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51
src/cm/cm.tex
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@ -20,54 +20,3 @@
}
\end{formula}
\Section[
\eng{Lattice vibrations}
\ger{Gitterschwingungen}
]{vib}
\begin{formula}{dispersion_1atom_basis}
\desc{Phonon dispersion of a lattice with a one-atom basis}{same as the dispersion of a linear chain}{$C_n$ force constants between layer $s$ and $s+n$, $M$ \qtyRef{mass} of the reference atom, $a$ \qtyRef{lattice_constant}, $q$ phonon \qtyRef{wavevector}, $u$ Ansatz for the atom displacement}
\desc[german]{Phonondispersion eines Gitters mit zweiatomiger Basis}{gleich der Dispersion einer linearen Kette}{$C_n$ Kraftkonstanten zwischen Ebene $s$ und $s+n$, $M$ \qtyRef{mass} des Referenzatoms, $a$ \qtyRef{lattice_constant}, $q$ Phonon \qtyRef{wavevector}, $u$ Ansatz für die Atomauslenkung}
\begin{gather}
\omega^2 = \frac{4C_1}{M}\left[\sin^2 \left(\frac{qa}{2}\right) + \frac{C2}{C1} \sin^2(qa)\right] \\
\intertext{\GT{with}}
u_{s+n} = U\e^{-i \left[\omega t - q(s+n)a \right]}
\end{gather}
\newFormulaEntry
\fig{img/cm_phonon_dispersion_one_atom_basis.pdf}
\end{formula}
\begin{formula}{dispersion_2atom_basis}
\desc{Phonon dispersion of a lattice with a two-atom basis}{}{$C$ force constant between layers, $M_i$ \qtyRef{mass} of the basis atoms, $a$ \qtyRef{lattice_constant}, $q$ phonon \qtyRef{wavevector}, $u, v$ Ansatz for the displacement of basis atom 1 and 2, respectively}
\desc[german]{Phonondispersion eines Gitters mit einatomiger Basis}{}{$C$ Kraftkonstanten zwischen Ebene $s$ und $s+n$, $M_i$ \qtyRef{mass} der Basisatome, $a$ \qtyRef{lattice_constant}, $q$ Phonon \qtyRef{wavevector}, $u, v$ jeweils Ansatz für die Atomauslenkung des Basisatoms 1 und 2}
\begin{gather}
\omega^2_{\txa,\txo} = C \left(\frac{1}{M_1}+\frac{1}{M_2}\right) \mp C \sqrt{\left(\frac{1}{M_1}+\frac{1}{M_2}\right)^2 - \frac{4}{M_1M_2} \sin^2 \left(\frac{qa}{2}\right)}
\intertext{\GT{with}}
u_{s} = U\e^{-i \left(\omega t - qsa \right)}, \quad
v_{s} = V\e^{-i \left(\omega t - qsa \right)}
\end{gather}
\newFormulaEntry
\fig{img/cm_phonon_dispersion_two_atom_basis.pdf}
\end{formula}
\begin{formula}{branches}
\desc{Vibration branches}{}{}
\desc[german]{Vibrationsmoden}{}{}
\ttxt{\eng{
\textbf{Acoustic}: 3 modes (1 longitudinal, 2 transversal), the two basis atoms oscillate in phase.
\\\textbf{Optical}: 3 modes, the two basis atoms oscillate in opposition. A dipole moment is created that can couple to photons.
}\ger{
\textbf{Akustisch}: 3 Moden (1 longitudinal, 2 transversal), die zwei Basisatome schwingen in Phase.
\\ \textbf{Optisch}: 3 Moden, die zwei Basisatome schwingen gegenphasig. Das dadurch entstehende Dipolmoment erlaubt die Wechselwirkung mit Photonen.
