Some Implications of an Alternate Equation for the BCS Energy Gap


A set of generalized-BCS equations (GBCSEs) was recently derived from a temperature-dependent Bethe-Salpeter equation and shown to deal satisfactorily with the experimental data comprising the Tcs and the multiple gaps of a variety of high-temperature superconductors (SCs). These equations are formulated in terms of the binding energies W1(T),W2(T),… of Cooper pairs (CPs) bound via one- and more than one-phonon exchange mechanisms; they contain no direct reference to the gap/s of an SC. Applications of these equations so far were based on the observation that for elemental SCs |W01|=0 at T = 0 inthe limit of the dimensionless BCS interaction parameter λ0. Here 0 is the zero-temperature gap whence it follows that the binding energy of a CP bound via one-phonon exchanges at T = 0 is 2|W01|. In this note we carry out a detailed comparison between the GBCSE-based W1(T) and the BCS-based energy gap (T) for all 0TTc and realistic, non-vanishingly-small values of λ. Our study is based on the experimental values of Tc Debye temperature , and ?0 of several selected elements including the “bad actors” such as Pb and Hg. It is thus established that the equation for W1(T) provides a viable alternative to the BCS equation for (T). This suggests the use of, when required, the equation for W2(T) which refers to CPs bound via two-phonon exchanges, for the larger of the two T-dependent gaps of a non-elemental SC. These considerations naturally lead one to the concept of T-dependent interaction parameters in the theory of superconductivity. It is pointed out that such a concept is needed both in the well-known approach of Suhl et al. to multi-gap superconductivity and the approach provided by the GBCSEs. Attention is drawn to diverse fields where T-dependent Hamiltonians have been fruitfully employed in the past.

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G. Malik and M. Llano, "Some Implications of an Alternate Equation for the BCS Energy Gap," Journal of Modern Physics, Vol. 4 No. 4A, 2013, pp. 6-12. doi: 10.4236/jmp.2013.44A002.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] J. Bardeen, L. N. Cooper and J. R. Schrieffer, “Theory of Superconductivity,” Physical Review, Vol. 108, No. 5, 1957, pp. 1175-1204. doi:10.1103/PhysRev.108.1175
[2] G. P. Malik, “On the Equivalence of the Binding Energy of a Cooper Pair and the BCS Energy Gap: A Framework for Dealing with Composite Superconductors,” International Journal of Modern Physics B, Vol. 24, No. 9, 2010, pp. 1159-1172. doi:10.1142/S0217979210055408
[3] G. P. Malik, “Generalized BCS Equations: Applications,” International Journal of Modern Physics B, Vol. 24, No. 19, 2010, pp. 3701-3712. doi:10.1142/S0217979210055858
[4] G. P. Malik and U. Malik, “A Study of the Thallium- and Bismuth-Based High-Temperature Superconductors in the Framework of the Generalized BCS Equations,” Journal of Superconductivity and Novel Magnetism, Vol. 24, No. 1-2, 2011, pp. 255-260. doi:10.1007/s10948-010-1009-0
[5] H. Suhl, B. T. Matthias and L. R. Walker, “Bardeen-Cooper-Schrieffer Theory of Super-Conductivity in the Case of Overlapping Bands,” Physical Review Letters, Vol. 3, 1959, pp. 552-554. doi:10.1103/PhysRevLett.3.552
[6] C. P. Poole, “Handbook of Superconductivity,” Academic Press, San Diego, 2000, p. 48.
[7] D. Pines, “Superconductivity in the Periodic System,” Physical Review, Vol. 109, No. 2, 1958, pp. 280-287. doi:10.1103/PhysRev.109.280
[8] T. Mamedov and M. de Llano, “Superconducting Pseudogap in a Boson-Fermion Model,” Journal of the Physical Society of Japan, Vol. 79, No. 4, 2010, Article ID: 044706.
[9] T. Mamedov and M. de Llano, “Generalized Superconducting Gap in an Anisotropic BosonFermion Mixture with a Uniform Coulomb Field,” Journal of the Physical Society of Japan, Vol. 80, No. 4, 2011, Article ID: 074718.
[10] G. P. Malik, “On Landau Quantization of Cooper Pairs in a Heat Bath,” Physica B: Condensed Matter, Vol. 405, No. 16, 2011, pp. 3475-3481. doi:10.1016/j.physb.2010.05.026
[11] J. M. Blatt, “Theory of Superconductivity,” Academic Press, New York, 1964, p. 206.
[12] T. P. Sheahan, “Effective Interaction Strength in Superconductors,” Physical Review, Vol. 149, No. 1, 1966, pp. 370-377. doi:10.1103/PhysRev.149.370
[13] S. Weinberg, “Gauge and Global Symmetries at High Temperature,” Physical Review D, Vol. 9, No. 12, 1974, pp. 3357-3378. doi:10.1103/PhysRevD.9.3357
[14] A. D. Linde, “Phase Transitions in Gauge Theories and Cosmology,” Reports on Progress in Physics, Vol. 42, No. 3, 1979, pp. 390-437. doi:10.1088/0034-4885/42/3/001
[15] L. Dolan and R. Jackiw, “Symmetry Behavior at Finite Temperture,” Physical Review D, Vol. 9, No. 12, 1974, pp. 3320-3341. doi:10.1103/PhysRevD.9.3320
[16] G. P. Malik and L. K. Pande, “Wick-Cutkosky Model in the Large-Temperature Limit,” Physical Review D, Vol. 37, No. 12, 1988, pp. 3742-3748. doi:10.1103/PhysRevD.37.3742
[17] G. P. Malik, L. K. Pande and V. S. Varma, “On Solar Emission Lines,” The Astrophysical Journal, Vol. 379, 1991, pp. 788-795. doi:10.1086/170554
[18] G. P. Malik, R. K. Jha and V. S. Varma, “Mass Spectrum of the Temperature-Dependent Bethe-Salpeter Equation for Composites of Quarks with a Coulomb plus a Linear Kernel,” The European Physical Journal A, Vol. 2, No. 1, 1998, pp. 105-110. doi:10.1007/s100500050096
[19] G. P. Malik, R. K. Jha and V. S. Varma, “Quarkonium Mass Spectra from the Temperature-Dependent Bethe-Salpeter Equation with Logarithmic and Coulomb plus Square-Root Kernels,” The European Physical Journal A, Vol. 3, No. 4, 1998, pp. 373-375. doi:10.1007/s100500050191
[20] B. T. Geilikman, “Thermal Conductivity of Super-Conductors,” Soviet Physics, Vol. 7, 1958, pp. 721-722.
[21] B. T. Geilikman and V. Z. Kresin, “Phonon Thermal Conductivity of Superconductors,” Soviet Physics Dolady, Vol. 3, No. 6, 1958, pp. 1161-1163.
[22] J. Bardeen, G. Rickayzen and L. Tewordt, “Theory of Thermal Conductivity of Superconductors,” Physical Review, Vol. 113, No. 4, 1959, pp. 982-994. doi:10.1103/PhysRev.113.982

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