\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}