}}
\end{formula}
\Subsection[
\eng{Debye model}
\ger{Debye-Modell}
]{debye}
\begin{ttext}
\eng{Atoms behave like coupled \fRef[quantum harmonic oscillators]{sec:qm:hosc}. The finite sample size leads to periodic boundary conditio. The finite sample size leads to periodic boundary conditions for the vibrations.}
\ger{Atome verhalten sich wie gekoppelte \fRef[quantenmechanische harmonische Oszillatoren]{sec:qm:hosc}. Die endliche Ausdehnung des Körpers führt zu periodischen Randbedingungen. }
\end{ttext}

32
src/cm/crystal.tex
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@ -17,7 +17,7 @@
\eng[bravais_lattices]{Bravais lattices}
\ger[bravais_lattices]{Bravais Gitter}
\newcommand\bvimg[1]{\begin{center}\includegraphics[width=0.1\textwidth]{img/bravais/#1.pdf}\end{center}}
\newcommand\bvimg[1]{\begin{center}\includegraphics[width=0.1\textwidth]{img_static/bravais/#1.pdf}\end{center}}
\renewcommand\tabularxcolumn[1]{m{#1}}
\newcolumntype{Z}{>{\centering\let\newline\\\arraybackslash\hspace{0pt}}X}
@ -170,18 +170,34 @@
\begin{formula}{zincblende}
\desc{Zincblende lattice}{}{}
\desc[german]{Zinkblende-Struktur}{}{}
\ttxt{
\includegraphics[width=0.5\textwidth]{img/cm_zincblende.png}
\eng{Like \fRef{:::diamond} but with different species on each basis}
\ger{Wie \fRef{:::diamond} aber mit unterschiedlichen Spezies auf den Basen}
\fsplit{
\centering
\includegraphics[width=0.9\textwidth]{img/cm_crystal_zincblende.png}
}{
\ttxt{
\eng{Like \fRef{:::diamond} but with different species on each basis}
\ger{Wie \fRef{:::diamond} aber mit unterschiedlichen Spezies auf den Basen}
}
}
\end{formula}
\begin{formula}{rocksalt}
\desc{Rocksalt structure}{\elRef{Na}\elRef{Cl}}{}
\desc[german]{Kochsalz-Struktur}{}{}
\fsplit{
\centering
\includegraphics[width=0.9\textwidth]{img/cm_crystal_NaCl.png}
}{
}
\end{formula}
\begin{formula}{wurtzite}
\desc{Wurtzite structure}{hP4}{}
\desc[german]{Wurtzite-Struktur}{hP4}{}
\ttxt{
\includegraphics[width=0.5\textwidth]{img/cm_wurtzite.png}
\TODO{Placeholder}
\fsplit{
\centering
\includegraphics[width=0.9\textwidth]{img/cm_crystal_wurtzite.png}
}{
}
\end{formula}

151
src/cm/superconductivity.tex
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@ -27,6 +27,7 @@
Has a single critical magnetic field, $\Bcth$.
\\$B < \Bcth$: \fRef{:::meissner_effect}
\\$B > \Bcth$: Normal conductor
\\ Very small usable current density because current only flows within the \fRef{cm:super:london:penetration_depth} of the surface.
}}
\end{formula}
@ -38,6 +39,7 @@
\\$B < B_\text{c1}$: \fRef{:::meissner_effect}
\\$B_\text{c1} < B < B_\text{c2}$: \fRef{:::shubnikov_phase}
\\$B > B_\text{c2}$: Normal conductor
\\ In \fRef{:::shubnikov_phase} larger usable current density because current flows within the \fRef{cm:super:london:penetration_depth} of the surface and the penetrating flux lines.
}}
\end{formula}
@ -75,7 +77,9 @@
\desc{Shubnikov phase}{in \fRef{::type2}}{}
\desc[german]{Shubnikov-Phase}{}{}
\ttxt{\eng{
Mixed phase in which some magnetic flux penetrates the superconductor.
}\ger{
Gemischte Phase in der der Supraleiter teilweise von magnetischem Fluss durchdrungen werden kann.
}}
\end{formula}
@ -92,9 +96,6 @@
\eng{London Theory}
\ger{London-Theorie}
]{london}
\begin{ttext}
\end{ttext}
\begin{formula}{description}
\desc{Description}{}{}
\desc[german]{Beschreibung}{}{}
@ -148,7 +149,27 @@
\eq{\lambda_\txL(T) = \lambda_\txL(0) \frac{1}{\sqrt{1- \left(\frac{T}{T_\txc}\right)^4}}}
\end{formula}
\TODO{macroscopic wavefunction approach, energy-phase relation, current-phase relation}
\Subsubsection[
\eng{Macroscopic wavefunction}
\ger{Makroskopische Wellenfunktion}
]{macro}
\begin{formula}{ansatz}
\desc{Ansatz}{}{}
\desc[german]{Ansatz}{}{}
\ttxt{\eng{Alternative derivation of London equations by assuming a macroscopic wavefunction which is uniform in space}\ger{Alternative Herleitung der London-Gleichungen durch Annahme einer makroskopischen Wellenfunktion, welche nicht Ortsabhängig ist}}
\eq{\Psi(\vecr,t) = \Psi_0(\vecr,t) \e^{\theta(\vecr,t)}}
\end{formula}
\begin{formula}{energy-phase_relation}
\desc{Energy-phase relation}{}{$\theta$ \qtyRef{phase}, $m_\txs$/$n_\txs$/$q_\txs$ \qtyRef{mass}/\qtyRef{charge_carrier_density}/\qtyRef{charge} \GT{of_sc_particle}, \QtyRef{current_density}, $\phi_\text{el}$ \qtyRef{electric_scalar_potential}, \QtyRef{chemical_potential}}
