Condensed Matter

1306 Submissions

[15] viXra:1306.0194 [pdf] submitted on 2013-06-22 23:49:35

Can CDW Physics Explain Ultra Fast Transitions, and Current Vs. Applied Electric Field Values Seen in the Laboratory?

Authors: A.W. Beckwith
Comments: 4 Pages. additional re do of part of my PhD dissertation

The tunneling Hamiltonian is a proven method to treat particle tunneling between different states represented as wavefunctions in many-body physics. Our problem is how to apply a wave functional formulation of tunneling Hamiltonians to a driven sine-Gordon system. We apply a generalization of the tunneling Hamiltonian to charge density wave (CDW) transport problems in which we consider tunneling between states that are wavefunctionals of a scalar quantum field. We present derived I-E curves that match Zenier curves used to fit data experimentally with wavefunctionals congruent with the false vacuum hypothesis. The open question is whether the coefficients picked in both the wavefunctionals and the magnitude of the coefficients of the driven sine Gordon physical system should be picked by topological charge arguments that in principle appear to assign values that have a tie in with the false vacuum hypothesis first presented by Sidney Coleman. Our supposition is that indeed this is useful and that the topological arguments give evidence as to a first order phase transition which gives credence to the observed and calculated I-E curve as evidence
Category: Condensed Matter

[14] viXra:1306.0172 [pdf] submitted on 2013-06-20 05:27:01

Crystal Cell and Space Lattice Symmetries in Clifford Geometric Algebra

Authors: Eckhard Hitzer, Christian Perwass
Comments: 4 Pages. 2 figures, 2 tables. Iin TE. Simos, G. Sihoyios, C. Tsitouras (eds.), International Conference on Numerical Analysis and Applied Mathematics 2005, Wiley-VCH, Weinheim, 2005, pp. 937-941 (2005).

The structure of crystal cells in two and three dimensions is fundamental for many material properties. In two dimensions atoms (or molecules) often group together in triangles, squares and hexagons (regular polygons). Crystal cells in three dimensions have triclinic, monoclinic, orthorhombic, hexagonal, rhombohedral, tetragonal and cubic shapes. The geometric symmetry of a crystal manifests itself in its physical properties, reducing the number of independent components of a physical property tensor, or forcing some components to zero values. There is therefore an important need to efficiently analyze the crystal cell symmetries. Mathematics based on geometry itself offers the best descriptions. Especially if elementary concepts like the relative directions of vectors are fully encoded in the geometric multiplication of vectors.
Category: Condensed Matter

[13] viXra:1306.0158 [pdf] submitted on 2013-06-18 22:51:41

Interactive 3D Space Group Visualization with CLUCalc and the Clifford Geometric Algebra Description of Space Groups

Authors: Eckhard Hitzer, Christian Perwass
Comments: 27 Pages. 21 figures, 14 tables. Adv. Appl. Clifford Alg., Vol. 20(3-4), pp. 631–658, (2010), DOI 10.1007/s00006-010-0214-z

A new interactive software tool is described, that visualizes 3D space group symmetries. The software computes with Clifford (geometric) algebra. The space group visualizer (SGV) originated as a script for the open source visual 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 first give some insights into the Clifford geometric algebra description of space groups. We relate the choice of symmetry vectors and the origin of cells in the geometric algebra description and its implementation in the SGV to the conventional crystal cell choices in the International Tables of Crystallography (T. Hahn, Springer, 2005). Finally we briefly explain how to use the SGV beginning with space group selection. The interactive computer graphics can be used to fully understand how reflections combine to generate all 230 three-dimensional space groups. Mathematics Subject Classification (2000). Primary 20H15; Secondary 15A66, 74N05, 76M27, 20F55 . Keywords. Clifford geometric algebra, interactive software, space groups, crystallography, visualization.
Category: Condensed Matter

[12] viXra:1306.0156 [pdf] submitted on 2013-06-19 01:42:14

Interactive 3D Space Group Visualization with CLUCalc and Crystallographic Subperiodic Groups in Geometric Algebra

