\Section[
    \eng{Crystals}
    \ger{Kristalle}
]{crystal}
\Subsection[
    \eng{Bravais lattice}
    \ger{Bravais-Gitter}
]{bravais}


    \Eng[lattice_system]{Lattice system}
    \Ger[lattice_system]{Gittersystem}
    \Eng[crystal_family]{Crystal system}
    \Ger[crystal_family]{Kristall-system}
    \Eng[point_group]{Point group}
    \Ger[point_group]{Punktgruppe}
    \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}}
    \renewcommand\tabularxcolumn[1]{m{#1}}
    \newcolumntype{Z}{>{\centering\let\newline\\\arraybackslash\hspace{0pt}}X}

    \begin{bigformula}{2d}
        \desc{2D}{In 2D, there are 5 different Bravais lattices}{}
        \desc[german]{2D}{In 2D gibt es 5 verschiedene Bravais-Gitter}{}
        \begin{adjustbox}{width=\textwidth}
            \begin{tabularx}{\textwidth}{||Z|c|Z|Z||}
                \hline
                \multirow{2}{*}{\GT{lattice_system}} & \multirow{2}{*}{\GT{point_group}} & \multicolumn{2}{c||}{5 \gt{bravais_lattices}} \\ \cline{3-4}
                                                     & & \GT{primitive} (p) & \GT{centered} (c) \\ \hline
                \GT{monoclinic} (m) & $\text{C}_\text{2}$   & \bvimg{mp} & \\ \hline
                \GT{orthorhombic} (o) & $\text{D}_\text{2}$ & \bvimg{op} & \bvimg{oc} \\ \hline
                \GT{tetragonal} (t) & $\text{D}_\text{4}$   & \bvimg{tp} & \\ \hline
                \GT{hexagonal} (h) & $\text{D}_\text{6}$    & \bvimg{hp} & \\ \hline
            \end{tabularx}
        \end{adjustbox}
    \end{bigformula}
    

    \begin{bigformula}{3d}
        \desc{3D}{In 3D, there are 14 different Bravais lattices}{}
        \desc[german]{3D}{In 3D gibt es 14 verschiedene Bravais-Gitter}{}
        % \newcolumntype{g}{>{\columncolor[]{0.8}}}
        \begin{adjustbox}{width=\textwidth}
            \begin{tabularx}{\textwidth}{||Z|Z|c|Z|Z|Z|Z||}
                \hline
                \multirow{2}{*}{\GT{crystal_family}} & \multirow{2}{*}{\GT{lattice_system}} & \multirow{2}{*}{\GT{point_group}} & \multicolumn{4}{c||}{14 \gt{bravais_lattices}} \\ \cline{4-7}
                                                     & & & \GT{primitive} (P) & \GT{base_centered} (S) & \GT{body_centered} (I) & \GT{face_centered} (F) \\ \hline
                \multicolumn{2}{||c|}{\GT{triclinic} (a)} & $\text{C}_\text{i}$ & \bvimg{tP}  &  &  &  \\  \hline
                \multicolumn{2}{||c|}{\GT{monoclinic} (m)} & $\text{C}_\text{2h}$ & \bvimg{mP} & \bvimg{mS}  &  &  \\ \hline
                \multicolumn{2}{||c|}{\GT{orthorhombic} (o)} & $\text{D}_\text{2h}$ & \bvimg{oP} & \bvimg{oS} & \bvimg{oI} & \bvimg{oF} \\ \hline
                \multicolumn{2}{||c|}{\GT{tetragonal} (t)} & $\text{D}_\text{4h}$ & \bvimg{tP} &  & \bvimg{tI} & \\ \hline
                \multirow{2}{*}{\GT{hexagonal} (h)} & \GT{rhombohedral} & $\text{D}_\text{3d}$ & \bvimg{hR} &  &  & \\ \cline{2-7}
                 & \GT{hexagonal} & $\text{D}_\text{6h}$ & \bvimg{hP} &  &  & \\ \hline
                \multicolumn{2}{||c|}{\GT{cubic} (c)} & $\text{O}_\text{h}$ & \bvimg{cP} &  & \bvimg{cI} & \bvimg{cF} \\ \hline
            \end{tabularx}
        \end{adjustbox}
    \end{bigformula}

    \begin{formula}{lattice_constant}
        \desc{Lattice constant}{Parameter (length or angle) describing the smallest unit cell}{}
        \desc[german]{Gitterkonstante}{Parameter (Länge oder Winkel) der die Einheitszelle beschreibt}{}
        \quantity{a}{}{s}
    \end{formula}

    \begin{formula}{lattice_vector}
        \desc{Lattice vector}{}{$n_i \in \Z$}
        \desc[german]{Gittervektor}{}{}
        \quantity{\vec{R}}{}{\angstrom}
        \eq{\vec{R} = n_1 \vec{a_1} + n_2 \vec{a_2} + n_3 \vec{a_3}}
    \end{formula}

    \TODO{primitive unit cell: contains one lattice point}\\
    \begin{formula}{miller}
        \desc{Miller index}{}{Miller family: planes that are equivalent due to crystal symmetry}
        \desc[german]{Millersche Indizes}{}{}
        \eq{
            (hkl) & \text{\GT{plane}}\\
            [hkl] & \text{\GT{direction}}\\
            \{hkl\} & \text{\GT{millerFamily}}
        }
    \end{formula}
    