\desc[german]{Energie-Phase Beziehung}{}{}
\eq{\hbar \pdv{\theta(\vecr,t)}{t} = - \left(\frac{m_\txs}{n_\txs^2 q_\txs^2} \vecj_\txs^2(\vecr,t) + q_\txs\phi_\text{el}(\vecr,t) + \mu(\vecr,t)\right)}
\end{formula}
\begin{formula}{current-phase_relation}
\desc{Current-phase relation}{}{$\theta$ \qtyRef{phase}, $m_\txs$/$n_\txs$/$q_\txs$ \qtyRef{mass}/\qtyRef{charge_carrier_density}/\qtyRef{charge} \GT{of_sc_particle}, \QtyRef{current_density}, \QtyRef{magnetic_vector_potential}}
\desc[german]{Strom-Phase Beziehung}{}{}
\eq{\vecj_\txs(\vecr,t) = \frac{q_\txs^2 n_\txs(\vecr,t)}{m_\txs} \left(\frac{\hbar}{q_\txs} \Grad\theta(\vecr,t) - \vecA(\vecr,t)\right) }
\end{formula}
\Subsubsection[
\eng{Josephson Effect}
@ -167,8 +188,8 @@
\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}
\desc{Josephson coupling energy}{}{$A$ junction \qtyRef{area}, \ConstRef{flux_quantum}, $\vecj_\txc$ \fRef[critical current density]{::1st_relation}, $\phi$ phase differnce accross junction}
\desc[german]{Josephson}{}{$A$ junction \qtyRef{area}, \ConstRef{flux_quantum}, $\vecj_\txc$ \fRef[kritische Stromdichte]{::1st_relation}, $\phi$ Phasendifferenz zwischen den Supraleitern}
\eq{\frac{E_\txJ}{A} = \frac{\Phi_0 \vecj_\txc}{2\pi}(1-\cos\phi)}
\end{formula}
@ -217,7 +238,7 @@
% \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
\\ \frac{1}{2\mu_0}(\vecB_\text{ext} -\vecB_\text{inside})^2 + \frac{1}{2m_\txs} \abs{ \left(-\I\hbar\Grad - q_\txs \vecA\right)\Psi}^2 + \dots
\end{multline}
\end{formula}
@ -254,9 +275,9 @@
\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}}
\desc{Boundary energy}{Negative for \fRef{:::type2}, positive for \fRef{:::type1}}{$\Delta E_\text{B}$ energy gained by expelling the external magnetic field, $\Delta E_\text{cond}$ \fRef{:::condensation_energy}}
\desc[german]{Grenzflächenenergie}{Negativ für \fRef{:::type2}, positiv für \fRef{:::type1}}{}
\eq{\Delta E_\text{boundary} = \Delta E_\text{con} - \Delta E_\txB = (\xi_\gl - \lambda_\gl) \frac{B_\text{c,th}^2}{2\mu_0}}
\end{formula}
\begin{formula}{parameter}
@ -276,14 +297,20 @@
\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}
\fig{img/cm_super_n_s_boundary.pdf}
% \TODO{plot, slide 106}
\end{formula}
\begin{formula}{bcth}
\desc{Thermodynamic critical field}{}{}
\desc[german]{Thermodynamisches kritisches Feld}{}{}
\eq{\Bcth = \frac{\Phi_0}{2\pi \sqrt{2} \xi_\gl \lambda_\gl}}
\end{formula}
\begin{formula}{bc1}
\desc{Lower critical magnetic field}{Above $B_\text{c1}$, flux starts to penetrate the superconducting phase}{\ConstRef{flux_quantum}, $\lambda\gl$ \fRef{::penetration_depth} $\kappa$ \fRef{::parameter}}
\desc{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}
\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}
@ -297,7 +324,6 @@
% \desc[german]{}{}{}
\ttxt{\eng{
Superconductor wavefunction extends into the normal conductor or isolator
\TODO{clarify}
}}
\end{formula}
@ -343,6 +369,7 @@
\end{itemize}
}}
\end{formula}
\def\BCS{{\text{BCS}}}
\begin{formula}{hamiltonian}
\desc{BCS Hamiltonian}{for $N$ interacting electrons}{
@ -388,11 +415,11 @@
\end{formula}
\begin{formula}{gap_at_t0}
\desc{BCS Gap at $T=0$}{}{\QtyRef{debye_frequency}, $V_0$ \fRef{::potential}, $D$ \qtyRef{dos}}
\desc{BCS Gap at $T=0$}{}{\QtyRef{debye_frequency}, $V_0$ \fRef{::potential}, $D$ \qtyRef{dos}, \TODO{gamma}}
\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
\frac{\Delta(T=0)}{\kB T_\txc} &= \frac{\pi}{\e^\gamma} = 1.764
}
\end{formula}
@ -400,30 +427,53 @@
\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)}}}
\eq{E \approx 2E_\txF - 2\hbar\omega_\txD \Exp{-\frac{4}{V_0 D(E_\txF)}}}
\end{formula}
\Subsubsection[
\eng{Excitations in BCS}
% \ger{}
\eng{Excitations and finite temperatures}
\ger{Anregungen und endliche Temperatur}
]{excite}
\begin{formula}{description}
\desc{Description}{}{}
\desc[german]{Beschreibung}{}{}
\ttxt{\eng{
The ground state consists of \fRef{cm:super:micro:cooper_pairs} and the excited state of Bogoliubov quasi-particles (electron-hole pairs).
The states are separated by an energy gap $\Delta$.
}\ger{
Den Grundzustand bilden \fRef{cm:super:micro:cooper_pairs} und den angeregten Zustands Bogoloiubons (Elektron-Loch Quasipartikel).
Die Zustände sind durch eine Energielücke $\Delta$ getrennt.
