f"$\\alpha={alpha2}$")
ax.plot(etas, i3, color="green", linestyle="dashed", label=f"$\\alpha={alpha3}$")
ax.plot(etas, i2, color="blue", linestyle="dashed", label=f"$\\alpha_\\text{{C}}={ac2}$")
ax.plot(etas, i3, color="green", linestyle="dashed", label=f"$\\alpha_\\text{{C}}={ac3}$")
# 0.5
ac = 0.5
irel_left = fbutler_volmer_left(ac, z, etas, T)
irel_right = fbutler_volmer_right(ac, z, etas, T)
ax.plot(etas, irel_left, color="gray")
ax.plot(etas, irel_right, color="gray")
ax.plot(etas, irel_right + irel_left, color="black", label=f"$\\alpha=0.5$")
irel_anode = fbutler_volmer_anode(ac, z, etas, T)
irel_cathode = fbutler_volmer_cathode(ac, z, etas, T)
ax.plot(etas, irel_anode, color="gray")
ax.plot(etas, irel_cathode, color="gray")
ax.plot(etas, irel_cathode + irel_anode, color="black", label=f"$\\alpha_\\text{{C}}=0.5$")
ax.grid()
ax.legend()
ylim = 6
ax.set_ylim(-ylim, ylim)
return fig
@np.vectorize
def ftafel_anode(ac, z, eta, T):
return 10**((1-ac)*z*Faraday*eta/(gas_constant*T*np.log(10)))
@np.vectorize
def ftafel_cathode(ac, z, eta, T):
return -10**(-ac*z*Faraday*eta/(gas_constant*T*np.log(10)))
def tafel():
i0 = 1
ac = 0.2
z = 1
T = 300
eta_max = 0.2
etas = np.linspace(-eta_max, eta_max, 400)
i = np.abs(fbutler_volmer(ac, z, etas ,T))
iright = i0 * np.abs(ftafel_cathode(ac, z, etas, T))
ileft = i0 * ftafel_anode(ac, z, etas, T)
fig, ax = plt.subplots(figsize=size_half_third)
ax.set_xlabel("$\\eta$ [V]")
ax.set_ylabel("$\\log_{10}\\left(\\frac{|j|}{j_0}\\right)$")
# ax.set_ylabel("$\\log_{10}\\left(|j|/j_0\\right)$")
ax.set_yscale("log")
# ax.plot(etas, linear, label="Tafel slope")
ax.plot(etas[etas >= 0], ileft[etas >= 0], linestyle="dashed", color="gray", label="Tafel Approximation")
ax.plot(etas[etas <= 0], iright[etas <= 0], linestyle="dashed", color="gray")
ax.plot(etas, i, label=f"Butler-Volmer $\\alpha_\\text{{C}}={ac:.1f}$")
ax.legend()
ax.grid()
return fig
if __name__ == '__main__':
export(butler_volmer(), "ch_butler_volmer")
export(tafel(), "ch_tafel")

2
scripts/cm_phonons.py
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@ -1,5 +1,5 @@
#!/usr/bin env python3
from formulasheet import *
from formulary import *
def fone_atom_basis(q, a, M, C1, C2):
return np.sqrt(4*C1/M * (np.sin(q*a/2)**2 + C2/C1 * np.sin(q*a)**2))

2
scripts/distributions.py
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@ -1,5 +1,5 @@
from numpy import fmax
from formulasheet import *
from formulary import *
import itertools

71
scripts/formulasheet.py
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@ -1,71 +0,0 @@
#!/usr/bin env python3
import os
import matplotlib.pyplot as plt
import numpy as np
import math
import scipy as scp
if __name__ == "__main__": # make relative imports work as described here: https://peps.python.org/pep-0366/#proposed-change
if __package__ is None:
__package__ = "formulasheet"
import sys
filepath = os.path.realpath(os.path.abspath(__file__))
sys.path.insert(0, os.path.dirname(os.path.dirname(filepath)))
from util.mpl_colorscheme import set_mpl_colorscheme
import util.colorschemes as cs
# SET THE COLORSCHEME
# hard white and black
# cs.p_gruvbox["fg0"] = "#000000"
# cs.p_gruvbox["bg0"] = "#ffffff"
COLORSCHEME = cs.gruvbox_dark()
# print(COLORSCHEME)
# COLORSCHEME = cs.GRUVBOX_DARK
tex_src_path = "../src/"
img_out_dir = os.path.join(tex_src_path, "img")
filetype = ".pdf"
skipasserts = False
full = 8
size_half_half = (full/2, full/2)
size_third_half = (full/3, full/2)
size_half_third = (full/2, full/3)
def assert_directory():
if not skipasserts:
assert os.path.abspath(".").endswith("scripts"), "Please run from the `scripts` directory"
def texvar(var, val, math=True):
s = "$" if math else ""
s += f"\\{var} = {val}"
if math: s += "$"
return s
def export(fig, name, notightlayout=False):
assert_directory()
filename = os.path.join(img_out_dir, name + filetype)
if not notightlayout:
fig.tight_layout()
fig.savefig(filename) #, bbox_inches="tight")
@np.vectorize
def smooth_step(x: float, left_edge: float, right_edge: float):
x = (x - left_edge) / (right_edge - left_edge)
if x <= 0: return 0.
elif x >= 1: return 1.
else: return 3*(x*2) - 2*(x**3)
# run even when imported
set_mpl_colorscheme(COLORSCHEME)
if __name__ == "__main__":
assert_directory()
s = \
"""% This file was generated by scripts/formulasheet.py\n% Do not edit it directly, changes will be overwritten\n""" + cs.generate_latex_colorscheme(COLORSCHEME)
filename = os.path.join(tex_src_path, "util/colorscheme.tex")
print(f"Writing tex colorscheme to {filename}")
with open(filename, "w") as file:
file.write(s)

2
scripts/qubits.py
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@ -1,4 +1,4 @@
from formulasheet import *
from formulary import *
import scqubits as scq
import qutip as qt

9
scripts/readme.md
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@ -1,9 +1,16 @@
# Scripts
Put all scripts that generate plots or tex files here.
You can run all files at once using `make scripts`
## Plots
For plots with `matplotlib`:
1. import `plot.py`
1. import `formulary.py`
2. use one of the preset figsizes
3. save the image using the `export` function in the `if __name__ == '__main__'` part
## Colorscheme
To ensure a uniform look of the tex source and the python plots,
the tex and matplotlib colorschemes are both handled in `formulary.py`.
Set the `COLORSCHEME` variable to the desired colors.
Importing `formulary.py` will automatically apply the colors to matplotlib,
and running it will generate `util/colorscheme.tex` for LaTeX.

2
scripts/stat-mech.py
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@ -1,5 +1,5 @@
#!/usr/bin env python3
from formulasheet import *
from formulary import *
def flennard_jones(r, epsilon, sigma):
return 4 * epsilon * ((sigma/r)**12 - (sigma/r)**6)

13
scripts/util/colorschemes.py
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@ -65,7 +65,7 @@ p_gruvbox = {
"alt-gray": "#7c6f64",
}
def grubox_light():
def gruvbox_light():
GRUVBOX_LIGHT = { "fg0": p_gruvbox["fg0-hard"], "bg0": p_gruvbox["bg0-hard"] } \
| {f"fg{n}": p_gruvbox[f"fg{n}"] for n in range(1,5)} \
| {f"bg{n}": p_gruvbox[f"bg{n}"] for n in range(1,5)} \
@ -177,14 +177,3 @@ def stupid():
| { f"fg-{n}": brightness(c, 2.0) for n,c in p_stupid.items() }
return LEGACY
# UTILITY
def color_latex_def(name, color):
# name = name.replace("-", "_")
color = color.strip("#")
return "\\definecolor{" + name + "}{HTML}{" + color + "}"
def generate_latex_colorscheme(palette, variant="light"):
s = ""
for n, c in palette.items():
s += color_latex_def(n, c) + "\n"
return s

302
src/ch/ch.tex
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@ -7,305 +7,3 @@
\ger{Periodensystem}
]{ptable}
\drawPeriodicTable
\Section[
\eng{Electrochemistry}
\ger{Elektrochemie}
]{el}
\eng[std_cell]{standard cell potential}
\ger[std_cell]{Standardzellpotential}
\eng[electrode_pot]{electrode potential}
\ger[electrode_pot]{Elektrodenpotential}
\begin{formula}{chemical_potential}
\desc{Chemical potential}{of species $i$\\Energy involved when the particle number changes}{\QtyRef{gibbs_free_energy}, \QtyRef{amount}}
\desc[german]{Chemisches Potential}{der Spezies $i$\\Involvierte Energie, wenn sich die Teilchenzahl ändert}{}
\quantity{\mu}{\joule\per\mol;\joule}{is}
\eq{
\mu_i \equiv \pdv{G}{n_i}_{n_j\neq n_i,p,T}
}
\end{formula}
\begin{formula}{standard_chemical_potential}
\desc{Standard chemical potential}{In equilibrium}{\QtyRef{chemical_potential}, \ConstRef{universal_gas}, \QtyRef{temperature}, \QtyRef{activity}}
\desc[german]{Standard chemisches Potential}{}{}
\eq{\mu_i = \mu_i^\theta + RT \Ln{a_i}}
\end{formula}
\begin{formula}{chemical_equilibrium}
\desc{Chemical equilibrium}{}{\QtyRef{chemical_potential}, \QtyRef{stoichiometric_coefficient}}
\desc[german]{Chemisches Gleichgewicht}{}{}
\eq{\sum_\text{\GT{products}} \nu_i \mu_i = \sum_\text{\GT{educts}} \nu_i \mu_i}
\end{formula}
\begin{formula}{activity}
\desc{Activity}{relative activity}{\QtyRef{chemical_potential}, \QtyRef{standard_chemical_potential}, \ConstRef{universal_gas}, \QtyRef{temperature}}
\desc[german]{Aktivität}{Relative Aktivität}{}
\quantity{a}{}{s}
\eq{a_i = \Exp{\frac{\mu_i-\mu_i^\theta}{RT}}}
\end{formula}
\begin{formula}{electrochemical_potential}
\desc{Electrochemical potential}{Chemical potential with electrostatic contributions}{\QtyRef{chemical_potential}, $z$ valency (charge), \ConstRef{faraday}, \QtyRef{electric_scalar_potential} (Galvani Potential)}
\desc[german]{Elektrochemisches Potential}{Chemisches Potential mit elektrostatischen Enegiebeiträgen}{\QtyRef{chemical_potential}, $z$ Ladungszahl, \ConstRef{faraday}, \QtyRef{electric_scalar_potential} (Galvanisches Potential)}
\quantity{\muecp}{\joule\per\mol;\joule}{is}
\eq{\muecp_i \equiv \mu_i + z_i F \phi}
\end{formula}
\begin{formula}{nernst_equation}
\desc{Nernst equation}{Elektrode potential for a half-cell reaction}{$E$ electrode potential, $E^\theta$ \gt{std_cell}, \ConstRef{universal_gas}, \ConstRef{temperature}, $z$ \qtyRef{charge_number}, \ConstRef{faraday}, \QtyRef{activity}, \QtyRef{stoichiometric_coefficient}}
\desc[german]{Nernst-Gleichung}{Elektrodenpotential für eine Halbzellenreaktion}{}
\eq{E = E^\theta + \frac{RT}{zF} \Ln{\frac{ \left(\prod_{i}(a_i)^{\abs{\nu_i}}\right)_\text{oxidized}}{\left(\prod_{i}(a_i)^{\abs{\nu_i}}\right)_\text{reduced}}}}
\end{formula}
\begin{formula}{cell}
\desc{Electrochemical cell}{}{}
\desc[german]{Elektrochemische Zelle}{}{}
\ttxt{
\eng{
\begin{itemize}
\item Electrolytic cell: Uses electrical energy to force a chemical reaction
\item Galvanic cell: Produces electrical energy through a chemical reaction
\end{itemize}
}
\ger{
\begin{itemize}
\item Elektrolytische Zelle: Nutzt elektrische Energie um eine Reaktion zu erzwingen
\item Galvanische Zelle: Produziert elektrische Energie durch eine chemische Reaktion
\end{itemize}
}
}
\end{formula}
\begin{formula}{standard_cell_potential}
\desc{Standard cell potential}{}{$\Delta_\txR G^\theta$ standard \qtyRef{gibbs_free_energy} of reaction, $n$ number of electrons, \ConstRef{faraday}}
\desc[german]{Standard Zellpotential}{}{$\Delta_\txR G^\theta$ Standard \qtyRef{gibbs_free_energy} der Reaktion, $n$ Anzahl der Elektronen, \ConstRef{faraday}}
\eq{E^\theta_\text{rev} = \frac{-\Delta_\txR G^\theta}{nF}}
\end{formula}
\begin{formula}{she}
\desc{Standard hydrogen electrode (SHE)}{}{}
\desc[german]{Standard Wasserstoffelektrode}{}{}
\ttxt{
\eng{Defined as reference for measuring half-cell potententials}
\ger{Definiert als Referenz für Messungen von Potentialen von Halbzellen}
}
$a_{\ce{H+}} =1 \, (\text{pH} = 0)$, $p_{\ce{H2}} = \SI{100}{\kilo\pascal}$
\end{formula}
\eng[galvanic]{galvanic}
\ger[galvanic]{galvanisch}
\eng[electrolytic]{electrolytic}
\ger[electrolytic]{electrolytisch}
\begin{formula}{cell_efficiency}
\desc{Thermodynamic cell efficiency}{}{$P$ \fqEqRef{ed:el:power}}
\desc[german]{Thermodynamische Zelleffizienz}{}{}
\eq{
\eta_\text{cell} &= \frac{P_\text{obtained}}{P_\text{maximum}} = \frac{E_\text{cell}}{E_\text{cell,rev}} & & \text{\gt{galvanic}} \\
\eta_\text{cell} &= \frac{P_\text{minimum}}{P_\text{applied}} = \frac{E_\text{cell,rev}}{E_\text{cell}} & & \text{\gt{electrolytic}}
}
\end{formula}
\Subsection[
\eng{Ionic conduction in electrolytes}
\ger{Ionische Leitung in Elektrolyten}
]{ion_cond}
\eng[z]{charge number}
\ger[z]{Ladungszahl}
\eng[of_i]{of ion $i$}
\ger[of_i]{des Ions $i$}
\begin{formula}{diffusion}
\desc{Diffusion}{caused by concentration gradients}{$z_i$ \qtyRef{charge_number} \gt{of_i}, \ConstRef{faraday}, \QtyRef{diffusion_constant} \gt{of_i}, \QtyRef{concentration} \gt{of_i}}
\desc[german]{Diffusion}{durch Konzentrationsgradienten}{}
\eq{ i_\text{diff} = \sum_i -z_i F D_i \left(\odv{c_i}{x}\right) }
\end{formula}
\begin{formula}{migration}
\desc{Migration}{caused by potential gradients}{$z_i$ \qtyRef{charge_number} \gt{of_i}, \ConstRef{faraday}, \QtyRef{concentration} \gt{of_i}, \QtyRef{mobility} \gt{of_i}, $\nabla\phi_\txs$ potential gradient in the solution}
\desc[german]{Migration}{durch Potentialgradienten}{$z_i$ \qtyRef{charge_number} \gt{of_i}, \ConstRef{faraday}, \QtyRef{concentration} \gt{of_i}, \QtyRef{mobility} \gt{of_i}, $\nabla\phi_\txs$ Potentialgradient in der Lösung}
\eq{ i_\text{mig} = \sum_i -z_i^2 F^2 \, c_i \, \mu_i \, \nabla\Phi_\txs }
\end{formula}
\begin{formula}{convection}
\desc{Convection}{caused by pressure gradients}{$z_i$ \qtyRef{charge_number} \gt{of_i}, \ConstRef{faraday}, \QtyRef{concentration} \gt{of_i}, $v_i^\text{flow}$ \qtyRef{velocity} \gt{of_i} in flowing electrolyte}
\desc[german]{Convection}{durch Druckgradienten}{$z_i$ \qtyRef{charge_number} \gt{of_i}, \ConstRef{faraday}, \QtyRef{concentration} \gt{of_i}, $v_i^\text{flow}$ \qtyRef{velocity} \gt{of_i} im fliessenden Elektrolyt}
\eq{ i_\text{conv} = \sum_i -z_i F \, c_i \, v_i^\text{flow} }
\end{formula}
\begin{formula}{ionic_conductivity}
\desc{Ionic conductivity}{}{\ConstRef{faraday}, $z_i$, $c_i$, $\mu_i$ charge number, \qtyRef{concentration} and \qtyRef{mobility} of the positive (+) and negative (-) ions}
\desc[german]{Ionische Leitfähigkeit}{}{\ConstRef{faraday}, $z_i$, $c_i$, $\mu_i$ Ladungszahl, \qtyRef{concentration} und \qtyRef{mobility} der positiv (+) und negativ geladenen Ionen}
\quantity{\kappa}{\per\ohm\cm=\siemens\per\cm}{}
\eq{\kappa = F^2 \left(z_+^2 \, c_+ \, \mu_+ + z_-^2 \, c_- \, \mu_-\right)}
\end{formula}
\begin{formula}{ionic_resistance}
\desc{Ohmic resistance of ionic current flow}{}{$L$ \qtyRef{length}, $A$ \qtyRef{area}, \QtyRef{ionic_conductivity}}
\desc[german]{Ohmscher Widerstand für Ionen-Strom}{}{}
\eq{R_\Omega = \frac{L}{A\,\kappa}}
\end{formula}
\begin{formula}{ionic_mobility}
\desc{Ionic mobility}{}{$v_\pm$ steady state drift \qtyRef{velocity}, $\phi$ \qtyRef{electric_scalar_potential}, $z$ \qtyRef{charge_number}, \ConstRef{faraday}, \ConstRef{charge}, \QtyRef{viscosity}, $r_\pm$ ion radius}
\desc[german]{Ionische Moblilität}{}{}
\quantity{u_\pm}{\cm^2\mol\per\joule\s}{}
% \eq{u_\pm = - \frac{v_\pm}{\nabla \phi \,z_\pm F} = \frac{e}{6\pi F \eta_\text{dyn} \r_\pm}}
\end{formula}
\begin{formula}{transference}
\desc{Transference number}{Ion transport number \\Fraction of the current carried by positive / negative ions}{$i_{+/-}$ current through positive/negative charges}
\desc[german]{Überführungszahl}{Anteil der positiv / negativ geladenen Ionen am Gesamtstrom}{$i_{+/-}$ Strom durch positive / negative Ladungn}
\eq{t_{+/-} = \frac{i_{+/-}}{i_+ + i_-}}
\end{formula}
\eng[csalt]{electrolyte \qtyRef{concentration}}
\eng[csalt]{\qtyRef{concentration} des Elektrolyts}
\begin{formula}{molar_conductivity}
\desc{Molar conductivity}{}{\QtyRef{ionic_conductivity}, $c_\text{salt}$ \gt{csalt}}
\desc[german]{Molare Leitfähigkeit}{}{\QtyRef{ionic_conductivity}, $c_\text{salt}$ \gt{salt}}
\quantity{\Lambda_\txM}{\siemens\cm^2\per\mol=\ampere\cm^2\per\volt\mol}{ievs}
\eq{\Lambda_\txM = \frac{\kappa}{c_\text{salt}}}
\end{formula}
\begin{formula}{kohlrausch_law}
\desc{Kohlrausch's law}{}{$\Lambda_\txM^0$ \qtyRef{molar_conductivity} at infinite dilution, $c_\text{salt}$ \gt{csalt}, $K$ \GT{constant}}
\desc[german]{}{}{$\Lambda_\txM^0$ \qtyRef{molar_conductivity} bei unendlicher Verdünnung, $\text{salt}$ \gt{csalt} $K$ \GT{constant}}
\eq{\Lambda_\txM = \Lambda_\txM^0 - K \sqrt{c_\text{salt}}}
\end{formula}
% Electrolyte conductivity
\begin{formula}{molality}
\desc{Molality}{}{\QtyRef{amount} of the solute, \QtyRef{mass} of the solvent}
\desc[german]{Molalität}{}{\QtyRef{amount} des gelösten Stoffs, \QtyRef{mass} des Lösungsmittels}
\quantity{b}{\mol\per\kg}{}
\eq{b = \frac{n}{m}}
\end{formula}
\begin{formula}{molarity}
\desc{Molarity}{\GT{see} \qtyRef{concentration}}{\QtyRef{amount} of the solute, \QtyRef{volume} of the solvent}
\desc[german]{Molarität}{}{\QtyRef{amount} des gelösten Stoffs, \QtyRef{volume} des Lösungsmittels}
\quantity{c}{\mol\per\litre}{}
\eq{c = \frac{n}{V}}
\end{formula}
\begin{formula}{ionic_strength}
\desc{Ionic strength}{Measure of the electric field in a solution through solved ions}{\QtyRef{molality}, \QtyRef{molarity}, $z$ \qtyRef{charge_number}}
\desc[german]{Ionenstärke}{Maß eienr Lösung für die elektrische Feldstärke durch gelöste Ionen}{}
\quantity{I}{\mol\per\kg;\mol\per\litre}{}
\eq{I_b &= \frac{1}{2} \sum_i b_i z_i^2 \\ I_c &= \frac{1}{2} \sum_i c_i z_i^2}
\end{formula}
\begin{formula}{debye_screening_length}
\desc{Debye screening length}{}{\ConstRef{avogadro}, \ConstRef{charge}, \QtyRef{ionic_strength}, \QtyRef{permittivity}, \ConstRef{boltzmann}, \QtyRef{temperature}}
\desc[german]{Debye-Länge / Abschirmlänge}{}{}
\eq{\lambda_\txD = \sqrt{\frac{\epsilon \kB T}{2\NA e^2 I_C}}}
\end{formula}
\begin{formula}{mean_ionic_activity}
\desc{Mean ionic activity coefficient}{Accounts for decreased reactivity because ions must divest themselves of their ion cloud before reacting}{}
\desc[german]{Mittlerer ionischer Aktivitätskoeffizient}{Berücksichtigt dass Ionen sich erst von ihrer Ionenwolke lösen müssen, bevor sie reagieren können}{}
\quantity{\gamma}{}{s}
\eq{\gamma_\pm = \left(\gamma_+^{\nu_+} \, \gamma_-^{\nu_-}\right)^{\frac{1}{\nu_+ + \nu_-}}}
\end{formula}
\begin{formula}{debye_hueckel_law}
\desc{Debye-Hückel limiting law}{For an infinitely dilute solution}{\QtyRef{mean_ionic_activity}, $A$ solvent dependant constant, $z$ \qtyRef{charge_number}, \QtyRef{ionic_strength} in [\si{\mol\per\kg}]}
\desc[german]{Debye-Hückel Gesetz}{Für eine unendlich verdünnte Lösung}{}
\eq{\Ln{\gamma_{\pm}} = -A \abs{z_+ \, z_-} \sqrt{I_b}}
\end{formula}
\Subsection[
\eng{Kinetics}
\ger{Kinetik}
]{kin}
\begin{formula}{overpotential}
\desc{Overpotential}{}{$E_\text{electrode}$ potential at which the reaction starts $E_\text{ref}$ thermodynamic potential of the reaction}
\desc[german]{Überspannung}{}{$E_\text{electrode}$ Potential bei der die Reaktion beginnt, $E_\text{ref}$ thermodynamisches Potential der Reaktion}
\eq{\eta_\text{act} = E_\text{electrode} - E_\text{ref}}
\end{formula}
\begin{formula}{activation_overpotential}
\desc{Activation overpotential}{}{}
\desc[german]{Aktivierungsüberspannung}{}{}
\eq{}
\end{formula}
\begin{formula}{concentration_overpotential}
\desc{Concentration overpotential}{}{}
\desc[german]{Konzentrationsüberspannung}{}{}
\eq{\eta_\text{conc} = -\frac{RT}{(1-\alpha) nF} \ln \left(\frac{c_\text{ox}^0}{c_\text{ox}^\txS}\right)}
\end{formula}
\begin{formula}{diffusion_overpotential}
\desc{Diffusionoverpotential}{}{}
\desc[german]{Diffusionsüberspannung}{}{}
\eq{}
\end{formula}
\begin{formula}{roughness_factor}
\desc{Roughness factor}{Surface area related to electrode geometry}{}
\eq{\rfactor}
\end{formula}
\begin{formula}{butler_volmer}
\desc{Butler-Volmer equation}{Reaction kinetics near the equilibrium potentential}
{$j$ \qtyRef{current_density}, $j_0$ exchange current density, $\eta$ \fqEqRef{ch:el:kin:overpotential}, \QtyRef{temperature}, $z$ \qtyRef{charge_number}, \ConstRef{faraday}, \ConstRef{universal_gas}, $\alpha_{\txc/\txa}$ cathodic/anodic charge transfer coefficient}
%Current through an electrode iof a unimolecular redox reaction with both anodic and cathodic reaction occuring on the same electrode
\desc[german]{Butler-Volmer-Gleichung}{Reaktionskinetik in der Nähe des Gleichgewichtspotentials}
{$j$ \qtyRef{current_density}, $j_0$ Austauschstromdichte, $\eta$ \fqEqRef{ch:el:kin:overpotential}, \QtyRef{temperature}, $z$ \qtyRef{charge_number}, \ConstRef{faraday}, \ConstRef{universal_gas}, $\alpha_{\txc/\txa}$ Ladungstransferkoeffizient an der Kathode/Anode}
\begin{gather}
j = j_0 \,\rfactor\, \left[ \Exp{\frac{(1-a_\txc) z F \eta}{RT}} - \Exp{-\frac{\alpha_\txc z F \eta}{RT}}\right]
\intertext{\GT{with}}
\alpha_\txa = 1 - \alpha_\txc
\end{gather}
\separateEntries
\fig{img/ch_butler_volmer.pdf}
\end{formula}
\Section[
\eng{misc}
\ger{misc}
]{misc}
\begin{formula}{std_condition}
\desc{Standard temperature and pressure}{}{}
\desc[german]{Standardbedingungen}{}{}
\eq{
T &= \SI{273.15}{\kelvin} = \SI{0}{\celsius} \\
p &= \SI{100000}{\pascal} = \SI{1.000}{\bar}
}
\end{formula}
\begin{formula}{ph}
\desc{pH definition}{}{$a_{\ce{H+}}$ hyrdrogen ion \qtyRef{activity}}
\desc[german]{pH-Wert definition}{}{$a_{\ce{H+}}$ Wasserstoffionen-\qtyRef{activity}}
\eq{\pH = -\log_{10}(a_{\ce{H+}})}
\end{formula}
\begin{formula}{ph_rt}
\desc{pH}{At room temperature \SI{25}{\celsius}}{}
\desc[german]{pH-Wert}{Bei Raumtemperatur \SI{25}{\celsius}}{}
\eq{
\pH > 7 &\quad\tGT{basic} \\
\pH < 7 &\quad\tGT{acidic} \\
\pH = 7 &\quad\tGT{neutral}
}
\end{formula}
\begin{formula}{covalent_bond}
\desc{Covalent bond}{}{}
\desc[german]{Kolvalente Bindung}{}{}
\ttxt{
\eng{Bonds that involve sharing of electrons to form electron pairs between atoms.}
\ger{Bindungen zwischen Atomen die durch geteilte Elektronen, welche Elektronenpaare bilden, gebildet werden.}
}
\end{formula}
\begin{formula}{grotthuss}
\desc{Grotthuß-mechanism}{}{}
\desc[german]{Grotthuß-Mechanismus}{}{}
\ttxt{
\eng{The mobility of protons in aqueous solutions is much higher than that of other ions because they can "move" by breaking and reforming covalent bonds of water molecules.}
\ger{The Moblilität von Protononen in wässrigen Lösungen ist wesentlich größer als die anderer Ionen, da sie sich "bewegen" können indem die Wassertsoffbrückenbindungen gelöst und neu gebildet werden.}
}
\end{formula}

