| Summary: | This thesis includes the generaliztution to all spin magnitudes of two fermion representations which were previously known only for the spin - half case. The resulting "2S" fermion representation is found to be useful for converting electronic exchange Fsmiltonians into Rnniltonians which involve only spin operators. This is used to exhibit the general correspondence between the Anderson and Kondo model Hamiltonians. The generalized drone - fermion representation is used to establish two new Wick's theorems for spin half andspin one operators. These results are both simple and easy to use. They extend the use of standard diagrammatic Quantum Field Theory techniques to those problems involving such spin operators. The properties of a metal which contains a single localized paramagnetic impurity (the Rondo model) are then investigated with these new methods. The results include a derivation of a log T term in the impurity g - shift, close agreement with previous low - order perturbation treatments, and a high - order equation for the resistivity, which is obtained by a selective resimmation of a complete sub - series of self - energy diagrams. The resistivity derived using this approximationl is shown to exhibit a resonance - like behaviour (rather than a divergence ) for both ferro - and anti - ferromagnetic coupling (although at different temperatures) in passing from the high to low temperature regions, through the Nagaoka - Suhl instability temperature Tk. At higher concentrations , the disappearance of the first - order Yosida g - shift is shown in a microscopic theory. The narrowing of the local - moment resonance line, which has recently been measured for such dilute paramagnetic alloys, is similarly demonstrated. An examination of the rare - earth ferromagnetic model results in equations whioh explicitly demonstrate the coupled spin - wave behaviour of both local - moments, and of the conduction electrons. This is a: more general result than previously demonstrated. Finally these methods are applied in an extensive investigation of the Heisenberg model in high and low temperature domains, where the expansion criteria of Stinchcombe et al are closely followed. Renormalization is now quite simple and straight-forward. At low temperatures, Dyson's T4 contribution to the free energy is obtained in the first Born approximation to spin wave scattering. Higher order spin-wave contributions give a damping term, which upon evaluation in the lowest approximation, is identical to that found by ter Haar and Tahir- Kheli.
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