\def\L{\text{L}}
\def\gl{\text{GL}}
\def\GL{Ginzburg-Landau }
\def\Tcrit{T_\text{c}}
\def\Bcrit{B_\text{c}}
\def\ssc{\text{s}}
\def\ssn{\text{n}}

\Section[
    \eng{Superconductivity}
    \ger{Supraleitung}
]{sc}
    \begin{ttext}
        \eng{
            Materials for which the electric resistance jumps to 0 under a critical temperature $\Tcrit$.
            Below $\Tcrit$ they have perfect conductivity and perfect diamagnetism, up until a critical magnetic field $\Bcrit$.
            \\\textbf{Type I}: Has a single critical magnetic field at which the superconuctor becomes a normal conductor.
            \\\textbf{Type II}: Has two critical 
        }
        \ger{
            Materialien, bei denen der elektrische Widerstand beim unterschreiten einer kritischen Temperatur $\Tcrit$ auf 0 springt.
            Sie verhalten sich dann wie ideale Leiter und ideale Diamagnete, bis zu einem kritischen Feld $\Bcrit$.

        }
    \end{ttext}

    \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.
                (\fqEqRef{ed:fields:mag:induction:lenz})
            }
            \ger{
                Im Gegensatz zu einem Supraleiter werden ideale Leiter nur dann diamagnetisch, wenn das externe magnetische Feld \textbf{nach} dem Abkühlen unter die kritische Temperatur eingeschaltet wird.
                (\fqEqRef{ed:fields:mag:induction:lenz})
            }
        }
    \end{formula}

    \begin{formula}{meissner_effect}
        \desc{Meißner-Ochsenfeld effect}{Perfect diamagnetism}{}
        \desc[german]{Meißner-Ochsenfeld Effekt}{Idealer Diamagnetismus}{}
        \ttxt{
            \eng{External magnetic field decays exponetially inside the superconductor below a critical temperature and a critical magnetic field.} 
            \ger{Externes Magnetfeld fällt im Supraleiter exponentiell unterhalb einer kritischen Temperatur und unterhalb einer kritischen Feldstärke ab.} 
        }
    \end{formula}
    \Subsection[
        \eng{London equations}
        \ger{London-Gleichungen}
    ]{london}
        \begin{ttext}
            \eng{
                Quantitative description of the \fqEqRef{cm:sc:meissner_effect}.
            }
            \ger{
                Quantitative Beschreibung des \fqEqRef{cm:sc:meissner_effect}s.
            }
            
        \end{ttext}
        % \begin{formula}{coefficient}
        %     \desc{London-coefficient}{}{}
        %     \desc[german]{London-Koeffizient}{}{}
        %     \eq{\Lambda = \frac{m_\ssc}{n_\ssc q_\ssc^2}}
        % \end{formula}
        \begin{formula}{first}
            % \vec{j} = \frac{nq\hbar}{m}\Grad S - \frac{nq^2}{m}\vec{A}
            \desc{First London Equation}{}{$\vec{j}$ current density, $n_\ssc$, $m_\ssc$, $q_\ssc$  density, mass and charge of superconduticng particles}
            \desc[german]{Erste London-Gleichun-}{}{$\vec{j}$ Stromdichte, $n_\ssc$, $m_\ssc$, $q_\ssc$ Dichte, Masse und Ladung der supraleitenden Teilchen}
            \eq{
                \pdv{\vec{j}_{\ssc}}{t} = \frac{n_\ssc q_\ssc^2}{m_\ssc}\vec{E} {\color{gray}- \Order{\vec{j}_\ssc^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 \fqEqRef{cm:sc:meissner_effect}}{$\vec{j}$ current density, $n_\ssc$, $m_\ssc$, $q_\ssc$ density, mass and charge of superconduticng particles}
            \desc[german]{Zweite London-Gleichung}{Beschreibt den \fqEqRef{cm:sc:meissner_effect}}{$\vec{j}$ Stromdichte, $n_\ssc$, $m_\ssc$, $q_\ssc$ Dichte, Masse und Ladung der supraleitenden Teilchen}
            \eq{
                \Rot \vec{j_\ssc} = -\frac{n_\ssc q_\ssc^2}{m_\ssc} \vec{B}
            }
        \end{formula}
        \begin{formula}{penetration_depth}
            \desc{London penetration depth}{}{}
            \desc[german]{London Eindringtiefe}{}{}
            \eq{\lambda_\L = \sqrt{\frac{m_\ssc}{\mu_0 n_\ssc q_\ssc^2}}}
        \end{formula}



    \Subsection[
        \eng{\GL Theory (GLAG)}
        \ger{\GL Theorie (GLAG)}
    ]{gl}
        \begin{ttext}
            \eng{
                \TODO{TODO}
            }
            