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\Section[
\eng{Electrochemistry}
\ger{Elektrochemie}
]{el}
\begin{formula}{chemical_potential}
\desc{Chemical potential}{of species $i$\\Energy involved when the particle number changes}{\QtyRef{gibbs_free_energy}, \QtyRef{amount}}
\desc[german]{Chemisches Potential}{der Spezies $i$\\Involvierte Energie, wenn sich die Teilchenzahl ändert}{}
\quantity{\mu}{\joule\per\mol;\joule}{is}
\eq{
\mu_i \equiv \pdv{G}{n_i}_{n_j\neq n_i,p,T}
}
\end{formula}
\begin{formula}{standard_chemical_potential}
\desc{Standard chemical potential}{In equilibrium}{\QtyRef{chemical_potential}, \ConstRef{universal_gas}, \QtyRef{temperature}, \QtyRef{activity}}
\desc[german]{Standard chemisches Potential}{}{}
\eq{\mu_i = \mu_i^\theta + RT \Ln{a_i}}
\end{formula}
\begin{formula}{chemical_equilibrium}
\desc{Chemical equilibrium}{}{\QtyRef{chemical_potential}, \QtyRef{stoichiometric_coefficient}}
\desc[german]{Chemisches Gleichgewicht}{}{}
\eq{\sum_\text{\GT{products}} \nu_i \mu_i = \sum_\text{\GT{educts}} \nu_i \mu_i}
\end{formula}
\begin{formula}{activity}
\desc{Activity}{relative activity}{\QtyRef{chemical_potential}, \QtyRef{standard_chemical_potential}, \ConstRef{universal_gas}, \QtyRef{temperature}}
\desc[german]{Aktivität}{Relative Aktivität}{}
\quantity{a}{}{s}
\eq{a_i = \Exp{\frac{\mu_i-\mu_i^\theta}{RT}}}
\end{formula}
\begin{formula}{electrochemical_potential}
\desc{Electrochemical potential}{Chemical potential with electrostatic contributions}{\QtyRef{chemical_potential}, $z$ valency (charge), \ConstRef{faraday}, \QtyRef{electric_scalar_potential} (Galvani Potential)}
\desc[german]{Elektrochemisches Potential}{Chemisches Potential mit elektrostatischen Enegiebeiträgen}{\QtyRef{chemical_potential}, $z$ Ladungszahl, \ConstRef{faraday}, \QtyRef{electric_scalar_potential} (Galvanisches Potential)}
\quantity{\muecp}{\joule\per\mol;\joule}{is}
\eq{\muecp_i \equiv \mu_i + z_i F \phi}
\end{formula}
\Subsection[
\eng{Electrochemical cell}
\ger{Elektrochemische Zelle}
]{cell}
\eng[galvanic]{galvanic}
\ger[galvanic]{galvanisch}
\eng[electrolytic]{electrolytic}
\ger[electrolytic]{electrolytisch}
\begin{formula}{schematic}
\desc{Schematic}{}{}
\desc[german]{Aufbau}{}{}
\begin{tikzpicture}
\pgfmathsetmacro{\width}{3}
\pgfmathsetmacro{\height}{4}
\pgfmathsetmacro{\elWidth}{\width/9}
\draw[thick] (0,0) rectangle (\width,\height);
\fill[bg-blue] (-2,-2) rectangle (2,0.5);
% Electrodes
\draw[thick, red] (-1,2) -- (-1,-1.2); % Reference electrode
\draw[thick, green] (0,2) -- (0,-1); % Counter electrode
\draw[thick, gray] (1,2) -- (1,-1.5); % Working electrode
% Labels
\node[left] at (-1,0) {Reference electrode};
\node[left] at (0,-0.5) {Counter electrode};
\node[right] at (1,-1) {Working electrode};
\node[left] at (-2,-1.5) {Electrolyte};
% Potentiostat
\draw[thick] (-2.5,3) rectangle (2.5,4);
\node at (0,3.5) {Potentiostat};
% Wires
\draw[thick] (-1,2) -- (-1,3);
\draw[thick] (0,2) -- (0,3);
\draw[thick] (1,2) -- (1,3);
% Ammeter and Voltmeter
\draw[thick] (-1,2) to[ammeter] (-1,3);
\draw[thick] (0,2) -- (0,3);
\draw[thick] (1,2) to[voltmeter] (1,3);
% Connecting to potentiostat
\draw[thick] (-1,3.8) -- (-1,4);
\draw[thick] (1,3.8) -- (1,4);
\end{tikzpicture}
\end{formula}
\begin{formula}{cell}
\desc{Electrochemical cell types}{}{}
\desc[german]{Arten der Elektrochemische Zelle}{}{}
\ttxt{
\eng{
\begin{itemize}
\item Electrolytic cell: Uses electrical energy to force a chemical reaction
\item Galvanic cell: Produces electrical energy through a chemical reaction
\end{itemize}
}
\ger{
\begin{itemize}
\item Elektrolytische Zelle: Nutzt elektrische Energie um eine Reaktion zu erzwingen
\item Galvanische Zelle: Produziert elektrische Energie durch eine chemische Reaktion
\end{itemize}
}
}
\end{formula}
% todo group together
\begin{formula}{faradaic}
\desc{Faradaic process}{}{}
\desc[german]{Faradäischer Prozess}{}{}
\ttxt{
\eng{Charge transfers between the electrode bulk and the electrolyte.}
\ger{Ladung wird zwischen Elektrode und dem Elektrolyten transferiert.}
}
\end{formula}
\begin{formula}{non-faradaic}
\desc{Non-Faradaic (capacitive) process}{}{}
\desc[german]{Nicht-Faradäischer (kapazitiver) Prozess}{}{}
\ttxt{
\eng{Charge is stored at the electrode-electrolyte interface.}
\ger{Ladung lagert sich am Elektrode-Elektrolyt Interface an.}
}
\end{formula}
\begin{formula}{electrode_potential}
\desc{Electrode potential}{}{}
\desc[german]{Elektrodenpotential}{}{}
\quantity{E}{\volt}{s}
\end{formula}
\begin{formula}{standard_cell_potential}
\desc{Standard cell potential}{}{$\Delta_\txR G^\theta$ standard \qtyRef{gibbs_free_energy} of reaction, $n$ number of electrons, \ConstRef{faraday}}
\desc[german]{Standard Zellpotential}{}{$\Delta_\txR G^\theta$ Standard \qtyRef{gibbs_free_energy} der Reaktion, $n$ Anzahl der Elektronen, \ConstRef{faraday}}
\eq{E^\theta_\text{rev} = \frac{-\Delta_\txR G^\theta}{nF}}
\end{formula}
\begin{formula}{nernst_equation}
\desc{Nernst equation}{Electrode potential for a half-cell reaction}{\QtyRef{electrode_potential}, $E^\theta$ \secEqRef{standard_cell_potential}, \ConstRef{universal_gas}, \ConstRef{temperature}, $z$ \qtyRef{charge_number}, \ConstRef{faraday}, \QtyRef{activity}, \QtyRef{stoichiometric_coefficient}}
\desc[german]{Nernst-Gleichung}{Elektrodenpotential für eine Halbzellenreaktion}{}
\eq{E = E^\theta + \frac{RT}{zF} \Ln{\frac{ \left(\prod_{i}(a_i)^{\abs{\nu_i}}\right)_\text{oxidized}}{\left(\prod_{i}(a_i)^{\abs{\nu_i}}\right)_\text{reduced}}}}
\end{formula}
\begin{formula}{cell_efficiency}
\desc{Thermodynamic cell efficiency}{}{$P$ \fqEqRef{ed:el:power}}
\desc[german]{Thermodynamische Zelleffizienz}{}{}
\eq{
\eta_\text{cell} &= \frac{P_\text{obtained}}{P_\text{maximum}} = \frac{E_\text{cell}}{E_\text{cell,rev}} & & \text{\gt{galvanic}} \\
\eta_\text{cell} &= \frac{P_\text{minimum}}{P_\text{applied}} = \frac{E_\text{cell,rev}}{E_\text{cell}} & & \text{\gt{electrolytic}}
}
\end{formula}
\Subsection[
\eng{Ionic conduction in electrolytes}
\ger{Ionische Leitung in Elektrolyten}
]{ion_cond}
\eng[z]{charge number}
\ger[z]{Ladungszahl}
\eng[of_i]{of ion $i$}
\ger[of_i]{des Ions $i$}
\begin{formula}{diffusion}
\desc{Diffusion}{caused by concentration gradients}{$z_i$ \qtyRef{charge_number} \gt{of_i}, \ConstRef{faraday}, \QtyRef{diffusion_constant} \gt{of_i}, \QtyRef{concentration} \gt{of_i}}
\desc[german]{Diffusion}{durch Konzentrationsgradienten}{}
\eq{ i_\text{diff} = \sum_i -z_i F D_i \left(\odv{c_i}{x}\right) }
\end{formula}
\begin{formula}{migration}
\desc{Migration}{caused by potential gradients}{$z_i$ \qtyRef{charge_number} \gt{of_i}, \ConstRef{faraday}, \QtyRef{concentration} \gt{of_i}, \QtyRef{mobility} \gt{of_i}, $\nabla\phi_\txs$ potential gradient in the solution}
\desc[german]{Migration}{durch Potentialgradienten}{$z_i$ \qtyRef{charge_number} \gt{of_i}, \ConstRef{faraday}, \QtyRef{concentration} \gt{of_i}, \QtyRef{mobility} \gt{of_i}, $\nabla\phi_\txs$ Potentialgradient in der Lösung}
\eq{ i_\text{mig} = \sum_i -z_i^2 F^2 \, c_i \, \mu_i \, \nabla\Phi_\txs }
\end{formula}
\begin{formula}{convection}
\desc{Convection}{caused by pressure gradients}{$z_i$ \qtyRef{charge_number} \gt{of_i}, \ConstRef{faraday}, \QtyRef{concentration} \gt{of_i}, $v_i^\text{flow}$ \qtyRef{velocity} \gt{of_i} in flowing electrolyte}
\desc[german]{Convection}{durch Druckgradienten}{$z_i$ \qtyRef{charge_number} \gt{of_i}, \ConstRef{faraday}, \QtyRef{concentration} \gt{of_i}, $v_i^\text{flow}$ \qtyRef{velocity} \gt{of_i} im fliessenden Elektrolyt}
\eq{ i_\text{conv} = \sum_i -z_i F \, c_i \, v_i^\text{flow} }
\end{formula}
\begin{formula}{ionic_conductivity}
\desc{Ionic conductivity}{}{\ConstRef{faraday}, $z_i$, $c_i$, $\mu_i$ charge number, \qtyRef{concentration} and \qtyRef{mobility} of the positive (+) and negative (-) ions}
\desc[german]{Ionische Leitfähigkeit}{}{\ConstRef{faraday}, $z_i$, $c_i$, $\mu_i$ Ladungszahl, \qtyRef{concentration} und \qtyRef{mobility} der positiv (+) und negativ geladenen Ionen}
\quantity{\kappa}{\per\ohm\cm=\siemens\per\cm}{}
\eq{\kappa = F^2 \left(z_+^2 \, c_+ \, \mu_+ + z_-^2 \, c_- \, \mu_-\right)}
\end{formula}
\begin{formula}{ionic_resistance}
\desc{Ohmic resistance of ionic current flow}{}{$L$ \qtyRef{length}, $A$ \qtyRef{area}, \QtyRef{ionic_conductivity}}
\desc[german]{Ohmscher Widerstand für Ionen-Strom}{}{}
\eq{R_\Omega = \frac{L}{A\,\kappa}}
\end{formula}
\begin{formula}{ionic_mobility}
\desc{Ionic mobility}{}{$v_\pm$ steady state drift \qtyRef{velocity}, $\phi$ \qtyRef{electric_scalar_potential}, $z$ \qtyRef{charge_number}, \ConstRef{faraday}, \ConstRef{charge}, \QtyRef{viscosity}, $r_\pm$ ion radius}
\desc[german]{Ionische Moblilität}{}{}
\quantity{u_\pm}{\cm^2\mol\per\joule\s}{}
\eq{u_\pm = - \frac{v_\pm}{\nabla \phi \,z_\pm F} = \frac{e}{6\pi F \eta_\text{dyn} r_\pm}}
\end{formula}
\begin{formula}{stokes_friction}
\desc{Stokes's law}{Frictional force exerted on spherical objects moving in a viscous fluid at low Reynolds numbers}{$r$ particle radius, \QtyRef{viscosity}, $v$ particle \qtyRef{velocity}}
\desc[german]{Gesetz von Stokes}{Reibungskraft auf ein sphärisches Objekt in einer Flüssigkeit bei niedriger Reynolds-Zahl}{$r$ Teilchenradius, \QtyRef{viscosity}, $v$ Teilchengeschwindigkeit}
\eq{F_\txR = 6\pi\,r \eta v}
\end{formula}
\begin{formula}{transference}
\desc{Transference number}{Ion transport number \\Fraction of the current carried by positive / negative ions}{$i_{+/-}$ current through positive/negative charges}
\desc[german]{Überführungszahl}{Anteil der positiv / negativ geladenen Ionen am Gesamtstrom}{$i_{+/-}$ Strom durch positive / negative Ladungn}
\eq{t_{+/-} = \frac{i_{+/-}}{i_+ + i_-}}
\end{formula}
\eng[csalt]{electrolyte \qtyRef{concentration}}
\eng[csalt]{\qtyRef{concentration} des Elektrolyts}
\begin{formula}{molar_conductivity}
\desc{Molar conductivity}{}{\QtyRef{ionic_conductivity}, $c_\text{salt}$ \gt{csalt}}
\desc[german]{Molare Leitfähigkeit}{}{\QtyRef{ionic_conductivity}, $c_\text{salt}$ \gt{salt}}
\quantity{\Lambda_\txM}{\siemens\cm^2\per\mol=\ampere\cm^2\per\volt\mol}{ievs}
\eq{\Lambda_\txM = \frac{\kappa}{c_\text{salt}}}
\end{formula}
\begin{formula}{kohlrausch_law}
\desc{Kohlrausch's law}{}{$\Lambda_\txM^0$ \qtyRef{molar_conductivity} at infinite dilution, $c_\text{salt}$ \gt{csalt}, $K$ \GT{constant}}
\desc[german]{}{}{$\Lambda_\txM^0$ \qtyRef{molar_conductivity} bei unendlicher Verdünnung, $\text{salt}$ \gt{csalt} $K$ \GT{constant}}
\eq{\Lambda_\txM = \Lambda_\txM^0 - K \sqrt{c_\text{salt}}}
\end{formula}
% Electrolyte conductivity
\begin{formula}{molality}
\desc{Molality}{}{\QtyRef{amount} of the solute, \QtyRef{mass} of the solvent}
\desc[german]{Molalität}{}{\QtyRef{amount} des gelösten Stoffs, \QtyRef{mass} des Lösungsmittels}
\quantity{b}{\mol\per\kg}{}
\eq{b = \frac{n}{m}}
\end{formula}
\begin{formula}{molarity}
\desc{Molarity}{\GT{see} \qtyRef{concentration}}{\QtyRef{amount} of the solute, \QtyRef{volume} of the solvent}
\desc[german]{Molarität}{}{\QtyRef{amount} des gelösten Stoffs, \QtyRef{volume} des Lösungsmittels}
\quantity{c}{\mol\per\litre}{}
\eq{c = \frac{n}{V}}
\end{formula}
\begin{formula}{ionic_strength}
\desc{Ionic strength}{Measure of the electric field in a solution through solved ions}{\QtyRef{molality}, \QtyRef{molarity}, $z$ \qtyRef{charge_number}}
\desc[german]{Ionenstärke}{Maß eienr Lösung für die elektrische Feldstärke durch gelöste Ionen}{}
\quantity{I}{\mol\per\kg;\mol\per\litre}{}
\eq{I_b &= \frac{1}{2} \sum_i b_i z_i^2 \\ I_c &= \frac{1}{2} \sum_i c_i z_i^2}
\end{formula}
\begin{formula}{debye_screening_length}
\desc{Debye screening length}{}{\ConstRef{avogadro}, \ConstRef{charge}, \QtyRef{ionic_strength}, \QtyRef{permittivity}, \ConstRef{boltzmann}, \QtyRef{temperature}}
\desc[german]{Debye-Länge / Abschirmlänge}{}{}
\eq{\lambda_\txD = \sqrt{\frac{\epsilon \kB T}{2\NA e^2 I_C}}}
\end{formula}
\begin{formula}{mean_ionic_activity}
\desc{Mean ionic activity coefficient}{Accounts for decreased reactivity because ions must divest themselves of their ion cloud before reacting}{}
\desc[german]{Mittlerer ionischer Aktivitätskoeffizient}{Berücksichtigt dass Ionen sich erst von ihrer Ionenwolke lösen müssen, bevor sie reagieren können}{}
\quantity{\gamma}{}{s}
\eq{\gamma_\pm = \left(\gamma_+^{\nu_+} \, \gamma_-^{\nu_-}\right)^{\frac{1}{\nu_+ + \nu_-}}}
\end{formula}
\begin{formula}{debye_hueckel_law}
\desc{Debye-Hückel limiting law}{For an infinitely dilute solution}{\QtyRef{mean_ionic_activity}, $A$ solvent dependant constant, $z$ \qtyRef{charge_number}, \QtyRef{ionic_strength} in [\si{\mol\per\kg}]}
\desc[german]{Debye-Hückel Gesetz}{Für eine unendlich verdünnte Lösung}{}
\eq{\Ln{\gamma_{\pm}} = -A \abs{z_+ \, z_-} \sqrt{I_b}}
\end{formula}
\Subsection[
\eng{Kinetics}
\ger{Kinetik}
]{kin}
\begin{formula}{transfer_coefficient}
\desc{Transfer coefficient}{}{}
\desc[german]{Durchtrittsfaktor}{Transferkoeffizient\\Anteil des Potentials der sich auf die freie Reaktionsenthalpie des anodischen Prozesses auswirkt}{}
\eq{
\alpha_\txA &= \alpha \\
\alpha_\txC &= 1-\alpha
}
\end{formula}
\begin{formula}{overpotential}
\desc{Overpotential}{}{}
\desc[german]{Überspannung}{}{}
\ttxt{
\eng{Potential deviation from the equilibrium cell potential}
\ger{Abweichung der Spannung von der Zellspannung im Gleichgewicht}
}
\end{formula}
\begin{formula}{activation_overpotential}
\desc{Activation verpotential}{}{$E_\text{electrode}$ potential at which the reaction starts $E_\text{ref}$ thermodynamic potential of the reaction}
\desc[german]{Aktivierungsüberspannung}{}{$E_\text{electrode}$ Potential bei der die Reaktion beginnt, $E_\text{ref}$ thermodynamisches Potential der Reaktion}
\eq{\eta_\text{act} = E_\text{electrode} - E_\text{ref}}
\end{formula}
\begin{formula}{concentration_overpotential}
\desc{Concentration overpotential}{Due to concentration gradient near the electrode, the ions need to \hyperref[f:ch:el:ion_cond:diffusion]{diffuse} to the electrode before reacting}{\ConstRef{universal_gas}, \QtyRef{temperature}, $\c_{0/\txS}$ ion concentration in the electrolyte / at the double layer, $z$ \qtyRef{charge_number}, \ConstRef{faraday}}
\desc[german]{Konzentrationsüberspannung}{Durch einen Konzentrationsgradienten an der Elektrode müssen Ionen erst zur Elektrode \hyperref[f:ch:el:ion_cond:diffusion]{diffundieren}, bevor sie reagieren können}{}
\eq{
\eta_\text{conc,anodic} &= -\frac{RT}{\alpha \,zF} \ln \left(\frac{c_\text{red}^0}{c_\text{red}^\txS}\right) \\
\eta_\text{conc,cathodic} &= -\frac{RT}{(1-\alpha) zF} \ln \left(\frac{c_\text{ox}^0}{c_\text{ox}^\txS}\right)
}
\end{formula}
\begin{formula}{diffusion_overpotential}
\desc{Diffusion overpotential}{}{}
\desc[german]{Diffusionsüberspannung}{}{}
\eq{\eta_\text{diff} = \frac{RT}{nF} \ln \left( \cfrac{\cfrac{c^\txs_\text{ox}}{c^0_\text{ox}}}{\cfrac{c^\txs_\text{red}}{c^0_\text{red}}} \right)}
\end{formula}
\begin{formula}{diffusion_layer}
\desc{Cell layers}{}{}
\desc[german]{Zellschichten}{}{}
\begin{tikzpicture}
\tikzset{
label/.style={color=fg1,anchor=center,rotate=90},
}
\pgfmathsetmacro{\tkW}{8} % Total width
\pgfmathsetmacro{\tkH}{5} % Total height
\pgfmathsetmacro{\edW}{1} % electrode width
\pgfmathsetmacro{\hhW}{1} % helmholtz width
\pgfmathsetmacro{\ndW}{2} % nernst diffusion with
\pgfmathsetmacro{\eyW}{\tkW-\edW-\hhW-\ndW} % electrolyte width
\pgfmathsetmacro{\edX}{0} % electrode width
\pgfmathsetmacro{\hhX}{\edW} % helmholtz width
\pgfmathsetmacro{\ndX}{\edW+\hhW} % nernst diffusion with
\pgfmathsetmacro{\eyX}{\tkW-\eyW} % electrolyte width
\draw[->] (0,0) -- (\tkW+0.2,0) node[anchor=north] {$x$};
\draw[->] (0,0) -- (0,\tkH+0.2) node[anchor=east] {$c$};
\path[fill=bg-orange] (\edX,0) rectangle (\edX+\edW,\tkH); \node[label] at (\edX+\edW/2,\tkH/2) {\GT{electrode}};
\path[fill=bg-green!90!bg0] (\hhX,0) rectangle (\hhX+\hhW,\tkH); \node[label] at (\hhX+\hhW/2,\tkH/2) {\GT{helmholtz_layer}};
\path[fill=bg-green!60!bg0] (\ndX,0) rectangle (\ndX+\ndW,\tkH); \node[label] at (\ndX+\ndW/2,\tkH/2) {\GT{nernst_layer}};
\path[fill=bg-green!20!bg0] (\eyX,0) rectangle (\eyX+\eyW,\tkH); \node[label] at (\eyX+\eyW/2,\tkH/2) {\GT{elektrolyte}};
\draw (\hhX,2) -- (\ndX,3) -- (\tkW,3);
\tkYTick{2}{$c^\txS$};
\tkYTick{3}{$c^0$};
\end{tikzpicture}
\end{formula}
\Eng[c_surface]{surface \qtyRef{concentration}}
\Eng[c_bulk]{bulk \qtyRef{concentration}}
\Ger[c_surface]{Oberflächen-\qtyRef{concentration}}
\Ger[c_bulk]{Bulk-\qtyRef{concentration}}
\begin{formula}{diffusion_layer_thickness}
\desc{Nerst Diffusion layer thickness}{}{$c^0$ \GT{c_bulk}, $c^\txs$ \GT{c_surface}}
\desc[german]{Dicke der Nernstschen Diffusionsschicht}{}{}
\eq{\delta_\txN = \frac{c^0 - c^\txs}{\odv{c}{x}_{x=0}}}
\end{formula}
\begin{formula}{roughness_factor}
\desc{Roughness factor}{Surface area related to electrode geometry}{}
\eq{\rfactor}
\end{formula}
\begin{formula}{butler_volmer}
\desc{Butler-Volmer equation}{Reaction kinetics near the equilibrium potentential}
{$j$ \qtyRef{current_density}, $j_0$ exchange current density, $\eta$ \fqEqRef{ch:el:kin:overpotential}, \QtyRef{temperature}, $z$ \qtyRef{charge_number}, \ConstRef{faraday}, \ConstRef{universal_gas}, $\alpha_{\txC/\txA}$ cathodic/anodic charge transfer coefficient}
%Current through an electrode iof a unimolecular redox reaction with both anodic and cathodic reaction occuring on the same electrode
\desc[german]{Butler-Volmer-Gleichung}{Reaktionskinetik in der Nähe des Gleichgewichtspotentials}
{$j$ \qtyRef{current_density}, $j_0$ Austauschstromdichte, $\eta$ \fqEqRef{ch:el:kin:overpotential}, \QtyRef{temperature}, $z$ \qtyRef{charge_number}, \ConstRef{faraday}, \ConstRef{universal_gas}, $\alpha_{\txC/\txA}$ Ladungstransferkoeffizient an der Kathode/Anode}
\begin{gather}
j = j_0 \,\rfactor\, \left[ \Exp{\frac{(1-a_\txC) z F \eta}{RT}} - \Exp{-\frac{\alpha_\txC z F \eta}{RT}}\right]
\intertext{\GT{with}}
\alpha_\txA = 1 - \alpha_\txC
\end{gather}
\separateEntries
\fig{img/ch_butler_volmer.pdf}
\end{formula}
% \Subsubsection[
% \eng{Tafel approximation}
% \ger{Tafel Näherung}
% ]{tafel}
% \begin{formula}{slope}
% \desc{Tafel slope}{}{}
% \desc[german]{Tafel Steigung}{}{}
% \eq{}
% \end{formula}
\begin{formula}{equation}
\desc{Tafel approximation}{For slow kinetics: $\abs{\eta} > \SI{0.1}{\volt}$}{}
\desc[german]{Tafel Näherung}{Für langsame Kinetik: $\abs{\eta} > \SI{0.1}{\volt}$}{}
\eq{
\Log{j} &\approx \Log{j_0} + \frac{\alpha_\txC zF \eta}{RT\ln(10)} && \eta \gg \SI{0.1}{\volt}\\
\Log{\abs{j}} &\approx \Log{j_0} - \frac{(1-\alpha_\txC) zF \eta}{RT\ln(10)} && \eta \ll -\SI{0.1}{\volt}
}
\fig{img/ch_tafel.pdf}
\end{formula}
\Subsection[
\eng{Techniques}
\ger{Techniken}
]{tech}
\Subsubsection[
\eng{Reference electrodes}
\ger{Referenzelektroden}
]{ref}
\begin{ttext}
\eng{Defined as reference for measuring half-cell potententials}
\ger{Definiert als Referenz für Messungen von Potentialen von Halbzellen}
\end{ttext}
\begin{formula}{she}
\desc{Standard hydrogen elektrode (SHE)}{}{$p=\SI{e5}{\pascal}$, $a_{\ce{H+}}=\SI{1}{\mol\per\litre}$ (\Rightarrow $\pH=0$)}
\desc[german]{Standardwasserstoffelektrode (SHE)}{}{}
\ttxt{
\eng{Potential of the reaction: \ce{2H^+ +2e^- <--> H2}}
\ger{Potential der Reaktion: \ce{2H^+ +2e^- <--> H2}}
}
\end{formula}
\begin{formula}{rhe}
\desc{Reversible hydrogen electrode (RHE)}{RHE Potential does not change with the pH value}{$E^0\equiv \SI{0}{\volt}$, \QtyRef{activity}, \QtyRef{pressure}, \GT{see} \fqEqRef{ch:el:cell:nernst_equation}}
\desc[german]{Reversible Wasserstoffelektrode (RHE)}{Potential ändert sich nicht mit dem pH-Wert}{}
\eq{
E_\text{RHE} &= E^0 + \frac{RT}{F} \Ln{\frac{a_{\ce{H^+}}}{p_{\ce{H2}}}} \\
&= \SI{0}{\volt} - \SI{0.059}{\volt}
}
\end{formula}
\Subsubsection[
\eng{Cyclic voltammetry}
\ger{Zyklische Voltammetrie}
]{cycl_v}
\begin{formula}{duck}
\desc{Cyclic voltammogram}{}{}
% \desc[german]{}{}{}
\begin{tikzpicture}
\pgfmathsetmacro{\Ax}{-2.3}
\pgfmathsetmacro{\Ay}{ 0.0}
\pgfmathsetmacro{\Bx}{ 0.0}
\pgfmathsetmacro{\By}{ 1.0}
\pgfmathsetmacro{\Cx}{ 0.4}
\pgfmathsetmacro{\Cy}{ 1.5}
\pgfmathsetmacro{\Dx}{ 2.0}
\pgfmathsetmacro{\Dy}{ 0.5}
\pgfmathsetmacro{\Ex}{ 0.0}
\pgfmathsetmacro{\Ey}{-1.5}
\pgfmathsetmacro{\Fx}{-0.4}
\pgfmathsetmacro{\Fy}{-2.0}
\pgfmathsetmacro{\Gx}{-2.3}
\pgfmathsetmacro{\Gy}{-0.3}
\begin{axis}[ymin=-3,ymax=3,xmax=3,xmin=-3,
% equal axis,
minor tick num=1,
xlabel={$U$}, xlabel style={at={(axis description cs:0.5,+0.02)}},
ylabel={$I$}, ylabel style={at={(axis description cs:0.1,0.5)}},
anchor=center, at={(0,0)},
axis equal image,clip=false,
]
% CV with beziers
\draw[thick, fg-blue] (axis cs:\Ax,\Ay) coordinate (A) node[left] {A}
..controls (axis cs:\Ax+1.8, \Ay+0.0) and (axis cs:\Bx-0.2, \By-0.4) .. (axis cs:\Bx,\By) coordinate (B) node[left] {B}
..controls (axis cs:\Bx+0.1, \By+0.2) and (axis cs:\Cx-0.3, \Cy+0.0) .. (axis cs:\Cx,\Cy) coordinate (C) node[above] {C}
..controls (axis cs:\Cx+0.5, \Cy+0.0) and (axis cs:\Dx-1.3, \Dy+0.1) .. (axis cs:\Dx,\Dy) coordinate (D) node[right] {D}
..controls (axis cs:\Dx-2.0, \Dy-0.1) and (axis cs:\Ex+0.3, \Ey+0.8) .. (axis cs:\Ex,\Ey) coordinate (E) node[right] {E}
..controls (axis cs:\Ex-0.1, \Ey-0.2) and (axis cs:\Fx+0.2, \Fy+0.0) .. (axis cs:\Fx,\Fy) coordinate (F) node[below] {F}
..controls (axis cs:\Fx-0.2, \Fy+0.0) and (axis cs:\Gx+1.5, \Gy-0.2) .. (axis cs:\Gx,\Gy) coordinate (G) node[left] {G};
\node[above] at (A) {\rightarrow};
\end{axis}
\end{tikzpicture}
\end{formula}
\begin{formula}{upd}
\desc{Underpotential deposition (UPD)}{}{}
\desc[german]{}{}{}
\ttxt{Reversible deposition of metal onto a foreign metal electrode at potentials positive of the Nernst potential \TODO{clarify}}
\end{formula}
\Subsubsection[
\eng{Rotating disk electrodes}
% \ger{}
]{rde}
\begin{formula}{viscosity}
\desc{Dynamic viscosity}{}{}
\desc[german]{Dynamisch Viskosität}{}{}
\quantity{\eta,\mu}{\pascal\s=\newton\s\per\m^2=\kg\per\m\s}{}
\end{formula}
\begin{formula}{kinematic_viscosity}
\desc{Kinematic viscosity}{\qtyRef{viscosity} related to density of a fluid}{\QtyRef{viscosity}, \QtyRef{density}}
\desc[german]{Kinematische Viskosität}{\qtyRef{viscosity} im Verhältnis zur Dichte der Flüssigkeit}{}
\quantity{\nu}{\cm^2\per\s}{}
\eq{\nu = \frac{\eta}{\rho}}
\end{formula}
\begin{formula}{diffusion_layer_thickness}
\desc{Diffusion layer thickness}{\TODO{Where does 1.61 come from}}{$D$ \qtyRef{diffusion_coefficient}, $\nu$ \qtyRef{kinematic_viscosity}, \QtyRef{angular_frequency}}
\desc[german]{Diffusionsshichtdicke}{}{}
\eq{\delta_\text{diff}= 1.61 D{^\frac{1}{3}} \nu^{\frac{1}{6}} \omega^{-\frac{1}{2}}}
\end{formula}
\begin{formula}{limiting_current}
\desc{Limiting current}{}{$n$ \QtyRef{charge_number}, \ConstRef{faraday}, $c^0$ \GT{c_bulk}, $D$ \qtyRef{diffusion_coefficient}, $\delta_\text{diff}$ \secEqRef{diffusion_layer_thickness}, $\nu$ \qtyRef{kinematic_viscosity}, \QtyRef{angular_frequency}}
% \desc[german]{Limitierender Strom}{}{}
\eq{j^\infty = nFD \frac{c^0}{\delta_\text{diff}} = \frac{1}{1.61} nFD^{\frac{2}{3}} v^{\frac{-1}{6}} c^0 \sqrt{\omega}}
\end{formula}

