From 4b3f4be3305d130c0435c269fd038eb049252f5d Mon Sep 17 00:00:00 2001 From: Nao Pross Date: Wed, 28 Apr 2021 21:52:38 +0200 Subject: Create slides --- vorlesungen/punktgruppen/slides.tex | 61 +++++++++++++++++++++++++++++++++++++ 1 file changed, 61 insertions(+) create mode 100644 vorlesungen/punktgruppen/slides.tex (limited to 'vorlesungen/punktgruppen/slides.tex') diff --git a/vorlesungen/punktgruppen/slides.tex b/vorlesungen/punktgruppen/slides.tex new file mode 100644 index 0000000..5c6982c --- /dev/null +++ b/vorlesungen/punktgruppen/slides.tex @@ -0,0 +1,61 @@ +\documentclass[12pt, xcolor, aspectratio=169]{beamer} + + +% Theme +\beamertemplatenavigationsymbolsempty + +% set look +\usetheme{default} +\usecolortheme{fly} + +%% Set font +\usepackage[p,osf]{scholax} +\usepackage{amsmath} +\usepackage[scaled=1.075,ncf,vvarbb]{newtxmath} + +% set colors +\definecolor{background}{HTML}{202020} + +\setbeamercolor{normal text}{fg=white, bg=background} +\setbeamercolor{structure}{fg=white} + +\setbeamercolor{item projected}{use=item,fg=background,bg=item.fg!35} + +\setbeamercolor*{palette primary}{use=structure,fg=white,bg=structure.fg} +\setbeamercolor*{palette secondary}{use=structure,fg=white,bg=structure.fg!75} +\setbeamercolor*{palette tertiary}{use=structure,fg=white,bg=structure.fg!50} +\setbeamercolor*{palette quaternary}{fg=white,bg=background} + +\setbeamercolor*{block title}{parent=structure} +\setbeamercolor*{block body}{fg=background, bg=} + +\setbeamercolor*{framesubtitle}{fg=white} + +% Metadata +\title{\LARGE \scshape Punktgruppen und Kristallen} +\author[N. Pross, T. T\"onz]{Naoki Pross, Tim T\"onz} +\institute{Hochschule f\"ur Technik OST, Rapperswil} +\date{10. Mai 2021} + +% Slides +\begin{document} +\begin{frame}{} +\titlepage +\end{frame} + +% \begin{frame}{} +% \tableofcontents +% \end{frame} + +\section{Einleitung} +\begin{frame}{Kristallen} +\[ + \psi +\] +\end{frame} + +\section{Geometrische Symmetrien} +\section{Algebraische Symmetrien} +\section{Kristallen} + +\end{document} -- cgit v1.2.1 From 152b7c463b1ae7cd7916dff075a11b306e5c22fc Mon Sep 17 00:00:00 2001 From: Nao Pross Date: Sat, 1 May 2021 19:43:15 +0200 Subject: Add section title pages --- vorlesungen/punktgruppen/slides.tex | 20 ++++++++++++++++++++ 1 file changed, 20 insertions(+) (limited to 'vorlesungen/punktgruppen/slides.tex') diff --git a/vorlesungen/punktgruppen/slides.tex b/vorlesungen/punktgruppen/slides.tex index 5c6982c..8394451 100644 --- a/vorlesungen/punktgruppen/slides.tex +++ b/vorlesungen/punktgruppen/slides.tex @@ -1,5 +1,8 @@ \documentclass[12pt, xcolor, aspectratio=169]{beamer} +% language +\usepackage{polyglossia} +\setmainlanguage{german} % Theme \beamertemplatenavigationsymbolsempty @@ -31,6 +34,15 @@ \setbeamercolor*{framesubtitle}{fg=white} +\setbeamertemplate{section page} +{ + \begin{center} + \Huge + \insertsection + \end{center} +} +\AtBeginSection{\frame{\sectionpage}} + % Metadata \title{\LARGE \scshape Punktgruppen und Kristallen} \author[N. Pross, T. T\"onz]{Naoki Pross, Tim T\"onz} @@ -55,7 +67,15 @@ \end{frame} \section{Geometrische Symmetrien} +%% Made in video + \section{Algebraische Symmetrien} +\begin{frame}{} + Hello +\end{frame} + \section{Kristallen} +\section{Anwendungen} + \end{document} -- cgit v1.2.1 From 6e507ac59576c7b7ac5f0c80eee4e3a36b599a7e Mon Sep 17 00:00:00 2001 From: Nao Pross Date: Sat, 1 May 2021 21:09:26 +0200 Subject: Add slide for 4th roots of 1 --- vorlesungen/punktgruppen/slides.tex | 73 +++++++++++++++++++++++++++++-------- 1 file changed, 58 insertions(+), 15 deletions(-) (limited to 'vorlesungen/punktgruppen/slides.tex') diff --git a/vorlesungen/punktgruppen/slides.tex b/vorlesungen/punktgruppen/slides.tex index 8394451..2199b26 100644 --- a/vorlesungen/punktgruppen/slides.tex +++ b/vorlesungen/punktgruppen/slides.tex @@ -10,6 +10,7 @@ % set look \usetheme{default} \usecolortheme{fly} +\usefonttheme{serif} %% Set font \usepackage[p,osf]{scholax} @@ -51,28 +52,70 @@ % Slides \begin{document} -\begin{frame}{} -\titlepage -\end{frame} - -% \begin{frame}{} -% \tableofcontents -% \end{frame} +\frame{\titlepage} +\frame{\tableofcontents} \section{Einleitung} -\begin{frame}{Kristallen} -\[ - \psi -\] -\end{frame} +\frame{ + \[ + \psi + \] +} \section{Geometrische Symmetrien} %% Made in video \section{Algebraische Symmetrien} -\begin{frame}{} - Hello -\end{frame} +\frame{ + \begin{columns} + \begin{column}{.3\textwidth} + Produkt mit \(i\) + \begin{align*} + 1 \cdot i &= i \\ + i \cdot i &= -1 \\ + -1 \cdot i &= -i \\ + -i \cdot i &= 1 + \end{align*} + \pause + % + Gruppe + \begin{align*} + G &= \left\{ + 1, i, -1, -i + \right\} \\ + &= \left\{ + 1, i, i^2, i^3 + \right\} \\ + Z_4 &= \left\{ + \mathbb{1}, r, r^2, r^3 + \right\} + \end{align*} + \pause + % + \end{column} + \begin{column}{.5\textwidth} + % + Darstellung + \[ + \phi : Z_4 \to G + \] + \begin{align*} + \phi(\mathbb{1}) &= 1 & \phi(r^2) &= i^2 \\ + \phi(r) &= i & \phi(r^3) &= i^3 + \end{align*} + \pause + % + Homomorphismus + \begin{align*} + \phi(r \circ \mathbb{1}) &= \phi(r) \cdot \phi(\mathbb{1}) \\ + &= i \cdot 1 + \end{align*} + \pause + % + \(\phi\) ist bijektiv \(\implies Z_4 \cong G\) + \end{column} + \end{columns} +} \section{Kristallen} -- cgit v1.2.1 From 6eed506199b5ff21b58af9f8db6477d5af56aa0b Mon Sep 17 00:00:00 2001 From: tim30b Date: Mon, 3 May 2021 18:45:24 +0200 Subject: Write Intro --- vorlesungen/punktgruppen/slides.