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refactor

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matthias@quintern.xyz 2025-03-02 00:32:49 +01:00
parent d9da44ac74
commit e60fa83593
40 changed files with 1261 additions and 841 deletions
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10
scripts/ch_elchem.py
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@ -18,7 +18,7 @@ def fbutler_volmer(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)
fig, ax = plt.subplots(figsize=size_formula_fill_default)
ax.set_xlabel("$\\eta$ [V]")
ax.set_ylabel("$j/j_0$")
etas = np.linspace(-0.1, 0.1, 400)
@ -62,7 +62,7 @@ def tafel():
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)
fig, ax = plt.subplots(figsize=size_formula_normal_default)
ax.set_xlabel("$\\eta$ [V]")
ax.set_ylabel("$\\log_{10}\\left(\\frac{|j|}{j_0}\\right)$")
# ax.set_ylabel("$\\log_{10}\\left(|j|/j_0\\right)$")
@ -91,7 +91,7 @@ def fZ_ind(L, omega):
def nyquist():
fig, ax = plt.subplots(figsize=(full/2, full/3))
fig, ax = plt.subplots(figsize=size_formula_fill_default)
split_z = lambda Z: (Z.real, -Z.imag)
ax.grid()
ax.set_xlabel("$\\text{Re}(Z)$ [\\si{\\ohm}]")
@ -146,7 +146,7 @@ def fZ_tlm(Rel, Rion, Rct, Cct, ws, N):
return Z
def nyquist_tlm():
fig, ax = plt.subplots(figsize=(full/2, full/4))
fig, ax = plt.subplots(figsize=(width_formula, width_formula*0.5))
split_z = lambda Z: (Z.real, -Z.imag)
ax.grid()
ax.set_xlabel("$\\text{Re}(Z)$ [\\si{\\ohm}]")
@ -168,7 +168,7 @@ def fkohlrausch(L0, K, c):
return L0 - K*np.sqrt(c)
def kohlrausch():
fig, ax = plt.subplots(figsize=(full/4, full/4))
fig, ax = plt.subplots(figsize=size_formula_small_quadratic)
ax.grid()
ax.set_xlabel("$c_\\text{salt}$")
ax.set_ylabel("$\\Lambda_\\text{M}$")

4
scripts/cm_phonons.py
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@ -11,7 +11,7 @@ def one_atom_basis():
M = 1.
qs = np.linspace(-2*np.pi/a, 2*np.pi/a, 300)
omega = fone_atom_basis(qs, a, M, C1, C2)
fig, ax = plt.subplots(figsize=size_half_third)
fig, ax = plt.subplots(figsize=size_formula_normal_default)
ax.set_xlabel(r"$q$")
ax.set_xticks([i * np.pi/a for i in range(-2, 3)])
ax.set_xticklabels([f"${i}\\pi/a$" if i != 0 else "0" for i in range(-2, 3)])
@ -41,7 +41,7 @@ def two_atom_basis():
qs = np.linspace(-2*np.pi/a, 2*np.pi/a, 300)
omega_a = ftwo_atom_basis_acoustic(qs, a, M1, M2, C)
omega_o = ftwo_atom_basis_optical(qs, a, M1, M2, C)
fig, ax = plt.subplots(figsize=size_half_third)
fig, ax = plt.subplots(figsize=size_formula_normal_default)
ax.plot(qs, omega_a, label="acoustic")
ax.plot(qs, omega_o, label="optical")
ax.text(0, 0.8, "1. BZ", ha='center')

50
scripts/cm_superconductivity.py Normal file
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@ -0,0 +1,50 @@
#!/usr/bin env python3
from formulary import *
# Define the functions
def psi_squared(x, xi):
return np.tanh(x/(np.sqrt(2)*xi))**2
def B_z(x, B0, lam):
return B0 * np.exp(-x/lam)
def n_s_boundary():
xs = np.linspace(0, 6, 400)
xn = np.linspace(-1, 0, 10)
B0 = 1.0
fig, ax = plt.subplots(figsize=size_formula_fill_default)
ax.axvline(x=0, color='gray', linestyle='--', linewidth=0.8)
ax.axhline(y=1, color='gray', linestyle='--', linewidth=0.8)
ax.axhline(y=0, color='gray', linestyle='--', linewidth=0.8)
ax.fill_between(xn, -2, 2 , color=COLORSCHEME["bg-yellow"], alpha=0.5)
ax.fill_between(xs, -2, 2 , color=COLORSCHEME["bg-blue"], alpha=0.5)
ax.text(-0.5, 0.9, 'N', color=COLORSCHEME["fg-yellow"], fontsize=14, ha="center", va="center")
ax.text(3, 0.9, 'S', color=COLORSCHEME["fg-blue"], fontsize=14, ha="center", va="center")
ax.set_xlabel("$x$")
ax.set_ylabel(r"$|\Psi|^2$, $B_z(x)/B_\text{ext}$")
ax.set_ylim(-0.1, 1.1)
ax.set_xlim(-1, 6)
ax.grid()
lines = []
for i, (xi, lam, color) in enumerate([(0.5, 2, "blue"), (2, 0.5, "red")]):
psi = psi_squared(xs, xi)
B = B_z(xs, B0, lam)
line, = ax.plot(xs, psi, color=color, linestyle="solid", label=f"$\\xi_\\text{{GL}}={xi}$, $\\lambda_\\text{{GL}}={lam}$")
lines.append(line)
ax.plot(xs, B, color=color, linestyle="dashed")
if i == 1:
ylam = 1/np.exp(1)
ax.plot([0, lam], [ylam, ylam], linestyle="dashed", color=COLORSCHEME["fg2"])
ax.text(lam/2, ylam, r'$\lambda_\text{GL}$', color=color, ha="center", va="bottom")
yxi = psi_squared(xi, xi)
ax.plot([0, xi], [yxi, yxi], linestyle="dotted", color=COLORSCHEME["fg2"])
ax.text(xi/2, yxi, r'$\xi_\text{GL}$', color=color, ha="center", va="bottom")
lines.append(mpl.lines.Line2D([], [], color="black", label=r"$\lvert\Psi\rvert^2$"))
lines.append(mpl.lines.Line2D([], [], color="black", linestyle="dashed", label=r"$B_z(x)/B_\text{ext}$"))
ax.legend(loc='center right', handles=lines)
return fig
if __name__ == "__main__":
export(n_s_boundary(), "cm_sc_n_s_boundary")

2
scripts/distributions.py
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@ -4,7 +4,7 @@ import itertools
def get_fig():
fig, ax = plt.subplots(figsize=size_half_half)
fig, ax = plt.subplots(figsize=size_bigformula_half_quadratic)
ax.grid()
ax.set_xlabel(f"$x$")
ax.set_ylabel("PDF")

23
scripts/formulary.py
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@ -37,10 +37,25 @@ 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 pt_2_inch(pt):
return 0.0138888889 * pt
def cm_2_inch(cm):
return 0.3937007874 * cm
# A4 - margins
width_line = cm_2_inch(21.0 - 2 * 2.0)
# width of a formula box, the prefactor has to match \eqwidth
width_formula = 0.69 * width_line
# arbitrary choice
height_default = width_line * 2 / 5
size_bigformula_fill_default = (width_line, height_default)
size_bigformula_half_quadratic = (width_line*0.5, width_line*0.5)
size_bigformula_small_quadratic = (width_line*0.33, width_line*0.33)
size_formula_fill_default = (width_formula, height_default)
size_formula_normal_default = (width_formula*0.8, height_default*0.8)
size_formula_half_quadratic = (width_formula*0.5, width_formula*0.5)
size_formula_small_quadratic = (width_formula*0.4, width_formula*0.4)
def assert_directory():
if not skipasserts:

6
scripts/qubits.py
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@ -39,7 +39,7 @@ def transmon_cpb(wavefunction=True):
ngs = np.linspace(-2, 2, 200)
nrows = 4 if wavefunction else 1
fig, axs = plt.subplots(nrows, 3, squeeze=False, figsize=(full,full/3))
fig, axs = plt.subplots(nrows, 3, squeeze=False, figsize=(width_line,height_default))
axs = axs.T
qubit.ng = 0
qubit.EJ = 0.1 * EC
@ -68,7 +68,7 @@ def transmon_cpb(wavefunction=True):
def flux_onium():
fig, axs = plt.subplots(1, 3, squeeze=True, figsize=(full,full/3))
fig, axs = plt.subplots(1, 3, squeeze=True, figsize=(width_line,height_default))
fluxs = np.linspace(0.4, 0.6, 50)
EJ = 35.0
alpha = 0.3
@ -97,7 +97,7 @@ def flux_onium():
# axs[0].set_xlim(0.4, 0.6)
fluxs = np.linspace(-1.1, 1.1, 101)
fluxonium = scq.Fluxonium(EJ=9, EC=3, EL=0.5, flux=1, cutoff=100)
fluxonium = scq.Fluxonium(EJ=9, EC=3, EL=0.5, flux=1, cutoff=30)
fluxonium.plot_evals_vs_paramvals("flux", fluxs, evals_count=5, subtract_ground=True, fig_ax=(fig, axs[2]))
axs[2].set_title("Fluxonium")
return fig

8
scripts/stat-mech.py
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@ -5,7 +5,7 @@ def flennard_jones(r, epsilon, sigma):
return 4 * epsilon * ((sigma/r)**12 - (sigma/r)**6)
def lennard_jones():
fig, ax = plt.subplots(figsize=size_half_half)
fig, ax = plt.subplots(figsize=size_formula_normal_default)
ax.grid()
ax.set_xlabel(r"$r$")
ax.set_ylabel(r"$V(r)$")
@ -29,7 +29,7 @@ def ffermi_dirac(x):
def id_qgas():
fig, ax = plt.subplots(figsize=size_half_half)
fig, ax = plt.subplots(figsize=size_formula_normal_default)
ax.grid()
ax.set_xlabel(r"$\beta(\epsilon-\mu)$")
ax.set_ylabel(r"$\langle n(\epsilon)\rangle$")
@ -51,7 +51,7 @@ def fstep(x):
return 1 if x >= 0 else 0
def fermi_occupation():
fig, ax = plt.subplots(figsize=size_half_third)
fig, ax = plt.subplots(figsize=size_formula_normal_default)
# ax.grid()
# ax.set_xlabel(r"$\beta(\epsilon-\mu)$")
ax.set_xticks([0])
@ -68,7 +68,7 @@ def fermi_occupation():
return fig
def fermi_heat_capacity():
fig, ax = plt.subplots(figsize=size_half_third)
fig, ax = plt.subplots(figsize=size_formula_normal_default)
# ax.grid()
# ax.set_xlabel(r"$\beta(\epsilon-\mu)$")
x = np.linspace(0, 4, 100)

22
src/ch/el.tex
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@ -3,7 +3,7 @@
\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{Chemical potential}{of species $i$\\Energy involved when the particle number changes}{\QtyRef{free_enthalpy}, \QtyRef{amount}}
\desc[german]{Chemisches Potential}{der Spezies $i$\\Involvierte Energie, wenn sich die Teilchenzahl ändert}{}
\quantity{\mu}{\joule\per\mol;\joule}{is}
\eq{
@ -140,14 +140,14 @@
\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}}
\desc{Standard cell potential}{}{$\Delta_\txR G^\theta$ standard \qtyRef{free_enthalpy} of reaction, $n$ number of electrons, \ConstRef{faraday}}
\desc[german]{Standard Zellpotential}{}{$\Delta_\txR G^\theta$ Standard \qtyRef{free_enthalpy} der Reaktion, $n$ Anzahl der Elektronen, \ConstRef{faraday}}
\eq{E^\theta_\text{rev} = \frac{-\Delta_\txR G^\theta}{nF}}
\end{formula}
\begin{formula}{nernst_equation}
\desc{Nernst equation}{Electrode potential for a half-cell reaction}{\QtyRef{electrode_potential}, $E^\theta$ \fRef{::standard_cell_potential}, \ConstRef{universal_gas}, \ConstRef{temperature}, $z$ \qtyRef{charge_number}, \ConstRef{faraday}, \QtyRef{activity}, \QtyRef{stoichiometric_coefficient}}
\desc{Nernst equation}{Electrode potential for a half-cell reaction}{\QtyRef{electrode_potential}, $E^\theta$ \fRef{::standard_cell_potential}, \ConstRef{universal_gas}, \QtyRef{temperature}, $z$ \qtyRef{charge_number}, \ConstRef{faraday}, \QtyRef{activity}, \QtyRef{stoichiometric_coefficient}}
\desc[german]{Nernst-Gleichung}{Elektrodenpotential für eine Halbzellenreaktion}{}
\eq{E = E^\theta + \frac{RT}{zF} \Ln{\frac{ \left(\prod_{i}(a_i)^{\abs{\nu_i}}\right)_\text{oxidized}}{\left(\prod_{i}(a_i)^{\abs{\nu_i}}\right)_\text{reduced}}}}
\end{formula}
@ -172,7 +172,7 @@
\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{Diffusion}{caused by concentration gradients}{$z_i$ \qtyRef{charge_number} \gt{of_i}, \ConstRef{faraday}, \QtyRef{diffusion_coefficient} \gt{of_i}, \QtyRef{concentration} \gt{of_i}}
\desc[german]{Diffusion}{durch Konzentrationsgradienten}{}
\eq{ i_\text{diff} = \sum_i -z_i F D_i \left(\odv{c_i}{x}\right) }
\end{formula}
@ -604,14 +604,18 @@
\begin{hiddenformula}{scan_rate}
\desc{Scan rate}{}{}
\desc[german]{Scanrate}{}{}
\quantity{v}{\volt\per\s}{s}
\hiddenQuantity{v}{\volt\per\s}{s}
\end{hiddenformula}
\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}}
% \desc[german]{}{}{}
\ttxt{\eng{
Reversible deposition of metal onto a foreign metal electrode at potentials positive of the Nernst potential.
}\ger{
Reversible Ablagerung von Metall auf eine Elektrode aus einem anderen Metall bei positiveren Potentialen als das Nernst-Potential.
}}
\end{formula}
\Subsubsection[
@ -632,7 +636,7 @@
\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{Diffusion layer thickness}{}{$D$ \qtyRef{diffusion_coefficient}, $\nu$ \qtyRef{kinematic_viscosity}, \QtyRef{angular_frequency}}
\desc[german]{Diffusionsschichtdicke}{}{}
\eq{\delta_\text{diff}= 1.61 D{^\frac{1}{3}} \nu^{\frac{1}{6}} \omega^{-\frac{1}{2}}}
\end{formula}

54
src/cm/charge_transport.tex
View File

@ -6,21 +6,24 @@
\eng{Drude model}
\ger{Drude-Modell}
]{drude}
\begin{ttext}
\eng{Classical model describing the transport properties of electrons in materials (metals):
\begin{formula}{description}
\desc{Description}{}{}
\desc[german]{Beschreibung}{}{}
\ttxt{\eng{
Classical model describing the transport properties of electrons in materials (metals):
The material is assumed to be an ion lattice and with freely moving electrons (electron gas). The electrons are
accelerated by an electric field and decelerated through collisions with the lattice ions.
The model disregards the Fermi-Dirac partition of the conducting electrons.
}
\ger{Ein klassisches Model zur Beschreibung der Transporteigenschaften von Elektronen in (v.a.) Metallen:
}\ger{
Ein klassisches Model zur Beschreibung der Transporteigenschaften von Elektronen in (v.a.) Metallen:
Der Festkörper wird als Ionenkristall mit frei beweglichen Elektronen (Elektronengas).
Die Elektronen werden durch ein Elektrisches Feld $E$ beschleunigt und durch Stöße mit den Gitterionen gebremst.
Das Modell vernachlässigt die Fermi-Dirac Verteilung der Leitungselektronen.
}
\end{ttext}
\begin{formula}{motion}
\desc{Equation of motion}{}{$v$ electron speed, $\vec{v}_\text{D}$ drift velocity, $\tau$ mean free time between collisions}
\desc[german]{Bewegungsgleichung}{}{$v$ Elektronengeschwindigkeit, $\vec{v}_\text{D}$ Driftgeschwindigkeit, $\tau$ Stoßzeit}
}}
\end{formula}
\begin{formula}{eom}
\desc{Equation of motion}{}{$v$ electron speed, $\vec{v}_\text{D}$ drift velocity, \QtyRef{scattering_time}}
\desc[german]{Bewegungsgleichung}{}{$v$ Elektronengeschwindigkeit, $\vec{v}_\text{D}$ Driftgeschwindigkeit, \QtyRef{scattering_time}}
\eq{\masse \odv{\vec{v}}{t} + \frac{\masse}{\tau} \vec{v}_\text{D} = -e \vec{\E}}
\end{formula}
\begin{formula}{scattering_time}
@ -28,35 +31,40 @@
\desc[german]{Streuzeit}{}{}
\quantity{\tau}{\s}{s}
\ttxt{
\eng{$\tau$\\ the average time between scattering events weighted by the characteristic momentum change cause by the scattering process.}
\eng{The average time between scattering events weighted by the characteristic momentum change cause by the scattering process.}
}
\end{formula}
\begin{formula}{current_density}
\desc{Current density}{Ohm's law}{$n$ charge particle density}
\desc[german]{Stromdichte}{Ohmsches Gesetz}{$n$ Ladungsträgerdichte}
\desc{Current density}{Ohm's law}{\QtyRef{charge_carrier_density}, \ConstRef{charge}, \QtyRef{drift_velocity}, \QtyRef{mobility}, \QtyRef{electric_field}}
\desc[german]{Stromdichte}{Ohmsches Gesetz}{}
\quantity{\vec{j}}{\ampere\per\m^2}{v}
\eq{\vec{j} = -ne\vec{v}_\text{D} = ne\mu \vec{\E}}
\end{formula}
\begin{formula}{conductivity}
\desc{Drude-conductivity}{}{}
\desc[german]{Drude-Leitfähigkeit}{}{}
\eq{\sigma = \frac{\vec{j}}{\vec{\E}} = \frac{e^2 \tau n}{\masse} = n e \mu}
\desc{Electrical conductivity}{Both from Drude model and Sommerfeld model}{\QtyRef{current_density}, \QtyRef{electric_field}, \QtyRef{charge_carrier_density}, \ConstRef{charge}, \QtyRef{scattering_time}, \ConstRef{electron_mass}, \QtyRef{mobility}}
\desc[german]{Elektrische Leitfähigkeit}{Aus dem Drude-Modell und dem Sommerfeld-Modell}{}
\quantity{\sigma}{\siemens\per\m=\per\ohm\m=\ampere^2\s^3\per\kg\m^3}{t}
\eq{\sigma = \frac{\vec{j}}{\vec{\E}} = \frac{n e^2 \tau}{\masse} = n e \mu}
\end{formula}
\Subsection[
\eng{Sommerfeld model}
\ger{Sommerfeld-Modell}
]{sommerfeld}
\begin{ttext}
\eng{Assumes a gas of free fermions underlying the pauli-exclusion principle. Only electrons in an energy range of $\kB T$ around the Fermi energy $\EFermi$ participate in scattering processes.}
\ger{Annahme eines freien Fermionengases, welches dem Pauli-Prinzip unterliegt. Nur Elektronen in einem Energiebereich von $\kB T$ um die Fermi Energe $\EFermi$ nehmen an Streuprozessen teil.}
\end{ttext}
\begin{formula}{description}
\desc{Description}{}{}
\desc[german]{Beschreibung}{}{}
\ttxt{
\eng{Assumes a gas of free fermions underlying the pauli-exclusion principle. Only electrons in an energy range of $\kB T$ around the Fermi energy $\EFermi$ participate in scattering processes. The \qtyRef{conductivity} is the same as in \fRef{::::drude}}
\ger{Annahme eines freien Fermionengases, welches dem Pauli-Prinzip unterliegt. Nur Elektronen in einem Energiebereich von $\kB T$ um die Fermi Energe $\EFermi$ nehmen an Streuprozessen teil. Die \qtyRef{conductivity} ist die selbe wie im \fRef{::::drude}}
}
\end{formula}
\begin{formula}{current_density}
\desc{Electrical current density}{}{}
\desc[german]{Elektrische Stromdichte}{}{}
\eq{\vec{j} = -en\braket{v} = -e n \frac{\hbar}{\masse}\braket{\vec{k}} = -e \frac{1}{V} \sum_{\vec{k},\sigma} \frac{\hbar \vec{k}}{\masse}}
\end{formula}
\TODO{The formula for the conductivity is the same as in the drude model?}
\Subsection[
\eng{Boltzmann-transport}
@ -67,7 +75,7 @@
\ger{Semiklassische Beschreibung, benutzt eine Wahrscheinlichkeitsverteilung (\fRef{stat:todo:fermi_dirac}).}
\end{ttext}
\begin{formula}{boltzmann_transport}
\desc{Boltzmann Transport equation}{for charge transport}{$f$ \ref{stat:todo:fermi-dirac}}
\desc{Boltzmann Transport equation}{for charge transport}{$f$ \fRef{stat:todo:fermi-dirac}}
\desc[german]{Boltzmann-Transportgleichung}{für Ladungstransport}{}
\eq{
\pdv{f(\vec{r},\vec{k},t)}{t} = -\vec{v} \cdot \Grad_{\vec{r}} f - \frac{e}{\hbar}(\vec{\mathcal{E}} + \vec{v} \times \vec{B}) \cdot \Grad_{\vec{k}} f + \left(\pdv{f(\vec{r},\vec{k},t)}{t}\right)_{\text{\GT{scatter}}}
@ -79,8 +87,8 @@
\ger{misc}
]{misc}
\begin{formula}{tsu_esaki}
\desc{Tsu-Esaki tunneling current}{Describes the current $I_{\txL \leftrightarrow \txR}$ through a barrier}{$\mu_i$ \qtyRef{chemical_pot} at left/right side, $U_i$ voltage on left/right side. Electrons occupy region between $U_i$ and $\mu_i$}
\desc[german]{Tsu-Esaki Tunnelstrom}{Beschreibt den Strom $I_{\txL \leftrightarrow \txR}$ durch eine Barriere }{$\mu_i$ \qtyRef{chemical_pot} links/rechts, $U_i$ Spannung links/rechts. Elektronen besetzen Bereich zwischen $U_i$ und $\mu_i$}
\desc{Tsu-Esaki tunneling current}{Describes the current $I_{\txL \leftrightarrow \txR}$ through a barrier}{$\mu_i$ \qtyRef{chemical_potential} at left/right side, $U_i$ voltage on left/right side. Electrons occupy region between $U_i$ and $\mu_i$}
\desc[german]{Tsu-Esaki Tunnelstrom}{Beschreibt den Strom $I_{\txL \leftrightarrow \txR}$ durch eine Barriere }{$\mu_i$ \qtyRef{chemical_potential} links/rechts, $U_i$ Spannung links/rechts. Elektronen besetzen Bereich zwischen $U_i$ und $\mu_i$}
\eq{
I_\text{T} = \frac{2e}{h} \int_{U_\txL}^\infty \left(f(E, \mu_\txL) -f(E, \mu_\txR)\right) T(E) \d E
}

