Low-Temperature Thermodynamic Properties of Vanadium. II. Mixed State

The specific heat of a single-crystal vanadium sample (resistivity ratio=140) has been measured in 12 different magnetic fields between 0.5 and 5.4°K to study the mixed state. The specific heat above approximately $H_c$ is found to be independent of sample history, which leads to a complete thermodynamic description of the mixed state. The upper critical field $H_{c2\mathrm{}}\mathrm{}\left(t\right)\mathrm{}$ is found from the specific heat at high fields and from heating and cooling curves at low fields and is used with $H_c\mathrm{}\left(t\right)\mathrm{}$ determined from specific-heat measurements in part I of this investigation to obtain the parameter ${\kappa }_1\mathrm{}\left(t\right)=\frac{H_{c2\mathrm{}}\mathrm{}\left(t\right)\mathrm{}}{\sqrt{2}{H}_c\mathrm{}\left(t\right)\mathrm{}}$. The Ginzburg-Landau parameter $\kappa \equiv {\kappa }_1\mathrm{}\left(1\right)\mathrm{}$ is found to be 0.979±0.010 for this sample and, when extrapolated to infinite electron mean free path, yields the value ${\kappa }_0=0.848\mathrm{}\pm 0.015>\mathrm{}\frac{1}{\sqrt{2}}$, which proves that vanadium, like niobium, is an intrinsic type-II superconductor. The behavior of $\frac{{\kappa }_1\mathrm{}\left(t\right)\mathrm{}}{\kappa }$ for vanadium is shown to be nearly identical to that for niobium and is higher at all temperatures than the theoretical prediction of Gor'kov. In particular $\frac{{\kappa }_1\mathrm{}\left(0\right)\mathrm{}}{\kappa }$ is 1.50±0.02 compared with the theoretical value 1.25, which points out the need for a refinement of the theory concerning intrinsic type-II superconductors. Magnetization curves are thermodynamically deduced from the specific-heat data and are used to find $H_{c1\mathrm{}}\mathrm{}\left(t\right)\mathrm{}$, given by $1150(1-t^2)$ Oe. In addition the curves appear to represent the average behavior of actual magnetization measurements in increasing and decreasing fields. The parameter ${\kappa }_2\mathrm{}\left(t\right)\mathrm{}$ calculated from the slopes of the deduced magnetization curves is in excellent agreement near $t=1\mathrm{}$ with the prediction of Maki and Tsuzuki, and at low temperatures approaches the value $\frac{{\kappa }_2\mathrm{}\left(0\right)\mathrm{}}{\kappa }=2.66\mathrm{}$. A reliable estimate of the Fermi-surface area, given by $\frac{S}{\mathrm{Sf}}=0.714\mathrm{}$, is obtained from the value of ${\kappa }_0$, which then leads to the values ${\lambda }_L\mathrm{}\left(0\right)=398\mathrm{}$ Å, ${\xi }_0=450\mathrm{}$ Å, and $l=2450\mathrm{}$ Å.