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authorLukaszogg <82384106+Lukaszogg@users.noreply.github.com>2021-07-08 20:10:11 +0200
committerLukaszogg <82384106+Lukaszogg@users.noreply.github.com>2021-07-08 20:10:11 +0200
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+\documentclass[12pt, xcolor, aspectratio=169, handout]{beamer}
+
+% language
+\usepackage{polyglossia}
+\setmainlanguage{german}
+
+% pretty drawings
+\usepackage{tikz}
+\usepackage{tikz-3dplot}
+
+\usetikzlibrary{positioning}
+\usetikzlibrary{arrows.meta}
+\usetikzlibrary{shapes.misc}
+\usetikzlibrary{calc}
+
+\usetikzlibrary{external}
+\tikzexternalize[
+ mode = graphics if exists,
+ figure list = true,
+ prefix=build/
+]
+
+% Theme
+\beamertemplatenavigationsymbolsempty
+
+% set look
+\usetheme{default}
+\usecolortheme{fly}
+\usefonttheme{serif}
+
+%% Set font
+\usepackage[p,osf]{scholax}
+\usepackage{amsmath}
+\usepackage[scaled=1.075,ncf,vvarbb]{newtxmath}
+
+% set colors
+\definecolor{background}{HTML}{202020}
+
+\setbeamercolor{normal text}{fg=white, bg=background}
+\setbeamercolor{structure}{fg=white}
+
+\setbeamercolor{item projected}{use=item,fg=background,bg=item.fg!35}
+
+\setbeamercolor*{palette primary}{use=structure,fg=white,bg=structure.fg}
+\setbeamercolor*{palette secondary}{use=structure,fg=white,bg=structure.fg!75}
+\setbeamercolor*{palette tertiary}{use=structure,fg=white,bg=structure.fg!50}
+\setbeamercolor*{palette quaternary}{fg=white,bg=background}
+
+\setbeamercolor*{block title}{parent=structure}
+\setbeamercolor*{block body}{fg=background, bg=}
+
+\setbeamercolor*{framesubtitle}{fg=white}
+
+\setbeamertemplate{section page}
+{
+ \begin{center}
+ \Huge
+ \insertsection
+ \end{center}
+}
+\AtBeginSection{\frame{\sectionpage}}
+
+% Macros
+\newcommand{\ten}[1]{#1}
+
+% Metadata
+\title{\LARGE \scshape Punktgruppen und Kristalle}
+\author[N. Pross, T. T\"onz]{Naoki Pross, Tim T\"onz}
+\institute{Hochschule f\"ur Technik OST, Rapperswil}
+\date{10. Mai 2021}
+
+% Slides
+\begin{document}
+\frame{
+ \titlepage
+ \vfill
+ \begin{center}
+ \small \color{gray}
+ Slides: \texttt{s.0hm.ch/ctBsD}
+ \end{center}
+}
+\frame{\tableofcontents}
+
+\frame{
+ \begin{itemize}
+ \item Was heisst \emph{Symmetrie} in der Mathematik? \pause
+ \item Wie kann ein Kristall modelliert werden? \pause
+ \item Aus der Physik: Licht, Piezoelektrizit\"at \pause
+ \end{itemize}
+ \begin{center}
+ \begin{tikzpicture}
+ \begin{scope}[
+ node distance = 0cm
+ ]
+ \node[
+ rectangle, fill = gray!40!background,
+ minimum width = 3cm, minimum height = 2cm,
+ ] (body) {\(\vec{E}_p = \vec{0}\)};
+
+ \node[
+ draw, rectangle, thick, white, fill = red!50,
+ minimum width = 3cm, minimum height = 1mm,
+ above = of body
+ ] (pos) {};
+
+ \node[
+ draw, rectangle, thick, white, fill = blue!50,
+ minimum width = 3cm, minimum height = 1mm,
+ below = of body
+ ] (neg) {};
+
+ \draw[white, very thick, -Circle] (pos.east) to ++ (1,0) node (p) {};
