Bulk Upper Critical Field of Clean Type-II Superconductors: V and Nb

The bulk upper critical field $H_{c2\mathrm{}}$ of single crystals of Nb and V has been measured for various crystallographic orientations within the temperature range between the respective critical temperature and $T\approx 0.06$ °K. The normalized critical field averaged over all crystallographic directions $〈h\left(t\right)\mathrm{}〉=\frac{〈H_{c2\mathrm{}}\mathrm{}\left(t\right)\mathrm{}〉}{{(-\frac{d{H}_{c2\mathrm{}}}{\mathrm{dt}})}_{t=1\mathrm{}}}$, where $t=\frac{T}{T_c}$, and the relative anisotropy of $H_{c2\mathrm{}}$ for the two metals are found to be similar. The measurements support predictions of the Hohenberg-Werthamer calculation that attributes the observed anisotropy of $H_{c2\mathrm{}}$ to the effects of an anisotropic Fermi surface. Some differences in the behavior of $H_{c2\mathrm{}}\mathrm{}\left(t\right)\mathrm{}$ for the present samples can be understood qualitatively to arise from the effects of a shorter electron-collision time $\tau$ for the less pure V specimen. At very low temperatures, $H_{c2\mathrm{}}\mathrm{}\left(0\right)-H_{c2\mathrm{}}\mathrm{}\left(t\right)\mathrm{}$ for V does not follow a $t^2\ln \mathrm{}\left(t\right)\mathrm{}$ temperature dependence, previously observed for Nb, but can be characterized by an empirical formula $t^2\ln (t+\alpha )$ in which the constant $\alpha$ is conjectured to be inversely proportional to $\tau$.