| Historically, our theoretical understanding of solid state physics has been based on the symmetry properties of perfect latices. That this foundation is more formal than physical is clear from the existence of band gaps in amorphous semiconductors and also from the striking similarities between the physical properties of liquid and crystalline metals. As long as the theorist's goal was to develop a formalism that would provide an understanding of the simplest solid systems, perfect crystals, the need to decouple that understanding from Bloch's theorem was unimportant. As our knowledge of perfect crystals progressed, however, both theoretical and experimental solid state physicists began to devote more of their attention to the non-periodic states of matter.;This change of emphasis took place in conjunction with two other recent developments. The first was the rapid growth in the power of computers, which provided essentially exact solutions to many previously intractable problems. The second was the implementation of the model Hamiltonian approach to a wide variety of physical problems. Here, rather than trying to give complete solutions to all the details of real physical systems, one attempts to show that the basic underlying features of the system can be understood in terms of the assumptions on which the model is based. The confluence of these two approaches has produced a rapid growth in our understanding of solid state physics.;In this thesis, I have applied the model Hamiltonian approach to two problems in the theory of non-crystalline solids. In particular, I will consider the low temperature properties of semi-infinite and compositionally disordered ferromagnets within the framework of the spin wave model Hamiltonian.;In the surface problem we are dealing with the simplest possible symmetry change. The perfect crystal is considered to be cleaved, creating a layer of atoms with neighbors to one side, but not the other. Using the spin wave approximation for ferromagnetic insulators, I investigated the changes in the magnetic spectrum, density of states, and magnetization induced by this cleavage. The origin of these changes is the fact that the exchange constant between the bulk and surface layers need not equal the perfect crystal value once the crystal is cleaved. We consider both a reduction of the ferromagnetic coupling between surface and bulk, and also the possibility of a surface layer antiferromagnetically bonded to the remainder of the crystal. Both effects can result from a change in the lattice spacing between the surface layer and the bulk crystal. This is, in turn, caused by the assymetrical crystal restoring force and leads, eventually, to a contraction in the magnitude of the surface spins.;The second problem discussed in this thesis involves the study of a binary alloy of ferromagnetic materials. The random nature of the alloy destroys the crystalline symmetry of the potential. Because of the formal similarity of the spin wave magnetic and tight binding electronic model Hamiltonians, we argue that the Coherent Potential Approximation (CPA) is applicable to this problem. However, it should be emphasized that the magnetic case differs strongly from the electronic case in one major respect. In the spin wave approximation to the Heisenberg Hamiltonian, the off-diagonal terms in the Hamiltonian are of the same order of magnitude as the diagonal ones. In the simplest models, they are identical. This necessitates the introduction of off-diagonal disorder into the Coherent Potential Approximation. This can be accomplished in the present case and we derive a closed system of equations which determines the magnon self energy. |
