\Section[ \eng{Crystals} \ger{Kristalle} ]{crystal} \Subsection[ \eng{Bravais lattice} \ger{Bravais-Gitter} ]{bravais} \eng[table2D]{In 2D, there are 5 different Bravais lattices} \ger[table2D]{In 2D gibt es 5 verschiedene Bravais-Gitter} \eng[table3D]{In 3D, there are 14 different Bravais lattices} \ger[table3D]{In 3D gibt es 14 verschiedene Bravais-Gitter} \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{table}[H] \centering \expandafter\caption\expandafter{\gt{table2D}} \label{tab:bravais2} \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{table} \begin{table}[H] \centering \caption{\gt{table3D}} \label{tab:bravais3} % \newcolumntype{g}{>{\columncolor[]{0.8}}} \begin{adjustbox}{width=\textwidth} % \begin{tabularx}{\textwidth}{|c|} % asdfasdfadslfasdfaasdofiuapsdoifuapodisufpaoidsufpaoidsufpaoisdfaoisdfpaosidfupaoidsufpaoidsufpaoidsufpaoisdufpaoidsufpoaiudsfpioaspdoifuaposidufpaoisudpfoiaupsdoifupasodf \\ % asdfasdfadslfasdfaasdofiuapsdoifuapodisufpaoidsufpaoidsufpaoisdfaoisdfpaosidfupaoidsufpaoidsufpaoidsufpaoisdufpaoidsufpoaiudsfpioaspdoifuaposidufpaoisudpfoiaupsdoifupasodf \\ % \end{tabularx} % \begin{tabular}{|c|} % asdfasdfadslfasdfaasdofiuapsdoifuapodisufpaoidsufpaoidsufpaoisdfaoisdfpaosidfupaoidsufpaoidsufpaoidsufpaoisdufpaoidsufpoaiudsfpioaspdoifuaposidufpaoisudpfoiaupsdoifupasodf \\ % asdfasdfadslfasdfaasdofiuapsdoifuapodisufpaoidsufpaoidsufpaoisdfaoisdfpaosidfupaoidsufpaoidsufpaoidsufpaoisdufpaoidsufpoaiudsfpioaspdoifuaposidufpaoisudpfoiaupsdoifupasodf \\ % \end{tabular} % \\ \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{table} \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}