7
src/cm/cm.tex
View File

@ -2,12 +2,12 @@
\eng{Condensed matter physics}
\ger{Festkörperphysik}
]{cm}
\TODO{Bonds, hybridized orbitals}
\TODO{Lattice vibrations, van hove singularities, debye frequency}
\TODO{van hove singularities, debye frequency}
\begin{formula}{dos}
\desc{Density of states (DOS)}{}{\QtyRef{volume}, $N$ number of energy levels, \QtyRef{energy}}
\desc[german]{Zustandsdichte (DOS)}{}{\QtyRef{volume}, $N$ Anzahl der Energieniveaus, \QtyRef{energy}}
\quantity{D}{\per\m^3}{s}
\eq{D(E) = \frac{1}{V}\sum_{i=1}^{N} \delta(E-E(\vec{k_i}))}
\end{formula}
\begin{formula}{dos_parabolic}
@ -33,9 +33,9 @@
\intertext{\GT{with}}
u_{s+n} = U\e^{-i \left[\omega t - q(s+n)a \right]}
\end{gather}
\newFormulaEntry
\fig{img/cm_phonon_dispersion_one_atom_basis.pdf}
\end{formula}
\TODO{Plots}
\begin{formula}{dispersion_2atom_basis}
\desc{Phonon dispersion of a lattice with a two-atom basis}{}{$C$ force constant between layers, $M_i$ \qtyRef{mass} of the basis atoms, $a$ \qtyRef{lattice_constant}, $q$ phonon \qtyRef{wavevector}, $u, v$ Ansatz for the displacement of basis atom 1 and 2, respectively}
\desc[german]{Phonondispersion eines Gitters mit einatomiger Basis}{}{$C$ Kraftkonstanten zwischen Ebene $s$ und $s+n$, $M_i$ \qtyRef{mass} der Basisatome, $a$ \qtyRef{lattice_constant}, $q$ Phonon \qtyRef{wavevector}, $u, v$ jeweils Ansatz für die Atomauslenkung des Basisatoms 1 und 2}
@ -45,6 +45,7 @@
u_{s} = U\e^{-i \left(\omega t - qsa \right)}, \quad
v_{s} = V\e^{-i \left(\omega t - qsa \right)}
\end{gather}
\newFormulaEntry
\fig{img/cm_phonon_dispersion_two_atom_basis.pdf}
\end{formula}

11
src/cm/crystal.tex
View File

@ -71,7 +71,12 @@
\eq{\vec{R} = n_1 \vec{a_1} + n_2 \vec{a_2} + n_3 \vec{a_3}}
\end{formula}
\TODO{primitive unit cell: contains one lattice point}\\
\begin{formula}{primitive_unit_cell}
\desc{Primitve unit cell}{}{}
\desc[german]{Primitive Einheitszelle}{}{}
\ttxt{\eng{Unit cell containing exactly one lattice point}\ger{Einheitszelle die genau einen Gitterpunkt enthält}}
\end{formula}
\begin{formula}{miller}
\desc{Miller index}{}{Miller family: planes that are equivalent due to crystal symmetry}
\desc[german]{Millersche Indizes}{}{}
@ -116,8 +121,8 @@
\desc{Matthiessen's rule}{Approximation, only holds if the processes are independent of each other}{\QtyRef{mobility}, \QtyRef{scattering_time}}
\desc[german]{Matthiessensche Regel}{Näherung, nur gültig wenn die einzelnen Streuprozesse von einander unabhängig sind}{}
\eq{
\frac{1}{\mu} &= \sum_{i = \textrm{\GT{\fqname}}} \frac{1}{\mu_i} \\
\frac{1}{\tau} &= \sum_{i = \textrm{\GT{\fqname}}} \frac{1}{\tau_i}
\frac{1}{\mu} &= \sum_{i = \textrm{\GT{:::scatter}}} \frac{1}{\mu_i} \\
\frac{1}{\tau} &= \sum_{i = \textrm{\GT{:::scatter}}} \frac{1}{\tau_i}
}
\end{formula}

248
src/cm/low_temp.tex
View File

@ -1,248 +0,0 @@
\def\txL{\text{L}}
\def\gl{\text{GL}}
\def\GL{Ginzburg-Landau }
\def\Tcrit{T_\text{c}}
\def\Bcth{B_\text{c}}
\Section[
\eng{Superconductivity}
\ger{Supraleitung}
]{sc}
\begin{ttext}
\eng{
Materials for which the electric resistance jumps to 0 under a critical temperature $\Tcrit$.
Below $\Tcrit$ they have perfect conductivity and perfect diamagnetism, up until a critical magnetic field $\Bcth$.
\\\textbf{Type I}:
\\\textbf{Type II}: Has two critical
}
\ger{
Materialien, bei denen der elektrische Widerstand beim unterschreiten einer kritischen Temperatur $\Tcrit$ auf 0 springt.
Sie verhalten sich dann wie ideale Leiter und ideale Diamagnete, bis zu einem kritischen Feld $\Bcth$.
}
\end{ttext}
\begin{formula}{type1}
\desc{Type-I superconductor}{}{}
\desc[german]{Typ-I Supraleiter}{}{}
\ttxt{\eng{
Has a single critical magnetic field, $\Bcth$.
\\$B < \Bcth$: \fRef{:::meissner_effect}
\\$B > \Bcth$: Normal conductor
}}
\end{formula}
\begin{formula}{type2}
\desc{Type-II superconductor}{}{}
\desc[german]{Typ-II Supraleiter}{}{}
\ttxt{\eng{
Has a two critical magnetic fields.
\\$B < B_\text{c1}$: \fRef{:::meissner_effect}
\\$B_\text{c1} < B < B_\text{c2}$: \fRef{:::shubnikov_phase}
\\$B > B_\text{c2}$: Normal conductor
}}
\end{formula}
\begin{formula}{perfect_conductor}
\desc{Perfect conductor}{}{}
\desc[german]{Ideale Leiter}{}{}
\ttxt{
\eng{
In contrast to a superconductor, perfect conductors become diamagnetic only when the external magnetic field is turned on \textbf{after} the material was cooled below the critical temperature.
(\fRef{ed:fields:mag:induction:lenz})
}
\ger{
Im Gegensatz zu einem Supraleiter werden ideale Leiter nur dann diamagnetisch, wenn das externe magnetische Feld \textbf{nach} dem Abkühlen unter die kritische Temperatur eingeschaltet wird.
(\fRef{ed:fields:mag:induction:lenz})
}
}
\end{formula}
\begin{formula}{meissner_effect}
\desc{Meißner-Ochsenfeld effect}{Perfect diamagnetism}{$\chi=-1$ \qtyRef{magnetic_susceptibility}}
\desc[german]{Meißner-Ochsenfeld Effekt}{Perfekter Diamagnetismus}{}
\ttxt{
\eng{External magnetic field decays exponetially inside the superconductor below a critical temperature and a critical magnetic field, path-independant.}
\ger{Externes Magnetfeld fällt im Supraleiter exponentiell unterhalb einer kritischen Temperatur und unterhalb einer kritischen Feldstärke ab, wegunabhängig.}
}
\end{formula}
\begin{formula}{condensation_energy}
\desc{Condensation energy}{}{\QtyRef{gibbs_energy}}
\desc[german]{Kondensationsenergie}{}{}
\eq{
\d G &= -S \d T + V \d p - V \vecM \cdot \d\vecB \\
G_\text{con} &= G_\txn(B=0,T) - G_\txs(B=0,T) = \frac{V \Bcth^2(T)}{2\mu_0}
}
\end{formula}
\Subsection[
\eng{London equations}
\ger{London-Gleichungen}
]{london}
\begin{ttext}
\eng{
Quantitative description of the \fRef{cm:sc:meissner_effect}.
}
\ger{
Quantitative Beschreibung des \fRef{cm:sc:meissner_effect}s.
}
\end{ttext}
% \begin{formula}{coefficient}
% \desc{London-coefficient}{}{}
% \desc[german]{London-Koeffizient}{}{}
% \eq{\txLambda = \frac{m_\txs}{n_\txs q_\txs^2}}
% \end{formula}
\Eng[of_sc_particle]{of the superconducting particle}
\Ger[of_sc_particle]{der Supraleitenden Teilchen}
\begin{formula}{first}
% \vec{j} = \frac{nq\hbar}{m}\Grad S - \frac{nq^2}{m}\vec{A}
\desc{First London Equation}{}{$\vec{j}$ \qtyRef{current_density}, $m_\txs$/$n_\txs$/$q_\txs$ \qtyRef{mass}/\qtyRef{charge_carrier_density}/\qtyRef{charge} \GT{of_sc_particle}, \QtyRef{electric_field}}
\desc[german]{Erste London-Gleichun-}{}{}
\eq{
\pdv{\vec{j}_{\txs}}{t} = \frac{n_\txs q_\txs^2}{m_\txs}\vec{\E} {\color{gray}- \Order{\vec{j}_\txs^2}}
% \\{\color{gray} = \frac{q}{m}\Grad \left(\frac{1}{2} \TODO{FActor} \vec{j}^2\right)}
}
\end{formula}
\begin{formula}{second}
\desc{Second London Equation}{Describes the \fRef{cm:sc:meissner_effect}}{$\vec{j}$ \qtyRef{current_density}, $m_\txs$/$n_\txs$/$q_\txs$ \qtyRef{mass}/\qtyRef{charge_carrier_density}/\qtyRef{charge} \GT{of_sc_particle}, \QtyRef{magnetic_field}}
\desc[german]{Zweite London-Gleichung}{Beschreibt den \fRef{cm:sc:meissner_effect}}{}
\eq{
\Rot \vec{j_\txs} = -\frac{n_\txs q_\txs^2}{m_\txs} \vec{B}
}
\end{formula}
\begin{formula}{penetration_depth}
\desc{London penetration depth}{Depth at which $B$ is $1/\e$ times the value of $B_\text{ext}$}{$m_\txs$/$n_\txs$/$q_\txs$ \qtyRef{mass}/\qtyRef{charge_carrier_density}/\qtyRef{charge} \GT{of_sc_particle}}
\desc[german]{London Eindringtiefe}{Tiefe bei der $B$ das $1/\e$-fache von $B_\text{ext}$ ist}{}
\eq{\lambda_\txL = \sqrt{\frac{m_\txs}{\mu_0 n_\txs q_\txs^2}}}
\end{formula}
\begin{formula}{penetration_depth_temp}
\desc{Temperature dependence of \fRef{::penetration_depth}}{}{}
\desc[german]{Temperaturabhängigkeit der \fRef{::penetration_depth}}{}{}
\eq{\lambda_\txL(T) = \lambda_\txL(0) \frac{1}{\sqrt{1- \left(\frac{T}{T_\txc}\right)^4}}}
\end{formula}
\Subsection[
\eng{\GL Theory (GLAG)}
\ger{\GL Theorie (GLAG)}
]{gl}
\begin{ttext}
\eng{
\TODO{TODO}
}
\end{ttext}
\begin{formula}{boundary_energy}
\desc{Boundary energy}{}{$\Delta E_\text{boundary}$ \TODO{TODO}}
\desc[german]{Grenzflächenenergie}{}{}
\eq{\Delta E_\text{boundary} = \Delta E_\text{con} - \Delta E_\txB = (\xi_\gl - \lambda) \frac{B_\text{c,th}^2}{2\mu_0}}
\end{formula}
\begin{formula}{coherence_length}
\desc{\GL Coherence Length}{}{}
\desc[german]{\GL Kohärenzlänge}{}{}
\eq{
\xi_\gl &= \frac{\hbar}{\sqrt{2m \abs{\alpha}}} \\
\xi_\gl(T) &= \xi_\gl(0) \frac{1}{\sqrt{1-\frac{T}{\Tcrit}}}
}
\end{formula}
\begin{formula}{penetration_depth}
\desc{\GL Penetration Depth / Field screening length}{}{}
\desc[german]{\GL Eindringtiefe}{}{}
\eq{
\lambda_\gl &= \sqrt{\frac{m_\txs\beta}{\mu_0 \abs{\alpha} q_s^2}} \\
\lambda_\gl(T) &= \lambda_\gl(0) \frac{1}{\sqrt{1-\frac{T}{\Tcrit}}}
}
\end{formula}
\begin{formula}{first}
\desc{First Ginzburg-Landau Equation}{}{$\xi_\gl$ \fRef{cm:sc:gl:coherence_length}, $\lambda_\gl$ \fRef{cm:sc:gl:penetration_depth}}
\desc[german]{Erste Ginzburg-Landau Gleichung}{}{}
\eq{\alpha\Psi + \beta\abs{\Psi}^2 \Psi + \frac{1}{2m} (-i\hbar \Grad + 2e\vec{A})^2\Psi = 0}
\end{formula}
\begin{formula}{second}
\desc{Second Ginzburg-Landau Equation}{}{}
\desc[german]{Zweite Ginzburg-Landau Gleichung}{}{}
\eq{\vec{j_\txs} = \frac{ie\hbar}{m}(\Psi^*\Grad\Psi - \Psi\Grad\Psi^*) - \frac{4e^2}{m}\abs{\Psi}^2 \vec{A}}
\end{formula}
\TODO{proximity effect}
\Subsection[
\eng{Microscopic theory}
\ger{Mikroskopische Theorie}
]{micro}
\begin{formula}{isotop_effect}
\desc{Isotope effect}{Superconducting behaviour depends on atomic mass and thereby of the lattice \Rightarrow Microscopic origin}{$\Tcrit$ critial temperature, $M$ isotope mass, $\omega_\text{ph}$}
\desc[german]{Isotopeneffekt}{Supraleitung hängt von der Atommasse und daher von den Gittereigenschaften ab \Rightarrow Mikroskopischer Ursprung}{$\Tcrit$ kritische Temperatur, $M$ Isotopen-Masse, $\omega_\text{ph}$}
\eq{
\Tcrit \propto \frac{1}{\sqrt{M}} \\
\omega_\text{ph} \propto \frac{1}{\sqrt{M}} \Rightarrow \Tcrit \propto \omega_\text{ph}
}
\end{formula}
\begin{formula}{cooper_pairs}
\desc{Cooper pairs}{}{}
\desc[german]{Cooper-Paars}{}{}
\ttxt{
\eng{Conduction electrons reduce their energy through an attractive interaction: One electron passing by atoms attracts the these, which creats a positive charge region behind the electron, which in turn attracts another electron. }
}
\end{formula}
\Subsubsection[
\eng{BCS-Theory}
\ger{BCS-Theorie}
]{bcs}
\begin{ttext}
\eng{
Electron pairs form bosonic quasi-particles called Cooper pairs which can condensate into the ground state.
The wave function spans the whole material, which makes it conduct without resistance.
The exchange bosons between the electrons are phonons.
}
\ger{
Elektronenpaar bilden bosonische Quasipartikel (Cooper Paare) welche in den Grundzustand kondensieren können.
Die Wellenfunktion übersoannt den gesamten Festkörper, was einen widerstandslosen Ladungstransport garantiert.
Die Austauschbosononen zwischen den Elektronen sind Bosonen.
}
\end{ttext}
\def\BCS{{\text{BCS}}}
\begin{formula}{hamiltonian}
\desc{BCS Hamiltonian}{for $N$ interacting electrons}{
$c_{\veck\sigma}$ creation/annihilation operators create/destroy at $\veck$ with spin $\sigma$ \\
First term: non-interacting free electron gas\\
Second term: interaction energy
}
\desc[german]{BCS Hamiltonian}{}{}
\eq{
\hat{H}_\BCS =
\sum_{\sigma} \sum_\veck \epsilon_\veck \hat{c}_{\veck\sigma}^\dagger \hat{c}_{\veck\sigma}
+ \sum_{\veck,\veck^\prime} V_{\veck,\veck^\prime}
\hat{c}_{\veck\uparrow}^\dagger \hat{c}_{-\veck\downarrow}^\dagger
\hat{c}_{-\veck^\prime\downarrow} \hat{c}_{\veck^\prime,\uparrow}
}
\end{formula}
\begin{formula}{bogoliubov-valatin}
\desc{Bogoliubov-Valatin transformation}{Diagonalization of the \fRef{cm:sc:micro:bcs:hamiltonian} to derive excitation energies}{}
\desc[german]{Bogoliubov-Valatin transformation}{}{}
\eq{
\hat{H}_\BCS - N\mu = \sum_\veck \big[\xi_\veck - E_\veck + \Delta_\veck g_\veck^*\big] + \sum_\veck \big[E_\veck a_\veck^\dagger a_\veck + E_\veck \beta_{-\veck}^\dagger \beta_{-\veck}\big]
}
\end{formula}
\begin{formula}{gap_equation}
\desc{BCS-gap equation}{}{}
\desc[german]{}{}{}
\eq{\Delta_\veck^* = -\sum_\veck^+\prime V_{\veck,\veck^\prime} \frac{\Delta_{\veck^\prime}}{2E_\veck} \tanh \left(\frac{E_{\veck^\prime}}{2\kB T}\right)}
\end{formula}
\begin{formula}{tcrit_temp}
\desc{Temperatur dependance of the crictial temperature}{}{}
\desc[german]{Temperaturabhängigkeit der kritischen Temperatur}{}{}
\eq{ \Bcth(T) = \Bcth(0) \left[1- \left(\frac{t}{T_\txc}\right) \right] }
\TODO{empirical relation, relate to BCS}
\end{formula}

