Calculation of the Higher Order Dipole-Dipole Effect in Paramagnetic Crystals

This report is an attempt to investigate the influence of dipole-dipole coupling in a paramagnetic spin system at low temperatures. It consists of two parts. The first part is a discussion of the use of C(M)=C(H=)(0) for a system with mutual interaction. It is pointed out that only if the external field is large compared to the internal field is this equation correct. The other part consists of a calculation of higher order correction of the dipole-dipole interaction on a system of paramagnetic spins which is subject to a crystalline field which we chose of the Y(2,0) type. The total Hamiltonian consists of a spin Hamiltonian in accordance with this symmetry, a term representing the external magnetic field and the dipole-dipole interaction between the spins. The partition function is calculated by means of the Schwinger trace formula considering a representation in which the first two terms of the Hamiltonian are diagonal. The trace of the density matrix can be expressed as the trace of a product, one factor is the density matrix of the noninteracting spins, the other factor consists of a sum of commutators. These commutators are worked out in detail and the result is given in the form of a finite series over the quantum number m. There seems to be no obvious way to perform these summations.