+ \draw[white, very thick, -Circle] (neg.east) to ++ (1,0) node (n) {};
+
+ \draw[white, thick, ->] (p) to[out = -70, in = 70] node[midway, right] {\(U = 0\)} (n);
+ \end{scope}
+ \begin{scope}[
+ node distance = 0cm,
+ xshift = 7cm
+ ]
+ \node[
+ rectangle, fill = gray!40!background,
+ minimum width = 3cm, minimum height = 1.5cm,
+ ] (body) {\(\vec{E}_p = \vec{0}\)};
+
+ \node[
+ draw, rectangle, thick, white, fill = red!50,
+ minimum width = 3cm, minimum height = 1mm,
+ above = of body
+ ] (pos) {};
+
+ \node[
+ draw, rectangle, thick, white, fill = blue!50,
+ minimum width = 3cm, minimum height = 1mm,
+ below = of body
+ ] (neg) {};
+
+ \draw[orange, very thick, <-] (pos.north) to node[near end, right] {\(\vec{F}\)} ++(0,1);
+ \draw[orange, very thick, <-] (neg.south) to node[near end, right] {\(\vec{F}\)} ++(0,-1);
+
+ \draw[white, very thick, -Circle] (pos.east) to ++ (1,0) node (p) {};
+ \draw[white, very thick, -Circle] (neg.east) to ++ (1,0) node (n) {};
+
+ \draw[white, thick, ->] (p) to[out = -70, in = 70] node[midway, right] {\(U > 0\)} (n);
+ \end{scope}
+ \end{tikzpicture}
+ \end{center}
+}
+
+\section{2D Symmetrien}
+%% Made in video
+{
+ \usebackgroundtemplate{
+ \includegraphics[height=\paperheight]{media/images/nosignal}}
+ \frame{}
+}
+
+\section{Algebraische Symmetrien}
+%% Made in video
+\frame{
+ \begin{columns}[T]
+ \begin{column}{.5\textwidth}
+ Produkt mit \(i\)
+ \begin{align*}
+ 1 \cdot i &= i \\
+ i \cdot i &= -1 \\
+ -1 \cdot i &= -i \\
+ -i \cdot i &= 1
+ \end{align*}
+ \pause
+ %
+ Gruppe
+ \begin{align*}
+ G &= \left\{
+ 1, i, -1, -i
+ \right\} \\
+ &= \left\{
+ 1, i, i^2, i^3
+ \right\} \\
+ C_4 &= \left\{
+ \mathbb{1}, r, r^2, r^3
+ \right\}
+ \end{align*}
+ \pause
+ \end{column}
+ \begin{column}{.5\textwidth}
+ Darstellung \(\phi : C_4 \to G\)
+ \begin{align*}
+ \phi(\mathbb{1}) &= 1 & \phi(r^2) &= i^2 \\
+ \phi(r) &= i & \phi(r^3) &= i^3
+ \end{align*}
+ \pause
+ %
+ Homomorphismus
+ \begin{align*}
+ \phi(r \circ \mathbb{1}) &= \phi(r) \cdot \phi(\mathbb{1}) \\
+ &= i \cdot 1
+ \end{align*}
+ \pause
+ %
+ \(\phi\) ist bijektiv \(\implies C_4 \cong G\)
+ \pause
+ %
+ \begin{align*}
+ \psi : C_4 &\to (\mathbb{Z}/4\mathbb{Z}, +) \\
+ \psi(\mathbb{1}\circ r^2) &= 0 + 2 \pmod{4}
+ \end{align*}
+ \end{column}
+ \end{columns}
+}
+
+\section{3D Symmetrien}
+%% Made in video
+{
+ \usebackgroundtemplate{
+ \includegraphics[height=\paperheight]{media/images/nosignal}}
+ \frame{}
+}
+
+\section{Matrizen}
+\frame{
+ \begin{columns}[T]
+ \begin{column}{.5\textwidth}
+ Symmetriegruppe
+ \[
+ G = \left\{\mathbb{1}, r, \sigma, \dots \right\}
+ \]
+ \pause
+ Matrixdarstellung
+ \begin{align*}
+ \Phi : G &\to O(3) \\
+ g &\mapsto \Phi_g
+ \end{align*}
+ \pause
+ Orthogonale Gruppe
+ \[
+ O(n) = \left\{ Q : QQ^t = Q^tQ = I \right\}
+ \]
+ \end{column}
+ \pause
+ \begin{column}{.5\textwidth}
+ \begin{align*}
+ \Phi_\mathbb{1} &= \begin{pmatrix}
+ 1 & 0 & 0 \\
