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-rw-r--r--vorlesungen/punktgruppen/script.pdfbin31136 -> 34480 bytes
-rw-r--r--vorlesungen/punktgruppen/shotcut/Punktgruppen/Punktgruppen.mlt96
-rw-r--r--vorlesungen/punktgruppen/slides.pdfbin53285 -> 148139 bytes
-rw-r--r--vorlesungen/punktgruppen/slides.tex233
4 files changed, 173 insertions, 156 deletions
diff --git a/vorlesungen/punktgruppen/script.pdf b/vorlesungen/punktgruppen/script.pdf
index 6b17967..70ea683 100644
--- a/vorlesungen/punktgruppen/script.pdf
+++ b/vorlesungen/punktgruppen/script.pdf
Binary files differ
diff --git a/vorlesungen/punktgruppen/shotcut/Punktgruppen/Punktgruppen.mlt b/vorlesungen/punktgruppen/shotcut/Punktgruppen/Punktgruppen.mlt
index 37d8b30..7275d5c 100644
--- a/vorlesungen/punktgruppen/shotcut/Punktgruppen/Punktgruppen.mlt
+++ b/vorlesungen/punktgruppen/shotcut/Punktgruppen/Punktgruppen.mlt
@@ -1,52 +1,24 @@
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- <property name="mlt_service">avformat-novalidate</property>
- <property name="seekable">1</property>
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<property name="mute_on_pause">0</property>
<property name="xml">was here</property>
- <property name="shotcut:hash">f5af3911322b3cb8d68246387eef0a48</property>
+ <property name="shotcut:hash">d255c7f03c50cb9dcfb2f9e0db73cb05</property>
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<playlist id="main_bin">
<property name="xml_retain">1</property>
- <entry producer="chain0" in="00:00:00.000" out="00:01:05.850"/>
- <entry producer="chain1" in="00:00:00.000" out="00:01:39.050"/>
- <entry producer="chain2" in="00:00:00.000" out="00:00:43.850"/>
+ <entry producer="chain0" in="00:00:00.000" out="00:01:09.983"/>
</playlist>
- <producer id="black" in="00:00:00.000" out="00:02:22.917">
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+ <producer id="black" in="00:00:00.000" out="00:01:09.983">
+ <property name="length">00:01:10.000</property>
<property name="eof">pause</property>
<property name="resource">0</property>
<property name="aspect_ratio">1</property>
@@ -55,47 +27,54 @@
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<playlist id="background">
- <entry producer="black" in="00:00:00.000" out="00:02:22.917"/>
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<property name="eof">pause</property>
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<property name="xml">was here</property>
- <property name="shotcut:hash">47ac11813561d23004945f1738115b32</property>
- <property name="shotcut:caption">Geometric2DSymmetries.mp4</property>
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+ <property name="shotcut:defaultAudioIndex">1</property>
</chain>
- <chain id="chain4" out="00:00:43.850">
- <property name="length">2632</property>
+ <playlist id="playlist0">
+ <property name="shotcut:video">1</property>
+ <property name="shotcut:name">V1</property>
+ <entry producer="chain1" in="00:00:00.000" out="00:01:09.983"/>
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<property name="mlt_service">avformat-novalidate</property>
<property name="seekable">1</property>
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<property name="xml">was here</property>
- <property name="shotcut:hash">f5af3911322b3cb8d68246387eef0a48</property>
- <property name="shotcut:caption">Geometric3DSymmetries.mp4</property>
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+ <property name="shotcut:caption">DSCF6392.MOV</property>
+ <property name="shotcut:defaultAudioIndex">1</property>
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- <playlist id="playlist0">
- <property name="shotcut:video">1</property>
- <property name="shotcut:name">V1</property>
- <entry producer="chain3" in="00:00:00.000" out="00:01:39.050"/>
- <entry producer="chain4" in="00:00:00.000" out="00:00:43.850"/>
+ <playlist id="playlist1">
+ <property name="shotcut:audio">1</property>
+ <property name="shotcut:name">A1</property>
+ <entry producer="chain2" in="00:00:00.000" out="00:01:09.983"/>
</playlist>
- <tractor id="tractor0" title="Shotcut version 21.05.01" in="00:00:00.000" out="00:02:22.917">
+ <tractor id="tractor0" title="Shotcut version 21.05.01" in="00:00:00.000" out="00:01:09.983">
<property name="shotcut">1</property>
- <property name="shotcut:scaleFactor">0.124449</property>
+ <property name="shotcut:scaleFactor">0.174581</property>
<property name="shotcut:projectAudioChannels">2</property>
<property name="shotcut:projectFolder">1</property>
<track producer="background"/>
<track producer="playlist0"/>
+ <track producer="playlist1" hide="video"/>
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<property name="b_track">1</property>
@@ -111,5 +90,12 @@
<property name="threads">0</property>
<property name="disable">1</property>
</transition>
+ <transition id="transition2">
+ <property name="a_track">0</property>
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+ <property name="sum">1</property>
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</tractor>
</mlt>
diff --git a/vorlesungen/punktgruppen/slides.pdf b/vorlesungen/punktgruppen/slides.pdf
index 45ae1dd..37ada17 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 d38065d..cc62969 100644
--- a/vorlesungen/punktgruppen/slides.tex
+++ b/vorlesungen/punktgruppen/slides.tex
@@ -64,12 +64,11 @@
\frame{\titlepage}
\frame{\tableofcontents}
-\section{Einleitung}
\frame{
\begin{itemize}
- \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}
}