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) (limited to 'vorlesungen/punktgruppen/slides.tex') diff --git a/vorlesungen/punktgruppen/slides.tex b/vorlesungen/punktgruppen/slides.tex index 2199b26..1c1b1d8 100644 --- a/vorlesungen/punktgruppen/slides.tex +++ b/vorlesungen/punktgruppen/slides.tex @@ -45,7 +45,7 @@ \AtBeginSection{\frame{\sectionpage}} % Metadata -\title{\LARGE \scshape Punktgruppen und Kristallen} +\title{\LARGE \scshape Punktgruppen und Kristalle} \author[N. Pross, T. T\"onz]{Naoki Pross, Tim T\"onz} \institute{Hochschule f\"ur Technik OST, Rapperswil} \date{10. Mai 2021} @@ -117,7 +117,7 @@ \end{columns} } -\section{Kristallen} +\section{Kristalle} \section{Anwendungen} -- cgit v1.2.1 From 2de3a8ffcfb661abfde05cf3611ad08d5a1160fc Mon Sep 17 00:00:00 2001 From: Nao Pross Date: Tue, 4 May 2021 20:08:33 +0200 Subject: Create slides for applications --- vorlesungen/punktgruppen/slides.tex | 202 ++++++++++++++++++++++++++++-------- 1 file changed, 160 insertions(+), 42 deletions(-) (limited to 'vorlesungen/punktgruppen/slides.tex') diff --git a/vorlesungen/punktgruppen/slides.tex b/vorlesungen/punktgruppen/slides.tex index 1c1b1d8..380dcec 100644 --- a/vorlesungen/punktgruppen/slides.tex +++ b/vorlesungen/punktgruppen/slides.tex @@ -4,6 +4,10 @@ \usepackage{polyglossia} \setmainlanguage{german} +% pretty drawings +\usepackage{tikz} +\usetikzlibrary{positioning} + % Theme \beamertemplatenavigationsymbolsempty @@ -44,6 +48,9 @@ } \AtBeginSection{\frame{\sectionpage}} +% Macros +\newcommand{\ten}[1]{#1} + % Metadata \title{\LARGE \scshape Punktgruppen und Kristalle} \author[N. Pross, T. T\"onz]{Naoki Pross, Tim T\"onz} @@ -66,59 +73,170 @@ %% Made in video \section{Algebraische Symmetrien} +%% Made in video + +\section{Kristalle} + +\section{Anwendungen} +\begin{frame}[fragile]{} + \centering + \begin{tikzpicture}[ + box/.style = { + rectangle, thick, draw = white, fill = darkgray!50!background, + minimum height = 1cm, outer sep = 2mm, + }, + ] + + \matrix [nodes = {box, align = center}, column sep = 1cm, row sep = 1.5cm] { + & \node (A) {32 Punktgruppe}; \\ + \node (B) {11 Mit\\ Inversionszentrum}; & \node (C) {21 Ohne\\ Inversionszentrum}; \\ + & \node[fill=red!20!background] (D) {20 Piezoelektrisch}; & \node (E) {1 Nicht\\ piezoelektrisch}; \\ + }; + + \draw[thick, ->] (A.west) to[out=180, in=90] (B.north); + \draw[thick, ->] (A.south) to (C); + \draw[thick, ->] (C.south) to (D.north); + \draw[thick, ->] (C.east) to[out=0, in=90] (E.north); + \end{tikzpicture} +\end{frame} + +\begin{frame}[fragile]{} + \begin{tikzpicture}[ + overlay, xshift = 1.5cm, yshift = 1.5cm, + node distance = 2mm, + charge/.style = { + circle, draw = white, thick, + minimum size = 5mm + }, + positive/.style = { fill = red!50 }, + negative/.style = { fill = blue!50 }, + ] + + \node[font = {\large\bfseries}, align = center] (title) at (6,0) {Mit und Ohne\\ Symmetriezentrum}; + \node[below = of title] {Polarisation Feld \(\vec{E}_p\)}; + + \begin{scope} + \matrix[nodes = { charge }, row sep = 8mm, column sep = 8mm] { + \node[positive] {}; & \node[negative] (N) {}; & \node [positive] {}; \\ + \node[negative] (W) {}; & \node[positive] {}; & \node [negative] (E) {}; \\ + \node[positive] {}; & \node[negative] (S) {}; & \node [positive] {}; \\ + }; + \draw[gray, dashed] (W) to (N) to (E) to (S) to (W); + \end{scope} + + \begin{scope}[yshift=-4.5cm] + \matrix[nodes = { charge }, row sep = 5mm, column sep = 1cm] { + \node[positive] (NW) {}; & \node[negative] (N) {}; & \node [positive] (NE) {}; \\ + \node[negative] (W) {}; & \node[positive] {}; & \node [negative] (E) {}; \\ + \node[positive] (SW) {}; & \node[negative] (S) {}; & \node [positive] (SE) {}; \\ + }; + + \foreach \d in {NW, N, NE} { + \draw[orange, very thick, <-] (\d) to ++(0,.7); + } + + \foreach \d in {SW, S, SE} { + \draw[orange, very thick, <-] (\d) to ++(0,-.7); + } + + \draw[gray, dashed] (W) to (N) to (E) to (S) to (W); + \end{scope} + + \begin{scope}[xshift=11cm] + \foreach \x/\t [count=\i] in {60/positive, 120/negative, 180/positive, 240/negative, 300/positive, 360/negative} { + \node[charge, \t] (C\i) at (\x: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); + \end{scope} + + \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) {}; + + \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); + } + + \node[white] (E) {\(\vec{E}_p\)}; + \begin{scope}[node distance = .5mm] + \node[blue!50, right = of E] {\(-\)}; + \node[red!50, left = of E] {\(+\)}; + \end{scope} + \end{scope} + + \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); + + \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] + \node[blue!50, right = of E] {\(-\)}; + \node[red!50, left = of E] {\(+\)}; + \end{scope} + \end{scope} + \end{tikzpicture} +\end{frame} + \frame{ - \begin{columns} - \begin{column}{.3\textwidth} - Produkt mit \(i\) + \begin{columns}[T] + \begin{column}{.5\textwidth} + Symmetriegruppe und Darstellung \begin{align*} - 1 \cdot i &= i \\ - i \cdot i &= -1 \\ - -1 \cdot i &= -i \\ - -i \cdot i &= 1 + G &= \left\{\mathbb{1}, r, \sigma, \dots \right\} \\ + &\Phi : G \to O(n) \end{align*} - \pause - % - Gruppe \begin{align*} - G &= \left\{ - 1, i, -1, -i - \right\} \\ - &= \left\{ - 1, i, i^2, i^3 - \right\} \\ - Z_4 &= \left\{ - \mathbb{1}, r, r^2, r^3 - \right\} + U_\lambda &= \left\{ v : \Phi v = \lambda v \right\} \\ + &= \mathrm{null}\left(\Phi - \lambda I\right) \end{align*} - \pause - % + Helmholtz Wellengleichung + \[ + \nabla^2 \vec{E} = \ten{\varepsilon}\mu + \frac{\partial^2}{\partial t^2} \vec{E} + \] \end{column} \begin{column}{.5\textwidth} - % - Darstellung + Ebene Welle \[ - \phi : Z_4 \to G + \vec{E} = \vec{E}_0 \exp\left[i + \left(\vec{k}\cdot\vec{r} - \omega t \right)\right] + \] + Anisotropisch Dielektrikum + \[ + \ten{R}\ten{\varepsilon}\vec{E} = \frac{\omega^2}{\mu k^2} \vec{E} + \] + \[ + \vec{E} \in U_\lambda \implies (\ten{R}\ten{\varepsilon}) \vec{E} = \lambda \vec{E} + \] + \"Ahenlich auch in der Mechanik + \[ + \vec{F} = \kappa \vec{x} \quad \text{(Hooke)} \] - \begin{align*} - \phi(\mathbb{1}) &= 1 & \phi(r^2) &= i^2 \\ - \phi(r) &= i & \phi(r^3) &= i^3 - \end{align*} - \pause - % - Homomorphismus - \begin{align*} - \phi(r \circ \mathbb{1}) &= \phi(r) \cdot \phi(\mathbb{1}) \\ - &= i \cdot 1 - \end{align*} - \pause - % - \(\phi\) ist bijektiv \(\implies Z_4 \cong G\) \end{column} \end{columns} } -\section{Kristalle} - -\section{Anwendungen} - \end{document} -- cgit v1.2.1 From 5d2f685df7b2d7691625937609cf92be9fc8c9da Mon Sep 17 00:00:00 2001 From: Nao Pross Date: Tue, 4 May 2021 20:36:41 +0200 Subject: Create intro slide --- vorlesungen/punktgruppen/slides.tex | 79 ++++++++++++++++++++++++++++++++++--- 1 file changed, 73 insertions(+), 6 deletions(-) (limited to 'vorlesungen/punktgruppen/slides.tex') diff --git a/vorlesungen/punktgruppen/slides.tex b/vorlesungen/punktgruppen/slides.tex index 380dcec..25761c0 100644 --- a/vorlesungen/punktgruppen/slides.tex +++ b/vorlesungen/punktgruppen/slides.tex @@ -7,6 +7,7 @@ % pretty drawings \usepackage{tikz} \usetikzlibrary{positioning} +\usetikzlibrary{arrows.meta} % Theme \beamertemplatenavigationsymbolsempty @@ -64,17 +65,82 @@ \section{Einleitung} \frame{ - \[ - \psi - \] + \begin{itemize} + \item Was heisst \emph{Symmetrie} in der Mathematik? + \item Wie kann ein Kristall modelliert werden? + \item Aus der Physik: Piezoelektrizit\"at + \end{itemize} + \begin{center} + \begin{tikzpicture} + \begin{scope}[ + node distance = 0cm + ] + \node[ + rectangle, fill = gray!40!background, + minimum width = 3cm, minimum height = 2cm, + ] (body) {\(\vec{E}_p = \vec{0}\)}; + + \node[ + draw, rectangle, thick, white, fill = red!50, + minimum width = 3cm, minimum height = 1mm, + above = of body + ] (pos) {}; + + \node[ + draw, rectangle, thick, white, fill = blue!50, + minimum width = 3cm, minimum height = 1mm, + below = of body + ] (neg) {}; + + \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); + \end{scope} + \begin{scope}[ + node distance = 0cm, + xshift = 7cm + ] + \node[ + rectangle, fill = gray!40!background, + minimum width = 3cm, minimum height = 1.5cm, + ] (body) {\(\vec{E}_p = \vec{0}\)}; + + \node[ + draw, rectangle, thick, white, fill = red!50, + minimum width = 3cm, minimum height = 1mm, + above = of body + ] (pos) {}; + + \node[ + draw, rectangle, thick, white, fill = blue!50, + minimum width = 3cm, minimum height = 1mm, + below = of body + ] (neg) {}; + + \draw[orange, very thick, <-] (pos.north) to node[near end, right] {\(\vec{F}\)} ++(0,1); + \draw[orange, very thick, <-] (neg.south) to node[near end, right] {\(\vec{F}\)} ++(0,-1); + + \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); + \end{scope} + \end{tikzpicture} + \end{center} } -\section{Geometrische Symmetrien} +\section{2D Symmetrien} %% Made in video \section{Algebraische Symmetrien} %% Made in video +\section{3D Symmetrien} +%% Made in video + +\section{Matrizen} + \section{Kristalle} \section{Anwendungen} @@ -201,6 +267,7 @@ \end{frame} \frame{ + \frametitle{Licht in Kristallen} \begin{columns}[T] \begin{column}{.5\textwidth} Symmetriegruppe und Darstellung @@ -226,10 +293,10 @@ \] Anisotropisch Dielektrikum \[ - \ten{R}\ten{\varepsilon}\vec{E} = \frac{\omega^2}{\mu k^2} \vec{E} + (\ten{K}\ten{\varepsilon})\vec{E} = \frac{\omega^2}{\mu k^2} \vec{E} \] \[ - \vec{E} \in U_\lambda \implies (\ten{R}\ten{\varepsilon}) \vec{E} = \lambda \vec{E} + \vec{E} \in U_\lambda \implies (\ten{K}\ten{\varepsilon}) \vec{E} = \lambda \vec{E} \] \"Ahenlich auch in der Mechanik \[ -- cgit v1.2.1 From 714f0685c47dfa46775859024936b22f610619c9 Mon Sep 17 00:00:00 2001 From: Nao Pross Date: Wed, 5 May 2021 13:03:09 +0200 Subject: Create crystal slides --- vorlesungen/punktgruppen/slides.tex | 115 +++++++++++++++++++++++++++++++++++- 1 file changed, 112 insertions(+), 3 deletions(-) (limited to 'vorlesungen/punktgruppen/slides.tex') 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}; \\ }; -- cgit v1.2.1 From cbcb1b7f273875a0faf6ffa3d4daf7de7d6e0490 Mon Sep 17 00:00:00 2001 From: Nao Pross Date: Wed, 5 May 2021 14:57:07 +0200 Subject: Add slides for matrices --- vorlesungen/punktgruppen/slides.tex | 55 +++++++++++++++++++++++++++++++------ 1 file changed, 47 insertions(+), 8 deletions(-) (limited to 'vorlesungen/punktgruppen/slides.tex') diff --git a/vorlesungen/punktgruppen/slides.tex b/vorlesungen/punktgruppen/slides.tex index e800a87..54798f1 100644 --- a/vorlesungen/punktgruppen/slides.tex +++ b/vorlesungen/punktgruppen/slides.tex @@ -141,6 +141,44 @@ %% Made in video \section{Matrizen} +\frame{ + \begin{columns}[T] + \begin{column}{.5\textwidth} + Symmetriegruppe + \[ + G = \left\{\mathbb{1}, r, \sigma, \dots \right\} + \] + Matrixdarstellung + \begin{align*} + \Phi : G &\to O(3) \\ + g &\mapsto \Phi_g + \end{align*} + Orthogonale Gruppe + \[ + O(n) = \left\{ Q \in \mathrm{GL}_n(\mathbb{R}) : QQ^t = Q^tQ = I \right\} + \] + \end{column} + \begin{column}{.5\textwidth} + \begin{align*} + \Phi_\mathbb{1} &= \begin{pmatrix} + 1 & 0 & 0 \\ + 0 & 1 & 0 \\ + 0 & 0 & 1 + \end{pmatrix} = I \\[1em] + \Phi_\sigma &= \begin{pmatrix} + 1 & 0 & 0 \\ + 0 & -1 & 0 \\ + 0 & 0 & 1 + \end{pmatrix} \\[1em] + \Phi_r &= \begin{pmatrix} + \cos \alpha & -\sin \alpha & 0 \\ + \sin \alpha & \cos \alpha & 0 \\ + 0 & 0 & 1 \\[1em] + \end{pmatrix} + \end{align*} + \end{column} + \end{columns} +} \section{Kristalle} \begin{frame}[fragile]{} @@ -178,11 +216,19 @@ \end{tikzpicture} \end{center} + \end{column} + \begin{column}{.5\textwidth} Invariant (symmetrisch) unten Translation \[ Q_i(\vec{r}) = \vec{r} + \vec{a}_i \] + M\"ogliche Kristallstrukturen \end{column} + \end{columns} +\end{frame} + +\begin{frame}[fragile]{} + \begin{columns}[T] \begin{column}{.5\textwidth} Wie kombiniert sich \(Q_i\) mit der anderen Symmetrien? \begin{center} @@ -233,11 +279,6 @@ n &= 1 - 2\cos\alpha \end{align*} \end{column} - \end{columns} -\end{frame} - -\frame{ - \begin{columns}[T] \begin{column}{.5\textwidth} Somit muss \[ @@ -247,10 +288,8 @@ \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} -} +\end{frame} \section{Anwendungen} \begin{frame}[fragile]{} -- cgit v1.2.1 From 262989abd634eddac1dae3974964c382c0301a94 Mon Sep 17 00:00:00 2001 From: Nao Pross Date: Wed, 5 May 2021 16:17:06 +0200 Subject: Add venn diagram and fix typos --- vorlesungen/punktgruppen/slides.tex | 59 +++++++++++++++++++++++++------------ 1 file changed, 40 insertions(+), 19 deletions(-) (limited to 'vorlesungen/punktgruppen/slides.tex') 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} -- cgit v1.2.1 From c9e25f8c35027bde9af6f2ab72071dbdd21fa444 Mon Sep 17 00:00:00 2001 From: Nao Pross Date: Wed, 5 May 2021 21:23:44 +0200 Subject: Add pauses and minor fixes --- vorlesungen/punktgruppen/slides.tex | 233 ++++++++++++++++++++---------------- 1 file changed, 132 insertions(+), 101 deletions(-) (limited to 'vorlesungen/punktgruppen/slides.tex') 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} } -- cgit v1.2.1 From 8f862d9be3117979c9694544cca0afa8d2e53cb3 Mon Sep 17 00:00:00 2001 From: Nao Pross Date: Wed, 5 May 2021 23:07:28 +0200 Subject: Add (back) slide for algebraic symmetries Probably won't be used but still. --- vorlesungen/punktgruppen/slides.tex | 51 +++++++++++++++++++++++++++++++++++++ 1 file changed, 51 insertions(+) (limited to 'vorlesungen/punktgruppen/slides.tex') diff --git a/vorlesungen/punktgruppen/slides.tex b/vorlesungen/punktgruppen/slides.tex index cc62969..8edaff6 100644 --- a/vorlesungen/punktgruppen/slides.tex +++ b/vorlesungen/punktgruppen/slides.tex @@ -135,6 +135,57 @@ \section{Algebraische Symmetrien} %% Made in video +\frame{ + \begin{columns}[T] + \begin{column}{.5\textwidth} + Produkt mit \(i\) + \begin{align*} + 1 \cdot i &= i \\ + i \cdot i &= -1 \\ + -1 \cdot i &= -i \\ + -i \cdot i &= 1 + \end{align*} + \pause + % + Gruppe + \begin{align*} + G &= \left\{ + 1, i, -1, -i + \right\} \\ + &= \left\{ + 1, i, i^2, i^3 + \right\} \\ + C_4 &= \left\{ + \mathbb{1}, r, r^2, r^3 + \right\} + \end{align*} + \pause + \end{column} + \begin{column}{.5\textwidth} + Darstellung \(\phi : C_4 \to G\) + \begin{align*} + \phi(\mathbb{1}) &= 1 & \phi(r^2) &= i^2 \\ + \phi(r) &= i & \phi(r^3) &= i^3 + \end{align*} + \pause + % + Homomorphismus + \begin{align*} + \phi(r \circ \mathbb{1}) &= \phi(r) \cdot \phi(\mathbb{1}) \\ + &= i \cdot 1 + \end{align*} + \pause + % + \(\phi\) ist bijektiv \(\implies C_4 \cong G\) + \pause + % + \begin{align*} + \psi : C_4 &\to (\mathbb{Z}/4\mathbb{Z}, +) \\ + \psi(\mathbb{1}\circ r^2) &= 0 + 2 \pmod{4} + \end{align*} + \end{column} + \end{columns} +} \section{3D Symmetrien} %% Made in video -- cgit v1.2.1 From 1d952ad204000bd84383564cbd4240dcb6982738 Mon Sep 17 00:00:00 2001 From: Nao Pross Date: Wed, 5 May 2021 23:51:37 +0200 Subject: Add placeholder slides for video parts and update gitignore --- vorlesungen/punktgruppen/slides.tex | 10 ++++++++++ 1 file changed, 10 insertions(+) (limited to 'vorlesungen/punktgruppen/slides.tex') diff --git a/vorlesungen/punktgruppen/slides.tex b/vorlesungen/punktgruppen/slides.tex index 8edaff6..3aa6d83 100644 --- a/vorlesungen/punktgruppen/slides.tex +++ b/vorlesungen/punktgruppen/slides.tex @@ -132,6 +132,11 @@ \section{2D Symmetrien} %% Made in video +{ + \usebackgroundtemplate{ + \includegraphics[height=\paperheight]{media/images/nosignal}} + \frame{} +} \section{Algebraische Symmetrien} %% Made in video @@ -189,6 +194,11 @@ \section{3D Symmetrien} %% Made in video +{ + \usebackgroundtemplate{ + \includegraphics[height=\paperheight]{media/images/nosignal}} + \frame{} +} \section{Matrizen} \frame{ -- cgit v1.2.1 From 965fe8cb04683c1efea1b955af93485e7860d235 Mon Sep 17 00:00:00 2001 From: Nao Pross Date: Thu, 6 May 2021 00:04:07 +0200 Subject: Add link to slides --- vorlesungen/punktgruppen/slides.tex | 9 ++++++++- 1 file changed, 8 insertions(+), 1 deletion(-) (limited to 'vorlesungen/punktgruppen/slides.tex') diff --git a/vorlesungen/punktgruppen/slides.tex b/vorlesungen/punktgruppen/slides.tex index 3aa6d83..da13003 100644 --- a/vorlesungen/punktgruppen/slides.tex +++ b/vorlesungen/punktgruppen/slides.tex @@ -61,7 +61,14 @@ % Slides \begin{document} -\frame{\titlepage} +\frame{ + \titlepage + \vfill + \begin{center} + \small \color{gray} + Slides: \texttt{s.0hm.ch/ctBsD} + \end{center} +} \frame{\tableofcontents} \frame{ -- cgit v1.2.1 From