Chapman, S. J. (1991) Macroscopic models of superconductivity. PhD thesis, University of Oxford.

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Abstract
After giving a description of the basic physical phenomena to be modelled, we begin by formulating a sharpinterface freeboundary model for the destruction of superconductivity by an applied magnetic field, under isothermal and anisothermal conditions, which takes the form of a vectorial Stefan model similar to the classical scalar Stefan model of solid/liquid phase transitions and identical in certain twodimensional situations. This model is found sometimes to have instabilities similar to those of the classical Stefan model.
We then describe the GinzburgLandau theory of superconductivity, in which the sharp interface is `smoothed out' by the introduction of an order parameter, representing the number density of superconducting electrons. By performing a formal asymptotic analysis of this model as various parameters in it tend to zero we find that the leading order solution does indeed satisfy the vectorial Stefan model. However, at the next order we find the emergence of terms analogous to those of `surface tension' and `kinetic undercooling' in the scalar Stefan model. Moreover, the `surface energy' of a normal/superconducting interface is found to take both positive and negative values, defining Type I and Type II superconductors respectively.
We discuss the response of superconductors to external influences by considering the nucleation of superconductivity with decreasing magnetic field and with decreasing temperature respectively, and find there to be a pitchfork bifurcation to a superconducting state which is subcritical for Type I superconductors and supercritical for Type II superconductors. We also examine the effects of boundaries on the nucleation field, and describe in more detail the nature of the superconducting solution in Type II superconductors  the socalled `mixed state'.
Finally, we present some open questions concerning both the modelling and analysis of superconductors.
Item Type:  Thesis (PhD) 

Subjects:  O  Z > Partial differential equations O  Z > Optics, electromagnetic theory O  Z > Statistical mechanics, structure of matter 
Research Groups:  Oxford Centre for Industrial and Applied Mathematics 
ID Code:  24 
Deposited By:  Eprints Administrator 
Deposited On:  09 Mar 2004 
Last Modified:  29 May 2015 18:15 
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