Authors: Eckhard Hitzer, Christian Perwass, Daisuke Ichikawa
Comments: 21 Pages. 11 figures, 7 tables. In G. Scheuermann, E. Bayro-Corrochano (eds.), Geometric Algebra Computing, Springer, New York, 2010, pp. 385-400. DOI: 10.1007/978-1-84996-108-0_18

The Space Group Visualizer (SGV) for all 230 3D space groups is a standalone PC application based on the visualization software CLUCalc. We first explain the unique geometric algebra structure behind the SGV. In the second part we review the main features of the SGV: The GUI, group and symmetry selection, mouse pointer interactivity, and visualization options.We further introduce the joint use with the International Tables of Crystallography, Vol. A [7]. In the third part we explain how to represent the 162 socalled subperiodic groups of crystallography in geometric algebra. We construct a new compact geometric algebra group representation symbol, which allows to read off the complete set of geometric algebra generators. For clarity we moreover state explicitly what generators are chosen. The group symbols are based on the representation of point groups in geometric algebra by versors.
Category: Condensed Matter

[11] viXra:1306.0154 [pdf] submitted on 2013-06-19 02:06:08

Crystallographic Space Groups: Representation and Interactive Visualization by Geometric Algebra

Authors: Eckhard Hitzer, Christian Perwass
Comments: 6 Pages. 2 figures, 3 tables. submitted to: Proceedings of the 26th Int. Conference on Group Theoretical Methods in Physics, New York, USA, 2006.

We treat the symmetries of crystal space lattices in geometric algebra (GA)~\cite{DH:PGSG}. All crystal cell point groups are generated by geometric multiplication of two or three physical cell vectors. Only one or two relative angles subtended by these vectors need to be known. This treatment extends to space groups by including translations. GA helps to identify optimal multivector generators. As example we take the monoclinic case. New free interactive OpenGL and GA based software visualizes these symmetries.
Category: Condensed Matter

[10] viXra:1306.0153 [pdf] submitted on 2013-06-19 02:44:44

The Hidden Beauty of Gold

Authors: Eckhard Hitzer, Christian Perwass
Comments: 11 Pages. 24 figures. Proc. of Int. Symp. on Adv. Mech. & Power Engin. 2007 (ISAMPE 2007) between Pukyong Nat. Univ. (Kor.), Univ. of Fukui (Jap.) and Univ. of Shanghai for Sci. & Tech. (PRC), Nov. 22-25, 2007, at Univ. of Fukui, pp. 157-167. Figs. 15,16,17,23 rv.

This paper first reviews the history, the economy, the material properties, and the applications of gold. Then the geometry of the face centered cubic (fcc) gold lattice is introduced. Based on the symmetric arrangement of atoms the gold lattice has a rich variety of symmetry transformations, that interchange the positions of atoms, but leave the lattice as a whole invariant. This begins with the point group symmetry of a single fcc lattice cell and is extended by combination with lattice translations to the full space group symmetry of the whole (practically infinite) lattice. We use the newly created interactive Space Group Visualizer (based on geometric algebra) in order to systematically picture all these symmetries. We can thus understand their origin and their relationships. In particular we give a full geometric explanation of the 192 screw symmetries passing through a single fcc cell of the gold lattice.
Category: Condensed Matter

[9] viXra:1306.0152 [pdf] submitted on 2013-06-19 02:52:40

The Space Group Visualizer

Authors: Eckhard Hitzer, Christian Perwass
Comments: 9 Pages. 18 figures, 1 table. Proc. of In. Symp. on Adv. Mech. Eng. 2006 between Pukyong Nat. Univ. (Kor.), Univ. of Fukui (Jap.) and Univ. of Shanghai for Sci. and Techn. (PRC), Oct. 26-29, 2006, at Univ. of Shanghai for Sci. and Techn. pp. 172-181 (2006).

