diff options
Diffstat (limited to 'vorlesungen/punktgruppen')
| -rw-r--r-- | vorlesungen/punktgruppen/slides.pdf | bin | 52627 -> 53285 bytes | |||
| -rw-r--r-- | vorlesungen/punktgruppen/slides.tex | 59 |
2 files changed, 40 insertions, 19 deletions
diff --git a/vorlesungen/punktgruppen/slides.pdf b/vorlesungen/punktgruppen/slides.pdf Binary files differindex 199498c..45ae1dd 100644 --- a/vorlesungen/punktgruppen/slides.pdf +++ b/vorlesungen/punktgruppen/slides.pdf 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} |