Superconductor disks and cylinders in an axial magnetic field: II. Nonlinear and linear ac susceptibilities

Abstract

The ac susceptibility $\chi ={\chi }^{\prime }-i{\chi }^{″}$ of superconductor cylinders of finite length in a magnetic field applied along the cylinder axis is calculated using the method developed in the preceding paper, part I. This method does not require any approximation of the infinitely extended magnetic field outside the cylinder or disk but directly computes the current density $J$ inside the superconductor. The material is characterized by a general current-voltage law $E\left(J\right),$ e.g., ${E\left(J\right)=E}_c\left[{J/J}_c\mathrm{}\right(B)\mathrm{}{]}^{n\left(B\right)\mathrm{}},$ where $E$ is the electric field, $B={\mu }_{0}H$ the magnetic induction, $E_c$ a prefactor, $J_c$ the critical current density, and $n>~1$ the creep exponent. For $n>\mathrm{}1,$ the nonlinear ac susceptibility is calculated from the hysteresis loops of the magnetic moment of the cylinder, which is obtained by time integration of the equation for $J(\mathbf{r},t).\mathrm{}$ For $n\gg 1$ these results go over into the Bean critical state model. For $n=1,$ and for any linear complex resistivity ${\rho }_{\mathrm{ac}}\left(\omega \right)=E/J,$ the linear ac susceptibility is calculated from an eigenvalue problem which depends on the aspect ratio $b/a$ of the cylinder or disk. In the limits $b/a\ll 1$ and $b/a\gg 1,$ the known results for thin disks in a perpendicular field and long cylinders in a parallel field are reproduced. For thin disks in a perpendicular field, at large frequencies $\chi \left(\omega \right)$ crosses over to the behavior of slabs in parallel geometry since the magnetic field lines are expelled and have to flow around the disk. The results presented may be used to obtain the nonlinear or linear resistivity from contact-free magnetic measurements on superconductors of realistic shape.