A new free interactive OpenGL software tool is demonstrated, that visualizes all monoclinic, and so far part of the orthorhombic, triclinic and hexagonal space group symmetries. The software computes with Clifford (geometric) algebra. The space group visualizer originated as a script for the open source visual CLUCalc, which fully supports geometric algebra computation. This paper briefly describes the historical and scientific developments leading to the space group visualizer project. Then we step by step demonstrate space group selection and the powerful set of interactive tools, including continuous free interactive 3D rotations, repositioning and resizing of the crystal domain in view. The most prominent feature of the space group visualizer is the full visualization of all spatial symmetries of a crystal domain. Beyond this the user can reduce the view to single symmetry operations or to certain classes of symmetries.
Category: Condensed Matter

[8] viXra:1306.0151 [pdf] submitted on 2013-06-19 02:58:51

Space Group Visualizer for Monoclinic Space Groups

Authors: Eckhard Hitzer, Christian Perwass
Comments: 2 Pages. 4 figures. Bulletin of the Society for Science on Form, 21(1), pp. 38,39 (2006).

A new free interactive OpenGL software tool is demonstrated, that visualizes all monoclinic space group symmetries described by geometric algebra.[1] Keywords: Crystal lattice, space group symmetry, geometric algebra, OpenGL, spacegroup visualizer.
Category: Condensed Matter

[7] viXra:1306.0149 [pdf] submitted on 2013-06-19 03:38:52

Interactive Visualization of Full Geometric Description of Crystal Space Groups

Authors: Christian Perwass, Eckhard Hitzer
Comments: 6 Pages. 9 figures. Proc. of the In. Symp. on Adv. Mech. Eng., between Univ. of Fukui (Japan), Pukyong Nat. Univ. (Korea) and Univ. of Shanghai for Sci. and Techn. (China), 23-26 Nov. 2005, pp. 276-282 (2005).

In this text we present a software tool that visualises the symmetry properties of the space groups of 3D Euclidean space, which play an important role in the investigation of crystalline materials. The main source that lists the properties of all space groups are the "International Tables For Crystallography, Volume A" [1], where the symmetries are shown in three orthographic projections. It is clearly much more intuitive to look at these symmetry properties in a 3D visualisation. The visualisation software presented here (for monoclinic crystals) allows the user to look at the space group symmetries from any view point and to modify lattice parameters in real time. The visualisation software is freely available from www.spacegroup.info.
Category: Condensed Matter

[6] viXra:1306.0148 [pdf] submitted on 2013-06-19 03:45:12

Full Geometric Description of All Symmetry Elements of Crystal Space Groups by the Suitable Choice of Only Three Vectors for Each Bravais Cell or Crystal Family

Authors: Eckhard Hitzer, Christian Perwass
Comments: 7 Pages. 7 figures, 2 tables. Proc. of the Int. Sym. on Adv. Mech. Eng., between Univ. of Fukui (Japan), Pukyong Nat. Univ. (Korea) and Univ. of Shanghai for Sci. and Techn. (China), 23-26 Nov. 2005, pp. 19-25 (2005).

This paper focuses on the symmetries of crystal cells and crystal space lattices. All two dimensional (2D) and three dimensional (3D) point groups of 2D and 3D crystal cells are exclusively described by vectors (two in 2D, three in 3D for one particular cell) taken from the physical cells. Geometric multiplication of these vectors completely generates all symmetries, including reflections, rotations, inversions, rotary-reflections and rotary-inversions. The sets of vectors necessary are illustrated in drawings. We then extend this treatment to 2D and 3D space groups by including translations, glide reflections and screw rotations. For 3D space groups we focus on the monoclinic case as an example. A companion paper [15] describes corresponding interactive visualization software.
Category: Condensed Matter

[5] viXra:1306.0146 [pdf] submitted on 2013-06-19 03:49:27

Crystal Cells in Geometric Algebra

Authors: Eckhard Hitzer, Christian Perwass
Comments: 6 Pages. 5 figures. Proceedings of the International Symposium on Advanced Mechanical Engineering, between University of Fukui (Japan) - Pukyong National University (Korea), 27 Nov. 2004, pp. 290-295 (2004).

