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a r A t'a critic al value, h0, o f the field, for a given value of*, the angle o f magnetization c from K to & (or from f 0 to f 0). The relation most readily obtained is that between ton Ko

I n t r o d u c t i o nThe essential contribution of Heisenberg to the theory of ferromagnetism was in showing th a t 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 th a t involved in the form ation of homopolar molecules. The Heisenberg m ethod of approach has, however, proved in m any ways less convenient in the detailed tre a tm e n t of ferrom agnetism th a n the method initiated by Bloch for the theory of m etallic properties generally, in which possible energy states are derived for electrons treated as waves travelling through the whole crystal. The first approxim ation in this collective electron treatm en t is th a t of free electrons, for which the energy is purely kinetic, the num ber of states per un it 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. E laborate calculation is necessary to determ ine the form of these bands w ith 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, b u t w ith a different pro portionality factor; this holds also near the top of a band, the energy being m easured downwards from th a t limit. The salient characteristics of metals depend on the electrons in unfilled bands. In particular, in the ferro m agnetic m etals, iron, cobalt and nickel, the ferromagnetism may be a ttrib u te d to the electrons in the partially filled band corresponding to the d electron states in the free atom s. The exchange interaction is such th a t, a t low tem peratures, instead of the electrons occupying the lowest states in balanced pairs, there is an excess of electrons w ith spins pointing in one direction, giving rise to a spontaneous m agnetization. The d,ecrease of energy due to the exchange effect w ith increase in the num ber 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 [ 372 ] on May 12, 2018 http://rspa.royalsocietypublishing.org/ Downloaded from

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The quantitative application of Fermi-Dirac statistics involves the evaluation of certain integrals which have not previously been tabulated. In this paper, tables are given of the values of the basic integrals most frequently required , with a view to placing Fermi-Dirrac statistics on as firm a numerical basis as is Maxwell-Boltzmann statistics. T e expression for the energy distribution of particles subject to Fermi-Dirrac statistics may be written in the form dN He) de e<*+Pe -)-1 ’ wherev(e) is the number of states per unit energy range, and dN is the number of particles in the energy range e to e--de. In the statistical treatment, the parameters ot and fi, which are usually introduced as undetermined multipliers in a variational equation, are to be determined from two equations expressing conditions imposed by the total number of particles, and the total energy of the system. By linking up the statistical and thermodynamical treatments, interpretation can be given to a and b this is expressed by P**:l IkT, a = -C lk T ,