\Subsection[
    \eng{Reciprocal lattice}
    \ger{Reziprokes Gitter}
]{reci}
    \begin{ttext}
        \eng{The reciprokal lattice is made up of all the wave vectors $\vec{k}$ that ressemble standing waves with the periodicity of the Bravais lattice.}
        \ger{Das rezioproke Gitter besteht aus dem dem Satz aller Wellenvektoren $\vec{k}$, die ebene Wellen mit der Periodizität des Bravais-Gitters ergeben.}
    \end{ttext}

    \begin{formula}{vectors}
        \desc{Reciprocal lattice vectors}{}{$a_i$ real-space lattice vectors, $V_c$ volume of the primitive lattice cell}
        \desc[german]{Reziproke Gittervektoren}{}{$a_i$ Bravais-Gitter Vektoren, $V_c$ Volumen der primitiven Gitterzelle}
        \eq{
            \vec{b_1} &= \frac{2\pi}{V_c} \vec{a_2} \times \vec{a_3} \\
            \vec{b_2} &= \frac{2\pi}{V_c} \vec{a_3} \times \vec{a_1} \\
            \vec{b_3} &= \frac{2\pi}{V_c} \vec{a_1} \times \vec{a_2}
        }
    \end{formula}
    \begin{formula}{reciprocal_lattice_vector}
        \desc{Reciprokal attice vector}{}{$n_i \in \Z$}
        \desc[german]{Reziproker Gittervektor}{}{}
        \quantity{\vec{G}}{}{\angstrom}
        \eq{\vec{G}_{{hkl}} = h \vec{b_1} + k \vec{b_2} + l \vec{b_3}}
    \end{formula}

    \Subsection[
        \eng{Scattering processes}
        \ger{Streuprozesse}
    ]{scatter}
        \begin{formula}{matthiessen}
            \desc{Matthiessen's rule}{Approximation, only holds if the processes are independent of each other}{\QtyRef{mobility}, \QtyRef{scattering_time}}
            \desc[german]{Matthiessensche Regel}{Näherung, nur gültig wenn die einzelnen Streuprozesse von einander unabhängig sind}{}
            \eq{
                \frac{1}{\mu} &= \sum_{i = \textrm{\GT{\fqname}}} \frac{1}{\mu_i} \\
                \frac{1}{\tau} &= \sum_{i = \textrm{\GT{\fqname}}} \frac{1}{\tau_i}
            }
        \end{formula}

\Subsection[
    \eng{Lattices}
    \ger{Gitter}
]{lat}
    \begin{formula}{sc}
        \desc{Simple cubic (SC)}{Reciprocal: Simple cubic}{\QtyRef{lattice_constant}}
        \desc[german]{Einfach kubisch (SC)}{Reziprok: Einfach kubisch}{}
        \eq{
            \vec{a}_{1}=a \begin{pmatrix} 1\\0\\0 \end{pmatrix},\, 
            \vec{a}_{2}=a \begin{pmatrix} 0\\1\\0 \end{pmatrix},\, 
            \vec{a}_{3}=a \begin{pmatrix} 0\\0\\1 \end{pmatrix} 
        }
    \end{formula}
    \begin{formula}{bcc}
        \desc{Body centered cubic (BCC)}{Reciprocal: \fqEqRef{cm:bravais:fcc}}{\QtyRef{lattice_constant}}
        \desc[german]{Kubisch raumzentriert (BCC)}{Reziprok: \fqEqRef{cm:bravais:fcc}}{}
        \eq{
            \vec{a}_{1}=\frac{a}{2} \begin{pmatrix} -1\\1\\1 \end{pmatrix},\, 
            \vec{a}_{2}=\frac{a}{2} \begin{pmatrix} 1\\-1\\1 \end{pmatrix},\, 
            \vec{a}_{3}=\frac{a}{2} \begin{pmatrix} 1\\1\\-1 \end{pmatrix} 
        }
    \end{formula}

    \begin{formula}{fcc}
        \desc{Face centered cubic (FCC)}{Reciprocal: \fqEqRef{cm:bravais:bcc}}{\QtyRef{lattice_constant}}
        \desc[german]{Kubisch flächenzentriert (FCC)}{Reziprok: \fqEqRef{cm:bravais:bcc}}{}
        \eq{
            \vec{a}_{1}=\frac{a}{2} \begin{pmatrix} 0\\1\\1 \end{pmatrix},\,
            \vec{a}_{2}=\frac{a}{2} \begin{pmatrix} 1\\0\\1 \end{pmatrix},\, 
            \vec{a}_{3}=\frac{a}{2} \begin{pmatrix} 1\\1\\0 \end{pmatrix}
        }
    \end{formula}

    \begin{formula}{diamond}
        \desc{Diamond lattice}{}{}
        \desc[german]{Diamantstruktur}{}{}
        \ttxt{
            \eng{\fqEqRef{cm:bravais:fcc} with basis $\begin{pmatrix} 0 & 0 & 0 \end{pmatrix}$ and $\begin{pmatrix} \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \end{pmatrix}$}
            \ger{\fqEqRef{cm:bravais:fcc} mit Basis  $\begin{pmatrix} 0 & 0 & 0 \end{pmatrix}$ und $\begin{pmatrix} \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \end{pmatrix}$}
        }
    \end{formula}
    \begin{formula}{zincblende}
        \desc{Zincblende lattice}{}{}
        \desc[german]{Zinkblende-Struktur}{}{}
        \ttxt{
            \includegraphics[width=0.5\textwidth]{img/cm_zincblende.png}
            \eng{Like \fqEqRef{cm:bravais:diamond} but with different species on each basis}
            \ger{Wie  \fqEqRef{cm:bravais:diamond} aber mit unterschiedlichen Spezies auf den Basen}
        }
    \end{formula}
    \begin{formula}{wurtzite}
        \desc{Wurtzite structure}{hP4}{}
        \desc[german]{Wurtzite-Struktur}{hP4}{}
        \ttxt{
            \includegraphics[width=0.5\textwidth]{img/cm_wurtzite.png}
            Placeholder
        }
    \end{formula}

