Distribution of fields, currents, and surface currents in type-II superconductors with square cross section

Simple analytic expressions with 90° rotation symmetry are obtained for $\overrightarrow{\mathrm{J}}$ and $\overrightarrow{\mathrm{B}}$ in a long type-II superconductor of square cross section carrying an axial current. For large squares $\frac{U_l}{I^2}\propto \lambda$ where $U_l$ is the magnetic energy per length, $I$ is the current, and $\lambda$ is the penetration depth; we know of no experimental data for square cross sections, but this is the reported behavior for thick rectangular films. For whiskers $\frac{U_l}{I^2}\propto {\lambda }^2$; this agrees with reported results. Corresponding but quite different expressions are then derived for $\overrightarrow{\mathrm{J}}$ and $\overrightarrow{\mathrm{B}}$ when a uniform parallel field ($H<\mathrm{}{H}_{c1\mathrm{}}$) is impressed. Here surface currents arise for any polygonal cross section because $\overrightarrow{\mathrm{J}}$ and $\overrightarrow{\mathrm{B}}$ can only have 180°, not 90°, rotation symmetry, and $\overrightarrow{\mathrm{J}}$ then has a normal component at some surface. For a square the entire internal $\overrightarrow{\mathrm{J}}$ is normal to two opposite faces; the surface $j$ guarantees that the divergence of the total current density vanishes everywhere. Simple expressions are given both for the internal and surface current densities and for the internal and external field distributions.