+ 0 & 1 & 0 \\
+ 0 & 0 & 1
+ \end{pmatrix} = I \\[1em]
+ \Phi_\sigma &= \begin{pmatrix}
+ 1 & 0 & 0 \\
+ 0 & -1 & 0 \\
+ 0 & 0 & 1
+ \end{pmatrix} \\[1em]
+ \Phi_r &= \begin{pmatrix}
+ \cos \alpha & -\sin \alpha & 0 \\
+ \sin \alpha & \cos \alpha & 0 \\
+ 0 & 0 & 1
+ \end{pmatrix}
+ \end{align*}
+ \end{column}
+ \end{columns}
+}
+
+\section{Kristalle}
+\begin{frame}[fragile]{}
+ \begin{columns}
+ \onslide<1->{
+ \begin{column}{.5\textwidth}
+ \begin{center}
+ \begin{tikzpicture}[
+ dot/.style = {
+ draw, circle, thick, white, fill = gray!40!background,
+ minimum size = 2mm,
+ inner sep = 0pt,
+ outer sep = 1mm,
+ },
+ ]
+
+ \begin{scope}
+ \clip (-2,-2) rectangle (3,4);
+ \foreach \y in {-7,-6,...,7} {
+ \foreach \x in {-7,-6,...,7} {
+ \node[dot, xshift=3mm*\y] (N\x\y) at (\x, \y) {};
+ }
+ }
+ \end{scope}
+ \draw[white, thick] (-2, -2) rectangle (3,4);
+
+ \draw[red!80!background, thick, ->]
+ (N00) to node[midway, below] {\(\vec{a}_1\)} (N10);
+ \draw[cyan!80!background, thick, ->]
+ (N00) to node[midway, left] {\(\vec{a}_2\)} (N01);
+ \end{tikzpicture}
+ \end{center}
+ \end{column}
+ }
+ \begin{column}{.5\textwidth}
+ \onslide<2->{
+ Kristallgitter:
+ \(n_i \in \mathbb{Z}\),
+ }
+ \onslide<3->{
+ \(\vec{a}_i \in \mathbb{R}^3\)
+ }
+ \onslide<2->{
+ \[
+ \vec{r} = n_1 \vec{a}_1 + n_2 \vec{a}_2 \onslide<3->{+ n_3 \vec{a}_3}
+ \]
+ }
+ \vspace{1cm}
+
+ \onslide<4->{
+ Invariant unter Translation
+ \[
+ Q_i(\vec{r}) = \vec{r} + \vec{a}_i
+ \]
+ }
+ \end{column}
+ \end{columns}
+\end{frame}
+
+\begin{frame}[fragile]{}
+ \begin{columns}[T]
+ \begin{column}{.5\textwidth}
+ \onslide<1->{
+ Wie kombiniert sich \(Q_i\) mit der anderen Symmetrien?
+ }
+ \begin{center}
+ \begin{tikzpicture}[
+ dot/.style = {
+ draw, circle, thick, white, fill = gray!40!background,
+ minimum size = 2mm,
+ inner sep = 0pt,
+ outer sep = 1mm,
+ },
+ ]
+
+ \onslide<2->{
+ \node[dot] (A1) at (0,0) {};
+ \node[below left] at (A1) {\(A\)};
+ }
+
+ \onslide<3->{
+ \node[dot] (A2) at (2.5,0) {};
+ \node[below right] at (A2) {\(A'\)};
+
+ \draw[red!80!background, thick, ->]
+ (A1) to node[midway, below] {\(\vec{Q}\)} (A2);
+ }
+
+ \onslide<4->{
+ \node[dot] (B1) at (120:2.5) {};
+ \node[above left] at (B1) {\(B\)};
+
+ \draw[green!70!background, thick, ->]
+ (A1) ++(.5,0) arc (0:120:.5)
+ node[midway, above, xshift=1mm] {\(C_n\)};
+ \draw[red!80!background, dashed, thick, ->] (A1) to (B1);
+ }
+
+ \onslide<5->{
+ \node[dot] (B2) at ($(A2)+(60:2.5)$) {};
+ \node[above right] at (B2) {\(B'\)};
+
+ \draw[green!70!background, thick, dashed, ->]
+ (A2) ++(-.5,0) arc (180:60:.5);
+ \draw[red!80!background, dashed, thick, ->] (A2) to (B2);
+ }
+
+ \onslide<6->{
+ \draw[yellow!80!background, thick, ->]
+ (B1) to node[above, midway] {\(\vec{Q}'\)} (B2);
+ }
+
+ \onslide<10->{
+ \draw[gray, dashed, thick] (A1) to (A1 |- B1) node (Xl) {};
+ \draw[gray, dashed, thick] (A2) to (A2 |- B2) node (Xr) {};
+ \node[above left, xshift=-2mm] at (Xl) {\(x\)};