107
src/ch/misc.tex Normal file
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@ -0,0 +1,107 @@
\Section[
\eng{Thermoelectricity}
\ger{Thermoelektrizität}
]{thermo}
\begin{formula}{seebeck}
\desc{Seebeck coefficient}{}{$V$ voltage, \QtyRef{temperature}}
\desc[german]{Seebeck-Koeffizient}{}{}
\quantity{S}{\micro\volt\per\kelvin}{s}
\eq{S = -\frac{\Delta V}{\Delta T}}
\end{formula}
\begin{formula}{seebeck_effect}
\desc{Seebeck effect}{Elecromotive force across two points of a material with a temperature difference}{\QtyRef{conductivity}, $V$ local voltage, \QtyRef{seebeck}, \QtyRef{temperature}}
\desc[german]{Seebeck-Effekt}{}{}
\eq{\vec{j} = \sigma(-\Grad V - S \Grad T)}
\end{formula}
\begin{formula}{thermal_conductivity}
\desc{Thermal conductivity}{Conduction of heat, without mass transport}{\QtyRef{heat}, \QtyRef{length}, \QtyRef{area}, \QtyRef{temperature}}
\desc[german]{Wärmeleitfähigkeit}{Leitung von Wärme, ohne Stofftransport}{}
\quantity{\kappa,\lambda,k}{\watt\per\m\K=\kg\m\per\s^3\kelvin}{s}
\eq{\kappa = \frac{\dot{Q} l}{A\,\Delta T}}
\eq{\kappa_\text{tot} = \kappa_\text{lattice} + \kappa_\text{electric}}
\end{formula}
\begin{formula}{wiedemann-franz}
\desc{Wiedemann-Franz law}{}{Electric \QtyRef{thermal_conductivity}, $L$ in \si{\watt\ohm\per\kelvin} Lorentz number, \QtyRef{conductivity}}
\desc[german]{Wiedemann-Franz Gesetz}{}{Elektrische \QtyRef{thermal_conductivity}, $L$ in \si{\watt\ohm\per\kelvin} Lorentzzahl, \QtyRef{conductivity}}
\eq{\kappa = L\sigma T}
\end{formula}
\begin{formula}{zt}
\desc{Thermoelectric figure of merit}{Dimensionless quantity for comparing different materials}{\QtyRef{seebeck}, \QtyRef{conductivity}, }
\desc[german]{Thermoelektrische Gütezahl}{Dimensionsoser Wert zum Vergleichen von Materialien}{}
\eq{zT = \frac{S^2\sigma}{\lambda} T}
\end{formula}
\Section[
\eng{misc}
\ger{misc}
]{misc}
% TODO: hide
\begin{formula}{stoichiometric_coefficient}
\desc{Stoichiometric coefficient}{}{}
\desc[german]{Stöchiometrischer Koeffizient}{}{}
\quantity{\nu}{}{s}
\end{formula}
\begin{formula}{std_condition}
\desc{Standard temperature and pressure}{}{}
\desc[german]{Standardbedingungen}{}{}
\eq{
T &= \SI{273.15}{\kelvin} = \SI{0}{\celsius} \\
p &= \SI{100000}{\pascal} = \SI{1.000}{\bar}
}
\end{formula}
\begin{formula}{ph}
\desc{pH definition}{}{$a_{\ce{H+}}$ hyrdrogen ion \qtyRef{activity}}
\desc[german]{pH-Wert definition}{}{$a_{\ce{H+}}$ Wasserstoffionen-\qtyRef{activity}}
\eq{\pH = -\log_{10}(a_{\ce{H+}})}
\end{formula}
\begin{formula}{ph_rt}
\desc{pH}{At room temperature \SI{25}{\celsius}}{}
\desc[german]{pH-Wert}{Bei Raumtemperatur \SI{25}{\celsius}}{}
\eq{
\pH > 7 &\quad\tGT{basic} \\
\pH < 7 &\quad\tGT{acidic} \\
\pH = 7 &\quad\tGT{neutral}
}
\end{formula}
\begin{formula}{covalent_bond}
\desc{Covalent bond}{}{}
\desc[german]{Kolvalente Bindung}{}{}
\ttxt{
\eng{Bonds that involve sharing of electrons to form electron pairs between atoms.}
\ger{Bindungen zwischen Atomen die durch geteilte Elektronen, welche Elektronenpaare bilden, gebildet werden.}
}
\end{formula}
\begin{formula}{grotthuss}
\desc{Grotthuß-mechanism}{}{}
\desc[german]{Grotthuß-Mechanismus}{}{}
\ttxt{
\eng{The mobility of protons in aqueous solutions is much higher than that of other ions because they can "move" by breaking and reforming covalent bonds of water molecules.}
\ger{The Moblilität von Protononen in wässrigen Lösungen ist wesentlich größer als die anderer Ionen, da sie sich "bewegen" können indem die Wassertsoffbrückenbindungen gelöst und neu gebildet werden.}
}
\end{formula}
\Eng[cyanide]{Cyanide}
\Ger[cyanide]{Zyanid}
\Eng[ammonia]{Ammonia}
\Ger[ammonia]{Ammoniak}
\begin{formula}{common_chemicals}
\desc{Common chemicals}{}{}
\desc[german]{Häufige Chemikalien}{}{}
\begin{tabular}{l|c}
\GT{name} & \GT{formula} \\ \hline\hline
\GT{cyanide} & \ce{CN} \\ \hline
\GT{ammonia} & \ce{NH3}
\end{tabular}
\end{formula}

77
src/cm/crystal.tex
View File

@ -6,11 +6,7 @@
\eng{Bravais lattice}
\ger{Bravais-Gitter}
]{bravais}
\eng[table2D]{In 2D, there are 5 different Bravais lattices}
\ger[table2D]{In 2D gibt es 5 verschiedene Bravais-Gitter}
\eng[table3D]{In 3D, there are 14 different Bravais lattices}
\ger[table3D]{In 3D gibt es 14 verschiedene Bravais-Gitter}
\Eng[lattice_system]{Lattice system}
\Ger[lattice_system]{Gittersystem}
@ -24,56 +20,43 @@
\newcommand\bvimg[1]{\begin{center}\includegraphics[width=0.1\textwidth]{img/bravais/#1.pdf}\end{center}}
\renewcommand\tabularxcolumn[1]{m{#1}}
\newcolumntype{Z}{>{\centering\let\newline\\\arraybackslash\hspace{0pt}}X}
\begin{table}[H]
\centering
\expandafter\caption\expandafter{\gt{table2D}}
\label{tab:bravais2}
\begin{bigformula}{2d}
\desc{2D}{In 2D, there are 5 different Bravais lattices}{}
\desc[german]{2D}{In 2D gibt es 5 verschiedene Bravais-Gitter}{}
\begin{adjustbox}{width=\textwidth}
\begin{tabularx}{\textwidth}{||Z|c|Z|Z||}
\hline
\multirow{2}{*}{\GT{lattice_system}} & \multirow{2}{*}{\GT{point_group}} & \multicolumn{2}{c||}{5 \gt{bravais_lattices}} \\ \cline{3-4}
& & \GT{primitive} (p) & \GT{centered} (c) \\ \hline
\GT{monoclinic} (m) & $\text{C}_\text{2}$ & \bvimg{mp} & \\ \hline
\GT{orthorhombic} (o) & $\text{D}_\text{2}$ & \bvimg{op} & \bvimg{oc} \\ \hline
\GT{tetragonal} (t) & $\text{D}_\text{4}$ & \bvimg{tp} & \\ \hline
\GT{hexagonal} (h) & $\text{D}_\text{6}$ & \bvimg{hp} & \\ \hline
\end{tabularx}
\begin{tabularx}{\textwidth}{||Z|c|Z|Z||}
\hline
\multirow{2}{*}{\GT{lattice_system}} & \multirow{2}{*}{\GT{point_group}} & \multicolumn{2}{c||}{5 \gt{bravais_lattices}} \\ \cline{3-4}
& & \GT{primitive} (p) & \GT{centered} (c) \\ \hline
\GT{monoclinic} (m) & $\text{C}_\text{2}$ & \bvimg{mp} & \\ \hline
\GT{orthorhombic} (o) & $\text{D}_\text{2}$ & \bvimg{op} & \bvimg{oc} \\ \hline
\GT{tetragonal} (t) & $\text{D}_\text{4}$ & \bvimg{tp} & \\ \hline
\GT{hexagonal} (h) & $\text{D}_\text{6}$ & \bvimg{hp} & \\ \hline
\end{tabularx}
\end{adjustbox}
\end{table}
\end{bigformula}
\begin{table}[H]
\centering
\caption{\gt{table3D}}
\label{tab:bravais3}
\begin{bigformula}{3d}
\desc{3D}{In 3D, there are 14 different Bravais lattices}{}
\desc[german]{3D}{In 3D gibt es 14 verschiedene Bravais-Gitter}{}
% \newcolumntype{g}{>{\columncolor[]{0.8}}}
\begin{adjustbox}{width=\textwidth}
% \begin{tabularx}{\textwidth}{|c|}
% asdfasdfadslfasdfaasdofiuapsdoifuapodisufpaoidsufpaoidsufpaoisdfaoisdfpaosidfupaoidsufpaoidsufpaoidsufpaoisdufpaoidsufpoaiudsfpioaspdoifuaposidufpaoisudpfoiaupsdoifupasodf \\
% asdfasdfadslfasdfaasdofiuapsdoifuapodisufpaoidsufpaoidsufpaoisdfaoisdfpaosidfupaoidsufpaoidsufpaoidsufpaoisdufpaoidsufpoaiudsfpioaspdoifuaposidufpaoisudpfoiaupsdoifupasodf \\
% \end{tabularx}
% \begin{tabular}{|c|}
% asdfasdfadslfasdfaasdofiuapsdoifuapodisufpaoidsufpaoidsufpaoisdfaoisdfpaosidfupaoidsufpaoidsufpaoidsufpaoisdufpaoidsufpoaiudsfpioaspdoifuaposidufpaoisudpfoiaupsdoifupasodf \\
% asdfasdfadslfasdfaasdofiuapsdoifuapodisufpaoidsufpaoidsufpaoisdfaoisdfpaosidfupaoidsufpaoidsufpaoidsufpaoisdufpaoidsufpoaiudsfpioaspdoifuaposidufpaoisudpfoiaupsdoifupasodf \\
% \end{tabular}
% \\
\begin{tabularx}{\textwidth}{||Z|Z|c|Z|Z|Z|Z||}
\hline
\multirow{2}{*}{\GT{crystal_family}} & \multirow{2}{*}{\GT{lattice_system}} & \multirow{2}{*}{\GT{point_group}} & \multicolumn{4}{c||}{14 \gt{bravais_lattices}} \\ \cline{4-7}
& & & \GT{primitive} (P) & \GT{base_centered} (S) & \GT{body_centered} (I) & \GT{face_centered} (F) \\ \hline
\multicolumn{2}{||c|}{\GT{triclinic} (a)} & $\text{C}_\text{i}$ & \bvimg{tP} & & & \\ \hline
\multicolumn{2}{||c|}{\GT{monoclinic} (m)} & $\text{C}_\text{2h}$ & \bvimg{mP} & \bvimg{mS} & & \\ \hline
\multicolumn{2}{||c|}{\GT{orthorhombic} (o)} & $\text{D}_\text{2h}$ & \bvimg{oP} & \bvimg{oS} & \bvimg{oI} & \bvimg{oF} \\ \hline
\multicolumn{2}{||c|}{\GT{tetragonal} (t)} & $\text{D}_\text{4h}$ & \bvimg{tP} & & \bvimg{tI} & \\ \hline
\multirow{2}{*}{\GT{hexagonal} (h)} & \GT{rhombohedral} & $\text{D}_\text{3d}$ & \bvimg{hR} & & & \\ \cline{2-7}
& \GT{hexagonal} & $\text{D}_\text{6h}$ & \bvimg{hP} & & & \\ \hline
\multicolumn{2}{||c|}{\GT{cubic} (c)} & $\text{O}_\text{h}$ & \bvimg{cP} & & \bvimg{cI} & \bvimg{cF} \\ \hline
\end{tabularx}
\begin{tabularx}{\textwidth}{||Z|Z|c|Z|Z|Z|Z||}
\hline
\multirow{2}{*}{\GT{crystal_family}} & \multirow{2}{*}{\GT{lattice_system}} & \multirow{2}{*}{\GT{point_group}} & \multicolumn{4}{c||}{14 \gt{bravais_lattices}} \\ \cline{4-7}
& & & \GT{primitive} (P) & \GT{base_centered} (S) & \GT{body_centered} (I) & \GT{face_centered} (F) \\ \hline
\multicolumn{2}{||c|}{\GT{triclinic} (a)} & $\text{C}_\text{i}$ & \bvimg{tP} & & & \\ \hline
\multicolumn{2}{||c|}{\GT{monoclinic} (m)} & $\text{C}_\text{2h}$ & \bvimg{mP} & \bvimg{mS} & & \\ \hline
\multicolumn{2}{||c|}{\GT{orthorhombic} (o)} & $\text{D}_\text{2h}$ & \bvimg{oP} & \bvimg{oS} & \bvimg{oI} & \bvimg{oF} \\ \hline
\multicolumn{2}{||c|}{\GT{tetragonal} (t)} & $\text{D}_\text{4h}$ & \bvimg{tP} & & \bvimg{tI} & \\ \hline
\multirow{2}{*}{\GT{hexagonal} (h)} & \GT{rhombohedral} & $\text{D}_\text{3d}$ & \bvimg{hR} & & & \\ \cline{2-7}
& \GT{hexagonal} & $\text{D}_\text{6h}$ & \bvimg{hP} & & & \\ \hline
\multicolumn{2}{||c|}{\GT{cubic} (c)} & $\text{O}_\text{h}$ & \bvimg{cP} & & \bvimg{cI} & \bvimg{cF} \\ \hline
\end{tabularx}
\end{adjustbox}
\end{table}
\end{bigformula}
\begin{formula}{lattice_constant}
\desc{Lattice constant}{Parameter (length or angle) describing the smallest unit cell}{}

