SEMINARI CHIMICI. Dr. Eckhard Hitzer Department of Applied Physics University of Fukui - Japan

Transcription

1 SEMINARI CHIMICI Data mercoledì 3 Marzo 2010, ore giovedì 4 Marzo 2010, ore Dipartimento di Chimica Strutturale e Stereochimica Inorganica Aula H dcssi.istm.cnr.it Via Venezian, 21 - MILANO Oratore Titolo Coordinatore Dr. Eckhard Hitzer Department of Applied Physics University of Fukui - Japan 3D Space Group Visualization with Interactive Crystal Symmetry Software Prof. Davide M. Proserpio davide.proserpio@unimi.it Dipartimento di Chimica Strutturale e Stereochimica Inorganica A new interactive software tool ( is described, that visualizes 3D space group symmetries, as tabulated in the Int. Tables of Crystallography (T. Hahn, Springer, 2005). The software computes with Clifford (geometric) algebra. The Space Group Visualizer (SGV) originated as a script for the geometric visualization platform CLUCalc, which fully supports geometric algebra computation. Selected generators (Hestenes and Holt, JMP, 2007) form a multivector generator basis of each space group. The approach corresponds to an algebraic implementation of groups generated by reflections (Coxeter and Moser, 4th ed., 1980). The basic operation is the reflection. Two reflections at non-parallel planes yield a rotation, two reflections at parallel planes a translation, etc. Combination of reflections corresponds to the geometric product of vectors describing the individual reflection planes. We begin by demonstrating how to use the SGV for 3D space group visualization, beginning with space group selection. The interactive computer graphics shows a perspective view of one cell (domain expandable) with all its symmetry elements and general positions. Mouse interaction allows to animate symmetry operations, move general positions, navigate through the cell, etc. Dropdown menus provide scene rotation (like a movie), orthographic projection, changes of background, lighting conditions, 3D stereo views (anaglyphs). There is also a multi functional toolbar. Next we demonstrate 3D point groups interactively rendered in 3D with a free CLUCalc script, and in the SGV. We then show how to use the SGV for visualizing the 17 plane groups (two dimensional space groups) and for many subperiodic groups. Finally, we will give some insights into the Clifford geometric algebra description of space groups. The software/scripts needed to follow/repeat what will be shown can be download at : More details on the lectures are reported in the following pages. Responsabili: Prof. Elena Selli (tel ), Dr. Alessandra Silvani

2 How to get to the Dipartimento di Chimica Strutturale e Stereochimica Inorganica

3 Abstracts for tutorials and presentations about the interactive crystal symmetry software Space Group Visualizer 1) Wednesday, 03 March 2010, 14:15 15:15 This tutorial introduces how to use the socalled Space Group Visualizer ( software for interactively visualizing all 230 crystallographic space groups in three dimensions. It describes how to obtain, install and open the Space Group Visualizer (SGV), see the Figure for a screenshot. It shows how to select a space group and display it and how to interactively change and animate the view. Fig. Space Group Visualizer (SGV) screenshot. Groups of symmetries and individual symmetries can be selected or excluded freely according to symmetry type, orientation, location, angle and mathematical generator expression. Axis vectors of a cell, cell angles, cell frame, general position symbols (asymmetric), etc. can all be manipulated interactively. The action of each symmetry can be shown by animation. Combination with the online edition of International Tables for Crystallography, Volume A by opening an online IT window. Features for 3D stereo vision, saving images (in several standard formats), changing background colors, lighting, and rotation center of the view, orthographic projection, etc. will be explained and demonstrated. The participants are encouraged to bring their own notebook PCs (Windows XP, or Windows Vista) and install the free demo version of the SGV (download: Fig. 90 degree screws of Fm 3m.