4
src/cm/semiconductors.tex
View File

@ -1,9 +1,9 @@
\Section[
\eng{Semiconductors}
\ger{Halbleiter}
]{semic}
]{sc}
\begin{formula}{types}
\desc{Intrinsic/extrinsic}{}{$n,p$ \fRef{cm:semic:charge_density_eq}}
\desc{Intrinsic/extrinsic}{}{$n,p$ \fRef{cm:sc:charge_density_eq}}
\desc[german]{Intrinsisch/Extrinsisch}{}{}
\ttxt{
\eng{

463
src/cm/superconductivity.tex Normal file
View File

@ -0,0 +1,463 @@
\def\txL{\text{L}}
\def\gl{\text{GL}}
\def\GL{Ginzburg-Landau }
\def\Tcrit{T_\text{c}}
\def\Bcth{B_\text{c,th}}
\Section[
\eng{Superconductivity}
\ger{Supraleitung}
]{super}
\begin{ttext}
\eng{
Materials for which the electric resistance jumps to 0 under a critical temperature $\Tcrit$.
Below $\Tcrit$ they have perfect conductivity and perfect diamagnetism, up until a critical magnetic field $\Bcth$.
}
\ger{
Materialien, bei denen der elektrische Widerstand beim unterschreiten einer kritischen Temperatur $\Tcrit$ auf 0 springt.
Sie verhalten sich dann wie ideale Leiter und ideale Diamagnete, bis zu einem kritischen Feld $\Bcth$.
}
\end{ttext}
\begin{formula}{type1}
\desc{Type-I superconductor}{}{}
\desc[german]{Typ-I Supraleiter}{}{}
\ttxt{\eng{
Has a single critical magnetic field, $\Bcth$.
\\$B < \Bcth$: \fRef{:::meissner_effect}
\\$B > \Bcth$: Normal conductor
}}
\end{formula}
\begin{formula}{type2}
\desc{Type-II superconductor}{}{}
\desc[german]{Typ-II Supraleiter}{}{}
\ttxt{\eng{
Has a two critical magnetic fields.
\\$B < B_\text{c1}$: \fRef{:::meissner_effect}
\\$B_\text{c1} < B < B_\text{c2}$: \fRef{:::shubnikov_phase}
\\$B > B_\text{c2}$: Normal conductor
}}
\end{formula}
\begin{formula}{perfect_conductor}
\desc{Perfect conductor}{}{}
\desc[german]{Ideale Leiter}{}{}
\ttxt{
\eng{
In contrast to a superconductor, perfect conductors become diamagnetic only when the external magnetic field is turned on \textbf{after} the material was cooled below the critical temperature.
(\fRef{ed:em:induction:lenz})
}
\ger{
Im Gegensatz zu einem Supraleiter werden ideale Leiter nur dann diamagnetisch, wenn das externe magnetische Feld \textbf{nach} dem Abkühlen unter die kritische Temperatur eingeschaltet wird.
(\fRef{ed:em:induction:lenz})
}
}
\end{formula}
\begin{formula}{meissner_effect}
\desc{Meißner-Ochsenfeld effect}{Perfect diamagnetism}{$\chi=-1$ \qtyRef{magnetic_susceptibility}}
\desc[german]{Meißner-Ochsenfeld Effekt}{Perfekter Diamagnetismus}{}
\ttxt{
\eng{External magnetic field decays exponetially inside the superconductor below a critical temperature and a critical magnetic field, path-independant.}
\ger{Externes Magnetfeld fällt im Supraleiter exponentiell unterhalb einer kritischen Temperatur und unterhalb einer kritischen Feldstärke ab, wegunabhängig.}
}
\end{formula}
\begin{formula}{bcth}
\desc{Thermodynamic cricitial field}{for \fRef[type I]{::type1} and \fRef[type II]{::type2}}{}
\desc[german]{Thermodynamisches kritische Feldstärke}{für \fRef[type I]{::type1} und \Ref[type II]{::type2}}{}
\eq{g_\txs - g_\txn = - \frac{\Bcth^2(T)}{2\mu_0}}
\end{formula}
\begin{formula}{shubnikov_phase}
\desc{Shubnikov phase}{in \fRef{::type2}}{}
\desc[german]{Shubnikov-Phase}{}{}
\ttxt{\eng{
}}
\end{formula}
\begin{formula}{condensation_energy}
\desc{Condensation energy}{}{\QtyRef{free_enthalpy}, \ConstRef{magnetic_vacuum_permeability}}
\desc[german]{Kondensationsenergie}{}{}
\eq{
\d G &= -S \d T + V \d p - V \vecM \cdot \d\vecB \\
G_\text{con} &= G_\txn(B=0,T) - G_\txs(B=0,T) = \frac{V \Bcth^2(T)}{2\mu_0}
}
\end{formula}
\Subsection[
\eng{London Theory}
\ger{London-Theorie}
]{london}
\begin{ttext}
\end{ttext}
\begin{formula}{description}
\desc{Description}{}{}
\desc[german]{Beschreibung}{}{}
\ttxt{\eng{
\begin{itemize}
\item Phenomenological theory
\item Quantitative description of the \fRef{cm:super:meissner_effect}.
\item Assumies uniform charge density $n(\vecr,t) = n(t)$ (London-approximation).
\item Does not work near $T_\txc$
\end{itemize}
}\ger{
\begin{itemize}
\item Phänomenologische Theorie
\item Quantitative Beschreibung des \fRef{cm:super:meissner_effect}s.
\item Annahme: uniforme Ladungsdichte $n(\vecr,t) = n(t)$ (London-Näherung)
\item Funktioniert nicht nahe $T_\txc$
\end{itemize}
}}
\end{formula}
% \begin{formula}{coefficient}
% \desc{London-coefficient}{}{}
% \desc[german]{London-Koeffizient}{}{}
% \eq{\txLambda = \frac{m_\txs}{n_\txs q_\txs^2}}
% \end{formula}
\Eng[of_sc_particle]{of the superconducting particle}
\Ger[of_sc_particle]{der Supraleitenden Teilchen}
\begin{formula}{first}
% \vec{j} = \frac{nq\hbar}{m}\Grad S - \frac{nq^2}{m}\vec{A}
\desc{First London Equation}{}{$\vec{j}$ \qtyRef{current_density}, $m_\txs$/$n_\txs$/$q_\txs$ \qtyRef{mass}/\qtyRef{charge_carrier_density}/\qtyRef{charge} \GT{of_sc_particle}, \QtyRef{electric_field}}
\desc[german]{Erste London-Gleichun-}{}{}
\eq{
\pdv{\vec{j}_{\txs}}{t} = \frac{n_\txs q_\txs^2}{m_\txs}\vec{\E} {\color{gray}- \Order{\vec{j}_\txs^2}}
% \\{\color{gray} = \frac{q}{m}\Grad \left(\frac{1}{2} \TODO{FActor} \vec{j}^2\right)}
}
\end{formula}
\begin{formula}{second}
\desc{Second London Equation}{Describes the \fRef{cm:super:meissner_effect}}{$\vec{j}$ \qtyRef{current_density}, $m_\txs$/$n_\txs$/$q_\txs$ \qtyRef{mass}/\qtyRef{charge_carrier_density}/\qtyRef{charge} \GT{of_sc_particle}, \QtyRef{magnetic_flux_density}}
\desc[german]{Zweite London-Gleichung}{Beschreibt den \fRef{cm:super:meissner_effect}}{}
\eq{
\Rot \vec{j_\txs} = -\frac{n_\txs q_\txs^2}{m_\txs} \vec{B}
}
\end{formula}
\begin{formula}{penetration_depth}
\desc{London penetration depth}{Depth at which $B$ is $1/\e$ times the value of $B_\text{ext}$}{$m_\txs$/$n_\txs$/$q_\txs$ \qtyRef{mass}/\qtyRef{charge_carrier_density}/\qtyRef{charge} \GT{of_sc_particle}}
\desc[german]{London Eindringtiefe}{Tiefe bei der $B$ das $1/\e$-fache von $B_\text{ext}$ ist}{}
\eq{\lambda_\txL = \sqrt{\frac{m_\txs}{\mu_0 n_\txs q_\txs^2}}}
\end{formula}
\begin{formula}{penetration_depth_temp}
\desc{Temperature dependence of \fRef{::penetration_depth}}{}{}
\desc[german]{Temperaturabhängigkeit der \fRef{::penetration_depth}}{}{}
\eq{\lambda_\txL(T) = \lambda_\txL(0) \frac{1}{\sqrt{1- \left(\frac{T}{T_\txc}\right)^4}}}
\end{formula}
\TODO{macroscopic wavefunction approach, energy-phase relation, current-phase relation}
\Subsubsection[
\eng{Josephson Effect}
\ger{Josephson Effekt}
]{josephson}
\begin{formula}{1st_relation}
\desc{1. Josephson relation}{Dissipationless supercurrent accros junction at zero applied voltage}{$\vecj_\text{C}=\frac{2e}{\hbar}E_\text{J}$ critical current, $\phi$ phase difference accross junction}
\desc[german]{1. Josephson Gleichung}{Dissipationsloser Suprastrom durch die Kreuzung ohne angelegte Spannung}{$\vecj_\text{C}=\frac{2e}{\hbar}E_\text{J}$ kritischer Strom, $\phi$ Phasendifferenz zwischen den Supraleitern}
\eq{\vecj_\txs(\vecr,t) = \vecj_\text{C}(\vecr,t) \sin\phi(\vecr,t)}
\end{formula}
\begin{formula}{2nd_relation}
\desc{2. Josephson relation}{Superconducting phase change is proportional to applied voltage}{$\phi$ phase differnce accross junction, \ConstRef{flux_quantum}, \QtyRef{voltage}}
\desc[german]{2. Josephson Gleichung}{Supraleitende Phasendifferenz is proportional zur angelegten Spannung}{$\phi$ Phasendifferenz, \ConstRef{flux_quantum}, \QtyRef{voltage}}
\eq{\odv{\phi(t)}{t} = \frac{2\pi}{\Phi_0} U(t)}
\end{formula}
\begin{formula}{coupling_energy}
\desc{Josephson coupling energy}{}{$A$ junction \qtyRef{area}, \ConstRef{flux_quantum}, \fRef[critical current density]{::1st_relation}, $\phi$ phase differnce accross junction}
\desc[german]{Josephson}{}{$A$ junction \qtyRef{area}, \ConstRef{flux_quantum}, \fRef[kritische Stromdichte]{::1st_relation}, $\phi$ Phasendifferenz zwischen den Supraleitern}
\eq{\frac{E_\txJ}{A} = \frac{\Phi_0 \vecj_\txc}{2\pi}(1-\cos\phi)}
\end{formula}
\Subsection[
\eng{\GL Theory (GLAG)}
\ger{\GL Theorie (GLAG)}
]{gl}
\begin{formula}{description}
\desc{Description}{}{}
\desc[german]{Beschreibung}{}{}
\ttxt{\eng{
\begin{itemize}
\item Phenomenological theory
\item Improvement on the Landau-Theory of 2nd order phase transitions
% which introduces an order parameter that is $0$ in the normal state and rises to saturation in the superconducting state.
\item Additional complex, position-dependent order parameter is introduced $\Psi(\vecr)$
\item Only valid close to $T_\txc$.
\item Does not have time dependancy
\end{itemize}
}\ger{
\begin{itemize}
\item Phänomenologische Theorie
\item Weiterentwicklung der Landau-Theorie für Phasenübergänge zweiter Ordnung,
% in der ein Ordnungsparameter in the normalen Phase 0 ist und ein der supraleitenden Phase bis zur Sättigung ansteigt.
\item Zusätzlicher, komplexer, ortsabhängiger Ordnungsparameter $\Psi(\vecr)$
\item Nur nahe $T_\txc$ gültig.
\item Beschreibt keine Zeitabhängigkeit
\end{itemize}
}}
\end{formula}
\begin{formula}{expansion}
\desc{Expansion}{Expansion of free enthalpy of superconducting state}{
$g_{\txs/\txn}$ specific \qtyRef{free_enthalpy} of superconducting/normal state,
$\Psi(\vecr) = \abs{\Psi_0(\vecr)} \e^{\I\theta(\vecr)}$ order parameter,
$n(\vecr) = \abs{\Psi}^2$ Cooper-Pair density,
\QtyRef{magnetic_flux_density},
\QtyRef{magnetic_vector_potential},
$\alpha(T) = -\bar{\alpha} \left(1-\frac{T}{T_\txc}\right)^2$,
% $\alpha > 0$ for $T > T_\txc$ and $\alpha < 0$ for $T< T_\txc$,
$\beta = \const > 0$
}
% \desc[german]{}{}{}
\begin{multline}
g_\txs = g_\txn + \alpha \abs{\Psi}^2 + \frac{1}{2}\beta \abs{\Psi}^4 +
\\ \frac{1}{2\mu_0}(\vecB_\text{ext} -\vecB_\text{inside})^2 + \frac{1}{2m_\txs} \abs{ \left(\frac{\hbar}{t}\Grad - q_\txs \vecA\right)\Psi}^2 + \dots
\end{multline}
\end{formula}
\begin{formula}{first}
\desc{First Ginzburg-Landau Equation}{Obtained by minimizing $g_\txs$ with respect to $\delta\Psi$ in \fRef{::expansion}}{
$\xi_\gl$ \fRef{cm:super:gl:coherence_length},
$\lambda_\gl$ \fRef{cm:super:gl:penetration_depth}
}
\desc[german]{Erste Ginzburg-Landau Gleichung}{}{}
\eq{\alpha\Psi + \beta\abs{\Psi}^2 \Psi + \frac{1}{2m} (-i\hbar \Grad + 2e\vec{A})^2\Psi = 0}
\end{formula}
\begin{formula}{second}
\desc{Second Ginzburg-Landau Equation}{Obtained by minimizing $g_\txs$ with respect to $\delta\vec{A}$ in \fRef{::expansion}}{}
\desc[german]{Zweite Ginzburg-Landau Gleichung}{}{}
\eq{\vec{j_\txs} = \frac{ie\hbar}{m}(\Psi^*\Grad\Psi - \Psi\Grad\Psi^*) - \frac{4e^2}{m}\abs{\Psi}^2 \vec{A}}
\end{formula}
\begin{formula}{coherence_length}
\desc{\GL Coherence Length}{Depth in the superconductor where $\abs{\Psi}$ goes from 0 to 1}{}
\desc[german]{\GL Kohärenzlänge}{Tiefe im Supraleiter, bei der $\abs{\Psi}$ von 0 auf 1 steigt}{}
\eq{
\xi_\gl &= \frac{\hbar}{\sqrt{2m \abs{\alpha}}} \\
\xi_\gl(T) &= \xi_\gl(0) \frac{1}{\sqrt{1-\frac{T}{\Tcrit}}}
}
\end{formula}
\begin{formula}{penetration_depth}
\desc{\GL Penetration Depth}{Field screening length\\Depth in the supercondcutor where $B_\text{ext}$ decays}{}
\desc[german]{\GL Eindringtiefe}{Tiefe im Supraleiter, bei der $B_\text{ext}$ abfällt}{}
\eq{
\lambda_\gl &= \sqrt{\frac{m_\txs\beta}{\mu_0 \abs{\alpha} q_s^2}} \\
\lambda_\gl(T) &= \lambda_\gl(0) \frac{1}{\sqrt{1-\frac{T}{\Tcrit}}}
}
\end{formula}
\begin{formula}{boundary_energy}
\desc{Boundary energy}{}{$\Delta E_\text{boundary}$ \TODO{TODO}, $\Delta E_\text{cond}$ \fRef{:::condensation_energy}}
\desc[german]{Grenzflächenenergie}{}{}
\eq{\Delta E_\text{boundary} = \Delta E_\text{con} - \Delta E_\txB = (\xi_\gl - \lambda) \frac{B_\text{c,th}^2}{2\mu_0}}
\end{formula}
\begin{formula}{parameter}
\desc{Ginzburg-Landau parameter}{}{}
\desc[german]{Ginzburg-Landau Parameter}{}{}
\eq{\kappa \equiv \frac{\lambda_\gl}{\xi_\gl}}
\eq{
\kappa \le \frac{1}{\sqrt{2}} &\quad\Rightarrow\quad\text{\fRef{cm:super:type1}} \\
\kappa \ge \frac{1}{\sqrt{2}} &\quad\Rightarrow\quad\text{\fRef{cm:super:type2}}
}
\end{formula}
\begin{formula}{ns_boundary}
\desc{Normal-superconductor boundary}{}{}
\desc[german]{Normal-Supraleiter Grenzfläche}{}{}
\eq{
\abs{\Psi(x)}^2 &= \frac{n_\txs(x)}{n_\txs(\infty)} = \tanh^2 \left(\frac{x}{\sqrt{2}\xi_\gl}\right) \\
B_z(x) &= B_z(0) \Exp{-\frac{x}{\lambda_\gl}}
}
\fig{img/cm_sc_n_s_boundary.pdf}
\TODO{plot, slide 106}
\end{formula}
\begin{formula}{bc1}
\desc{Lower critical magnetic field}{Above $B_\text{c1}$, flux starts to penetrate the superconducting phase}{\ConstRef{flux_quantum}, $\lambda\gl$ \fRef{::penetration_depth} $\kappa$ \fRef{::parameter}}
\desc[german]{Unteres kritisches Magnetfeld}{Über $B_\text{c1}$ dringt erstmals Fluss in die supraleitende Phase ein}{}
\eq{B_\text{c1} = \frac{\Phi_0}{4\pi\lambda\gl^2}(\ln\kappa+0.08) = \frac{1}{\sqrt{2}\kappa}(\ln\kappa + 0.08) \Bcth}
\end{formula}
\begin{formula}{bc2}
\desc{Upper critical magnetic field}{Above $B_\text{c2}$, superconducting phase is is destroyed}{\ConstRef{flux_quantum}, $\xi_\gl$ \fRef{::coherence_length}}
\desc[german]{Oberes kritisches Magnetfeld}{Über $B_\text{c2}$ ist die supraleitende Phase zerstört}{}
\eq{B_\text{c2} = \frac{\Phi_0}{2\pi\xi_\gl^2}}
\end{formula}
\begin{formula}{proximity_effect}
\desc{Proximity-Effect}{}{}
% \desc[german]{}{}{}
\ttxt{\eng{
Superconductor wavefunction extends into the normal conductor or isolator
\TODO{clarify}
}}
\end{formula}
\Subsection[
\eng{Microscopic theory}
\ger{Mikroskopische Theorie}
]{micro}
\begin{formula}{isotop_effect}
\desc{Isotope effect}{Superconducting behaviour depends on atomic mass and thereby on the lattice \Rightarrow Microscopic origin}{$\Tcrit$ critial temperature, $M$ isotope mass, $\omega_\text{ph}$}
\desc[german]{Isotopeneffekt}{Supraleitung hängt von der Atommasse und daher von den Gittereigenschaften ab \Rightarrow Mikroskopischer Ursprung}{$\Tcrit$ kritische Temperatur, $M$ Isotopen-Masse, $\omega_\text{ph}$}
\eq{
\Tcrit &\propto \frac{1}{\sqrt{M}} \\
\omega_\text{ph} &\propto \frac{1}{\sqrt{M}} \Rightarrow \Tcrit \propto \omega_\text{ph}
}
\end{formula}
\begin{formula}{cooper_pairs}
\desc{Cooper pairs}{}{}
\desc[german]{Cooper-Paars}{}{}
\ttxt{
\eng{Conduction electrons reduce their energy through an attractive interaction: One electron passing by atoms attracts the these, which creats a positive charge region behind the electron, which in turn attracts another electron. }
}
\end{formula}
\Subsubsection[
\eng{BCS-Theory}
\ger{BCS-Theorie}
]{bcs}
\begin{formula}{description}
\desc{Description}{}{}
\desc[german]{Beschreibung}{}{}
\ttxt{\eng{
\begin{itemize}
\item Electron pairs form bosonic quasi-particles called Cooper pairs which can condensate into the ground state
\item The wave function spans the whole material, which makes it conduct without resistance
\item The exchange bosons between the electrons are phonons
\end{itemize}
}\ger{
\begin{itemize}
\item Elektronenpaar bilden bosonische Quasipartikel (Cooper Paare) welche in den Grundzustand kondensieren können.
\item Die Wellenfunktion übersoannt den gesamten Festkörper, was einen widerstandslosen Ladungstransport garantiert
\item Die Austauschbosononen zwischen den Elektronen sind Bosonen
\end{itemize}
}}
\end{formula}
\def\BCS{{\text{BCS}}}
\begin{formula}{hamiltonian}
\desc{BCS Hamiltonian}{for $N$ interacting electrons}{
$c_{\veck\sigma}$ creation/annihilation operators create/destroy at $\veck$ with spin $\sigma$ \\
First term: non-interacting free electron gas\\
Second term: interaction energy
}
\desc[german]{BCS Hamiltonian}{}{}
\eq{
\hat{H}_\BCS =
\sum_{\sigma} \sum_\veck \epsilon_\veck \hat{c}_{\veck\sigma}^\dagger \hat{c}_{\veck\sigma}
+ \sum_{\veck,\veck^\prime} V_{\veck,\veck^\prime}
\hat{c}_{\veck\uparrow}^\dagger \hat{c}_{-\veck\downarrow}^\dagger
\hat{c}_{-\veck^\prime\downarrow} \hat{c}_{\veck^\prime,\uparrow}
}
\end{formula}
\begin{formula}{ansatz}
\desc{BCS ground state wave function Ansatz}{\fRef{comp:est:mean_field} approach\\Coherent fermionic state}{}
\desc[german]{BCS Grundzustandswellenfunktion-Ansatz}{\fRef{comp:est:mean_field} Ansatz\\Kohärenter, fermionischer Zustand}{}
\eq{\Ket{\Psi_\BCS} = \prod_{\veck=\veck_1,\dots,\veck_M} \left(u_\veck + v_\veck \hat{c}_{\veck\uparrow}^\dagger \hat{c}_{-\veck\downarrow}^\dagger\right) \ket{0} }
\end{formula}
\begin{formula}{coherence_factors}
\desc{BCS coherence factors}{}{$\abs{u_\veck}^2$/$\abs{v_\veck}^2$ probability that pair state is $(\veck\uparrow,\,-\veck\downarrow)$ is empty/occupied, $\abs{u_\veck}^2+\abs{v_\veck}^2 = 1$}
\desc[german]{BCS Kohärenzfaktoren}{}{$\abs{u_\veck}^2$/$\abs{v_\veck}^2$ Wahrscheinlichkeit, dass Paarzustand $(\veck\uparrow,\,-\veck\downarrow)$ leer/besetzt ist, $\abs{u_\veck}^2+\abs{v_\veck}^2 = 1$}
\eq{
u_\veck &= \frac{1}{\sqrt{1+\abs{\alpha_\veck}^2}} \\
v_\veck &= \frac{\alpha_\veck}{\sqrt{1+\abs{\alpha_\veck}^2}}
}
\end{formula}
\begin{formula}{potential}
\desc{BCS potential approximation}{}{}
\desc[german]{BCS Potentialnäherung}{}{}
\eq{
V_{\veck,\veck^\prime} =
\left\{ \begin{array}{rc}
-V_0 & k^\prime > k_\txF,\, k<k_\txF + \Delta k\\
0 & \tGT{else}
\end{array}\right.
}
\end{formula}
\begin{formula}{gap_at_t0}
\desc{BCS Gap at $T=0$}{}{\QtyRef{debye_frequency}, $V_0$ \fRef{::potential}, $D$ \qtyRef{dos}}
\desc[german]{BCS Lücke bei $T=0$}{}{}
\eq{
\Delta(T=0) &= \frac{\hbar\omega_\txD}{\Sinh{\frac{2}{V_0\.D(E_\txF)}}} \approx 2\hbar \omega_\txD\\
\frac{\Delta(T=0)}{\kB T_\txc} &= = \frac{\pi}{\e^\gamma} = 1.764
}
\end{formula}
\begin{formula}{cooper_pair_binding_energy}
\desc{Binding energy of Cooper pairs}{}{$E_\txF$ \absRef{fermi_energy}, \QtyRef{debye_frequency}, $V_0$ retarded potential, $D$ \qtyRef{dos}}
\desc[german]{Bindungsenergie von Cooper-Paaren}{}{}
\eq{E \approx 2E_\txF - 2\hbar\omega_\txD \Exp{-\frac{4}{V_0D(E_t\txF)}}}
\end{formula}
\Subsubsection[
\eng{Excitations in BCS}
% \ger{}
]{excite}
\begin{formula}{bogoliubov-valatin}
\desc{Bogoliubov-Valatin transformation}{Diagonalization of the \fRef{cm:super:micro:bcs:hamiltonian} to derive excitation energies}{}
\desc[german]{Bogoliubov-Valatin transformation}{}{}
\eq{
\hat{H}_\BCS - N\mu = \sum_\veck \big[\xi_\veck - E_\veck + \Delta_\veck g_\veck^*\big] + \sum_\veck \big[E_\veck a_\veck^\dagger a_\veck + E_\veck \beta_{-\veck}^\dagger \beta_{-\veck}\big]
}
\end{formula}
\begin{formula}{pairing_amplitude}
\desc{Pairing amplitude}{Excitation energy}{\TODO{gamma}}
\desc[german]{Paarungsamplitude}{Anregungsenergie}{}
\eq{E_\veck = \pm \sqrt{\xi^2_\veck + \abs{\Delta_\veck}^2}}
\end{formula}
\begin{formula}{coherence_factors_energy}
\desc{Energy dependance of the \fRef{:::coherence_factors}}{}{$E_\veck$ \fRef{::pairing_amplitude}, \GT{see} \fRef{:::coherence_facotrs}}
\desc[german]{Energieabhängigkeit der \fRef{:::coherence_factors}}{}{}
\eq{
\abs{u_\veck}^2 &= \frac{1}{2} \left(1+\frac{\xi_\veck}{E_\veck}\right) \\
\abs{v_\veck}^2 &= \frac{1}{2} \left(1-\frac{\xi_\veck}{E_\veck}\right) \\
u_\veck^* v_\veck &= \frac{\Delta_\veck}{2E_\veck}
}
\end{formula}
\begin{formula}{gap_equation}
\desc{BCS-gap equation}{}{}
\desc[german]{BCS Energielückengleichung}{}{}
\eq{\Delta_\veck^* = -\sum_\veck^\prime V_{\veck,\veck^\prime} \frac{\Delta_{\veck^\prime}}{2E_\veck} \tanh \left(\frac{E_{\veck^\prime}}{2\kB T}\right)}
\end{formula}
\begin{formula}{gap_t}
\desc{Temperature dependence of the BCS gap}{}{}
\desc[german]{Temperaturabhängigkeit der BCS-Lücke}{}{}
\eq{\frac{\Delta(T)}{\Delta(T=0)} \approx 1.74 \frac{1}{\sqrt{1-\frac{T}{T_\txC}}}}
\end{formula}
\begin{formula}{dos}
\desc{Quasiparticle density of states}{}{}
\desc[german]{Quasiteilchen Zustandsdichte}{}{}
\eq{D_\txs(E_\veck) = D_\txn(\xi_\veck) \pdv{\xi_\veck}{E_\veck} = \left\{
\begin{array}{ll}
D_\txn(E_\txF) \frac{E_\veck}{\sqrt{E^2_\veck -\Delta^2}} & E_\veck > \Delta \\
& E_\veck < \Delta
\end{array}
\right.}
\end{formula}
\begin{formula}{Bcth_temp}
\desc{Temperature dependance of the crictial magnetic field}{}{}
\desc[german]{Temperaturabhängigkeit des kritischen Magnetfelds}{}{}
\eq{ \Bcth(T) = \Bcth(0) \left[1- \left(\frac{T}{T_\txc}\right)^2 \right] }
\TODO{empirical relation, relate to BCS}
\end{formula}

