Susceptibility of superconductor disks and rings with and without flux creep

Abstract

First some consequences of the Bean assumption of constant critical current ${\mathrm{J}}_{\mathrm{c}}$ in type-II superconductors are listed and the Bean ac susceptibility of narrow rings is derived. Then flux creep is described by a nonlinear current-voltage law E\ensuremath{\propto}${\mathrm{J}}^{\mathrm{n}}$, from which the saturated magnetic moment at constant ramp rate H-|${\mathrm{Ap}}_{\mathrm{a}}$(t) is derived for rings with general hole radius ${\mathrm{a}}_{1}$ and general creep exponent n. Next the exact formulation for rings in a perpendicular applied field ${\mathrm{H}}_{\mathrm{a}}$(t) is presented in the form of an equation of motion for the current density in thick rings and disks or the sheet current in thin rings and disks. This method is used to compute general magnetization curves m(${\mathrm{H}}_{\mathrm{a}}$) and ac susceptibilities \ensuremath{\chi} of rings with and without creep, accounting also for nonconstant ${\mathrm{J}}_{\mathrm{c}}$(B). Typical current and field (B) profiles are depicted. The initial slope of m(${\mathrm{H}}_{\mathrm{a}}$) (the ideal diamagnetic moment) and the field of full penetration are expressed as functions of the inner and outer ring radii ${\mathrm{a}}_{1}$ and a. A scaling law is derived which states that for arbitrary creep exponent n the complex nonlinear ac susceptibility \ensuremath{\chi}(${\mathrm{H}}_{0}$,\ensuremath{\omega}) depends only on the combination ${\mathrm{H}}_{0}^{\mathrm{n}\mathrm{\ensuremath{-}}1}$/\ensuremath{\omega} of the ac amplitude ${\mathrm{H}}_{0}$ and the ac frequency \ensuremath{\omega}/2\ensuremath{\pi}. This scaling law thus connects the known dependencies \ensuremath{\chi}=\ensuremath{\chi}(\ensuremath{\omega}) in the ohmic limit (n=1) and \ensuremath{\chi}=\ensuremath{\chi}(${\mathrm{H}}_{0}$) in the Bean limit (n\ensuremath{\rightarrow}\ensuremath{\infty}).