Theoretical Investigation of the Electronic Energy Band Structure of Solids

Publisher Summary This chapter discusses the space groups and their representations. Space groups are special groups of operators of the type {α|}. They are characterized by the fact that they possess an invariant subgroup of translations of particular form. Once one knows that a space group corresponds to a given crystal class one gains information about the possible invariant subgroups of primitive translations that it can possess. Crystals occur naturally and are recognized by the fact that they are bounded by plane faces. The plane faces are an outward indication of the underlying regular arrangement of the atoms and molecules in the crystal. In spite of this regularity, it is well known that a crystal of a given substance is not an exact scale model of another crystal of the same substance. It is also well known that the external macroscopic symmetry of the crystal of a given substance may not display the symmetry of the point group associated with the space group to which the substance belongs.

Space Groups and Their Representations

Publisher Summary This chapter discusses electron energy bands in solids. A convenient assumption to make in beginning the discussion of the behavior of electrons in crystals is that one may consider the motion of only one particle in a periodic potential. This is an oversimplification, for one really has a many-electron problem. It should be apparent that the E(k) relation is one of the most important properties of a material. It is involved in a basic way in any calculation of the electric, optical, or thermal characteristics of the substance. The calculation of E(k) relations for actual materials is extremely complicated. Many considerations of electrical properties, for example, are based on extremely simplified models of the real band structure. Progress has been made, however, in determining energy bands in a number of the simpler metals and semiconductors. Although a theory for finding the exact energy to be associated with a model state exists, it has not yet been possible to make calculations for real systems. All existing calculations of energy bands are based on the use of one model function. The best model function is that for which the expectation value of the energy is a minimum, subject to the conditions that the one-particle functions which compose it be orthonormal.

Electron Energy Bands in Solids

The Electronic Structure of Solids

The described method starts from the assumption that each valency electron in a crystal is subject to the following potential: inside the atomic sphere surrounding each atom this potential is spherically symmetrical while outside this sphere it is constant. The corresponding Schrodinger equation is solved by means of the perturbation theory with the help of certain approximate wave functions. The parameters determining the spherically symmetrical potential inside the sphere are determined from the values of energy for the lowest energy level, calculated by statistical methods.

A new method for calculating the energy levels of electrons in solids

In a few papers which will be published in the near future the two authors deal with the electron theory of metallic magnesium and aluminium and the interpretation of some of the properties of these elements or their solid solutions. This paper deals with the group analysis of the wave functions of valency electrons in the first Brillouin zone of both metals and shows how it is possible in general to construct an irreducible representation of a space group for the case that this group contains screw axes and gliding planes.

E. Antončík

Papers

A note on the group analysis of the wave functions of valency electrons in a crystal