+ \node[above right, xshift= 2mm] at (Xr) {\(x\)};
+ }
+ \end{tikzpicture}
+ \end{center}
+ \end{column}
+ \begin{column}{.5\textwidth}
+ \onslide<7->{
+ Sei \(q = |\vec{Q}|\), \(\alpha = 2\pi/n\) und \(n \in \mathbb{N}\)
+ }
+ \begin{align*}
+ \onslide<9->{q' = n q \onslide<10->{&= q + 2x \\}}
+ \onslide<11->{nq &= q + 2q\sin(\alpha - \pi/2) \\}
+ \onslide<12->{n &= 1 - 2\cos\alpha}
+ \end{align*}
+ \onslide<13->{
+ Somit muss
+ \begin{align*}
+ \alpha &= \cos^{-1}\left(\frac{1-n}{2}\right) \\[1em]
+ \alpha &\in \left\{ 0, 60^\circ, 90^\circ, 120^\circ, 180^\circ \right\} \\
+ n &\in \left\{ 1, 2, 3, 4, 6 \right\}
+ \end{align*}
+ }
+ \end{column}
+ \end{columns}
+\end{frame}
+
+\begin{frame}[fragile]{M\"ogliche Kristallstrukturen}
+ \begin{center}
+ \begin{tikzpicture}[]
+ \node[circle, dashed, draw = gray,
+ thick, fill = background,
+ minimum size = 4cm] {};
+ \node[gray] at (.9,-1.2) {674};
+
+ \node[circle, draw = white, thick,
+ fill = orange!40!background,
+ xshift = -3mm, yshift = 2mm,
+ minimum size = 2.75cm,
+ outer sep = 1mm] (A) {};
+ \node[white, yshift = 2mm] at (A) {230};
+ \node[white, font=\large, above right = of A] (Al) {Raumgruppe};
+ \draw[white, thick, ->] (Al.west) to[out=180, in=60] (A);
+
+ \node[circle, draw = white, thick,
+ fill = red!20!background,
+ xshift = -5mm, yshift = -5mm,
+ minimum size = 1cm,
+ outer sep = 1mm] (B) {32};
+ \node[white, font=\large, below left = of B, xshift=-4mm] (Bl) {Kristallklassen};
+ \draw[white, thick, ->] (Bl.east) to[out = 0, in = 180] (B);
+ \end{tikzpicture}
+ \end{center}
+\end{frame}
+
+{
+ \usebackgroundtemplate[fragile]{
+ \begin{tikzpicture}[
+ overlay,
+ xshift = .45\paperwidth,
+ yshift = .47\paperheight,
+ classcirc/.style = {
+ draw = gray, thick, circle,
+ minimum size = 12mm,
+ inner sep = 0pt, outer sep = 0pt,
+ },
+ classlabel/.style = {
+ below right = 5mm
+ },
+ round/.style = {
+ draw = yellow, thick, circle,
+ minimum size = 1mm,
+ inner sep = 0pt, outer sep = 0pt,
+ },
+ cross/.style = {
+ cross out, draw = magenta, thick,
+ minimum size = 1mm,
+ inner sep = 0pt, outer sep = 0pt
+ },
+ ]
+ \matrix [row sep = 3mm, column sep = 0mm] {
+ \node[classcirc] (C1) {} node[classlabel] {\(C_{1}\)}; &
+ \node[classcirc] (C2) {} node[classlabel] {\(C_{2}\)}; &
+ \node[classcirc] (C3) {} node[classlabel] {\(C_{3}\)}; &
+ \node[classcirc] (Ci) {} node[classlabel] {\(C_{i}\)}; &
+
+ \node[classcirc] (Cs) {} node[classlabel] {\(C_{s}\)}; &
+ \node[classcirc] (C3i) {} node[classlabel] {\(C_{3i}\)}; &
+ \node[classcirc] (C2h) {} node[classlabel] {\(C_{2h}\)}; &
+ \node[classcirc] (D2) {} node[classlabel] {\(D_{2}\)}; \\
+
+ \node[classcirc] (D3d) {} node[classlabel] {\(D_{3d}\)}; &
+ \node[classcirc] (C2v) {} node[classlabel] {\(C_{2v}\)}; &
+ \node[classcirc] (D2h) {} node[classlabel] {\(D_{2h}\)}; &
+ \node[classcirc] (D3) {} node[classlabel] {\(D_{3}\)}; &
+
+ \node[classcirc] (C4) {} node[classlabel] {\(C_{4}\)}; &
