Superfluidity is a special quantum state of matter in which a substance (called superfluid) flows with zero viscosity (without loss of kinetic energy), by the absence of entropy and by having infinite thermal conductivity. The superfluids, if placed in a closed path, can flow infinitely without friction. Superfluidity is the phenomenon whereby some systems have zero viscosity and can therefore flow without dissipating energy. Superfluidity occurs at flow rates and temperatures below certain critical values, specific to each system.

Superfluidity occurs for temperatures and transport rates below certain specific critical values. Helium with isotopic number 4 (4He) in the liquid state was the first system to exhibit superfluidity below the critical temperature Tλ = 2.17 K, as was first observed from some peculiarities in thermodynamic properties and then, in the years 1936-38, from more strictly dynamical and transport aspects. Superfluidity can, at least partially, be traced back to a condensation analogous to the Bose-Einstein one, and all liquids composed, like 4He, by integer spin atoms, i.e., obeying the Bose-Einstein statistics, should at temperatures close to absolute zero become superfluid; but only 4He remains liquid at these temperatures.

The superfluidity of helium with isotopic number 3 (3He) was discovered only in the early 1970s, as it occurs at much lower temperatures, Tc = 2.7 mK (because reaching such low temperatures required the use of more advanced technologies), immediately near the coexistence curve with the solid. Since the atoms of 3He have half-integer spin, and therefore obey to Fermi-Dirac statistics, the mechanism of superfluidity is in this case more complex: under the action of very weak interatomic van der Waals attractive forces, pairs of atoms are formed very weakly bound (hence the extremely low value of the critical temperature), which, having whole spin, obey to Bose-Einstein statistics.

To the absence of viscosity in superfluids corresponds the absence of electrical resistance for the motion of electrons of many metals, below critical temperatures with typical values Tc ≈ 1÷10 K, which are inversely proportional to the root of the isotopic mass of the ions that constitute the crystal lattice (isotopic effect). The phenomenon, called superconductivity, was discovered by Heike Kamerlingh Onnes in mercury, with Tc = 4 K. Higher critical temperatures have been obtained in metal alloys and subsequently in some copper oxides that, normally insulating, present the phenomenon of superconductivity at high temperature (Tc ≈ 30÷150 K) if properly treated. The superconductive state, in addition to the null electrical resistance, is characterized by the expulsion of the magnetic field: the magnetic induction field B inside a superconductor immersed in a magnetic field H, of intensity lower than a critical value Hc, is null (Meissner-Ochsenfeld effect).

Helium is the simplest element after hydrogen and presents an interesting variety of behaviors at low temperature. Because of the small mass and the weakness of the attractive part of the van der Waals interaction between the atoms, with diameter d = 2.7 Å, He condenses in the liquid phase at very low temperatures (4He is liquid below 4.2 K) and is the only element that remains liquid until the lowest temperatures reached and solidifies only under high pressures. The two isotopes 3He and 4He have very different behaviors that cannot be traced to the isotopic mass difference alone. In fact 3He has half-integer spin and obeys, like electrons, to Fermi-Dirac statistic: each single particle quantum state can be occupied by only one particle (Pauli exclusion principle). The 4He, instead, has zero spin and obeys to Bose-Einstein statistic: each quantum state can be occupied by an arbitrary number of particles and this can give rise, in appropriate conditions, to Bose-Einstein condensation. In this case, we have a macroscopic occupation of a single particle state.

To understand superfluidity and superconductivity we must first explain why the respective flow states are stable (or rather metastable on extremely long times), even if, for example in He, the interactions between the atoms of the liquid and the walls of the tube in which the flow occurs are more intense than the intermolecular forces and should tend to brake the ordered motion. There must be a mechanism that prevents the viscous or resistive process, that is, the conversion of kinetic energy, associated with the ordered motion of the fluid, into thermal energy of the disordered motion of the individual constituents.

