Magnetization of Ellipsoidal Superconductors

Abstract

A well-known result which can be rigorously deduced from Maxwell's equations and potential theory is that the magnetization M of any homogeneous, isotropic ellipsoid is related to the local field B and the external applied magnetic field ${\mathbf{H}}_0$ by $\mathbf{H}=\mathbf{B}-4\mathrm{}\pi \mathbf{M}={\mathbf{H}}_0-4\mathrm{}\pi \mathbf{N}·\mathbf{M}\left(\mathbf{H}\right)$. The configuration matrix N, whose elements are the demagnetization coefficients, depends only on the shape of the ellipsoid. In particular, this relation provides a means for computing the magnetization M (as a function of ${\mathbf{H}}_0$), if it is known for the special case of an infinitely along cylinder whose axis is aligned with ${\mathbf{H}}_0$. This transformation has been commonly applied to systems for which the M(H) relationship is linear: $\mathbf{M}=\chi \mathbf{H}$, where $\chi$ is a constant. We find, more generally, when the ellipsoid is homogeneous and isotropic, that the transformation depends only on the assumption that M is a smooth, single-valued, and otherwise arbitrary function of the local field B. We apply the above result to the well-known magnetization functions for superconductors and find for spheroids whose symmetry axis is parallel to ${\mathbf{H}}_0$: $4\pi \mathbf{M}=-\frac{{\mathbf{H}}_0}{\mathrm{}(1-n)\mathrm{}} \left(\mathrm{Meissner}\mathrm{state}\right)$ and $4\pi \mathbf{M}=-\left[\frac{(H_{c2\mathrm{}}-H_0)}{(\gamma +n)H_0}\right]{\mathbf{H}}_0 \left(\mathrm{mixed}\mathrm{state}\right),$ where $\gamma =(2\mathrm{}{{\kappa }_2}^2-1)\beta$, and $n$ is the element of N associated with the symmetry direction. We also compute the torque exerted on a superconducting spheroid whose symmetry axis is not aligned with ${\mathbf{H}}_0$.