14
src/cm/mat.tex
View File

@ -12,3 +12,17 @@
\tau &= \frac{l}{L}
}
\end{formula}
\begin{formula}{stress}
\desc{Stress}{Force per area}{\QtyRef{force}, \QtyRef{area}}
\desc[german]{Spannung}{(Engl. "stress") Kraft pro Fläche}{}
\quantity{\sigma}{\newton\per\m^2}{v}
\eq{\ten{\sigma}_{ij} = \frac{F_i}{A_j}}
\end{formula}
\begin{formula}{strain}
\desc{Strain}{}{$\Delta x$ distance from reference position $x_0$}
\desc[german]{Dehnung}{(Engl. "strain")}{$\Delta x$ Auslenkung aus der Referenzposition $x_0$}
\quantity{\epsilon}{}{s}
\eq{\epsilon = \frac{\Delta x}{x_0}}
\end{formula}

162
src/cm/techniques.tex
View File

@ -13,54 +13,53 @@
\eng{Raman spectroscopy}
\ger{Raman Spektroskopie}
]{raman}
% \begin{minipagetable}{raman}
% \entry{name}{
% \eng{Raman spectroscopy}
% \ger{Raman-Spektroskopie}
% }
% \entry{application}{
% \eng{Vibrational modes, Crystal structure, Doping, Band Gaps, Layer thickness in \fqEqName{cm:misc:vdw_material}}
% \ger{Vibrationsmoden, Kristallstruktur, Dotierung, Bandlücke, Schichtdicke im \fqEqName{cm:misc:vdw_material}}
% }
% % \entry{how}{
% % \eng{Monochromatic light (\fqEqRef{Laser}) shines on sample, inelastic scattering because of rotation-, vibration-, phonon and spinflip-processes, plot spectrum as shift of the laser light (in \si{\per\cm})}
% % \ger{Monochromatisches Licht (\fqEqRef{Laser}) bestrahlt Probe, inelastische Streuung durch Rotations-, Schwingungs-, Phonon und Spin-Flip-Prozesse, plotte Spektrum als Verschiebung gegen das Laser Licht (in \si{\per\cm}) }
% % }
% \end{minipagetable}
\begin{minipage}{0.5\textwidth}
% TODO remove fqname from minipagetable?
\begin{bigformula}{raman}
\desc{Raman spectroscopy}{}{}
\desc[german]{Raman-Spektroskopie}{}{}
\begin{minipagetable}{raman}
\tentry{application}{
\eng{Vibrational modes, Crystal structure, Doping, Band Gaps, Layer thickness in \fqEqRef{cm:misc:vdw_material}}
\ger{Vibrationsmoden, Kristallstruktur, Dotierung, Bandlücke, Schichtdicke im \fqEqRef{cm:misc:vdw_material}}
}
\tentry{how}{
\eng{Monochromatic light (\fqEqRef{Laser}) shines on sample, inelastic scattering because of rotation-, vibration-, phonon and spinflip-processes, plot spectrum as shift of the laser light (in \si{\per\cm})}
\ger{Monochromatisches Licht (\fqEqRef{Laser}) bestrahlt Probe, inelastische Streuung durch Rotations-, Schwingungs-, Phonon und Spin-Flip-Prozesse, plotte Spektrum als Verschiebung gegen das Laser Licht (in \si{\per\cm}) }
}
\end{minipagetable}
\begin{minipage}{0.45\textwidth}
\begin{figure}[H]
\centering
% \includegraphics[width=0.8\textwidth]{img/cm_amf.pdf}
% \caption{\cite{Bian2021}}
\end{figure}
\end{minipage}
\end{bigformula}
\expandafter\detokenize\expandafter{\fqname}
\GT{cm:meas:raman:raman:application}
\separateEntries
% \begin{minipagetable}{pl}
% \entry{name}{
% \eng{Photoluminescence spectroscopy}
% \ger{Photolumeszenz-Spektroskopie}
% }
% \entry{application}{
% \eng{Crystal structure, Doping, Band Gaps, Layer thickness in \fqEqName{cm:misc:vdw_material}}
% \ger{Kristallstruktur, Dotierung, Bandlücke, Schichtdicke im \fqEqName{cm:misc:vdw_material}}
% }
% \entry{how}{
% \eng{Monochromatic light (\fqEqRef{Laser}) shines on sample, electrons are excited, relax to the conduction band minimum and finally accross the band gap under photon emission}
% \ger{Monochromatisches Licht (\fqEqRef{Laser}) bestrahlt Probe, Elektronen werden angeregt und relaxieren in das Leitungsband-Minimum und schließlich über die Bandlücke unter Photonemission}
% }
% \end{minipagetable}
\begin{minipage}{0.5\textwidth}
\begin{bigformula}{pl}
\desc{Photoluminescence spectroscopy}{}{}
\desc[german]{Photolumeszenz-Spektroskopie}{}{}
\begin{minipagetable}{pl}
\tentry{application}{
\eng{Crystal structure, Doping, Band Gaps, Layer thickness in \fqEqRef{cm:misc:vdw_material}}
\ger{Kristallstruktur, Dotierung, Bandlücke, Schichtdicke im \fqEqRef{cm:misc:vdw_material}}
}
\tentry{how}{
\eng{Monochromatic light (\fqEqRef{Laser}) shines on sample, electrons are excited, relax to the conduction band minimum and finally accross the band gap under photon emission}
\ger{Monochromatisches Licht (\fqEqRef{Laser}) bestrahlt Probe, Elektronen werden angeregt und relaxieren in das Leitungsband-Minimum und schließlich über die Bandlücke unter Photonemission}
}
\end{minipagetable}
\begin{minipage}{0.45\textwidth}
\begin{figure}[H]
\centering
% \includegraphics[width=0.8\textwidth]{img/cm_amf.pdf}
% \caption{\cite{Bian2021}}
\end{figure}
\end{minipage}
\end{bigformula}
\Subsection[
@ -82,63 +81,63 @@
\end{ttext}
\begin{bigformula}{amf}
\desc{Atomic force microscopy (AMF)}{}{}
\desc[german]{Atomare Rasterkraftmikroskopie (AMF)}{}{}
\begin{minipagetable}{amf}
\entry{name}{
\eng{Atomic force microscopy (AMF)}
\ger{Atomare Rasterkraftmikroskopie (AMF)}
}
\entry{application}{
\tentry{application}{
\eng{Surface stuff}
\ger{Oberflächenzeug}
}
\entry{how}{
\tentry{how}{
\eng{With needle}
\ger{Mit Nadel}
}
\end{minipagetable}
\begin{minipage}{0.5\textwidth}
\begin{minipage}{0.45\textwidth}
\begin{figure}[H]
\centering
\includegraphics[width=0.8\textwidth]{img/cm_amf.pdf}
\caption{\cite{Bian2021}}
\end{figure}
\end{minipage}
\end{bigformula}
\begin{bigformula}{stm}
\desc{Scanning tunneling microscopy (STM)}{}{}
\desc[german]{Rastertunnelmikroskop (STM)}{}{}
\begin{minipagetable}{stm}
\entry{name}{
\eng{Scanning tunneling microscopy (STM)}
\ger{Rastertunnelmikroskop (STM)}
}
\entry{application}{
\tentry{application}{
\eng{Surface stuff}
\ger{Oberflächenzeug}
}
\entry{how}{
\tentry{how}{
\eng{With TUnnel}
\ger{Mit TUnnel}
}
\end{minipagetable}
\begin{minipage}{0.5\textwidth}
\begin{minipage}{0.45\textwidth}
\begin{figure}[H]
\centering
\includegraphics[width=0.8\textwidth]{img/cm_stm.pdf}
\caption{\cite{Bian2021}}
\end{figure}
\end{minipage}
\end{bigformula}
\Section[
\eng{Fabrication techniques}
\ger{Herstellungsmethoden}
]{fab}
]{fab}
\begin{bigformula}{cvd}
\desc{Chemical vapor deposition (CVD)}{}{}
\desc[german]{Chemische Gasphasenabscheidung (CVD)}{}{}
\begin{minipagetable}{cvd}
\entry{name}{
\eng{Chemical vapor deposition (CVD)}
\ger{Chemische Gasphasenabscheidung (CVD)}
}
\entry{how}{
\tentry{how}{
\eng{
A substrate is exposed to volatile precursors, which react and/or decompose on the heated substrate surface to produce the desired deposit.
By-products are removed by gas flow through the chamber.
@ -148,7 +147,7 @@
Nebenprodukte werden durch den Gasfluss durch die Kammer entfernt.
}
}
\entry{application}{
\tentry{application}{
\eng{
\begin{itemize}
\item Polysilicon \ce{Si}
@ -167,10 +166,11 @@
}
}
\end{minipagetable}
\begin{minipage}{0.5\textwidth}
\begin{minipage}{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{img/cm_cvd_english.pdf}
\end{minipage}
\end{bigformula}
\Subsection[
@ -182,31 +182,31 @@
\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}
\begin{bigformula}{mbe}
\desc{Molecular Beam Epitaxy (MBE)}{}{}
\desc[german]{Molekularstrahlepitaxie (MBE)}{}{}
\begin{minipagetable}{mbe}
\tentry{how}{
\eng{In a ultra-high vacuum, the elements are heated until they slowly sublime. The gases then condensate on the substrate surface}
\ger{Die Elemente werden in einem Ultrahochvakuum erhitzt, bis sie langsam sublimieren. Die entstandenen Gase kondensieren dann auf der Oberfläche des Substrats}
}
\ger{
\begin{itemize}
\item Galliumarsenid \ce{GaAs}
\end{itemize}
\tentry{application}{
\eng{
\begin{itemize}
\item Gallium arsenide \ce{GaAs}
\end{itemize}
\TODO{Link to GaAs}
}
\ger{
\begin{itemize}
\item Galliumarsenid \ce{GaAs}
\end{itemize}
}
}
}
\end{minipagetable}
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{img/cm_mbe_english.pdf}
\end{minipage}
\end{minipagetable}
\begin{minipage}{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{img/cm_mbe_english.pdf}
\end{minipage}
\end{bigformula}

28
src/cm/topo.tex
View File

@ -56,17 +56,17 @@
\eq{\gamma_n = \oint_C \d \vec{R} \cdot A_n(\vec{R}) = \int_S \d\vec{S} \cdot \vec{\Omega}_n(\vec{R})}
\end{formula}
\begin{ttext}[chern_number_desc]
\eng{The Berry flux through any 2D closed surface is quantized by the \textbf{Chern number}.
If there is time-reversal symmetry, the Chern-number is 0.
}
\ger{Der Berry-Fluß durch eine geschlossene 2D Fl[cher is quantisiert durch die \textbf{Chernzahl}
Bei erhaltener Zeitumkehrungssymmetrie ist die Chernzahl 0.
}
\end{ttext}
\begin{formula}{chern_number}
\desc{Chern number}{Eg. number of Berry curvature monopoles in the Brillouin zone (then $\vec{R} = \vec{k}$)}{$\vec{S}$ closed surface in $\vec{R}$-space. A \textit{Chern insulator} is a 2D insulator with $C_n \neq 0$}
\desc[german]{Chernuzahl}{Z.B. Anzahl der Berry-Krümmungs-Monopole in der Brilouinzone (dann ist $\vec{R} = \vec{k}$). Ein \textit{Chern-Isolator} ist ein 2D Isolator mit $C_n\neq0$}{$\vec{S}$ geschlossene Fläche im $\vec{R}$-Raum}
\ttxt{
\eng{The Berry flux through any 2D closed surface is quantized by the \textbf{Chern number}.
If there is time-reversal symmetry, the Chern-number is 0.
}
\ger{Der Berry-Fluß durch eine geschlossene 2D Fl[cher is quantisiert durch die \textbf{Chernzahl}
Bei erhaltener Zeitumkehrungssymmetrie ist die Chernzahl 0.
}
}
\eq{C_n = \frac{1}{2\pi} \oint \d \vec{S}\ \cdot \vec{\Omega}_n(\vec{R})}
\end{formula}
@ -76,10 +76,14 @@
\eq{\vec{\sigma}_{xy} = \sum_n \frac{e^2}{h} \int_\text{\GT{occupied}} \d^2k\, \frac{\Omega_{xy}^n}{2\pi} = \sum_n C_n \frac{e^2}{h}}
\end{formula}
\begin{ttext}
\eng{A 2D insulator with a non-zero Chern number is called a \textbf{topological insulator}.}
\ger{Ein 2D Isolator mit einer Chernzahl ungleich 0 wird \textbf{topologischer Isolator} genannt.}
\end{ttext}
\begin{formula}{topological_insulator}
\desc{Topological insulator}{}{}
\desc[german]{Topologischer Isolator}{}{}
\ttxt{
\eng{A 2D insulator with a non-zero Chern number is called a \textbf{topological insulator}.}
\ger{Ein 2D Isolator mit einer Chernzahl ungleich 0 wird \textbf{topologischer Isolator} genannt.}
}
\end{formula}

