formelsammlung/src/ed/misc.tex

% TODO move
\Section[
    \eng{Hall-Effect}
    \ger{Hall-Effekt}
    ]{hall}

    \begin{formula}{cyclotron}
        \desc{Cyclontron frequency}{}{}
        \desc[german]{Zyklotronfrequenz}{}{}
        \eq{\omega_\text{c} = \frac{e B}{\masse}}
    \end{formula}
    \TODO{Move}


    \Subsection[
        \eng{Classical Hall-Effect}
        \ger{Klassischer Hall-Effekt}
        ]{classic}
        \begin{ttext}
            \eng{Current flowing in $x$ direction in a conductor ($l \times b \times d$) with a magnetic field $B$ in $z$ direction leads to a hall voltage $U_\text{H}$ in $y$ direction.}
            \ger{Fließt in einem Leiter ($l \times b \times d$) ein Strom in $x$ Richtung, während der Leiter von einem Magnetfeld $B$ in $z$-Richtung durchdrungen, wird eine Hallspannung $U_\text{H}$ in $y$-Richtung induziert.}
        \end{ttext}
        \begin{formula}{voltage}
            \desc{Hall voltage}{}{$n$ charge carrier density}
            \desc[german]{Hallspannung}{}{$n$ Ladungsträgerdichte}
            \eq{U_\text{H} = \frac{I B}{ne d}}
        \end{formula}

        \begin{formula}{coefficient}
            \desc{Hall coefficient}{Sometimes $R_\txH$}{}
            \desc[german]{Hall-Koeffizient}{Manchmal $R_\txH$}{}
            \eq{A_\text{H} := -\frac{E_y}{j_x B_z} \explOverEq{\GT{metals}} \frac{1}{ne} = \frac{\rho_{xy}}{B_z}}
        \end{formula}

        \begin{formula}{resistivity}
            \desc{Resistivity}{}{}
            \desc[german]{Spezifischer Widerstand}{}{}
            \eq{\rho_{xx} &= \frac{\masse}{ne^2\tau} \\ \rho_{xy} &= \frac{B}{ne}}
        \end{formula}


    \Subsection[
        \eng{Integer quantum hall effect}
        \ger{Ganzahliger Quantenhalleffekt}
    ]{quantum}  

        \begin{formula}{conductivity}
            \desc{Conductivity tensor}{}{}
            \desc[german]{Leitfähigkeitstensor}{}{}
            \eq{\sigma = \begin{pmatrix} \sigma_{xy} & \sigma_{xy} \\ \sigma_{yx} & \sigma_{yy} \end{pmatrix} }
        \end{formula}

        \begin{formula}{resistivity_tensor}
            \desc{Resistivity tensor}{}{}
            \desc[german]{Spezifischer Widerstands-tensor}{}{}
            \eq{
                \rho = \sigma^{-1}
                % \sigma = \begin{pmatrix} \sigma_{xy} & \sigma_{xy} \\ \sigma_{yx} & \sigma_{yy} \end{pmatrix} }
            }
        \end{formula}

        \begin{formula}{resistivity}
            \desc{Resistivity}{}{$\nu \in \mathbb{Z}$ filing factor}
            \desc[german]{Spezifischer Hallwiderstand}{}{$\nu \in \mathbb{Z}$ Füllfaktor}
            \eq{\rho_{xy} = \frac{2\pi\hbar}{e^2} \frac{1}{\nu}}
        \end{formula}

        % \begin{formula}{qhe}
        %     \desc{Integer quantum hall effect}{}{}
        %     \desc[german]{Ganzahliger Quanten-Hall-Effekt}{}{}
        %     \fig{img/qhe-klitzing.jpeg}
        % \end{formula}

        \begin{formula}{fqhe}
            \desc{Fractional quantum hall effect}{}{$\nu$ fraction of two numbers without shared divisors}
            \desc[german]{Fraktionaler Quantum-Hall-Effekt}{}{$\nu$ Bruch aus Zahlen ohne gemeinsamen Teiler}
            \eq{\nu = \frac{1}{3},\frac{2}{5},\frac{3}{7},\frac{2}{3}...}
        \end{formula}

        \begin{ttext}
            \eng{
                \begin{itemize}
                    \item \textbf{Integer} (QHE): filling factor $\nu$ is an integer
                    \item \textbf{Fractional} (FQHE): filling factor $\nu$ is a fraction
                    \item \textbf{Spin} (QSHE): spin currents instead of charge currents
                    \item \textbf{Anomalous} (QAHE): symmetry breaking by internal effects instead of external magnetic fields
                \end{itemize}
            }
            \ger{
                \begin{itemize}
                    \item \textbf{Integer} (QHE): Füllfaktor $\nu$ ist ganzzahlig
                    \item \textbf{Fractional} (FQHE): Füllfaktor $\nu$ ist ein Bruch
                    \item \textbf{Spin} (QSHE): Spin Ströme anstatt Ladungsströme
                    \item \textbf{Anomalous} (QAHE): Symmetriebruch durch interne Effekte anstatt druch ein externes Magnetfeld
                \end{itemize}
            }
        \end{ttext}
    

    \TODO{sort}


\Section[
    \eng{Dipole-stuff}
    \ger{Dipol-zeug}
]{dipole}

    \begin{formula}{poynting}
        \desc{Dipole radiation Poynting vector}{}{}
        \desc[german]{Dipolsrahlung Poynting-Vektor}{}{}
        \eq{\vec{S} = \left(\frac{\mu_0 p_0^2 \omega^4}{32\pi^2 c}\right)\frac{\sin^2\theta}{r^2} \vec{r}}
    \end{formula}

    \begin{formula}{power}
        \desc{Time-average power}{}{}
        \desc[german]{Zeitlich mittlere Leistung}{}{}
        \eq{P = \frac{\mu_0\omega^4 p_0^2}{12\pi c}}
    \end{formula}

\Section[
    \eng{misc}
    \ger{misc}
]{misc}
    \begin{formula}{impedance_r}
        \desc{Impedance of an ohmic resistor}{}{\QtyRef{resistance}}
        \desc[german]{Impedanz eines Ohmschen Widerstands}{}{}
        \eq{Z_{R} = R}
    \end{formula}
    \begin{formula}{impedance_c}
        \desc{Impedance of a capacitor}{}{\QtyRef{capacity}, \QtyRef{angular_velocity}}
        \desc[german]{Impedanz eines Kondensators}{}{}
        \eq{Z_{C} = \frac{1}{\I\omega C}}
    \end{formula}

    \begin{formula}{impedance_l}
        \desc{Impedance of an inductor}{}{\QtyRef{inductance}, \QtyRef{angular_velocity}}
        \desc[german]{Impedanz eines Induktors}{}{}
        \eq{Z_{L} = \I\omega L}
    \end{formula}

    \TODO{impedance addition for parallel / linear}