The Theoretical and Experimental Status of the Collective Electron Theory of Ferromagnetism

This paper is intended to supplement the review articles published by Stoner in 1948 and 1951, firstly by considering in greater detail the quantum-mechanical and statistical-mechanical foundations of the collective electron theory and secondly by considering briefly a wider range of relevant experimental results.

In Sec. 1 previous theoretical work is recalled. In Sec. 2 the difficulties of a rigorous quantum mechanical derivation of the internal energy of a ferromagnetic metal at absolute zero are outlined. In order to determine at least the form of the expressions, a calculation based on the tight binding approximation is described for a crystal containing $N$ singly charged ions, which are fairly widely separated, and $N$ electrons. The forms of the Coulomb and exchange contributions to the energy are discussed in the two instances of maximum and minimum multiplicity. The need for a correlation correction is stressed, and the effects of this correction are discussed with special reference to the state of affairs at infinite ionic separation. The fundamental difficulties involved in calculating the energy as function of magnetization are considered below; it is shown that they are probably less serious for tightly bound than for free electrons, so that the approximation of neglecting them in the first instance is not too unreasonable. The dependence of the exchange energy on the relative magnetization $\zeta$ is then of the form $\Sigma n^{}\frac{A_n{\zeta }^{2n}}{2n}$, and the relative orders of magnitude of $A_1$ and $A_2$ are considered. A previous calculation by Slater is critically reviewed.

In an application of statistical mechanics the difficulty arises that only if $\zeta =1\mathrm{} or 0$ will the zero-order wave function be in general a single Slater determinant. A calculation by Lidiard on this problem, that of spin degeneracy, is referred to. The relevance of the dependence of the interaction energies on wave vector is stressed. If these energy contributions are constant, then the free energy expression is that derived by Stoner, based on Fermi-Dirac statistics, even if spin degeneracy is taken into account. Several related difficulties are exemplified by a discussion of the properties of the free-electron gas, for which the exchange energy varies rapidly with wave vector ${\mathbf{k}}_i$, ${\mathbf{k}}_j$ when $|{\mathbf{k}}_i-{\mathbf{k}}_j|\to 0$. For tightly bound electrons the dependence on wave vector is much less rapid, and it is suggested that spin degeneracy effects are here less serious. A simple calculation of the low temperature electronic heat of tightly bound electrons, including exchange and correlation, is referred to in exemplification.

In Sec. 3 several experimental results are briefly discussed, including the following: (1) The variation with composition of the Curie temperature and saturation magnetization, and the variation with temperature of the susceptibility of nickel alloys; (2) the variation with temperature of the magnetization, electronic heat, and magnetocaloric effect of cubic cobalt, nickel, and nickel alloys below the Curie point; (3) the high temperature electronic heat of nickel and palladium; (4) the electronic properties of chromium and some other transition metals; (5) the electronic properties of palladium and its alloys.