small changes

src/cm/cm.tex

@@ -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}
-
-

src/cm/crystal.tex

@@ -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{
+        \fsplit{
-            \includegraphics[width=0.5\textwidth]{img/cm_zincblende.png}
+            \centering
-            \eng{Like \fRef{:::diamond} but with different species on each basis}
+            \includegraphics[width=0.9\textwidth]{img/cm_crystal_zincblende.png}
-            \ger{Wie  \fRef{:::diamond} aber mit unterschiedlichen Spezies auf den Basen}
+        }{
+            \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{
+        \fsplit{
-            \includegraphics[width=0.5\textwidth]{img/cm_wurtzite.png}
+            \centering
-            \TODO{Placeholder}
+            \includegraphics[width=0.9\textwidth]{img/cm_crystal_wurtzite.png}
+        }{
+
         }
     \end{formula}
 

src/cm/superconductivity.tex

@@ -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{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}, \fRef[kritische Stromdichte]{::1st_relation}, $\phi$ Phasendifferenz zwischen den Supraleitern}
+                \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{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}{}{}
+            \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) \frac{B_\text{c,th}^2}{2\mu_0}}
+            \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}
+            \fig{img/cm_super_n_s_boundary.pdf}
-            \TODO{plot, slide 106}
+            % \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}
+                \eng{Excitations and finite temperatures}
-                % \ger{}
+                \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{Pairing amplitude}{}{}
-                    \desc[german]{Paarungsamplitude}{Anregungsenergie}{}
+                    \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{Energy dependance of the \fRef{:::bcs:coherence_factors}}{}{$E_\veck$ \fRef{::pairing_amplitude}, \GT{see} \fRef{:::bcs:coherence_factors}}
-                    \desc[german]{Energieabhängigkeit der \fRef{:::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{Self-consistend gap equation}{}{}
-                    \desc[german]{BCS Energielückengleichung}{}{}
+                    \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)}
+                    \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{Temperature dependance of the crictial magnetic field}{Jump at $T_\txc$, then exponential decay}{}
-                    \desc[german]{Temperaturabhängigkeit des kritischen Magnetfelds}{}{}
+                    \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}

src/cm/techniques.tex

@@ -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}
 

src/cm/vib.tex

@@ -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}
+
+
+

src/main.tex

@@ -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}

src/pkg/mqformula.sty

@@ -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}%
-            \raggedright
+        \IfTranslationExists{#3}{%
-            \GT{#2}
+            \\ {\color{fg1} \GT{#3}}%
-        }{\detokenize{#2}}
-        \IfTranslationExists{#3}{
-            \\ {\color{fg1} \GT{#3}}
         }{}
     \end{minipage}
 }
@@ -152,12 +149,16 @@
     }
 
     \newcommand{\fsplit}[3][0.5]{
-        \begin{minipage}{##1\linewidth}
+        \begingroup
-            ##2
+            \renewcommand{\newFormulaEntry}{}
-        \end{minipage}
+            \begin{minipage}{##1\linewidth}
-        \begin{minipage}{\luavar{0.99-##1}\linewidth}
+                ##2
-            ##3
+            \end{minipage}
-        \end{minipage}
+            \begin{minipage}{\luavar{0.99-##1}\linewidth}
+                ##3
+            \end{minipage}
+        \endgroup
+        \newFormulaEntry
     }
 }{
     \mqfqname@leave

src/pkg/mqperiodictable.sty

@@ -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,

src/pkg/mqref.sty

@@ -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][]{%

src/qm/atom.tex

@@ -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{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 \ref{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}

src/qm/qm.tex

@@ -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.}
+            \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 unitär \ref{sec:linalg:unitary}, da die Norm eines Zustands invariant unter Raum-, Zeit- und Spin-Transformationen sein muss.}
+            \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}{}

src/readme.md

@@ -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

src/statistical_mechanics.tex

@@ -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}

src/util/colors.tex

@@ -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
 }