From 2de3a8ffcfb661abfde05cf3611ad08d5a1160fc Mon Sep 17 00:00:00 2001
From: Nao Pross <np@0hm.ch>
Date: Tue, 4 May 2021 20:08:33 +0200
Subject: Create slides for applications

---
 vorlesungen/punktgruppen/slides.tex | 202 ++++++++++++++++++++++++++++--------
 1 file changed, 160 insertions(+), 42 deletions(-)

(limited to 'vorlesungen/punktgruppen/slides.tex')

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}
-- 
cgit v1.2.1