        \end{ttext}
        \begin{formula}{coherence_length}
            \desc{\GL Coherence Length}{}{}
            \desc[german]{\GL Kohärenzlänge}{}{}
            \eq{
                \xi_\gl &= \frac{\hbar}{\sqrt{2m \abs{\alpha}}} \\
                \xi_\gl(T) &= \xi_\gl(0) \frac{1}{\sqrt{1-\frac{T}{\Tcrit}}}
            }
        \end{formula}
        \begin{formula}{penetration_depth}
            \desc{\GL Penetration Depth / Field screening length}{}{}
            \desc[german]{\GL Eindringtiefe}{}{}
            \eq{
                \lambda_\gl &= \sqrt{\frac{m_\ssc\beta}{\mu_0 \abs{\alpha} q_s^2}} \\
                \lambda_\gl(T) &= \lambda_\gl(0) \frac{1}{\sqrt{1-\frac{T}{\Tcrit}}}
            }
        \end{formula}
        \begin{formula}{first}
            \desc{First Ginzburg-Landau Equation}{}{$\xi_\gl$ \fqEqRef{cm:sc:gl:coherence_length}, $\lambda_\gl$ \fqEqRef{cm:sc:gl:penetration_depth}}
            \desc[german]{Erste Ginzburg-Landau Gleichung}{}{}
            \eq{\alpha\Psi + \beta\abs{\Psi}^2 \Psi + \frac{1}{2m} (-i\hbar \Grad + 2e\vec{A})^2\Psi = 0}
        \end{formula}
        \begin{formula}{second}
            \desc{Second Ginzburg-Landau Equation}{}{}
            \desc[german]{Zweite Ginzburg-Landau Gleichung}{}{}
            \eq{\vec{j_\ssc} = \frac{ie\hbar}{m}(\Psi^*\Grad\Psi - \Psi\Grad\Psi^*) - \frac{4e^2}{m}\abs{\Psi}^2 \vec{A}}
        \end{formula}

        \TODO{proximity effect}

    \Subsection[
        \eng{Microscopic theory}
        \ger{Mikroskopische Theorie}
    ]{micro}

        \begin{formula}{isotop_effect}
            \desc{Isotope effect}{Superconducting behaviour depends on atomic mass and thereby of the lattice \Rightarrow Microscopic origin}{$\Tcrit$ critial temperature, $M$ isotope mass, $\omega_\text{ph}$}
            \desc[german]{Isotopeneffekt}{Supraleitung hängt von der Atommasse und daher von den Gittereigenschaften ab \Rightarrow Mikroskopischer Ursprung}{$\Tcrit$ kritische Temperatur, $M$ Isotopen-Masse, $\omega_\text{ph}$}
            \eq{
                \Tcrit \propto \frac{1}{\sqrt{M}} \\
                \omega_\text{ph} \propto \frac{1}{\sqrt{M}} \Rightarrow \Tcrit \propto \omega_\text{ph}
            }
        \end{formula}

        \begin{formula}{cooper_pairs}
            \desc{Cooper pairs}{}{}
            \desc[german]{Cooper-Paars}{}{}
            \ttxt{
                \eng{Conduction electrons reduce their energy through an attractive interaction: One electron passing by atoms attracts the these, which creats a positive charge region behind the electron, which in turn attracts another electron. }
            }
        \end{formula}
    
        \Subsubsection[
            \eng{BCS-Theory}
            \ger{BCS-Theorie}
        ]{bcs}
            \begin{ttext}
                \eng{
                    Electron pairs form bosonic quasi-particles called Cooper pairs which can condensate into the ground state. 
                    The wave function spans the whole material, which makes it conduct without resistance.
                    The exchange bosons between the electrons are phonons.
                }
                \ger{
                    Elektronenpaar bilden bosonische Quasipartikel (Cooper Paare) welche in den Grundzustand kondensieren können. 
                    Die Wellenfunktion übersoannt den gesamten Festkörper, was einen widerstandslosen Ladungstransport garantiert.
                    Die Austauschbosononen zwischen den Elektronen sind Bosonen.
                }
            \end{ttext}
            \def\BCS{{\text{BCS}}}
            \begin{formula}{hamiltonian}
                \desc{BCS Hamiltonian}{for $N$ interacting electrons}{
                    $c_{\veck\sigma}$ creation/annihilation operators create/destroy at $\veck$ with spin $\sigma$ \\
                    First term: non-interacting free electron gas\\
                    Second term: interaction energy
                }
                \desc[german]{BCS Hamiltonian}{}{}
                \eq{
                    \hat{H}_\BCS = 
                    \sum_{\sigma} \sum_\veck \epsilon_\veck \hat{c}_{\veck\sigma}^\dagger \hat{c}_{\veck\sigma}
                    + \sum_{\veck,\veck^\prime} V_{\veck,\veck^\prime} 
                        \hat{c}_{\veck\uparrow}^\dagger \hat{c}_{-\veck\downarrow}^\dagger
                        \hat{c}_{-\veck^\prime\downarrow} \hat{c}_{\veck^\prime,\uparrow}
                }
            \end{formula}

            \begin{formula}{bogoliubov-valatin}
                \desc{Bogoliubov-Valatin transformation}{Diagonalization of the \fqEqRef{cm:sc:micro:bcs:hamiltonian} to derive excitation energies}{}
                \desc[german]{Bogoliubov-Valatin transformation}{}{}
                \eq{
                    \hat{H}_\BCS  - N\mu = \sum_\veck \big[\xi_\veck - E_\veck + \Delta_\veck g_\veck^*\big] + \sum_\veck \big[E_\veck a_\veck^\dagger a_\veck + E_\veck \beta_{-\veck}^\dagger \beta_{-\veck}\big]
                }
            \end{formula}

            \begin{formula}{gap_equation}
                \desc{BCS-gap equation}{}{}
                \desc[german]{}{}{}
                \eq{\Delta_\veck^* = -\sum_\veck^+\prime V_{\veck,\veck^\prime} \frac{\Delta_{\veck^\prime}}{2E_\veck} \tanh \left(\frac{E_{\veck^\prime}}{2\kB T}\right)}
            \end{formula}
