Nonorthogonality and Ferromagnetism

The calculations in Heisenberg's theory of ferromagnetism have been questioned by Inglis and others on the ground that the error resulting from the nonorthogonality of the wave functions may possibly increase without limit when the number of atoms becomes arbitrarily large. In the present paper it is proved that this difficulty does not really arise. Semiquantitative formulas are given to correct for the error due to nonorthogonality, which is shown to be of the order 2z^2 relative to unity, where z is the number of neighbors and is the overlap integral (1). A supplementary note is included on a new method of approximating the partition function in Heisenberg's theory. This approximation should be somewhat better than the assumption of a Gaussian distribution, but agrees even worse with experiment, provided one assumes orthogonality. Actually, the influence of nonorthogonality is sufficiently large to render uncertain any attempt to deduce exactly the critical conditions (minimum number of neighbors, etc. ) necessary for ferromagnetism.