Molecular Field in the Spherical Model

The Heisenberg model of ferromagnetism is replaced by a classical model in which the interaction between a pair of neighboring atoms is -2JS (S+1) ₈₉, where S is the spin of any atom, J is the exchange integral, and the ₈ are classical unit vectors. The spherical model is then used to evaluate the molecular field acting on any atom i. This effective field is found to have the generalized Weiss form, H+W (T) M+W^' (T) M^'{y₈}^, where H is the magnetic field, M the magnetization, and M^' the antiferromagnetic order (in units of magnetization). The coefficient {y₈}^ changes sign from one sublattice to another. The "Weiss" coefficients W (T) and W^' (T) are slowly temperature-dependent and obey dWdT>0; dW^{'}dT>0. A phase transition is found in three dimensions, but not in one or two dimensions. For spin lattices of simple cubic, body-centered cubic, and face-centered cubic type, the ferromagnetic transition temperature T₂ in units kT₂J is found to be 1. 98, 2. 87, and 4. 45, respectively. Corresponding values for the simple cubic case due to P. R. Weiss and V. Zehler are 1. 85 and 1. 93 respectively. The susceptibility and the paramagnetic temperature are found for the three lattices. The ratio {T₂} is found to be independent of spin with the values 1. 52, 1. 39, and 1. 34 for the s. c. c. , b. c. , and f. c. c. lattices respectively. Corresponding results for the transition temperatures, susceptibility, etc. , are obtained for the antiferromagnetic cubic lattices.