+ \node[classcirc] (C6) {} node[classlabel] {\(C_{6}\)}; &
+ \node[classcirc] (D3dP) {} node[classlabel] {\(D_{3d}\)}; &
+ \node[classcirc] (S4) {} node[classlabel] {\(S_{4}\)}; \\
+
+ \node[classcirc] (S3) {} node[classlabel] {\(S_{3}\)}; &
+ \node[classcirc, dashed] (T) {} node[classlabel] {\(T_{}\)}; &
+ \node[classcirc] (C4h) {} node[classlabel] {\(C_{4h}\)}; &
+ \node[classcirc] (C6h) {} node[classlabel] {\(C_{6h}\)}; &
+
+ \node[classcirc, dashed] (Th) {} node[classlabel] {\(T_{h}\)}; &
+ \node[classcirc] (C4v) {} node[classlabel] {\(C_{4v}\)}; &
+ \node[classcirc] (C6v) {} node[classlabel] {\(C_{6v}\)}; &
+ \node[classcirc, dashed] (Td) {} node[classlabel] {\(T_{d}\)}; \\
+
+ \node[classcirc] (D2d) {} node[classlabel] {\(D_{2d}\)}; &
+ \node[classcirc] (D3h) {} node[classlabel] {\(D_{3h}\)}; &
+ \node[classcirc, dashed] (O) {} node[classlabel] {\(O_{}\)}; &
+ \node[classcirc] (D4) {} node[classlabel] {\(D_{4}\)}; &
+
+ \node[classcirc] (D6) {} node[classlabel] {\(D_{6}\)}; &
+ \node[classcirc, dashed] (Oh) {} node[classlabel] {\(O_{h}\)}; &
+ \node[classcirc] (D4h) {} node[classlabel] {\(D_{4h}\)}; &
+ \node[classcirc] (D6h) {} node[classlabel] {\(D_{6h}\)}; \\
+ };
+
+
+ \node[cross] at ($(C1)+(4mm,0)$) {};
+
+
+ \node[cross] at ($(C2)+(4mm,0)$) {};
+ \node[cross] at ($(C2)-(4mm,0)$) {};
+
+
+ \node[cross] at ($(C3)+( 0:4mm)$) {};
+ \node[cross] at ($(C3)+(120:4mm)$) {};
+ \node[cross] at ($(C3)+(240:4mm)$) {};
+
+
+ \node[cross] at ($(Ci)+(4mm,0)$) {};
+ \node[round] at ($(Ci)-(4mm,0)$) {};
+
+
+ \node[cross] at ($(Cs)+(4mm,0)$) {};
+ \node[round] at ($(Cs)+(4mm,0)$) {};
+
+
+ \node[cross] at ($(C3i)+( 0:4mm)$) {};
+ \node[cross] at ($(C3i)+(120:4mm)$) {};
+ \node[cross] at ($(C3i)+(240:4mm)$) {};
+ \node[round] at ($(C3i)+( 60:4mm)$) {};
+ \node[round] at ($(C3i)+(180:4mm)$) {};
+ \node[round] at ($(C3i)+(300:4mm)$) {};
+
+
+ \node[cross] at ($(C2h)+(4mm,0)$) {};
+ \node[cross] at ($(C2h)-(4mm,0)$) {};
+ \node[round] at ($(C2h)+(4mm,0)$) {};
+ \node[round] at ($(C2h)-(4mm,0)$) {};
+
+
+ \node[cross] at ($(D2)+( 20:4mm)$) {};
+ \node[cross] at ($(D2)+(200:4mm)$) {};
+ \node[round] at ($(D2)+(160:4mm)$) {};
+ \node[round] at ($(D2)+(340:4mm)$) {};
+
+
+ \foreach \x in {0, 120, 240} {
+ \node[cross] at ($(D3d)+({\x+15}:4mm)$) {};
+ \node[cross] at ($(D3d)+({\x-15}:4mm)$) {};
+ }
+
+
+ \foreach \x in {0, 180} {
+ \node[cross] at ($(C2v)+({\x+15}:4mm)$) {};
+ \node[cross] at ($(C2v)+({\x-15}:4mm)$) {};
+ }
+
+
+ \foreach \x in {0, 180} {
+ \node[cross] at ($(D2h)+({\x+15}:4mm)$) {};
+ \node[cross] at ($(D2h)+({\x-15}:4mm)$) {};
+ \node[round] at ($(D2h)+({\x+15}:4mm)$) {};
+ \node[round] at ($(D2h)+({\x-15}:4mm)$) {};
+ }
+
+
+ \foreach \x in {0, 120, 240} {
+ \node[cross] at ($(D3)+({\x+15}:4mm)$) {};
+ \node[round] at ($(D3)+({\x-15}:4mm)$) {};
+ }
+
+
+ \foreach \x in {0, 90, 180, 270} {
+ \node[cross] at ($(C4)+(\x:4mm)$) {};
+ }
+
+
+ \foreach \x in {0, 60, 120, 180, 240, 300} {
