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UBC Theses and Dissertations

Phase transitions in multiply-connected superconductors Fillmore, Keith Geddes


The first two chapters present a critical review of the derivation of the Ginzburg-Landau macroscopic equations and their application to the determination of the critical fields and temperatures for superconducting-normal phase transitions in simply connected bodies. Second order phase transition criteria are obtained in the form of volume integrals which do not require prior solution of the field equations. With the G-L effective wave function ψ in the London gauge for doubly-connected bodies, we obtain several equivalent expressions for the electromagnetic Free Energy which do not assume uniform |ψ|. One of these leads to a systematic method for expanding F[sub H] in powers of Z = |ψ|² and this method is applied to the long hollow circular cylinder. The expansion obtained is used to determine second order transition criteria for all possible fluxoid states. A closed expression for F[sub H] in the hollow cylinder which does not assume uniform |ψ| is obtained. A detailed analysis of the thin hollow cylinder yields curves for the equilibrium values of Free Energy, Super Electron Density, Magnetic Moment, and Super Electron Momentum as functions of a general field variable. Critical points on the curves are identified and the loci of critical points traced as the geometric parameter varies. These functions are extended to surfaces of the variables of field and temperature. Phase transition criteria are obtained for increasing temperature at constant field. Reversible and irreversible curves are obtained describing the variation and destruction of persistent currents at varying temperature, and the results are compared with experiment. The full range of possible fluxoid states is determined and it is shown that the families of curves obtained for low fluxoid states may be applied to high fluxoid states by suitable scaling. All thin cylinders undergo second order transitions at sufficiently high fluxoid. The correction due to finite coherence length is of the order of the fourth power of the ratio of the cylinder wall thickness to the coherence length. Other thin doubly connected superconductors are described by the same functions applicable to circular cylinders by making the appropriate correspondence of parameters. The exact second order critical field for a torus is calculated. Systems of two co-axial magnetically coupled loops are analysed and the critical conditions for one loop are found to depend in a complicated way on the presence of the other. If a thin loop is closely coupled to a thick one, the behaviour of the former is dominated by that of the latter. Second order phase transition criteria are obtained for systems of n co-axial loops of equal radius. In the final chapter a comparison is made of the results obtained in this thesis with those appearing previously in the literature.

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