Critical properties of random systems

Calculations are presented for a series of interrelated problems in the theory of disordered solids. The simple mean field theory of tricriticality in the layered Ising metamagnet is modified by inclusion of the Bethe-Peierls equation of state for the planar interactions. The approximation allows for a model of dilution with a finite percolation concentration for the layers. The calculated behaviour as a function of dilution for anisotropic coupling strengths allows comparison with experimental results on dilute ferrous chloride. Estimation is made of the effects of treating fluctuations by theories with mean field singularities. A discussion is given of first order phase transitions in disordered systems. A mean field theory of the implications of introducing quenched disorder to a system undergoing a transition of first order is effected by reformulating the problem in terms of a translationally invariant one via the replica method. The conclusions are examined in terms of a simple scaling theory and criteria derived for smearing of the singularities. Dynamic and thermodynamic properties of diluted magnetic insulators near the percolation concentration are considered in terms of simple geometric models of the percolating cluster as introduced by de Gennes. New scaling relations for the spin wave stiffness and conductivity exponents are derived and differences from previous relations interpreted geometrically. The scaling models of the percolating cluster so found, in which correlations propagate locally via effectively one-dimensional paths, are applied to the determination of the mobility edge for spin wave excitations in a dilute Heisenberg magnet near the percolation threshold. A prediction for the functional form of the mobility edge is made by means of results known for the Anderson model and a localisation length derived from an exact solution of a random problem by Dyson.