+ \node[cross] at ($(C6)+(\x:4mm)$) {};
+ }
+
+
+ \foreach \x in {0, 120, 240} {
+ \node[cross] at ($(D3dP)+({\x+15}:4mm)$) {};
+ \node[cross] at ($(D3dP)+({\x-15}:4mm)$) {};
+ \node[round] at ($(D3dP)+({\x+15+60}:4mm)$) {};
+ \node[round] at ($(D3dP)+({\x-15+60}:4mm)$) {};
+ }
+
+
+ \node[cross] at ($(S4)+(4mm,0)$) {};
+ \node[cross] at ($(S4)-(4mm,0)$) {};
+ \node[round] at ($(S4)+(0,4mm)$) {};
+ \node[round] at ($(S4)-(0,4mm)$) {};
+
+
+ \foreach \x in {0, 120, 240} {
+ \node[cross] at ($(S3)+(\x:4mm)$) {};
+ \node[round] at ($(S3)+(\x:4mm)$) {};
+ }
+
+
+ %% TODO: T
+
+
+ \foreach \x in {0, 90, 180, 270} {
+ \node[cross] at ($(C4h)+(\x:4mm)$) {};
+ \node[round] at ($(C4h)+(\x:4mm)$) {};
+ }
+
+
+ \foreach \x in {0, 60, 120, 180, 240, 300} {
+ \node[cross] at ($(C6h)+(\x:4mm)$) {};
+ \node[round] at ($(C6h)+(\x:4mm)$) {};
+ }
+
+
+ %% TODO: Th
+
+
+ \foreach \x in {0, 90, 180, 270} {
+ \node[cross] at ($(C4v)+(\x+15:4mm)$) {};
+ \node[cross] at ($(C4v)+(\x-15:4mm)$) {};
+ }
+
+
+
+ \foreach \x in {0, 60, 120, 180, 240, 300} {
+ \node[cross] at ($(C6v)+(\x+10:4mm)$) {};
+ \node[cross] at ($(C6v)+(\x-10:4mm)$) {};
+ }
+
+
+ %% TODO: Td
+
+
+ \foreach \x in {0, 180} {
+ \node[cross] at ($(D2d)+({\x+15}:4mm)$) {};
+ \node[round] at ($(D2d)+({\x-15}:4mm)$) {};
+
+ \node[round] at ($(D2d)+({\x+15+90}:4mm)$) {};
+ \node[cross] at ($(D2d)+({\x-15+90}:4mm)$) {};
+ }
+
+
+ \foreach \x in {0, 120, 240} {
+ \node[cross] at ($(D3h)+({\x+15}:4mm)$) {};
+ \node[cross] at ($(D3h)+({\x-15}:4mm)$) {};
+ \node[round] at ($(D3h)+({\x+15}:4mm)$) {};
+ \node[round] at ($(D3h)+({\x-15}:4mm)$) {};
+ }
+
+
+ %% TODO: O
+
+
+ \foreach \x in {0, 90, 180, 270} {
+ \node[cross] at ($(D4)+({\x+15}:4mm)$) {};
+ \node[round] at ($(D4)+({\x-15}:4mm)$) {};
+ }
+
+ \foreach \x in {0, 60, 120, 180, 240, 300} {
+ \node[cross] at ($(D6)+({\x+10}:4mm)$) {};
+ \node[round] at ($(D6)+({\x-10}:4mm)$) {};
+ }
+
+
+ % TODO Oh
+
+
+ \foreach \x in {0, 90, 180, 270} {
+ \node[cross] at ($(D4h)+(\x+15:4mm)$) {};
+ \node[cross] at ($(D4h)+(\x-15:4mm)$) {};
+ \node[round] at ($(D4h)+(\x+15:4mm)$) {};
+ \node[round] at ($(D4h)+(\x-15:4mm)$) {};
+ }
+
+
+ \foreach \x in {0, 60, 120, 180, 240, 300} {
+ \node[cross] at ($(D6h)+({\x+10}:4mm)$) {};
+ \node[cross] at ($(D6h)+({\x-10}:4mm)$) {};
+ \node[round] at ($(D6h)+({\x+10}:4mm)$) {};
+ \node[round] at ($(D6h)+({\x-10}:4mm)$) {};
+ }
+ \end{tikzpicture}
+ }
+ \begin{frame}[fragile]{}
+ \end{frame}
+}
+
+\section{Anwendungen}
+\begin{frame}[fragile]{}
+ \centering
+ \begin{tikzpicture}[
+ box/.style = {
+ rectangle, thick, draw = white, fill = darkgray!50!background,
+ minimum height = 1cm, outer sep = 2mm,
+ },
+ ]
+
+ \matrix [nodes = {box, align = center}, column sep = 1cm, row sep = 1.5cm] {
+ & \node (A) {32 Kristallklassen}; \\
+ \node (B) {11 Mit\\ Inversionszentrum}; & \node (C) {21 Ohne\\ Inversionszentrum}; \\