4 2) Wednesday, 03 March 2010, 15:30 16:15 In this tutorial we learn about interactive computer software for exploring and teaching the 32 three-dimensional point groups of crystallography. Two types of interactive software will be used: A) A socalled CLUScript called Point Groups (Version 2.0, 29 Jan. 2010, by C. Perwass & E. Hitzer). This CLUScript works on the geometric visualization platform CLUCalc (version 6.1 and higher, by C. Perwass). The software can be freely downloaded from and can easily be installed on Windows OS computers (Windows XP, Windows Vista). A screenshot of the CLUScript Point Groups is shown in the Figure below. One can select a crystal system (triclinic to cubic) and then a list of the corresponding point groups together with generators appears in the left browser panel. The visualization window shows two crystal cells with color coded vertices. A small lower left corner inset shows the group generator elements. By selecting a point group and clicking on the point group generators in the left browser panel, Fig. Screenshot of the CLUScript Point Groups showing group 6mmm. the symmetry operations will be applied successively to the second cell and graphically displayed in the original cell. An interactive toolbar to further manipulate the generator sequence appears at the bottom, the sequence of applied point group generators is shown at the top. B) The second software to be introduced is the socalled Space Group Visualizer (SGV). A free demo version with three space groups can be downloaded from Teaching versions with a limited set of space groups (the same set of space groups used in the teaching edition of International Tables of Crystallography, Vol. A) come at an affordable price. Since a point group manifest itself in symmorphic space groups as the general element symmetry group of a vertex, the interactive SGV can also be used to visualize 3D point group symmetries. Concrete instructions on how to achieve this will be given. All symmetry operations can be selected or excluded and animated. Mouse motion of the general position elements allows to illustrate special positions. The fully asymmetric general position elements allow to see the effect of a symmetry on polar and axial vectors. Fig. Cloud of general positions of point group m 3m.

5 3) Thursday, 04 March 2010, 14:15 15:10 In this tutorial we will learn how to use the Space Group Visualizer ( for the Interactive Visualization of Plane Space Groups, and of many subperiodic groups. We learn how to successfully display the 17 plane two-dimensional (2D) space groups in the interactive crystal symmetry software Space Group Visualizer (SGV). The SGV is based on a new type of powerful geometric algebra visualization platform CLUCalc. The principle is to select in the SGV a three-dimensional super space group and by orthogonal projection produce a view of the desired plane 2D space group. The choice of 3D super space group is conveniently summarized in a lookup table. The direction of view for the orthographic projection needs to be adapted only for displaying the plane 2D space groups Nos. 3, 4 and 5. In all other cases space group selection followed by orthographic projection immediately displays one cell of the desired plane 2D space group, see e.g. plane space group p3m1 in the Figure. Fig. All symmetries of plane space group p3m1. The full symmetry selection, interactivity and animation features for 3D space groups offered by the SGV software become thus also available for plane 2D space groups. A special advantage of this visualization method is, that by canceling the orthographic projection (remove the tick mark of Orthographic View in drop down menu Visualization), every plane 2D space group is seen to be a subgroup of a corresponding 3D super space group. A free complete detailed script is available for this part of the tutorial on plane 2D space groups. Fig. General position diagram of plane space group p3m1. The interactive visualization of some subperiodic groups (including all layer groups) relies on the SGV features of view reduction to 1D and 2D cells, as well as orthographic projection. It will be explained how to visualize subperiodic groups, together with concrete examples. See the Figure below. Fig. (Left) Frieze groups p11m, p11g, p211. (Center) Rod group p6 1. (Right) Layer group p6mmm.

6 4) Thursday, 04 March 2010, 15:20 16:15 Most matter in nature and technology is composed of crystals and crystal grains. A full understanding of the inherent symmetry is vital. We approach this problem using Clifford s geometric algebra. Geometric algebra systematically includes all of real numbers, complex numbers, quaternions, Pauli spin algebra, Dirac algebra, Grassmann algebra, Lie algebras, etc. as subalgebras. Geometric algebra provides a clear intuitive geometric interpretation of these algebras. Geometric algebra has thus become an important unifying mathematical language in diverse fields as computer graphics, elementary particle physics, robotics, crystal symmetry, hypercomplex analysis, etc. We introduce the general foundations of geometric algebra. Scalars, vectors, bivectors (axial vector duals), and pseudoscalars are seen to represent all subspaces of three-dimensional (3D) Euclidean space algebraically. A geometric algebra of a nd vector space is the 2 n D algebra of all its 2 n subspaces. As a concrete example we discuss the Space Group Visualizer ( software, which computes with geometric algebra. The Space Group Visualizer (SGV) is a script for the visual computation platform CLUCalc, which fully supports Clifford geometric algebra computation. Selected generators (Hestenes & Holt, JMP, 2007) form a multivector generator basis of each space group. The approach corresponds to an algebraic implementation of groups generated by reflections (Coxeter and Moser, 4 th ed., 1980). The basic operation is the reflection. Two reflections at non-parallel planes yield a rotation, two reflections at parallel planes a translation, etc. Combination of reflections corresponds to the geometric product of vectors describing the individual reflection planes. In our presentation we will give insights into the Clifford geometric algebra description of space groups. Fig. Reflection at plane with normal a. Fig. Double reflection at planes with normals a and b, respectively. The result is a rotation in the a,b plane by 2 <)(a,b).