Excitations in impure magnetic crystals

In this thesis we consider two types of magnetic system, the ferromagnet and the antiferromagnet, containing substitutional impurities. The Heisenberg Hamiltonian is used and the spin-waves are examined by means of Green function equations of motion; the coherent potential approximation (CPA) is employed to obtain expressions for self-energies which are approximately valid for all impurity concentrations. In each case, an extension of the usual CPA is made: in Part I the impurities are described by scattering potentials with matrix elements on several lattice sites, and in Part II the effects of clusters of defects are included. In Part I, we discuss the dilute ferromagnet, and attempt to obtain an estimate of the critical concentration of magnetic ions by considering the long-wavelength spin-waves. The self-energy is calculated in such a way as to sum coherently the scattering of spin-waves from all sites affected by one impurity, although correlations between pairs of defects are neglected. We examine solutions for a single non-magnetic impurity and for small impurity concentrations; it emerges from these considerations that it is essential that the resonance which the non-magnetic ions produce at zero energy should be kept there by any approximation which is to be of use in describing the low-energy spin-waves. A generalized form of the CPA, suitable for extended defect matrices, is developed and used to derive expressions for the self-energy; detailed numerical results are presented. We find that although the CPA behaves well over most of the spin-wave band, it fails to describe the zero-energy resonance correctly, and this failure prevents us from deducing a value for the critical concentration. Two attempts to avoid this difficulty are described: the first uses defect Green functions to fix the energy of the resonance, the second moves it away from zero and outside the spin-wave band, but neither enables us to estimate the critical concentration. We also discuss a magnetic analogue of the percolation bond problem, to which the application of the CPA is successful. In Part II we apply the CPA to a calculation of the spin-wave energies in antiferromagnets containing either magnetic or non-magnetic impurities. Each site is allowed to have any possible number of impurities among its nearest neighbours; in order to deal with this we extend the CPA to deal with a large number of different defect types. Variations in the transverse interactions are not included in the CPA, but are treated in an approximate way afterwards. Expressions for the neutron scattering and Raman cross sections are derived in order to permit a detailed comparison of theory with experiment. The results of numerical calculations for three systems are presented: (Mn,Co)F2, K(Mn,Co)F3 and (Mn,Zn)F2. These are compared with experimental data, and excellent agreement is found; in particular, the neutron scattering results agree extremely well, the line-shapes and widths being accurately described by the theory.