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Calculation of the Superconductivity Gap of Metal from Its Parameters in Normal State

A previously considered model interpreted a superconductor as an electron gas immersed in a medium with the dielectric constant $ε$ and a certain elasticity, which could be determined by measured sonic speed in the metal. The obtained expression of effective electron-electron interaction (EEI) potential unambiguously implied that, contrary to the suggestions of BCS theory, it is its long-wave limit which is responsible for the emergence of bound two-electron states and, consequently, for gap formation in one-electron spectrum of the metal. However, the existence of singularities in the EEI potential expression continued to pose a problem, which did not allow a calculation of the gap value for specific superconducting materials, first of all, for metals belonging to periodic table (PT). In the present work, I suggest taking into account matrix elements traditionally attributed to electron scattering in EEI effective potential calculations. For superconductors that has been made on the basis of a semiconductor material by implanting electro-active defects into it, this inclusion results in the appearance of an uncertainty of electron momentum $δp \sim l^{- 1}$, where $l$ is the electron free path. When considering pure PT metals, Hamiltonian terms relating to creation and annihilaton of phonons should be taken into account, which also produces an uncertainty of electron momentum. This uncertainty results in a regularization of EEI potential expression and, therefore, in a possibility of examination of physical properties of specific superconductors. Results of calculation of superconductivity gap value for a range of simple metals (Al, Zn, Pb, Sn) confirm the consistency of the developed approach.