b879ffd58aa2c74b8f7b199edfa4b108b88c6e0b Mon Sep 17 00:00:00 2001 From: Nao Pross Date: Fri, 7 May 2021 00:12:11 +0200 Subject: Create slide to show all point groups --- vorlesungen/punktgruppen/slides.tex | 265 +++++++++++++++++++++++++++++++++++- 1 file changed, 263 insertions(+), 2 deletions(-) (limited to 'vorlesungen/punktgruppen/slides.tex') diff --git a/vorlesungen/punktgruppen/slides.tex b/vorlesungen/punktgruppen/slides.tex index da13003..08b6ecb 100644 --- a/vorlesungen/punktgruppen/slides.tex +++ b/vorlesungen/punktgruppen/slides.tex @@ -1,4 +1,4 @@ -\documentclass[12pt, xcolor, aspectratio=169]{beamer} +\documentclass[12pt, xcolor, aspectratio=169, handout]{beamer} % language \usepackage{polyglossia} @@ -8,8 +8,16 @@ \usepackage{tikz} \usetikzlibrary{positioning} \usetikzlibrary{arrows.meta} +\usetikzlibrary{shapes.misc} \usetikzlibrary{calc} +\usetikzlibrary{external} +\tikzexternalize[ + mode = graphics if exists, + figure list = true, + prefix=build/ +] + % Theme \beamertemplatenavigationsymbolsempty @@ -399,13 +407,266 @@ 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\} + \alpha &\in \left\{ 0, 60^\circ, 90^\circ, 120^\circ, 180^\circ \right\} \\ + n &\in \left\{ 1, 2, 3, 4, 6 \right\} \end{align*} } \end{column} \end{columns} \end{frame} +{ + \usebackgroundtemplate[fragile]{ + \begin{tikzpicture}[ + overlay, + xshift = .45\paperwidth, + yshift = .47\paperheight, + classcirc/.style = { + draw = gray, thick, circle, + minimum size = 12mm, + inner sep = 0pt, outer sep = 0pt, + }, + classlabel/.style = { + below right = 5mm + }, + round/.style = { + draw = yellow, thick, circle, + minimum size = 1mm, + inner sep = 0pt, outer sep = 0pt, + }, + cross/.style = { + cross out, draw = magenta, thick, + minimum size = 1mm, + inner sep = 0pt, outer sep = 0pt + }, + ] + \matrix [row sep = 3mm, column sep = 0mm] { + \node[classcirc] (C1) {} node[classlabel] {\(C_{1}\)}; & + \node[classcirc] (C2) {} node[classlabel] {\(C_{2}\)}; & + \node[classcirc] (C3) {} node[classlabel] {\(C_{3}\)}; & + \node[classcirc] (Ci) {} node[classlabel] {\(C_{i}\)}; & + + \node[classcirc] (Cs) {} node[classlabel] {\(C_{s}\)}; & + \node[classcirc] (C3i) {} node[classlabel] {\(C_{3i}\)}; & + \node[classcirc] (C2h) {} node[classlabel] {\(C_{2h}\)}; & + \node[classcirc] (D2) {} node[classlabel] {\(D_{2}\)}; \\ + + \node[classcirc] (D3d) {} node[classlabel] {\(D_{3d}\)}; & + \node[classcirc] (C2v) {} node[classlabel] {\(C_{2v}\)}; & + \node[classcirc] (D2h) {} node[classlabel] {\(D_{2h}\)}; & + \node[classcirc] (D3) {} node[classlabel] {\(D_{3}\)}; & + + \node[classcirc] (C4) {} node[classlabel] {\(C_{4}\)}; & + \node[classcirc] (C6) {} node[classlabel] {\(C_{6}\)}; & + \node[classcirc] (D3dP) {} node[classlabel] {\(D_{3d}\)}; & + \node[classcirc] (S4) {} node[classlabel] {\(S_{4}\)}; \\ + + \node[classcirc] (S3) {} node[classlabel] {\(S_{3}\)}; & + \node[classcirc, dashed] (T) {} node[classlabel] {\(T_{}\)}; & + \node[classcirc] (C4h) {} node[classlabel] {\(C_{4h}\)}; & + \node[classcirc] (C6h) {} node[classlabel] {\(C_{6h}\)}; & + + \node[classcirc, dashed] (Th) {} node[classlabel] {\(T_{h}\)}; & + \node[classcirc] (C4v) {} node[classlabel] {\(C_{4v}\)}; & + \node[classcirc] (C6v) {} node[classlabel] {\(C_{6v}\)}; & + \node[classcirc, dashed] (Td) {} node[classlabel] {\(T_{d}\)}; \\ + + \node[classcirc] (D2d) {} node[classlabel] {\(D_{2d}\)}; & + \node[classcirc] (D3h) {} node[classlabel] {\(D_{3h}\)}; & + \node[classcirc, dashed] (O) {} node[classlabel] {\(O_{}\)}; & + \node[classcirc] (D4) {} node[classlabel] {\(D_{4}\)}; & + + \node[classcirc] (D6) {} node[classlabel] {\(D_{6}\)}; & + \node[classcirc, dashed] (Oh) {} node[classlabel] {\(O_{h}\)}; & + \node[classcirc] (D4h) {} node[classlabel] {\(D_{4h}\)}; & + \node[classcirc] (D6h) {} node[classlabel] {\(D_{6h}\)}; \\ + }; + + + \node[cross] at ($(C1)+(4mm,0)$) {}; + + + \node[cross] at ($(C2)+(4mm,0)$) {}; + \node[cross] at ($(C2)-(4mm,0)$) {}; + + + \node[cross] at ($(C3)+( 0:4mm)$) {}; + \node[cross] at ($(C3)+(120:4mm)$) {}; + \node[cross] at ($(C3)+(240:4mm)$) {}; + + + \node[cross] at ($(Ci)+(4mm,0)$) {}; + \node[round] at ($(Ci)-(4mm,0)$) {}; + + + \node[cross] at ($(Cs)+(4mm,0)$) {}; + \node[round] at ($(Cs)+(4mm,0)$) {}; + + + \node[cross] at ($(C3i)+( 0:4mm)$) {}; + \node[cross] at ($(C3i)+(120:4mm)$) {}; + \node[cross] at ($(C3i)+(240:4mm)$) {}; + \node[round] at ($(C3i)+( 60:4mm)$) {}; + \node[round] at ($(C3i)+(180:4mm)$) {}; + \node[round] at ($(C3i)+(300:4mm)$) {}; + + + \node[cross] at ($(C2h)+(4mm,0)$) {}; + \node[cross] at ($(C2h)-(4mm,0)$) {}; + \node[round] at ($(C2h)+(4mm,0)$) {}; + \node[round] at ($(C2h)-(4mm,0)$) {}; + + + \node[cross] at ($(D2)+( 20:4mm)$) {}; + \node[cross] at ($(D2)+(200:4mm)$) {}; + \node[round] at ($(D2)+(160:4mm)$) {}; + \node[round] at ($(D2)+(340:4mm)$) {}; + + + \foreach \x in {0, 120, 240} { + \node[cross] at ($(D3d)+({\x+15}:4mm)$) {}; + \node[cross] at ($(D3d)+({\x-15}:4mm)$) {}; + } + + + \foreach \x in {0, 180} { + \node[cross] at ($(C2v)+({\x+15}:4mm)$) {}; + \node[cross] at ($(C2v)+({\x-15}:4mm)$) {}; + } + + + \foreach \x in {0, 180} { + \node[cross] at ($(D2h)+({\x+15}:4mm)$) {}; + \node[cross] at ($(D2h)+({\x-15}:4mm)$) {}; + \node[round] at ($(D2h)+({\x+15}:4mm)$) {}; + \node[round] at ($(D2h)+({\x-15}:4mm)$) {}; + } + + + \foreach \x in {0, 120, 240} { + \node[cross] at ($(D3)+({\x+15}:4mm)$) {}; + \node[round] at ($(D3)+({\x-15}:4mm)$) {}; + } + + + \foreach \x in {0, 90, 