429
src/comp/ad.tex
View File

@ -2,86 +2,373 @@
\eng{Atomic dynamics}
% \ger{}
]{ad}
\Subsection[
\eng{Born-Oppenheimer Approximation}
\ger{Born-Oppenheimer Näherung}
]{bo}
\begin{formula}{hamiltonian}
\desc{Electron Hamiltonian}{}{$\hat{T}$ \fqEqRef{comp:elsth:kinetic_energy}, $\hat{V}$ \fqEqRef{comp:elsth:potential_energy}, $\txe$ \GT{electrons}, $\txn$ \GT{nucleons}}
\desc[german]{Hamiltonian der Elektronen}{}{}
\eq{\hat{H}_\txe = \hat{T}_\txe + V_{\txe \leftrightarrow \txe} + V_{\txn \leftrightarrow \txe}}
\begin{formula}{hamiltonian}
\desc{Electron Hamiltonian}{}{$\hat{T}$ \fqEqRef{comp:est:kinetic_energy}, $\hat{V}$ \fqEqRef{comp:est:potential_energy}, $\txe$ \GT{electrons}, $\txn$ \GT{nucleons}}
\desc[german]{Hamiltonian der Elektronen}{}{}
\eq{\hat{H}_\txe = \hat{T}_\txe + V_{\txe \leftrightarrow \txe} + V_{\txn \leftrightarrow \txe}}
\end{formula}
\begin{formula}{ansatz}
\desc{Wave function ansatz}{}{$\psi_\text{en}^n$ eigenstate $n$ of \fqEqRef{comp:est:hamiltonian}, $\psi_\txe^i$ eigenstate $i$ of \fqEqRef{comp:ad:bo:hamiltonian}, $\vecr,\vecR$ electron/nucleus positions, $\sigma$ electron spin, $c^{ni}$ coefficients}
\desc[german]{Wellenfunktion Ansatz}{}{}
\eq{\psi_\text{en}^n\big(\{\vecr,\sigma\},\{\vecR\}\big) = \sum_i c^{ni}\big(\{\vecR\}\big)\, \psi_\txe^i\big(\{\vecr,\sigma\},\{\vecR\}\big)}
\end{formula}
\begin{formula}{equation}
\desc{Equation}{}{}
% \desc[german]{}{}{}
\eq{
\label{eq:\fqname}
\left[E_\txe^j\big(\{\vecR\}\big) + \hat{T}_\txn + V_{\txn \leftrightarrow \txn} - E^n \right]c^{nj} = -\sum_i \Lambda_{ij} c^{ni}\big(\{\vecR\}\big)
}
\end{formula}
\begin{formula}{coupling_operator}
\desc{Exact nonadiabtic coupling operator}{Electron-phonon couplings / electron-vibrational couplings}{$\psi^i_\txe$ electronic states, $\vecR$ nucleus position, $M$ nucleus \qtyRef{mass}}
% \desc[german]{}{}{}
\begin{multline}
\Lambda_{ij} = \int \d^3r (\psi_\txe^j)^* \left(-\sum_I \frac{\hbar^2\nabla_{\vecR_I}^2}{2M_I}\right) \psi_\txe^i \\
+ \sum_I \frac{1}{M_I} \int\d^3r \left[(\psi_\txe^j)^* (-i\hbar\nabla_{\vecR_I})\psi_\txe^i\right](-i\hbar\nabla_{\vecR_I})
\end{multline}
\end{formula}
\Subsection[
\eng{Born-Oppenheimer Approximation}
\ger{Born-Oppenheimer Näherung}
]{bo}
\begin{formula}{adiabatic_approx}
\desc{Adiabatic approximation}{Electronic configuration remains the same when atoms move (\absRef{adiabatic_theorem})}{$\Lambda_{ij}$ \fqEqRef{comp:ad:coupling_operator}}
\desc[german]{Adiabatische Näherung}{Elektronenkonfiguration bleibt gleich bei Bewegung der Atome gleichl (\absRef{adiabatic_theorem})}{}
\eq{\Lambda_{ij} = 0 \quad \text{\GT{for} } i\neq j}
\end{formula}
\begin{formula}{approx}
\desc{Born-Oppenheimer approximation}{Electrons are not influenced by the movement of the atoms}{\GT{see} \fqEqRef{comp:ad:equation}, $V_{\txn \leftrightarrow \txn} = \const$ absorbed into $E_\txe^j$}
\desc[german]{Born-Oppenheimer Näherung}{Elektronen werden nicht durch die Bewegung der Atome beeinflusst}{}
\begin{gather}
\Lambda_{ij} = 0
\shortintertext{\fqEqRef{comp:ad:bo:equation} \Rightarrow}
\left[E_e^i\big(\{\vecR\}\big) + \hat{T}_\txn - E^n\right]c^{ni}\big(\{\vecR\}\big) = 0
\end{gather}
\end{formula}
\begin{formula}{surface}
\desc{Born-Oppenheimer surface}{Potential energy surface (PES)\\ The nuclei follow Newtons equations of motion on the BO surface if the system is in the electronic ground state}{$E_\txe^0, \psi_\txe^0$ lowest eigenvalue/eigenstate of \fqEqRef{comp:ad:bo:hamiltonian}}
\desc[german]{Born-Oppenheimer Potentialhyperfläche}{Die Nukleonen Newtons klassichen Bewegungsgleichungen auf der BO Hyperfläche wenn das System im elektronischen Grundzustand ist}{$E_\txe^0, \psi_\txe^0$ niedrigster Eigenwert/Eigenzustand vom \fqEqRef{comp:ad:bo:hamiltonian}}
\begin{gather}
V_\text{BO}\big(\{\vecR\}\big) = E_\txe^0\big(\{\vecR\}\big) \\
M_I \ddot{\vecR}_I(t) = - \Grad_{\vecR_I} V_\text{BO}\big(\{\vecR(t)\}\big)
\end{gather}
\end{formula}
\begin{formula}{ansatz}
\desc{Ansatz for \secEqRef{approx}}{Product of single electronic and single nuclear state}{}
\desc[german]{Ansatz für \secEqRef{approx}}{Produkt aus einem einzelnen elektronischen Zustand und einem Nukleus-Zustand}{}
\eq{
\psi_\text{BO} = c^{n0} \big(\{\vecR\}\big) \,\psi_\txe^0 \big(\{\vecr,\sigma\},\{\vecR\}\big)
}
\end{formula}
\begin{formula}{limitations}
\desc{Limitations}{}{$\tau$ passage of time for electrons/nuclei, $L$ characteristic length scale of atomic dynamics, $\dot{\vec{R}}$ nuclear velocity, $\Delta E$ difference between two electronic states}
\desc[german]{Limitationen}{}{}
\ttxt{
\eng{
\begin{itemize}
\item Nuclei velocities must be small and electron energy state differences large
\item Nuclei need spin for effects like spin-orbit coupling
\item Nonadiabitc effects in photochemistry, proteins
\end{itemize}
Valid when Massey parameter $\xi \gg 1$
}
}
\eq{
\xi = \frac{\tau_\txn}{\tau_\txe} = \frac{L \Delta E}{\hbar \abs{\dot{\vecR}}}
}
\end{formula}
\Subsection[
\eng{Structure optimization}
\ger{Strukturoptimierung}
]{opt}
\begin{formula}{forces}
\desc{Forces}{}{}
\desc[german]{Kräfte}{}{}
\eq{\vec{F}_I = -\Grad_{\vecR_I} E \explOverEq{\fqEqRef{qm:se:hellmann_feynmann}} -\Braket{\psi(\vecR_I) | \left(\Grad_{\vecR_I} \hat{H}(\vecR_I)\right) | \psi(\vecR) }}
\end{formula}
\begin{formula}{ionic_cycle}
\desc{Ionic cycle}{\fqEqRef{comp:est:dft:ks:scf} for geometry optimization}{}
\desc[german]{}{}{}
\ttxt{
\eng{
\begin{enumerate}
\item Initial guess for $n(\vecr)$
\begin{enumerate}
\item Calculate effective potential $V_\text{eff}$
\item Solve \fqEqRef{comp:est:dft:ks:equation}
\item Calculate density $n(\vecr)$
\item Repeat b-d until self consistent
\end{enumerate}
\item Calculate \secEqRef{forces}
\item If $F\neq0$, get new geometry by interpolating $R$ and restart
\end{enumerate}
}
}
\end{formula}
\begin{formula}{transformation}
\desc{Transformation of atomic positions under stress}{}{$\alpha,\beta=1,2,3$ position components, $R$ position, $R(0)$ zero-strain position, $\ten{\epsilon}$ \qtyRef{strain} tensor}
\desc[german]{Transformation der Atompositionen unter Spannung}{}{$\alpha,\beta=1,2,3$ Positionskomponenten, $R$ Position, $R(0)$ Position ohne Dehnung, $\ten{\epsilon}$ \qtyRef{strain} Tensor}
\eq{R_\alpha(\ten{\epsilon}_{\alpha\beta}) = \sum_\beta \big(\delta_{\alpha\beta} + \ten{\epsilon}_{\alpha\beta}\big)R_\beta(0)}
\end{formula}
\begin{formula}{stress_tensor}
\desc{Stress tensor}{}{$\Omega$ unit cell volume, \ten{\epsilon} \qtyRef{strain} tensor}
\desc[german]{Spannungstensor}{}{}
\eq{\ten{\sigma}_{\alpha,\beta} = \frac{1}{\Omega} \pdv{E_\text{total}}{\ten{\epsilon}_{\alpha\beta}}_{\ten{\epsilon}=0}}
\end{formula}
\begin{formula}{pulay_stress}
\desc{Pulay stress}{}{}
\desc[german]{Pulay-Spannung}{}{}
\eq{
N_\text{PW} \propto E_\text{cut}^\frac{3}{2} \propto \abs{\vec{G}_\text{max}}^3
}
\ttxt{\eng{
Number of plane waves $N_\text{PW}$ depends on $E_\text{cut}$.
If $G$ changes during optimization, $N_\text{PW}$ may change, thus the basis set can change.
This typically leads to too small volumes.
}}
\end{formula}
\Subsection[
\eng{Lattice vibrations}
\ger{Gitterschwingungen}
]{latvib}
\begin{formula}{force_constant_matrix}
\desc{Force constant matrix}{}{}
% \desc[german]{}{}{}
\eq{\Phi_{IJ}^{\mu\nu} = \pdv{V(\{\vecR\})}{R_I^\mu,R_J^\nu}_{\{\vecR_I\}=\{\vecR_I^0\}}}
\end{formula}
\begin{formula}{harmonic_approx}
\desc{Harmonic approximation}{Hessian matrix, 2nd order Taylor expansion of the \fqEqRef{comp:ad:bo:surface} around every nucleus position $\vecR_I^0$}{$\Phi_{IJ}^{\mu\nu}$ \secEqRef{force_constant_matrix}, $s$ displacement}
\desc[german]{Harmonische Näherung}{Hesse matrix, Taylor Entwicklung der \fqEqRef{comp:ad:bo:surface} in zweiter Oddnung um Atomposition $\vecR_I^0$}{}
\eq{ V^\text{BO}(\{\vecR_I\}) \approx V^\text{BO}(\{\vecR_I^0\}) + \frac{1}{2} \sum_{I,J}^N \sum_{\mu,\nu}^3 s_I^\mu s_J^\nu \Phi_{IJ}^{\mu\nu} }
\end{formula}
% solving difficult becaus we need to calculate (3N)^2 derivatives, Hellmann-Feynman cant be applied directly
% -> DFPT
% finite-difference method
\Subsubsection[
\eng{Finite difference method}
% \ger{}
]{fin_diff}
\begin{formula}{approx}
\desc{Approximation}{Assume forces in equilibrium structure vanish}{$\Delta s$ displacement of atom $J$}
% \desc[german]{}{}{}
\eq{\Phi_{IJ}^{\mu\nu} \approx \frac{\vecF_I^\mu(\vecR_1^0, \dots, \vecR_J^0+\Delta s_J^\nu,\dots, \vecR_N^0)}{\Delta s_J^\nu}}
\end{formula}
\begin{formula}{ansatz}
\desc{Wave function ansatz}{}{$\psi_\text{en}^n$ eigenstate $n$ of \fqEqRef{comp:elst:hamiltonian}, $\psi_\txe^i$ eigenstate $i$ of \fqEqRef{comp:ad:bo:hamiltonian}, $\vecr,\vecR$ electron/nucleus positions, $\sigma$ electron spin, $c^{ni}$ coefficients}
\desc[german]{Wellenfunktion Ansatz}{}{}
\eq{\psi_\text{en}^n\big(\{\vecr,\sigma\},\{\vecR\}\big) = \sum_i c^{ni}\big(\{\vecR\}\big)\, \psi_\txe^i\big(\{\vecr,\sigma\},\{\vecR\}\big)}
\begin{formula}{dynamical_matrix}
\desc{Dynamical matrix}{Mass reduced \absRef[fourier transform]{fourier_transform} of the \fqEqRef{comp:ad:latvib:force_constant_matrix}}{$\vec{L}$ vector from origin to unit cell $n$, $\alpha/\beta$ atom index in th unit cell, $\vecq$ \qtyRef{wave_vector}, $\Phi$ \fqEqRef{comp:ad:latvib:force_constant_matrix}, $M$ \qtyRef{mass}}
% \desc[german]{}{}{}
\eq{D_{\alpha\beta}^{\mu\nu} = \frac{1}{\sqrt{M_\alpha M_\beta}} \sum_{n^\prime} \Phi_{\alpha\beta}^{\mu\nu}(n-n^\prime) \e^{\I \vec{q}(\vec{L}_n - \vec{L}_{n^\prime})}}
\end{formula}
\begin{formula}{equation}
\desc{Equation}{}{}
\begin{formula}{eigenvalue_equation}
\desc{Eigenvalue equation}{For a periodic crystal, reduces number of equations from $3N_p\times N$ to $3N_p$. Eigenvalues represent phonon band structure.}{$N_p$ number of atoms per unit cell, $\vecc$ displacement amplitudes, $\vecq$ \qtyRef{wave_vector}, $\mat{D}$ \secEqRef{dynamical_matrix}}
\desc[german]{Eigenwertgleichung}{}{}
\eq{\omega^2 \vecc(\vecq) = \mat{D}(\vecq) \vecc(\vecq) }
\end{formula}
\Subsubsection[
\eng{Anharmonic approaches}
\ger{Anharmonische Ansätze}
]{anharmonic}
\begin{formula}{qha}
\desc{Quasi-harmonic approximation}{}{}
\desc[german]{}{}{}
\ttxt{\eng{
Include thermal expansion by assuming \fqEqRef{comp:ad:bo:surface} is volume dependant.
}}
\end{formula}
\begin{formula}{pertubative}
\desc{Pertubative approaches}{}{}
% \desc[german]{Störungs}{}{}
\ttxt{\eng{
Expand \fqEqRef{comp:ad:latvib:force_constant_matrix} to third order.
}}
\end{formula}
\Subsection[
\eng{Molecular Dynamics}
\ger{Molekulardynamik}
]{md} \abbrLink{md}{MD}
\begin{formula}{desc}
\desc{Description}{}{}
\desc[german]{Beschreibung}{}{}
\ttxt{\eng{
\begin{itemize}
\item Exact (within previous approximations) approach to treat anharmonic effects in materials.
\item Computes time-dependant observables.
\item Assumes fully classical nuclei.
\item Macroscropical observables from statistical ensembles
\item System evolves in time (ehrenfest). Number of points to consider does NOT scale with system size.
\item Exact because time dependance is studied explicitly, not via harmonic approx.
\end{itemize}
\TODO{cleanup}
}}
\end{formula}
\begin{formula}{procedure}
\desc{MD simulation procedure}{}{}
\desc[german]{Ablauf von MD Simulationen}{}{}
\ttxt{\eng{
\begin{enumerate}
\item Initialize with optimized geometry, interaction potential, ensemble, integration scheme, temperature/pressure control
\item Equilibrate to desired temperature/pressure (eg with statistical starting velocities)
\item Production run, run MD long enough to calculate desired observables
\end{enumerate}
}}
\end{formula}
\Subsubsection[
\eng{Ab-initio molecular dynamics}
\ger{Ab-initio molecular dynamics}
]{ab-initio}
\begin{formula}{bomd}
\abbrLabel{BOMD}
\desc{Born-Oppenheimer MD (BOMD)}{}{}
\desc[german]{Born-Oppenheimer MD (BOMD)}{}{}
\ttxt{\eng{
\begin{enumerate}
\item Calculate electronic ground state of current nucleui configuration $\{\vecR(t)\}$ with \abbrRef{ksdft}
\item \hyperref[f:comp:ad:opt:forces]{Calculate forces} from the \fqEqRef{comp:ad:bo:surface}
\item Update positions and velocities
\end{enumerate}
\begin{itemize}
\gooditem "ab-inito" - no empirical information required
\baditem Many expensive \abbrRef{dft} calculations
\end{itemize}
}}
\end{formula}
\begin{formula}{cpmd}
\desc{Car-Parrinello MD (CPMD)}{}{$\mu$ electron orbital mass, $\varphi_i$ \abbrRef{ksdft} eigenststate, $\lambda_{ij}$ Lagrange multiplier}
\desc[german]{Car-Parrinello MD (CPMD)}{}{}
\ttxt{\eng{
Evolve electronic wave function $\varphi$ (adiabatically) along with the nuclei \Rightarrow only one full \abbrRef{ksdft}
}}
\begin{gather}
M_I \odv[2]{\vecR_I}{t} = -\Grad_{\vecR_I} E[\{\varphi_i\},\{\vecR_I\}] \\
% not using pdv because of comma in parens
% E[\{\varphi_i\}\{\vecR_I\}] = \Braket{\psi_0|H_\text{el}^\text{KS}|\psi_0}
\mu \odv[2]{\varphi_i(\vecr,t)}{t} = - \frac{\partial}{\partial\varphi_i^*(\vecr,t)} E[\{\varphi_i\},\{\vecR_I\}] + \sum_j \lambda_{ij} \varphi_j(\vecr,t)
\end{gather}
\end{formula}
\Subsubsection[
\eng{Force-field MD}
\ger{Force-field MD}
]{ff}
\begin{formula}{ffmd}
\desc{Force field MD (FFMD)}{}{}
% \desc[german]{}{}{}
\ttxt{\eng{
\begin{itemize}
\item Use empirical interaction potential instead of electronic structure
\baditem Force fields need to be fitted for specific material \Rightarrorw not transferable
\gooditem Faster than \abbrRef{bomd}
\item Example: \absRef{lennard_jones}
\end{itemize}
}}
\end{formula}
\Subsubsection[
\eng{Integration schemes}
% \ger{}
]{scheme}
\begin{ttext}
\eng{Procedures for updating positions and velocities to obey the equations of motion.}
\end{ttext}
\begin{formula}{euler}
\desc{Euler method}{First-order procedure for solving \abbrRef{ode}s with a given initial value.\\Taylor expansion of $\vecR/\vecv (t+\Delta t)$}{}
\desc[german]{Euler-Verfahren}{Prozedur um gewöhnliche DGLs mit Anfangsbedingungen in erster Ordnung zu lösen.\\Taylor Entwicklung von $\vecR/\vecv (t+\Delta t)$}{}
\eq{
\vecR(t+\Delta t) &= \vecR(t) + \vecv(t) \Delta t + \Order{\Delta t^2} \\
\vecv(t+\Delta t) &= \vecv(t) + \veca(t) \Delta t + \Order{\Delta t^2}
}
\ttxt{\eng{
Cumulative error scales linearly $\Order{\Delta t}$. Not time reversible.
}}
\end{formula}
\begin{formula}{verlet}
\desc{Verlet integration}{Preverses time reversibility, does not require velocity updates}{}
\desc[german]{Verlet-Algorithmus}{Zeitumkehr-symmetrisch}{}
\eq{
\vecR(t+\Delta t) = 2\vecR(t) -\vecR(t-\Delta t) + \veca(t) \Delta t^2 + \Order{\Delta t^4}
}
\end{formula}
\begin{formula}{velocity-verlet}
\desc{Velocity-Verlet integration}{}{}
% \desc[german]{}{}{}
\eq{
\label{eq:\fqname}
\left[E_\txe^j\big(\{\vecR\}\big) + \hat{T}_\txn + V_{\txn \leftrightarrow \txn} - E^n \right]c^{nj} = -\sum_i \Lambda_{ij} c^{ni}\big(\{\vecR\}\big)
\vecR(t+\Delta t) &= \vecR(t) + \vecv(t)\Delta t + \frac{1}{2} \veca(t) \Delta t^2 + \Order{\Delta t^4} \\
\vecv(t+\Delta t) &= \vecv(t) + \frac{\veca(t) + \veca(t+\Delta t)}{2} \Delta t + \Order{\Delta t^4}
}
\end{formula}
\begin{formula}{coupling_operator}
\desc{Exact nonadiabtic coupling operator}{Electron-phonon couplings / electron-vibrational couplings}{$\psi^i_\txe$ electronic states, $\vecR$ nucleus position, $M$ nucleus \qtyRef{mass}}
\TODO{leapfrog}
\Subsubsection[
\eng{Thermostats and barostats}
\ger{Thermostate und Barostate}
]{stats}
\begin{formula}{velocity_rescaling}
\desc{Velocity rescaling}{Thermostat, keep temperature at $T_0$ by rescaling velocities. Does not allow temperature fluctuations and thus does not obey the \absRef{c_ensemble}}{$T$ target \qtyRef{temperature}, $M$ \qtyRef{mass} of nucleon $I$, $\vecv$ \qtyRef{velocity}, $f$ number of degrees of freedom, $\lambda$ velocity scaling parameter, \ConstRef{boltzmann}}
% \desc[german]{}{}{}
\begin{multline}
\Lambda_{ij} = \int \d^3r (\psi_\txe^j)^* \left(-\sum_I \frac{\hbar^2\nabla_{\vecR_I}^2}{2M_I}\right) \psi_\txe^i \\
+ \sum_I \frac{1}{M_I} \int\d^3r \left[(\psi_\txe^j)^* (-i\hbar\nabla_{\vecR_I})\psi_\txe^i\right](-i\hbar\nabla_{\vecR_I})
\end{multline}
\end{formula}
\begin{formula}{adiabatic_approx}
\desc{Adiabatic approximation}{Electronic configuration remains the same when atoms move}{$\Lambda_{ij}$ \fqEqRef{comp:ad:bo:coupling_operator}}
\desc[german]{Adiabatische Näherung}{Elektronenkonfiguration bleibt gleich bei Bewegung der Atome gleich}{}
\eq{\Lambda_{ij} = 0 \quad \text{\GT{for} } i\neq j}
\end{formula}
\begin{formula}{approx}
\desc{Born-Oppenheimer approximation}{}{\GT{see} \fqEqRef{comp:ad:bo:equation}}
\desc[german]{Born-Oppenheimer Näherung}{}{}
\begin{gather}
\Lambda_{ij} = 0
\shortintertext{\fqEqRef{comp:ad:bo:equation} \Rightarrow}
\left[E_e^i\big(\{\vecR\}\big) + \hat{T}_\txn - E^n\right]c^{ni}\big(\{\vecR\}\big) = 0
\end{gather}
\end{formula}
\begin{formula}{surface}
\desc{Born-Oppenheimer surface}{Potential energy surface (PES)\\ The nuclei follow Newtons equations of motion on the BO surface if the system is in the electronic ground state}{$E_\txe^0, \psi_\txe^0$ lowest eigenvalue/eigenstate of \fqEqRef{comp:ad:bo:hamiltonian}}
\desc[german]{Born-Oppenheimer Potentialhyperfläche}{Die Nukleonen Newtons klassichen Bewegungsgleichungen auf der BO Hyperfläche wenn das System im elektronischen Grundzustand ist}{$E_\txe^0, \psi_\txe^0$ niedrigster Eigenwert/Eigenzustand vom \fqEqRef{comp:ad:bo:hamiltonian}}
\begin{gather}
V_\text{BO}\big(\{\vecR\}\big) = E_\txe^0\big(\{\vecR\}\big) \\
M_I \ddot{\vecR}_I(t) = - \Grad_{\vecR_I} V_\text{BO}\big(\{\vecR(t)\}\big)
\shortintertext{\GT{ansatz} \GT{for} \fqEqRef{comp:ad:bo:approx}}
\psi_\text{BO} = c^{n0} \big(\{\vecR\}\big) \,\psi_\txe^0 \big(\{\vecr,\sigma\},\{\vecR\}\big)
\end{gather}
\end{formula}
\begin{formula}{limitations}
\desc{Limitations}{}{}
\desc[german]{Limitationen}{}{}
\ttxt{
\eng{
\begin{itemize}
\item Nuclei velocities must be small and electron energy state differences large
\item Nuclei need spin for effects like spin-orbit coupling
\item Nonadiabitc effects in photochemistry, proteins
\end{itemize}
}
\eq{
\Delta T(t) &= T_0 - T(t) \\
&= \sum_I^N \frac{M_I\,(\lambda \vecv_I(t))^2}{f\kB} - \sum_I^N \frac{M_I\,\vecv_I(t)^2}{f\kB} \\
&= (\lambda^2 - 1) T(t)
}
\end{formula}
\TODO{geometry optization?, lattice vibrations (harmionic approx, dynamical matrix)}
\eq{\lambda = \sqrt{\frac{T_0}{T(t)}}}
\end{formula}
\begin{formula}{berendsen}
\desc{Berendsen thermostat}{Does not obey \absRef{c_ensemble} but efficiently brings system to target temperature}{}
% \desc[german]{}{}{}
\eq{\odv{T}{t} = \frac{T_0-T}{\tau}}
\end{formula}
\Subsection[
\eng{Molecular Dynamics}
\ger{Molekulardynamik}
]{md}
\begin{ttext}
\eng{Statistical method}
\end{ttext}
\begin{formula}{nose-hoover}
\desc{Nosé-Hoover thermostat}{Control the temperature with by time stretching with an associated mass.\\Compliant with \absRef{c_ensemble}}{$s$ scaling factor, $Q$ associated "mass", $\mathcal{L}$ \absRef{lagrangian}, $g$ degrees of freedom}
\desc[german]{Nosé-Hoover Thermostat}{}{}
\begin{gather}
\d\tilde{t} = \tilde{s}\d t \\
\mathcal{L} = \sum_{I=1}^N \frac{1}{2} M_I \tilde{s}^2 v_i^2 - V(\tilde{\vecR}_1, \ldots, \tilde{\vecR}_I, \ldots, \tilde{\vecR}_N) + \frac{1}{2} Q \dot{\tilde{s}}^2 - g \kB T_0 \ln \tilde{s}
\end{gather}
\end{formula}
\TODO{ab-initio MD, force-field MD}
\Subsubsection[
\eng{Calculating observables}
\ger{Berechnung von Observablen}
]{obs}
\begin{formula}{spectral_density}
\desc{Spectral density}{Wiener-Khinchin theorem\\\absRef{fourier_transform} of \absRef{autocorrelation}}{$C$ \absRef{autocorrelation}}
\desc[german]{Spektraldichte}{Wiener-Khinchin Theorem\\\absRef{fourier_transform} of \absRef{autocorrelation}}{}
\eq{S(\omega) = \int_{-\infty}^\infty \d\tau C(\tau) \e^{-\I\omega t} }
\end{formula}
\begin{formula}{vdos} \abbrLabel{VDOS}
\desc{Vibrational density of states (VDOS)}{}{$S_{v_i}$ velocity \secEqRef{spectral_density} of particle $I$}
\desc[german]{Vibrationszustandsdicht (VDOS)}{}{}
\eq{g(\omega) \sim \sum_{I=1}^N M_I S_{v_I}(\omega)}
\end{formula}

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@ -1,183 +0,0 @@
\Section[
\eng{Electronic structure theory}
% \ger{}
]{elst}
\begin{formula}{kinetic_energy}
\desc{Kinetic energy}{of species $i$}{$i$ = nucleons/electrons, $N$ number of particles, $m$ \qtyRef{mass}}
\desc[german]{Kinetische Energie}{von Spezies $i$}{$i$ = Nukleonen/Elektronen, $N$ Teilchenzahl, $m$ \qtyRef{mass}}
\eq{\hat{T}_i &= -\sum_{n=1}^{N_i} \frac{\hbar^2}{2 m_i} \vec{\nabla}^2_n}
\end{formula}
\begin{formula}{potential_energy}
\desc{Electrostatic potential}{between species $i$ and $j$}{$i,j$ = nucleons/electrons, $r$ particle position, $Z_i$ charge of species $i$, \ConstRef{charge}}
\desc[german]{Elektrostatisches Potential}{zwischen Spezies $i$ und $j$}{}
\eq{\hat{V}_{i \leftrightarrow j} &= -\sum_{k,l} \frac{Z_i Z_j e^2}{\abs{\vecr_k - \vecr_l}}}
\end{formula}
\begin{formula}{hamiltonian}
\desc{Electronic structure Hamiltonian}{}{$\hat{T}$ \fqEqRef{comp:elsth:kinetic_energy}, $\hat{V}$ \fqEqRef{comp:elsth:potential_energy}, $\txe$ \GT{electrons}, $\txn$ \GT{nucleons}}
\eq{\hat{H} &= \hat{T}_\txe + \hat{T}_\txn + V_{\txe \leftrightarrow \txe} + V_{\txn \leftrightarrow \txe} + V_{\txn \leftrightarrow \txn}}
\end{formula}
\begin{formula}{mean_field}
\desc{Mean field approximation}{Replaces 2-particle operator by 1-particle operator}{Example for Coulumb interaction between many electrons}
\desc[german]{Molekularfeldnäherung}{Ersetzt 2-Teilchen Operator durch 1-Teilchen Operator}{Beispiel für Coulumb Wechselwirkung zwischen Elektronen}
\eq{
\frac{1}{2}\sum_{i\neq j} \frac{e^2}{\abs{\vec{r}_i - \vec{r}_j}} \approx \sum_{i} V_\text{eff}(\vec{r}_i)
}
\end{formula}
\Subsection[
\eng{Tight-binding}
\ger{Modell der stark gebundenen Elektronen / Tight-binding}
]{tb}
\begin{formula}{assumptions}
\desc{Assumptions}{}{}
\desc[german]{Annahmen}{}{}
\ttxt{
\eng{
\begin{itemize}
\item Atomic wave functions are localized \Rightarrow Small overlap, interaction cutoff
\end{itemize}
}
}
\end{formula}
\begin{formula}{hamiltonian}
\desc{Tight-binding Hamiltonian}{in second quantized form}{$\hat{a}_i^\dagger$, $\hat{a}_i$ \GT{creation_annihilation_ops} create/destory an electron on site $i$, $\epsilon_i$ on-site energy, $t_{i,j}$ hopping amplitude, usually $\epsilon$ and $t$ are determined from experiments or other methods}
\desc[german]{Tight-binding Hamiltonian}{in zweiter Quantisierung}{$\hat{a}_i^\dagger$, $\hat{a}_i$ \GT{creation_annihilation_ops} erzeugen/vernichten ein Elektron auf Platz $i$, $\epsilon_i$ on-site Energie, $t_{i,j}$ hopping Amplitude, meist werden $\epsilon$ und $t$ aus experimentellen Daten oder anderen Methoden bestimmt}
\eq{\hat{H} = \sum_i \epsilon_i \hat{a}_i^\dagger \hat{a}_i - \sum_{i,j} t_{i,j} \left(\hat{a}_i^\dagger \hat{a}_j + \hat{a}_j^\dagger \hat{a}_i\right)}
\end{formula}
\Subsection[
\eng{Density functional theory (DFT)}
\ger{Dichtefunktionaltheorie (DFT)}
]{dft}
\Subsubsection[
\eng{Hartree-Fock}
\ger{Hartree-Fock}
]{hf}
\begin{formula}{description}
\desc{Description}{}{}
\desc[german]{Beschreibung}{}{}
\begin{ttext}
\eng{
\begin{itemize}
\item \fqEqRef{comp:elst:mean_field} theory obeying the Pauli principle
\item Self-interaction free: Self interaction is cancelled out by the Fock-term
\end{itemize}
}
\end{ttext}
\end{formula}
\begin{formula}{equation}
\desc{Hartree-Fock equation}{}{
$\varphi_\xi$ single particle wavefunction of $\xi$th orbital,
$\hat{T}$ kinetic electron energy,
$\hat{V}_{\text{en}}$ electron-nucleus attraction,
$\hat{V}_{\text{HF}}$ \fqEqRef{comp:dft:hf:potential},
}
\desc[german]{Hartree-Fock Gleichung}{}{
$\varphi_\xi$ ein-Teilchen Wellenfunktion des $\xi$-ten Orbitals,
$\hat{T}$ kinetische Energie der Elektronen,
$\hat{V}_{\text{en}}$ Electron-Kern Anziehung,
$\hat{V}_{\text{HF}}$ \fqEqRef{comp:dft:hf:potential}
}
\eq{
\left(\hat{T} + \hat{V}_{\text{en}} + \hat{V}_{\text{HF}}^\xi\right)\varphi_\xi(x) = \epsilon_\xi \varphi_\xi(x)
}
\end{formula}
\begin{formula}{potential}
\desc{Hartree-Fock potential}{}{}
\desc[german]{Hartree Fock Potential}{}{}
\eq{
V_{\text{HF}}^\xi(\vecr) =
\sum_{\vartheta} \int \d x'
\frac{e^2}{\abs{\vecr - \vecr'}}
\left(
\underbrace{\abs{\varphi_\xi(x')}^2}_{\text{Hartree-Term}}
- \underbrace{\frac{\varphi_{\vartheta}^*(x') \varphi_{\xi}(x') \varphi_{\vartheta}(x)}{\varphi_\xi(x)}}_{\text{Fock-Term}}
\right)
}
\end{formula}
\begin{formula}{scf}
\desc{Self-consistent field cycle}{}{}
% \desc[german]{}{}{}
\ttxt{
\eng{
\begin{enumerate}
\item Initial guess for $\psi$
\item Solve SG for each particle
\item Make new guess for $\psi$
\end{enumerate}
}
}
\end{formula}
\Subsubsection[
\eng{Hohenberg-Kohn Theorems}
\ger{Hohenberg-Kohn Theoreme}
]{hk}
\begin{formula}{hk1}
\desc{Hohenberg-Kohn theorem (HK1)}{}{}
\desc[german]{Hohenberg-Kohn Theorem (HK1)}{}{}
\ttxt{
\eng{For any system of interacting electrons, the ground state electron density $n(\vecr)$ determines $\hat{V}_\text{ext}$ uniquely up to a trivial constant. }
\ger{Die Elektronendichte des Grundzustandes $n(\vecr)$ bestimmt ein einzigartiges $\hat{V}_{\text{ext}}$ eines Systems aus interagierenden Elektronen bis auf eine Konstante.}
}
\end{formula}
\begin{formula}{hk2}
\desc{Hohenberg-Kohn theorem (HK2)}{}{}
\desc[german]{Hohenberg-Kohn Theorem (HK2)}{}{}
\ttxt{
\eng{Given the energy functional $E[n(\vecr)]$, the ground state density and energy can be obtained variationally. The density that minimizes the total energy is the ecxact ground state density. }
\ger{Für ein Energiefunktional $E[n(\vecr)]$ kann die Grundzustandsdichte und Energie durch systematische Variation bestimmt werden. Die Dichte, welche die Gesamtenergie minimiert ist die exakte Grundzustandsichte. }
}
\end{formula}
\Subsubsection[
\eng{Kohn-Sham DFT}
\ger{Kohn-Sham DFT}
]{ks}
\begin{formula}{map}
\desc{Kohn-Sham map}{}{}
\desc[german]{Kohn-Sham Map}{}{}
\ttxt{
\eng{Maps fully interacting system of electrons to a system of non-interacting electrons with the same ground state density $n^\prime(\vecr) = n(\vecr)$}
}
\eq{n(\vecr) = \sum_{i=1}^N \abs{\phi_i(\vecr)}^2}
\end{formula}
\begin{formula}{functional}
\desc{Kohn-Sham functional}{}{$T_\text{KS}$ kinetic enery, $V_\text{ext}$ external potential, $E_\txH$ \hyperref[f:comp:elst:dft:hf:potential]{Hartree term}, $E_\text{XC}$ exchange correlation functional}
\desc[german]{Kohn-Sham Funktional}{}{}
\eq{E_\text{KS}[n(\vecr)] = T_\text{KS}[n(\vecr)] + V_\text{ext}[n(\vecr)] + E_\text{H}[n(\vecr)] + E_\text{XC}[n(\vecr)] }
\end{formula}
\begin{formula}{equation}
\desc{Kohn-Sham equation}{Solving it uses up a large portion of supercomputer resources}{$\phi_i^\text{KS}$ KS orbitals}
\desc[german]{Kohn-Sham Gleichung}{Die Lösung der Gleichung macht einen großen Teil der Supercomputer Ressourcen aus}{}
\begin{multline}
\biggr\{
-\frac{\hbar^2\nabla^2}{2m}
+ v_\text{ext}(\vecr)
+ e^2 \int\d^3 \vecr^\prime \frac{n(\vecr^\prime)}{\abs{\vecr-\vecr^\prime}} \\
+ \pdv{E_\txX[n(\vecr)]}{n(\vecr)}
+ \pdv{E_\txC[n(\vecr)]}{n(\vecr)}
\biggr\} \phi_i^\text{KS}(\vecr) =\\
= \epsilon_i^\text{KS} \phi_i^\text{KS}(\vecr)
\end{multline}
\end{formula}
\begin{formula}{scf}
\desc{Self-consistent field cycle for Kohn-Sham}{}{}
% \desc[german]{}{}{}
\ttxt{
\itemsep=\parsep
\eng{
\begin{enumerate}
\item Initial guess for $n(\vecr)$
\item Calculate effective potential $V_\text{eff}$
\item Solve \fqEqRef{comp:elst:dft:ks:equation}
\item Calculate density $n(\vecr)$
\item Repeat 2-4 until self consistent
\end{enumerate}
}
}
\end{formula}