13
src/comp/ad.tex
View File

@ -170,7 +170,7 @@
\eq{\Phi_{IJ}^{\mu\nu} \approx \frac{\vecF_I^\mu(\vecR_1^0, \dots, \vecR_J^0+\Delta s_J^\nu,\dots, \vecR_N^0)}{\Delta s_J^\nu}}
\end{formula}
\begin{formula}{dynamical_matrix}
\desc{Dynamical matrix}{Mass reduced \absRef[fourier transform]{fourier_transform} of the \fRef{comp:ad:latvib:force_constant_matrix}}{$\vec{L}$ vector from origin to unit cell $n$, $\alpha/\beta$ atom index in th unit cell, $\vecq$ \qtyRef{wave_vector}, $\Phi$ \fRef{comp:ad:latvib:force_constant_matrix}, $M$ \qtyRef{mass}}
\desc{Dynamical matrix}{Mass reduced \absRef[fourier transform]{fourier_transform} of the \fRef{comp:ad:latvib:force_constant_matrix}}{$\vec{L}$ vector from origin to unit cell $n$, $\alpha/\beta$ atom index in th unit cell, $\vecq$ \qtyRef{wavevector}, $\Phi$ \fRef{comp:ad:latvib:force_constant_matrix}, $M$ \qtyRef{mass}}
% \desc[german]{}{}{}
\eq{D_{\alpha\beta}^{\mu\nu} = \frac{1}{\sqrt{M_\alpha M_\beta}} \sum_{n^\prime} \Phi_{\alpha\beta}^{\mu\nu}(n-n^\prime) \e^{\I \vec{q}(\vec{L}_n - \vec{L}_{n^\prime})}}
\end{formula}
@ -214,14 +214,13 @@
\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 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.
\item Number of points to consider does NOT scale with system size
\item System evolves in time (\absRef{ehrenfest_theorem})
\item Computes time-dependant observables
\item Does not use \fRef{comp:ad:latvib:harmonic_approx} \Rightarrow Anharmonic effects included
\end{itemize}
\TODO{cleanup}
}}
\end{formula}

2
src/comp/est.tex
View File

@ -225,8 +225,6 @@
\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}

9
src/comp/ml.tex
View File

@ -19,22 +19,22 @@
\begin{formula}{accuracy}
\desc{Accuracy}{}{}
\desc[german]{Genauigkeit}{}{}
\eq{a = \frac{\tgt{cp}}{\tgt{fp} + \tgt{cp}}}
\eq{a = \frac{\tGT{::cp}}{\tGT{::fp} + \tGT{::cp}}}
\end{formula}
\eng{n_desc}{Number of data points}
\ger{n_desc}{Anzahl der Datenpunkte}
\begin{formula}{mean_abs_error}
\desc{Mean absolute error (MAE)}{}{$y$ \gt{y}, $\hat{y}$ \gt{yhat}, $n$ \gt{n_desc}}
\desc{Mean absolute error (MAE)}{}{$y$ \GT{::y}, $\hat{y}$ \GT{::yhat}, $n$ \GT{::n_desc}}
\desc[german]{Mittlerer absoluter Fehler (MAE)}{}{}
\eq{\text{MAE} = \frac{1}{n} \sum_{i=1}^n \abs{y_i - \hat{y}_i}}
\end{formula}
\begin{formula}{mean_square_error}
\desc{Mean squared error (MSE)}{}{$y$ \gt{y}, $\hat{y}$ \gt{yhat}, $n$ \gt{n_desc}}
\desc{Mean squared error (MSE)}{}{$y$ \GT{::y}, $\hat{y}$ \GT{::yhat}, $n$ \GT{::n_desc}}
\desc[german]{Methode der kleinsten Quadrate (MSE)}{Quadratwurzel des mittleren quadratischen Fehlers (SME)}{}
\eq{\text{MSE} = \frac{1}{n} \sum_{i=1}^n \left(y_i - \hat{y}_i\right)^2}
\end{formula}
\begin{formula}{root_mean_square_error}
\desc{Root mean squared error (RMSE)}{}{$y$ \gt{y}, $\hat{y}$ \gt{yhat}, $n$ \gt{n_desc}}
\desc{Root mean squared error (RMSE)}{}{$y$ \GT{::y}, $\hat{y}$ \GT{::yhat}, $n$ \GT{::n_desc}}
\desc[german]{Standardfehler der Regression}{Quadratwurzel des mittleren quadratischen Fehlers (RSME)}{}
\eq{\text{RMSE} = \sqrt{\frac{1}{n} \sum_{i=1}^n \left(y_i - \hat{y}_i\right)^2}}
\end{formula}
@ -135,7 +135,6 @@
\desc{Optimal weights}{for ridge regression}{$\lambda = \frac{\sigma^2}{\xi^2}$ shrinkage parameter, $\xi$ \absRef{variance} of the gaussian \fRef{math:pt:bayesian:prior}, $\sigma$ \absRef{variance} of the gaussian likelihood of the data}
\desc[german]{Optimale Gewichte}{für Ridge Regression}{}
\eq{\vec{\beta} = \left(\mat{X}^\T \mat{X} + \lambda \mathcal{1} \right)^{-1} \mat{X}^\T \vecy}
\TODO{Does this only work for gaussian data?}
\end{formula}
\begin{formula}{lasso}

1
src/ed/el.tex
View File

@ -34,6 +34,7 @@
\eq{
\epsilon(\omega)_\txr = \frac{\epsilon(\omega)}{\epsilon_0}
}
\hiddenQuantity{\epsilon_\txr}{}{s}
\end{formula}
\begin{formula}{vacuum_permittivity}

13
src/ed/em.tex
View File

@ -2,7 +2,7 @@
\eng{Electromagnetism}
\ger{Elektromagnetismus}
]{em}
\begin{formula}{speed_of_light}
\begin{formula}{vacuum_speed_of_light}
\desc{Speed of light}{in the vacuum}{}
\desc[german]{Lightgeschwindigkeit}{in the vacuum}{}
\constant{c}{exp}{
@ -10,7 +10,7 @@
}
\end{formula}
\begin{formula}{vacuum_relations}
\desc{Vacuum permittivity - permeability relation}{\TODO{Does this have a name?}}{\ConstRef{vacuum_permittivity}, \ConstRef{magnetic_vacuum_permeability}, \ConstRef{speed_of_light}}
\desc{Vacuum permittivity - permeability relation}{\TODO{Does this have a name?}}{\ConstRef{vacuum_permittivity}, \ConstRef{magnetic_vacuum_permeability}, \ConstRef{vacuum_speed_of_light}}
\desc[german]{Vakuum Permittivität - Permeabilität Beziehung}{}{}
\eq{
\epsilon_0 \mu_0 = \frac{1}{c^2}
@ -25,8 +25,9 @@
\end{formula}
\begin{formula}{poynting}
\desc{Poynting vector}{Directional energy flux or power flow of an electromagnetic field [$\si{\W\per\m^2}$]}{}
\desc{Poynting vector}{Directional energy flux or power flow of an electromagnetic field}{}
\desc[german]{Poynting-Vektor}{Gerichteter Energiefluss oder Leistungsfluss eines elektromgnetischen Feldes [$\si{\W\per\m^2}$]}{}
\quantity{\vecS}{\W\per\m^2}{v}
\eq{\vec{S} = \vec{E} \times \vec{H}}
\end{formula}
@ -37,8 +38,8 @@
\end{formula}
\begin{formula}{hamiltonian}
\desc{Hamiltonian of a particle in an electromagnetic field}{In the \fRef{ed:em:gauge:coulomb}}{\QtyRef{mass}, $\hat{p}$ \fRef{qm:se:momentum_operator}, \QtyRef{charge}, \QtyRef{magnetic_vector_potential}, \ConstRef{speed_of_light}}
\desc[german]{Hamiltonian eines Teilchens im elektromagnetischen Feld}{In der \fRef{ed:em:gauge:coulomb}}{}
\desc{Hamiltonian of a particle in an electromagnetic field}{In the \fRef{ed:em:maxwell:gauge:coulomb}}{\QtyRef{mass}, $\hat{p}$ \fRef{qm:se:momentum_operator}, \QtyRef{charge}, \QtyRef{magnetic_vector_potential}, \ConstRef{vacuum_speed_of_light}}
\desc[german]{Hamiltonian eines Teilchens im elektromagnetischen Feld}{In der \fRef{ed:em:maxwell:gauge:coulomb}}{}
\eq{
\hat{H} = \frac{1}{2m} \left[\hat{p} \ \frac{e \vec{A}}{c}\right]^2
}
@ -48,7 +49,7 @@
\Subsection[
\eng{Maxwell-Equations}
\ger{Maxwell-Gleichungen}
]{Maxwell}
]{maxwell}
\begin{formula}{vacuum}
\desc{Vacuum}{microscopic formulation}{}
\desc[german]{Vakuum}{Mikroskopische Formulierung}{}

