Dipoles on a Lattice : the Spherical Model

The statistical treatment of a lattice of classical dipoles, electric or magnetic, has been intractable because of the ``strong'' condition that all dipoles have a fixed length. The spherical model consists in replacing the above strong condition by the weaker condition that the sum of the squares of the dipole vectors have the correct value, thus permitting some, but not much, fluctuation from the true situation. The previous work of Berlin and Thomsen using the spherical model and nearest neighbor interactions is generalized to include a mixture of long and short range interactions. The evaluation of the partition sum in the presence of long-range forces requires care because some of the eigenvalues depend on specimen shape. A formula has been derived for the susceptibility, however, in which the shape dependence is explicitly shown to cancel. In fact, it is proven that in the spherical model a permanent dipole lattice is equivalent to an induced dipole lattice with a certain effective polarizability. Expansion of this polarizability in inverse powers of the temperature shows precise agreement to terms of order (1/T)3 with an exact evaluation of the corresponding expansion coefficients. All qualitative results are in agreement with those of Berlin and Thomsen. One- and two-dimensional lattices show no transition. A simple cubic lattice (the case they treat) shows an antiferroelectric transition bounded by a critical field curve E=Ec1—(T/Tc)½. The effective field and its application to an elementary quantum mechanical treatment of the ferromagnetic problem are described.