Ebook: Primer for Point and Space Groups

This text stems from a course I have taught a number of times, attended by students of material science, electrical engineering, physics, chemistry, physical chemistry and applied mathematics. It is intended as an intro­ ductory discourse to give the reader a first encounter with group theory. The work concentrates on point and space groups as these groups have the principal application in technology. Here is an outline of the salient features of the chapters. In Chapter 1, basic notions and definitions are introduced including that of Abelian groups, cyclic groups, Sylow's theorems, Lagrange's subgroup theorem and the rearrangement theorem. In Chapter 2, the concepts of classes and direct products are discussed. Applications of point groups to the Platonic solids and non-regular dual polyhedra are described. In Chapter 3, matrix representation of operators are introduced leading to the notion of irreducible representations ('irreps'). The Great Orthogonal­ ity Theorem (GOT) is also introduced, followed by six important rules relating to dimensions of irreps. Schur's lemma and character tables are described. Applications to quantum mechanics are discussed in Chapter 4 including descriptions of the rotation groups in two and three dimensions, the symmetric group, Cayley's theorem and Young diagrams. The relation of degeneracy of a quantum state of a system to dimensions of irreps of the group of symmetries of the system are discussed, as well as the basis properties of related eigenfunctions.

---

Written in the spirit of Liboff’s acclaimed text on quantum mechanics, Primer for Point and Space Groups is an ideal introductory text for undergraduates in physics, engineering, materials science, chemistry. With its shrewd selection and arrangement of examples, the book traces at every turn the physical implications of abstract concepts. The book will therefore provide a solid background for those students who expect to use group theory in nuclear and particle physics and other specific applications. Among the many introductions to group theory currently available, few are as closely attuned to the needs of the applied scientist without compromise to the logic and lucidity of the presentation.

Here are some of the applications covered in Liboff’s Primer for Point and Space Groups:

Applications to Quantum Mechanics:

- Irreducible representations and degeneracy

- Full rotation group; SU(2) group; angular momentum

- Symmetry group and the wavefunction

- Young diagrams and the wavefunction

- Degenerate perturbation theory

- Great Orthogonality Theorem

Applications to Solid-State Physics:

- Translation and Crystallographic Point Groups

- Holohedral groups

- Bloch waves and space groups

- Bravais lattice

- Energy -band eigenenergies

- Seitz operator, (Translation and Rotation)

- Glide-plane and screw operators

- Diamond structure

- Group of {bf k} and Star of $bf k $.

- Space group of {bf k}

- Time reversal and Space inversion effects on the

- wavefunction and eigenenergies

- Symmorphic group

- Factor Group theorem

Applications to Material Media:

- Splitting of electron levels in crystals with symmetry

- Correlation diagrams

- Neumann's Principle

- Polarizability

- Piezoelectric effect

- Classification of magnetic crystals

- Black and white groups

---

Written in the spirit of Liboff’s acclaimed text on quantum mechanics, Primer for Point and Space Groups is an ideal introductory text for undergraduates in physics, engineering, materials science, chemistry. With its shrewd selection and arrangement of examples, the book traces at every turn the physical implications of abstract concepts. The book will therefore provide a solid background for those students who expect to use group theory in nuclear and particle physics and other specific applications. Among the many introductions to group theory currently available, few are as closely attuned to the needs of the applied scientist without compromise to the logic and lucidity of the presentation.

Here are some of the applications covered in Liboff’s Primer for Point and Space Groups:

Applications to Quantum Mechanics:

- Irreducible representations and degeneracy

- Full rotation group; SU(2) group; angular momentum

- Symmetry group and the wavefunction

- Young diagrams and the wavefunction

- Degenerate perturbation theory

- Great Orthogonality Theorem

Applications to Solid-State Physics:

- Translation and Crystallographic Point Groups

- Holohedral groups

- Bloch waves and space groups

- Bravais lattice

- Energy -band eigenenergies

- Seitz operator, (Translation and Rotation)

- Glide-plane and screw operators

- Diamond structure

- Group of {bf k} and Star of $bf k $.

- Space group of {bf k}

- Time reversal and Space inversion effects on the

- wavefunction and eigenenergies

- Symmorphic group

- Factor Group theorem

Applications to Material Media:

- Splitting of electron levels in crystals with symmetry

- Correlation diagrams

- Neumann's Principle

- Polarizability

- Piezoelectric effect

- Classification of magnetic crystals

- Black and white groups

Content:
Front Matter....Pages i-xiii
Groups and Subgroups....Pages 1-17
Classes and Platonic Solids....Pages 18-35
Matrices, Irreps and the Great Orthogonality Theorem....Pages 36-53
Quantum Mechanics, the Full Rotation Group, and Young Diagrams....Pages 54-88
Space Groups, Brillouin Zone and the Group of k....Pages 89-135
Atoms in Crystals and Correlation Diagrams....Pages 136-159
Elements of Abstract Algebra and the Galois Group....Pages 160-193
Back Matter....Pages 194-220

---

Written in the spirit of Liboff’s acclaimed text on quantum mechanics, Primer for Point and Space Groups is an ideal introductory text for undergraduates in physics, engineering, materials science, chemistry. With its shrewd selection and arrangement of examples, the book traces at every turn the physical implications of abstract concepts. The book will therefore provide a solid background for those students who expect to use group theory in nuclear and particle physics and other specific applications. Among the many introductions to group theory currently available, few are as closely attuned to the needs of the applied scientist without compromise to the logic and lucidity of the presentation.

Here are some of the applications covered in Liboff’s Primer for Point and Space Groups:

Applications to Quantum Mechanics:

- Irreducible representations and degeneracy

- Full rotation group; SU(2) group; angular momentum

- Symmetry group and the wavefunction

- Young diagrams and the wavefunction

- Degenerate perturbation theory

- Great Orthogonality Theorem

Applications to Solid-State Physics:

- Translation and Crystallographic Point Groups

- Holohedral groups

- Bloch waves and space groups

- Bravais lattice

- Energy -band eigenenergies

- Seitz operator, (Translation and Rotation)

- Glide-plane and screw operators

- Diamond structure

- Group of {bf k} and Star of $bf k $.

- Space group of {bf k}

- Time reversal and Space inversion effects on the

- wavefunction and eigenenergies

- Symmorphic group

- Factor Group theorem

Applications to Material Media:

- Splitting of electron levels in crystals with symmetry

- Correlation diagrams

- Neumann's Principle

- Polarizability

- Piezoelectric effect

- Classification of magnetic crystals

- Black and white groups

Content:
Front Matter....Pages i-xiii
Groups and Subgroups....Pages 1-17
Classes and Platonic Solids....Pages 18-35
Matrices, Irreps and the Great Orthogonality Theorem....Pages 36-53
Quantum Mechanics, the Full Rotation Group, and Young Diagrams....Pages 54-88
Space Groups, Brillouin Zone and the Group of k....Pages 89-135
Atoms in Crystals and Correlation Diagrams....Pages 136-159
Elements of Abstract Algebra and the Galois Group....Pages 160-193
Back Matter....Pages 194-220
....