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

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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