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ORIGINAL PAPER

Thermal Stability of Superconductors Jacob Szeftel1

· Nicolas Sandeau2 · Michel Abou Ghantous3 · Muhammad El-Saba4

Received: 5 August 2020 / Accepted: 22 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020

Abstract A stability criterion is worked out for the superconducting phase. The validity of a prerequisite, established previously for persistent currents, is thereby confirmed. Temperature dependence is given for the specific heat and concentration of superconducting electrons in the vicinity of the critical temperature Tc . The isotope effect, mediated by electron-phonon interaction and hyperfine coupling, is analyzed. Several experiments, intended at validating this analysis, are presented, including one giving access to the specific heat of high-Tc compounds. Keywords High-Tc superconductivity · Thermodynamics · Isotope effect

1 Introduction In the mainstream view [1–3], the thermal properties of superconductors are discussed within the framework of the phenomenological equation by Ginzburg and Landau [4] (GL) and the BCS theory [5]. However, since this work is aimed at accounting for the stability of the superconducting state with respect to the normal one, we shall develop an alternative approach, based on thermodynamics [6], and the properties of the Fermi gas [7], and recent results [8, 9], claimed to be valid for all superconductors, including low and high Tc materials. The outline is as follows: the specific heat of the superconducting phase is calculated in Section 2, which enables us to assess its binding energy and thereby to confirm and refine a necessary condition, established previously for the existence of persistent currents [8]. Section 3 is concerned with the inter-electron coupling, mediated by the electron-phonon and hyperfine interactions; new experiments, dedicated at validating this analysis, are

discussed in Section 4 and the results are summarized in the “Conclusion.”

2 Binding Energy As in our previous work [8–13], the present analysis will proceed within the framework of the two-fluid model, for which the conduction electrons comprise bound and independent electrons, in respective temperature dependent concentration cs (T ), cn (T ). They are organized, respectively, as a many bound electron [9] (MBE) state, characterized by its chemical potential μ(cs ), and a Fermi gas [7] of Fermi energy EF (T , cn ). The Helmholz free energy of independent electrons per unit volume Fn and EF on the one hand, and the eigenenergy per unit volume Es (cs ) of bound electrons and μ on the other hand, are ∂ Es n related [6, 7], respectively, by EF = ∂F ∂cn and μ = ∂cs . At last, according to Gibbs and Duhem’s law [6], the two-fluid model fulfils [8] at thermal equilibrium: EF (T , cn (T )) = μ(cs (T )),

 Jacob Szeftel

[email protected] 1

ENS Cachan, LPQM, 61 avenue du Pr´esident Wilson, 94230, Cachan, France

2

CNRS, Centrale Marseille, Institut Fresnel, Aix Marseille Univ, F-13013 Marseille, France

3

American University of Technology,