}}
\end{formula}
\begin{formula}{bogoliubov-valatin}
\desc{Bogoliubov-Valatin transformation}{Diagonalization of the \fRef{cm:super:micro:bcs:hamiltonian} to derive excitation energies}{}
\desc{Bogoliubov-Valatin transformation}{Diagonalization of the \fRef{cm:super:micro:bcs:hamiltonian} to derive excitation energies}{
$\xi_\veck = \epsilon_\veck-\mu$ Energy relative to the \qtyRef{chemical_potential},
\\ $E_\veck$ \fRef{::excitation_energy},
\\ $\Delta$ Gap
\\ $g_\veck$ \fRef{::pairing_amplitude},
\\ $\alpha / \beta$ create and destroy symmetric/antisymmetric Bogoliubov quasiparticles
}
\desc[german]{Bogoliubov-Valatin transformation}{}{}
\eq{
\hat{H}_\BCS - N\mu = \sum_\veck \big[\xi_\veck - E_\veck + \Delta_\veck g_\veck^*\big] + \sum_\veck \big[E_\veck a_\veck^\dagger a_\veck + E_\veck \beta_{-\veck}^\dagger \beta_{-\veck}\big]
\hat{H}_\BCS - N\mu = \sum_\veck \big[\xi_\veck - E_\veck + \Delta_\veck g_\veck^*\big] + \sum_\veck \big[E_\veck \alpha_\veck^\dagger \alpha_\veck + E_\veck \beta_{-\veck}^\dagger \beta_{-\veck}\big]
}
\end{formula}
\begin{formula}{pairing_amplitude}
\desc{Pairing amplitude}{Excitation energy}{\TODO{gamma}}
\desc[german]{Paarungsamplitude}{Anregungsenergie}{}
\desc{Pairing amplitude}{}{}
\desc[german]{Paarungsamplitude}{}{}
\eq{g_\veck \equiv \Braket{\hat{c}_{-\veck\downarrow} \hat{c}_{\veck\uparrow}}}
\end{formula}
\begin{formula}{excitation_energy}
\desc{Excitation energy}{}{}
\desc[german]{Anregungsenergie}{}{}
\eq{E_\veck = \pm \sqrt{\xi^2_\veck + \abs{\Delta_\veck}^2}}
\end{formula}
\begin{formula}{coherence_factors_energy}
\desc{Energy dependance of the \fRef{:::coherence_factors}}{}{$E_\veck$ \fRef{::pairing_amplitude}, \GT{see} \fRef{:::coherence_facotrs}}
\desc[german]{Energieabhängigkeit der \fRef{:::coherence_factors}}{}{}
\desc{Energy dependance of the \fRef{:::bcs:coherence_factors}}{}{$E_\veck$ \fRef{::pairing_amplitude}, \GT{see} \fRef{:::bcs:coherence_factors}}
\desc[german]{Energieabhängigkeit der \fRef{:::bcs:coherence_factors}}{}{}
\eq{
\abs{u_\veck}^2 &= \frac{1}{2} \left(1+\frac{\xi_\veck}{E_\veck}\right) \\
\abs{v_\veck}^2 &= \frac{1}{2} \left(1-\frac{\xi_\veck}{E_\veck}\right) \\
@ -431,17 +481,18 @@
}
\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)}
\desc{Self-consistend gap equation}{}{}
\desc[german]{Selbstkonsitente Energielückengleichung}{}{}
\eq{\Delta_\veck^* = -\sum_{\veck^\prime} V_{\veck,\veck^\prime} \frac{\Delta_{\veck^\prime}}{2E_\veck} \tanh \left(\frac{E_{\veck^\prime}}{2\kB T}\right)}
\end{formula}
\begin{formula}{gap_t}
\desc{Temperature dependence of the BCS gap}{}{}
\desc[german]{Temperaturabhängigkeit der BCS-Lücke}{}{}
\eq{\frac{\Delta(T)}{\Delta(T=0)} \approx 1.74 \frac{1}{\sqrt{1-\frac{T}{T_\txC}}}}
\eq{\frac{\Delta(T)}{\Delta(T=0)} \approx 1.74 \sqrt{1-\frac{T}{T_\txC}}}
\end{formula}
\begin{formula}{dos}
@ -456,8 +507,40 @@
\end{formula}
\begin{formula}{Bcth_temp}
\desc{Temperature dependance of the crictial magnetic field}{}{}
\desc[german]{Temperaturabhängigkeit des kritischen Magnetfelds}{}{}
\desc{Temperature dependance of the crictial magnetic field}{Jump at $T_\txc$, then exponential decay}{}
\desc[german]{Temperaturabhängigkeit des kritischen Magnetfelds}{Sprung bei $T_\txc$, denn exponentieller Abfall}{}
\eq{ \Bcth(T) = \Bcth(0) \left[1- \left(\frac{T}{T_\txc}\right)^2 \right] }
\TODO{empirical relation, relate to BCS}
% \TODO{empirical relation, relate to BCS}
\end{formula}
\begin{formula}{heat_capacity}
\desc{Heat capacity in superconductors}{}{}
\desc[german]{Wärmekapazität in Supraleitern}{}{}
\fsplit{
\fig{img/cm_super_heat_capacity.pdf}
}{
\eq{c_\txs \propto T^{-\frac{3}{2}} \e^{\frac{\Delta(0)}{\kB T}}}
}
\end{formula}
\Subsubsection[
\eng{Flux pinning}
\ger{Haftung von Flusslinien}
]{pinning}
\begin{formula}{description}
\desc{Description}{}{}
\desc[german]{Beschreibung}{}{}
\ttxt{\eng{
If a current flows in a \fRef{cm:super:type2}s in the \fRef{cm:super:shubnikov_phase} perpendicular to the penetrating flux lines,
the lines experience a Lorentz force. This leads to ohmic behaviour of the superconductor.
