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Boris V. Vasiliev Supercondustivity Superfluidity

Superconductivity and Superfluidity relative to the ion A and r 2 is the radius-vector of the second electron relative to the ion B. Following the Born-Oppenheimer approximation, slowly oscillating ions are assumed fixed. Let the temperature be low enough (T → 0), so only zero-point fluctuations of electrons would be taken into consideration. In this case, the Hamiltonian of the system can be written as: H = H 0 + H ′ ( ) H 0 = − 2 4m e ∇ 2 1 + ∇ 2 2 − 4e 2 H ′ = 4e2 L + 4e2 r 12 − 4e2 r 1B − 4e2 r 1 − 4e2 r 2 (6.2) r 2A Eigenfunctions of the unperturbed Hamiltonian describes two ions surrounded by electronic clouds without interactions between them. Due to the fact that the distance between the ions is large compared with the size of the electron clouds L ≫ r , the additional term H ′ characterizing the interaction can be regarded as a perturbation. If we are interested in the leading term of the interaction energy for L, the function H ′ can be expanded in a series in powers of 1/L and we can write the first term: { [ ] −1/2 H ′ = 4e2 L 1 + 1 + 2(z2−z1) L + (x2−x1)2 +(y 2−y 1) 2 +(z 2−z 1) 2 L 2 ( ) −1/2 ( ) −1/2 − 1 − 2z1 L + r2 1 L − 1 + 2z2 2 L + r2 2 L }. 2 (6.3) After combining the terms in this expression, we get: H ′ ≈ 4e2 L 3 (x 1x 2 + y 1 y 2 − 2z 1 z 2 ) . (6.4) This expression describes the interaction of two dipoles d 1 and d 2 , which are formed by fixed ions and electronic clouds of the corresponding instantaneous configuration. Let us determine the displacements of electrons which lead to an attraction in the system . Let zero-point fluctuations of the dipole moments formed by ions with their electronic clouds occur with the frequency Ω 0 , whereas each dipole moment can be decomposed 70 Science Publishing Group

Chapter 6 Superconductivity as a Consequence of Ordering of Zero-Point Oscillations in Electron Gas into three orthogonal projection d x = ex, d y = ey and d z = ez, and fluctuations of the second clouds are shifted in phase on ϕ x , ϕ y and ϕ z relative to fluctuations of the first. As can be seen from Eq.(6.4), the interaction of z-components is advantageous at inphase zero-point oscillations of clouds, i.e., when ϕ z = 2π. Since the interaction of oscillating electric dipoles is due to the occurrence of oscillating electric field generated by them, the phase shift on 2π means that attracting dipoles are placed along the z-axis on the wavelength Λ 0 : L z = Λ 0 = c 2πΩ 0 . (6.5) As follows from (6.4), the attraction of dipoles at the interaction of the x and y-component will occur if these oscillations are in antiphase, i.e. separated along these axes on the distance equals to half of the wavelength: if the dipoles are L x,y = Λ 0 2 = c 4πΩ 0 . (6.6) In this case H ′ = −4e 2 ( x1 x 2 L 3 x + y 1y 2 L 3 y + 2 z ) 1z 2 L 3 . (6.7) z Assuming that the electronic clouds have isotropic oscillations with amplitude a 0 for each axis x 1 = x 2 = y 1 = y 2 = z 1 = z 2 = a 0 (6.8) we obtain H ′ = 576π 3 e2 c 3 Ω3 0a 2 0. (6.9) 6.4 The Zero-Point Oscillations Amplitude The principal condition for the superconducting state formation, that is the ordering of zero-point oscillations, is realized due to the fact that the paired electrons, which obey Bose-Einstein statistics, interact with each other. Science Publishing Group 71

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References [1] W. Gilbert: De magne

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References [33] Bethe H., Sommerfel