2
src/ed/mag.tex
View File

@ -60,6 +60,7 @@
\eq{
\mu_\txr = \frac{\mu}{\mu_0}
}
\hiddenQuantity{\mu_\txr}{ }{}
\end{formula}
\begin{formula}{gauss_law}
@ -91,6 +92,7 @@
\desc{Susceptibility}{}{$\mu_\txr$ \fRef{ed:mag:relative_permeability}}
\desc[german]{Suszeptibilität}{}{}
\eq{\chi_\txm = \pdv{M}{B} = \mu_\txr - 1}
\hiddenQuantity{\chi}{}{}
\end{formula}

47
src/ed/misc.tex
View File

@ -40,9 +40,30 @@
\Subsection[
\eng{Integer quantum hall effect}
\ger{Ganzahliger Quantenhalleffekt}
\eng{Quantum hall effects}
\ger{Quantenhalleffekte}
]{quantum}
\begin{formula}{types}
\desc{Types of quantum hall effects}{}{}
\desc[german]{Arten von Quantenhalleffekten}{}{}
\ttxt{\eng{
\begin{itemize}
\item \textbf{Integer} (QHE): filling factor $\nu$ is an integer
\item \textbf{Fractional} (FQHE): filling factor $\nu$ is a fraction
\item \textbf{Spin} (QSHE): spin currents instead of charge currents
\item \textbf{Anomalous} (QAHE): symmetry breaking by internal effects instead of external magnetic fields
\end{itemize}
}\ger{
\begin{itemize}
\item \textbf{Integer} (QHE): Füllfaktor $\nu$ ist ganzzahlig
\item \textbf{Fractional} (FQHE): Füllfaktor $\nu$ ist ein Bruch
\item \textbf{Spin} (QSHE): Spin Ströme anstatt Ladungsströme
\item \textbf{Anomalous} (QAHE): Symmetriebruch durch interne Effekte anstatt druch ein externes Magnetfeld
\end{itemize}
}}
\end{formula}
\begin{formula}{conductivity}
\desc{Conductivity tensor}{}{}
@ -77,28 +98,6 @@
\eq{\nu = \frac{1}{3},\frac{2}{5},\frac{3}{7},\frac{2}{3}...}
\end{formula}
\begin{ttext}
\eng{
\begin{itemize}
\item \textbf{Integer} (QHE): filling factor $\nu$ is an integer
\item \textbf{Fractional} (FQHE): filling factor $\nu$ is a fraction
\item \textbf{Spin} (QSHE): spin currents instead of charge currents
\item \textbf{Anomalous} (QAHE): symmetry breaking by internal effects instead of external magnetic fields
\end{itemize}
}
\ger{
\begin{itemize}
\item \textbf{Integer} (QHE): Füllfaktor $\nu$ ist ganzzahlig
\item \textbf{Fractional} (FQHE): Füllfaktor $\nu$ ist ein Bruch
\item \textbf{Spin} (QSHE): Spin Ströme anstatt Ladungsströme
\item \textbf{Anomalous} (QAHE): Symmetriebruch durch interne Effekte anstatt druch ein externes Magnetfeld
\end{itemize}
}
\end{ttext}
\TODO{sort}
\Section[
\eng{Dipole-stuff}

9
src/main.tex
View File

@ -79,15 +79,14 @@
% \def\lambda{\temoji{sheep}}
% \def\psi{\temoji{pickup-truck}}
% \def\pi{\temoji{birthday-cake}}
% \def\Pi{\temoji{hospital}}
% \def\rho{\temoji{rhino}}
% % \def\Pi{\temoji{hospital}}
% % \def\rho{\temoji{rhino}}
% \def\nu{\temoji{unicorn}}
% \def\mu{\temoji{mouse}}
\newcommand{\TODO}[1]{{\color{fg-red}TODO:#1}}
\newcommand{\ts}{\textsuperscript}
\input{circuit.tex}
\input{util/macros.tex}
\input{util/math-macros.tex}
@ -146,7 +145,7 @@
\Input{cm/crystal}
\Input{cm/egas}
\Input{cm/charge_transport}
\Input{cm/low_temp}
\Input{cm/superconductivity}
\Input{cm/semiconductors}
\Input{cm/misc}
\Input{cm/techniques}
@ -190,7 +189,7 @@
]{elements}
\printAllElements
\newpage
% \Input{test}
\Input{test}
% \bibliographystyle{plain}
% \bibliography{ref}

1
src/math/calculus.tex
View File

@ -51,7 +51,6 @@
b_k &= \I(c_k - c_{-k}) \quad\text{\GT{for}}\,k\ge1
}
\end{formula}
\TODO{cleanup}
\Subsubsection[

6
src/math/probability_theory.tex
View File

@ -108,16 +108,16 @@
\begin{bigformula}{multivariate_normal}
\absLabel[multivariate_normal_distribution]
\desc{Multivariate normal distribution}{Multivariate Gaussian distribution}{$\vec{mu}$ \absRef{mean}, $\mat{\Sigma}$ \absRef{covariance}}
\desc{Multivariate normal distribution}{Multivariate Gaussian distribution}{$\vec{\mu}$ \absRef{mean}, $\mat{\Sigma}$ \absRef{covariance}}
\desc[german]{Mehrdimensionale Normalverteilung}{Multivariate Normalverteilung}{}
\TODO{k-variate normal plot}
\begin{distribution}
\disteq{parameters}{\vec{\mu} \in \R^k,+\quad \mat{\Sigma} \in \R^{k\times k}}
\disteq{support}{\vec{x} \in \vec{\mu} + \text{span}(\mat{\Sigma})}
\disteq{pdf}{\mathcal{N}(\vec{mu}, \mat{\Sigma}) = \frac{1}{(2\pi)^{k/2}} \frac{1}{\sqrt{\det{\Sigma}}} \Exp{-\frac{1}{2} \left(\vecx-\vec{\mu}\right)^\T \mat{\Sigma}^{-1} \left(\vecx-\vec{\mu}\right)}}
\disteq{pdf}{\mathcal{N}(\vec{\mu}, \mat{\Sigma}) = \frac{1}{(2\pi)^{k/2}} \frac{1}{\sqrt{\det{\Sigma}}} \Exp{-\frac{1}{2} \left(\vecx-\vec{\mu}\right)^\T \mat{\Sigma}^{-1} \left(\vecx-\vec{\mu}\right)}}
\disteq{mean}{\vec{\mu}}
\disteq{variance}{\mat{\Sigma}}
\end{distribution}
\TODO{k-variate normal plot}
\end{bigformula}
\begin{formula}{laplace}

66
src/pkg/mqconstant.sty
View File

@ -2,38 +2,56 @@
\RequirePackage{mqlua}
\RequirePackage{etoolbox}
\directLuaAux{
if constants == nil then
constants = {}
\begin{luacode}
constants = {}
function constantAdd(key, symbol, exp_or_def, fqname)
constants[key] = {
["symbol"] = symbol,
["units"] = units,
["exp_or_def"] = exp_or_def,
["values"] = {} -- array of {value, unit}
}
if fqname == "" then
constants[key]["fqname"] = fqnameGet()
else
constants[key]["fqname"] = fqname
end
end
}
function constantAddValue(key, value, unit)
table.insert(constants[key]["values"], { value = value, unit = unit })
end
function constantGetSymbol(key)
local const = constants[key]
if const == nil then return "???" end
local symbol = const["symbol"]
if symbol == nil then return "???" end
return symbol
end
function constantGetFqname(key)
local const = constants[key]
if const == nil then return "const:"..key end
local fqname_ = const["fqname"]
if fqname_ == nil then return "const:"..key end
return fqname_
end
\end{luacode}
% [1]: label to point to
% 2: key
% 3: symbol
% 4: either exp or def; experimentally or defined constant
\newcommand{\constant@new}[4][\relax]{
\directLuaAux{
constants["#2"] = {}
constants["#2"]["symbol"] = [[\detokenize{#3}]]
constants["#2"]["exp_or_def"] = [[\detokenize{#4}]]
constants["#2"]["values"] = {} %-- array of {value, unit}
}
\ifstrempty{#1}{}{
\directLuaAuxExpand{
constants["#2"]["linkto"] = [[#1]] %-- fqname required for getting the translation key
}
}
\newcommand{\constant@new}[4][]{%
\directLuaAuxExpand{constantAdd(\luastring{#2}, \luastringN{#3}, \luastringN{#4}, \luastring{#1})}%
}
% 1: key
% 2: value
% 3: units
\newcommand{\constant@addValue}[3]{
\directlua{
table.insert(constants["#1"]["values"], { value = [[\detokenize{#2}]], unit = [[\detokenize{#3}]] })
}
\newcommand{\constant@addValue}[3]{%
\directlua{constantAddValue(\luastring{#1}, \luastringN{#2}, \luastringN{#3})}%
}
% 1: key
\newcommand{\constant@getSymbol}[1]{\luavar{constantGetSymbol(\luastring{#1})}}
% 1: key
\newcommand\constant@print[1]{
@ -50,13 +68,5 @@
%--tex.sprint("VALUE ", i, v)
end
}
% label it only once
\directlua{
if constants["#1"]["labeled"] == nil then
constants["#1"]["labeled"] = true
tex.print("\\label{const:#1}")
end
}
\endgroup
}
\newcounter{constant}

27
src/pkg/mqformula.sty
View File

@ -1,5 +1,8 @@
\ProvidesPackage{mqformula}
\def\descwidth{0.3\textwidth}
\def\eqwidth{0.69\textwidth}
\RequirePackage{mqfqname}
\RequirePackage{mqconstant}
\RequirePackage{mqquantity}
@ -36,14 +39,13 @@
\begin{minipage}{#1}
}{
\IfTranslationExists{\ContentFqName}{%
\smartnewline
\noindent
\begingroup
\color{fg1}
\GT{\ContentFqName}
% \edef\temp{\GT{#1_defs}}
% \expandafter\StrSubstitute\expandafter{\temp}{:}{\\}
\endgroup
\smartnewline%
\noindent%
\begingroup%
\color{fg1}%
\raggedright%
\GT{\ContentFqName}%
\endgroup%
}{}
\end{minipage}
\end{lrbox}
@ -148,6 +150,15 @@
\newFormulaEntry
\constant@print{#1}
}
\newcommand{\fsplit}[3][0.5]{
\begin{minipage}{##1\linewidth}
##2
\end{minipage}
\begin{minipage}{\luavar{0.99-##1}\linewidth}
##3
\end{minipage}
}
}{
\mqfqname@leave
}

33
src/pkg/mqfqname.sty
View File

@ -1,4 +1,5 @@
\ProvidesPackage{mqfqname}
\edef\fqname{NULL}
\RequirePackage{mqlua}
\RequirePackage{etoolbox}
@ -48,8 +49,37 @@
end
\end{luacode}
% Allow using :<key>, ::<key> and so on
% where : points to current fqname, :: to the upper one and so on
\begin{luacode*}
function translateRelativeFqname(target)
local relN = 0
local relTarget = ""
warning('translateRelativeFqname', '(target=' .. target .. ') ');
for i = 1, #target do
local c = target:sub(i,i)
if c == ":" then
relN = relN + 1
else
relTarget = target:sub(i,#target)
break
end
end
if relN == 0 then
return target
end
local N = fqnameGetDepth()
local newtarget = fqnameGetN(N - relN + 1) .. ":" .. relTarget
warning('translateRelativeFqname', '(relN=' .. relN .. ') ' .. newtarget);
return newtarget
end
\end{luacode*}
\newcommand{\mqfqname@update}{%
\edef\fqname{\luavar{fqnameGet()}}
\edef\fqname{\luavar{fqnameGet()}} %
}
\newcommand{\mqfqname@enter}[1]{%
\directlua{fqnameEnter("\luaescapestring{#1}")}%
@ -104,6 +134,5 @@
\subsubsection{\fqnameText}
\mqfqname@label
}
\edef\fqname{NULL}
\newcommand\printFqName{\expandafter\detokenize\expandafter{\fqname}}

19
src/pkg/mqlua.sty
View File

@ -64,6 +64,7 @@ end
end
\end{luacode*}
% Write directlua command to aux and run it as well
% THESE CAN ONLY BE RUN BETWEEN \begin{document} and \end{document}
% This one expands the argument in the aux file:
\newcommand\directLuaAuxExpand[1]{
\immediate\write\luaAuxFile{\noexpand\directlua{#1}}
@ -76,15 +77,17 @@ end
}
% read
\IfFileExists{\jobname.lua.aux}{%
\input{\jobname.lua.aux}%
}{%
% \@latex@warning@no@line{"Lua aux not loaded!"}
\AtBeginDocument{
\IfFileExists{\jobname.lua.aux}{%
\input{\jobname.lua.aux}%
}{%
% \@latex@warning@no@line{"Lua aux not loaded!"}
}
% write
\newwrite\luaAuxFile
\immediate\openout\luaAuxFile=\jobname.lua.aux
\immediate\write\luaAuxFile{\noexpand\def\noexpand\luaAuxLoaded{True}}%
}
\def\luaAuxLoaded{False}
% write
\newwrite\luaAuxFile
\immediate\openout\luaAuxFile=\jobname.lua.aux
\immediate\write\luaAuxFile{\noexpand\def\noexpand\luaAuxLoaded{True}}%
\AtEndDocument{\immediate\closeout\luaAuxFile}

53
src/pkg/mqperiodictable.sty
View File

@ -5,13 +5,27 @@
% Print as list or as periodic table
% The data is taken from https://pse-info.de/de/data as json and parsed by the scripts/periodic_table.py
% INFO
\directLuaAux{
if elements == nil then
elements = {} %-- Symbol: {symbol, atomic_number, properties, ... }
elementsOrder = {} %-- Number: Symbol
\begin{luacode}
elements = {}
elementsOrder = {}
function elementAdd(symbol, nr, period, column)
elementsOrder[nr] = symbol
elements[symbol] = {
symbol = symbol,
atomic_number = nr,
period = period,
column = column,
properties = {}
}
end
function elementAddProperty(symbol, key, value)
if elements[symbol] and elements[symbol].properties then
elements[symbol].properties[key] = value
end
}
end
\end{luacode}
% 1: symbol
% 2: nr
@ -23,30 +37,22 @@
% 3: description
% 4: definitions/links
\newcommand{\desc}[4][english]{
% language, name, description, definitions
\ifblank{##2}{}{\DT[el:#1]{##1}{##2}}
\ifblank{##3}{}{\DT[el:#1_desc]{##1}{##3}}
\ifblank{##4}{}{\DT[el:#1_defs]{##1}{##4}}
}
\directLuaAux{
elementsOrder[#2] = "#1";
elements["#1"] = {};
elements["#1"]["symbol"] = [[\detokenize{#1}]];
elements["#1"]["atomic_number"] = [[\detokenize{#2}]];
elements["#1"]["period"] = [[\detokenize{#3}]];
elements["#1"]["column"] = [[\detokenize{#4}]];
elements["#1"]["properties"] = {};
\directLuaAuxExpand{
elementAdd(\luastring{#1}, \luastring{#2}, \luastring{#3}, \luastring{#4})
}
% 1: key
% 2: value
\newcommand{\property}[2]{
\directlua{ %-- writing to aux is only needed for references for now
elements["#1"]["properties"]["##1"] = "\luaescapestring{\detokenize{##2}}" %-- cant use [[ ]] because electron_config ends with ]
\directlua{
elementAddProperty(\luastring{#1}, \luastringN{##1}, \luastringN{##2})
}
}
\edef\lastElementName{#1}
}{
% \expandafter\printElement{\lastElementName}
\ignorespacesafterend
}
@ -56,9 +62,7 @@
\par\noindent\ignorespaces
\vspace{0.5\baselineskip}
\begingroup
% label it only once
% \detokenize{\label{el:#1}}
\directlua{
\directlua{
if elements["#1"]["labeled"] == nil then
elements["#1"]["labeled"] = true
tex.print("\\phantomsection\\label{el:#1}")
@ -70,12 +74,8 @@
\directlua{
tex.sprint("Symbol: \\ce{"..elements["#1"]["symbol"].."}")
tex.sprint("\\\\Number: "..elements["#1"]["atomic_number"])
}
\directlua{
%--tex.sprint("Hier steht Luatext" .. ":", #elementVals)
for key, value in pairs(elements["#1"]["properties"]) do
tex.sprint("\\\\\\hspace*{1cm}{\\GT{", key, "}: ", value, "}")
%--tex.sprint("VALUE ", i, v)
tex.sprint("\\\\\\hspace*{1cm}{\\GT{"..key.."}: "..value.."}")
end
}
\end{ContentBoxWithExplanation}
@ -84,6 +84,7 @@
\vspace{0.5\baselineskip}
\ignorespacesafterend
}
\newcommand{\printAllElements}{
\directlua{
%-- tex.sprint("\\printElement{"..val.."}")