The flux lines can be pinned to defects, in which the superconducting order parameter is reduced.
To move the flux line out of the defect, work would have to be spent overcoming the \fRef{cm:super:micro:pinning:potential}.
This restores the superconductivity.
}\ger{
Wenn ein Strom in einem \fRef{cm:super:type2}s in der \fRef{cm:super:shubnikov_phase} senkrecht zu den eindringenden Flusslinien fließt, erfahren die Linien eine Lorentzkraft.
Dies führt zu einem ohmschen Verhalten des Supraleiters.
Die Flusslinien können an Defekten festgehalten werden, in denen der supraleitende Ordnungsparameter reduziert ist.
Um die Flusslinie aus dem Defekt zu bewegen, müsste Arbeit aufgewendet werden, um das \fRef{cm:super:micro:pinning:potential} zu überwinden.
Dies stellt die Supraleitfähigkeit wieder her.
}}
\end{formula}

8
src/cm/techniques.tex
View File

@ -55,7 +55,7 @@
\begin{minipage}{0.45\textwidth}
\begin{figure}[H]
\centering
% \includegraphics[width=0.8\textwidth]{img/cm_amf.pdf}
% \includegraphics[width=0.8\textwidth]{img_static/cm_amf.pdf}
% \caption{\cite{Bian2021}}
\end{figure}
\end{minipage}
@ -97,7 +97,7 @@
\begin{minipage}{0.45\textwidth}
\begin{figure}[H]
\centering
\includegraphics[width=0.8\textwidth]{img/cm_amf.pdf}
\includegraphics[width=0.8\textwidth]{img_static/cm_amf.pdf}
\caption{\cite{Bian2021}}
\end{figure}
\end{minipage}
@ -122,7 +122,7 @@
\begin{minipage}{0.45\textwidth}
\begin{figure}[H]
\centering
\includegraphics[width=0.8\textwidth]{img/cm_stm.pdf}
\includegraphics[width=0.8\textwidth]{img_static/cm_stm.pdf}
\caption{\cite{Bian2021}}
\end{figure}
\end{minipage}
@ -168,7 +168,7 @@
\end{minipagetable}
\begin{minipage}{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{img/cm_cvd_english.pdf}
\includegraphics[width=\textwidth]{img_static/cm_cvd_english.pdf}
\end{minipage}
\end{bigformula}

102
src/cm/vib.tex Normal file
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@ -0,0 +1,102 @@
\Section[
\eng{Lattice vibrations}
\ger{Gitterschwingungen}
]{vib}
\begin{formula}{dispersion_1atom_basis}
\desc{Phonon dispersion of a lattice with a one-atom basis}{same as the dispersion of a linear chain}{$C_n$ force constants between layer $s$ and $s+n$, $M$ \qtyRef{mass} of the reference atom, $a$ \qtyRef{lattice_constant}, $q$ phonon \qtyRef{wavevector}, $u$ Ansatz for the atom displacement}
\desc[german]{Phonondispersion eines Gitters mit zweiatomiger Basis}{gleich der Dispersion einer linearen Kette}{$C_n$ Kraftkonstanten zwischen Ebene $s$ und $s+n$, $M$ \qtyRef{mass} des Referenzatoms, $a$ \qtyRef{lattice_constant}, $q$ Phonon \qtyRef{wavevector}, $u$ Ansatz für die Atomauslenkung}
\begin{gather}
\omega^2 = \frac{4C_1}{M}\left[\sin^2 \left(\frac{qa}{2}\right) + \frac{C2}{C1} \sin^2(qa)\right] \\
\intertext{\GT{with}}
u_{s+n} = U\e^{-i \left[\omega t - q(s+n)a \right]}
\end{gather}
\newFormulaEntry
\fig{img/cm_vib_dispersion_one_atom_basis.pdf}
\end{formula}
\begin{formula}{dispersion_2atom_basis}
\desc{Phonon dispersion of a lattice with a two-atom basis}{}{$C$ force constant between layers, $M_i$ \qtyRef{mass} of the basis atoms, $a$ \qtyRef{lattice_constant}, $q$ phonon \qtyRef{wavevector}, $u, v$ Ansatz for the displacement of basis atom 1 and 2, respectively}
\desc[german]{Phonondispersion eines Gitters mit einatomiger Basis}{}{$C$ Kraftkonstanten zwischen Ebene $s$ und $s+n$, $M_i$ \qtyRef{mass} der Basisatome, $a$ \qtyRef{lattice_constant}, $q$ Phonon \qtyRef{wavevector}, $u, v$ jeweils Ansatz für die Atomauslenkung des Basisatoms 1 und 2}
\begin{gather}
\omega^2_{\txa,\txo} = C \left(\frac{1}{M_1}+\frac{1}{M_2}\right) \mp C \sqrt{\left(\frac{1}{M_1}+\frac{1}{M_2}\right)^2 - \frac{4}{M_1M_2} \sin^2 \left(\frac{qa}{2}\right)}
\intertext{\GT{with}}
u_{s} = U\e^{-i \left(\omega t - qsa \right)}, \quad
v_{s} = V\e^{-i \left(\omega t - qsa \right)}
\end{gather}
\newFormulaEntry
\fig{img/cm_vib_dispersion_two_atom_basis.pdf}
\end{formula}
\begin{formula}{branches}
\desc{Vibration branches}{}{}
\desc[german]{Vibrationsmoden}{}{}
\ttxt{\eng{
\textbf{Acoustic}: 3 modes (1 longitudinal, 2 transversal), the two basis atoms oscillate in phase.
