diff options
Diffstat (limited to 'vorlesungen')
| -rw-r--r-- | vorlesungen/punktgruppen/script.pdf | bin | 31136 -> 34480 bytes | |||
| -rw-r--r-- | vorlesungen/punktgruppen/shotcut/Punktgruppen/Punktgruppen.mlt | 96 | ||||
| -rw-r--r-- | vorlesungen/punktgruppen/slides.pdf | bin | 53285 -> 148139 bytes | |||
| -rw-r--r-- | vorlesungen/punktgruppen/slides.tex | 233 |
4 files changed, 173 insertions, 156 deletions
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\item Was heisst \emph{Symmetrie} in der Mathematik? - \item Wie kann ein Kristall modelliert werden? - \item Aus der Physik: Piezoelektrizit\"at + \item Was heisst \emph{Symmetrie} in der Mathematik? \pause + \item Wie kann ein Kristall modelliert werden? \pause + \item Aus der Physik: Piezoelektrizit\"at \pause \end{itemize} \begin{center} \begin{tikzpicture} @@ -125,7 +124,7 @@ \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 = -70, in = 70] node[midway, right] {\(U \neq 0\)} (n); + \draw[white, thick, ->] (p) to[out = -70, in = 70] node[midway, right] {\(U > 0\)} (n); \end{scope} \end{tikzpicture} \end{center} @@ -148,16 +147,19 @@ \[ G = \left\{\mathbb{1}, r, \sigma, \dots \right\} \] + \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} @@ -181,15 +183,31 @@ } \section{Kristalle} +\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] (A) {}; + \node[white, yshift = 2mm] at (A) {230}; + + \node[circle, draw = white, thick, + fill = red!20!background, + xshift = -5mm, yshift = -5mm, + minimum size = 1cm] {32}; + \end{tikzpicture} + \end{center} +\end{frame} + \begin{frame}[fragile]{} - \begin{columns}[T] + \begin{columns} \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 = { @@ -201,14 +219,14 @@ ] \begin{scope} - \clip (-1,-1) rectangle (4,3); - \foreach \y in {-5,-4,...,5} { - \foreach \x in {-5,-4,...,5} { + \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] (-1, -1) rectangle (4,3); + \draw[white, thick] (-2, -2) rectangle (3,4); \draw[red!80!background, thick, ->] (N00) to node[midway, below] {\(\vec{a}_1\)} (N10); @@ -217,31 +235,21 @@ \end{tikzpicture} \end{center} \end{column} + \pause \begin{column}{.5\textwidth} - Invariant (symmetrisch) unter Translation + Kristallgitter: + \(n_i \in \mathbb{Z}\), + \(\vec{a}_i \in \mathbb{R}^3\) + \[ + \vec{r} = n_1 \vec{a}_1 + n_2 \vec{a}_2 + n_3 \vec{a}_3 + \] + \vspace{1cm} + \pause + + Invariant unter Translation \[ Q_i(\vec{r}) = \vec{r} + \vec{a}_i \] - 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] (A) {}; - \node[white, yshift = 2mm] at (A) {230}; - - \node[circle, draw = white, thick, - fill = red!20!background, - xshift = -5mm, yshift = -5mm, - minimum size = 1cm] {32}; - \end{tikzpicture} - \end{center} \end{column} \end{columns} \end{frame} @@ -249,7 +257,9 @@ \begin{frame}[fragile]{} \begin{columns}[T] \begin{column}{.5\textwidth} - Wie kombiniert sich \(Q_i\) mit der anderen Symmetrien? + \onslide<1->{ + Wie kombiniert sich \(Q_i\) mit der anderen Symmetrien? + } \begin{center} \begin{tikzpicture}[ dot/.style = { @@ -260,50 +270,70 @@ }, ] - \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); + \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) {\(A'\)}; - \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[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<7->{ \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) {\(x\)}; + \onslide<8->{ + \node[above left, xshift=-2mm] at (X) {\(x\)}; + } \end{tikzpicture} \end{center} \end{column} \begin{column}{.5\textwidth} - 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*} - Somit muss + \onslide<9->{ + Sei \(q = |\vec{Q}|\), \(\alpha = 2\pi/n\) und \(n \in \mathbb{N}\) + } \begin{align*} - \alpha &= \cos^{-1}\left(\frac{n-1}{2}\right) \\[1em] - \alpha &\in \left\{ 0, 60^\circ, 90^\circ, 120^\circ, 180^\circ \right\} + \onslide<10->{q' = n q &= 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\} + \end{align*} + } \end{column} \end{columns} \end{frame} @@ -343,8 +373,9 @@ negative/.style = { fill = blue!50 }, ] - \node[font = {\large\bfseries}, align = center] (title) at (6,0) {Mit und Ohne\\ Symmetriezentrum}; + \node[font = {\large\bfseries}, align = center] (title) at (5.5,0) {Mit und Ohne\\ Symmetriezentrum}; \node[below = of title] {Polarisation Feld \(\vec{E}_p\)}; + \pause \begin{scope} \matrix[nodes = { charge }, row sep = 8mm, column sep = 8mm] { @@ -354,6 +385,7 @@ }; \draw[gray, dashed] (W) to (N) to (E) to (S) to (W); \end{scope} + \pause \begin{scope}[yshift=-4.5cm] \matrix[nodes = { charge }, row sep = 5mm, column sep = 1cm] { @@ -372,6 +404,7 @@ \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} { @@ -379,27 +412,23 @@ } \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[gray, dashed] (C2) to (C4) to (C6) to (C2); \end{scope} + \pause - \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) {}; + \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); - } + \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] @@ -409,20 +438,26 @@ \draw[gray, thick, dotted] (E) to ++(0,2); \draw[gray, thick, dotted] (E) to ++(0,-2); \end{scope} + \pause - \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) {}; + \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); + \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] @@ -462,15 +497,11 @@ \] Anisotropisch Dielektrikum \[ - (\ten{K}\ten{\varepsilon})\vec{E} = \frac{\omega^2}{\mu k^2} \vec{E} + (\ten{K}\ten{\varepsilon})\vec{E} = \frac{k^2}{\mu \omega^2} \vec{E} \] \[ \vec{E} \in U_\lambda \implies (\ten{K}\ten{\varepsilon}) \vec{E} = \lambda \vec{E} \] - \"Ahenlich auch in der Mechanik - \[ - \vec{F} = \kappa \vec{x} \quad \text{(Hooke)} - \] \end{column} \end{columns} } |