From 0cc117a7171009c2e42bad5998d85a7048c40476 Mon Sep 17 00:00:00 2001 From: Ralf Hielscher Date: Fri, 25 Sep 2015 09:08:48 +0200 Subject: [PATCH] reorganize --- latex/geometry/.gitignore | 9 + latex/geometry/CrystalGeometry.tex | 2636 +++++++++++++++++ latex/geometry/pic/.gitignore | 1 + latex/geometry/pic/2mm.pdf | Bin 0 -> 18415 bytes latex/geometry/pic/432.pdf | Bin 0 -> 40453 bytes latex/geometry/pic/43m.pdf | Bin 0 -> 35562 bytes latex/geometry/pic/HemPDF111.pdf | Bin 0 -> 28291 bytes latex/geometry/pic/MagPDF110.pdf | Bin 0 -> 21689 bytes latex/geometry/pic/MillerPlane.pdf | Bin 0 -> 14765 bytes latex/geometry/pic/MillerSymmetrised.pdf | Bin 0 -> 27203 bytes latex/geometry/pic/MillerUVW.pdf | Bin 0 -> 12870 bytes latex/geometry/pic/angleDistri.pdf | Bin 0 -> 7571 bytes latex/geometry/pic/annotationq.pdf | Bin 0 -> 155369 bytes latex/geometry/pic/annotationv.pdf | Bin 0 -> 162633 bytes latex/geometry/pic/axisDistri.pdf | Bin 0 -> 281326 bytes 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100644 latex/mtex.sty create mode 100644 scripts/ExercisesWithSolutions.m rename {15_brasil/matlab => scripts}/Forsterite.ctf (100%) rename {15_brasil/matlab => scripts}/Gaussian.m (100%) rename {15_brasil/matlab => scripts}/colorCodingExample.m (100%) rename {15_brasil/matlab => scripts}/ebsdExample.m (100%) rename {15_brasil/matlab => scripts}/exercises.m (100%) rename {15_brasil/matlab => scripts}/grainExample.m (100%) rename {15_brasil/matlab => scripts}/grainExercises.m (100%) rename {15_brasil/matlab => scripts}/misorientationExample.m (100%) rename {15_brasil/matlab => scripts}/odfExample.m (100%) rename {15_brasil/matlab => scripts}/pm-17_1.cpr (100%) rename {15_brasil/matlab => scripts}/pm-17_1.crc (100%) rename {15_brasil/matlab => scripts}/poleFigureExample.m (100%) rename {15_brasil/matlab => scripts}/randomPoints.m (100%) rename {15_brasil/matlab => scripts}/reduce.m (100%) rename {15_brasil/matlab => scripts}/tensorExample.m (100%) rename {15_brasil/matlab => scripts}/twinning.ctf (100%) rename {15_brasil/matlab => scripts}/twinningExample.m (100%) diff --git a/latex/geometry/.gitignore b/latex/geometry/.gitignore new file mode 100644 index 0000000..79f29c6 --- /dev/null +++ b/latex/geometry/.gitignore @@ -0,0 +1,9 @@ +*.aux +*.log +*.nav +*.out +/CrystalGeometry.pdf +*.snm +*.synctex.gz +*.toc +*.vrb diff --git a/latex/geometry/CrystalGeometry.tex b/latex/geometry/CrystalGeometry.tex new file mode 100644 index 0000000..8241f4a --- /dev/null +++ b/latex/geometry/CrystalGeometry.tex @@ -0,0 +1,2636 @@ +% presentation +\documentclass[compress]{beamer} + +% handout +%\documentclass[handout]{beamer} + +\usepackage{../mtex,fancyvrb,booktabs} +\usepackage{tikz} +%\usepackage{subfigure} +\usepackage[labelformat=empty]{subcaption} +\captionsetup{compatibility=false} + +\author{R. Hielscher} + +\title{Crystal Geometry} +%\subtitle{Chemnitz {\bf{\color{red}M}TEX} Workshop 2015} + +\institute{Faculty of Mathematics,\\ + Chemnitz University of Technology, Germany} + +\date{{\bf{\color{red}M}TEX} Workshop 2015} + + +\newcommand{\crystal}[5]{% + \draw[->, >=latex, #5] (#1) --+ (#2);% + \draw[->, >=latex, #5] (#1) --+ (#3);% + \draw[->, >=latex, #5] (#1) --+ (#4);% +} + +\newcommand{\fillcrystal}[5]{% + \filldraw[fill=red!25,draw=red!35,opacity=0.5] +% (0,0) + (1,0) -- (0,0) --++ (0,1) --+ (1,0); + (#1) +(#2) -- (#1) --++ (#3) -- +(#2) ;% + \filldraw[fill=red!45,draw=red!35,opacity=0.5] + (#1) +(#3) -- (#1) --++ (#4) -- +(#3) ;% + \filldraw[fill=red!65,draw=red!35,opacity=0.5] + (#1) +(#4) -- (#1) --++ (#2) -- +(#4) ;% + \draw[->, >=latex, #5] (#1) --+ (#2);% + \draw[->, >=latex, #5] (#1) --+ (#3);% + \draw[->, >=latex, #5] (#1) --+ (#4);% +} + +\newcommand{\filldirection}[5]{% + \filldraw[fill=red!25,draw=red!35,opacity=0.5] +% (0,0) + (1,0) -- (0,0) --++ (0,1) --+ (1,0); + (#1) +(#2) -- (#1) --++ (#3) -- +(#2) ;% + \filldraw[fill=red!45,draw=red!35,opacity=0.5] + (#1) +(#3) -- (#1) --++ (#4) -- +(#3) ;% + \filldraw[fill=red!65,draw=red!35,opacity=0.5] + (#1) +(#4) -- (#1) --++ (#2) -- +(#4) ;% + \draw[->, >=latex, #5] (#1) --+ (#2);% + \draw[->, >=latex, #5] (#1) --+ (#3);% + \draw[->, >=latex,color=red, very thick] (#1) --+ (#4);% +} + + +\begin{document} + +\begin{frame} + \maketitle{} +\end{frame} + +\begin{frame} + \frametitle{Table of Content} + +\tableofcontents{} + +\end{frame} + + +\section{Vectors} + +\subsection*{Vector3d} + +\begin{frame}[fragile] + \frametitle{Three dimensional vectors - The \MTEX Class \texttt{\bf vector3d}} + + \begin{overlayarea}{\textwidth}{8cm} + Three dimensional vectors are given by there coordinates with respect to a + orthogonal coordinate system $\vec X, \vec Y, \vec Z$ + \begin{equation*} + \vec r = x \cdot \vec X + y \cdot \vec Y + z \cdot \vec Z + \end{equation*} + + \pause + + For general vectors, \MTEX does not care about the coordinate system, but + works only with the coordinates. + + \begin{columns} + \begin{column}{8cm} + + \begin{lstlisting}[style=input] +r = vector3d(1,2,3) + \end{lstlisting} + \begin{onlyenv}<2> + \vspace{-0.3cm} +\begin{lstlisting}[style=output] +r = /+vector3d+/ (show methods, plot) + size: 1 x 1 + x y z + 1 2 3 +\end{lstlisting} + \end{onlyenv} + + + \pause + \medskip + + The alignment of the coordinate system is only important when plotting + data + \begin{onlyenv}<3|handout> +\begin{lstlisting}[style=input] +plotx2north, plotzOutOfPlane +plot(r) +\end{lstlisting} + \end{onlyenv} + \begin{onlyenv}<4> +\begin{lstlisting}[style=input] +plotx2east, plotzOutOfPlane +plot(r) +\end{lstlisting} + \end{onlyenv} + + \end{column} + \begin{column}{3.5cm} + \only<3|handout>{\includegraphics[width=3.5cm]{pic/vectorNorth}} + \only<4>{\includegraphics[width=3.5cm]{pic/vectorEast}} + \end{column} + \end{columns} + + \pause + \medskip + + Only for directions relative to the crystal coordinate system the reference + frame is considered. + +\end{overlayarea} +\end{frame} + +\subsection*{Definition} + +\begin{frame}[fragile,fragile,fragile] + \frametitle{Defining vectors} + + \begin{overlayarea}{\textwidth}{8cm} + + polar coordinates $\vec r = (\sin \theta \cos \rho,\sin \theta \sin \rho,\cos \theta)^{t}$ + +\begin{lstlisting}[style=input] +r = vector3d('theta',theta,'rho',rho) +\end{lstlisting} + + \pause \medskip + + predefined vectors +\begin{lstlisting}[style=input] +r = xvector, r = yvector, r = zvector +\end{lstlisting} + + \pause \medskip + + combine vectors +\begin{lstlisting}[style=input] +r = [xvector, yvector, vector3d(1,1,1)]; +\end{lstlisting} + +\begin{onlyenv}<3> + \vspace{-.3cm} +\begin{lstlisting}[style=output] +r = /+vector3d+/ (show methods, plot) + size: 1 x 3 + x y z + 1 0 0 + 0 1 0 + 1 1 1 +\end{lstlisting} +\end{onlyenv} + + \pause \medskip + + importing vectors +\begin{lstlisting}[style=input] +r = loadVector3d('file','ColumnNames',{'x','y','z'}) +\end{lstlisting} +\begin{onlyenv}<4> + \vspace{-.3cm} +\begin{lstlisting}[style=output] +r = /+vector3d+/ (show methods, plot) + size: 200 x 1 +\end{lstlisting} +\end{onlyenv} + +\end{overlayarea} +\end{frame} + +\subsection*{Calculations} + +\begin{frame}[fragile] + \frametitle{Vector Calculations} + + simple algebra +\begin{lstlisting}[style=input] +r = 2*xvector - yvector; +\end{lstlisting} + + \pause \medskip + + + basic operations +\begin{lstlisting}[style=input] +dot(v1,v2) % dot product +cross(v1,v2) % cross product +angle(v1,v2) % angle between two vectors +\end{lstlisting} + + \pause \medskip + + extract properties +\begin{lstlisting}[style=input] +r.theta % polar angle in radiant +r.rho % azimuth