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-rw-r--r--vorlesungen/punktgruppen/slides.pdfbin25800 -> 32512 bytes
-rw-r--r--vorlesungen/punktgruppen/slides.tex202
2 files changed, 160 insertions, 42 deletions
diff --git a/vorlesungen/punktgruppen/slides.pdf b/vorlesungen/punktgruppen/slides.pdf
index 66c44e8..d732296 100644
--- a/vorlesungen/punktgruppen/slides.pdf
+++ b/vorlesungen/punktgruppen/slides.pdf
Binary files differ
diff --git a/vorlesungen/punktgruppen/slides.tex b/vorlesungen/punktgruppen/slides.tex
index 1c1b1d8..380dcec 100644
--- a/vorlesungen/punktgruppen/slides.tex
+++ b/vorlesungen/punktgruppen/slides.tex
@@ -4,6 +4,10 @@
\usepackage{polyglossia}
\setmainlanguage{german}
+% pretty drawings
+\usepackage{tikz}
+\usetikzlibrary{positioning}
+
% Theme
\beamertemplatenavigationsymbolsempty
@@ -44,6 +48,9 @@
}
\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}
@@ -66,59 +73,170 @@
%% Made in video
\section{Algebraische Symmetrien}
+%% Made in video
+
+\section{Kristalle}
+
+\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 Punktgruppe}; \\
+ \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 (6,0) {Mit und Ohne\\ Symmetriezentrum};
+ \node[below = of title] {Polarisation Feld \(\vec{E}_p\)};
+
+ \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}
+
+ \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}
+
+ \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);
+ \draw[gray, dashed] (C2) to (C4) to (C6) to (C2);
+ \end{scope}
+
+ \begin{scope}[xshift=6cm, 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[blue!50, right = of E] {\(-\)};
+ \node[red!50, left = of E] {\(+\)};
+ \end{scope}
+ \end{scope}
+
+ \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);
+
+ \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}
+ \end{scope}
+ \end{tikzpicture}
+\end{frame}
+
\frame{
- \begin{columns}
- \begin{column}{.3\textwidth}
- Produkt mit \(i\)
+ \begin{columns}[T]
+ \begin{column}{.5\textwidth}
+ Symmetriegruppe und Darstellung
\begin{align*}
- 1 \cdot i &= i \\
- i \cdot i &= -1 \\
- -1 \cdot i &= -i \\
- -i \cdot i &= 1
+ G &= \left\{\mathbb{1}, r, \sigma, \dots \right\} \\
+ &\Phi : G \to O(n)
\end{align*}
- \pause
- %
- Gruppe
\begin{align*}
- G &= \left\{
- 1, i, -1, -i
- \right\} \\
- &= \left\{
- 1, i, i^2, i^3
- \right\} \\
- Z_4 &= \left\{
- \mathbb{1}, r, r^2, r^3
- \right\}
+ U_\lambda &= \left\{ v : \Phi v = \lambda v \right\} \\
+ &= \mathrm{null}\left(\Phi - \lambda I\right)
\end{align*}
- \pause
- %
+ Helmholtz Wellengleichung
+ \[
+ \nabla^2 \vec{E} = \ten{\varepsilon}\mu
+ \frac{\partial^2}{\partial t^2} \vec{E}
+ \]
\end{column}
\begin{column}{.5\textwidth}
- %
- Darstellung
+ Ebene Welle
\[
- \phi : Z_4 \to G
+ \vec{E} = \vec{E}_0 \exp\left[i
+ \left(\vec{k}\cdot\vec{r} - \omega t \right)\right]
+ \]
+ Anisotropisch Dielektrikum
+ \[
+ \ten{R}\ten{\varepsilon}\vec{E} = \frac{\omega^2}{\mu k^2} \vec{E}
+ \]
+ \[
+ \vec{E} \in U_\lambda \implies (\ten{R}\ten{\varepsilon}) \vec{E} = \lambda \vec{E}
+ \]
+ \"Ahenlich auch in der Mechanik
+ \[
+ \vec{F} = \kappa \vec{x} \quad \text{(Hooke)}
\]
- \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 \(\implies Z_4 \cong G\)
\end{column}
\end{columns}
}
-\section{Kristalle}
-
-\section{Anwendungen}
-
\end{document}