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-rw-r--r--vorlesungen/punktgruppen/slides.pdfbin52627 -> 53285 bytes
-rw-r--r--vorlesungen/punktgruppen/slides.tex59
2 files changed, 40 insertions, 19 deletions
diff --git a/vorlesungen/punktgruppen/slides.pdf b/vorlesungen/punktgruppen/slides.pdf
index 199498c..45ae1dd 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 54798f1..d38065d 100644
--- a/vorlesungen/punktgruppen/slides.tex
+++ b/vorlesungen/punktgruppen/slides.tex
@@ -155,7 +155,7 @@
\end{align*}
Orthogonale Gruppe
\[
- O(n) = \left\{ Q \in \mathrm{GL}_n(\mathbb{R}) : QQ^t = Q^tQ = I \right\}
+ O(n) = \left\{ Q : QQ^t = Q^tQ = I \right\}
\]
\end{column}
\begin{column}{.5\textwidth}
@@ -173,7 +173,7 @@
\Phi_r &= \begin{pmatrix}
\cos \alpha & -\sin \alpha & 0 \\
\sin \alpha & \cos \alpha & 0 \\
- 0 & 0 & 1 \\[1em]
+ 0 & 0 & 1
\end{pmatrix}
\end{align*}
\end{column}
@@ -208,21 +208,40 @@
}
}
\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);
-
+ \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}
\end{column}
\begin{column}{.5\textwidth}
- Invariant (symmetrisch) unten Translation
+ Invariant (symmetrisch) 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}
@@ -272,21 +291,19 @@
\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*}
- \end{column}
- \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*}
+ \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\}
+ \end{align*}
\end{column}
\end{columns}
\end{frame}
@@ -386,9 +403,11 @@
\node[white] (E) {\(\vec{E}_p\)};
\begin{scope}[node distance = .5mm]
- \node[blue!50, right = of E] {\(-\)};
- \node[red!50, left = of E] {\(+\)};
+ \node[red!50, right = of E] {\(+\)};
+ \node[blue!50, left = of E] {\(-\)};
\end{scope}
+ \draw[gray, thick, dotted] (E) to ++(0,2);
+ \draw[gray, thick, dotted] (E) to ++(0,-2);
\end{scope}
\begin{scope}[xshift=11cm, yshift=-4.5cm]
@@ -407,9 +426,11 @@
\node[white] (E) {\(\vec{E}_p\)};
\begin{scope}[node distance = .5mm]
- \node[blue!50, right = of E] {\(-\)};
- \node[red!50, left = of E] {\(+\)};
+ \node[red!50, right = of E] {\(+\)};
+ \node[blue!50, left = of E] {\(-\)};
\end{scope}
+ \draw[gray, thick, dotted] (E) to ++(0,2);
+ \draw[gray, thick, dotted] (E) to ++(0,-2);
\end{scope}
\end{tikzpicture}
\end{frame}