In 1941 Lev Davidovič Landau established a criterion of superfluidity under the assumption that the systems in question could exchange energy with the outside world only through the creation or destruction of elementary excitations (quasi-particles) with energy ε and momentum p well defined and linked together by a certain relationship ε = ε(p) (law of dispersion), specific to each system. In isotropic liquids, the energy of excitations depends only on the modulus of momentum p ≡ |p|. The laws of conservation of energy and momentum associated with the possibility of quasi-particle excitation imply that the fluid motion can be damped only if the flow velocity is greater than a critical velocity vc = [ε(p)/p]min i.e. equal to the minimum of the function ε(p)/p on the quasi-particle dispersion curve. The existence of a non-zero minimum value is a necessary condition for superfluidity A gaseous system cannot be a model superfluid, because for the single particle excitation spectrum ε(p) = p2/2m, we have vc = 0, where m is the particle mass.

Phenomenology of 4He

The specific heat tends to increase as a function of temperature when approaching the critical value for the superfluid transition from both lower and higher temperatures, acquiring the peculiar form of the Greek letter λ, hence the name Tλ for the critical temperature of this transition.

The trend is that of phase transitions disorder-order of the second kind: passing, therefore, for the temperature Tλ the system is ordered. This is confirmed by the fact that, based on the trend of the coexistence curve between the superfluid liquid and the solid (which tends to be horizontal at low temperatures), it can be deduced that helium in the two phases has about the same entropy, i.e. the superfluid liquid is ordered as much as the solid.

However, this ordering cannot be spatial in nature as with the solid, because X-ray diffraction experiments show no difference between the structure of normal liquid helium above Tλ and superfluid helium below Tλ. On the contrary, immediately below Tλ the helium density decreases instead of increasing as the temperature decreases, indicating a preference for an enlarged spatial distribution.

The values of the dynamic viscosity η of helium below Tλ, as measured by the ability of helium to pass through thin capillaries (η < 10-12 Pa∙s), are at least a million times smaller than those measured by the damping of oscillations of a torsion pendulum immersed in liquid (η ≃ 10-6 Pa∙s). This enormous difference, together with numerous other experimental indications, led to the model of the two fluids: it is assumed that the liquid consists of two non-separable interpenetrating components, one of which, called normal, is viscous and entropy-bearing, the other, superfluid, is non-viscous, with zero entropy and therefore highly ordered.

In the capillary only the superfluid component passes through and therefore we measure a very low viscosity; with the torsion pendulum, however, the effect of the normal component is felt and therefore we measure the highest value of viscosity. At absolute zero, there is only the superfluid component, while, as the temperature increases, the normal component is formed, which behaves similarly to a gas.

The density ρn of the normal component increases slowly and then rapidly, until at Tλ it coincides with the total density ρ. Conversely, the superfluid density ρs goes to zero at the critical point and coincides with the total density ρ at absolute zero. If we consider helium, in a first approximation, as a gas of noninteracting bosons, we have Bose-Einstein condensation, consisting in the fact that at temperatures below a certain critical value Tc (which depends on the density) macroscopic occupation of a single state takes place.

At this point it would seem natural to identify the superfluid component of helium with the fraction (equal to 1 at absolute zero and 0 at the critical temperature) of helium atoms that have undergone the Bose-Einstein condensation. Moreover, if we insert the experimental value of the helium density in the formula for the critical temperature of a Bose gas, we obtain Tc ≃ 3.4 K, value not too different, given the simplicity of the model, from the experimental value of Tλ.

The Bose-Einstein condensation would also give rise to the ordering of the system in a space other than ordinary space, i.e., in the space of momentum, as required by the experimental data. However, a Bose gas cannot be a realistic model of superfluid because it does not predict the existence of a critical transport velocity above which superfluidity cannot occur. One must, therefore, take into account the interaction between the atoms as well as their statistics.

Interacting helium atoms must have a fundamental state that gives rise to quasiparticle excitations compatible with the existence of the critical velocity and, at the same time, presents the Bose-Einstein condensation phenomenon (generalizing its properties to interacting systems).

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