This paper focuses on the symmetries of space lattice crystal cells. All 32 point groups of three dimensional crystal cells are exclusively described by vectors (three for one particular cell) taken from the physical cell. Geometric multiplication of these vectors completely generates all symmetries, including reflections, rotations, inversions, rotary-reflections and rotary-inversions. The sets of vectors necessary are illustrated in drawings and all symmetry group elements are listed explicitly as geometric vector products. Finally a new free interactive software tool is introduced, that visualizes all symmetry transformations in the way described in the main geometrical part of this paper.
Category: Condensed Matter

[4] viXra:1306.0145 [pdf] submitted on 2013-06-19 03:58:59

Algorithm for Conversion Between Geometric Algebra Versor Notation and Conventional Crystallographic Symmetry-Operation Symbols

Authors: Eckhard Hitzer, Christian Perwass
Comments: 14 Pages. 6 tables. Preprint (2009).

This paper establishes an algorithm for the conversion of conformal geometric algebra (GA) [3, 4] versor symbols of space group symmetry-operations [6–8, 10] to standard symmetry-operation symbols of crystallography [5]. The algorithm is written in the mathematical language of geometric algebra [2–4], but it takes up basic algorithmic ideas from [1]. The geometric algebra treatment simplifies the algorithm, due to the seamless use of the geometric product for operations like intersection, projection, rejection; and the compact conformal versor notation for all symmetry operations and for geometric elements like lines and planes. The transformations between the set of three geometric symmetry vectors a,b,c, used for generating multivector versors, and the set of three conventional crystal cell vectors a,b,c of [5] have already been fully specified in [8] complete with origin shift vectors. In order to apply the algorithm described in the present work, all locations, axis vectors and trace vectors must be computed and oriented with respect to the conventional crystall cell, i.e. its origin and its three cell vectors.
Category: Condensed Matter

[3] viXra:1306.0135 [pdf] submitted on 2013-06-17 05:02:34

Interactive Visualization of Plane Space Groups with the Space Group Visualizer

Authors: Eckhard Hitzer
Comments: 36 Pages. 32 figures, 1 table.

This set of instructions shows how to successfully display the 17 two-dimensional (2D) space groups in the interactive crystal symmetry software Space Group Visualizer (SGV) [6]. The SGV is described in [4]. It is based on a new type of powerful geometric algebra visualization platform [5]. 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 summarized in the lookup table Table 1. 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. 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.
Category: Condensed Matter

[2] viXra:1306.0129 [pdf] submitted on 2013-06-17 01:41:53

Interactive Visualization of Plane Groups

Authors: Eckhard Hitzer
Comments: 4 Pages. 3 figs, 1 tab. Symm.: Art + Sci., Spec. Iss. of The Jour. of the Int. Soc. For the Interdisc. Study of Symmetry (ISIS): G. Lugosi, D. Nagy (eds.), Proc. of Symm.: Art + Sci., 8th Congr. ISIS, Days of Harmonics, Austr., Aug. 2010, v 2010 (1-4), pp. 80-83.

This contribution shows how to successfully display the 17 two-dimensional space groups (wallpaper groups) in the interactive crystal symmetry software Space Group Visualizer (SGV) (Perwass & Hitzer, 2005). We show examples of four wallpaper groups that contain (as sub patterns, i.e as subgroups) all other 13 wallpaper groups. The SGV is described in (Hitzer & Perwass, 2010). It is based on a new type of powerful geometric algebra visualization platform (Perwass, 2000).
Category: Condensed Matter

[1] viXra:1306.0123 [pdf] submitted on 2013-06-17 03:00:32

Symmetry of Orthorhombic Materials and Interactive 3D Visualization in Geometric Algebra

Authors: Daisuke Ichikawa, Eckhard Hitzer
Comments: 11 Pages. 12 figures, 4 tables. Proc. of the Int. Symp. on Adv. Mechanical and Power Engineering 2007, between Univ. of Fukui (Japan), Pukyong Nat. Univ. (Korea) and Univ. of Shanghai for Science and Technology (China), 22--25 Nov. 2007, 302-312 (2007).

The Space Group Visualizer is the main software that we use in this work to show the symmetry of orthorhombic space groups as interactive computer graphics in three dimensions. For that it is necessary to know the features and the classification of orthorhombic point groups and space groups. For representing the symmetry transformations of point groups and space groups, we employ (Clifford) geometric algebra. This algebra results from applying the associative geometric product to the vectors of a vector space. Some major features of the software implementation are discussed. Finally a brief overview of interactive functions of the Space Group Visualizer is given.
Category: Condensed Matter