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Boris V. Vasiliev Supercondustivity Superfluidity

Superconductivity and Superfluidity At they interact, their amplitudes, frequencies and phases of zero-point oscillations become ordered. Let an electron gas has density n e and its Fermi-energy be E F . Each electron of this gas can be considered as fixed inside a cell with linear dimension λ F : 1 which corresponds to the de Broglie wavelength: λ 3 F = 1 n e (6.10) λ F = 2π p F . (6.11) as Having taken into account (6.11), the Fermi energy of the electron gas can be written E F = p2 F = 2π 2 e2 a B . (6.12) 2m e However, a free electron interacts with the ion at its zero-point oscillations. If we consider the ions system as a positive background uniformly spread over the cells, the electron inside one cell has the potential energy: λ 2 F E p ≃ − e2 λ F . (6.13) As zero-point oscillations of the electron pair are quantized by definition, their frequency and amplitude are related m e a 2 0Ω 0 ≃ 2 . (6.14) Therefore, the kinetic energy of electron undergoing zero-point oscillations in a limited region of space, can be written as: 2 E k ≃ 2m e a 2 . (6.15) 0 1 Of course, the electrons are quantum particles and their fixation cannot be considered too literally. Due to the Coulomb forces of ions, it is more favorable for collectivized electrons to be placed near the ions for the shielding of ions fields. At the same time, collectivized electrons are spread over whole metal. It is wrong to think that a particular electron is fixed inside a cell near to a particular ion. But the spread of the electrons does not play a fundamental importance for our further consideration, since there are two electrons near the node of the lattice in the divalent metal at any given time. They can be considered as located inside the cell as averaged. 72 Science Publishing Group

Chapter 6 Superconductivity as a Consequence of Ordering of Zero-Point Oscillations in Electron Gas In accordance with the virial theorem [45], if a particle executes a finite motion, its potential energy E p should be associated with its kinetic energy E k through the simple relation |E p | = 2E k . In this regard, we find that the amplitude of the zero-point oscillations of an electron in a cell is: a 0 ≃ √ 2λ F a B . (6.16) 6.5 The Condensation Temperature Hence the interaction energy, which unites particles into the condensate of ordered zero-point oscillations where α = 1 137 is the fine structure constant. ∆ 0 ≡ H ′ = 18π 3 α 3 e2 a B λ 2 , (6.17) F Comparing this association energy with the Fermi energy (6.12), we obtain ∆ 0 E F = 9πα 3 ≃ 1.1 · 10 −5 . (6.18) Assuming that the critical temperature below which the possible existence of such condensate is approximately equal T c ≃ 1 ∆ 0 (6.19) 2 k (the coefficient approximately equal to 1/2 corresponds to the experimental data, discussed below in the section (7.6)). After substituting obtained parameters, we have T c ≃ 5.5 · 10 −6 T F (6.20) The experimentally measured ratios Tc T F Table 7.1 and in Figure 7.1. for I-type superconductors are given in The straight line on this figure is obtained from Eq.(6.20), which as seen defines an upper limit of critical temperatures of I-type superconductors. Science Publishing Group 73

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References [1] W. Gilbert: De magne

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References [33] Bethe H., Sommerfel