Phenomenology of Shape Effects in Amperian Magnetic Systems with Application to Superconductors

This paper is concerned with the effects of sample shape and orientation on the magnetic properties of superconductors. The absence of adequate theoretical formulas is probably the main reason that these effects have been largely ignored in the past. The present discussion begins with the formulation of a thermodynamic potential appropriate to an Amperian system and expressed in a form adapted to the Landau theory of phase transitions. The shape dependence is expressed explicitly in terms of the usual demagnetization factors, and the dependence on $H_0$ and $T$ (the applied magnetic field and temperature) is expressed implicitly through the magnetization which is defined uniquely and self-consistently in terms of the currents and the magnetic moment. The formulation finds a natural application in the description of superconductors and leads to a vector generalization of Abrikosov's constituitive relation for the superconducting mixed state. Explicit formulas are derived for the magnetic moment per unit volume, specific heat, force, and torque for the Meissner state and mixed state of a spheroid of arbitrary shape and orientation in an applied field. It is found that the sample shape is always important for the Meissner state, and is important for the mixed state whenever the inequality $(2\mathrm{}{\kappa }^2-1)\beta \gg 1$ is not satisfied. Here $\kappa$ is the Ginzburg-Landau parameter, and $\beta$ is expected to have the value 1.16 for fluxoid lattices carrying a single flux quantum per unit cell.