\\\textbf{Optical}: 3 modes, the two basis atoms oscillate in opposition. A dipole moment is created that can couple to photons.
}\ger{
\textbf{Akustisch}: 3 Moden (1 longitudinal, 2 transversal), die zwei Basisatome schwingen in Phase.
\\ \textbf{Optisch}: 3 Moden, die zwei Basisatome schwingen gegenphasig. Das dadurch entstehende Dipolmoment erlaubt die Wechselwirkung mit Photonen.
}}
\end{formula}
\Subsection[
\eng{Einstein model}
\ger{Einstein-Modell}
]{einstein}
\begin{formula}{description}
\desc{Description}{}{}
\desc[german]{Beschreibung}{}{}
\ttxt{\eng{
All lattice vibrations have the \fRef[same frequency]{:::frequency}.
Underestimates the \fRef{:::heat_capacity} for low temperatures.
}\ger{
Alle Gittereigenschwingungen haben die \fRef[selbe Frequenz]{:::frequency}
Sagt zu kleine \fRef[Wärmekapazitäten]{:::heat_capacity} für tiefe Temperaturen voraus.
}}
\end{formula}
\begin{formula}{frequency}
\desc{Einstein frequency}{}{}
\desc[german]{Einstein-Frequenz}{}{}
\eq{\omega_\txE}
\end{formula}
\begin{formula}{heat_capacity}
\desc{\qtyRef{heat_capacity}}{according to the Einstein model}{}
\desc[german]{}{nach dem Einstein-Modell}{}
\eq{C_V^\txE = 3N\kB \left( \frac{\hbar\omega_\txE}{\kB T}\right)^2 \frac{\e^{\frac{\hbar\omega_\txE}{\kB T}}}{ \left(\e^{\frac{\hbar\omega_\txE}{\kB T}} - 1\right)^2}}
\end{formula}
\Subsection[
\eng{Debye model}
\ger{Debye-Modell}
]{debye}
\begin{formula}{description}
\desc{Description}{}{}
\desc[german]{Beschreibung}{}{}
\ttxt{\eng{
Atoms behave like coupled \fRef[quantum harmonic oscillators]{sec:qm:hosc}. The finite sample size leads to periodic boundary conditio. The finite sample size leads to periodic boundary conditions for the vibrations.
}\ger{
Atome verhalten sich wie gekoppelte \fRef[quantenmechanische harmonische Oszillatoren]{sec:qm:hosc}. Die endliche Ausdehnung des Körpers führt zu periodischen Randbedingungen.
