Some computations on the band structure of metals

• Main Author: Cracknell, A. P.
• Published: University of Oxford 1964
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.732585

The work is in two parts, the first part is concerned with the theory of symmetry adapted functions for all the cubic space groups and the second pert is concerned with an application of this theory to band structure calculations, using the cellular method, for metals of one particular space group (Fm3m) and the physical interpretation of the bands computed. <strong>PART 1.</strong> In the first chapter a brief introduction is given to the concepts and notation which are used extensively in the following chapters. In these chapters lattice harmonics are given for all the irreducible representations of the simple cubic (Pm3m), face-centred cubic (Fm3m) and bod-centred cubic (Im3m) space groups for 1 &leq; 12. A general discussion of the treatment of asymmorphic space groups is given and one example is considered by way of illustration of the general method. The irreducible representations and lattice harmonics are discussed but the complete results which have been obtained for all the cubic and tetragonal groups have not been reproduced here for lack of space. All the expansions are given in polar coordinates and care has been taken that different bases corresponding to the same representation span identical rather equivalent representations, which are given in full. Moreover, all the different expansions listed in the tables are fully orthogonal. <strong>PART 2.</strong> Part 2 is concerned with the use of some of the lattice harmonics deduced in Part 1 in a cellular calculation on several face centred cubic metals. A programme has been written for a Ferranti "Mercury" computer which can be used for any face-centred cubic metal provided the lattice spacing and the potential due to a free ion of the metal are available. The programme, which is fully automatic, calculates energies and wave functions at the points of symmetry and along the lines of symmetry in the Brillouin zone and from this the band structure of the metal is reconstructed. Chapters I - IV describe the details of the theory and of the programme, and some of the difficulties and errors are considered. The results of the computations are patented in Chapter V. From these band structures the density of states and Fermi surface are computed and in Chapter VI the observable quantities deducible from them are considered; these include the temperature coefficient of the electronic specific heat, the Pauli spin paramagnetic susceptibility, the de Haas-van Alphen effect, the high pressure behaviour of the electrical resistance of calcium, the magnetoresistance, the Knight shift and the effective mass. In chapter VII a self-consistent potential field for calcium is considered; this is a Hartree self-consistent field among the conduction electrons with on approximate consideration of the inner 3s and 3p electrons. A good review, by Callaway (1964), of the present state of band theory has just been published since the writing of this thesis.