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\Section[
\eng{Electronic structure theory}
% \ger{}
]{est}
\begin{formula}{kinetic_energy}
\desc{Kinetic energy}{of species $i$}{$i$ = nucleons/electrons, $N$ number of particles, $m$ \qtyRef{mass}}
\desc[german]{Kinetische Energie}{von Spezies $i$}{$i$ = Nukleonen/Elektronen, $N$ Teilchenzahl, $m$ \qtyRef{mass}}
\eq{\hat{T}_i &= -\sum_{n=1}^{N_i} \frac{\hbar^2}{2 m_i} \vec{\nabla}^2_n}
\end{formula}
\begin{formula}{potential_energy}
\desc{Electrostatic potential}{between species $i$ and $j$}{$i,j$ = nucleons/electrons, $r$ particle position, $Z_i$ charge of species $i$, \ConstRef{charge}}
\desc[german]{Elektrostatisches Potential}{zwischen Spezies $i$ und $j$}{}
\eq{\hat{V}_{i \leftrightarrow j} &= -\sum_{k,l} \frac{Z_i Z_j e^2}{\abs{\vecr_k - \vecr_l}}}
\end{formula}
\begin{formula}{hamiltonian}
\desc{Electronic structure Hamiltonian}{}{$\hat{T}$ \fqEqRef{comp:est:kinetic_energy}, $\hat{V}$ \fqEqRef{comp:est:potential_energy}, $\txe$ \GT{electrons}, $\txn$ \GT{nucleons}}
\eq{\hat{H} &= \hat{T}_\txe + \hat{T}_\txn + V_{\txe \leftrightarrow \txe} + V_{\txn \leftrightarrow \txe} + V_{\txn \leftrightarrow \txn}}
\end{formula}
\begin{formula}{mean_field}
\desc{Mean field approximation}{Replaces 2-particle operator by 1-particle operator}{Example for Coulomb interaction between many electrons}
\desc[german]{Molekularfeldnäherung}{Ersetzt 2-Teilchen Operator durch 1-Teilchen Operator}{Beispiel für Coulomb Wechselwirkung zwischen Elektronen}
\eq{
\frac{1}{2}\sum_{i\neq j} \frac{e^2}{\abs{\vec{r}_i - \vec{r}_j}} \approx \sum_{i} V_\text{eff}(\vec{r}_i)
}
\end{formula}
\Subsection[
\eng{Tight-binding}
\ger{Modell der stark gebundenen Elektronen / Tight-binding}
]{tb}
\begin{formula}{assumptions}
\desc{Assumptions}{}{}
\desc[german]{Annahmen}{}{}
\ttxt{
\eng{
\begin{itemize}
\item Atomic wave functions are localized \Rightarrow Small overlap, interaction cutoff
\end{itemize}
}
}
\end{formula}
\begin{formula}{hamiltonian}
\desc{Tight-binding Hamiltonian}{in second quantized form}{$\hat{a}_i^\dagger$, $\hat{a}_i$ \GT{creation_annihilation_ops} create/destory an electron on site $i$, $\epsilon_i$ on-site energy, $t_{i,j}$ hopping amplitude, usually $\epsilon$ and $t$ are determined from experiments or other methods}
\desc[german]{Tight-binding Hamiltonian}{in zweiter Quantisierung}{$\hat{a}_i^\dagger$, $\hat{a}_i$ \GT{creation_annihilation_ops} erzeugen/vernichten ein Elektron auf Platz $i$, $\epsilon_i$ on-site Energie, $t_{i,j}$ hopping Amplitude, meist werden $\epsilon$ und $t$ aus experimentellen Daten oder anderen Methoden bestimmt}
\eq{\hat{H} = \sum_i \epsilon_i \hat{a}_i^\dagger \hat{a}_i - \sum_{i,j} t_{i,j} \left(\hat{a}_i^\dagger \hat{a}_j + \hat{a}_j^\dagger \hat{a}_i\right)}
\end{formula}
\Subsection[
\eng{Density functional theory (DFT)}
\ger{Dichtefunktionaltheorie (DFT)}
]{dft}
\abbrLink{dft}{DFT}
\Subsubsection[
\eng{Hartree-Fock}
\ger{Hartree-Fock}
]{hf}
\begin{formula}{description}
\desc{Description}{}{}
\desc[german]{Beschreibung}{}{}
\begin{ttext}
\eng{
\begin{itemize}
\item Assumes wave functions are \fqEqRef{qm:other:slater_det} \Rightarrow Approximation
\item \fqEqRef{comp:est:mean_field} theory obeying the Pauli principle
\item Self-interaction free: Self interaction is cancelled out by the Fock-term
\end{itemize}
}
\end{ttext}
\end{formula}
\begin{formula}{equation}
\desc{Hartree-Fock equation}{}{
$\varphi_\xi$ single particle wavefunction of $\xi$th orbital,
$\hat{T}$ kinetic electron energy,
$\hat{V}_{\text{en}}$ electron-nucleus attraction,
$\hat{V}_{\text{HF}}$ \fqEqRef{comp:dft:hf:potential},
}
\desc[german]{Hartree-Fock Gleichung}{}{
$\varphi_\xi$ ein-Teilchen Wellenfunktion des $\xi$-ten Orbitals,
$\hat{T}$ kinetische Energie der Elektronen,
$\hat{V}_{\text{en}}$ Electron-Kern Anziehung,
$\hat{V}_{\text{HF}}$ \fqEqRef{comp:dft:hf:potential}
}
\eq{
\left(\hat{T} + \hat{V}_{\text{en}} + \hat{V}_{\text{HF}}^\xi\right)\varphi_\xi(x) = \epsilon_\xi \varphi_\xi(x)
}
\end{formula}
\begin{formula}{potential}
\desc{Hartree-Fock potential}{}{}
\desc[german]{Hartree Fock Potential}{}{}
\eq{
V_{\text{HF}}^\xi(\vecr) =
\sum_{\vartheta} \int \d x'
\frac{e^2}{\abs{\vecr - \vecr'}}
\left(
\underbrace{\abs{\varphi_\xi(x')}^2}_{\text{Hartree-Term}}
- \underbrace{\frac{\varphi_{\vartheta}^*(x') \varphi_{\xi}(x') \varphi_{\vartheta}(x)}{\varphi_\xi(x)}}_{\text{Fock-Term}}
\right)
}
\end{formula}
\begin{formula}{scf}
\desc{Self-consistent field cycle}{}{}
% \desc[german]{}{}{}
\ttxt{
\eng{
\begin{enumerate}
\item Initial guess for $\psi$
\item Solve SG for each particle
\item Make new guess for $\psi$
\end{enumerate}
}
}
\end{formula}
\Subsubsection[
\eng{Hohenberg-Kohn Theorems}
\ger{Hohenberg-Kohn Theoreme}
]{hk}
\begin{formula}{hk1}
\desc{Hohenberg-Kohn theorem (HK1)}{}{}
\desc[german]{Hohenberg-Kohn Theorem (HK1)}{}{}
\ttxt{
\eng{For any system of interacting electrons, the ground state electron density $n(\vecr)$ determines $\hat{V}_\text{ext}$ uniquely up to a trivial constant. }
\ger{Die Elektronendichte des Grundzustandes $n(\vecr)$ bestimmt ein einzigartiges $\hat{V}_{\text{ext}}$ eines Systems aus interagierenden Elektronen bis auf eine Konstante.}
}
\end{formula}
\begin{formula}{hk2}
\desc{Hohenberg-Kohn theorem (HK2)}{}{}
\desc[german]{Hohenberg-Kohn Theorem (HK2)}{}{}
\ttxt{
\eng{Given the energy functional $E[n(\vecr)]$, the ground state density and energy can be obtained variationally. The density that minimizes the total energy is the ecxact ground state density. }
\ger{Für ein Energiefunktional $E[n(\vecr)]$ kann die Grundzustandsdichte und Energie durch systematische Variation bestimmt werden. Die Dichte, welche die Gesamtenergie minimiert ist die exakte Grundzustandsichte. }
}
\end{formula}
\begin{formula}{density}
\desc{Ground state electron density}{}{}
\desc[german]{Grundzustandselektronendichte}{}{}
\eq{n(\vecr) = \Braket{\psi_0|\sum_{i=1}^N \delta(\vecr-\vecr_i)|\psi_0}}
\end{formula}
\Subsubsection[
\eng{Kohn-Sham DFT}
\ger{Kohn-Sham DFT}
]{ks}
\abbrLink{ksdft}{KS-DFT}
\begin{formula}{map}
\desc{Kohn-Sham map}{}{}
\desc[german]{Kohn-Sham Map}{}{}
\ttxt{
\eng{Maps fully interacting system of electrons to a system of non-interacting electrons with the same ground state density $n^\prime(\vecr) = n(\vecr)$}
}
\eq{n(\vecr) = \sum_{i=1}^N \abs{\phi_i(\vecr)}^2}
\end{formula}
\begin{formula}{functional}
\desc{Kohn-Sham functional}{}{$T_\text{KS}$ kinetic enery, $V_\text{ext}$ external potential, $E_\txH$ \hyperref[f:comp:est:dft:hf:potential]{Hartree term}, $E_\text{XC}$ \fqEqRef{comp:est:dft:xc:xc}}
\desc[german]{Kohn-Sham Funktional}{}{}
\eq{E_\text{KS}[n(\vecr)] = T_\text{KS}[n(\vecr)] + V_\text{ext}[n(\vecr)] + E_\text{H}[n(\vecr)] + E_\text{XC}[n(\vecr)] }
\end{formula}
\begin{formula}{equation}
\desc{Kohn-Sham equation}{Exact single particle \abbrRef{schroedinger_equation} (though often exact $E_\text{XC}$ is not known)\\ Solving it uses up a large portion of supercomputer resources}{$\phi_i^\text{KS}$ KS orbitals, $\int\d^3r v_\text{ext}(\vecr)n(\vecr)=V_\text{ext}[n(\vecr)]$}
\desc[german]{Kohn-Sham Gleichung}{Exakte Einteilchen-\abbrRef{schroedinger_equation} (allerdings ist das exakte $E_\text{XC}$ oft nicht bekannt)\\ Die Lösung der Gleichung macht einen großen Teil der Supercomputer Ressourcen aus}{}
\begin{multline}
\biggr\{
-\frac{\hbar^2\nabla^2}{2m}
+ v_\text{ext}(\vecr)
+ e^2 \int\d^3 \vecr^\prime \frac{n(\vecr^\prime)}{\abs{\vecr-\vecr^\prime}} \\
+ \pdv{E_\txX[n(\vecr)]}{n(\vecr)}
+ \pdv{E_\txC[n(\vecr)]}{n(\vecr)}
\biggr\} \phi_i^\text{KS}(\vecr) =\\
= \epsilon_i^\text{KS} \phi_i^\text{KS}(\vecr)
\end{multline}
\end{formula}
\begin{formula}{scf}
\desc{Self-consistent field cycle for Kohn-Sham}{}{}
% \desc[german]{}{}{}
\ttxt{
\itemsep=\parsep
\eng{
\begin{enumerate}
\item Initial guess for $n(\vecr)$
\item Calculate effective potential $V_\text{eff}$
\item Solve \fqEqRef{comp:est:dft:ks:equation}
\item Calculate density $n(\vecr)$
\item Repeat 2-4 until self consistent
\end{enumerate}
}
}
\end{formula}
\Subsubsection[
\eng{Exchange-Correlation functionals}
\ger{Exchange-Correlation Funktionale}
]{xc}
\begin{formula}{xc}
\desc{Exchange-Correlation functional}{}{}
\desc[german]{Exchange-Correlation Funktional}{}{}
\eq{ E_\text{XC}[n(\vecr)] = \Braket{\hat{T}} - T_\text{KS}[n(\vecr)] + \Braket{\hat{V}_\text{int}} - E_\txH[n(\vecr)] }
\ttxt{\eng{
Accounts for:
\begin{itemize}
\item Kinetic energy difference between interaction and non-interacting system
\item Exchange energy due to Pauli principle
\item Correlation energy due to many-body Coulomb interaction (not accounted for in mean field Hartree term $E_\txH$)
\end{itemize}
}}
\end{formula}
\begin{formula}{lda}
\desc{Local density approximation (LDA)}{Simplest DFT functionals}{$\epsilon_\txX$ calculated exchange energy from \hyperref[f:comp:qmb:models:heg]{HEG model}, $\epsilon_\txC$ correlation energy calculated with \fqSecRef{comp:qmb:methods:qmonte-carlo}}
\desc[german]{}{}{}
\abbrLabel{LDA}
\eq{E_\text{XC}^\text{LDA}[n(\vecr)] = \int \d^3r\,n(r) \Big[\epsilon_\txX[n(\vecr)] + \epsilon_\txC[n(\vecr)]\Big]}
\end{formula}
\begin{formula}{gga}
\desc{Generalized gradient approximation (GGA)}{}{$\epsilon_\txX$ calculated exchange energy from \hyperref[f:comp:qmb:models:heg]{HEG model}, $F_\text{XC}$ function containing exchange-correlation energy dependency on $n$ and $\Grad n$}
\desc[german]{}{}{}
\abbrLabel{GGA}
\eq{E_\text{XC}^\text{GGA}[n(\vecr)] = \int \d^3r\,n(r) \epsilon_\txX[n(\vecr)]\,F_\text{XC}[n(\vecr), \Grad n(\vecr)]}
\end{formula}
\TODO{PBE}
\begin{formula}{hybrid}
\desc{Hybrid functionals}{}{}
\desc[german]{Hybride Funktionale}{}{$\alpha$ mixing paramter, $E_\txX$ exchange energy, $E_\txC$ correlation energy}
\eq{\alpha E_\txX^\text{HF} + (1-\alpha) E_\txX^\text{GGA} + E_\txC^\text{GGA}}
\ttxt{\eng{
Include \hyperref[f:comp:dft:hf:potential]{Fock term} (exact exchange) in other functional, like \abbrRef{gga}. Computationally expensive
}}
\end{formula}
\begin{formula}{range-separated-hybrid}
\desc{Range separated hyrid functionals (RSH)}{Here HSE as example}{$\alpha$ mixing paramter, $E_\txX$ exchange energy, $E_\txC$ correlation energy}
% \desc[german]{}{}{}
\begin{gather}
\frac{1}{r} = \frac{\erf(\omega r)}{r} + \frac{\erfc{\omega r}}{r} \\
E_\text{XC}^\text{HSE} = \alpha E_\text{X,SR}^\text{HF}(\omega) + (1-\alpha)E_\text{X,SR}^\text{GGA}(\omega) + E_\text{X,LR}^\text{GGA}(\omega) + E_\txC^\text{GGA}
\end{gather}
\separateEntries
\ttxt{\eng{
Use \abbrRef{gga} and \hyperref[comp:est:dft:hf:potential]{Fock} exchange for short ranges (SR) and only \abbrRef{GGA} for long ranges (LR).
\abbrRef{GGA} correlation is always used. Useful when dielectric screening reduces long range interactions, saves computational cost.
}}
\end{formula}
\begin{formula}{comparison}
\desc{Comparison of DFT functionals}{}{}
\desc[german]{Vergleich von DFT Funktionalen}{}{}
\begin{tabular}{l|c}
\hyperref[f:comp:est:dft:hf:potential]{Hartree-Fock} & only exchange, no correlation \Rightarrow upper bound of GS energy \\
\abbrRef{lda} & understimates e repulsion \Rightarrow Overbinding \\
\abbrRef{gga} & underestimate band gap \\
hybrid & underestimate band gap
\end{tabular}
\end{formula}
\Subsubsection[
\eng{Basis sets}
\ger{Basis-Sets}
]{basis}
\begin{formula}{plane_wave}
\desc{Plane wave basis}{Plane wave ansatz in \fqEqRef{comp:est:dft:ks:equation}\\Good for periodic structures, allows computation parallelization over a sample points in the \abbrRef{brillouin_zone}}{}
\desc[german]{Ebene Wellen als Basis}{}{}
\eq{\sum_{\vecG^\prime} \left[\frac{\hbar^2 \abs{\vecG+\veck}^2}{2m} \delta_{\vecG,\vecG^\prime} + V_\text{eff}(\vecG-\vecG^\prime)\right] c_{i,\veck,\vecG^\prime} = \epsilon_{i,\veck} c_{i,\veck,\vecG}}
\end{formula}
\begin{formula}{plane_wave_cutoff}
\desc{Plane wave cutoff}{Number of plane waves included in the calculation must be finite}{}
% \desc[german]{}{}{}
\eq{E_\text{cutoff} = \frac{\hbar^2 \abs{\veck+\vecG}^2}{2m}}
\end{formula}
\Subsubsection[
\eng{Pseudo-Potential method}
\ger{Pseudopotentialmethode}
]{pseudo}
\begin{formula}{ansatz}
\desc{Ansatz}{}{}
\desc[german]{Ansatz}{}{}
\ttxt{\eng{
Core electrons are absorbed into the potential since they do not contribute much to interesting properties.
}}
\end{formula}

2
src/comp/ml.tex
View File

@ -80,5 +80,5 @@
\eng{Gradient descent}
\ger{Gradientenverfahren}
]{gd}
\TODO{TODO}
\TODO{in lecture 30 CMP}

22
src/comp/qmb.tex
View File

@ -2,6 +2,28 @@
\eng{Quantum many-body physics}
\ger{Quanten-Vielteilchenphysik}
]{qmb}
\Subsection[
\eng{Quantum many-body models}
\ger{Quanten-Vielteilchenmodelle}
]{models}
\begin{formula}{heg}
\desc{Homogeneous electron gas (HEG)}{Also "Jellium"}{}
\desc[german]{}{}{}
\ttxt{
\eng{Both positive (nucleus) and negative (electron) charges are distributed uniformly.}
}
\end{formula}
\Subsection[
\eng{Methods}
\ger{Methoden}
]{methods}
\Subsubsection[
\eng{Quantum Monte-Carlo}
\ger{Quantum Monte-Carlo}
]{qmonte-carlo}
\TODO{TODO}
\Subsection[
\eng{Importance sampling}

2
src/constants.tex
View File

@ -40,7 +40,7 @@
\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}{}
\constant{F}{def}{
\val{9.64853321233100184}{\coulomb\per\mol}
\val{9.64853321233100184\xE{4}}{\coulomb\per\mol}
\val{\NA\,e}{}
}
\end{formula}

34
src/ed/misc.tex
View File

@ -98,19 +98,7 @@
\TODO{sort}
\begin{formula}{impedance_c}
\desc{Impedance of a capacitor}{}{}
\desc[german]{Impedanz eines Kondesnators}{}{}
\eq{Z_{C} = \frac{1}{i\omega C}}
\end{formula}
\begin{formula}{impedance_l}
\desc{Impedance of an inductor}{}{}
\desc[german]{Impedanz eines Induktors}{}{}
\eq{Z_{L} = i\omega L}
\end{formula}
\TODO{impedance addition for parallel / linear}
\Section[
\eng{Dipole-stuff}
@ -129,3 +117,25 @@
\eq{P = \frac{\mu_0\omega^4 p_0^2}{12\pi c}}
\end{formula}
\Section[
\eng{misc}
\ger{misc}
]{misc}
\begin{formula}{impedance_r}
\desc{Impedance of an ohmic resistor}{}{\QtyRef{resistance}}
\desc[german]{Impedanz eines Ohmschen Widerstands}{}{}
\eq{Z_{R} = R}
\end{formula}
\begin{formula}{impedance_c}
\desc{Impedance of a capacitor}{}{\QtyRef{capacity}, \QtyRef{angular_velocity}}
\desc[german]{Impedanz eines Kondensators}{}{}
\eq{Z_{C} = \frac{1}{\I\omega C}}
\end{formula}
\begin{formula}{impedance_l}
\desc{Impedance of an inductor}{}{\QtyRef{inductance}, \QtyRef{angular_velocity}}
\desc[german]{Impedanz eines Induktors}{}{}
\eq{Z_{L} = \I\omega L}
\end{formula}
\TODO{impedance addition for parallel / linear}