66
src/pkg/mqquantity.sty
View File

@ -1,41 +1,59 @@
\ProvidesPackage{mqquantity}
\RequirePackage{mqlua}
\RequirePackage{mqfqname}
\RequirePackage{etoolbox}
\directLuaAux{
quantities = quantities or {}
}
% TODO: MAYBE:
% store the fqname where the quantity is defined
% In qtyRef then use the stored label to reference it, instead of linking to qty:<name>
% Use the mqlua hyperref function
% [1]: label to point to
\begin{luacode}
quantities = {}
function quantityAdd(key, symbol, units, comment, fqname)
quantities[key] = {
["symbol"] = symbol,
["units"] = units,
["comment"] = comment
}
if fqname == "" then
quantities[key]["fqname"] = fqnameGet()
else
quantities[key]["fqname"] = fqname
end
end
function quantityGetSymbol(key)
local qty = quantities[key]
if qty == nil then return "???" end
local symbol = qty["symbol"]
if symbol == nil then return "???" end
return symbol
end
function quantityGetFqname(key)
local qty = quantities[key]
if qty == nil then return "qty:"..key end
local fqname_ = qty["fqname"]
if fqname_ == nil then return "qty:"..key end
return fqname_
end
\end{luacode}
% [1]: label to point to, if not given use current fqname
% 2: key - must expand to a valid lua string!
% 3: symbol
% 4: units
% 5: comment key to translation
\newcommand{\quantity@new}[5][\relax]{%
\directLuaAux{
quantities["#2"] = {}
quantities["#2"]["symbol"] = [[\detokenize{#3}]]
quantities["#2"]["units"] = [[\detokenize{#4}]]
quantities["#2"]["comment"] = [[\detokenize{#5}]]
}
\ifstrempty{#1}{}{
\directLuaAuxExpand{
quantities["#2"]["linkto"] = [[#1]] %-- fqname required for getting the translation key
}
}
\newcommand{\quantity@new}[5][]{%
\directLuaAuxExpand{quantityAdd(\luastring{#2}, \luastringN{#3}, \luastringN{#4}, \luastringN{#5}, \luastring{#1})}
}
% 1: key
\newcommand{\quantity@getSymbol}[1]{\luavar{quantityGetSymbol(\luastring{#1})}}
% 1: key
\newcommand\quantity@print[1]{
\begingroup % for label
Symbol: $\luavar{quantities["#1"]["symbol"]}$
Symbol: $\luavar{quantityGetSymbol(\luastring{#1})}$
\hfill Unit: $\directlua{split_and_print_units(quantities["#1"]["units"])}$ %
% label it only once
\directlua{
if quantities["#1"]["labeled"] == nil then
quantities["#1"]["labeled"] = true
tex.print("\\label{qty:#1}")
end
}%
\endgroup%
}

159
src/pkg/mqref.sty
View File

@ -1,6 +1,7 @@
\ProvidesPackage{mqref}
\RequirePackage{mqlua}
\RequirePackage{mqfqname}
\RequirePackage{mqquantity}
\newcommand\luaDoubleFieldValue[3]{%
\directlua{
@ -102,7 +103,7 @@
else -- mark as missing and referenced in current section
missingLabels[target] = fqnameGet()
end
if text == "" then
if text == nil or text == "" then
tex.sprint(s .. "{" .. tlGetFallbackCurrent(target) .. "}")
else
tex.sprint(s .. "{" .. text .. "}")
@ -117,31 +118,6 @@
\directlua{hyperref(\luastring{#2}, \luastring{#1})}%
}
\begin{luacode*}
function translateRelativeFqname(target)
local relN = 0
local relTarget = ""
warning('translateRelativeFqname', '(target=' .. target .. ') ');
for i = 1, #target do
local c = target:sub(i,i)
if c == ":" then
relN = relN + 1
else
relTarget = target:sub(i,#target)
break
end
end
if relN == 0 then
return target
end
local N = fqnameGetDepth()
local newtarget = fqnameGetN(N - relN + 1) .. ":" .. relTarget
warning('translateRelativeFqname', '(relN=' .. relN .. ') ' .. newtarget);
return newtarget
end
\end{luacode*}
\newcommand{\fRef}[2][]{
\directlua{hyperref(translateRelativeFqname(\luastring{#2}), \luastring{#1})}
}
@ -166,23 +142,25 @@
% Quantities
% <symbol>
\newrobustcmd{\qtyRef}[1]{%
\edef\tempname{\luaDoubleFieldValue{quantities}{"#1"}{"linkto"}}%
\hyperref[qty:#1]{\GT{\tempname:#1}}%
\newrobustcmd{\qtyRef}[2][]{%
% \edef\tempname{\luaDoubleFieldValue{quantities}{"#1"}{"fqname"}}%
% \hyperref[qty:#1]{\GT{\tempname}}%
\directlua{hyperref(quantityGetFqname(\luastring{#2}), \luastring{#1})}
}
% <symbol> <name>
\newrobustcmd{\QtyRef}[1]{%
$\luaDoubleFieldValue{quantities}{"#1"}{"symbol"}$ \qtyRef{#1}%
\newrobustcmd{\QtyRef}[2][]{%
$\quantity@getSymbol{#2}$ \qtyRef{#2}{}%
}
% Constants
% <name>
\newrobustcmd{\constRef}[1]{%
\edef\tempname{\luaDoubleFieldValue{constants}{"#1"}{"linkto"}}%
\hyperref[const:#1]{\GT{\tempname:#1}}%
\newrobustcmd{\constRef}[2][]{%
% \edef\tempname{\luaDoubleFieldValue{constants}{"#1"}{"linkto"}}%
% \hyperref[const:#1]{\GT{\tempname}}%
\directlua{hyperref(constantGetFqname(\luastring{#2}), \luastring{#1})}
}
% <symbol> <name>
\newrobustcmd{\ConstRef}[1]{%
$\luaDoubleFieldValue{constants}{"#1"}{"symbol"}$ \constRef{#1}%
\newrobustcmd{\ConstRef}[2][]{%
$\constant@getSymbol{#2}$ \constRef{#2}%
}
% Element from periodic table
% <symbol>
@ -199,60 +177,95 @@
% "LABELS"
% These currently do not place a label,
% instead they provide an alternative way to reference an existing label
\directLuaAux{
absLabels = absLabels or {}
abbrLabels = abbrLabel or {}
}
\begin{luacode}
absLabels = absLabels or {}
abbrLabels = abbrLabels or {}
function absLabelAdd(key, target, translationKey)
absLabels[key] = {
fqname = (target == "") and fqnameGet() or target,
translation = translationKey or ""
}
end
function absLabelGetTarget(key)
if absLabels[key] then
return absLabels[key].fqname or "abs:" .. key
else
return "abs:" .. key
end
end
function absLabelGetTranslationKey(key)
if absLabels[key] then
return absLabels[key].translation or ""
else
return ""
end
end
function abbrLabelAdd(key, target, label)
abbrLabels[key] = {
abbr = label,
fqname = (target == "") and fqnameGet() or target
}
end
function abbrLabelGetTarget(key)
if abbrLabels[key] then
return abbrLabels[key].fqname or "abbr:" .. key
else
return "abbr:" .. key
end
end
function abbrLabelGetAbbr(key)
if abbrLabels[key] then
return abbrLabels[key].abbr or ""
else
return ""
end
end
\end{luacode}
% [1]: translation key, if different from target
% 2: target (fqname to point to)
% 2: target (fqname to point to), if left empty will use current fqname
% 3: key
\newcommand{\absLink}[3][\relax]{
\newcommand{\absLink}[3][]{
\directLuaAuxExpand{
absLabels["#3"] = {}
absLabels["#3"]["fqname"] = [[#2]]
absLabels["#3"]["translation"] = [[#1]] or [[#2]]
% if [[#1]] == "" then
% absLabels["#3"]["translation"] = [[#2]]
% else
% absLabels["#3"]["translation"] = [[#1]]
% end
}
absLabelAdd(\luastring{#3}, \luastring{#2}, \luastring{#1})
}
}
% [1]: target (fqname to point to)
% 2: key
% 3: label (abbreviation)
\newcommand{\abbrLink}[3][sec:\fqname]{
\newcommand{\abbrLink}[3][]{
\directLuaAuxExpand{
abbrLabels["#2"] = {}
abbrLabels["#2"]["abbr"] = [[#3]]
abbrLabels["#2"]["fqname"] = [[#1]]
}
abbrLabelAdd(\luastring{#2}, \luastring{#1}, \luastring{#3})
}
}
\newcommand{\fLabel}
% [1]: text
% 2: key
\newcommand{\absRef}[2][]{%
\directlua{
if absLabels["#2"] == nil then
tex.sprint(string.sanitize(\luastring{#2}) .. "???")
else
if \luastring{#1} == "" then %-- if [#1] is not given, use translation of key as text, else us given text
tex.sprint("\\hyperref[" .. absLabels["#2"]["fqname"] .. "]{\\GT{" .. absLabels["#2"]["translation"] .. "}}")
else
tex.sprint("\\hyperref[" .. absLabels["#2"]["fqname"] .. "]{\luaescapestring{#1}}")
end
local text = (\luastring{#1} == "") and absLabelGetTranslationKey(\luastring{#2}) or \luastring{#1}
if text \string~= "" then
text = tlGetFallbackCurrent(text)
end
}
hyperref(absLabelGetTarget(\luastring{#2}, text))
}%
}
\newrobustcmd{\abbrRef}[1]{%
\directlua{
if abbrLabels["#1"] == nil then
tex.sprint(string.sanitize(\luastring{#1}) .. "???")
else
tex.sprint("\\hyperref[" .. abbrLabels["#1"]["fqname"] .. "]{" .. abbrLabels["#1"]["abbr"] .. "}")
end
}
\directlua{hyperref(abbrLabelGetTarget(\luastring{#1}), abbrLabelGetAbbr(\luastring{#1}))}
% if abbrLabels["#1"] == nil then
% tex.sprint(string.sanitize(\luastring{#1}) .. "???")
% else
% tex.sprint("\\hyperref[" .. abbrLabels["#1"]["fqname"] .. "]{" .. abbrLabels["#1"]["abbr"] .. "}")
% end
% }
}

7
src/pkg/mqtranslation.sty
View File

@ -8,8 +8,9 @@
\begin{luacode}
translations = translations or {}
-- string to append to missing translations
-- string to append to missing translations, for debugging
-- unknownTranslation = "???"
-- unknownTranslation = "!UT!"
unknownTranslation = ""
-- language that is set in usepackage[<lang>]{babel}
language = "\languagename"
@ -117,8 +118,8 @@
% shortcuts for translations
% 1: key
\newcommand{\gt}[1]{\luavar{tlGetFallbackCurrent(\luastring{\fqname:#1})}}
\newrobustcmd{\robustGT}[1]{\luavar{tlGetFallbackCurrent(\luastring{#1})}}
\newcommand{\GT}[1]{\luavar{tlGetFallbackCurrent(\luastring{#1})}}
\newrobustcmd{\robustGT}[1]{\luavar{tlGetFallbackCurrent(translateRelativeFqname(\luastring{#1}))}}
\newcommand{\GT}[1]{\luavar{tlGetFallbackCurrent(translateRelativeFqname(\luastring{#1}))}}
% text variants for use in math mode
\newcommand{\tgt}[1]{\text{\gt{#1}}}

22
src/qm/atom.tex
View File

@ -50,7 +50,7 @@
\end{formula}
\begin{formula}{rydberg_constant_heavy}
\desc{Rydberg constant}{for heavy atoms}{\ConstRef{electron_mass}, \ConstRef{elementary_charge}, \QtyRef{vacuum_permittivity}, \ConstRef{planck}, \ConstRef{vacuum_speed_of_light}}
\desc{Rydberg constant}{for heavy atoms}{\ConstRef{electron_mass}, \ConstRef{charge}, \ConstRef{vacuum_permittivity}, \ConstRef{planck}, \ConstRef{vacuum_speed_of_light}}
\desc[german]{Rydberg-Konstante}{für schwere Atome}{}
\constant{R_\infty}{exp}{
\val{10973731.568157(12)}{\per\m}
@ -61,7 +61,7 @@
\end{formula}
\begin{formula}{rydberg_constant_corrected}
\desc{Rydberg constant}{corrected for nucleus mass $M$}{\QtyRef{rydberg_constant_heavy}, $\mu = \left(\frac{1}{m_\txe} + \frac{1}{M}\right)^{-1}$ \GT{reduced_mass}, \ConstRef{electron_mass}}
\desc{Rydberg constant}{corrected for nucleus mass $M$}{\ConstRef{rydberg_constant_heavy}, $\mu = \left(\frac{1}{m_\txe} + \frac{1}{M}\right)^{-1}$ \GT{reduced_mass}, \ConstRef{electron_mass}}
\desc[german]{Rydberg Konstante}{korrigiert für Kernmasse $M$}{}
\eq{R_\txM = \frac{\mu}{m_\txe} R_\infty}
\end{formula}
@ -85,15 +85,15 @@
\Subsection[
\eng{Corrections}
\ger{Korrekturen}
]{corrections}
]{corrections}
\Subsubsection[
\eng{Darwin term}
\ger{Darwin-Term}
]{darwin}
]{darwin}
\begin{ttext}[desc]
\eng{Relativisitc correction: Because of the electrons zitterbewegung, it is not entirely localised. \TODO{fact check}}
\ger{Relativistische Korrektur: Elektronen führen eine Zitterbewegung aus und sind nicht vollständig lokalisiert.}
\eng{Relativisitc correction: Accounts for interaction with nucleus (non-zero wavefunction at nucleaus position)}
\ger{Relativistische Korrektur: Berücksichtigt die Interatkion mit dem Kern (endliche Wellenfunktion bei der Kernposition)}
\end{ttext}
\begin{formula}{energy_shift}
\desc{Energy shift}{}{}
@ -110,7 +110,7 @@
\Subsubsection[
\eng{Spin-orbit coupling (LS-coupling)}
\ger{Spin-Bahn-Kopplung (LS-Kopplung)}
]{ls_coupling}
]{ls_coupling}
\begin{ttext}[desc]
\eng{The interaction of the electron spin with the electrostatic field of the nuclei lead to energy shifts.}
\ger{The Wechselwirkung zwischen dem Elektronenspin und dem elektrostatischen Feld des Kerns führt zu Energieverschiebungen.}
@ -131,10 +131,10 @@
\Subsubsection[
\eng{Fine-structure}
\ger{Feinstruktur}
]{fine_structure}
]{fine_structure}
\begin{ttext}[desc]
\eng{The fine-structure combines relativistic corrections \ref{sec:qm:h:corrections:darwin} and the spin-orbit coupling \ref{sec:qm:h:corrections:ls_coupling}.}
\ger{Die Feinstruktur vereint relativistische Korrekturen \ref{sec:qm:h:corrections:darwin} und die Spin-Orbit-Kupplung \ref{sec:qm:h:corrections:ls_coupling}.}
\eng{The fine-structure combines \fRef[relativistic corrections]{qm:h:corrections:darwin} and \fRef{qm:h:corrections:ls_coupling}.}
\ger{Die Feinstruktur vereint \fRef[relativistische Korrekturen]{qm:h:corrections:darwin} und \fRef{qm:h:corrections:ls_coupling}.}
\end{ttext}
\begin{formula}{energy_shift}
\desc{Energy shift}{}{}
@ -146,7 +146,7 @@
\Subsubsection[
\eng{Lamb-shift}
\ger{Lamb-Shift}
]{lamb_shift}
]{lamb_shift}
\begin{ttext}[desc]
\eng{The interaction of the electron with virtual photons emitted/absorbed by the nucleus leads to a (very small) shift in the energy level.}
\ger{The Wechselwirkung zwischen dem Elektron und vom Kern absorbierten/emittierten virtuellen Photonen führt zu einer (sehr kleinen) Energieverschiebung.}

7
src/qm/qm.tex
View File

@ -145,7 +145,6 @@
\desc[german]{Kommutatorrelationen}{}{}
\eq{[A, BC] = [A, B]C - B[A,C]}
\end{formula}
\TODO{add some more?}
\begin{formula}{function}
\desc{Commutator involving a function}{}{given $[A,[A,B]] = 0$}
@ -288,7 +287,8 @@
\Subsubsection[
\eng{Ehrenfest theorem}
\ger{Ehrenfest-Theorem}
]{ehrenfest_theorem}
]{ehrenfest_theorem}
\absLink{}{ehrenfest_theorem}
\GT{see_also} \ref{sec:qm:basics:schroedinger_equation:correspondence_principle}
\begin{formula}{ehrenfest_theorem}
\desc{Ehrenfest theorem}{applies to both pictures}{}
@ -486,11 +486,10 @@
\ger{Aharanov-Bohm Effekt}
]{aharanov_bohm}
\begin{formula}{phase}
\desc{Acquired phase}{Electron along a closed loop aquires a phase proportional to the enclosed magnetic flux}{}
\desc{Acquired phase}{Electron along a closed loop aquires a phase proportional to the enclosed magnetic flux}{\QtyRef{magnetic_vector_potential}, \QtyRef{magnetic_flux}}
\desc[german]{Erhaltene Phase}{Elektron entlang eines geschlossenes Phase erhält eine Phase die proportional zum eingeschlossenen magnetischem Fluss ist}{}
\eq{\delta = \frac{2 e}{\hbar} \oint \vec{A}\cdot \d\vec{s} = \frac{2 e}{\hbar} \Phi}
\end{formula}
\TODO{replace with loop intergral symbol and add more info}
\Section[
\eng{Periodic potentials}
\ger{Periodische Potentiale}

58
src/quantities.tex
View File

@ -98,10 +98,16 @@
\desc[german]{Volumen}{$d$ dimensionales Volumen}{}
\quantity{V}{\m^d}{}
\end{formula}
\begin{formula}{heat_capacity}
\desc{Heat capacity}{}{}
\desc[german]{Wärmekapazität}{}{}
\quantity{C}{\joule\per\kelvin}{}
\begin{formula}{heat}
\desc{Heat}{}{}
\desc[german]{Wärme}{}{}
\quantity{Q}{\joule}{}
\end{formula}
\begin{formula}{density}
\desc{Density}{}{}
\desc[german]{Dichte}{}{}
\quantity{\rho}{\kg\per\m^3}{s}
\end{formula}
\Subsection[
@ -125,6 +131,12 @@
\quantity{\rho}{\coulomb\per\m^3}{s}
\end{formula}
\begin{formula}{charge_carrier_density}
\desc{Charge carrier density}{Number of charge carriers per volume}{}
\desc[german]{Ladungsträgerdichte}{Anzahl der Ladungsträger pro Volumen}{}
\quantity{n}{\per\m^3}{s}
\end{formula}
\begin{formula}{frequency}
\desc{Frequency}{}{}
\desc[german]{Frequenz}{}{}
@ -136,6 +148,12 @@
\quantity{\omega}{\radian\per\s}{s}
\eq{\omega = \frac{2\pi/T}{2\pi f}}
\end{formula}
\begin{formula}{angular_velocity}
\desc{Angular velocity}{}{\QtyRef{time_period}, \QtyRef{frequency}}
\desc[german]{Kreisgeschwindigkeit}{}{}
\quantity{\vec{\omega}}{\radian\per\s}{v}
\eq{\vec{\omega} = \frac{\vecr \times \vecv}{r^2}}
\end{formula}
\begin{formula}{time_period}
\desc{Time period}{}{\QtyRef{frequency}}
@ -144,19 +162,41 @@
\eq{T = \frac{1}{f}}
\end{formula}
\begin{formula}{conductivity}
\desc{Conductivity}{}{}
\desc[german]{Leitfähigkeit}{}{}
\quantity{\sigma}{\per\ohm\m}{}
\begin{formula}{wavelength}
\desc{Wavelength}{}{}
\desc[german]{Wellenlänge}{}{}
\quantity{\lambda}{\per\m}{s}
\end{formula}
\begin{formula}{wave_vector}
\begin{formula}{angular_wavenumber}
\desc{Wavenumber}{Angular wavenumber}{\QtyRef{wavelength}}
\desc[german]{Wellenzahl}{}{}
\eq{k = \frac{2\pi}{\lambda}}
\quantity{k}{\radian\per\m}{s}
\end{formula}
\begin{formula}{wavevector}
\desc{Wavevector}{Vector perpendicular to the wavefront}{}
\desc[german]{Wellenvektor}{Vektor senkrecht zur Wellenfront}{}
\eq{\abs{k} = \frac{2\pi}{\lambda}}
\quantity{\vec{k}}{1\per\m}{v}
\end{formula}
\begin{formula}{impedance}
\desc{Impedance}{}{}
\desc[german]{Impedanz}{}{}
\quantity{Z}{\ohm}{s}
\end{formula}
\begin{formula}{resistance}
\desc{Resistance}{}{}
\desc[german]{Widerstand}{}{}
\quantity{R}{\ohm}{s}
\end{formula}
\begin{formula}{inductance}
\desc{Inductance}{}{}
\desc[german]{Induktivität}{}{}
\quantity{L}{\henry=\kg\m^2\per\s^2\ampere^2=\weber\per\ampere=\volt\s\per\ampere=\ohm\s}{s}
\end{formula}
\Subsection[
\eng{Others}
\ger{Sonstige}

