\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_static/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}

	\begin{formula}{primitive_unit_cell}
		\desc{Primitve unit cell}{}{}
		\desc[german]{Primitive Einheitszelle}{}{}
		\ttxt{\eng{Unit cell containing exactly one lattice point}\ger{Einheitszelle die genau einen Gitterpunkt enthält}}
	\end{formula}

	\Eng[miller-point]{Point}
	\Ger[miller-point]{Punkt}
	\Eng[miller-direction]{Direction}
	\Ger[miller-direction]{Richtung}
	\Eng[miller-direction-family]{Family of directions}
	\Ger[miller-direction-family]{Familie von Richtungen}
	\Eng[miller-plane]{Plane}
	\Ger[miller-plane]{Ebene}
	\Eng[miller-plane-family]{Family of planes}
	\Ger[miller-plane-family]{Familie von Ebenen}
	\begin{formula}{miller}
		\desc{Miller indices}{}{
		  Miller planes: $(hkl)$, $\frac{1}{h}$/$\frac{1}{k}$/$\frac{1}{l}$ give intersection with $x$/$y$/$z$ axes\\
		  Miller family: planes that are equivalent due to crystal symmetry
		}
		\desc[german]{Millersche Indizes}{}{
		  Miller-Ebenen: $(hkl)$, $\frac{1}{h}$/$\frac{1}{k}$/$\frac{1}{l}$ geben die Schnittpunkte mit den $x$/$y$/$z$-Achsen\\
		  Miller-Familien: Ebenen, die durch Kristallsymmetrie äquivalent sind
		}
		\centering
		\newFormulaEntry
		\begin{tabularx}{\textwidth}{clcl}
			$(h,k,l)$ & \GT{miller-point} &  & \\
			$hkl$ & \GT{miller-direction} & $\langle hkl \rangle$ & \GT{miller-direction-family} \\
			$(hkl)$ & \GT{miller-plane} & $\{hkl\}$ & \GT{miller-plane-family}
		\end{tabularx}
		\pgfmathsetmacro{\rectX}{2}
		\pgfmathsetmacro{\rectZ}{2}
		\newFormulaEntry
		\begin{tikzpicture}[3d view={100}{20},perspective={p={(-55,0,0)},q={(0,25,0)},r={(0,0,-30)}}]
			% <100> direction family
		   \begin{scope}
				\drawRectCS{1.4*\rectX}{1.4*\rectZ}
				\setRectPoints{R1}{(0.5*\rectX,0.5*\rectX,0)}{\rectX}{\rectX}
				\setRectPoints{R2}{(0.5*\rectX,0.5*\rectX,\rectZ)}{\rectX}{\rectX}
				\drawRectBack{R1}
				\drawRectConnectionsBack{R1}{R2}
				\draw[miller dir] (0,0,0) -- ++( \rectX,0,0) node[anchor=east] {$[100]$};
				\draw[miller dir] (0,0,0) -- ++(-\rectX,0,0) node[anchor=west] {$[\bar{1}00]$};
				\draw[miller dir] (0,0,0) -- ++(0, \rectX,0) node[anchor=south] {$[010]$};
				\draw[miller dir] (0,0,0) -- ++(0,-\rectX,0) node[anchor=south] {$[0\bar{1}0]$};
				\draw[miller dir] (0,0,0) -- ++(0,0, \rectX) node[anchor=east] {$[001]$};
				\draw[miller dir] (0,0,0) -- ++(0,0,-\rectX) node[anchor=west] {$[00\bar{1}]$};
				\drawRectFront{R1}
				\drawRectBack{R2}
				\drawRectConnectionsFront{R1}{R2}
				\drawRectFront{R2}
				\node at (1.5*\rectX,1.5*\rectX, 0) {$\langle100\rangle$};
			\end{scope}
			\pgfmathsetmacro{\rectDistance}{4.5}
			% {100} plane family
			\begin{scope}[shift={(0,\rectDistance,0)}]
				\drawRectCS{1.4*\rectX}{1.4*\rectZ}
				\setRectPoints{R1}{(0.5*\rectX,0.5*\rectX,0)}{\rectX}{\rectX}
				\setRectPoints{R2}{(0.5*\rectX,0.5*\rectX,\rectZ)}{\rectX}{\rectX}
				\drawRectBack{R1}
				\drawRectConnectionsBack{R1}{R2}
				\drawRectFront{R1}
				\drawRectBack{R2}
				\drawRectConnectionsFront{R1}{R2}
				\drawRectFront{R2}
				\fill[miller plane] (R1-C) -- (R1-D) node[anchor=north,midway] {$(100)$} -- (R2-D) -- (R2-C) -- cycle;
				\fill[miller plane] (R1-A) -- (R1-D) node[anchor=west,midway] {$(010)$} -- (R2-D) -- (R2-A) -- cycle node[anchor=north east] {$(010)$};
				\fill[miller plane] (R2-A) -- (R2-B) node[midway,anchor=south] {$(001)$} -- (R2-C) -- (R2-D) -- cycle;
				\node at (1.5*\rectX,1.5*\rectX, 0) {$\{100\}$};
			\end{scope}
		\end{tikzpicture}
		% describe how to construct miller planes
	\end{formula}