127
src/main.tex
View File

@ -25,7 +25,7 @@
\setlist{noitemsep} % no vertical space between items
\setlist[1]{labelindent=\parindent} % < Usually a good idea
\setlist[itemize]{leftmargin=*}
\setlist[enumerate]{labelsep=*, leftmargin=1.5pc} % horizontal indent of items
% \setlist[enumerate]{labelsep=*, leftmargin=1.5pc} % horizontal indent of items
\usepackage{titlesec} % colored titles
\usepackage{array} % more array options
@ -37,11 +37,14 @@
\input{util/colorscheme.tex}
\input{util/colors.tex} % after colorscheme
% GRAPHICS
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}
\usepackage{tikz} % drawings
\usetikzlibrary{decorations.pathmorphing}
\usetikzlibrary{decorations.pathreplacing} % braces
\usetikzlibrary{calc}
\usetikzlibrary{patterns}
\usetikzlibrary{patterns}
\input{util/tikz_macros}
% speed up compilation by externalizing figures
% \usetikzlibrary{external}
@ -90,113 +93,6 @@
\newcommand{\TODO}[1]{{\color{fg-red}TODO:#1}}
\newcommand{\ts}{\textsuperscript}
\newcommand\printFqName{\expandafter\detokenize\expandafter{\fqname}}
% "automate" sectioning
% start <section>, get heading from translation, set label
% fqname is the fully qualified name: the keys of all previous sections joined with a ':'
% [1]: code to run after setting \fqname, but before the \part, \section etc
% 2: key
\newcommand{\Part}[2][desc]{
\newpage
\def\partName{#2}
\def\sectionName{}
\def\subsectionName{}
\def\subsubsectionName{}
\edef\fqname{\partName}
#1
\edef\fqnameText{\expandafter\GetTranslation\expandafter{\fqname}}
\part{\fqnameText}
\label{sec:\fqname}
}
\newcommand{\Section}[2][]{
\def\sectionName{#2}
\def\subsectionName{}
\def\subsubsectionName{}
\edef\fqname{\partName:\sectionName}
#1
% this is necessary so that \section takes the fully expanded string. Otherwise the pdf toc will have just the fqname
\edef\fqnameText{\expandafter\GetTranslation\expandafter{\fqname}}
\section{\fqnameText}
\label{sec:\fqname}
}
% \newcommand{\Subsection}[1]{\Subsection{#1}{}}
\newcommand{\Subsection}[2][]{
\def\subsectionName{#2}
\def\subsubsectionName{}
\edef\fqname{\partName:\sectionName:\subsectionName}
#1
\edef\fqnameText{\expandafter\GetTranslation\expandafter{\fqname}}
\subsection{\fqnameText}
\label{sec:\fqname}
}
\newcommand{\Subsubsection}[2][]{
\def\subsubsectionName{#2}
\edef\fqname{\partName:\sectionName:\subsectionName:\subsubsectionName}
#1
\edef\fqnameText{\expandafter\GetTranslation\expandafter{\fqname}}
\subsubsection{\fqnameText}
\label{sec:\fqname}
}
\edef\fqname{NULL}
\newcommand\luaDoubleFieldValue[3]{%
\directlua{
if #1 \string~= nil and #1[#2] \string~= nil and #1[#2][#3] \string~= nil then
tex.sprint(#1[#2][#3])
return
end
luatexbase.module_warning('luaDoubleFieldValue', 'Invalid indices to `#1`: `#2` and `#3`');
tex.sprint("???")
}%
}
% 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]{%
\newrobustcmd{\fqEqRef}[1]{%
% \edef\fqeqrefname{\GT{#1}}
% \hyperref[eq:#1]{\fqeqrefname}
\hyperref[f:#1]{\GT{#1}}%
}
% Section
% <name>
\newrobustcmd{\fqSecRef}[1]{%
\hyperref[sec:#1]{\GT{#1}}%
}
% Quantities
% <symbol>
\newrobustcmd{\qtyRef}[1]{%
\edef\tempname{\luaDoubleFieldValue{quantities}{"#1"}{"fqname"}}%
\hyperref[qty:#1]{\expandafter\GT\expandafter{\tempname:#1}}%
}
% <symbol> <name>
\newrobustcmd{\QtyRef}[1]{%
$\luaDoubleFieldValue{quantities}{"#1"}{"symbol"}$ \qtyRef{#1}%
}
% Constants
% <name>
\newrobustcmd{\constRef}[1]{%
\edef\tempname{\luaDoubleFieldValue{constants}{"#1"}{"fqname"}}%
\hyperref[const:#1]{\expandafter\GT\expandafter{\tempname:#1}}%
}
% <symbol> <name>
\newrobustcmd{\ConstRef}[1]{%
$\luaDoubleFieldValue{constants}{"#1"}{"symbol"}$ \constRef{#1}%
}
% Element from periodic table
% <symbol>
\newrobustcmd{\elRef}[1]{%
\hyperref[el:#1]{{\color{fg0}#1}}%
}
% <name>
\newrobustcmd{\ElRef}[1]{%
\hyperref[el:#1]{\GT{el:#1}}%
}
% \usepackage{xstring}
@ -218,6 +114,7 @@
\immediate\write\luaAuxFile{\noexpand\directlua{\detokenize{#1}}}
\directlua{#1}
}
% read
\IfFileExists{\jobname.lua.aux}{%
\input{\jobname.lua.aux}%
@ -240,6 +137,8 @@
}
\AtEndDocument{\immediate\closeout\labelsFile}
\input{util/fqname.tex}
\input{circuit.tex}
\input{util/macros.tex}
\input{util/environments.tex} % requires util/translation.tex to be loaded first
@ -284,7 +183,7 @@
\input{util/translations.tex}
% \InputOnly{ch}
% \InputOnly{comp}
\Input{math/math}
\Input{math/linalg}
@ -314,6 +213,7 @@
\Input{cm/misc}
\Input{cm/techniques}
\Input{cm/topo}
\Input{cm/mat}
\Input{particle}
@ -322,18 +222,25 @@
\Input{comp/comp}
\Input{comp/qmb}
\Input{comp/elsth}
\Input{comp/est}
\Input{comp/ad}
\Input{comp/ml}
\Input{ch/periodic_table} % only definitions
\Input{ch/ch}
\Input{ch/el}
\Input{ch/misc}
\newpage
\Part[
\eng{Appendix}
\ger{Anhang}
]{appendix}
\begin{formula}{world}
\desc{World formula}{}{}
\desc[german]{Weltformel}{}{}
\eq{E = mc^2 +\text{AI}}
\end{formula}
\Input{quantities}
\Input{constants}

4
src/math/calculus.tex
View File

@ -20,7 +20,7 @@
\eng{Fourier series}
\ger{Fourierreihe}
]{series}
\begin{formula}{series}
\begin{formula}{series} \absLabel[fourier_series]
\desc{Fourier series}{Complex representation}{$f\in \Lebesgue^2(\R,\C)$ $T$-\GT{periodic}}
\desc[german]{Fourierreihe}{Komplexe Darstellung}{}
\eq{f(t) = \sum_{k=-\infty}^{\infty} c_k \Exp{\frac{2\pi \I kt}{T}}}
@ -58,7 +58,7 @@
\eng{Fourier transformation}
\ger{Fouriertransformation}
]{trafo}
\begin{formula}{transform}
\begin{formula}{transform} \absLabel[fourier_transform]
\desc{Fourier transform}{}{$\hat{f}:\R^n \mapsto \C$, $\forall f\in L^1(\R^n)$}
\desc[german]{Fouriertransformierte}{}{}
\eq{\hat{f}(k) \coloneq \frac{1}{\sqrt{2\pi}^n} \int_{\R^n} \e^{-\I kx}f(x)\d x}

8
src/math/probability_theory.tex
View File

@ -54,10 +54,10 @@
\eq{p_X(x) = P(X = x)}
\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)}}
\begin{formula}{autocorrelation} \absLabel
\desc{Autocorrelation}{Correlation of $f$ to itself at an earlier point in time, $C$ is a covariance function}{$\tau$ lag-time}
\desc[german]{Autokorrelation}{Korrelation vonn $f$ zu sich selbst zu einem früheren Zeitpunkt. $C$ ist auch die Kovarianzfunktion}{$\tau$ Zeitverschiebung}
\eq{C_A(\tau) &= \lim_{T \to \infty} \frac{1}{2T}\int_{-T}^{T} f(t+\tau) f(t) \d t) \\ &= \braket{f(t+\tau)\cdot f(t)}}
\end{formula}
\begin{formula}{binomial_coefficient}

2
src/mechanics.tex
View File

@ -66,7 +66,7 @@
Zum Beispiel findet man für ein 2D Pendel die generalisierte Koordinate $q=\varphi$, mit $\vec{x} = \begin{pmatrix} \cos\varphi \\ \sin\varphi \end{pmatrix}$.
}
\end{ttext}
\begin{formula}{lagrangian}
\begin{formula}{lagrangian} \absLabel
\desc{Lagrange function}{}{$T$ kinetic energy, $V$ potential energy }
\desc[german]{Lagrange-Funktion}{}{$T$ kinetische Energie, $V$ potentielle Energie}
\eq{\lagrange = T - V}

15
src/particle.tex
View File

@ -11,6 +11,17 @@
}
\end{formula}
\begin{formula}{spin}
\desc{Spin}{}{}
\desc[german]{Spin}{}{}
\quantity{\sigma}{}{v}
\end{formula}
\begin{bigformula}{standard_model}
\desc{Standard model}{}{}
\desc[german]{Standartmodell}{}{}
\centering
\tikzset{%
label/.style = { black, midway, align=center },
toplabel/.style = { label, above=.5em, anchor=south },
@ -82,7 +93,7 @@
\draw [->] (-0.7, 0.35) node [legend] {\qtyRef{mass}} -- (-0.5, 0.35);
\draw [->] (-0.7, 0.20) node [legend] {\qtyRef{spin}} -- (-0.5, 0.20);
\draw [->] (-0.7, 0.05) node [legend] {\qtyRef{charge}} -- (-0.5, 0.05);
\draw [->] (-0.7,-0.10) node [legend] {\qtyRef{colors}} -- (-0.5,-0.10);
\draw [->] (-0.7,-0.10) node [legend] {\GT{colors}} -- (-0.5,-0.10);
\draw [brace,draw=\colorQuarks] (-0.55, 0.5) -- (-0.55,-1.5) node[leftlabel,color=\colorQuarks] {\gt{quarks}};
\draw [brace,draw=\colorLepton] (-0.55,-1.5) -- (-0.55,-3.5) node[leftlabel,color=\colorLepton] {\gt{leptons}};
@ -99,3 +110,5 @@
\node at (2,0.85) [generation] {\small III};
\node at (1,1.05) [generation] {\small generation};
\end{tikzpicture}
\end{bigformula}

42
src/qm/qm.tex
View File

@ -206,9 +206,18 @@
\begin{formula}{schroedinger_equation}
\desc{Schrödinger equation}{}{}
\desc[german]{Schrödingergleichung}{}{}
\abbrLabel{SE}
\eq{i\hbar\frac{\partial}{\partial t}\psi(x, t) = (- \frac{\hbar^2}{2m} \vec{\nabla}^2 + \vec{V}(x)) \psi(x)}
\end{formula}
\begin{formula}{hellmann_feynmann} \absLabel
\desc{Hellmann-Feynman-Theorem}{Derivative of the energy to a parameter}{}
\desc[german]{Hellmann-Feynman-Theorem}{Abletiung der Energie nach einem Parameter}{}
\eq{
\odv{E_\lambda}{\lambda} = \int \d^3r \psi^*_\lambda \odv{\hat{H}_\lambda}{\lambda} \psi_\lambda = \Braket{\psi(\lambda)|\odv{\hat{H}_{\lambda}}{\lambda}|\psi(\lambda)}
}
\end{formula}
\Subsection[
\eng{Time evolution}
\ger{Zeitentwicklug}
@ -232,13 +241,6 @@
\eq{\dot{\rho} = \underbrace{-\frac{i}{\hbar} [\hat{H}, \rho]}_\text{reversible} + \underbrace{\sum_{n.m} h_{nm} \left(\hat{A}_n\rho \hat{A}_{m^\dagger} - \frac{1}{2}\left\{\hat{A}_m^\dagger \hat{A}_n,\rho \right\}\right)}_\text{irreversible}}
\end{formula}
\begin{formula}{hellmann_feynmann}
\desc{Hellmann-Feynman-Theorem}{Derivative of the energy to a parameter}{}
\desc[german]{Hellmann-Feynman-Theorem}{Abletiung der Energie nach einem Parameter}{}
\eq{
\odv{E_\lambda}{\lambda} = \int \d^3r \psi^*_\lambda \odv{\hat{H}_\lambda}{\lambda} \psi_\lambda = \Braket{\psi(\lambda)|\odv{\hat{H}_{\lambda}}{\lambda}|\psi(\lambda)}
}
\end{formula}
\TODO{unitary transformation of time dependent H}
@ -292,14 +294,15 @@
\end{formula}
% \eq{Time evolution}{\hat{H}\ket{\psi} = E\ket{\psi}}{sg_time}
\Subsection[
\ger{Korrespondenzprinzip}
\eng{Correspondence principle}
]{correspondence_principle}
\begin{ttext}[desc]
% TODO: wo gehört das hin?
\begin{formula}{correspondence_principle}
\desc{Correspondence principle}{}{}
\desc[german]{Korrespondenzprinzip}{}{}
\ttxt{
\ger{Die klassischen Bewegungsgleichungen lassen sich als Grenzfall (große Quantenzahlen) aus der Quantenmechanik ableiten.}
\eng{The classical mechanics can be derived from quantum mechanics in the limit of large quantum numbers.}
\end{ttext}
}
\end{formula}
@ -308,8 +311,8 @@
\ger{Störungstheorie}
]{qm_pertubation}
\begin{ttext}
\eng[desc]{The following holds true if the pertubation $\hat{H_1}$ is sufficently small and the $E^{(0)}_n$ levels are not degenerate.}
\ger[desc]{Die folgenden Gleichungen gelten wenn $\hat{H_1}$ ausreichend klein ist und die $E_n^{(0)}$ Niveaus nicht entartet sind.}
\eng{The following holds true if the pertubation $\hat{H_1}$ is sufficently small and the $E^{(0)}_n$ levels are not degenerate.}
\ger{Die folgenden Gleichungen gelten wenn $\hat{H_1}$ ausreichend klein ist und die $E_n^{(0)}$ Niveaus nicht entartet sind.}
\end{ttext}
\begin{formula}{pertubation_hamiltonian}
\desc{Hamiltonian}{}{}
@ -566,6 +569,15 @@
\eq{\Delta\omega \coloneq \abs{\omega_0 - \omega_\text{L}} \ll \abs{\omega_0 + \omega_\text{L}} \approx 2\omega_0}
\end{formula}
\begin{formula}{adiabatic_theorem} \absLabel
\desc{Adiabatic theorem}{}{}
\desc[german]{Adiabatentheorem}{}{}
\ttxt{
\eng{A physical system remains in its instantaneous eigenstate if a given perturbation is acting on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian's spectrum.}
\ger{Ein quantenmechanisches System bleibt in im derzeitigen Eigenzustand falls eine Störung langsam genug wirkt und der Eigenwert durch eine Lücke vom Rest des Spektrums getrennt ist.}
}
\end{formula}
\begin{formula}{slater_det}
\desc{Slater determinant}{Construction of a fermionic (antisymmetric) many-particle wave function from single-particle wave functions}{}
\desc[german]{Slater Determinante}{Konstruktion einer fermionischen (antisymmetrischen) Vielteilchen Wellenfunktion aus ein-Teilchen Wellenfunktionen}{}

2
src/quantities.tex
View File

@ -124,7 +124,7 @@
\end{formula}
\begin{formula}{angular_frequency}
\desc{Angular frequency}{}{\QtyRef{time_period}, \QtyRef{frequency}}
\desc[german]{Winkelgeschwindigkeit}{}{}
\desc[german]{Kreisfrequenz}{}{}
\quantity{\omega}{\radian\per\s}{s}
\eq{\omega = \frac{2\pi/T}{2\pi f}}
\end{formula}

2
src/quantum_computing.tex
View File

@ -433,7 +433,7 @@
\begin{formula}{rabi_oscillation}
\desc{Rabi oscillations}{}{$\omega_{21}$ resonance frequency of the energy transition, $\Omega$ Rabi frequency}
\desc[german]{Rabi-Oszillationen}{}{$\omega_{21}$ Resonanzfrequenz des Energieübergangs, $\Omega$ Rabi-Frequenz}
\eq{\Omega_ TODO}
\eq{\Omega_ \text{\TODO{TODO}}}
\end{formula}
\Subsection[

84
src/statistical_mechanics.tex
View File

@ -346,24 +346,76 @@
\eng{Ensembles}
\ger{Ensembles}
]{ensembles}
\Eng[const_variables]{Constant variables}
\Ger[const_variables]{Konstante Variablen}
\begin{bigformula}{nve} \absLabel[mc_ensemble]
\desc{Microcanonical ensemble}{}{}
\desc[german]{Mikrokanonisches Ensemble}{}{}
\begin{minipagetable}{nve}
\entry{const_variables} {$E$, $V,$ $N$ }
\entry{partition_sum} {$\Omega = \sum_n 1$ }
\entry{probability} {$p_n = \frac{1}{\Omega}$}
\entry{td_pot} {$S = \kB\ln\Omega$ }
\entry{pressure} {$p = T \pdv{S}{V}_{E,N}$}
\entry{entropy} {$S = \kB = \ln\Omega$ }
\end{minipagetable}
\end{bigformula}
\begin{bigformula}{nvt} \absLabel[c_ensemble]
\desc{Canonical ensemble}{}{}
\desc[german]{Kanonisches Ensemble}{}{}
\begin{minipagetable}{nvt}
\entry{const_variables} {$T$, $V$, $N$ }
\entry{partition_sum} {$Z = \sum_n \e^{-\beta E_n}$ }
\entry{probability} {$p_n = \frac{\e^{-\beta E_n}}{Z}$}
\entry{td_pot} {$F = - \kB T \ln Z$ }
\entry{pressure} {$p = -\pdv{F}{V}_{T,N}$ }
\entry{entropy} {$S = -\pdv{F}{T}_{V,N}$ }
\end{minipagetable}
\end{bigformula}
\begin{bigformula}{mvt} \absLabel[gc_ensemble]
\desc{Grand canonical ensemble}{}{}
\desc[german]{Grosskanonisches Ensemble}{}{}
\begin{minipagetable}{mvt}
\entry{const_variables} {$T$, $V$, $\mu$ }
\entry{partition_sum} {$Z_\text{g} = \sum_{n} \e^{-\beta(E_n - \mu N_n)}$ }
\entry{probability} {$p_n = \frac{\e^{-\beta (E_n - \mu N_n}}{Z_\text{g}}$}
\entry{td_pot} {$ \Phi = - \kB T \ln Z$ }
\entry{pressure} {$p = -\pdv{\Phi}{V}_{T,\mu} = -\frac{\Phi}{V}$ }
\entry{entropy} {$S = -\pdv{\Phi}{T}_{V,\mu}$ }
\end{minipagetable}
\end{bigformula}
\begin{bigformula}{npt}
\desc{Isobaric-isothermal}{Gibbs ensemble}{}
% \desc[german]{Kanonisches Ensemble}{}{}
\begin{minipagetable}{npt}
\entry{const_variables} {$N$, $p$, $T$}
\entry{partition_sum} {}
\entry{probability} {$p_n ? \frac{\e^{-\beta(E_n + pV_n)}}{Z}$}
\entry{td_pot} {}
\entry{pressure} {}
\entry{entropy} {}
\end{minipagetable}
\end{bigformula}
\begin{bigformula}{nph}
\desc{Isonthalpic-isobaric ensemble}{Enthalpy ensemble}{}
% \desc[german]{Kanonisches Ensemble}{}{}
\begin{minipagetable}{nph}
\entry{const_variables} {}
\entry{partition_sum} {}
\entry{probability} {}
\entry{td_pot} {}
\entry{pressure} {}
\entry{entropy} {}
\end{minipagetable}
\end{bigformula}
\TODO{complete, link potentials}
\begin{table}[H]
\centering
\caption{caption}
\label{tab:\fqname}
\begin{tabular}{l|c|c|c}
& \gt{mk} & \gt{k} & \gt{gk} \\ \hline
\GT{variables} & $E$, $V,$ $N$ & $T$, $V$, $N$ & $T$, $V$, $\mu$ \\ \hline
\GT{partition_sum} & $\Omega = \sum_n 1$ & $Z = \sum_n \e^{-\beta E_n}$ & $Z_\text{g} = \sum_{n} \e^{-\beta(E_n - \mu N_n)}$ \\ \hline
\GT{probability} & $p_n = \frac{1}{\Omega}$ & $p_n = \frac{\e^{-\beta E_n}}{Z}$ & $p_n = \frac{\e^{-\beta (E_n - \mu N_n}}{Z_\text{g}}$ \\ \hline
\GT{td_pot} & $S = \kB\ln\Omega$ & $F = - \kB T \ln Z$ & $ \Phi = - \kB T \ln Z$ \\ \hline
\GT{pressure} & $p = T \pdv{S}{V}_{E,N}$ &$p = -\pdv{F}{V}_{T,N}$ & $p = -\pdv{\Phi}{V}_{T,\mu} = -\frac{\Phi}{V}$ \\ \hline
\GT{entropy} & $S = \kB = \ln\Omega$ & $S = -\pdv{F}{T}_{V,N}$ & $S = -\pdv{\Phi}{T}_{V,\mu}$ \\ \hline
\end{tabular}
\end{table}
\begin{formula}{ergodic_hypo}
\desc{Ergodic hypothesis}{Over a long periode of time, all accessible microstates in the phase space are equiprobable}{$A$ Observable}
@ -560,7 +612,7 @@
% b - \frac{a}{\kB T}}
\end{formula}
\begin{formula}{lennard_jones}
\begin{formula}{lennard_jones} \absLabel
\desc{Lennard-Jones potential}{Potential between two molecules. Attractive for $r > \sigma$, repulsive for $r < \sigma$.\\ In condensed matter: Attraction due to Landau Dispersion \TODO{verify} and repulsion due to Pauli exclusion principle.}{}
\desc[german]{Lennard-Jones-Potential}{Potential zwischen zwei Molekülen. Attraktiv für $r > \sigma$, repulsiv für $r < \sigma$.\\ In Festkörpern: Anziehung durch Landau-Dispersion und Abstoßung durch Pauli-Prinzip.}{}
\fig[0.7]{img/potential_lennard_jones.pdf}