180, 270} { + \node[cross] at ($(C4)+(\x:4mm)$) {}; + } + + + \foreach \x in {0, 60, 120, 180, 240, 300} { + \node[cross] at ($(C6)+(\x:4mm)$) {}; + } + + + \foreach \x in {0, 120, 240} { + \node[cross] at ($(D3dP)+({\x+15}:4mm)$) {}; + \node[cross] at ($(D3dP)+({\x-15}:4mm)$) {}; + \node[round] at ($(D3dP)+({\x+15+60}:4mm)$) {}; + \node[round] at ($(D3dP)+({\x-15+60}:4mm)$) {}; + } + + + \node[cross] at ($(S4)+(4mm,0)$) {}; + \node[cross] at ($(S4)-(4mm,0)$) {}; + \node[round] at ($(S4)+(0,4mm)$) {}; + \node[round] at ($(S4)-(0,4mm)$) {}; + + + \foreach \x in {0, 120, 240} { + \node[cross] at ($(S3)+(\x:4mm)$) {}; + \node[round] at ($(S3)+(\x:4mm)$) {}; + } + + + %% TODO: T + + + \foreach \x in {0, 90, 180, 270} { + \node[cross] at ($(C4h)+(\x:4mm)$) {}; + \node[round] at ($(C4h)+(\x:4mm)$) {}; + } + + + \foreach \x in {0, 60, 120, 180, 240, 300} { + \node[cross] at ($(C6h)+(\x:4mm)$) {}; + \node[round] at ($(C6h)+(\x:4mm)$) {}; + } + + + %% TODO: Th + + + \foreach \x in {0, 90, 180, 270} { + \node[cross] at ($(C4v)+(\x+15:4mm)$) {}; + \node[cross] at ($(C4v)+(\x-15:4mm)$) {}; + } + + + + \foreach \x in {0, 60, 120, 180, 240, 300} { + \node[cross] at ($(C6v)+(\x+10:4mm)$) {}; + \node[cross] at ($(C6v)+(\x-10:4mm)$) {}; + } + + + %% TODO: Td + + + \foreach \x in {0, 180} { + \node[cross] at ($(D2d)+({\x+15}:4mm)$) {}; + \node[round] at ($(D2d)+({\x-15}:4mm)$) {}; + + \node[round] at ($(D2d)+({\x+15+90}:4mm)$) {}; + \node[cross] at ($(D2d)+({\x-15+90}:4mm)$) {}; + } + + + \foreach \x in {0, 120, 240} { + \node[cross] at ($(D3h)+({\x+15}:4mm)$) {}; + \node[cross] at ($(D3h)+({\x-15}:4mm)$) {}; + \node[round] at ($(D3h)+({\x+15}:4mm)$) {}; + \node[round] at ($(D3h)+({\x-15}:4mm)$) {}; + } + + + %% TODO: O + + + \foreach \x in {0, 90, 180, 270} { + \node[cross] at ($(D4)+({\x+15}:4mm)$) {}; + \node[round] at ($(D4)+({\x-15}:4mm)$) {}; + } + + \foreach \x in {0, 60, 120, 180, 240, 300} { + \node[cross] at ($(D6)+({\x+10}:4mm)$) {}; + \node[round] at ($(D6)+({\x-10}:4mm)$) {}; + } + + + % TODO Oh + + + \foreach \x in {0, 90, 180, 270} { + \node[cross] at ($(D4h)+(\x+15:4mm)$) {}; + \node[cross] at ($(D4h)+(\x-15:4mm)$) {}; + \node[round] at ($(D4h)+(\x+15:4mm)$) {}; + \node[round] at ($(D4h)+(\x-15:4mm)$) {}; + } + + + \foreach \x in {0, 60, 120, 180, 240, 300} { + \node[cross] at ($(D6h)+({\x+10}:4mm)$) {}; + \node[cross] at ($(D6h)+({\x-10}:4mm)$) {}; + \node[round] at ($(D6h)+({\x+10}:4mm)$) {}; + \node[round] at ($(D6h)+({\x-10}:4mm)$) {}; + } + \end{tikzpicture} + } + \begin{frame}[fragile]{} + \end{frame} +} + \section{Anwendungen} \begin{frame}[fragile]{} \centering -- cgit v1.2.1 From 84f60f0f078481b05f1305ceff9565f2e7dd7471 Mon Sep 17 00:00:00 2001 From: Nao Pross Date: Fri, 7 May 2021 17:54:43 +0200 Subject: Striptease --- vorlesungen/punktgruppen/slides.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) (limited to 'vorlesungen/punktgruppen/slides.tex') diff --git a/vorlesungen/punktgruppen/slides.tex b/vorlesungen/punktgruppen/slides.tex index 08b6ecb..dbcf40f 100644 --- a/vorlesungen/punktgruppen/slides.tex +++ b/vorlesungen/punktgruppen/slides.tex @@ -1,4 +1,4 @@ -\documentclass[12pt, xcolor, aspectratio=169, handout]{beamer} +\documentclass[12pt, xcolor, aspectratio=169]{beamer} % language \usepackage{polyglossia} -- cgit v1.2.1 From f751b6fa47c197216ef5d3faa617dbf9be4dd695 Mon Sep 17 00:00:00 2001 From: Nao Pross Date: Sat, 8 May 2021 01:09:30 +0200 Subject: Minor changes to slides --- vorlesungen/punktgruppen/slides.tex | 156 +++++++++++++++++++----------------- 1 file changed, 84 insertions(+), 72 deletions(-) (limited to 'vorlesungen/punktgruppen/slides.tex') diff --git a/vorlesungen/punktgruppen/slides.tex b/vorlesungen/punktgruppen/slides.tex index dbcf40f..01d59f4 100644 --- a/vorlesungen/punktgruppen/slides.tex +++ b/vorlesungen/punktgruppen/slides.tex @@ -259,73 +259,59 @@ } \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} - \begin{column}{.5\textwidth} - \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 (-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) {}; + \onslide<1->{ + \begin{column}{.5\textwidth} + \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 (-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] (-2, -2) rectangle (3,4); - - \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} - \pause + \end{scope} + \draw[white, thick] (-2, -2) rectangle (3,4); + + \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} - Kristallgitter: - \(n_i \in \mathbb{Z}\), - \(\vec{a}_i \in \mathbb{R}^3\) - \[ - \vec{r} = n_1 \vec{a}_1 + n_2 \vec{a}_2 + n_3 \vec{a}_3 - \] + \onslide<2->{ + Kristallgitter: + \(n_i \in \mathbb{Z}\), + } + \onslide<3->{ + \(\vec{a}_i \in \mathbb{R}^3\) + } + \onslide<2->{ + \[ + \vec{r} = n_1 \vec{a}_1 + n_2 \vec{a}_2 \onslide<3->{+ n_3 \vec{a}_3} + \] + } \vspace{1cm} - \pause - - Invariant unter Translation - \[ - Q_i(\vec{r}) = \vec{r} + \vec{a}_i - \] + + \onslide<4->{ + Invariant unter Translation + \[ + Q_i(\vec{r}) = \vec{r} + \vec{a}_i + \] + } \end{column} \end{columns} \end{frame} @@ -383,23 +369,21 @@ (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); - } - - \onslide<8->{ - \node[above left, xshift=-2mm] at (X) {\(x\)}; + \onslide<10->{ + \draw[gray, dashed, thick] (A1) to (A1 |- B1) node (Xl) {}; + \draw[gray, dashed, thick] (A2) to (A2 |- B2) node (Xr) {}; + \node[above left, xshift=-2mm] at (Xl) {\(x\)}; + \node[above right, xshift= 2mm] at (Xr) {\(x\)}; } \end{tikzpicture} \end{center} \end{column} \begin{column}{.5\textwidth} - \onslide<9->{ + \onslide<7->{ Sei \(q = |\vec{Q}|\), \(\alpha = 2\pi/n\) und \(n \in \mathbb{N}\) } \begin{align*} - \onslide<10->{q' = n q &= q + 2x \\} + \onslide<9->{q' = n q \onslide<10->{&= q + 2x \\}} \onslide<11->{nq &= q + 2q\sin(\alpha - \pi/2) \\} \onslide<12->{n &= 1 - 2\cos\alpha} \end{align*} @@ -415,6 +399,34 @@ \end{columns} \end{frame} +\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, + outer sep = 1mm] (A) {}; + \node[white, yshift = 2mm] at (A) {230}; + \node[white, font=\large, above right = of A] (Al) {Raumgruppe}; + \draw[white, thick, ->] (Al.west) to[out=180, in=60] (A); + + \node[circle, draw = white, thick, + fill = red!20!background, + xshift = -5mm, yshift = -5mm, + minimum size = 1cm, + outer sep = 1mm] (B) {32}; + \node[white, font=\large, below left = of B, xshift=-4mm] (Bl) {Kristallklassen}; + \draw[white, thick, ->] (Bl.east) to[out = 0, in = 180] (B); + \end{tikzpicture} + \end{center} +\end{frame} + { \usebackgroundtemplate[fragile]{ \begin{tikzpicture}[ @@ -678,7 +690,7 @@ ] \matrix [nodes = {box, align = center}, column sep = 1cm, row sep = 1.5cm] { - & \node (A) {32 Punktgruppen}; \\ + & \node (A) {32 Kristallklassen}; \\ \node (B) {11 Mit\\ Inversionszentrum}; & \node (C) {21 Ohne\\ Inversionszentrum}; \\ & \node[fill=red!20!background] (D) {20 Piezoelektrisch}; & \node (E) {1 Nicht\\ piezoelektrisch}; \\ }; -- cgit v1.2.1 From f200f1bdb1b14bfcd9c3056d6d0a65f11a7a5ecc Mon Sep 17 00:00:00 2001 From: Nao Pross Date: Sat, 8 May 2021 14:27:54 +0200 Subject: Little fix --- vorlesungen/punktgruppen/slides.tex | 49 +++++++++++++++++++------------------ 1 file changed, 25 insertions(+), 24 deletions(-) (limited to 'vorlesungen/punktgruppen/slides.tex') diff --git a/vorlesungen/punktgruppen/slides.tex b/vorlesungen/punktgruppen/slides.tex index 01d59f4..ddc7ed1 100644 --- a/vorlesungen/punktgruppen/slides.tex +++ b/vorlesungen/punktgruppen/slides.tex @@ -715,7 +715,6 @@ ] \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} @@ -728,35 +727,18 @@ \end{scope} \pause - \begin{scope}[yshift=-4.5cm] - \matrix[nodes = { charge }, row sep = 5mm, column sep = 1cm] { - \node[positive] (NW) {}; & \node[negative] (N) {}; & \node [positive] (NE) {}; \\ - \node[negative] (W) {}; & \node[positive] {}; & \node [negative] (E) {}; \\ - \node[positive] (SW) {}; & \node[negative] (S) {}; & \node [positive] (SE) {}; \\ - }; - - \foreach \d in {NW, N, NE} { - \draw[orange, very thick, <-] (\d) to ++(0,.7); - } - - \foreach \d in {SW, S, SE} { - \draw[orange, very thick, <-] (\d) to ++(0,-.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} { \node[charge, \t] (C\i) at (\x:1.5cm) {}; } \draw[white] (C1) to (C2) to (C3) to (C4) to (C5) to (C6) to (C1); + \node[circle, draw=gray, fill=gray, outer sep = 0, inner sep = 0, minimum size = 3mm] {}; % \draw[gray, dashed] (C2) to (C4) to (C6) to (C2); \end{scope} \pause + \node[below = of title] {Polarisation Feld \(\vec{E}_p\)}; \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) {}; @@ -776,8 +758,27 @@ \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); + % \draw[gray, thick, dotted] (E) to ++(0,2); + % \draw[gray, thick, dotted] (E) to ++(0,-2); + \end{scope} + \pause + + \begin{scope}[yshift=-4.5cm] + \matrix[nodes = { charge }, row sep = 5mm, column sep = 1cm] { + \node[positive] (NW) {}; & \node[negative] (N) {}; & \node [positive] (NE) {}; \\ + \node[negative] (W) {}; & \node[positive] {}; & \node [negative] (E) {}; \\ + \node[positive] (SW) {}; & \node[negative] (S) {}; & \node [positive] (SE) {}; \\ + }; + + \foreach \d in {NW, N, NE} { + \draw[orange, very thick, <-] (\d) to ++(0,.7); + } + + \foreach \d in {SW, S, SE} { + \draw[orange, very thick, <-] (\d) to ++(0,-.7); + } + + \draw[gray, dashed] (W) to (N) to (E) to (S) to (W); \end{scope} \pause @@ -805,8 +806,8 @@ \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); + % \draw[gray, thick, dotted] (E) to ++(0,2); + % \draw[gray, thick, dotted] (E) to ++(0,-2); \end{scope} \end{tikzpicture} \end{frame} -- cgit v1.2.1 From 5ea80792f7fea3d05b62239cb5d6220fb43b3d41 Mon Sep 17 00:00:00 2001 From: Nao Pross Date: Sat, 8 May 2021 14:58:28 +0200 Subject: Piezoelectricity is tricky --- vorlesungen/punktgruppen/slides.tex | 81 ++++++++++++++++++++----------------- 1 file changed, 43 insertions(+), 38 deletions(-) (limited to 'vorlesungen/punktgruppen/slides.tex') diff --git a/vorlesungen/punktgruppen/slides.tex b/vorlesungen/punktgruppen/slides.tex index ddc7ed1..e15dd9a 100644 --- a/vorlesungen/punktgruppen/slides.tex +++ b/vorlesungen/punktgruppen/slides.tex @@ -738,20 +738,28 @@ \end{scope} \pause + %% \node[below = of title] {Polarisation Feld \(\vec{E}_p\)}; + + %% hex with vertical pressure \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) {}; + \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] @@ -761,8 +769,9 @@ % \draw[gray, thick, dotted] (E) to ++(0,2); % \draw[gray, thick, dotted] (E) to ++(0,-2); \end{scope} - \pause + \pause + %% square with vertical pressure \begin{scope}[yshift=-4.5cm] \matrix[nodes = { charge }, row sep = 5mm, column sep = 1cm] { \node[positive] (NW) {}; & \node[negative] (N) {}; & \node [positive] (NE) {}; \\ @@ -782,33 +791,32 @@ \end{scope} \pause + %% hex with horizontal pressure \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) {}; + \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] - \node[red!50, right = of E] {\(+\)}; - \node[blue!50, left = of E] {\(-\)}; + \node[blue!50, right = of E] {\(-\)}; + \node[red!50, left = of E] {\(+\)}; \end{scope} % \draw[gray, thick, dotted] (E) to ++(0,2); % \draw[gray, thick, dotted] (E) to ++(0,-2); \end{scope} + \pause + + \end{tikzpicture} \end{frame} @@ -816,22 +824,11 @@ \frametitle{Licht in Kristallen} \begin{columns}[T] \begin{column}{.5\textwidth} - Symmetriegruppe und Darstellung - \begin{align*} - G &= \left\{\mathbb{1}, r, \sigma, \dots \right\} \\ - &\Phi : G \to O(n) - \end{align*} - \begin{align*} - U_\lambda &= \left\{ v : \Phi v = \lambda v \right\} \\ - &= \mathrm{null}\left(\Phi - \lambda I\right) - \end{align*} Helmholtz Wellengleichung \[ \nabla^2 \vec{E} = \ten{\varepsilon}\mu \frac{\partial^2}{\partial t^2} \vec{E} \] - \end{column} - \begin{column}{.5\textwidth} Ebene Welle \[ \vec{E} = \vec{E}_0 \exp\left[i @@ -841,9 +838,17 @@ \[ (\ten{K}\ten{\varepsilon})\vec{E} = \frac{k^2}{\mu \omega^2} \vec{E} \] - \[ - \vec{E} \in U_\lambda \implies (\ten{K}\ten{\varepsilon}) \vec{E} = \lambda \vec{E} - \] + \end{column} + \begin{column}{.5\textwidth} + Symmetriegruppe und Darstellung + \begin{align*} + G &= \left\{\mathbb{1}, r, \sigma, \dots \right\} \\ + &\Phi : G \to O(n) + \end{align*} + \begin{align*} + U_\lambda &= \left\{ v : \Phi v = \lambda v \right\} \\ + &= \mathrm{null}\left(\Phi - \lambda I\right) + \end{align*} \end{column} \end{columns} } -- cgit v1.2.1 From 941187a21b3ff980e2dc742e13108f066822cdfa Mon Sep 17 00:00:00 2001 From: tim30b Date: Sun, 9 May 2021 17:30:54 +0200 Subject: git will das ich commite --- vorlesungen/punktgruppen/slides.tex | 91 ++++++++++++++++++++++++++----------- 1 file changed, 65 insertions(+), 26 deletions(-) (limited to 'vorlesungen/punktgruppen/slides.tex') diff --git a/vorlesungen/punktgruppen/slides.tex b/vorlesungen/punktgruppen/slides.tex index e15dd9a..5e12fa9 100644 --- a/vorlesungen/punktgruppen/slides.tex +++ b/vorlesungen/punktgruppen/slides.tex @@ -6,6 +6,8 @@ % pretty drawings \usepackage{tikz} +\usepackage{tikz-3dplot} + \usetikzlibrary{positioning} \usetikzlibrary{arrows.meta} \usetikzlibrary{shapes.misc} @@ -83,7 +85,7 @@ \begin{itemize} \item Was heisst \emph{Symmetrie} in der Mathematik? \pause \item Wie kann ein Kristall modelliert werden? \pause - \item Aus der Physik: Piezoelektrizit\"at \pause + \item Aus der Physik: Licht, Piezoelektrizit\"at \pause \end{itemize} \begin{center} \begin{tikzpicture} @@ -823,34 +825,71 @@ \frame{ \frametitle{Licht in Kristallen} \begin{columns}[T] - \begin{column}{.5\textwidth} - Helmholtz Wellengleichung - \[ - \nabla^2 \vec{E} = \ten{\varepsilon}\mu - \frac{\partial^2}{\partial t^2} \vec{E} - \] - Ebene Welle - \[ - \vec{E} = \vec{E}_0 \exp\left[i - \left(\vec{k}\cdot\vec{r} - \omega t \right)\right] - \] - Anisotropisch Dielektrikum - \[ - (\ten{K}\ten{\varepsilon})\vec{E} = \frac{k^2}{\mu \omega^2} \vec{E} - \] + \begin{column}{.45\textwidth} + \onslide<2->{ + Helmholtz Wellengleichung + \[ + \nabla^2 \vec{E} = \ten{\varepsilon}\mu + \frac{\partial^2}{\partial t^2} \vec{E} + \] + } + \onslide<3->{ + Ebene Welle + \[ + \vec{E} = \vec{E}_0 \exp\left[i + \left(\vec{k}\cdot\vec{r} - \omega t \right)\right] + \] + } + \onslide<4->{ + Anisotropisch Dielektrikum + \[ + (\ten{K}\ten{\varepsilon})\vec{E} + = \frac{k^2}{\mu \omega^2} \vec{E} + \implies + \Phi \vec{E} = \lambda \vec{E} + \] + } \end{column} - \begin{column}{.5\textwidth} - Symmetriegruppe und Darstellung - \begin{align*} - G &= \left\{\mathbb{1}, r, \sigma, \dots \right\} \\ - &\Phi : G \to O(n) - \end{align*} - \begin{align*} - U_\lambda &= \left\{ v : \Phi v = \lambda v \right\} \\ - &= \mathrm{null}\left(\Phi - \lambda I\right) - \end{align*} + \begin{column}{.55\textwidth} + \onslide<5->{ + Eingenraum + \begin{align*} + U_\lambda &= \left\{ v : \Phi v = \lambda v \right\} + = \mathrm{null}\left(\Phi - \lambda I\right) + \end{align*} + }\onslide<6->{ + Symmetriegruppe und Darstellung + \begin{align*} + G &= \left\{\mathbb{1}, r, \sigma, \dots \right\} \\ + &\Phi : G \to O(n) + \end{align*} + }\onslide<7->{ + Kann man \(U_\lambda\) von \(G\) herauslesen? + \only<7>{ + \[ + U_\lambda \stackrel{?}{=} f\left(\bigoplus_{g \in G} \Phi_g\right) + \] + }\only<8>{ + \begin{align*} + \mathrm{Tr}\left[\Phi_r(g)\right] + &= \sum_i n_i \mathrm{Tr}\left[\Psi_i(g)\right] \\ + |G| &= \sum_i\mathrm{Tr}\left[\Psi_i(\mathbb{1})\right] + \end{align*} + } + } \end{column} \end{columns} } +% \begin{frame}[fragile] +% \centering +% \tdplotsetmaincoords{70}{110} +% \begin{tikzpicture}[scale=2, tdplot_main_coords] +% \node[draw=white, thick, minimum size = 3cm, circle] {}; +% % \foreach \x in {0, 120, 240} { +% % } +% \end{tikzpicture} +% \end{frame} + + \end{document} -- cgit v1.2.1 From dbeaa2f31d51c098caf46b83ca286cbef358f996 Mon Sep 17 00:00:00 2001 From: Nao Pross Date: Mon, 10 May 2021 13:14:42 +0200 Subject: Recompile slides for handout --- vorlesungen/punktgruppen/slides.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) (limited to 'vorlesungen/punktgruppen/slides.tex') diff --git a/vorlesungen/punktgruppen/slides.tex b/vorlesungen/punktgruppen/slides.tex index 5e12fa9..cd3d7d7 100644 --- a/vorlesungen/punktgruppen/slides.tex +++ b/vorlesungen/punktgruppen/slides.tex @@ -1,4 +1,4 @@ -\documentclass[12pt, xcolor, aspectratio=169]{beamer} +\documentclass[12pt, xcolor, aspectratio=169, handout]{beamer} % language \usepackage{polyglossia} -- cgit v1.2.1