Low-Temperature Thermodynamic Properties of Vanadium. I. Superconducting and Normal States

Ray Radebaugh; P. H. Keesom

doi:10.1103/PhysRev.149.209

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Low-Temperature Thermodynamic Properties of Vanadium. I. Superconducting and Normal States

Ray Radebaugh^* and P. H. Keesom

Department of Physics, Purdue University, Lafayette, Indiana

^*Present address: Institute for Materials Research, National Bureau of Standards, Boulder, Colorado.

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Phys. Rev. 149, 209 – Published 9 September, 1966

DOI: https://doi.org/10.1103/PhysRev.149.209

Abstract

The specific heat of vanadium (resistivity ratio = 140) has been measured in the superconducting and normal states between 0.5 and 5.4°K. The normal-state specific heat is given by $C_{n} = 9.82 T + 0.035 T^{3} + C_{nuc}$ mJ ${mole}^{- 1}$ $\deg^{- 1}$ , where the term $C_{nuc}$ arises from the interaction of the nuclear magnetic moments with the applied magnetic field. The coefficient of the cubic term corresponds to a Debye temperature $⊖_{0}$ at 0°K of (382±10)°K, which is slightly less than the value 399.3°K obtained from elastic measurements. The superconducting specific heat contains a term linear in $T$ which is 0.52% of the normal-state linear term. This indicates the presence of a very small energy gap at the Fermi surface in addition to the normal gap. At all but the lowest temperatures the specific heat is governed by the normal energy gap and is in fair agreement with the BCS prediction. The agreement becomes excellent if the normal energy gap is assumed to be anisotropic with a maximum value of $3.52 k T_{c}$ and a minimum of $3.20 k T_{c}$ , which is consistent with ultrasonic measurements. The superconducting transition temperature for this sample is (5.379±0.004) °K with a total transition width of only about 1 mdeg. The intrinsic transition temperature for vanadium is estimated to be (5.414±0.010) °K.

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[c1] R. D. Worley, M. W. Zemansky, and H. A. Boorse, Phys. Rev. 99, 447 (1955)

[c2] W. S. Corak, B. B. Goodman, C. B. Satterthwaite, and A. Wexler, Phys. Rev. 102, 656 (1956)

[c3] N. M. Wolcott, Conference de Physique des Basses Temperatures, Paris, 1955 (Institut International du Froid, Paris, 1955), p. 286

[c4] G. A. Alers, Phys. Rev. 119, 1532 (1960)

[c5] A. Wexler and W. S. Corak, Phys. Rev. 85, 85 (1952)

[c6] G. Busch and J. Müller, Helv. Phys. Acta 30, 230 (1957)

[c7] J. Müller, Helv. Phys. Acta 32, 141 (1959)

[c8] B. B. Goodman, Compt. Rend. 244, 2899 (1957)

[c9] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957)

[c10] H. V. Bohm and N. H. Horwitz, in Proceedings of the Eighth International Conference on Low Temperature Physics, edited by R. O. Davies (Butterworths Scientific Publications Ltd., London, 1963), p. 191

[c11] P. H. Keesom and Ray Radebaugh, Phys. Rev. Letters 13, 685 (1964)

[c12] L. Y. L. Shen, N. M. Senozan, and N. E. Phillips, Phys. Rev. Letters 14, 1025 (1965) N. E. Phillips (private communication)

[c13] Ray Radebaugh and P. H. Keesom, following paper, Phys. Rev. 149, 217 (1966)

[c14] G. M. Seidel and P. H. Keesom, Rev. Sci. Instr. 29, 606 (1958)

[c15] M. P. Mathur, Ph.D. thesis, Department of Physics, Purdue University, Lafayette, Indiana, 1966 (unpublished)

[c16] J. C. Ho, H. R. O'Neal, and N. E. Phillips, Rev. Sci. Instr. 34, 782 (1963)

[c17] R. H. Sherman, S. G. Sydoriak, and T. R. Roberts, J. Res. Natl. Bur. Std. (U. S.) 68A, 579 (1964)

[c18] F. G. Brickwedde, H. van Dijk, M. Durieux, J. R. Clement, and J. K. Logan, J. Res. Natl. Bur. Std. (U. S.) 64A, 1 (1960)

[c19] D. E. Moody and P. Rhodes, Cryogenics 3, 77 (1963)

[c20] B. J. C. van der Hoeven, Jr., and P. H. Keesom, Phys. Rev. 134, A1320 (1964)

[c21] H. A. Leupold and H. A. Boorse, Phys. Rev. 134, A1322 (1964)

[c22] W. Marshall, Phys. Rev. 110, 1280 (1958)

[c23] H. Suhl, B. T. Matthias, and L. R. Walker, Phys. Rev. Letters 3, 552 (1959)

[c24] J. W. Garland, Jr., Phys. Rev. Letters 11, 111 (1963)

[c25] P. W. Anderson, J. Phys. Chem. Solids 11, 26 (1959)

[c26] C. C. Sung and L. Yun Lung Shen, Phys. Letters 19, 101 (1965)

[c27] [26]

[c28] D. O. Van Ostenburg, D. J. Lam, H. D. Trapp, D. W. Pracht, and T. J. Rowland, Phys. Rev. 135, A455 (1964)

[c29] J. L. Brewster, M. Levy, and I. Rudnick, Phys. Rev. 132, 1062 (1963)

[c30] P. L. Richards and M. Tinkham, Phys. Rev. 119, 575 (1960)

[c31] I. Giaever, in Proceedings of the Eighth International Conference on Low Temperature Physics, edited by R. O. Davies (Butterworths Scientific Publications Ltd., London, 1963), p. 171

[c32] Arnold M. Toxen, Phys. Rev. Letters 15, 462 (1965)

[c33] B. J. C. van der Hoeven, Jr., and P. H. Keesom, Phys. Rev. 137, A103 (1965)

[c34] B. B. Goodman, Rev. Mod. Phys. 36, 198 (1964)

[c35] D. Markowitz and L. P. Kadanoff, Phys. Rev. 131, 563 (1963)

[c36] C. Caroli, P. G. de Gennes, and J. Matricon, J. Phys. Radium 23, 707 (1962)

[c37] T. Tsuneto, Progr. Theoret. Phys. (Kyoto) 28, 857 (1962)

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Low-Temperature Thermodynamic Properties of Vanadium. I. Superconducting and Normal States

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

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