54
src/util/colorscheme.tex
View File

@ -1,28 +1,28 @@
% This file was generated by scripts/formulasheet.py
% This file was generated by scripts/formulary.py
% Do not edit it directly, changes will be overwritten
\definecolor{fg0}{HTML}{f9f5d7}
\definecolor{bg0}{HTML}{1d2021}
\definecolor{fg1}{HTML}{ebdbb2}
\definecolor{fg2}{HTML}{d5c4a1}
\definecolor{fg3}{HTML}{bdae93}
\definecolor{fg4}{HTML}{a89984}
\definecolor{bg1}{HTML}{3c3836}
\definecolor{bg2}{HTML}{504945}
\definecolor{bg3}{HTML}{665c54}
\definecolor{bg4}{HTML}{7c6f64}
\definecolor{fg-red}{HTML}{fb4934}
\definecolor{fg-orange}{HTML}{f38019}
\definecolor{fg-yellow}{HTML}{fabd2f}
\definecolor{fg-green}{HTML}{b8bb26}
\definecolor{fg-aqua}{HTML}{8ec07c}
\definecolor{fg-blue}{HTML}{83a598}
\definecolor{fg-purple}{HTML}{d3869b}
\definecolor{fg-gray}{HTML}{a89984}
\definecolor{bg-red}{HTML}{cc241d}
\definecolor{bg-orange}{HTML}{d65d0e}
\definecolor{bg-yellow}{HTML}{d79921}
\definecolor{bg-green}{HTML}{98971a}
\definecolor{bg-aqua}{HTML}{689d6a}
\definecolor{bg-blue}{HTML}{458588}
\definecolor{bg-purple}{HTML}{b16286}
\definecolor{bg-gray}{HTML}{928374}
\definecolor{fg0}{HTML}{1d2021}
\definecolor{bg0}{HTML}{f9f5d7}
\definecolor{fg1}{HTML}{3c3836}
\definecolor{fg2}{HTML}{504945}
\definecolor{fg3}{HTML}{665c54}
\definecolor{fg4}{HTML}{7c6f64}
\definecolor{bg1}{HTML}{ebdbb2}
\definecolor{bg2}{HTML}{d5c4a1}
\definecolor{bg3}{HTML}{bdae93}
\definecolor{bg4}{HTML}{a89984}
\definecolor{fg-red}{HTML}{9d0006}
\definecolor{fg-orange}{HTML}{af3a03}
\definecolor{fg-yellow}{HTML}{b57614}
\definecolor{fg-green}{HTML}{79740e}
\definecolor{fg-aqua}{HTML}{427b58}
\definecolor{fg-blue}{HTML}{076678}
\definecolor{fg-purple}{HTML}{8f3f71}
\definecolor{fg-gray}{HTML}{7c6f64}
\definecolor{bg-red}{HTML}{fb4934}
\definecolor{bg-orange}{HTML}{f38019}
\definecolor{bg-yellow}{HTML}{fabd2f}
\definecolor{bg-green}{HTML}{b8bb26}
\definecolor{bg-aqua}{HTML}{8ec07c}
\definecolor{bg-blue}{HTML}{83a598}
\definecolor{bg-purple}{HTML}{d3869b}
\definecolor{bg-gray}{HTML}{a89984}

40
src/util/environments.tex
View File

@ -83,6 +83,19 @@
\ifblank{##4}{}{\dt[#1_defs]{##1}{##4}}
}
\directlua{n_formulaEntries = 0}
% makes this formula referencable with \abbrRef{<name>}
% [1]: label to use
% 2: Abbreviation to use for references
\newcommand{\abbrLabel}[2][#1]{
\abbrLink[f:\fqname]{##1}{##2}
}
% makes this formula referencable with \absRef{<name>}
% [1]: label to use
\newcommand{\absLabel}[1][#1]{
\absLink[f:\fqname]{##1}
}
\newcommand{\newFormulaEntry}{
\directlua{
if n_formulaEntries > 0 then
@ -229,11 +242,13 @@
\par\noindent\ignorespaces
% \textcolor{gray}{\hrule}
% \vspace{0.5\baselineskip}
\IfTranslationExists{\fqname:#1}{%
\raggedright
\GT{\fqname:#1}
\textbf{
\IfTranslationExists{\fqname:#1}{%
\raggedright
\GT{\fqname:#1}
}{\detokenize{#1}}
\IfTranslationExists{\fqname:#1_desc}{
}
\IfTranslationExists{\fqname:#1_desc}{
: {\color{fg1} \GT{\fqname:#1_desc}}
}{}
\hfill
@ -414,15 +429,24 @@
entries = {}
}
% Normal entry
% 1: field name (translation key)
% 2: entry text
\newcommand{\entry}[2]{
\directlua{
table.insert(entries, {key = "\luaescapestring{##1}", value = [[\detokenize{##2}]]})
}
}
% Translation entry
% 1: field name (translation key)
% 2: translation define statements (field content)
\newcommand{\entry}[2]{
\newcommand{\tentry}[2]{
% temporarily set fqname so that the translation commands dont need an explicit key
\edef\fqname{\tmpFqname:#2:##1}
##2
\edef\fqname{\tmpFqname}
\directlua{
table.insert(entries, "\luaescapestring{##1}")
table.insert(entries, {key = "\luaescapestring{##1}", value = "\\gt{" .. table_name .. ":\luaescapestring{##1}}"})
}
}
}{
@ -436,8 +460,8 @@
\begin{tabularx}{\textwidth}{|l|X|}
\hline
\directlua{
for _, k in ipairs(entries) do
tex.print("\\GT{" .. k .. "} & \\gt{"..table_name..":"..k .."}\\\\")
for _, kv in ipairs(entries) do
tex.print("\\GT{" .. kv.key .. "} & " .. kv.value .. "\\\\")
end
}
\hline

180
src/util/fqname.tex Normal file
View File

@ -0,0 +1,180 @@
% Everything related to referencing stuff
\newcommand\printFqName{\expandafter\detokenize\expandafter{\fqname}}
% SECTIONING
% start <section>, get heading from translation, set label
% secFqname is the fully qualified name of sections: the keys of all previous sections joined with a ':'
% fqname is secFqname:<key> where <key> is the key/id of some environment, like formula
% [1]: code to run after setting \fqname, but before the \part, \section etc
% 2: key
\newcommand{\Part}[2][desc]{
\newpage
\def\partName{#2}
\def\sectionName{}
\def\subsectionName{}
\def\subsubsectionName{}
\edef\fqname{\partName}
\edef\secFqname{\fqname}
#1
\edef\fqnameText{\expandafter\GetTranslation\expandafter{\fqname}}
\part{\fqnameText}
\label{sec:\fqname}
}
\newcommand{\Section}[2][]{
\def\sectionName{#2}
\def\subsectionName{}
\def\subsubsectionName{}
\edef\fqname{\partName:\sectionName}
\edef\secFqname{\fqname}
#1
% this is necessary so that \section takes the fully expanded string. Otherwise the pdf toc will have just the fqname
\edef\fqnameText{\expandafter\GetTranslation\expandafter{\fqname}}
\section{\fqnameText}
\label{sec:\fqname}
}
% \newcommand{\Subsection}[1]{\Subsection{#1}{}}
\newcommand{\Subsection}[2][]{
\def\subsectionName{#2}
\def\subsubsectionName{}
\edef\fqname{\partName:\sectionName:\subsectionName}
\edef\secFqname{\fqname}
#1
\edef\fqnameText{\expandafter\GetTranslation\expandafter{\fqname}}
\subsection{\fqnameText}
\label{sec:\fqname}
}
\newcommand{\Subsubsection}[2][]{
\def\subsubsectionName{#2}
\edef\fqname{\partName:\sectionName:\subsectionName:\subsubsectionName}
\edef\secFqname{\fqname}
#1
\edef\fqnameText{\expandafter\GetTranslation\expandafter{\fqname}}
\subsubsection{\fqnameText}
\label{sec:\fqname}
}
\edef\fqname{NULL}
\newcommand\luaDoubleFieldValue[3]{%
\directlua{
if #1 \string~= nil and #1[#2] \string~= nil and #1[#2][#3] \string~= nil then
tex.sprint(#1[#2][#3])
return
end
luatexbase.module_warning('luaDoubleFieldValue', 'Invalid indices to `#1`: `#2` and `#3`');
tex.sprint("???")
}%
}
% 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
% \newrobustcmd{\fqEqRef}[1]{%
\newrobustcmd{\fqEqRef}[1]{%
% \edef\fqeqrefname{\GT{#1}}
% \hyperref[eq:#1]{\fqeqrefname}
\hyperref[f:#1]{\GT{#1}}%
}
% Formula in the current section
\newrobustcmd{\secEqRef}[1]{%
% \edef\fqeqrefname{\GT{#1}}
% \hyperref[eq:#1]{\fqeqrefname}
\hyperref[f:\secFqname:#1]{\GT{\secFqname:#1}}%
}
% Section
% <name>
\newrobustcmd{\fqSecRef}[1]{%
\hyperref[sec:#1]{\GT{#1}}%
}
% Quantities
% <symbol>
\newrobustcmd{\qtyRef}[1]{%
\edef\tempname{\luaDoubleFieldValue{quantities}{"#1"}{"fqname"}}%
\hyperref[qty:#1]{\expandafter\GT\expandafter{\tempname:#1}}%
}
% <symbol> <name>
\newrobustcmd{\QtyRef}[1]{%
$\luaDoubleFieldValue{quantities}{"#1"}{"symbol"}$ \qtyRef{#1}%
}
% Constants
% <name>
\newrobustcmd{\constRef}[1]{%
\edef\tempname{\luaDoubleFieldValue{constants}{"#1"}{"fqname"}}%
\hyperref[const:#1]{\expandafter\GT\expandafter{\tempname:#1}}%
}
% <symbol> <name>
\newrobustcmd{\ConstRef}[1]{%
$\luaDoubleFieldValue{constants}{"#1"}{"symbol"}$ \constRef{#1}%
}
% Element from periodic table
% <symbol>
\newrobustcmd{\elRef}[1]{%
\hyperref[el:#1]{{\color{fg0}#1}}%
}
% <name>
\newrobustcmd{\ElRef}[1]{%
\hyperref[el:#1]{\GT{el:#1}}%
}
% "LABELS"
% These currently do not place a label,
% instead they provide an alternative way to reference an existing label
\directLuaAux{
if absLabels == nil then
absLabels = {}
end
}
% [1]: target (fqname to point to)
% 2: key
\newcommand{\absLink}[2][sec:\fqname]{
\directLuaAuxExpand{
absLabels["#2"] = [[#1]]
}
}
\directLuaAux{
if abbrLabels == nil then
abbrLabels = {}
end
}
% [1]: target (fqname to point to)
% 2: key
% 3: label (abbreviation)
\newcommand{\abbrLink}[3][sec:\fqname]{
\directLuaAuxExpand{
abbrLabels["#2"] = {}
abbrLabels["#2"]["abbr"] = [[#3]]
abbrLabels["#2"]["fqname"] = [[#1]]
}
}
% [1]:
\newrobustcmd{\absRef}[2][\relax]{%
\directlua{
if absLabels["#2"] == nil then
tex.sprint("\\detokenize{#2}???")
else
if "#1" == "" then %-- if [#1] is not given, use translation of key as text, else us given text
tex.sprint("\\hyperref[" .. absLabels["#2"] .. "]{\\GT{" .. absLabels["#2"] .. "}}")
else
tex.sprint("\\hyperref[" .. absLabels["#2"] .. "]{\luaescapestring{#1}}")
end
end
}
}
\newrobustcmd{\abbrRef}[1]{%
\directlua{
if abbrLabels["#1"] == nil then
tex.sprint("\\detokenize{#1}???")
else
tex.sprint("\\hyperref[" .. abbrLabels["#1"]["fqname"] .. "]{" .. abbrLabels["#1"]["abbr"] .. "}")
end
}
}

191
src/util/macros.tex
View File

@ -1,6 +1,7 @@
% use \newcommand instead of \def because we want to throw an error if a command gets redefined
\newcommand\smartnewline[1]{\ifhmode\\\fi} % newline only if there in horizontal mode
\def\gooditem{\item[{$\color{fg-red}\bullet$}]}
\def\baditem{\item[{$\color{fg-green}\bullet$}]}
\newcommand\gooditem{\item[{$\color{fg-green}\bullet$}]}
\newcommand\baditem{\item[{$\color{fg-red}\bullet$}]}
% Functions with (optional) paranthesis
% 1: The function (like \exp, \sin etc.)
@ -24,11 +25,11 @@
% COMMON SYMBOLS WITH SUPER/SUBSCRIPTS, VECTOR ARROWS ETC.
% \def\laplace{\Delta} % Laplace operator
\def\laplace{\bigtriangleup} % Laplace operator
\newcommand\laplace{\bigtriangleup} % Laplace operator
% symbols
\def\Grad{\vec{\nabla}}
\def\Div {\vec{\nabla} \cdot}
\def\Rot {\vec{\nabla} \times}
\newcommand\Grad{\vec{\nabla}}
\newcommand\Div {\vec{\nabla} \cdot}
\newcommand\Rot {\vec{\nabla} \times}
% symbols with parens
\newcommand\GradS[1][\relax]{\CmdInParenthesis{\Grad}{#1}}
\newcommand\DivS [1][\relax]{\CmdInParenthesis{\Div} {#1}}
@ -37,100 +38,96 @@
\newcommand\GradT[1][\relax]{\CmdWithParenthesis{\text{grad}\,}{#1}}
\newcommand\DivT[1][\relax] {\CmdWithParenthesis{\text{div}\,} {#1}}
\newcommand\RotT[1][\relax] {\CmdWithParenthesis{\text{rot}\,} {#1}}
\def\vecr{\vec{r}}
\def\vecR{\vec{R}}
\def\veck{\vec{k}}
\def\vecx{\vec{x}}
\def\kB{k_\text{B}} % boltzmann
\def\NA{N_\text{A}} % avogadro
\def\EFermi{E_\text{F}} % fermi energy
\def\Efermi{E_\text{F}} % fermi energy
\def\Evalence{E_\text{v}} % val vand energy
\def\Econd{E_\text{c}} % cond. band nergy
\def\Egap{E_\text{gap}} % band gap energy
\def\Evac{E_\text{vac}} % vacuum energy
\def\masse{m_\text{e}} % electron mass
\def\Four{\mathcal{F}} % Fourier transform
\def\Lebesgue{\mathcal{L}} % Lebesgue
\def\O{\mathcal{O}} % order
\def\PhiB{\Phi_\text{B}} % mag. flux
\def\PhiE{\Phi_\text{E}} % electric flux
\def\nreal{n^{\prime}} % refraction real part
\def\ncomplex{n^{\prime\prime}} % refraction index complex part
\def\I{i} % complex unit
\def\crit{\text{crit}} % crit (for subscripts)
\def\muecp{\overline{\mu}} % electrochemical potential
\def\pH{\text{pH}} % pH
\def\rfactor{\text{rf}} % rf roughness_factor
\newcommand\kB{k_\text{B}} % boltzmann
\newcommand\NA{N_\text{A}} % avogadro
\newcommand\EFermi{E_\text{F}} % fermi energy
\newcommand\Efermi{E_\text{F}} % fermi energy
\newcommand\Evalence{E_\text{v}} % val vand energy
\newcommand\Econd{E_\text{c}} % cond. band nergy
\newcommand\Egap{E_\text{gap}} % band gap energy
\newcommand\Evac{E_\text{vac}} % vacuum energy
\newcommand\masse{m_\text{e}} % electron mass
\newcommand\Four{\mathcal{F}} % Fourier transform
\newcommand\Lebesgue{\mathcal{L}} % Lebesgue
% \newcommand\O{\mathcal{O}} % order
\newcommand\PhiB{\Phi_\text{B}} % mag. flux
\newcommand\PhiE{\Phi_\text{E}} % electric flux
\newcommand\nreal{n^{\prime}} % refraction real part
\newcommand\ncomplex{n^{\prime\prime}} % refraction index complex part
\newcommand\I{i} % complex/imaginary unit
\newcommand\crit{\text{crit}} % crit (for subscripts)
\newcommand\muecp{\overline{\mu}} % electrochemical potential
% \newcommand\pH{\text{pH}} % pH, already defined by one of the chem packages
\newcommand\rfactor{\text{rf}} % rf roughness_factor
% SYMBOLS
\def\R{\mathbb{R}}
\def\C{\mathbb{C}}
\def\Z{\mathbb{Z}}
\def\N{\mathbb{N}}
\def\id{\mathbb{1}}
\newcommand\R{\mathbb{R}}
\newcommand\C{\mathbb{C}}
\newcommand\Z{\mathbb{Z}}
\newcommand\N{\mathbb{N}}
\newcommand\id{\mathbb{1}}
% caligraphic
\def\E{\mathcal{E}} % electric field
% upright
\def\txA{\text{A}}
\def\txB{\text{B}}
\def\txC{\text{C}}
\def\txD{\text{D}}
\def\txE{\text{E}}
\def\txF{\text{F}}
\def\txG{\text{G}}
\def\txH{\text{H}}
\def\txI{\text{I}}
\def\txJ{\text{J}}
\def\txK{\text{K}}
\def\txL{\text{L}}
\def\txM{\text{M}}
\def\txN{\text{N}}
\def\txO{\text{O}}
\def\txP{\text{P}}
\def\txQ{\text{Q}}
\def\txR{\text{R}}
\def\txS{\text{S}}
\def\txT{\text{T}}
\def\txU{\text{U}}
\def\txV{\text{V}}
\def\txW{\text{W}}
\def\txX{\text{X}}
\def\txY{\text{Y}}
\def\txZ{\text{Z}}
\def\txa{\text{a}}
\def\txb{\text{b}}
\def\txc{\text{c}}
\def\txd{\text{d}}
\def\txe{\text{e}}
\def\txf{\text{f}}
\def\txg{\text{g}}
\def\txh{\text{h}}
\def\txi{\text{i}}
\def\txj{\text{j}}
\def\txk{\text{k}}
\def\txl{\text{l}}
\def\txm{\text{m}}
\def\txn{\text{n}}
\def\txo{\text{o}}
\def\txp{\text{p}}
\def\txq{\text{q}}
\def\txr{\text{r}}
\def\txs{\text{s}}
\def\txt{\text{t}}
\def\txu{\text{u}}
\def\txv{\text{v}}
\def\txw{\text{w}}
\def\txx{\text{x}}
\def\txy{\text{y}}
\def\txz{\text{z}}
\newcommand\E{\mathcal{E}} % electric field
% upright, vector
\newcommand\txA{\text{A}} \newcommand\vecA{\vec{A}}
\newcommand\txB{\text{B}} \newcommand\vecB{\vec{B}}
\newcommand\txC{\text{C}} \newcommand\vecC{\vec{C}}
\newcommand\txD{\text{D}} \newcommand\vecD{\vec{D}}
\newcommand\txE{\text{E}} \newcommand\vecE{\vec{E}}
\newcommand\txF{\text{F}} \newcommand\vecF{\vec{F}}
\newcommand\txG{\text{G}} \newcommand\vecG{\vec{G}}
\newcommand\txH{\text{H}} \newcommand\vecH{\vec{H}}
\newcommand\txI{\text{I}} \newcommand\vecI{\vec{I}}
\newcommand\txJ{\text{J}} \newcommand\vecJ{\vec{J}}
\newcommand\txK{\text{K}} \newcommand\vecK{\vec{K}}
\newcommand\txL{\text{L}} \newcommand\vecL{\vec{L}}
\newcommand\txM{\text{M}} \newcommand\vecM{\vec{M}}
\newcommand\txN{\text{N}} \newcommand\vecN{\vec{N}}
\newcommand\txO{\text{O}} \newcommand\vecO{\vec{O}}
\newcommand\txP{\text{P}} \newcommand\vecP{\vec{P}}
\newcommand\txQ{\text{Q}} \newcommand\vecQ{\vec{Q}}
\newcommand\txR{\text{R}} \newcommand\vecR{\vec{R}}
\newcommand\txS{\text{S}} \newcommand\vecS{\vec{S}}
\newcommand\txT{\text{T}} \newcommand\vecT{\vec{T}}
\newcommand\txU{\text{U}} \newcommand\vecU{\vec{U}}
\newcommand\txV{\text{V}} \newcommand\vecV{\vec{V}}
\newcommand\txW{\text{W}} \newcommand\vecW{\vec{W}}
\newcommand\txX{\text{X}} \newcommand\vecX{\vec{X}}
\newcommand\txY{\text{Y}} \newcommand\vecY{\vec{Y}}
\newcommand\txZ{\text{Z}} \newcommand\vecZ{\vec{Z}}
\newcommand\txa{\text{a}} \newcommand\veca{\vec{a}}
\newcommand\txb{\text{b}} \newcommand\vecb{\vec{b}}
\newcommand\txc{\text{c}} \newcommand\vecc{\vec{c}}
\newcommand\txd{\text{d}} \newcommand\vecd{\vec{d}}
\newcommand\txe{\text{e}} \newcommand\vece{\vec{e}}
\newcommand\txf{\text{f}} \newcommand\vecf{\vec{f}}
\newcommand\txg{\text{g}} \newcommand\vecg{\vec{g}}
\newcommand\txh{\text{h}} \newcommand\vech{\vec{h}}
\newcommand\txi{\text{i}} \newcommand\veci{\vec{i}}
\newcommand\txj{\text{j}} \newcommand\vecj{\vec{j}}
\newcommand\txk{\text{k}} \newcommand\veck{\vec{k}}
\newcommand\txl{\text{l}} \newcommand\vecl{\vec{l}}
\newcommand\txm{\text{m}} \newcommand\vecm{\vec{m}}
\newcommand\txn{\text{n}} \newcommand\vecn{\vec{n}}
\newcommand\txo{\text{o}} \newcommand\veco{\vec{o}}
\newcommand\txp{\text{p}} \newcommand\vecp{\vec{p}}
\newcommand\txq{\text{q}} \newcommand\vecq{\vec{q}}
\newcommand\txr{\text{r}} \newcommand\vecr{\vec{r}}
\newcommand\txs{\text{s}} \newcommand\vecs{\vec{s}}
\newcommand\txt{\text{t}} \newcommand\vect{\vec{t}}
\newcommand\txu{\text{u}} \newcommand\vecu{\vec{u}}
\newcommand\txv{\text{v}} \newcommand\vecv{\vec{v}}
\newcommand\txw{\text{w}} \newcommand\vecw{\vec{w}}
\newcommand\txx{\text{x}} \newcommand\vecx{\vec{x}}
\newcommand\txy{\text{y}} \newcommand\vecy{\vec{y}}
\newcommand\txz{\text{z}} \newcommand\vecz{\vec{z}}
% SPACES
\def\sdots{\,\dots\,}
\def\qdots{\quad\dots\quad}
\def\qRarrow{\quad\Rightarrow\quad}
\newcommand\sdots{\,\dots\,}
\newcommand\qdots{\quad\dots\quad}
\newcommand\qRarrow{\quad\Rightarrow\quad}
% ANNOTATIONS
% put an explanation above an equal sign
@ -188,12 +185,12 @@
\newcommand\Order[1]{\CmdWithParenthesis{\mathcal{O}}{#1}}
% VECTOR, MATRIX and TENSOR
% use vecA to force an arrow
\NewCommandCopy{\vecA}{\vec}
% use vecAr to force an arrow
\NewCommandCopy{\vecAr}{\vec}
% extra {} assure they can b directly used after _
%% arrow/underline
\newcommand\mat[1]{{\ensuremath{\underline{#1}}}}
\renewcommand\vec[1]{{\ensuremath{\vecA{#1}}}}
\renewcommand\vec[1]{{\ensuremath{\vecAr{#1}}}}
\newcommand\ten[1]{{\ensuremath{[#1]}}}
\newcommand\complex[1]{{\ensuremath{\tilde{#1}}}}
%% bold

3
src/util/periodic_table.tex
View File

@ -54,10 +54,11 @@
\vspace{0.5\baselineskip}
\begingroup
% label it only once
% \detokenize{\label{el:#1}}
\directlua{
if elements["#1"]["labeled"] == nil then
elements["#1"]["labeled"] = true
tex.print("\\label{el:#1}")
tex.print("\\phantomsection\\label{el:#1}")
end
}
\NameWithDescription[\descwidth]{\elementName}{\elementName_desc}

9
src/util/tikz_macros.tex
View File

@ -103,3 +103,12 @@
\draw[->] (0,0) -- (\tkW+0.2,0) node[anchor=north] {$x$};
\draw[->] (0,0) -- (0,\tkH+0.2) node[anchor=east] {$E$};
}
\newcommand\tkXTick[2]{
\pgfmathsetmacro{\tickwidth}{0.1}
\draw (#1, -\tickwidth/2) -- (#1, \tickwidth/2) node[anchor=north] {#2};
}
\newcommand\tkYTick[2]{
\pgfmathsetmacro{\tickwidth}{0.1}
\draw (-\tickwidth/2, #1) -- (\tickwidth/2,#1) node[anchor=east] {#2};
}