Scientific Research

An Academic Publisher

Some Implications of an Alternate Equation for the BCS Energy Gap ()

**Author(s)**Leave a comment

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 *T _{c}s* and the

*multiple*gaps of a variety of high-temperature superconductors (SCs). These equations are formulated in terms of the binding energies

*W*

_{1}(

*T*),

*W*

_{2}(

*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 |

*W*

_{01}|=△

_{0 }at

*T*= 0 inthe limit of the dimensionless BCS interaction parameter

*λ*→0. 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|

*W*

_{01}|. In this note we carry out a detailed comparison between the GBCSE-based

*W*

_{1}(

*T*) and the BCS-based energy gap △(

*T*) for all 0≤

*T*≤

*T*

_{c}and realistic, non-vanishingly-small values of

*λ*. Our study is based on the experimental values of

*T*Debye temperature , and ?

_{c}_{0}of several selected elements including the “bad actors” such as Pb and Hg. It is thus established that the equation for

*W*

_{1}(

*T*) provides a viable alternative to the BCS equation for △(

*T*). This suggests the use of, when required, the equation for

*W*

_{2}(

*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.

Cite this paper

*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 |

Copyright © 2020 by authors and Scientific Research Publishing Inc.

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.