+ & \node[fill=red!20!background] (D) {20 Piezoelektrisch}; & \node (E) {1 Nicht\\ piezoelektrisch}; \\
+ };
+
+ \draw[thick, ->] (A.west) to[out=180, in=90] (B.north);
+ \draw[thick, ->] (A.south) to (C);
+ \draw[thick, ->] (C.south) to (D.north);
+ \draw[thick, ->] (C.east) to[out=0, in=90] (E.north);
+ \end{tikzpicture}
+\end{frame}
+
+\begin{frame}[fragile]{}
+ \begin{tikzpicture}[
+ overlay, xshift = 1.5cm, yshift = 1.5cm,
+ node distance = 2mm,
+ charge/.style = {
+ circle, draw = white, thick,
+ minimum size = 5mm
+ },
+ positive/.style = { fill = red!50 },
+ negative/.style = { fill = blue!50 },
+ ]
+
+ \node[font = {\large\bfseries}, align = center] (title) at (5.5,0) {Mit und Ohne\\ Symmetriezentrum};
+ \pause
+
+ \begin{scope}
+ \matrix[nodes = { charge }, row sep = 8mm, column sep = 8mm] {
+ \node[positive] {}; & \node[negative] (N) {}; & \node [positive] {}; \\
+ \node[negative] (W) {}; & \node[positive] {}; & \node [negative] (E) {}; \\
+ \node[positive] {}; & \node[negative] (S) {}; & \node [positive] {}; \\
+ };
+ \draw[gray, dashed] (W) to (N) to (E) to (S) to (W);
+ \end{scope}
+ \pause
+
+ \begin{scope}[xshift=11cm]
+ \foreach \x/\t [count=\i] in {60/positive, 120/negative, 180/positive, 240/negative, 300/positive, 360/negative} {
+ \node[charge, \t] (C\i) at (\x:1.5cm) {};
+ }
+
+ \draw[white] (C1) to (C2) to (C3) to (C4) to (C5) to (C6) to (C1);
+ \node[circle, draw=gray, fill=gray, outer sep = 0, inner sep = 0, minimum size = 3mm] {};
+ % \draw[gray, dashed] (C2) to (C4) to (C6) to (C2);
+ \end{scope}
+ \pause
+
+ %%
+ \node[below = of title] {Polarisation Feld \(\vec{E}_p\)};
+
+ %% hex with vertical pressure
+ \begin{scope}[xshift=11cm, yshift=-4.5cm]
+ \node[charge, positive, yshift=-2.5mm] (C1) at ( 60:1.5cm) {};
+ \node[charge, negative, yshift=-2.5mm] (C2) at (120:1.5cm) {};
+ \node[charge, positive, xshift=-2.5mm] (C3) at (180:1.5cm) {};
+ \node[charge, negative, yshift= 2.5mm] (C4) at (240:1.5cm) {};
+ \node[charge, positive, yshift= 2.5mm] (C5) at (300:1.5cm) {};
+ \node[charge, negative, xshift= 2.5mm] (C6) at (360:1.5cm) {};
+
+ \draw[white] (C1) to (C2) to (C3) to (C4) to (C5) to (C6) to (C1);
+ % \draw[gray, dashed] (C2) to (C4) to (C6) to (C2);
+
+ \foreach \d in {C1, C2} {
+ \draw[orange, very thick, <-] (\d) to ++(0,.7);
+ }
+
+ \foreach \d in {C4, C5} {
+ \draw[orange, very thick, <-] (\d) to ++(0,-.7);
+ }
+
+ \node[white] (E) {\(\vec{E}_p\)};
+ \begin{scope}[node distance = .5mm]
+ \node[red!50, right = of E] {\(+\)};
+ \node[blue!50, left = of E] {\(-\)};
+ \end{scope}
+ % \draw[gray, thick, dotted] (E) to ++(0,2);
+ % \draw[gray, thick, dotted] (E) to ++(0,-2);
+ \end{scope}
+ \pause
+
+ %% square with vertical pressure
+ \begin{scope}[yshift=-4.5cm]
+ \matrix[nodes = { charge }, row sep = 5mm, column sep = 1cm] {
+ \node[positive] (NW) {}; & \node[negative] (N) {}; & \node [positive] (NE) {}; \\
+ \node[negative] (W) {}; & \node[positive] {}; & \node [negative] (E) {}; \\
