Effect of geometry and anisotropy on the magnetic moment of type II superconductors.

Abstract

Formulae for the magnetic moment $\vec\mu$ of anisotropic platelets of high $T\sb{c}$ superconductors developed by Gyorgy et al and Peterson are frequently exploited by these and other researchers to estimate $j\sbsp{c}{c}$ and $j\sbsp{c}{ab}$, the critical current densities along the c axis and in the ab plane taken to be independent of the magnetic flux density B. These formulae were derived using the basic definition, $\ = ( -\ \mu\sb0H\sb{a})/\mu\sb0$ and ignoring end effects, (i.e. any demagnetizing fields), hence implied that the aspect ratio along the magnetizing field $H\sb{a}$ is large. This approximation is inappropriate for platelets penetrated by $H\sb{a}$. We develop these formulae using the alternative basic definition of a magnetic moment, $\vec\mu = 1/2\int(\vec{R}\times \vec{j})dV$. Now however, for the approach to be valid, $\vec{j} = j\sbsp{c}{c}$ or $\vec{j} = j\sbsp{c}{ab}$ must be independent of B (Bean approximation) and fill the entire volume of the specimen (i.e. a saturated critical state must be established). We show that these formulae are correct under these restrictions regardless of the configuration of $\vec{B}(x,y,z)$ and the neglect of end effects and attendant demagnetizing fields. Pursuing this framework and the latter definition we develop formulae for $\vec\mu$ for isotropic parallelepipeds of various aspect ratios as a function of their inclination $\theta$ with respect to the magnetizing field $\vec H\sb{a}$. We maintain throughout the critical assumption that the induced persistent currents circulate transverse to $\vec H\sb{a}$. The graphs of computations with these formulae are useful in identifying the role of geometry on the magnitude of $\vec\mu$. We also envisage two simple but basic regimes of anisotropy of the critical current densities. (Abstract shortened by UMI.)