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| -rw-r--r-- | vorlesungen/punktgruppen/slides.pdf | bin | 36038 -> 50470 bytes | |||
| -rw-r--r-- | vorlesungen/punktgruppen/slides.tex | 115 |
2 files changed, 112 insertions, 3 deletions
diff --git a/vorlesungen/punktgruppen/slides.pdf b/vorlesungen/punktgruppen/slides.pdf Binary files differindex bd7afc9..0851fd8 100644 --- a/vorlesungen/punktgruppen/slides.pdf +++ b/vorlesungen/punktgruppen/slides.pdf diff --git a/vorlesungen/punktgruppen/slides.tex b/vorlesungen/punktgruppen/slides.tex index 25761c0..e800a87 100644 --- a/vorlesungen/punktgruppen/slides.tex +++ b/vorlesungen/punktgruppen/slides.tex @@ -8,6 +8,7 @@ \usepackage{tikz} \usetikzlibrary{positioning} \usetikzlibrary{arrows.meta} +\usetikzlibrary{calc} % Theme \beamertemplatenavigationsymbolsempty @@ -95,7 +96,7 @@ \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 = -80, in = 80] node[midway, right] {\(U = 0\)} (n); + \draw[white, thick, ->] (p) to[out = -70, in = 70] node[midway, right] {\(U = 0\)} (n); \end{scope} \begin{scope}[ node distance = 0cm, @@ -124,7 +125,7 @@ \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 = -80, in = 80] node[midway, right] {\(U \neq 0\)} (n); + \draw[white, thick, ->] (p) to[out = -70, in = 70] node[midway, right] {\(U \neq 0\)} (n); \end{scope} \end{tikzpicture} \end{center} @@ -142,6 +143,114 @@ \section{Matrizen} \section{Kristalle} +\begin{frame}[fragile]{} + \begin{columns}[T] + \begin{column}{.5\textwidth} + Kristallgitter: + \(n_i \in \mathbb{Z}\), + \(\vec{a}_i \in \mathbb{R}^3\) + \[ + \vec{r} = n_1 \vec{a}_1 + n_2 \vec{a}_2 + n_3 \vec{a}_3 + \] + \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 (-1,-1) rectangle (4,3); + \foreach \y in {-5,-4,...,5} { + \foreach \x in {-5,-4,...,5} { + \node[dot, xshift=3mm*\y] (N\x\y) at (\x, \y) {}; + } + } + \end{scope} + + \draw[white, thick] (-1, -1) rectangle (4,3); + + \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} + Invariant (symmetrisch) unten Translation + \[ + Q_i(\vec{r}) = \vec{r} + \vec{a}_i + \] + \end{column} + \begin{column}{.5\textwidth} + 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, + }, + ] + + \node[dot] (A1) at (0,0) {}; + \node[below left] at (A1) {\(A\)}; + + \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); + + \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); + + + \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); + + \draw[yellow!80!background, thick, ->] (B1) to node[above, midway] {\(\vec{Q}'\)} (B2); + + \draw[gray, dashed, thick] (A1) to (A1 |- B1) node (X) {}; + \draw[gray, dashed, thick] (A2) to (A2 |- B2); + + \node[above left, xshift=-2mm] at (X) {\(x\)}; + \end{tikzpicture} + \end{center} + Sei \(q = |\vec{Q}|\), \(\alpha = 2\pi/n\) und \(n \in \mathbb{N}\) + \begin{align*} + q' = n q &= q + 2x \\ + nq &= q + 2q\sin(\alpha - \pi/2) \\ + n &= 1 - 2\cos\alpha + \end{align*} + \end{column} + \end{columns} +\end{frame} + +\frame{ + \begin{columns}[T] + \begin{column}{.5\textwidth} + Somit muss + \[ + \alpha = \cos^{-1}\left(\frac{m-1}{2}\right) + \] + \begin{gather*} + \alpha \in \left\{ 0, 60^\circ, 90^\circ, 120^\circ, 180^\circ \right\} + \end{gather*} + \end{column} + \begin{column}{.5\textwidth} + \end{column} + \end{columns} +} \section{Anwendungen} \begin{frame}[fragile]{} @@ -154,7 +263,7 @@ ] \matrix [nodes = {box, align = center}, column sep = 1cm, row sep = 1.5cm] { - & \node (A) {32 Punktgruppe}; \\ + & \node (A) {32 Punktgruppen}; \\ \node (B) {11 Mit\\ Inversionszentrum}; & \node (C) {21 Ohne\\ Inversionszentrum}; \\ & \node[fill=red!20!background] (D) {20 Piezoelektrisch}; & \node (E) {1 Nicht\\ piezoelektrisch}; \\ }; |