Calculation of Physical Properties of Co-Operative Assemblies.

Much of the work this year has been concerned with the detailed exploration of the critical region for different models. The form of the function h(x) in the scaling-law equation of state for the two and three dimensional Ising models was calculated. The results of the calculation have been compared with experiment and with empirical formulae. It was found possible to account theoretically for the spurious unphysical singularities in the low temperature series for the Ising model which mask the true critical behaviour. Low temperature series for the spin 1 Ising model have been derived. The numerical results are consistent with the hypothesis that the indices beta and delta are spin-independent. For the problem of non close packed dimers on a lattice, series expansions have been derived as in the Ising problem and they have been used to establish the behaviour of monomer-dimer equilibrium. Detailed research has been undertaken into the Green's function for random walks on the bcc lattice. A number of exact formulae have been derived using generalized hypergeometric functions which could be of use in several branches of solid state physics. Finally, a careful analysis of the form of correction terms to critical behaviour in the Ising model has been carried out. This enables a precise definition of the critical region to be made. (Author)