Collective electron ferromagnetism

Abstract The essential contribution of Heisenberg to the theory of ferromagnetism was in showing that those effects which had been correlated by means of the formal molecular field hypothesis of Weiss could be interpreted as arising from interchange interaction between electrons in atoms, of the same type as that involved in the formation of homopolar molecules. The Heisenberg method of approach has, however, proved in many ways less convenient in the detailed treatment of ferromagnetism than the method initiated by Bloch for the theory of metallic properties generally, in which possible energy states are derived for electrons treated as waves travelling through the whole crystal. The first approximation in this collective electron treatment is that of free electrons, for which the energy is purely kinetic, the number of states per unit energy range then being proportional to the square root of the energy. The effect of the periodic field of the lattice is to modify the distribution of states, giving rise to a series of energy bands, separate or overlapping. Elaborate calculation is necessary to determine the form of these bands with any precision, though in general near the bottom of a band the energy density of states depends on the energy in the same way as for free electrons, but with a different pro­portionality factor; this holds also near the top of a band, the energy being measured downwards from that limit. The salient characteristics of metals depend on the electrons in unfilled bands. In particular, in the ferro­magnetic metals, iron, cobalt and nickel, the ferromagnetism may be attributed to the electrons in the partially filled band corresponding to the d electron states in the free atoms. The exchange interaction is such that, at low temperatures, instead of the electrons occupying the lowest states in balanced pairs, there is an excess of electrons with spins pointing in one direction, giving rise to a spontaneous magnetization. The decrease of energy due to the exchange effect with increase in the number of excess parallel spins is accompanied by an increase due to the electrons moving to states of higher energy in the band. The equilibrium magnetization depends on the number of electrons, the form of the band, the magnitude of the exchange interaction, and the temperature, and must be calculated on the basis of Fermi-Dirac statistics. The primary purpose in this paper is the determination of the form of the magnetization temperature curves for bands of the standard type and for a range of values of the exchange interaction energy. Before discussing the particular problem more fully, the relation between the present and some of the previous work will be briefly indicated. The advantages of a collective electron treatment for ferromagnetism were pointed out some years ago in a paper (Stoner 1933) in which it was shown that such a treatment enabled an immediate interpretation to be given to the non-integral values of the atomic moments of the ferromagnetic metals, and of the variation of the moment with small additions of non­ferromagnetic metals in alloys. Little was then known about the form of the electronic energy bands in transition metals, and the treatment was necessarily qualitative. Considerably greater precision in formulation became possible on the basis of a suggestion by Mott (1935), having a general justification, on the form of the energy bands in the ferromagnetic metals. In nickel, in particular, it was suggested that a narrow d band was overlapped by a much wider s band, and that the top of the Fermi distri­bution came at a point corresponding to 0·6 electron/atom in the s band, with a deficit of the same number in the d band; this number corresponding to the observed saturation moment of nickel at low temperatures. (It may be noted that “holes” in bands are to a large extent magnetically equi­valent to the same number of electrons in otherwise empty bands, as is more fully discussed elsewhere (Stoner 1936a).) This general idea of Mott has been developed successfully in a number of directions (cf. Mott and Jones 1936). A first step in the quantitative treatment of the effect of temperature on the magnetic properties of metals was the determination of the temperature dependence of free electron susceptibility (Stoner 1935, 1936b); the results obtained are applicable not only to free electrons but also to electrons in unfilled bands with the same type of energy distri­bution of states. Series expressions were derived appropriate to high and low temperatures, and values for the intermediate temperature range were found by graphical interpolation. In a subsequent discussion on spin paramagnetism in metals (Stoner 1936a) a simple method of taking into account the effect of interchange interaction was described, the method being applicable to the determination of the temperature variation of the susceptibility of a ferromagnetic above the Curie point. Shortly after­ wards the treatment was extended to deal with the spontaneous magnetization below the Curie point and the general character of the modifications resulting from the use of Fermi-Dirac, in place of classical, statistics, was determined. It was found, however, that the method previously used, involving series calculations for high and low temperatures, and graphical interpolation, was inadequate to give numerical results of satisfactory precision, the lack of precision being most marked for the temperature range which was often of greatest interest as including the Curie point region. For precise numerical results, an accurate evaluation of a series of the basic Fermi-Dirac functions was indispensable. An extensive table of Fermi-Dirac functions is now available (McDougall and Stoner 1938) and this provides the necessary starting point for much of the computational work involved in the present paper.