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Some Implications of an Alternate Equation for the BCS Energy Gap

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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.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

*Journal of Modern Physics*, Vol. 4 No. 4A, 2013, pp. 6-12. doi: 10.4236/jmp.2013.44A002.

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