angle in radiant +r.x, r.y, r.z +\end{lstlisting} + +\end{frame} + + +\subsection*{Indexing} + +\begin{frame}[fragile] + \frametitle{Indexing of Vectors} + + \begin{overlayarea}{\textwidth}{8cm} + + consider a list of vectors + \begin{lstlisting}[style=input] +r = vector3d([0 0 1 1],[1 0 1 1],[1 1 1 0]); + \end{lstlisting} + + \begin{onlyenv}<1> + \vspace{-.3cm} + \begin{lstlisting}[style=output] +r = /+vector3d+/ (show methods, plot) + size: 1 x 4 + x y z + 0 1 1 + 0 0 1 + 1 1 1 + 1 1 0 + \end{lstlisting} + \end{onlyenv} + + \pause \medskip + + single out the second vector + \begin{lstlisting}[style=input] +r(2) + \end{lstlisting} + + \begin{onlyenv}<2> + \vspace{-0.3cm} + \begin{lstlisting}[style=output] +r = /+vector3d+/ (show methods, plot) + size: 1 x 1 + x y z + 0 0 1 + \end{lstlisting} + \end{onlyenv} + + \pause \medskip + + single out the second and the fourth vector + \begin{lstlisting}[style=input] +r([2 4]) + \end{lstlisting} + \begin{onlyenv}<3> + \vspace{-0.3cm} + \begin{lstlisting}[style=output] +r = /+vector3d+/ (show methods, plot) + size: 1 x 2 + x y z + 0 0 1 + 1 1 0 + \end{lstlisting} + \end{onlyenv} + + \pause \medskip + +single out vectors by a logical condition + \begin{lstlisting}[style=input] +r(r.x>0) + \end{lstlisting} + \begin{onlyenv}<4> + \vspace{-0.3cm} + \begin{lstlisting}[style=output] +r = /+vector3d+/ (show methods, plot) + size: 1 x 2 + x y z + 1 1 1 + 1 1 0 + \end{lstlisting} + \end{onlyenv} + + \pause \medskip + + \alert{The above techniques applies also to lists of + orientations, EBSD data, grains, boundary segments, etc.} + \end{overlayarea} +\end{frame} + + +\subsection*{Changing Vectors} + +\begin{frame}[fragile] + \frametitle{Changing Vectors} + + \begin{overlayarea}{\textwidth}{8cm} + + consider again the list of vectors + \begin{lstlisting}[style=input] +r = vector3d([0 0 1 1],[1 0 1 1],[1 1 1 0]); + \end{lstlisting} + + \begin{onlyenv}<1> + \vspace{-0.3cm} + \begin{lstlisting}[style=output] +r = /+vector3d+/ (show methods, plot) + size: 1 x 4 + x y z + 0 1 1 + 0 0 1 + 1 1 1 + 1 1 0 + \end{lstlisting} + \end{onlyenv} + + \pause \medskip + + replace the second vector by another vector + \begin{lstlisting}[style=input] +r(2) = yvector + \end{lstlisting} + + \begin{onlyenv}<2> + \vspace{-0.3cm} + \begin{lstlisting}[style=output] +r = /+vector3d+/ (show methods, plot) + size: 1 x 4 + x y z + 0 1 1 + 0 1 0 + 1 1 1 + 1 1 0 + \end{lstlisting} + \end{onlyenv} + + \pause \medskip + + remove the second vector completely + \begin{lstlisting}[style=input] +r(2) = [] + \end{lstlisting} + \begin{onlyenv}<3> + \vspace{-0.3cm} + \begin{lstlisting}[style=output] +r = /+vector3d+/ (show methods, plot) + size: 1 x 3 + 0 1 1 + 1 1 1 + 1 1 0 + \end{lstlisting} + \end{onlyenv} + + \pause \medskip + + change the x coordinate of all vectors + \begin{lstlisting}[style=input] +r.x = 0 + \end{lstlisting} + \begin{onlyenv}<4> + \vspace{-0.3cm} + \begin{lstlisting}[style=output] +r = /+vector3d+/ (show methods, plot) + size: 1 x 3 + 0 1 1 + 0 1 1 + 0 1 0 + \end{lstlisting} + \end{onlyenv} + + \pause \medskip + + \alert{The above techniques applies also to pole figure data, + orientations, EBSD data, grains, etc.} + \end{overlayarea} + + +\end{frame} + + +\subsection*{Plotting} +\label{sec:plotting} + +\begin{frame}[fragile] + \frametitle{Plotting Vectors} + + \begin{columns} + \begin{column}{8.5cm} + \begin{overlayarea}{\textwidth}{8cm} + + spherical projections: \texttt{earea, edist, eangle} + hemisphere: \texttt{upper, lower} + + \begin{onlyenv}<1> + \begin{lstlisting}[style=input] +plot(r,'projection','eangle','upper') +\end{lstlisting} + \end{onlyenv} + + \pause + + \begin{onlyenv}<2-> + \begin{lstlisting}[style=input] +plot(r,'projection','earea','upper') +\end{lstlisting} + \end{onlyenv} + +\pause \medskip + +combined plots + +\begin{lstlisting}[style=input] +plot(vector3d(1,1,1),'upper'); +hold all +plot(vector3d(1,2,3),'label','B'); +plot(vector3d(-1,2,1),'label','A'); +hold off +\end{lstlisting} + + \pause \medskip + + data plots + + \begin{lstlisting}[style=input] +r = randv(100) % 100 random points +plot(r,1:100) % colored by number +quiver(r,orth(r)) % a vector field + \end{lstlisting} + + \end{overlayarea} + \end{column} + + \begin{column}{3.5cm} + + \begin{overlayarea}{\textwidth}{8cm} + \includegraphics<1-2>[width=3.5cm]{pic/vectoreangle} + + \includegraphics<2>[width=3.5cm]{pic/vectorearea} + + \includegraphics<3>[width=3.5cm]{pic/vectorCombined} + + \includegraphics<4>[width=3.5cm]{pic/vectorColor} + + \includegraphics<4>[width=3.5cm]{pic/quiver} + \end{overlayarea} + \end{column} + \end{columns} + +\end{frame} + +\subsection*{Annotations} + +\begin{frame}[fragile] + \frametitle{Customize Plots} + +\begin{columns} + \begin{column}{8.5cm} + +\begin{overprint} + \onslide<1|handout:0>% + General Syntax% +\begin{lstlisting}[style=input] +plot(vector3d,) +\end{lstlisting} + \onslide<2|handout:1>% + General Syntax +\begin{lstlisting}[style=input] +annotate(orientation,) +\end{lstlisting} +\end{overprint} + +Options +\begin{lstlisting}[style=input] +Marker % marker shape +MarkerSize % marker size +MarkerFaceColor % face color +MarkerEdgeColor % edge color +label % a label text +color, background % text colors +\end{lstlisting} + +\begin{overprint} + \onslide<1|handout:0>% + Example +\begin{lstlisting}[style=input] +plot([xvector,yvector,zvector], + 'Backgroundcolor','w','Marker','s', + 'MarkerEdgeColor','w','labeled', + 'MarkerFaceColor','k') +\end{lstlisting} + + \onslide<2|handout:1> +Example +\begin{lstlisting}[style=input] +plot(q0,'label','A',... + 'Backgroundcolor','w','Marker','s', + 'MarkerEdgeColor','w',... + 'MarkerFaceColor','r') +\end{lstlisting} + +\end{overprint} + +\end{column} + +\begin{column}{3.5cm} + \onslide<1-> + \only<1|handout:0>{% + \includegraphics[width=3.5cm]{pic/annotationv}% + }% + \only<2>{% + \includegraphics[width=3.3cm]{pic/annotationq}% + }% +\end{column} + +\end{columns} + +\end{frame} + +\subsection*{Axes} + +\begin{frame}[fragile] + \frametitle{Axes} + + \begin{columns} + \begin{column}{8.5cm} + \begin{overlayarea}{8.5cm}{8cm} + Axes are three dimensional vectors where we do not care about length and + direction, e.g. plane normals. + +\begin{lstlisting}[style=input] +r = vector3d(1,1,1,'antipodal') +\end{lstlisting} + +\medskip +\pause + +Then \textbf{\texttt{r}} and \textbf{\texttt{-r}} represent the same axis +\begin{lstlisting}[style=input] +eq(r, -r) +\end{lstlisting} +\begin{onlyenv}<2> + \vspace{-0.3cm} +\begin{lstlisting}[style=output] + 1 +\end{lstlisting} +\end{onlyenv} + +\medskip +\pause + +The angle to an axis is always less then $90^{\degree}$ +\begin{lstlisting}[style=input] +angle(r,-xvector) / degree +\end{lstlisting} +\begin{onlyenv}<3> + \vspace{-0.3cm} +\begin{lstlisting}[style=output] + 54.7 +\end{lstlisting} +\end{onlyenv} + +\medskip +\pause + +The option \textbf{antipodal} in a contour plot +\begin{onlyenv}<4> +\begin{lstlisting}[style=input] +r = randv(1000) +plot(r,'contourf') +\end{lstlisting} +\end{onlyenv} +\begin{onlyenv}<5> +\begin{lstlisting}[style=input] +r = randv(1000) +plot(r,'contourf','antipodal') +\end{lstlisting} +\end{onlyenv} +\end{overlayarea} +\end{column} + \begin{column}{3.5cm} + \only<1-3>{\includegraphics[width=3.5cm]{pic/vectorAntipodal}} + \only<4>{\includegraphics[width=3.5cm]{pic/vectorContour}} + \only<5>{\includegraphics[width=3.5cm]{pic/vectorContourAntipodal}} + \end{column} +\end{columns} + + +\end{frame} + +\section{Rotations} + +\subsection*{Basics} + +\begin{frame}[fragile] + \frametitle{Rotations} + + A