}}
\end{formula}
\begin{formula}{phonon_dos}
\desc{Phonon density of states}{}{\QtyRef{volume}, $v$ \qtyRef{speed_of_sound} of the phonon mode, $\omega$ phonon frequency}
\desc[german]{Phononenzustandsdichte}{}{\QtyRef{volume}, $v$ \qtyRef{speed_of_sound} des Dispersionszweigs, $\omega$ Phononfrequenz}
\eq{D(\omega) \d \omega = \frac{V}{2\pi^2} \frac{\omega^2}{v^3} \d\omega}
\end{formula}
\begin{formula}{debye_frequency}
\desc{Debye frequency}{Maximum phonon frequency}{$v$ \qtyRef{speed_of_sound}, $N/V$ atom density}
\desc[german]{Debye-Frequenz}{Maximale Phononenfrequenz}{$v$ \qtyRef{speed_of_sound}, $N/V$ Atomdichte}
\eq{\omega_\txD = v \left(6\pi^2 \frac{N}{V}\right)^{1/3}}
\hiddenQuantity{\omega_\txD}{\per\s}{s}
\end{formula}
\begin{formula}{heat_capacity}
\desc{\qtyRef{heat_capacity}}{according to the Debye model}{$N$ number of atoms, \ConstRef{boltzmann}, \QtyRef{debye_frequency}}
\desc[german]{}{nach dem Debye-Modell}{$N$ Anzahl der Atome, \ConstRef{boltzmann}, \QtyRef{debye_frequency}}
\eq{C_V^\txD = 9N\kB \left(\frac{\kB T}{\hbar \omega_\txD}\right)^3 \int_0^{\frac{\hbar\omega_\txD}{\kB T}} \d x \frac{x^4 \e^x}{(\e^x-1)^2} }
\end{formula}

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3
src/main.tex
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@ -3,7 +3,7 @@
\documentclass[11pt, a4paper]{article}
% SET LANGUAGE HERE
\usepackage[english]{babel}
\usepackage[left=2cm,right=2cm,top=2cm,bottom=2cm]{geometry}
\usepackage[left=1.6cm,right=1.6cm,top=2cm,bottom=2cm]{geometry}
% ENVIRONMENTS etc
\usepackage{adjustbox}
\usepackage{colortbl} % color table
@ -145,6 +145,7 @@
\Input{cm/crystal}
\Input{cm/egas}
\Input{cm/charge_transport}
\Input{cm/vib}
\Input{cm/superconductivity}
\Input{cm/semiconductors}
\Input{cm/misc}

29
src/pkg/mqformula.sty
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@ -1,7 +1,7 @@
\ProvidesPackage{mqformula}
\def\descwidth{0.3\textwidth}
\def\eqwidth{0.69\textwidth}
\def\eqwidth{0.65\textwidth}
\RequirePackage{mqfqname}
\RequirePackage{mqconstant}
@ -16,14 +16,11 @@
% [1]: minipage width
% 2: fqname of name
% 3: fqname of a translation that holds the explanation
\newcommand{\NameWithDescription}[3][\descwidth]{
\newcommand{\NameWithDescription}[3][\descwidth]{%
\begin{minipage}{#1}
\IfTranslationExists{#2}{
\raggedright
\GT{#2}
}{\detokenize{#2}}
\IfTranslationExists{#3}{
\\ {\color{fg1} \GT{#3}}
\raggedright\GT{#2}%
\IfTranslationExists{#3}{%
\\ {\color{fg1} \GT{#3}}%
}{}
\end{minipage}
}
@ -152,12 +149,16 @@
}
\newcommand{\fsplit}[3][0.5]{
\begin{minipage}{##1\linewidth}
##2
\end{minipage}
\begin{minipage}{\luavar{0.99-##1}\linewidth}
##3
\end{minipage}
\begingroup
\renewcommand{\newFormulaEntry}{}
\begin{minipage}{##1\linewidth}
##2
\end{minipage}
\begin{minipage}{\luavar{0.99-##1}\linewidth}
##3
\end{minipage}
\endgroup
\newFormulaEntry
}
}{
\mqfqname@leave

3
src/pkg/mqperiodictable.sty
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@ -10,7 +10,8 @@ elements = {}
elementsOrder = {}
function elementAdd(symbol, nr, period, column)
elementsOrder[nr] = symbol
--elementsOrder[nr] = symbol
table.insert(elementsOrder, symbol)
elements[symbol] = {
symbol = symbol,
atomic_number = nr,

6
src/pkg/mqref.sty
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@ -119,7 +119,7 @@
}
\newcommand{\fRef}[2][]{
\directlua{hyperref(translateRelativeFqname(\luastring{#2}), \luastring{#1})}
\directlua{hyperref(translateRelativeFqname(\luastring{#2}), \luastring{#1})}%
}
% [1]: link text
% 2: number of steps to take up
@ -145,7 +145,7 @@
\newrobustcmd{\qtyRef}[2][]{%
% \edef\tempname{\luaDoubleFieldValue{quantities}{"#1"}{"fqname"}}%
% \hyperref[qty:#1]{\GT{\tempname}}%
\directlua{hyperref(quantityGetFqname(\luastring{#2}), \luastring{#1})}
\directlua{hyperref(quantityGetFqname(\luastring{#2}), \luastring{#1})}%
}
% <symbol> <name>
\newrobustcmd{\QtyRef}[2][]{%
@ -156,7 +156,7 @@
\newrobustcmd{\constRef}[2][]{%
% \edef\tempname{\luaDoubleFieldValue{constants}{"#1"}{"linkto"}}%
% \hyperref[const:#1]{\GT{\tempname}}%
\directlua{hyperref(constantGetFqname(\luastring{#2}), \luastring{#1})}
\directlua{hyperref(constantGetFqname(\luastring{#2}), \luastring{#1})}%
}
% <symbol> <name>
\newrobustcmd{\ConstRef}[2][]{%

6
src/qm/atom.tex
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@ -4,7 +4,7 @@
\Section[
\eng{Hydrogen Atom}
\ger{Wasserstoffatom}
]{h}
]{h}
