Thermodynamics of Volume and Pressure Effects for Type-II Superconductors

Some thermodynamic relationships which include volume and pressure effects are presented for the case of the ideal bulk type-II superconductor. On the basis of published data, it is assumed that the upper-critical-field transition at ($H_{c2\mathrm{}},T_s$) occurs without discontinuities in the entropy and magnetization and without infinite discontinuities in the second-order derivatives of the Gibbs free energy. For an ellipsoidal specimen in an applied field H directed along a principal axis at ($H_{c2\mathrm{}},T_s$), Clapeyron- and Ehrenfest-type equations yield $\Delta V=0\mathrm{}$, $\Delta K=\left(\frac{S_0}{4\pi V}\right){{\left(\frac{\partial H_{c2\mathrm{}}}{\partial P}\right)}^2}_T$, and $\Delta \beta =(-\frac{S_0}{4\pi V}){\left(\frac{\partial H_{c2\mathrm{}}}{\partial P}\right)}_T{\left(\frac{\partial H_{c2\mathrm{}}}{\partial T}\right)}_P$. Here $\Delta$ indicates the difference between the superconducting- and normal-state values at constant $H$; $V$ is the specimen volume (which is shown to be field-dependent in the superconducting type-II mixed state); $K=-V^{-1\mathrm{}}{\left(\frac{\partial V}{\partial P}\right)}_{H,T}$ is the isothermal compressibility; $\beta \equiv V^{-1\mathrm{}}{\left(\frac{\partial V}{\partial T}\right)}_{H,P}$ is the thermal expansivity; and $S_o\equiv {\left\{\partial \frac{\left[4\mathrm{}\pi \right(I_s-I_n\left)\right]}{\partial H}\right\}}_{P,T}$, where $I_s$ and $I_n$ are the total superconducting- and normal-state magnetic moments and the derivative is taken at ($H_{c2\mathrm{}},T_s$).