The Band Structure of Body-centred Cubic Transition Metals

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Abstract
The theory of the magnetic properties of iron put forward by Mott and Stevens made a number of assumptions about the band structures of the body-centred cubic 3d transition metals. The purpose of the present paper is to give an account of a theoretical study which has been made to see what could reasonably be expected on fairly general grounds about the band structures. It then becomes possible to see more clearly what innovations are required for the Mott and Stevens theory. The method used is essentially a tight binding method, with the feature that the functions used are not located at lattice sites but are on the links joining nearest neighbour lattice points. It leads to an 8 × 8 matrix for the lattice potential for each value of K. A discussion of the diagonalization of this 8 × 8 matrix is given for a number of symmetry points and directions in K-space, and it is shown that a good deal of valuable information can be obtained from relatively simple ideas. It is also shown that the band structure so obtained is incomplete because two of the d-functions are omitted, and these are then included by perturbation methods. The Mott and Stevens model of iron is shown to follow from assumptions about the results of this perturbation treatment. A discussion is given of the ground state determinantal functions for vanadium, chromium and iron which it is hoped will be of use in interpreting the physical properties of these elements. Apart from the value the theory may have in its application to the above metals, the algebraic treatment provides an elegant example of the effects of symmetry on band structures, previously thoroughly discussed in terms of groups by Bouckaert, Smoluchowski and Wigner.

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