rotation is a transformation that maps a right handed coordinate system + $(\vec X_{1}, \vec Y_{1}, \vec Z_{1})$ onto another right handed coordinate + system $(\vec X_{2}, \vec Y_{2}, \vec Z_{2})$. It is given by the rotation + matrix + + % \begin{equation*} + % \bf R = \left( + % \begin{matrix} + % \vec X_{1} \cdot \vec X_{2} & \vec Y_{1} \cdot \vec X_{2}& \vec Z_{1} \cdot \vec X_{2} \\ + % \vec X_{1} \cdot \vec Y_{2} & \vec Y_{1} \cdot \vec Y_{2}& \vec Z_{1} \cdot \vec Y_{2} \\ + % \vec X_{1} \cdot \vec Z_{2} & \vec Y_{1} \cdot \vec Z_{2}& \vec Z_{1} \cdot \vec Z_{2} + % \end{matrix} + % \right). + % \end{equation*} + \begin{equation*} + \mathbf R = (\vec X_{2}, \vec Y_{2}, \vec Z_{2}) \cdot (\vec X_{1}, \vec Y_{1}, \vec Z_{1})^{t} + \end{equation*} + + \medskip + + We have $\mathbf R \vec X_{1} = \vec X_{2}$, $\mathbf R \vec Y_{1} = \vec + Y_{2}$ and $\mathbf R \vec Z_{1} = \vec Z_{2}$. + + \medskip + \pause + + On the other hand, $\bf R$ transforms coordinates with respect to $(\vec + X_{2}, \vec Y_{2}, \vec Z_{2})$ into coordinates with respect to $(\vec + X_{1}, \vec Y_{1}, \vec Z_{1})$. I.e. for + + \begin{equation*} + \vec r + = x_{1} \vec X_{1} + y_{1} \vec Y_{1} + z_{1} \vec Z_{1} + = x_{2} \vec X_{2} + y_{2} \vec Y_{2} + z_{2} \vec Z_{2} + \end{equation*} + + we have + + \begin{equation*} + \mathbf R + \left( + \begin{matrix} + x_{2}\\ + y_{2}\\ + z_{2} + \end{matrix}\right) + = \left( + \begin{matrix} + x_{1}\\ + y_{1}\\ + z_{1} + \end{matrix}\right) + \end{equation*} + +\end{frame} + +\subsection*{Euler Angles} +\label{sec:euler-angles} + +\begin{frame}[fragile] + \frametitle{Euler Angles} + \begin{overlayarea}{\textwidth}{8cm} + + Most commonly, rotations are given by Bunge Euler angles. + + \begin{lstlisting}[style=input] +R = rotation('Euler',10*degree,20*degree,30*degree) + \end{lstlisting} + \begin{onlyenv}<1> + \vspace{-0.3cm} + \begin{lstlisting}[style=output] +R = /+rotation+/ (show methods, plot) + size: 1 x 1 + + Bunge Euler angles in degree + phi1 Phi phi2 Inv. + 10 20 30 0 + \end{lstlisting} + \end{onlyenv} + + \pause + \medskip + + \begin{lstlisting}[style=input] +R = rotation('Euler',... + 10*degree,20*degree,30*degree,'Roe') + \end{lstlisting} + \begin{onlyenv}<2> + \vspace{-0.3cm} + \begin{lstlisting}[style=output] +R = /+rotation+/ (show methods, plot) + size: 1 x 1 + + Bunge Euler angles in degree + phi1 Phi phi2 Inv. + 100 20 300 0 + \end{lstlisting} +\end{onlyenv} + +\pause +\medskip + +Supported conventions are \texttt{Bunge}, \texttt{Matthies}, \texttt{Roe}, +\texttt{Kocks}, \texttt{Canova}. + \begin{lstlisting}[style=input] +setMTEXpref('EulerAngleConvention','Roe') + \end{lstlisting} + + \begin{onlyenv}<3> + \vspace{-0.3cm} + \begin{lstlisting}[style=output] +R = /+rotation+/ (show methods, plot) + size: 1 x 1 + + Roe Euler angles in degree + Psi Theta Phi Inv. + 10 20 30 0 + \end{lstlisting} +\end{onlyenv} +\end{overlayarea} +\end{frame} + +\subsection*{Axis Angle Representation} +\label{sec:axis-angle-repr} + +\begin{frame}[fragile] + \frametitle{Other Ways to Define a Rotation} + + A rotation is uniquely defined by its rotation axis and its rotation angle + + \begin{lstlisting}[style=input] +R = rotation('axis',xvector,'angle',45*degree) + \end{lstlisting} + + \pause + \medskip + + Conversely, one can compute axis / angle from a rotation + + \begin{lstlisting}[style=input] +R.axis, R.angle + \end{lstlisting} + + \pause + \medskip + +Given four vectors $\vec u_{1}, \vec u_{2}, \vec v_{1}, \vec v_{2}$ there is a +unique rotation $\mathbf R$ such that $\mathbf R \vec u_{1} = \vec v_{1}$ and +$\mathbf R \vec u_{2} = \vec v_{2}$ + +\begin{lstlisting}[style=input] +R = rotation('map',u1,v1,u2,v2) +\end{lstlisting} + + \pause + \medskip + +Of course one can also define a rotation by its $3 \times 3$ matrix + + \begin{lstlisting}[style=input] +R = rotation('matrix',A) + \end{lstlisting} + +or by quaternions + + \begin{lstlisting}[style=input] +R = rotation('quaternion',a,b,c,d) + \end{lstlisting} + +\end{frame} + +\subsection*{Basic Calculations} +\label{sec:euler-angles} + +\begin{frame}[fragile] + \frametitle{Basic Calculations} + + rotate a vector + \begin{lstlisting}[style=input] +R = rotation('axis',xvector,'angle',-45*degree); +R * vector3d(0,1,1) + \end{lstlisting} + + + \begin{columns} + \begin{column}{8cm} + \begin{overlayarea}{8cm}{8cm} +\begin{onlyenv}<1> + \vspace{-0.3cm} +\begin{lstlisting}[style=output] +ans = /+vector3d+/ (show methods, plot) + size: 1 x 1 + x y z + 0 1.41421 0 + \end{lstlisting} +\end{onlyenv} + +\pause + \medskip + + the inverse rotation + \begin{lstlisting}[style=input] + inv(R) + \end{lstlisting} + \begin{onlyenv}<2> + \vspace{-0.3cm} + \begin{lstlisting}[style=output] +ans = /+rotation+/ (show methods, plot) + size: 1 x 1 + + Bunge Euler angles in degree + phi1 Phi phi2 Inv. + 0 45 0 0 + \end{lstlisting} + \end{onlyenv} + + \pause + \medskip + + combine rotations + \begin{lstlisting}[style=input] +R * inv(R) +\end{lstlisting} + +\begin{onlyenv}<3> + \vspace{-0.3cm} + \begin{lstlisting}[style=output] +ans = /+rotation+/ (show methods, plot) + size: 1 x 1 + + Bunge Euler angles in degree + phi1 Phi phi2 Inv. + 0 0 0 0 + \end{lstlisting} +\end{onlyenv} + +\pause +\medskip + +plotting +\begin{lstlisting}[style=input] +R = randq(100) +scatter(R) % Rodriguez space +\end{lstlisting} + +\end{overlayarea} +\end{column} + +\begin{column}{3.7cm} + \includegraphics[width=3.7cm]{pic/rotationScatter} +\end{column} + +\end{columns} + +\end{frame} + +\subsection*{Improper Rotations} + +\begin{frame}[fragile] + \frametitle{Improper Rotations} + + \begin{overlayarea}{\textwidth}{7cm} + + An improper rotation is a rotation followed by an inversion + \begin{lstlisting}[style=input] +I = -rotation('Euler',10*degree,20*degree,30*degree) + \end{lstlisting} + \begin{onlyenv}<1> + \vspace{-0.3cm} + \begin{lstlisting}[style=output] +I = /+rotation+/ (show methods, plot) + size: 1 x 1 + + Bunge Euler angles in degree + phi1 Phi phi2 Inv. + 10 20 30 1 + \end{lstlisting} + \end{onlyenv} + + \pause + \bigskip + + reflections + \begin{lstlisting}[style=input] +R = reflection(xvector+yvector) + \end{lstlisting} + \begin{onlyenv}<2> + \vspace{-0.3cm} + \begin{lstlisting}[style=output] +R = /+rotation+/ (show methods, plot) + size: 1 x 1 + + Bunge Euler angles in degree + phi1 Phi phi2 Inv. + 45 180 315 1 + \end{lstlisting} + \end{onlyenv} + + \pause + \bigskip + + angles between proper and improper rotations + \begin{lstlisting}[style=input] +angle(I,[R,-R]) + \end{lstlisting} + \begin{onlyenv}<3> + \vspace{-0.3cm} + \begin{lstlisting}[style=output] + 168.5677 180.0000 + \end{lstlisting} + \end{onlyenv} + + \pause + \bigskip + + check for improper rotations +\begin{lstlisting}[style=input] +I.isImproper + \end{lstlisting} + \begin{onlyenv}<4> + \vspace{-0.3cm} + \begin{lstlisting}[style=output] + 1 + \end{lstlisting} + \end{onlyenv} + +\end{overlayarea} +\end{frame} + +\section{Crystal Symmetries} +\label{sec:symmetries} + +\subsection*{Crystal Symmetries} +\label{sec:crystal-symmetries} + +\begin{frame}[fragile,fragile] + \frametitle{Crystal Symmetry} + + \begin{columns} + \begin{column}{8cm} + + \begin{overlayarea}{8cm}{8cm} + The \alert{point group} $\mathbf C$ of a crystal are all rotations + $\mathbf R$ that keep the crystal lattice invariant. + +\begin{lstlisting}[style=input] +CS = crystalSymmetry('m-3m') +\end{lstlisting} + \begin{onlyenv}<1> +\vspace{-.3cm}\begin{lstlisting}[style=output] +CS = /+crystalSymmetry+/ (show methods, plot) + + symmetry: m-3m + a, b, c : 1, 1, 1 +\end{lstlisting} + \end{onlyenv} + + \pause \medskip + + extract the rotations of a point group +\begin{lstlisting}[style=input] + rotation(crystalSymmetry('222')) +\end{lstlisting} + \begin{onlyenv}<2> +\vspace{-.3cm}\begin{lstlisting}[style=output] +ans = /+rotation+/ (show methods, plot) + size: 4 x 1 + + Bunge Euler angles in degree + phi1 Phi phi2 Inv. + 0 0 0 0 + 180 0 0 0 + 45 180 45 0 + 45 180 225 0 +\end{lstlisting} + \end{onlyenv} + + \pause \medskip + + import data from crystal information files +\begin{lstlisting}[style=input] +CS = loadCIF('Quarz.cif') +\end{lstlisting} + \begin{onlyenv}<3> +\vspace{-.3cm}\begin{lstlisting}[style=output] +CS = /+crystalSymmetry+/ (show methods, plot) + + mineral : Quartz + symmetry : P 32 2 1 (321) + a, b, c : 4.9, 4.9, 5.4 + alpha, beta, gamma: 90°, 90°, 120° + reference frame : X||a*, Y||b, Z||c* +\end{lstlisting} +\end{onlyenv} + + \pause + \vspace{-.3cm} + +\begin{lstlisting}[style=input] +CS = loadPHL('minerals.phl') +\end{lstlisting} + \begin{onlyenv}<4> +\vspace{-.3cm}\begin{lstlisting}[style=output] +CS{1} = /+crystalSymmetry+/ (show methods, plot) + + mineral : Magnetite + density : 5.054 + symmetry: m-3m + a, b, c : 8.4, 8.4, 8.4 +\end{lstlisting} +\end{onlyenv} + +\pause +\medskip + + switch to Laue / purely rotational group +\begin{lstlisting}[style=input] +CS.Laue +CS.properGroup +\end{lstlisting} + + \end{overlayarea} +\end{column} + \begin{column}{3.7cm} + \includegraphics<1->[width=3.7cm]{pic/m-3m} + + \medskip + + \includegraphics<3->[width=3.7cm]{pic/quartz} + + %\includegraphics<1-2>[width=3.7cm]{pic/m-3mHKL} + %\includegraphics<3>[width=3.7cm]{pic/quartzHKL} + \end{column} +\end{columns} +\end{frame} + +\subsection*{Laue and Proper Symmetry Groups} +\begin{frame}[fragile] + + \begin{figure}[H] + \centering + \begin{subfigure}{0.3\textwidth} + \includegraphics[width=\textwidth]{pic/432} + \subcaption{$432$} + \end{subfigure} + \begin{subfigure}{0.3\textwidth} + \includegraphics[width=\textwidth]{pic/43m} + \subcaption{$\bar 43m$} + \end{subfigure} + \begin{subfigure}{0.3\textwidth} + \includegraphics[width=\textwidth]{pic/m-3m} + \subcaption{$m\bar 3 m$} + \end{subfigure} + + \begin{subfigure}{0.3\textwidth} + \centering + \includegraphics[width=0.95\textwidth]{pic/2mm} + \caption{$2mm$} + \end{subfigure} + \begin{subfigure}{0.3\textwidth} + \centering + \includegraphics[width=0.95\textwidth]{pic/m2m} + \caption{$m2m$} + \end{subfigure} + \begin{subfigure}{0.3\textwidth} + \centering + \includegraphics[width=0.95\textwidth]{pic/mm2} + \caption{$mm2$} + \end{subfigure} + \end{figure} + +\end{frame} + +\subsection*{trigonal} + +\begin{frame} + + \begin{figure}[H] + \centering + \begin{subfigure}{0.3\textwidth} + \includegraphics[width=\textwidth]{pic/sym321} + \subcaption{$321$} + \end{subfigure} + \begin{subfigure}{0.3\textwidth} + \includegraphics[width=\textwidth]{pic/sym3m1} + \subcaption{$3m1$} + \end{subfigure} + \begin{subfigure}{0.3\textwidth} + \includegraphics[width=\textwidth]{pic/sym-3m1} + \caption{$\bar 3m1$} + \end{subfigure} + \begin{subfigure}{0.3\textwidth} + \includegraphics[width=\textwidth]{pic/sym312} + \caption{$312$} + \end{subfigure} + \begin{subfigure}{0.3\textwidth} + \includegraphics[width=\textwidth]{pic/sym31m} + \caption{$31m$} + \end{subfigure} + \begin{subfigure}{0.3\textwidth} + \includegraphics[width=\textwidth]{pic/sym-31m} + \caption{$\bar 31m$} + \end{subfigure} + \end{figure} + +\end{frame} + + +\subsection*{Crystal Symmetries} +\label{sec:crystal-symmetries} + +\begin{frame}[fragile] + \frametitle{Unit Cell, Reciprocal and Orthogonal Coordinate System} + + \begin{overlayarea}{\textwidth}{8cm} + The unit cell of a crystal is specified by the length of its three edges + $\vec a, \vec b, \vec c$ and by angles $\alpha, \beta, \gamma$ they enclose. + + \begin{lstlisting}[style=input] +C = crystalSymmetry('1',[a b c],[alpha beta gamma]) + \end{lstlisting} + + \begin{columns} + \begin{column}{7.5cm} + + \begin{uncoverenv}<2-> + The axes of the reciprocal lattice are defined + orthogonal to $\vec a, \vec b, \vec c$, i.e. +\vspace{-0.2cm} +\begin{equation*} + \vec a^{*} = \frac{\vec b \times \vec c}{V}, \; + \vec b^{*} = \frac{\vec c \times \vec a}{V}, \; + \vec c^{*} = \frac{\vec a \times \vec b}{V} +\end{equation*} + +\vspace{-0.4cm} + +with $V = \vec a \cdot (\vec b \times \vec c)$ volume of the unit cell + \end{uncoverenv} + + \medskip + + \begin{uncoverenv}<3-> + We will need also an orthogonal coordinate system $(\vec x, \vec y, + \vec z)$ fixed to the crystal. + \end{uncoverenv} + + \medskip + + \begin{uncoverenv}<4-> + \alert{There are different conventions.} + \end{uncoverenv} + + \end{column} + + \begin{column}{4.5cm} + \begin{tikzpicture}[x = {(-0.5cm,-0.5cm)}, + y = {(0.9659cm,-0.25882cm)}, + z = {(0cm,1cm)}, + scale = 2] + +\draw[draw=green, dashed] (0,0,0) -- (1.2,0,0) -- (1,1,0) -- (-0.2,1,0) -- +(0,1.2,1) -- (0.2,0.2,1) -- (1.4,0.2,1) -- (1.2,0,0); + +\draw[draw=green, dashed] (0,1.2,1) -- (1.2,1.2,1) -- (1,1,0); +\draw[draw=green, dashed] (1.2,1.2,1) -- (1.4,0.2,1); + + +\draw[draw=green,very thick] (0,0,0) -- (1.2,0,0) node [anchor=north,color=green!70!black] {$\vec a$}; +\draw[draw=green,very thick] (0,0,0) -- (-0.2,1,0) node [anchor=west,color=green!70!black] {$\vec b$}; +\draw[draw=green,very thick] (0,0,0) -- (0.2,0.2,1) node[anchor=south,color=green!70!black] {$\vec c$}; + +\only<1>{ +\draw[draw=green] (0.3,0,0) arc (250:329:0.3cm);% node {$\alpha$}; +\draw[draw=green] (-0.03,0.2,0) arc (0:68:0.3cm);% node {$\alpha$}; +\draw[draw=green] (0.3,0,0) arc (200:112:0.3cm);% node {$\alpha$}; +\node at (0.15,0.01,0.1)[color=green!70!black]{$\beta$}; +\node at (0.01,0.15,0.1)[color=green!70!black]{$\alpha$}; +\node at (0.15,0.1,0)[color=green!70!black]{$\gamma$}; +} + + +\only<2->{ +\draw[->,draw=red,very thick] (0,0,0) -- (0,0,1.2) node [anchor=west,color=red!70!black] {$\vec c^{*}$}; +\draw[->,draw=red,very thick] (0,0,0) -- (1,0.2,-0.24) node [anchor=west,color=red!70!black] {$\vec a^{*}$}; +\draw[->,draw=red,very thick] (0,0,0) -- (0,1.2,-0.24) node [anchor=north,color=red!70!black] {$\vec b^{*}$}; +} + +\only<3|handout>{ +\draw[->,draw=black,ultra thick] (0,0,0) -- (1,0,0) node +[anchor=north,color=black] {$\vec{\mathbf x}$}; +\draw[->,draw=black,ultra thick] (0,0,0) -- (0,1,0) node +[anchor=north,color=black] {$\vec{\mathbf y}$}; +\draw[->,draw=black,ultra thick] (0,0,0) -- (0,0,1) node +[anchor=east,color=black] {$\vec{\mathbf z}$}; +} + +\only<4>{ +\draw[->,draw=black,ultra thick] (0,0,0) -- (1,0.2,0) node +[anchor=north,color=black] {$\vec{\mathbf x}$}; +\draw[->,draw=black,ultra thick] (0,0,0) -- (-0.2,1,0) node +[anchor=north,color=black] {$\vec{\mathbf y}$}; +\draw[->,draw=black,ultra thick] (0,0,0) -- (0,0,1) node +[anchor=east,color=black] {$\vec{\mathbf z}$}; +} + +\end{tikzpicture} + +\end{column} +\end{columns} + +\begin{onlyenv}<3|handout> + \begin{lstlisting}[style=input] +CS = crystalSymmetry('321',[a b c],'X||a','Z||c*') + \end{lstlisting} +\end{onlyenv} + +\begin{onlyenv}<4-> + \begin{lstlisting}[style=input] +CS = crystalSymmetry('321',[a b c],'X||b','Z||c*') +\end{lstlisting} +The alignment of $\vec x$, $\vec y$, $\vec z$ is important as the Euler angles +refer to them. +\end{onlyenv} + +\end{overlayarea} + +\end{frame} + +\section{Miller Indices} +\label{sec:miller-indices} + +\subsection*{definition} +\label{sec:definition} + +\begin{frame}[fragile] + \frametitle{Miller Indices - Crystal Fixed Directions} + + A direction with respect to the crystal coordinate system $\mathbf C$ + + \begin{columns} + \begin{column}{7cm} + + \begin{overlayarea}{8cm}{8cm} + + \begin{equation*} + \vec m = u \cdot \vec a+ v \cdot \vec b + w \cdot \vec c. + \end{equation*} + + \begin{onlyenv}<2-> +\begin{lstlisting}[style=input] +CS = crystalSymmetry('mmm',[1 2 3]) +m = Miller(1,1,1,CS,'uvw') +\end{lstlisting} +\end{onlyenv} +\begin{onlyenv}<2> +\vspace{-.3cm}\begin{lstlisting}[style=output] +m = /+Miller+/ (show methods, plot) + size: 1 x 1 + symmetry: mmm + u 1 + v 1 + w 1 +\end{lstlisting} +\end{onlyenv} + + \begin{onlyenv}<3-> +A direction in reciprocal coordinates +\begin{equation*} + \vec r = h \cdot \vec a^{*}+ k \cdot \vec b^{*} + \ell \cdot \vec c^{*}. +\end{equation*} +\begin{lstlisting}[style=input] +m = Miller(1,1,1,CS,'hkl') +\end{lstlisting} +\end{onlyenv} +\begin{onlyenv}<3> + \vspace{-.3cm} + \begin{lstlisting}[style=output] +m = /+Miller+/ (show methods, plot) + size: 1 x 1 + symmetry: mmm + h 1 + k 1 + l 1 +\end{lstlisting} +\end{onlyenv} + +\medskip + +\begin{onlyenv}<4-> + A direction in the orthogonal coordinate system + \begin{lstlisting}[style=input] +m = Miller(xvector,CS) +\end{lstlisting} +\end{onlyenv} +\end{overlayarea} +\end{column} + \begin{column}{4cm} + + \begin{overlayarea}{4cm}{7cm} + + \begin{center} + \includegraphics<2>[width=4cm]{pic/MillerUVW} +% \includegraphics<3>[width=4cm]{pic/MillerHKL} + \includegraphics<3->[width=4cm]{pic/MillerPlane} + \end{center} + \vspace{-0.2cm} + \begin{onlyenv}<2-|handout> + \begin{lstlisting}[style=input] +plot(m,'labeled') + \end{lstlisting}% + \end{onlyenv}% + \vspace{-0.25cm} + \begin{onlyenv}<3->% + \begin{lstlisting}[style=input] +hold all +plot(m,'labeled') + \end{lstlisting}% + \end{onlyenv}% + \vspace{-0.25cm} + \begin{onlyenv}<3->% + \begin{lstlisting}[style=input] +plot(m,'plane') + \end{lstlisting} + \end{onlyenv} + \end{overlayarea} + \end{column} + \end{columns} + +\end{frame} + +\subsection*{The trigonal case} + +\begin{frame}[fragile] + \frametitle{Trigonal and Hexagonal Symmetries} + + \vspace{-.3cm} + + \begin{center} + \includegraphics[width=4cm]{pic/hkl} + \includegraphics[width=4cm]{pic/uvtw} + \includegraphics[width=4cm]{pic/uvw} + \end{center} + + \vspace{-.5cm} + + define lists of crystal directions + \vspace{-.2cm} + \begin{lstlisting}[style=input] +m = Miller({1 0 -1 0},{1 1 -2 1},CS,'hkil') +m = Miller({1 0 -1 0},{1 1 -2 2},CS,'UVTW') +m = Miller({1 0 0},{1 0 1},CS,'uvw') + \end{lstlisting}% + + \medskip + + Plot all symmetrically equivalent directions + \vspace{-.2cm} + \begin{lstlisting}[style=input] +plot(m,'symmetrised','labeled') + \end{lstlisting} + + + +\end{frame} + +\subsection*{Calculations} + +\begin{frame}[fragile] + \frametitle{Calculating with Crystal Directions} + + \begin{columns} + \begin{column}{8.6cm} + \begin{overlayarea}{\textwidth}{8cm} + Find all symmetrically equivalent directions + \begin{lstlisting}[style=input] +CS = loadCIF('quartz') +m = Miller(1,1,-2,1,CS,'hkil') +symmetrise(m) + \end{lstlisting} + + \begin{onlyenv}<1> + \vspace{-.3cm} + \begin{lstlisting}[style=output] +ans = /+Miller+/ (show methods, plot) + size: 6 x 1 + mineral: Quartz (P 32 2 1, X||a*, Y||b, Z||c*) + h 1 -2 1 1 -2 1 + k 1 1 -2 -2 1 1 + i -2 1 1 1 1 -2 + l 1 1 1 -1 -1 -1 + \end{lstlisting} + \end{onlyenv} + + \pause + \medskip + + Change from reciprocal coordinate system to $\vec a$, $\vec b$, $\vec c$ + \begin{lstlisting}[style=input] +m.dispStyle = 'UVTW' + \end{lstlisting} + \begin{onlyenv}<2> + \vspace{-.3cm} + \begin{lstlisting}[style=output] +m = /+Miller+/ (show methods, plot) + size: 6 x 1 + mineral: Quartz (P 32 2 1, X||a*, Y||b, Z||c*) + U 0.0828 + V 0.0828 + T -0.1655 + W 0.1027 + \end{lstlisting} + \end{onlyenv} + + \pause + \vspace{-.3cm} + + \begin{lstlisting}[style=input] +round(m) + \end{lstlisting} + \begin{onlyenv}<3> + \vspace{-.3cm} + \begin{lstlisting}[style=output] +ans = /+Miller+/ (show methods, plot) + size: 6 x 1 + mineral: Quartz (P 32 2 1, X||a*, Y||b, Z||c*) + U 4 + V 4 + T -8 + W 5 + \end{lstlisting} + \end{onlyenv} + + \pause + \medskip + + Access the coordinates and properties + \begin{lstlisting}[style=input] +m.U, m.hkl, m.uvw, m.UVTW +m.dspacing % d-spacing of planes + \end{lstlisting} + + \pause + \medskip + +angle modulo symmetry + \begin{lstlisting}[style=input] +angle(m1,m2) / degree +\end{lstlisting} + + + \end{overlayarea} + + \end{column} + \begin{column}{3.5cm} + \includegraphics<1->[width=3.5cm]{pic/MillerSymmetrised}%\\ +% \includegraphics<1->[width=4cm]{pic/Symmetry} + \end{column} + \end{columns} +\end{frame} + +\section{Orientations} +\label{sec:orientations} + +\subsection*{Definition} + +\begin{frame} + \frametitle{Crystal Orientations} + + \begin{columns} + + \begin{column}{6.3cm} + + \begin{overlayarea}{6.4cm}{8cm} + Let a vector $\vec v$ be given by specimen coordinates + $(r_{1},r_{2},r_{3})^{t}$ \underline{and} crystal coordinates + $(h_{1},h_{2},h_{3})^{t}$, i.e., + \begin{equation*} + \vec v = r_{1} \vec X + r_{2} \vec Y + r_{3} \vec Y + = h_{1} \vec x + h_{2} \vec y + h_{3} \vec z. + \end{equation*} + + \pause + + The coordinate transform $\mathbf O$ with + \begin{equation*} + \left(r_{1},r_{2},r_{3}\right)^{t} + = + \mathbf O + \left( + h_{1},h_{2},h_{3} \right)^{t} + \end{equation*} + is called \alert{crystal orientation}. + + \pause + \medskip + + The orientation $\mathbf O$ maps the specimen coordinate system $\vec X, + \vec Y, \vec Z$ onto the crystal coordinate systems $\vec x, \vec y, + \vec z$. + + \pause + \medskip + + The orientation $\mathbf O$ is well defined only up the crystal + symmetry. + + \end{overlayarea} + \end{column} + + \begin{column}{5cm} + + \onslide<1-> + \begin{tikzpicture}[scale=0.5] + + \draw[->, >=latex, color=green!80, thick] (0,0) -- (0,12) node[left] {$\vec Z$}; + \draw[->, >=latex, color=green!80, thick] (0,0) -- (6,0) node[right] {$\vec Y$}; + \draw[->, >=latex, color=green!80, thick] (0,0) -- (-2,-2) node[left] {$\vec X$}; +% \draw (6,11) node[color=green]{$K_S$}; + + \foreach \xpos/\ypos/\xx/\xy/\yx/\yy/\zx/\zy + in { +0.20/1.63/-0.20/0.13/1.09/0.10/-0.31/-2.21, +3.11/7.11/0.80/0.87/-0.20/0.55/1.50/-0.90, +3.70/4.70/-0.03/0.87/0.36/-0.55/-2.12/-0.90, +-1.91/1.15/-0.81/-0.59/0.59/-0.81/-1.00/-1.00, +2.90/9.73/-0.06/0.62/-1.11/-0.17/-0.22/-1.83, +%-0.67/6.44/-0.86/0.34/0.59/1.05/0.81/-0.37, +%1.13/9.19/-0.79/-0.78/0.37/-0.79/-1.40/0.21, +%2.28/0.74/0.44/1.02/-0.70/0.43/1.50/0.28, +%0.70/0.07/-0.06/1.08/-0.83/-0.25/1.50/0.28, +3.81/2.27/1.04/0.35/-0.40/0.54/-0.22/-1.83, +%-1.71/7.06/-0.16/0.06/0.61/1.04/1.85/-0.81, +%3.37/1.43/-0.61/-0.60/0.93/-0.23/-0.22/-1.83, +%-0.49/9.19/0.24/0.73/0.27/0.84/-2.12/0.28, +-1.21/9.29/1.07/-0.04/0.29/0.64/-0.22/-1.83, +%2.21/8.17/1.01/0.53/0.27/0.34/0.81/-1.85, +-0.74/3.46/-0.93/-0.71/0.05/0.48/-1.23/1.43, +-1.51/8.21/-0.68/-0.33/-0.53/-0.87/-1.43/1.23, +%1.37/1.83/1.07/0.51/-0.29/0.99/-0.22/0.21, +0.25/6.41/0.24/1.01/0.90/-0.27/-1.23/-0.81, +0.23/-1.12/-0.77/-0.57/0.31/-0.95/1.50/0.28 + } + \fillcrystal{\xpos,\ypos}{\xx,\xy}{\yx,\yy}{\zx,\zy}{color=red}; + + \end{tikzpicture} + \end{column} +\end{columns} +\end{frame} + +\subsection*{Defining Orientations} +\label{sec:defin-orient} + +\begin{frame}[fragile] + \frametitle{Defining Orientations} + + \begin{overlayarea}{\textwidth}{8cm} + define an orientation by Euler angles + \begin{lstlisting}[style=input] +CS = crystalSymmetry('321') +SS = specimenSymmetry('1') +O = orientation('Euler',10*degree,5*degree,0,CS,SS) + \end{lstlisting} + \begin{onlyenv}<1> + \vspace{-.3cm} + \begin{lstlisting}[style=output] +O = /+orientation+/ (show