517
src/quantum_computing.tex
View File

@ -73,7 +73,7 @@
\end{formula}
\begin{formula}{2nd_josephson_relation}
\desc{2. Josephson relation}{superconducting phase change is proportional to applied voltage}{$\varphi_0=\frac{\hbar}{2e}$ reduced flux quantum}
\desc{2. Josephson relation}{Superconducting phase change is proportional to applied voltage}{$\varphi_0=\frac{\hbar}{2e}$ reduced flux quantum}
\desc[german]{2. Josephson Gleichung}{Supraleitende Phasendifferenz is proportional zur angelegten Spannung}{$\varphi_0=\frac{\hbar}{2e}$ reduziertes Flussquantum}
\eq{\odv{\hat{\delta}}{t}=\frac{1}{i\hbar}[\hat{H},\hat{\delta}] = -\frac{2eU}{i\hbar}[\hat{n},\hat{\delta}] = \frac{1}{\varphi_0} U}
\end{formula}
@ -108,29 +108,24 @@
\end{formula}
\Subsection[
\eng{Josephson Qubit??}
\ger{TODO}
]{josephson_qubit}
\begin{tikzpicture}
\draw (0,0) to[capacitor] (0,2);
\draw (0,0) to (2,0);
\draw (0,2) to (2,2);
\draw (2,0) to[josephson] (2,2);
\eng{Josephson junction based qubits}
\ger{Qubits mit Josephson-Junctions}
]{josephson_qubit}
\draw[->] (3,1) -- (4,1);
\draw (5,0) to[josephsoncap=$C_\text{J}$] (5,2);
\end{tikzpicture}
\TODO{Include schaltplan}
\begin{tikzpicture}
\draw (0,0) to[sV=$V_\text{g}$] (0,2);
\draw (0,2) to[capacitor=$C_\text{g}$] (2,2);
\draw (2,2) to (4,2);
\draw (2,0) to[josephsoncap=$C_\text{J}$] (2,2);
\draw (4,0) to[capacitor=$C_C$] (4,2);
\draw (0,0) to (2,0);
\draw (2,0) to (4,0);
\end{tikzpicture}
\begin{formula}{circuit}
\desc{General circuit}{}{}
\desc[german]{Allgemeiner Schaltplan}{}{}
\centering
\begin{tikzpicture}
\draw (0,0) to[sV=$V_\text{g}$] (0,2);
\draw (0,2) to[capacitor=$C_\text{g}$] (2,2);
\draw (2,2) to (4,2);
\draw (2,0) to[josephsoncap=$C_\text{J}$] (2,2);
\draw (4,0) to[capacitor=$C_C$] (4,2);
\draw (0,0) to (2,0);
\draw (2,0) to (4,0);
\end{tikzpicture}
\end{formula}
\begin{formula}{charging_energy}
\desc{Charging energy / electrostatic energy}{}{}
@ -140,10 +135,9 @@
\begin{formula}{josephson_energy}
\desc{Josephson energy}{}{}
\desc[german]{Josephson-Energie?}{}{}
\desc[german]{Josephson-Energie}{}{}
\eq{E_\text{J} = \frac{I_0 \phi_0}{2\pi}}
\end{formula}
\TODO{Was ist I0}
\begin{formula}{inductive_energy}
\desc{Inductive energy}{}{}
@ -164,262 +158,265 @@
\end{formula}
\begin{minipage}{0.8\textwidth}
\begingroup
\setlength{\tabcolsep}{0.9em} % horizontal
\renewcommand{\arraystretch}{2} % vertical
\begin{tabular}{ p{0.5cm} |p{0.8cm}||p{2.2cm}|p{1.9cm}|p{1.9cm}|p{1.8cm}|}
\multicolumn{1}{c}{}& \multicolumn{1}{c}{} &\multicolumn{4}{c}{$E_L/(E_J-E_L)$} \\
\cline{3-6}
\multicolumn{1}{c}{} & & $0$ & $\ll$ 1 & $\sim 1$ & $\gg 1$\\
\hhline{~|=====|}
\multirow{4}{*}{$\frac{E_J}{E_C}$} & $\ll 1$ & cooper-pair box & & & \\
\cline{2-6}
& $\sim 1$ & quantronium & fluxonium & &\\
\cline{2-6}
& $\gg 1$ &transmon & & & flux qubit\\
\cline{2-6}
& $\ggg 1$ & & & phase qubit & \\
\cline{2-6}
\end{tabular}
\endgroup
\end{minipage}
\begin{minipage}{0.2\textwidth}
\begin{tikzpicture}[scale=2]
\draw[-latex,line width=2pt] (0,1)--++(0,1) node[midway,above,sloped] () {charge noise};
\draw[-latex,line width=2pt] (0,1)--++(0,1) node[midway,below,sloped] () {sensitivity};
\draw[-latex,line width=2pt] (0,0)--++(1,1) node[midway,above,sloped] () {flux noise};
\draw[-latex,line width=2pt] (0,0)--++(1,1) node[midway,below,sloped] () {sensitivity};
\draw[-latex,line width=2pt] (0,0)--++(1,-1) node[midway,above,sloped] () {critical current};
\draw[-latex,line width=2pt] (0,0)--++(1,-1) node[midway,below,sloped] () {noise sensitivity};
\end{tikzpicture}
\end{minipage}
\begin{bigformula}{comparison}
\desc{Comparison of superconducting qubits}{}{$E_C$ \fRef{::charging_energy}, $E_L$ \fRef{::inductive_energy}, $E_{\txJ}$ \fRef{::josephson_energy}}
\desc[german]{Vergleich supraleitender Qubits}{}{}
\begin{minipage}{0.8\textwidth}
\begingroup
\setlength{\tabcolsep}{0.9em} % horizontal
\renewcommand{\arraystretch}{2} % vertical
\begin{tabular}{ p{0.5cm} |p{0.8cm}||p{2.2cm}|p{1.9cm}|p{1.9cm}|p{1.8cm}|}
\multicolumn{1}{c}{}& \multicolumn{1}{c}{} &\multicolumn{4}{c}{$E_L/(E_J-E_L)$} \\
\cline{3-6}
\multicolumn{1}{c}{} & & $0$ & $\ll$ 1 & $\sim 1$ & $\gg 1$\\
\hhline{~|=====|}
\multirow{4}{*}{$\frac{E_J}{E_C}$} & $\ll 1$ & cooper-pair box & & & \\
\cline{2-6}
& $\sim 1$ & quantronium & fluxonium & &\\
\cline{2-6}
& $\gg 1$ &transmon & & & flux qubit\\
\cline{2-6}
& $\ggg 1$ & & & phase qubit & \\
\cline{2-6}
\end{tabular}
\endgroup
\end{minipage}
\begin{minipage}{0.19\textwidth}
\begin{tikzpicture}[scale=2]
\draw[-latex,line width=2pt] (0,1)--++(0,1) node[midway,above,sloped] () {charge noise};
\draw[-latex,line width=2pt] (0,1)--++(0,1) node[midway,below,sloped] () {sensitivity};
\draw[-latex,line width=2pt] (0,0)--++(1,1) node[midway,above,sloped] () {flux noise};
\draw[-latex,line width=2pt] (0,0)--++(1,1) node[midway,below,sloped] () {sensitivity};
\draw[-latex,line width=2pt] (0,0)--++(1,-1) node[midway,above,sloped] () {critical current};
\draw[-latex,line width=2pt] (0,0)--++(1,-1) node[midway,below,sloped] () {noise sensitivity};
\end{tikzpicture}
\end{minipage}
\end{bigformula}
\Subsection[
\eng{Cooper Pair Box (CPB) qubit}
\ger{Cooper Paar Box (QPB) Qubit}
]{cpb}
\begin{ttext}
\eng{
= voltage bias junction\\= charge qubit?
}
\ger{}
\end{ttext}
\begin{formula}{circuit}
\desc{Cooper Pair Box / Charge qubit}{
\begin{itemize}
\gooditem large anharmonicity
\baditem sensitive to charge noise
\end{itemize}
}{}
\desc[german]{Cooper Pair Box / Charge Qubit}{
\begin{itemize}
\gooditem Große Anharmonizität
\baditem Sensibel für charge noise
\end{itemize}
}{}
\centering
\begin{tikzpicture}
\draw (0,0) to[sV=$V_\text{g}$] (0,2);
% \draw (0,0) to (2,0);
\draw (0,2) to[capacitor=$C_\text{g}$] (2,2);
\draw (2,0) to[josephsoncap=$C_\text{J}$] (2,2);
\draw (0,0) to (2,0);
\end{tikzpicture}
\end{formula}
\begin{formula}{hamiltonian}
\desc{Hamiltonian}{}{}
\desc[german]{Hamiltonian}{}{}
\eq{\hat{H} &= 4 E_C(\hat{n} - n_\text{g})^2 - E_\text{J} \cos\hat{\phi} \\
&=\sum_n \left[4 E_C (n-n_\text{g})^2 \ket{n}\bra{n} - \frac{E_\text{J}}{2}\ket{n}\bra{n+1}+\ket{n+1}\bra{n}\right] }
\end{formula}
\Subsection[
\eng{Transmon qubit}
\ger{Transmon Qubit}
]{transmon}
\begin{formula}{circuit}
\desc{Transmon qubit}{
Josephson junction with a shunt \textbf{capacitance}.
\begin{itemize}
\gooditem charge noise insensitive
\baditem small anharmonicity
\end{itemize}
}{}
\desc[german]{Transmon Qubit}{
Josephson-Kontakt mit einem parallelen \textbf{kapzitiven Element}.
\begin{itemize}
\gooditem Charge noise resilient
\baditem Geringe Anharmonizität $\alpha$
\end{itemize}
}{}
\centering
\begin{tikzpicture}
% \draw (0,0) to[sV=$V_\text{g}$] ++(0,3)
% to[capacitor=$C_\text{g}$] ++(2,0)
\draw (0,0) to ++(2,0) to ++(0,-0.5) to[josephsoncap=$C_\text{J}$] ++(-0,-2) to ++(0,-0.5) to ++(-2,0)
to[capacitor=$C_C$] ++(0,3);
\end{tikzpicture}
\end{formula}
\begin{formula}{hamiltonian}
\desc{Hamiltonian}{}{}
\desc[german]{Hamiltonian}{}{}
\eq{\hat{H} &= 4 E_C\hat{n}^2 - E_\text{J} \cos\hat{\phi}}
\end{formula}
\eng{Charge based qubits}
\ger{Ladungsbasierte Qubits}
]{charge}
\begin{bigformula}{comparison}
\desc{Comparison of charge qubit states}{}{}
\desc[german]{Vergleich der Zustände von Ladungsbasierten Qubits}{}{}
\fig{img/qubit_transmon.pdf}
\end{bigformula}
\Subsubsection[
\eng{Tunable Transmon qubit}
\ger{Tunable Transmon Qubit}
]{tunable}
\eng{Cooper Pair Box (CPB) qubit}
\ger{Cooper Paar Box (QPB) Qubit}
]{cpb}
\begin{ttext}
\eng{
= voltage bias junction\\= charge qubit?
}
\ger{}
\end{ttext}
\begin{formula}{circuit}
\desc{Frequency tunable transmon}{By using a \fRef{qc:scq:elements:squid} instead of a \fRef{qc:scq:elements:josephson_junction}, the qubit is frequency tunable through an external field}{}
\desc[german]{}{Durch Nutzung eines \fRef{qc:scq:elements:squid} anstatt eines \fRef{qc:scq:elements:josephson_junction}s, ist die Frequenz des Qubits durch ein externes Magnetfeld einstellbar}{}
\desc{Cooper Pair Box / Charge qubit}{
\begin{itemize}
\gooditem large anharmonicity
\baditem sensitive to charge noise
\end{itemize}
}{}
\desc[german]{Cooper Pair Box / Charge Qubit}{
\begin{itemize}
\gooditem Große Anharmonizität
\baditem Sensibel für charge noise
\end{itemize}
}{}
\centering
\begin{tikzpicture}
\draw (0,0) to[sV=$V_\text{g}$] (0,2);
% \draw (0,0) to (2,0);
\draw (0,2) to[capacitor=$C_\text{g}$] (2,2);
\draw (2,0) to[josephsoncap=$C_\text{J}$] (2,2);
\draw (0,0) to (2,0);
\end{tikzpicture}
\end{formula}
\begin{formula}{hamiltonian}
\desc{Hamiltonian}{}{}
\desc[german]{Hamiltonian}{}{}
\eq{\hat{H} &= 4 E_C(\hat{n} - n_\text{g})^2 - E_\text{J} \cos\hat{\phi} \\
&=\sum_n \left[4 E_C (n-n_\text{g})^2 \ket{n}\bra{n} - \frac{E_\text{J}}{2}\ket{n}\bra{n+1}+\ket{n+1}\bra{n}\right] }
\end{formula}
\Subsubsection[
\eng{Transmon qubit}
\ger{Transmon Qubit}
]{transmon}
\begin{formula}{circuit}
\desc{Transmon qubit}{
Josephson junction with a shunt \textbf{capacitance}.
\begin{itemize}
\gooditem charge noise insensitive
\baditem small anharmonicity
\end{itemize}
}{}
\desc[german]{Transmon Qubit}{
Josephson-Kontakt mit einem parallelen \textbf{kapzitiven Element}.
\begin{itemize}
\gooditem Charge noise resilient
\baditem Geringe Anharmonizität $\alpha$
\end{itemize}
}{}
\centering
\begin{tikzpicture}
% \draw (0,0) to[sV=$V_\text{g}$] ++(0,3)
% to[capacitor=$C_\text{g}$] ++(2,0)
\draw (0,0) to ++(-2,0)
to ++(3,0) to ++(0,-0.5) \squidloop{loop}{SQUID} to ++(0,-0.5) to ++(-3,0)
to[capacitor=$C_C$] ++(0,3);
\draw (0,0) to ++(2,0) to ++(0,-0.5) to[josephsoncap=$C_\text{J}$] ++(-0,-2) to ++(0,-0.5) to ++(-2,0)
to[capacitor=$C_C$] ++(0,3);
\end{tikzpicture}
\end{formula}
\begin{formula}{energy}
\desc{Josephson energy}{}{$d=(E_\text{J1}-E_\text{J2})/(E_\text{J1}+E_\text{J2})$ asymmetry}
\desc[german]{Josephson Energie}{}{$d=(E_\text{J1}-E_\text{J2})/(E_\text{J1}+E_\text{J2})$ Asymmetrie}
\eq{E_\text{J,eff}(\Phi_\text{ext}) = (E_\text{J1}+E_\text{J2}) \sqrt{\cos^2\left(\pi\frac{\Phi_\text{ext}}{\Phi_0}\right) + d^2 \sin \left(\pi\frac{\Phi_\text{ext}}{\Phi_0}\right)}}
\end{formula}
\begin{formula}{hamiltonian}
\desc{Hamiltonian}{}{}
\desc[german]{Hamiltonian}{}{}
\eq{\hat{H} = 4E_C \hat{n}^2 - \frac{1}{2} E_\text{J,eff}(\Phi_\text{ext}) \sum_{n}\left[\ket{n}\bra{n+1} + \ket{n+1}\bra{n}\right]}
\eq{\hat{H} &= 4 E_C\hat{n}^2 - E_\text{J} \cos\hat{\phi}}
\end{formula}
\begin{figure}[h]
\centering
\includegraphics[width=0.8\textwidth]{img/qubit_transmon.pdf}
\caption{Transmon and so TODO}
\label{fig:img-qubit_transmon-pdf}
\end{figure}
\Subsubsection[
\eng{Tunable Transmon qubit}
\ger{Tunable Transmon Qubit}
]{tunable}
\begin{formula}{circuit}
\desc{Frequency tunable transmon}{By using a \fRef{qc:scq:elements:squid} instead of a \fRef{qc:scq:elements:josephson_junction}, the qubit is frequency tunable through an external field}{}
\desc[german]{}{Durch Nutzung eines \fRef{qc:scq:elements:squid} anstatt eines \fRef{qc:scq:elements:josephson_junction}s, ist die Frequenz des Qubits durch ein externes Magnetfeld einstellbar}{}
\centering
\begin{tikzpicture}
% \draw (0,0) to[sV=$V_\text{g}$] ++(0,3)
% to[capacitor=$C_\text{g}$] ++(2,0)
\draw (0,0) to ++(-2,0)
to ++(3,0) to ++(0,-0.5) \squidloop{loop}{SQUID} to ++(0,-0.5) to ++(-3,0)
to[capacitor=$C_C$] ++(0,3);
\end{tikzpicture}
\end{formula}
\Subsection[
\eng{Phase qubit}
\ger{Phase Qubit}
]{phase}
\begin{formula}{circuit}
\desc{Phase qubit}{}{}
\desc[german]{Phase Qubit}{}{}
\centering
\begin{tikzpicture}
% \draw (0,0) to[sV=$V_\text{g}$] ++(0,3)
% to ++(2,0) coordinate(top1)
% to ++(2,0) coordinate(top2)
% to ++(2,0) coordinate(top3);
% \draw (0,0)
% to ++(2,0) coordinate(bot1)
% to ++(2,0) coordinate(bot2)
% to ++(2,0) coordinate(bot3);
\draw[color=gray] (0,0) to[capacitor=$C_C$] (0,-2);
% \draw (top1) to ++(0,-0.5) to[josephsoncap=$C_\text{J}$] ++(-0,-2) to (bot2);
\draw(0,0) to ++(2,0) to[josephsoncap=$C_\text{J}$] ++(0,-2) to ++(-2,0);
\draw (2,0) to ++(2,0) to[cute inductor=$E_L$] ++(0,-2) to ++(-2,0);
\node at (3,-1.5) {$\Phi_\text{ext}$};
\end{tikzpicture}
\\\TODO{Ist beim Fluxonium noch die Voltage source dran?}
\end{formula}
\begin{formula}{hamiltonian}
\desc{Hamiltonian}{}{$\delta = \frac{\phi}{\phi_0}$}
\desc[german]{Hamiltonian}{}{}
\eq{\hat{H} = E_C \hat{n}^2 - E_J \cos \hat{\delta} + E_L(\hat{\delta} - \delta_s)^2}
\end{formula}
\Eng[TESTT]{This is only a test}
\Ger[TESTT]{}
\GT{TESTT}
\begin{formula}{energy}
\desc{Josephson energy}{}{$d=(E_\text{J1}-E_\text{J2})/(E_\text{J1}+E_\text{J2})$ asymmetry}
\desc[german]{Josephson Energie}{}{$d=(E_\text{J1}-E_\text{J2})/(E_\text{J1}+E_\text{J2})$ Asymmetrie}
\eq{E_\text{J,eff}(\Phi_\text{ext}) = (E_\text{J1}+E_\text{J2}) \sqrt{\cos^2\left(\pi\frac{\Phi_\text{ext}}{\Phi_0}\right) + d^2 \sin \left(\pi\frac{\Phi_\text{ext}}{\Phi_0}\right)}}
\end{formula}
\begin{formula}{hamiltonian}
\desc{Hamiltonian}{}{}
\desc[german]{Hamiltonian}{}{}
\eq{\hat{H} = 4E_C \hat{n}^2 - \frac{1}{2} E_\text{J,eff}(\Phi_\text{ext}) \sum_{n}\left[\ket{n}\bra{n+1} + \ket{n+1}\bra{n}\right]}
\end{formula}
\Subsection[
\eng{Flux qubit}
\ger{Flux Qubit}
]{flux}
\TODO{TODO}
\begin{formula}{circuit}
\desc{Flux qubit / Persistent current qubit}{}{}
\desc[german]{Flux Qubit / Persistent current qubit}{}{}
\centering
\begin{tikzpicture}
\draw (0,0) to[josephsoncap=$\alpha E_\text{J}$, scale=0.8, transform shape] (0,-3);
\draw (0,0) to ++(3,0)
to[josephsoncap=$E_\text{J}$] ++(0,-1.5)
to[josephsoncap=$E_\text{J}$] ++(0,-1.5)
to ++(-3,0);
\node at (1.5,-1.5) {$\Phi_\text{ext}$};
\end{tikzpicture}
% \begin{tikzpicture}
% \draw (0,0) to[sV=$V_\text{g}$] ++(0,3)
% to ++(2,0) coordinate(top1)
% to ++(2,0) coordinate(top2)
% to ++(2,0) coordinate(top3);
% \draw (0,0)
% to ++(2,0) coordinate(bot1)
% to ++(2,0) coordinate(bot2)
% to ++(2,0) coordinate(bot3);
% \draw[color=gray] (top1) to[capacitor=$C_C$] (bot1);
% % \draw (top1) to ++(0,-0.5) to[josephsoncap=$C_\text{J}$] ++(-0,-2) to (bot2);
% \draw[scale=0.8, transform shape] (top2) to[josephsoncap=$\alpha E_\text{J}$] (bot2);
% \draw (top3)
% to[josephsoncap=$E_\text{J}$] ++(0,-1.5)
% to[josephsoncap=$E_\text{J}$] (bot3);
% \node at (5,0.5) {$\Phi_\text{ext}$};
% \end{tikzpicture}
\end{formula}
\Subsection[
\eng{Inductive qubits}
\ger{Induktive Qubits}
]{inductive}
\begin{bigformula}{comparison}
\desc{Comparison of other qubit states}{}{}
\desc[german]{Vergleich der Zustände von anderen Qubits}{}{}
\fig{img/qubit_flux_onium.pdf}
\end{bigformula}
\Subsubsection[
\eng{Phase qubit}
\ger{Phase Qubit}
]{phase}
\begin{formula}{circuit}
\desc{Phase qubit}{}{}
\desc[german]{Phase Qubit}{}{}
\centering
\begin{tikzpicture}
% \draw (0,0) to[sV=$V_\text{g}$] ++(0,3)
% to ++(2,0) coordinate(top1)
% to ++(2,0) coordinate(top2)
% to ++(2,0) coordinate(top3);
% \draw (0,0)
% to ++(2,0) coordinate(bot1)
% to ++(2,0) coordinate(bot2)
% to ++(2,0) coordinate(bot3);
\draw[color=gray] (0,0) to[capacitor=$C_C$] (0,-2);
% \draw (top1) to ++(0,-0.5) to[josephsoncap=$C_\text{J}$] ++(-0,-2) to (bot2);
\draw(0,0) to ++(2,0) to[josephsoncap=$C_\text{J}$] ++(0,-2) to ++(-2,0);
\draw (2,0) to ++(2,0) to[cute inductor=$E_L$] ++(0,-2) to ++(-2,0);
\node at (3,-1.5) {$\Phi_\text{ext}$};
\end{tikzpicture}
\end{formula}
\begin{formula}{hamiltonian}
\desc{Hamiltonian}{}{$\delta = \frac{\phi}{\phi_0}$}
\desc[german]{Hamiltonian}{}{}
\eq{\hat{H} = E_C \hat{n}^2 - E_J \cos \hat{\delta} + E_L(\hat{\delta} - \delta_s)^2}
\end{formula}
\Subsubsection[
\eng{Flux qubit}
\ger{Flux Qubit}
]{flux}
\begin{formula}{circuit}
\desc{Flux qubit / Persistent current qubit}{}{}
\desc[german]{Flux Qubit / Persistent current qubit}{}{}
\centering
\begin{tikzpicture}
\draw (0,0) to[josephsoncap=$\alpha E_\text{J}$, scale=0.8, transform shape] (0,-3/0.8);
\draw (0,0) to ++(3,0)
to[josephsoncap=$E_\text{J}$] ++(0,-1.5)
to[josephsoncap=$E_\text{J}$] ++(0,-1.5)
to ++(-3,0);
\node at (1.5,-1.5) {$\Phi_\text{ext}$};
\end{tikzpicture}
% \begin{tikzpicture}
% \draw (0,0) to[sV=$V_\text{g}$] ++(0,3)
% to ++(2,0) coordinate(top1)
% to ++(2,0) coordinate(top2)
% to ++(2,0) coordinate(top3);
% \draw (0,0)
% to ++(2,0) coordinate(bot1)
% to ++(2,0) coordinate(bot2)
% to ++(2,0) coordinate(bot3);
% \draw[color=gray] (top1) to[capacitor=$C_C$] (bot1);
% % \draw (top1) to ++(0,-0.5) to[josephsoncap=$C_\text{J}$] ++(-0,-2) to (bot2);
% \draw[scale=0.8, transform shape] (top2) to[josephsoncap=$\alpha E_\text{J}$] (bot2);
% \draw (top3)
% to[josephsoncap=$E_\text{J}$] ++(0,-1.5)
% to[josephsoncap=$E_\text{J}$] (bot3);
% \node at (5,0.5) {$\Phi_\text{ext}$};
% \end{tikzpicture}
\end{formula}
\Subsection[
\eng{Fluxonium qubit}
\ger{Fluxonium Qubit}
]{fluxonium}
\begin{formula}{circuit}
\desc{Fluxonium qubit}{
Josephson junction with a shunt \textbf{inductance}. Instead of having to tunnel, cooper pairs can move to the island via the inductance.
The inductance consists of many parallel Josephson Junctions to avoid parasitic capacitances.
}{}
\desc[german]{Fluxonium Qubit}{
Josephson-Kontakt mit einem parallelen \textbf{induktiven Element}.
Anstatt zu tunneln, können die Cooper-Paare über das induktive Element auf die Insel gelangen.
Das induktive Element besteht aus sehr vielen parallelen Josephson-Kontakten um parisitische Kapazitäten zu vermeiden.
}{}
\centering
\begin{tikzpicture}
% \draw (0,0) to[sV=$V_\text{g}$] ++(0,3)
% to ++(2,0) coordinate(top1);
\draw[color=gray] (0,0) to ++(-2,0) to[capacitor=$C_C$] ++(0,-3) to ++(2,0);
\draw (0,0) to[josephsoncap=$C_\text{J}$] ++(-0,-3);
\draw (0,0) to ++(2,0) to[cute inductor=$E_L$] ++(0,-3) to ++(-2,0);
\node at (1,-0.5) {$\Phi_\text{ext}$};
\end{tikzpicture}
\\\TODO{Ist beim Fluxonium noch die Voltage source dran?}
\end{formula}
\Subsubsection[
\eng{Fluxonium qubit}
\ger{Fluxonium Qubit}
]{fluxonium}
\begin{formula}{circuit}
\desc{Fluxonium qubit}{
Josephson junction with a shunt \textbf{inductance}. Instead of having to tunnel, cooper pairs can move to the island via the inductance.
The inductance consists of many parallel Josephson Junctions to avoid parasitic capacitances.
}{}
\desc[german]{Fluxonium Qubit}{
Josephson-Kontakt mit einem parallelen \textbf{induktiven Element}.
Anstatt zu tunneln, können die Cooper-Paare über das induktive Element auf die Insel gelangen.
Das induktive Element besteht aus sehr vielen parallelen Josephson-Kontakten um parisitische Kapazitäten zu vermeiden.
}{}
\centering
\begin{tikzpicture}
% \draw (0,0) to[sV=$V_\text{g}$] ++(0,3)
% to ++(2,0) coordinate(top1);
\draw[color=gray] (0,0) to ++(-2,0) to[capacitor=$C_C$] ++(0,-3) to ++(2,0);
\draw (0,0) to[josephsoncap=$C_\text{J}$] ++(-0,-3);
\draw (0,0) to ++(2,0) to[cute inductor=$E_L$] ++(0,-3) to ++(-2,0);
\node at (1,-0.5) {$\Phi_\text{ext}$};
\end{tikzpicture}
\\\TODO{Ist beim Fluxonium noch die Voltage source dran?}
\end{formula}
\def\temp{$E_\text{C} = \frac{(2e)^2}{2C}, E_\text{L} = \frac{\varphi_0^2}{2L}, \delta_\text{s} = \frac{\varphi_\text{s}}{\varphi_0}$}
\begin{formula}{hamiltonian}
\desc{Hamiltonian}{}{\temp}
\desc[german]{Hamiltonian}{}{\temp}
\eq{\hat{H} = 4E_\text{C} \hat{n}^2 - E_\text{J} \cos \hat{\delta} + E_\text{L}(\hat{\delta} - \delta_\text{s})^2}
\end{formula}
\def\temp{$E_\text{C} = \frac{(2e)^2}{2C}, E_\text{L} = \frac{\varphi_0^2}{2L}, \delta_\text{s} = \frac{\varphi_\text{s}}{\varphi_0}$}
\begin{formula}{hamiltonian}
\desc{Hamiltonian}{}{\temp}
\desc[german]{Hamiltonian}{}{\temp}
\eq{\hat{H} = 4E_\text{C} \hat{n}^2 - E_\text{J} \cos \hat{\delta} + E_\text{L}(\hat{\delta} - \delta_\text{s})^2}
\end{formula}
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{img/qubit_flux_onium.pdf}
\caption{img/}
\label{fig:img-}
\end{figure}
\Section[
@ -432,7 +429,6 @@
\desc[german]{Ressonanzfrequenz}{}{}
\eq{\omega_{21} = \frac{E_2 - E_1}{\hbar}}
\end{formula}
\TODO{sollte das nicht 10 sein?}
\begin{formula}{rabi_oscillation}
\desc{Rabi oscillations}{}{$\omega_{21}$ resonance frequency of the energy transition, $\Omega$ Rabi frequency}
@ -440,15 +436,14 @@
\eq{\Omega_ \text{\TODO{TODO}}}
\end{formula}
\Subsection[
\eng{Ramsey interferometry}
\ger{Ramsey Interferometrie}
]{ramsey}
\begin{formula}{ramsey}
\desc{Ramsey interferometry}{}{}
\desc[german]{Ramsey Interferometrie}{}{}
\begin{ttext}
\eng{$\ket{0} \xrightarrow{\frac{\pi}{2}\,\text{pulse}}$ precession in $xy$ plane for time $\tau$ $\xrightarrow{\frac{\pi}{2}\,\text{pulse}}$ measurement}
\ger{q}
\end{ttext}
\end{formula}
\Section[
\eng{Noise and decoherence}

