Superfluidity as sequence of an ordering of zero-point oscillations

The phenomenon of superconductivity was explained as a consequence of ordering of zero-point oscillations. Superfluidity are related phenomenon. The consideration of interaction zero-point oscillations in liquid helium permit to obtain quantitative estimations of main characteristic of superfluid helium which are in a good agreement with measuring data. So both related phenomena, superconductivity and superfluidity, get explanation which is based on the same physical mechanism - they both are consequences of the ordering of zero-point oscillations.

The main features of superfluidity of liquid helium became clear few decades ago [1], [2]. L.D.Landau explains this phenomenon as the manifestation of a quantum behavior of the macroscopic object. However, the causes and mechanism of the formation of superfluidity are not clear still. There is not an explanation why the λ-transition in helium-4 occurs at about 2 K, ie at almost exactly twice less than its boiling point: while for helium-3, this transition is observed only at temperatures of about a thousand times smaller. The related phenomenon -superconductivity -can be regarded as superfluidity of a charged liquid. It can be quantitatively described at the consideration of it as the consequence of ordering of the electron gas zero-point oscillations [3].
Therefore it seems as appropriate to consider superfluidity from the same point of view.
The atoms in liquid helium-4 are electrically neutral, 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 with changes of energy of the same scale).
F.London was showing yet in the 30's of last century [?], that there is an interaction between the atoms in the ground state, which has 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 called this interaction of atoms in the ground state as the dispersion interaction. Following F.Londonu [4], let consider two spherically symmetric atoms without nonzero average dipole moments. Let suppose that at some time the charges of these atoms are fluctuationally displaced from the of equilibrium states: If atoms are located along the Z-axis at the distance L of each other, their potential energy can be written as: where α is the atom polarizability. The Hamiltonian can be diagonalized by using the normal coordinates of symmetric and antisymmetric displacements: As the result of this change of variables we obtain: Thus frequencies of oscillators depend on their orientation and they are determined by the equations: where is natural frequency of the electron shell of the atom (at L → ∞). The energy of zero-point oscillations is It is easy to see that the interaction between neutral atoms do not contain terms 1 L 3 , which are characteristic for the interaction of zero-point oscillations in the electron gas.
It is important to emphasize that the interaction energy of the zero-point oscillations are different for different dipole orientations. So the interaction zero-point oscillations oriented along the direction connecting the atoms leads to their attraction with energy: while the summary energy of the attraction of the oscillators of the perpendicular directions (x and y) is equal to one the half of it: (the minus sign is taken here because for this case the opposite direction of dipoles is energetically favorable).

The estimation of ordering energy of zero-point oscillations in helium-4
The first term in Eq.(2) is the zero-point energy of two isolated oscillators. The second term is the energy of attraction of two oscillators, arising due to the ordering of their zero-point fluctuations. Zero-point oscillations have quantum nature, so their interaction is also the quantum phenomenon. That is explicitly reflected by the presence of the Planck constant in the right-hand side of this equation. Following F.London, we will assume that the energy that characterizes the electron shell zero-point oscillations of helium atom is equal to its ionization energy of Ω 0 (4) = I 4 ≃ 3.9 · 10 −11 erg.
Given the fact that the density of liquid helium at the boiling point [6]: the helium electric polarizability α can be calculated with using of the Clausius-Mossotti relation [5]: Due to the fact that for helium-4 at T → 0 the permittivity ε ≈ 1.055 [6], we get from Eq.(11): In the result Eq.(8) for helium-4 gives: This value agrees well with the boiling point of helium-4. This can be considered as a proof that the liquefaction of helium is the result of action of the attractive forces between the atoms, which vibrations are oriented along the connecting them line.
The existence of the superfluid state needs in the total ordering of atoms zero-point oscillations.
The total ordering in the system happens if the component of zero-point oscillations which is perpendicular to the vector joining the interacting atoms will be ordered. Thus, a total ordering in the system zero-point fluctuations of the helium atoms occurs at twice lower temperature: It is in satisfactory agreement with the measurements of the transition temperature to the superfluid state. The ratio of these temperatures explain the reason why the measured ratio of the boiling temperature to the temperature of λ-transition is obtained close to two (Eq.(1)).

The estimation of characteristic temperatures of He-3
With using of F.London formula, one can estimate parameters of the interaction of zero-point oscillations in liquid helium-3. The density of atoms in He-3 n 3 ≈ 1.17 · 10 22 cm −3 .
There are no data on the ionization energy and the dielectric constant of liquid helium-3 in the literature. However, if to assume that these parameters for He-3 are the same as for He-4, the formula (8) gives possibility to obtain the estimation: which is about half the value of the boiling temperature of He-3 T boiling ≃ 3.19K.
As for the He-3 transition to the superfluid state, there is a radical difference from the He-4. Superfluidity occurs if complete ordering exists in the atomic system. For superfluidity of He-3 beside ordering zero-point vibrations of atoms, electromagnetic interaction should order the magnetic moments of the nuclei.
It is important to note that all characteristic dimensions in this task -the amplitude of the zero-point oscillations, the atomic radius, the distance between atoms in liquid helium -all equal on the order of value to the Bohr radius a B . Due to this fact we can estimate on the order of value of the oscillating magnetic field, which a fluctuating electron shell creates on "its" nucleus: where µ B = e 2mec is the Bohr magneton, α 3 is the electric polarizability of helium-3 atom.
Because the value magnetic moments of the nuclei He-3 is approximately equal to the nuclear Bohr magneton µ nB = e 2mpc , the ordering in their system must occur below the critical temperature This finding is in agreement with the measurement data. Like superfluidity, the consideration of the superconductivity phenomenon as a consequence of the zero-point oscillations ordering gives the possibility to construct the theory of superconductors [3]. The estimations of critical parameters of superconductors obtained in frames of this theory are in a good agreement with measurement data (see Fig(1)).  [3]. Circles are related to I-type superconductors, squares show II-type superconductors. On abscissa the measured values of critical temperatures are plotted, on ordinate the calculated estimations are plotted.