+ \node[positive] (SW) {}; & \node[negative] (S) {}; & \node [positive] (SE) {}; \\
+ };
+
+ \foreach \d in {NW, N, NE} {
+ \draw[orange, very thick, <-] (\d) to ++(0,.7);
+ }
+
+ \foreach \d in {SW, S, SE} {
+ \draw[orange, very thick, <-] (\d) to ++(0,-.7);
+ }
+
+ \draw[gray, dashed] (W) to (N) to (E) to (S) to (W);
+ \end{scope}
+ \pause
+
+ %% hex with horizontal pressure
+ \begin{scope}[xshift=5.5cm, yshift=-4.5cm]
+ \node[charge, positive, yshift= 2.5mm] (C1) at ( 60:1.5cm) {};
+ \node[charge, negative, yshift= 2.5mm] (C2) at (120:1.5cm) {};
+ \node[charge, positive, xshift= 2.5mm] (C3) at (180:1.5cm) {};
+ \node[charge, negative, yshift=-2.5mm] (C4) at (240:1.5cm) {};
+ \node[charge, positive, yshift=-2.5mm] (C5) at (300:1.5cm) {};
+ \node[charge, negative, xshift=-2.5mm] (C6) at (360:1.5cm) {};
+
+ \draw[white] (C1) to (C2) to (C3) to (C4) to (C5) to (C6) to (C1);
+ % \draw[gray, dashed] (C2) to (C4) to (C6) to (C2);
+
+ \draw[orange, very thick, <-] (C6) to ++(.7,0);
+ \draw[orange, very thick, <-] (C3) to ++(-.7,0);
+
+ \node[white] (E) {\(\vec{E}_p\)};
+ \begin{scope}[node distance = .5mm]
+ \node[blue!50, right = of E] {\(-\)};
+ \node[red!50, left = of E] {\(+\)};
+ \end{scope}
+ % \draw[gray, thick, dotted] (E) to ++(0,2);
+ % \draw[gray, thick, dotted] (E) to ++(0,-2);
+ \end{scope}
+ \pause
+
+
+ \end{tikzpicture}
+\end{frame}
+
+\frame{
+ \frametitle{Licht in Kristallen}
+ \begin{columns}[T]
+ \begin{column}{.45\textwidth}
+ \onslide<2->{
+ Helmholtz Wellengleichung
+ \[
+ \nabla^2 \vec{E} = \ten{\varepsilon}\mu
+ \frac{\partial^2}{\partial t^2} \vec{E}
+ \]
+ }
+ \onslide<3->{
+ Ebene Welle
+ \[
+ \vec{E} = \vec{E}_0 \exp\left[i
+ \left(\vec{k}\cdot\vec{r} - \omega t \right)\right]
+ \]
+ }
+ \onslide<4->{
+ Anisotropisch Dielektrikum
+ \[
+ (\ten{K}\ten{\varepsilon})\vec{E}
+ = \frac{k^2}{\mu \omega^2} \vec{E}
+ \implies
+ \Phi \vec{E} = \lambda \vec{E}
+ \]
+ }
+ \end{column}
+ \begin{column}{.55\textwidth}
+ \onslide<5->{
+ Eingenraum
+ \begin{align*}
+ U_\lambda &= \left\{ v : \Phi v = \lambda v \right\}
+ = \mathrm{null}\left(\Phi - \lambda I\right)
+ \end{align*}
+ }\onslide<6->{
+ Symmetriegruppe und Darstellung
+ \begin{align*}
+ G &= \left\{\mathbb{1}, r, \sigma, \dots \right\} \\
+ &\Phi : G \to O(n)
+ \end{align*}
+ }\onslide<7->{
+ Kann man \(U_\lambda\) von \(G\) herauslesen?
+ \only<7>{
+ \[
+ U_\lambda \stackrel{?}{=} f\left(\bigoplus_{g \in G} \Phi_g\right)
+ \]
+ }\only<8>{
+ \begin{align*}
+ \mathrm{Tr}\left[\Phi_r(g)\right]
+ &= \sum_i n_i \mathrm{Tr}\left[\Psi_i(g)\right] \\
+ |G| &= \sum_i\mathrm{Tr}\left[\Psi_i(\mathbb{1})\right]
+ \end{align*}
+ }
+ }
+ \end{column}
+ \end{columns}
+}
+
+% \begin{frame}[fragile]
+% \centering
+% \tdplotsetmaincoords{70}{110}
+% \begin{tikzpicture}[scale=2, tdplot_main_coords]
+% \node[draw=white, thick, minimum size = 3cm, circle] {};
+% % \foreach \x in {0, 120, 240} {
+% % }
+% \end{tikzpicture}
+% \end{frame}
+
+
+\end{document}