46
src/statistical_mechanics.tex
View File

@ -220,25 +220,26 @@
\ger{Materialeigenschaften}
]{props}
\begin{formula}{heat_cap}
\desc{Heat capacity}{}{$Q$ heat}
\desc[german]{Wärmekapazität}{}{$Q$ Wärme}
\desc{Heat capacity}{}{\QtyRef{heat}}
\desc[german]{Wärmekapazität}{}{}
\quantity{c}{\joule\per\kelvin}{}
\eq{c = \frac{Q}{\Delta T}}
\end{formula}
\begin{formula}{heat_cap_V}
\desc{Isochoric heat capacity}{}{$U$ internal energy}
\desc[german]{Isochore Wärmekapazität}{}{$U$ innere Energie}
\desc{Isochoric heat capacity}{}{\QtyRef{heat}, \QtyRef{internal_energy} \QtyRef{temperature}, \QtyRef{volume}}
\desc[german]{Isochore Wärmekapazität}{}{}
\eq{c_v = \pdv{Q}{T}_V = \pdv{U}{T}_V}
\end{formula}
\begin{formula}{heat_cap_p}
\desc{Isobaric heat capacity}{}{$H$ enthalpy}
\desc[german]{Isobare Wärmekapazität}{}{$H$ Enthalpie}
\eq{c_p = \pdv{Q}{T}_P = \pdv{H}{T}_P}
\desc{Isobaric heat capacity}{}{\QtyRef{heat}, \QtyRef{enthalpy} \QtyRef{temperature}, \QtyRef{pressure}}
\desc[german]{Isobare Wärmekapazität}{}{}
\eq{c_p = \pdv{Q}{T}_p = \pdv{H}{T}_p}
\end{formula}
\begin{formula}{bulk_modules}
\desc{Bulk modules}{}{$p$ pressure, $V$ initial volume}
\desc[german]{Kompressionsmodul}{}{$p$ Druck, $V$ Anfangsvolumen}
\desc{Bulk modules}{}{\QtyRef{pressure}, $V$ initial \qtyRef{volume}}
\desc[german]{Kompressionsmodul}{}{\QtyRef{pressure}, $V$ Anfangsvolumen}
\eq{K = -V \odv{p}{V} }
\end{formula}
@ -435,8 +436,8 @@
\hiddenQuantity{U}{\joule}{s}
\end{formula}
\begin{formula}{free_energy}
\desc{Free energy / Helmholtz energy }{}{}
\desc[german]{Freie Energie / Helmholtz Energie}{}{}
\desc{Free energy}{Helmholtz energy}{}
\desc[german]{Freie Energie}{Helmholtz Energie}{}
\eq{\d F(T,V,N) = -S\d T -p\d V + \mu\d N}
\hiddenQuantity{F}{\joule}{s}
\end{formula}
@ -446,9 +447,9 @@
\eq{\d H(S,p,N) = T\d S +V\d p + \mu\d N}
\hiddenQuantity{H}{\joule}{s}
\end{formula}
\begin{formula}{gibbs_energy}
\desc{Free enthalpy / Gibbs energy}{}{}
\desc[german]{Freie Entahlpie / Gibbs-Energie}{}{}
\begin{formula}{free_enthalpy}
\desc{Free enthalpy}{Gibbs energy}{}
\desc[german]{Freie Entahlpie}{Gibbs-Energie}{}
\eq{\d G(T,p,N) = -S\d T + V\d p + \mu\d N}
\hiddenQuantity{G}{\joule}{s}
\end{formula}
@ -459,11 +460,11 @@
\hiddenQuantity{\Phi}{\joule}{s}
\end{formula}
\TODO{Maxwell Relationen, TD Quadrat}
\TODO{Maxwell Relationen}
\begin{formula}{td-square}
\desc{Thermodynamic squre}{}{}
\desc{Thermodynamic square}{}{}
\desc[german]{Themodynamisches Quadrat}{Guggenheim Quadrat}{}
\begin{minipage}{0.3\textwidth}
\fsplit[0.3]{
\begin{tikzpicture}
\draw[thick] (0,0) grid (3,3);
\node at (0.5, 2.5) {$-S$};
@ -475,11 +476,12 @@
\node at (1.5, 0.5) {\color{blue}$G$};
\node at (2.5, 0.5) {$T$};
\end{tikzpicture}
\end{minipage}
\begin{ttext}
\eng{The corners opposite from the potential are the coefficients and each coefficients differential is opposite to it.}
\ger{Die Ecken gegenüber des Potentials sind die Koeffizienten, das Differential eines Koeffizienten ist in der Ecke gegenüber.}
\end{ttext}
}{
\begin{ttext}
\eng{The corners opposite from the potential are the coefficients and each coefficients differential is opposite to it.}
\ger{Die Ecken gegenüber des Potentials sind die Koeffizienten, das Differential eines Koeffizienten ist in der Ecke gegenüber.}
\end{ttext}
}
\end{formula}
\Section[

3
src/test.tex
View File

@ -1,5 +1,8 @@
\part{Testing}
Textwidth: \the\textwidth
\\Linewidth: \the\linewidth
% \directlua{tex.sprint("Compiled in directory: \\detokenize{" .. lfs.currentdir() .. "}")} \\
% \directlua{tex.sprint("Jobname: " .. tex.jobname)} \\
% \directlua{tex.sprint("Output directory \\detokenize{" .. os.getenv("TEXMF_OUTPUT_DIRECTORY") .. "}")} \\

2
src/util/environments.tex
View File

@ -1,5 +1,3 @@
\def\descwidth{0.3\textwidth}
\def\eqwidth{0.6\textwidth}
\newcommand\separateEntries{
\vspace{0.5\baselineskip}

2
src/util/math-macros.tex
View File

@ -177,6 +177,8 @@
\newcommand\Exp[1]{\CmdWithParenthesis{\exp}{#1}}
\newcommand\Sin[1]{\CmdWithParenthesis{\sin}{#1}}
\newcommand\Cos[1]{\CmdWithParenthesis{\cos}{#1}}
\newcommand\Sinh[1]{\CmdWithParenthesis{\sinh}{#1}}
\newcommand\Cosh[1]{\CmdWithParenthesis{\cosh}{#1}}
\newcommand\Ln[1]{\CmdWithParenthesis{\ln}{#1}}
\newcommand\Log[1]{\CmdWithParenthesis{\log}{#1}}
\newcommand\Order[1]{\CmdWithParenthesis{\mathcal{O}}{#1}}