Entropy of the Mixed State of Some Low-κType-II Superconductors

The entropy of the mixed state of type-II superconductors with low $\kappa \left(\sim 1\right)$ is studied for the alloys ${\mathrm{Pb}}_{0.96}$ ${\mathrm{Tl}}_{0.04}$ and ${\mathrm{In}}_{0.88}$ ${\mathrm{Sn}}_{0.12}$. The current theories are reviewed for approximations valid near $H_{c1\mathrm{}}$ and near $H_{c2\mathrm{}}$. The experiments show for the most reversible sample (${\mathrm{Pb}}_{0.96}$ ${\mathrm{Tl}}_{0.04}$) that ${\left(\frac{\partial S}{\partial B}\right)}_T$ of the mixed state depends only on temperature, and that ${\left(\frac{\partial S}{\partial T}\right)}_B$ increases linearly with $B$ from its value for the superconducting state until it drops discontinuously to its normal state value at $H_{c2\mathrm{}}$. There is no abrupt increase in the specific heat at constant magnetic induction near $H_{c1\mathrm{}}$ as is observed in the specific heat at constant applied magnetic field. A method of determining the thermodynamic critical field of an irreversible sample without assuming a parabolic form is discussed and applied to the data. The thermodynamic critical field determined in this way is used to calculate ${\kappa }_1$ and ${\kappa }_2$. The parameters are very nearly equal for ${\mathrm{Pb}}_{0.96}$ ${\mathrm{Tl}}_{0.04}$ with a temperature dependence of ${(1+t^2)}^{\frac{-1\mathrm{}}{2}}$. Measurements of ${\left(\frac{\partial S}{\partial B}\right)}_T$ are also reported on the type-I superconductors Sn and Pb. The conclusion, as expected, is that ${\left(\frac{\partial S}{\partial B}\right)}_T$ in the intermediate state depends on the temperature only.