\begin{formula}{reduced_mass}
\desc{Reduced mass}{}{}
@ -188,8 +188,8 @@
\eq{f &= j \pm i \\ m_f &= -f,-f+1,\dots,f-1,f}
\end{formula}
\begin{formula}{constant}
\desc{Hyperfine structure constant}{}{$B_\textrm{HFS}$ hyperfine field, $\mu_\textrm{K}$ nuclear magneton, $g_i$ nuclear g-factor \ref{qm:h:lande}}
\desc[german]{Hyperfeinstrukturkonstante}{}{$B_\textrm{HFS}$ Hyperfeinfeld, $\mu_\textrm{K}$ Kernmagneton, $g_i$ Kern-g-Faktor \ref{qm:h:lande}}
\desc{Hyperfine structure constant}{}{$B_\textrm{HFS}$ hyperfine field, $\mu_\textrm{K}$ nuclear magneton, $g_i$ nuclear g-factor \fRef{qm:h:lande}}
\desc[german]{Hyperfeinstrukturkonstante}{}{$B_\textrm{HFS}$ Hyperfeinfeld, $\mu_\textrm{K}$ Kernmagneton, $g_i$ Kern-g-Faktor \fRef{qm:h:lande}}
\eq{A = \frac{g_i \mu_\textrm{K} B_\textrm{HFS}}{\sqrt{j(j+1)}}}
\end{formula}
\begin{formula}{energy_shift}

9
src/qm/qm.tex
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@ -289,7 +289,7 @@
\ger{Ehrenfest-Theorem}
]{ehrenfest_theorem}
\absLink{}{ehrenfest_theorem}
\GT{see_also} \ref{sec:qm:basics:schroedinger_equation:correspondence_principle}
\GT{see_also} \fRef{qm:se:time:ehrenfest_theorem:correspondence_principle}
\begin{formula}{ehrenfest_theorem}
\desc{Ehrenfest theorem}{applies to both pictures}{}
\desc[german]{Ehrenfest-Theorem}{gilt für beide Bilder}{}
@ -386,8 +386,6 @@
\eq{E_n = \hbar\omega \Big(\frac{1}{2} + n\Big)}
\end{formula}
\GT{see_also} \ref{sec:qm:hosc:c_a_ops}
\Subsection[
\ger{Erzeugungs und Vernichtungsoperatoren / Leiteroperatoren}
\eng{Creation and Annihilation operators / Ladder operators}
@ -525,8 +523,8 @@
\ger{Symmetrien}
]{symmetry}
\begin{ttext}[desc]
\eng{Most symmetry operators are unitary \ref{sec:linalg:unitary} because the norm of a state must be invariant under transformations of space, time and spin.}
\ger{Die meisten Symmetrieoperatoren sind unitär \ref{sec:linalg:unitary}, da die Norm eines Zustands invariant unter Raum-, Zeit- und Spin-Transformationen sein muss.}
\eng{Most symmetry operators are \fRef[unitary]{math:linalg:matrix:unitary} because the norm of a state must be invariant under transformations of space, time and spin.}
\ger{Die meisten Symmetrieoperatoren sind \fRef[unitär]{math:linalg:matrix:unitary}, da die Norm eines Zustands invariant unter Raum-, Zeit- und Spin-Transformationen sein muss.}
\end{ttext}
\begin{formula}{invariance}
\desc{Invariance}{$\hat{H}$ is invariant under a symmetrie described by $\hat{U}$ if this holds}{}
@ -587,6 +585,7 @@
}
\end{formula}
\begin{formula}{slater_det}
\desc{Slater determinant}{Construction of a fermionic (antisymmetric) many-particle wave function from single-particle wave functions}{}
\desc[german]{Slater Determinante}{Konstruktion einer fermionischen (antisymmetrischen) Vielteilchen Wellenfunktion aus ein-Teilchen Wellenfunktionen}{}

18
src/readme.md Normal file
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@ -0,0 +1,18 @@
# Formulary tex source
## Special directories
- `pkg`: My custom Latex packages
- `img`: Images generated by `../scripts/`
- `img_static`: Downloaded or other images not generated by me
- `img_static_svgs`: Downloaded or other images not generated by me that need to be converted to pdf
## Subject directories
- `bib`: bibliography files
- `ch`: chemistry
- `cm`: condensed matter
- `comp`: computational
- `ed`: electrodynamics
- `math`: mathematics
- `qm`: quantum mechanics

8
src/statistical_mechanics.tex
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@ -219,19 +219,19 @@
\eng{Material properties}
\ger{Materialeigenschaften}
]{props}
\begin{formula}{heat_cap}
\begin{formula}{heat_capacity}
\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}
\begin{formula}{heat_capacity_V}
\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}
\begin{formula}{heat_capacity_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}
@ -832,7 +832,7 @@
}
\end{formula}
\begin{formula}{heat_cap}
\begin{formula}{heat_capacity}
\desc{Heat capacity}{\gt{low_temps}}{differs from \fRef{td:TODO:petit_dulong}}
\desc[german]{Wärmecapacity}{\gt{low_temps}}{weicht ab vom \fRef{td:TODO:petit_dulong}}
\fig{img/td_fermi_heat_capacity.pdf}

4
src/util/colors.tex
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@ -30,8 +30,8 @@
\hypersetup{
colorlinks=true,
linkcolor=fg-purple,
linkcolor=fg-blue,
citecolor=fg-green,
filecolor=fg-blue,
filecolor=fg-purple,
urlcolor=fg-orange
}