methods, plot) + size: 1 x 1 + crystal symmetry: 321, X||a*, Y||b, Z||c* + sample symmetry : 1 + + Bunge Euler angles in degree + Psi Theta Phi Inv. + 10 5 0 0 + \end{lstlisting} + \end{onlyenv} + + \pause + import orientations + \begin{lstlisting}[style=input] +O = loadOrientation('filename',CS,SS,... +'ColumnNames',{'phi1','Phi','phi2'}) + \end{lstlisting} + \begin{onlyenv}<2> + \vspace{-.3cm} + \begin{lstlisting}[style=output] +O = /+orientation+/ (show methods, plot) + size: 1000 x 1 + crystal symmetry: 321, X||a*, Y||b, Z||c* + sample symmetry : 1 + \end{lstlisting} + \end{onlyenv} + +\pause + +define orientations by Miller indices + \begin{lstlisting}[style=input] +O = orientation('Miller',[h k l],[u,v,w],CS,SS) + \end{lstlisting} + +\pause + +standard orientations: \texttt{Cube, CubeND22, CubeND45, CubeRD, Goss, +invGoss, Copper, Copper2, SR, SR2, SR3, SR4, Brass, Brass2, PLage, PLage2, +QLage, QLage2, QLage3, QLage4} + +\begin{lstlisting}[style=input] +O = brassOrientation(CS,SS) +\end{lstlisting} +\end{overlayarea} +\end{frame} + +\subsection*{Calculating with Orientations} +\label{sec:calc-with-orient} + +\begin{frame}[fragile,fragile,fragile,fragile] + \frametitle{Calculating with Orientations} + + + \begin{overlayarea}{\textwidth}{8cm} + find all symmetrically equivalent orientations +\begin{lstlisting}[style=input] +symmetrise(O) +\end{lstlisting} + \begin{onlyenv}<1> + \vspace{-.3cm} + \begin{lstlisting}[style=output] +ans = /+orientation+/ (show methods, plot) + size: 6 x 1 + crystal symmetry: 321, X||a*, Y||b, Z||c* + sample symmetry : 1 + + Roe Euler angles in degree + Psi Theta Phi Inv. + 10 5 0 0 + 10 5 120 0 + 10 5 240 0 + 190 175 60 0 + 190 175 180 0 + 190 175 300 0 +\end{lstlisting} + \end{onlyenv} + + \pause + \medskip + + convert crystal into specimen coordinates +\begin{lstlisting}[style=input] +h = Miller(1,0,-1,0,CS); +r = O * h +\end{lstlisting} + \begin{onlyenv}<2> + \vspace{-.3cm} + \begin{lstlisting}[style=output] +r = /+vector3d+/ (show methods, plot) + size: 1 x 1 + x y z + 0.984808 0.173648 0 +\end{lstlisting} + \end{onlyenv} + + \pause + \medskip + + convert specimen into crystal coordinates +\begin{lstlisting}[style=input] +inv(O) * r +\end{lstlisting} + \begin{onlyenv}<3> + \vspace{-.3cm} + \begin{lstlisting}[style=output] +ans = /+Miller+/ (show methods, plot) + size: 1 x 1 + symmetry: 321, X||a*, Y||b, Z||c* + h 1 + k 0 + i -1 + l 0 +\end{lstlisting} + \end{onlyenv} + + \pause + \medskip + + change specimen coordinates + \begin{lstlisting}[style=input] +R = rotation('axis',zvector,'angle',90*degree) +O2 = R * O1 +\end{lstlisting} +\end{overlayarea} + +\end{frame} + + +\subsection*{Pole Figures} + +\begin{frame}[fragile] + \frametitle{Pole Figures and Inverse Pole Figures} + + \begin{columns} + \begin{column}{8.5cm} + Lattice planes in specimen coordinates + \begin{lstlisting}[style=input] +h = Miller({1 0 0},{0 1 0},ori.CS) +plot(symmetrise(ori) * h(1)) +plot(ori * symmetrise(h(1))) + \end{lstlisting} + + \pause + \medskip + + \begin{lstlisting}[style=input] +plotPDF(ori,h) + \end{lstlisting} + \pause + \vspace{-0.3cm} + \begin{lstlisting}[style=input] +plotPDF(ori,h,'contourf') + \end{lstlisting} + + \pause + \medskip + + specimen directions in crystal coordinates + \begin{lstlisting}[style=input] +plot(symmetrise(inv(ori)) * xvector) +plot(symmetrise(inv(ori) * xvector)) + \end{lstlisting} + \pause + \vspace{-0.3cm} + \begin{lstlisting}[style=input] +v = [xvector,yvector] +plotIPDF(ori,v) + \end{lstlisting} + \pause + \vspace{-0.3cm} + \begin{lstlisting}[style=input] +plotIPDF(ori,v,'contourf') + \end{lstlisting} + + + \end{column} + \begin{column}{3.5cm} + + \begin{onlyenv}<1> + \includegraphics[width=\textwidth]{pic/pfSimple} + \end{onlyenv} + + \begin{onlyenv}<2> + \includegraphics[width=0.95\textwidth]{pic/pfSimple100} + + \medskip + + \includegraphics[width=0.95\textwidth]{pic/pfSimple010} + \end{onlyenv} + + \begin{onlyenv}<3> + \includegraphics[width=0.95\textwidth]{pic/pfSimple100Smooth} + + \medskip + + \includegraphics[width=0.95\textwidth]{pic/pfSimple010Smooth} + \end{onlyenv} + + \begin{onlyenv}<4> + \includegraphics[width=\textwidth]{pic/ipfSimple} + \end{onlyenv} + + \begin{onlyenv}<5> + \includegraphics[width=0.95\textwidth]{pic/ipfSimple100} + + \medskip + + \includegraphics[width=0.95\textwidth]{pic/ipfSimple010} + \end{onlyenv} + + \begin{onlyenv}<6> + \includegraphics[width=0.95\textwidth]{pic/ipfSimple100Smooth} + + \medskip + + \includegraphics[width=0.95\textwidth]{pic/ipfSimple010Smooth} + \end{onlyenv} + \end{column} + \end{columns} + +\end{frame} + +\subsection*{ODF Plots} + +\begin{frame}[fragile] + \frametitle{Plotting in Orientation Space} + + \begin{onlyenv}<1> + \begin{lstlisting}[style=input] +plotSection(ori,'phi2',(0:30:150)*degree) + \end{lstlisting} + + \includegraphics[width=\textwidth]{pic/odfOri} + \end{onlyenv} + + \begin{onlyenv}<2> + \begin{lstlisting}[style=input] +plotSection(ori,'phi2',(0:30:150)*degree,'contourf',... + 'halfwidth',10*degree) + \end{lstlisting} + + \includegraphics[width=0.95\textwidth]{pic/odfOriSmooth} + \end{onlyenv} + + \begin{onlyenv}<3> + \begin{lstlisting}[style=input] +plotSection(ori,'sigma',(0:30:150)*degree,'contourf') + \end{lstlisting} + \vspace{-0.3cm} + \begin{center} + \includegraphics[width=0.9\textwidth]{pic/odfOriSmoothSigma} + \end{center} + + \end{onlyenv} + +\end{frame} + +\subsection*{Operations on Orientations} + +\begin{frame}[fragile] + \frametitle{Operations on Orientations} + +% the misorientation angle and axis +% \begin{lstlisting}[style=input] +% angle(ori1,ori2) / degree +% axis(ori1,ori2) +% \end{lstlisting} + +% \begin{onlyenv}<1> +% \begin{lstlisting}[style=output] +% ans = /+Miller+/ (show methods, plot) +% size: 1 x 1 +% mineral: Forsterite (mmm) +% h 2.729 +% k 7.6909 +% l 1.9193 +% \end{lstlisting} +% \end{onlyenv} + +% \pause +% \medskip + +mean orientation + \begin{lstlisting}[style=input] +mean(ori) + \end{lstlisting} + + \begin{onlyenv}<2> + \begin{lstlisting}[style=output] +ans = /+orientation+/ (show methods, plot) + size: 1 x 1 + crystal symmetry : Forsterite (mmm) + specimen symmetry: 1 + + Bunge Euler angles in degree + phi1 Phi phi2 Inv. + 342.532 68.3179 284.955 0 + \end{lstlisting} + \end{onlyenv} + +\pause +\medskip + + mean orientation spread + \begin{lstlisting}[style=input] +mean(angle(ori,mean(ori)))./degree + \end{lstlisting} + + \begin{onlyenv}<3> + \begin{lstlisting}[style=output] +ans = + + 47.2287 +\end{lstlisting} + \end{onlyenv} + +\pause +\medskip + +volume portions +\begin{lstlisting}[style=input] +volume(ori,mean(ori),10*degree) +fibreVolume(ori,Miller(1,0,0,ori.CS),xvector) +\end{lstlisting} + +\pause +\medskip + +export to ASCII file +\begin{lstlisting}[style=input] +export(ori,'file.txt','bunge','degree') +\end{lstlisting} + +\end{frame} + + + + +\section{Misorientations} + + +\subsection*{Coordinate Transforms} +\label{sec:orientations} + + +\begin{frame}[fragile] + \frametitle{Coordinate Transforms} + + \begin{overlayarea}{\textwidth}{8cm} + Remember, an orientation converts crystal into specimen + coordinates + \vspace{-.2cm} + \begin{lstlisting}[style=input] +O1 = orientation('Euler',0,0,0,CS_Mag) +r = O1 * Miller(1,1,1,CS_Mag) +\end{lstlisting} + \begin{onlyenv}<1> +\vspace{-.3cm}\begin{lstlisting}[style=output] +r = /+vector3d+/ (show methods, plot) + size: 1 x 1 + x y z + 0.119107 0.119107 0.119107 +\end{lstlisting} + \end{onlyenv} + + \pause + \medskip + + The inverse of an orientation converts specimen into crystal coordinates + \vspace{-.2cm} + \begin{lstlisting}[style=input] +O2 = orientation('Euler',... + 135*degree,55*degree,60*degree,CS_Hem) +inv(O2) * r +\end{lstlisting} + \begin{onlyenv}<2> + \vspace{-.3cm} + \begin{lstlisting}[style=output] +ans = /+Miller+/ (show methods, plot) + size: 1 x 1 + mineral: Hematite (-3m1, X||a*, Y||b, Z||c) + h 0 + k 0 + i 0 + l 1 + +\end{lstlisting} + \end{onlyenv} + + \pause + \medskip + + Hence, $\text{\bf inv}(\mathbf{O2}) \cdot \mathbf{O1}$ converts crystal into + crystal coordinates + \vspace{-.2cm} + \begin{lstlisting}[style=input] +O2 = inv(O2) * O1 * Miller(1,1,1,0,CS_Mag) + \end{lstlisting} + \begin{onlyenv}<3> + \vspace{-.3cm} + \begin{lstlisting}[style=output] +ans = /+Miller+/ (show methods, plot) + size: 1 x 1 + mineral: Hematite (-3m1, X||a*, Y||b, Z||c) + h 0 + k 0 + i 0 + l 1 + +\end{lstlisting} + \end{onlyenv} + +\end{overlayarea} + +\end{frame} + + + +\subsection*{Misorientations} +\label{sec:orientations2} + +\begin{frame}[fragile] + \frametitle{Misorientations} + + \begin{overlayarea}{\textwidth}{8cm} + + A \alert{misorientation} {\bf MO} transforms coordinates with respect to + crystal A into coordinates with respect to crystal B. +\vspace{-0.1cm} +\begin{lstlisting}[style=input] +MO = inv(O1) * O2 +\end{lstlisting} + +\begin{onlyenv}<1> + \vspace{-.3cm} + \begin{lstlisting}[style=output] +MO = /+misorientation+/ (show methods, plot) + size: 1 x 1 + crystal symmetry : Magnetite (m-3m) + crystal symmetry : Hematite (-3m1, X||a*, Y||b, Z||c) + + Bunge Euler angles in degree + phi1 Phi phi2 Inv. + 120 55 45 0 +\end{lstlisting} +\end{onlyenv} + + \pause + \medskip + +As \textbf{O1} and \textbf{O2} are suspect to crystal symmetry, there are many +symmetrically equivalent misorientations to \textbf{oM}. + +\vspace{-0.1cm} +\begin{lstlisting}[style=input] +oM.symmetrise +\end{lstlisting} + +\begin{onlyenv}<2> + \vspace{-.3cm} + \begin{lstlisting}[style=output] +MO = /+misorientation+/ (show methods, plot) + size: 576 x 1 + crystal symmetry : Magnetite (m-3m) + crystal symmetry : Hematite (-3m1, X||a*, Y||b, Z||c) +\end{lstlisting} +\end{onlyenv} + + \pause + +\vspace{-0.3cm} +\begin{lstlisting}[style=input] +plotPDF(MO,Miller({0 0 0 1},{1 1 -2 1},CS_Hem)) +plotIPDF(MO,Miller({0 0 1},{1 1 1},CS_Mag)) +\end{lstlisting} + +\vspace{-0.5cm} + +\begin{onlyenv}<3> + \begin{center} + \includegraphics[height=3cm]{pic/HemPDF111.pdf} + \quad + \includegraphics[height=3cm]{pic/MagPDF110.pdf} + \end{center} +\end{onlyenv} + +\end{overlayarea} +\end{frame} + +\begin{frame}[fragile] + \frametitle{Misorientation angel and axis} + + \begin{overlayarea}{\textwidth}{8cm} + + the smallest rotational angle of all symmetrically equivalent + misorientations to \textbf{MO} is called \structure{misorientation angle} + \begin{lstlisting}[style=input] +angle(O1,O2) / degree +angle(MO) /degree + \end{lstlisting} + + \pause + \medskip + + the misorientation realizing this minimum angle is computed by + \vspace{-0.2cm} + \begin{lstlisting}[style=input] +MO.project2FundamentalRegion + \end{lstlisting} +\begin{onlyenv}<2> + \vspace{-.3cm} + \begin{lstlisting}[style=output] +MO = /+misorientation+/ (show methods, plot) + size: 1 x 1 + crystal symmetry : Magnetite (m-3m) + crystal symmetry : Hematite (-3m1, X||a*, Y||b, Z||c) + + Bunge Euler angles in degree + phi1 Phi phi2 Inv. + 60.1624 54.6037 314.719 0 +\end{lstlisting} +\end{onlyenv} + +\pause +\medskip + + its rotational axis is called the \structure{misorientation axis} of + \textbf{MO} + \vspace{-0.2cm} +\begin{lstlisting}[style=input] +axis(MO) +\end{lstlisting} +\vspace{-.3cm} + + \begin{onlyenv}<3> +\begin{lstlisting}[style=output] +ans = /+Miller+/ (show methods, plot) + size: 1 x 1 + mineral: Hematite (-3m1, X||a*, Y||b, Z||c) + h 0.6191 + k 3.8882 + i -4.5073 + l 3.3513 +\end{lstlisting} +\end{onlyenv} + + \pause +\begin{lstlisting}[style=input] +axis(inv(MO)) +\end{lstlisting} +\vspace{-.3cm} + \begin{onlyenv}<4> +\begin{lstlisting}[style=output] +ans = /+Miller+/ (show methods, plot) + size: 1 x 1 + mineral: Magnetite (m-3m) + h 4.9324 + k -6.4797 + l 2.0431 +\end{lstlisting} +\end{onlyenv} + +\pause + +\begin{lstlisting}[style=input] +axis(O1,O2) +\end{lstlisting} + + \begin{onlyenv}<5> +\vspace{-.3cm} +\begin{lstlisting}[style=output] +ans = /+vector3d+/ (show methods, plot) + size: 1 x 1 + x y z + 0.0495585 0.992437 -0.112305 +\end{lstlisting} +\end{onlyenv} + + \end{overlayarea} + +\end{frame} + + +\subsection*{Statistics} + +\begin{frame}[fragile] + \frametitle{Misorientation Statistics} + + \begin{columns} + \begin{column}{8cm} + \begin{lstlisting}[style=input] +plotAngleDistribution(MO) +hold on +plotAngleDistribution(MO.CS) +hold off + \end{lstlisting} + + \includegraphics[width=\textwidth]{pic/angleDistri} + + \pause + + \begin{lstlisting}[style=input] +plotAxisDistribution(MO) + \end{lstlisting} + + \pause +\vspace{-0.3cm} + \begin{lstlisting}[style=input] +plotAxisDistribution(MO,'contourf') + \end{lstlisting} +\end{column} +\begin{column}{4cm} + \begin{overlayarea}{\textwidth}{8cm} + \begin{onlyenv}<2-> + \includegraphics[width = \textwidth]{pic/axisDistri} + \end{onlyenv} + + \medskip + + \begin{onlyenv}<3> + \includegraphics[width = \textwidth]{pic/axisDistriSmooth} + \end{onlyenv} + \end{overlayarea} + +\end{column} + +\end{columns} + +\end{frame} + + +\subsection*{Phase Transitions} + + +\begin{frame}[fragile] + \frametitle{Phase Transitions} + + \begin{overlayarea}{\textwidth}{8cm} + We want to model the phase transition from magnetite to hematite given by + the orientation relationship $\{111\}_{\text{m}} || \{0001\}_{\text{h}}$ + and $\{\overline{1}01\}_{\text{m}} || \{00\overline{1}0\}_{\text{h}}$ + \begin{lstlisting}[style=input] +CS_Mag = loadCIF('Magnetite') +CS_Hem = loadCIF('Hematite') + \end{lstlisting} + \begin{onlyenv}<1> + \vspace{-0.3cm} + \begin{lstlisting}[style=output] +cs_Magnetite = /+crystalSymmetry+/ (show methods, plot) + mineral : Magnetite + symmetry: m-3m + a, b, c : 8.4, 8.4, 8.4 + +cs_Hematite = /+crystalSymmetry+/ (show methods, plot) + mineral : Hematite + symmetry : -3m1 + a, b, c : 5, 5, 14 + reference frame: X||a*, Y||b, Z||c + \end{lstlisting} + \end{onlyenv} + + \pause + \medskip + + \begin{lstlisting}[style=input] +Mag2Hem = orientation('map',... + Miller(1,1,1,CS_Mag),Miller(0,0,0,1,CS_Hem),... + Miller(-1,0,1,CS_Mag),Miller(1,0,-1,0,CS_Hem),... + CS_Mag,CS_Hem) + \end{lstlisting} + \begin{onlyenv}<2> + \vspace{-0.3cm} + \begin{lstlisting}[style=output] +Mag2Hem = /+misorientation+/ (show methods, plot) + size: 1 x 1 + crystal symmetry : Magnetite (m-3m) + crystal symmetry : Hematite (-3m1, X||a*, Y||b, Z||c) + + Bunge Euler angles in degree + phi1 Phi phi2 Inv. + 120 54.7356 45 0 + \end{lstlisting} + \end{onlyenv} +\end{overlayarea} +\end{frame} + + +\subsection*{Phase Transition II} + +\begin{frame}[fragile] + \frametitle{Phase Transition} + + \begin{overlayarea}{\textwidth}{8cm} + For measured hematite orientation + \begin{lstlisting}[style=input] +ori_Hem = orientation('Euler',0,0,0,CS_Hem) + \end{lstlisting} + \begin{onlyenv}<1> + \vspace{-0.3cm} + \begin{lstlisting}[style=output] +ori_Hem = /+orientation+/ (show methods, plot) + size: 1 x 1 + crystal symmetry : Hematite (-3m1, X||a*, Y||b, Z||c) + specimen symmetry: 1 + + Bunge Euler angles in degree + phi1 Phi phi2 Inv. + 0 0 0 0 + \end{lstlisting} + \end{onlyenv} + we can compute the initial magnetite orientation by + \begin{lstlisting}[style=input] +ori_Mag = ori_Hem * Mag2Hem + \end{lstlisting} + +\pause + + \begin{onlyenv}<2> + \vspace{-0.3cm} + \begin{lstlisting}[style=output] +ori_Mag = /+orientation+/ (show methods, plot) + size: 1 x 1 crystal + symmetry : Magnetite (m-3m) + specimen symmetry: 1 + + Bunge Euler angles in degree + phi1 Phi phi2 Inv. + 120 54.7356 45 0 + \end{lstlisting} + \end{onlyenv} + + \pause + \medskip + + We should care about symmetric equivalence + \begin{lstlisting}[style=input] +ori_Mag = ori_Hem * symmetrise(Mag2Hem) + \end{lstlisting} + \begin{onlyenv}<3-> + \vspace{-0.3cm} + \begin{lstlisting}[style=output] +ori_Mag = /+orientation+/ (show methods, plot) + size: 1 x 96 + crystal symmetry : Magnetite (m-3m) + specimen symmetry: 1 + \end{lstlisting} + \end{onlyenv} + +\pause + +\alert{MTEX keeps track about the symmetries throughout all computations and + warns in case of mismatch.