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| -rw-r--r-- | vorlesungen/punktgruppen/slides.pdf | bin | 25800 -> 32512 bytes | |||
| -rw-r--r-- | vorlesungen/punktgruppen/slides.tex | 202 |
2 files changed, 160 insertions, 42 deletions
diff --git a/vorlesungen/punktgruppen/slides.pdf b/vorlesungen/punktgruppen/slides.pdf Binary files differindex 66c44e8..d732296 100644 --- a/vorlesungen/punktgruppen/slides.pdf +++ b/vorlesungen/punktgruppen/slides.pdf 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} |