The Influence of Crystalline Fields on the Susceptibilities of Salts of Paramagnetic Ions. I. The Rare Earths, Especially Pr and Nd

The paramagnetic susceptibilities of the salts of the rare earth elements Pr and Nd are considered on the assumption that it is permissible to represent the potential of the electric field of the atoms surrounding the metallic ion by a Taylor's expansion. This amounts to applying to the whole crystal the method of the "self-consistent field, " and consequently neglects exchange effects between different crystal atoms. Hund has calculated the susceptibilities on the assumption that the ion can be regarded as free and that the multiplet intervals are so large compared with kT that only the lowest level need be considered. The introduction of an electric field causes a splitting of the levels and a redistribution of magnetic moment, with a consequent change in the susceptibility. The theoretical interpretation of the Curie-Weiss law =C (T+) is considered. At temperatures so high that kT is large compared with the splitting produced by the crystal field, the susceptibility can be expanded in the form of a series of inverse powers of T. It is shown that the susceptibility of a crystal powder, or the average susceptibility over all directions in a single crystal is of the form =CT+C₂{T^3}+ the term in 1{T^2} vanishing rigorously. Thus at sufficiently high temperatures, the susceptibility of a crystal powder obeys the simple Curie law up to and including terms in 1{T^2}. However, the curious result emerges that at ordinary temperatures kT is of the same order as the energy separations produced by the crystal field, and the behaviour of the susceptibility actually simulates the Curie-Weiss law closely over a large range of temperatures. The hydrated sulphates of Pr and Nd are considered in detail. Excellent agreement is obtained with the experimental results of Gorter and de Haas for the variation of susceptibility with temperature on the assumption that the crystal field has cubic symmetry, and can be represented by a potential D (x^4+y^4+z^4). In this connection the matrix elements of the squares and fourth powers of the coordinates for a many-electron system are given. The over-all splitting produced by this field in the hydrated sulphates is 389 cm^-1 for Pr and 834 cm^-1 for Nd, the detailed appearance of the energy spectrum being shown in Fig. 1. A comparison of the constant D in the two cases gives a value for Nd nearly four times that for Pr.