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Boris V. Vasiliev Supercondustivity Superfluidity

Superconductivity and Superfluidity The related phenomenon, superconductivity, can be regarded as superfluidity of a charged liquid. It can be quantitatively described considering it as the consequence of ordering of zero-point oscillations of electron gas. Therefore it seems appropriate to consider superfluidity from the same point of view [54]. Atoms in liquid helium-4 are electrically neutral, as they have no dipole moments and do not form molecules. Yet some electromagnetic mechanism should be responsible for phase transformations of liquid helium (as well as in other condensed substance where phase transformations are related to the changes of energy of the same scale). F. London has demonstrated already in the 1930’s [53], that there is an interaction between atoms in the ground state, and this interaction is of a quantum nature. It can be considered as a kind of the Van-der-Waals interaction. Atoms in their ground state (T = 0) perform zero-point oscillations. F.London was considering vibrating atoms as three-dimensional oscillating dipoles which are connected to each other by the electromagnetic interaction. He proposed to call this interaction as the dispersion interaction of atoms in the ground state. 11.2 The Dispersion Effect in Interaction of Atoms in the Ground State Following F. London [53], let us consider two spherically symmetric atoms without non-zero average dipole moments. Let us suppose that at some time the charges of these atoms are fluctuationally displaced from the equilibrium states: { r 1 = (x 1 , y 1 , z 1 ) r 2 = (x 2 , y 2 , z 2 ) (11.2) If atoms are located along the Z-axis at the distance L of each other, their potential energy can be written as: e 2 r1 2 H = 2a + e2 r2 2 } {{ 2a } elastic dipoles energy where a is the atom polarizability. + e2 L 3 (x 1x 2 + y 1 y 2 − 2z 1 z 2 ) . } {{ } elastic dipoles interaction (11.3) 108 Science Publishing Group

Chapter 11 Superfluidity as a Subsequence of Ordering of Zero-Point Oscillations The Hamiltonian can be diagonalized by using the normal coordinates of symmetric and antisymmetric displacements: ⎧ ⎪⎨ x s = √ 1 2 (x 1 + x 2 ) r s ≡ y s = √ 1 2 (y 1 + y 2 ) ⎪⎩ z s = √ 1 2 (z 1 + z 2 ) and and This yields ⎧ ⎪⎨ x a = √ 1 2 (x 1 − x 2 ) r a ≡ y a = √ 1 2 (y 1 − y 2 ) ⎪⎩ z a = √ 1 2 (z 1 − z 2 ) x 1 = 1 √ 2 (x s + x a ) y 1 = 1 √ 2 (y s + y a ) z 1 = 1 √ 2 (z s + z a ) x 2 = 1 √ 2 (x s − x a ) y 2 = 1 √ 2 (y s − y a ) z 2 = 1 √ 2 (z s − z a ) As the result of this change of variables we obtain: H = [ e2 2a (r2 s + ra) 2 + e2 2L (x 2 3 s + ys 3 − 2zs 2 − x 2 a − ya 2 + 2za) 2 (1 ) = e2 2a + a L (x 2 3 s + ys) 2 + ( ) 1 − a L (x 2 3 a + ya) 2 (11.4) + ( 1 − 2 a L 3 ) z 2 s + ( 1 + 2 a L 3 ) z 2 a ]. Consequently, frequencies of oscillators depend on their orientation and they are determined by the equations: Ω s a 0x = Ω s √ ( ) a 0y = Ω 0 1 ± a L ≈ Ω 3 0 1 ± a L − a2 3 8L ± ... , 6 where ( ) Ω s a 0z = Ω 0 √1 ∓ 2a L ≈ Ω 3 0 1 ∓ a L − a2 3 2L ∓ ... , 6 (11.5) Ω 0 = 2πe √ ma (11.6) Science Publishing Group 109

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