	\begin{formula}{miller-hexagon}
		\desc{Hexagonal miller indices}{}{}
		\desc[german]{Hexagonale Millersche Indizes}{}{}
		\eq{ (hkil) && \tGT{with}\quad i = h + k }
		\centering
		\newFormulaEntry
		\begin{tikzpicture}[3d view={0}{20}]
			\pgfmathsetmacro{\hexxY}{1.5}
			\begin{scope}
				\drawHexagonCS{1}{\hexxY}
				\setHexagonPoints{H1}{(0,0,0)}{1}{1}{1}
				\setHexagonPoints{H2}{(0,0,\hexxY)}{1}{1}{1}
				\drawHexagonBack{H1}
				\drawHexagonConnectionsBack{H1}{H2}
				\drawHexagonFront{H1}
				\drawHexagonBack{H2}
				\drawHexagonConnectionsFront{H1}{H2}
				\drawHexagonFront{H2}
			\end{scope}
			\pgfmathsetmacro{\hexDistance}{3.5}
			% 1121
			\begin{scope}[shift={(\hexDistance,0,0)}]
				\drawHexagonCS{1}{\hexxY}
				\setHexagonPoints{H1}{(0,0,0)}{1}{1}{1}
				\setHexagonPoints{H2}{(0,0,\hexxY)}{1}{1}{1}
				\drawHexagonBack{H1}
				\drawHexagonConnectionsBack{H1}{H2}
				\fill[miller plane] (H1-A) -- (H2-M) -- (H1-E) -- cycle;
				\drawHexagonFront{H1}
				\drawHexagonBack{H2}
				\drawHexagonConnectionsFront{H1}{H2}
				\drawHexagonFront{H2}
				\node[anchor=north] at (xyz cylindrical cs:radius=1.5,angle=270) {$(1211)$};
			\end{scope}
			% 1010
			\begin{scope}[shift={(2*\hexDistance,0,0)}]
				\drawHexagonCS{1}{\hexxY}
				\setHexagonPoints{H1}{(0,0,0)}{1}{1}{1}
				\setHexagonPoints{H2}{(0,0,\hexxY)}{1}{1}{1}
				\drawHexagonBack{H1}
				\drawHexagonConnectionsBack{H1}{H2}
				\drawHexagonFront{H1}
				\drawHexagonBack{H2}
				\drawHexagonConnectionsFront{H1}{H2}
				\drawHexagonFront{H2}
				\fill[miller plane] (H1-F) -- (H2-F) -- (H2-E) -- (H1-E) -- cycle;
				\node[anchor=north] at (xyz cylindrical cs:radius=1.5,angle=270) {$(1010)$};
			\end{scope}
		\end{tikzpicture}
	\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{:::scatter}}} \frac{1}{\mu_i} \\
				\frac{1}{\tau} &= \sum_{i = \textrm{\GT{:::scatter}}} \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: \fRef{::fcc}}{\QtyRef{lattice_constant}}
		\desc[german]{Kubisch raumzentriert (BCC)}{Reziprok: \fRef{::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: \fRef{::bcc}}{\QtyRef{lattice_constant}}
		\desc[german]{Kubisch flächenzentriert (FCC)}{Reziprok: \fRef{::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{\fRef{:::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{\fRef{:::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}{}{}
		\fsplit{
			\centering
			\includegraphics[width=0.9\textwidth]{img/cm_crystal_zincblende.png}
		}{
			\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}{}
		\fsplit{
			\centering
			\includegraphics[width=0.9\textwidth]{img/cm_crystal_wurtzite.png}
		}{

		}
	\end{formula}