} +\end{overlayarea} +\end{frame} + + +\subsection*{phase transistion III} + +\begin{frame}[fragile] + \frametitle{Relationship Between Lattice Planes} + + \begin{overlayarea}{\textwidth}{8cm} + + Compute the minimal angle between two lattice planes of two + different grains having different orientation and different phase. + \begin{lstlisting}[style=input] +m_Mag = Miller(1,0,0,cs_Magnetite); +m_Hem = Miller(1,1,-2,0,cs_Hematite); + \end{lstlisting} + \begin{onlyenv}<1> + \vspace{-.3cm} + \begin{lstlisting}[style=output] +m_Mag = /+Miller+/ (show methods, plot) + size: 1 x 1 + mineral: Magnetite (432) + h 1 + k 0 + l 0 + +m_Hem = Miller (show methods, plot) + size: 1 x 1 + mineral: Hematite (321, X||a*, Y||b, Z||c) + h 1 + k 1 + i -2 + l 0 +\end{lstlisting} + \end{onlyenv} + + \pause + \medskip + + The orientation relation should be given by + \begin{lstlisting}[style=input] +Mag2Hem = orientation('map',... + Miller(1,1,1,CS_Mag),Miller(0,0,0,1,CS_Hem),... + Miller(-1,0,1,CS_Mag),Miller(1,0,-1,0,CS_Hem),... + CS_Mag,CS_Hem) + \end{lstlisting} + \begin{onlyenv}<2> + \vspace{-0.3cm} + \begin{lstlisting}[style=output] +Mag2Hem = /+misorientation+/ (show methods, plot) + size: 1 x 1 + crystal symmetry : Magnetite (m-3m) + crystal symmetry : Hematite (-3m1, X||a*, Y||b, Z||c) + + Bunge Euler angles in degree + phi1 Phi phi2 Inv. + 120 54.7356 45 0 + \end{lstlisting} + \end{onlyenv} + + \pause + \medskip + + The minimum angle + \begin{lstlisting}[style=input] +min(angle(Mag2Hem * m_Mag.symmetrise,m_Hem ))/degree + \end{lstlisting} + \begin{onlyenv}<3> + \vspace{-.3cm} + \begin{lstlisting}[style=output] + 35.2644 + \end{lstlisting} + \end{onlyenv} + +\end{overlayarea} + +\end{frame} + +% \section{Exercises} + +% \subsection*{Exercises2} + +% \begin{frame} + +% \begin{Exercise} +% Consider trigonal crystal symmetry. + +% \begin{enumerate}[a)] +% \item Find all crystallographic directions symmetrically equivalent to $h +% = (1, 0, \bar 1, 0)$ (Miller indices)! +% \item Find crystallographic directions such that the number of their +% crystallographic equivalent directions on the upper hemisphere (without +% equator) is 1, 3, and 6, when including antipodal symmetry! +% \item Consider the orientation given by the Euler angles $(30\degree, +% 90\degree, 90\degree)$ in Bunge convention. Give the Euler angles of +% all symmetrically equivalent orientations! +% \item Which positions in the (0,0,0,1) - pole figure corresponds to the +% above orientation. Which crystal direction is rotated by this +% orientation to the specimen direction (0,0,1)? +% \item Construct an orientation that rotates the crystallographic +% directions $(0,0,0,1)$ and $(2,\bar 1,\bar 1,0)$ onto the specimen +% directions $(1,0,0)$ and $(0,1,0)$, respectively. Describe the rotation +% by axis and angle. +% \end{enumerate} + +% \end{Exercise} + +% \end{frame} + +\end{document} + + +% \begin{frame}[fragile] +% \begin{lstlisting}[mathescape=true] +% rot = rotation('Euler',$\phi_1$,$\Phi$,$\phi_2$,'ZXZ') +% rot = rotation('Euler',$\alpha$,$\beta$,$\gamma$,'ABG') +% rot = rotation('axis',v,'angle',omega) +% rot = rotation('map',$u_1$, $v_1$, $u_2$, $v_2$) % $\mathbf{gu}_1=\mathbf{v}_1,\mathbf{gu}_2=\mathbf{v}_2$ +% \end{lstlisting} + +% \begin{columns} +% \begin{column}{8.5cm} + +% Calculations: + +% \begin{lstlisting} +% v = rot * u % apply rot to u +% rot = rot1 * rot2 +% \end{lstlisting} + +% Basic Functions: + +% \begin{lstlisting}[mathescape=true] +% angle(rot), axis(rot) +% angle(rot1,rot2), inv(rot) +% [$\phi_1$,$\Phi$,$\phi_2$] = Euler(rot) +% [$\alpha$,$\beta$,$\gamma$] = Euler(rot,'ABG') +% \end{lstlisting} + +% \end{column} + +% \begin{column}{3cm} +% \includegraphics[width=3cm]{pic/quaternion} +% \end{column} + +% \end{columns} +% \end{frame} + + +% \subsection*{Symmetry} +% \begin{frame}[fragile] +% \frametitle{Crystal Symmetries - The \MTEX Class \texttt{\bf symmetry}} + +% Definition: + +% \begin{lstlisting} +% S = symmetry('triclinic',[a,b,c],[alpha,beta,gamma]) +% S = symmetry('-3m',[a,b,c],/+'X||a*'+/); +% S = symmetry('-3m',[a,b,c],'X||a','Z||c*'); +% S = symmetry('O'); +% \end{lstlisting} + +% \medskip + +% \begin{columns} +% \begin{column}{8.5cm} + +% Load Symmetry from CIF file: + +% \begin{lstlisting} +% symmetry('quartz.cif') +% \end{lstlisting} + +% \medskip + +% Basic Functions: + +% \begin{lstlisting} +% symmetrise(v,S) +% symmetrise(rot,CS,SS) +% rotation(S) +% project2FundamentalRegion(v,CS) +% project2FundamentalRegion(rot,CS,SS) +% \end{lstlisting} +% \end{column} + +% \begin{column}{3cm} +% \includegraphics[width=3cm]{pic/sym} +% \end{column} + +% \end{columns} + +% \end{frame} + +% \subsection*{Miller} + +% \begin{frame}[fragile] +% \frametitle{Crystal Directions - The \MTEX Class \texttt{\bf Miller}} + +% Definition: + +% \begin{lstlisting} +% m = Miller(u,v,w,CS,'uvw'); +% m = Miller(h,k,i,l,CS,'hkl'); +% m = [Miller(1,1,-2,3,CS),Miller(0,1,-1,0,CS)] +% \end{lstlisting} + +% \medskip + +% \begin{columns} +% \begin{column}{8.5cm} + +% Calculations: + +% \begin{lstlisting} +% h1 + h2 +% rot * h % apply rot on h +% \end{lstlisting} + +% \medskip + +% Basic Functions: + +% \begin{onlyenv}<1> +% \begin{lstlisting} +% eq(h1,h2), angle(h1,h2) +% symmetrise(h) %get all equivalent +% plot([h1,h2],'all') +% \end{lstlisting} +% \end{onlyenv} + +% \end{column} + +% \begin{column}{3cm} +% \includegraphics[width=3cm]{pic/miller} +% \end{column} +% \end{columns} + + +% \begin{onlyenv}<2> + +% \lstset{stringstyle=\color{red},emph={antipodal},emphstyle=\em\color{red}} + +% \begin{lstlisting} +% get(h,'hkl') +% eq(h1,h2,`antipodal`), angle(h1,h2,`antipodal`) +% symmetrise(h,`antipodal`) +% plot([h1,h2],'all',`antipodal`) +% \end{lstlisting} + + +% \end{onlyenv} + + +% \end{frame} + + + +% \begin{frame}[fragile] +% \frametitle{Orientations - The \MTEX Class \texttt{\bf orientation}} + +% Definition: + +% \begin{lstlisting}[mathescape=true] +% ori = orientation(rot,CS,SS) +% ori = orientation('Euler',$\phi_1$,$\Phi$,$\phi_2$,CS,SS) +% ori = orientation('Euler',$\alpha$,$\beta$,$\gamma$,'ABG',CS,SS) +% ori = orientation('Miller',[h k l],[u v w],CS,SS) +% ori = orientation('brass',CS,SS) +% \end{lstlisting} + +% \begin{columns} +% \begin{column}{8.5cm} + +% Calculations: + +% \begin{lstlisting} +% r = ori * h, h = ori \ r +% ori = rot * ori +% \end{lstlisting} + +% Basic Functions: + +% \begin{lstlisting}[mathescape=true] +% eq(ori1,ori2), angle(ori1,ori2) +% symmetrise(ori), angle(ori) +% project2FundamentalRegion(ori,ref_ori) +% [$\phi_1$,$\Phi$,$\phi_2$] = Euler(ori) +% \end{lstlisting} + +% \end{column} + +% \begin{column}{3cm} +% \includegraphics[width=3cm]{pic/quaternion} +% \end{column} + +% \end{columns} +% \end{frame} + +2732626 diff --git 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