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2018 | Book

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Computer Algebra and Materials Physics

A Practical Guidebook to Group Theoretical Computations in Materials Science

Author: Dr. Akihito Kikuchi

Publisher: Springer International Publishing

Book Series : Springer Series in Materials Science

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This book is intended as an introductory lecture in material physics, in which the modern computational group theory and the electronic structure calculation are in collaboration.
The first part explains how to use computer algebra for applications in solid-state simulation, based on the GAP computer algebra package. Computer algebra enables us to easily obtain various group theoretical properties, such as the representations, character tables, and subgroups. Furthermore it offers a new perspective on material design, which could be executed in a mathematically rigorous and systematic way.
The second part then analyzes the relation between the structural symmetry and the electronic structure in C60 (as an example of a system without periodicity). The principal object of the study was to illustrate the hierarchical change in the quantum-physical properties of the molecule, which correlates to the reduction in the symmetry (as it descends down in the ladder of subgroups).
The book also presents the computation of the vibrational modes of the C60 by means of the computer algebra. In order to serve the common interests of researchers, the details of the computations (the required initial data and the small programs developed for the purpose) are explained in as much detail as possible.

Table of Contents

Frontmatter
Chapter 1. Introduction
Abstract
In the history of physics, the cooperation of physicists and mathematicians has yielded great harvest in the first half of the twentieth century, as the installation of group theory in quantum mechanics (Weyl 1950). One of the main applications is realized in the field of solid-state physics (Inui et al. 1990; Tinkham 2003; Dresselhaus 2008). In subsequent years, however, such productive relationship between physics and mathematics became enfeebled, as physicists and mathematicians were pursuing their own interests separately. In material science, the typical tool of study has turned into “first principle electronic structure computation”, in which rapid computers are intensively used so that the quantitative simulation could be achieved. In contrast, the group theoretical view in the quantum physics is rather a qualitative one, which could explain the likeness in similar material structures, but could not illuminate the origin of subtle but distinct differences. The standpoints of group theoretical analysis, and of first principles simulation, are located at cross-purposes. This is one of the reasons which brought about the breaking-off between the group theory and the first principles electronic structure computation.
Akihito Kikuchi
Chapter 2. Computation of Group Theoretical Properties Using “GAP”
Abstract
The computational discrete algebra package GAP is one of powerful tools for the computation in the group theory (The Gap Group 2017). It can be applied to the determination of the group theoretical properties which are necessary for solid-state electronic structure computation. For this purpose, the point group of the crystal must be determined. The group operations are to be listed up; the multiplication table is to be prepared; through which the character table is computed.
Akihito Kikuchi
Chapter 3. Some Preliminaries
Abstract
This chapter gives a short account of some preliminaries (such as the definition of projection operators, crystallographic groups, Wickoff positions) which will be necessary in the later part of this monograph.
Akihito Kikuchi
Chapter 4. Application 1: Identification of Wavefunctions to Irreducible Representations
Abstract
The application in this chapter is the classical example in group theory in quantum physics. As the example of the application of computer algebra, this chapter demonstrates how to allot wavefunctions (obtained by the first-principles electronic structure computations) to the irreducible representations (obtained by the group-theoretical computation by GAP). The special care for the computation of this kind in the super-cells (the stacks of the duplicated minimal primitive lattice cells) is discussed.
Akihito Kikuchi
Chapter 5. Application 2: A Systematic Way of the Material Designing
Abstract
This chapter, based on the group theoretical view, proposes a plan for the purpose of managing quantum dynamical properties. The reduction of the symmetry (for example, by means of placements of atoms or defects on Wickoff positions) would produce the graded changes in the electronic structures. Concerning this, as a benefit of the computer algebra, the whole possible reductions of the symmetry can be listed up in the tree structure of the subgroups. We can follow this structure (so-called the subgroup lattice) and inspect the systematic change in the electronic structures. This would be useful in the material design in the artificial super-lattice structure.
Akihito Kikuchi
Chapter 6. Technical Details
Abstract
This chapter describes technical topics, such as the determination of crystal symmetry, the symmetry operations in the reciprocal space, the computation of the compatibility relation (which is related to the splitting of degenerated levels from a special point to another in a band dispersion), the character table at the boundary of the Brillouin zone, the special treatment for non-symmorphic crystal, the computation of characters of the group of the super cell by means of “semi-direct product” (which is an effective way to extend groups), the crystallography from the theory of homology.
Akihito Kikuchi
Chapter 7. Symmetry in C
Abstract
The C\(_{60}\) molecule is characterized by the peculiar structure, that of soccer-ball, composed of 60 atoms, 90 bonds, and 12 pentagons, and 60 hexagons. The group theoretical symmetry is of the icosahedral type, which contains 120 symmetric operations and 10 irreducible representations, according to which, the eigenstates are classified and the origins of degeneracies in energy spectrum are explained. We begin from the construction of the symmetric operations by means of permutations of atoms, generate the icosahedral group by GAP (The Gap Group 2017), compute the irreducible representations, and compose the projection operators. We execute the electronic structure calculations of the molecule, and analyze the wave-functions from the viewpoint of their irreducible representation. Furthermore, some concepts of symmetries (to which physicists pays little notice) are presented also by the analysis of the computer algebra. Throughout the study, it is necessary that whole subgroups of icosahedral symmetry should be extracted and classified by equivalence, and we will see that the utilization of the computer algebra is effective for this purpose.
Akihito Kikuchi
Chapter 8. Analysis of Vibrational Mode in C
Abstract
In this chapter, the computation of the vibrational modes of the C\(_{60}\) by means of the computer algebra is demonstrated. The C\(_{60}\) molecule has the unique symmetry, comparable to one soccer-ball (see Fig. 7.​1). The symmetry of the molecule is described by the icosahedral group, and this group is represented by the permutation of 60 vertexes or the group of rotation matrices operating on them. Owing to this high symmetry, the molecule shows peculiar electronic and dynamic properties, both of scientific and industrial interest, and the vibrational mode might be one of the principal phenomena in governing the quantum dynamics in it. As a tool of the computation, we utilize the software GAP, developed in the field of the pure mathematics (The Gap Group 2017). This software can construct the symmetry group and compute the irreducible representations and other group-theoretical properties and enable us to put into practice the application in the material science, possibly in a modernized way. The computation presented hereafter will be useful to the working physicists and the students, who want to deepen the understanding of the group theory that are given in the textbooks (Dresselhaus et al. 2008; Inui et al. 1990; Burnside 1897; Serre 1977; Fulton and Harris 2004).
Akihito Kikuchi
Chapter 9. Final Remarks
Abstract
This chapter reviews the content of this monograph.
Akihito Kikuchi
Backmatter
Metadata
Title
Computer Algebra and Materials Physics
Author
Dr. Akihito Kikuchi
Copyright Year
2018
Electronic ISBN
978-3-319-94226-1
Print ISBN
978-3-319-94225-4
DOI
https://doi.org/10.1007/978-3-319-94226-1

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