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Some Problems About The 3He Superfluid

Ting-Hang Pei

Published Jul 1, 2019• 0 citations
OpenAlex

Abstract

We discuss the stability problem of the atomic pair in the 3He superfluid. It is similar to the bound-state problem of the hydrogen molecule because both of them are composed of two spin-1/2 nuclei. The difference is the cause for bonding. What is of interest here is whether the nuclear spin coupling between two 3He nuclei is large enough to form a stable atomic pair in the superfluid. According to our calculations, the energy difference between the excited state F=1 and the ground state F=0 caused by the nuclear spin coupling of the two hydrogen nuclei is about 5.6x10^-9 eV when the intermolecular distance is 0.74 A. When the two 3He nuclei are separated by 2.9 A, the same energy difference caused by the nuclear spin coupling is 4.8x10^-10 eV. This value is much less than the binding energy of two 3He atoms about 1.0 eV, and is also much smaller than the Lennard-Jones potential of 8.74x10^-4 eV between the two helium atoms. Therefore, two 3He atoms cannot form a stable atomic pair due to the nuclear spin coupling even the spin wave.

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We discuss the stability problem of the atomic pair in the 3He superfluid. It is similar to the bound-state problem of the hydrogen molecule because both of them are composed of two spin-1/2 nuclei. The difference is the cause for bonding. What is of interest here is whether the nuclear spin coupling between two 3He nuclei is large enough to form a stable atomic pair in the superfluid. According to our calculations, the energy difference between the excited state F=1 and the ground state F=0 caused by the nuclear spin coupling of the two hydrogen nuclei is about 5.6x10^-9 eV when the intermolecular distance is 0.74 A. When the two 3He nuclei are separated by 2.9 A, the same energy difference caused by the nuclear spin coupling is 4.8x10^-10 eV. This value is much less than the binding energy of two 3He atoms about 1.0 eV, and is also much smaller than the Lennard-Jones potential of 8.74x10^-4 eV between the two helium atoms. Therefore, two 3He atoms cannot form a stable atomic pair due to the nuclear spin coupling even the spin wave.
A variational calculation is performed for the (3He)4 system interacting pairwise through the Lennard-Jones He–He potential. Use of an integrodifferential Euler equation is made to determine an optimum wave function, leading to results showing that the existence of this molecule is unlikely. It is confirmed that states of the system with nonzero orbital angular momentum yield lower energies. Also, it appears that the number of 3He atoms needed to form self-bound molecules must be larger than 5.
Abstract The energy of interaction of a normal helium atom and one excited to the first triplet or singlet metastable state is calculated over a range of nuclear separations from a0 to 12a0. The Heitler-London method is followed, with inclusion of all non-orthogonality integrals, and using analytic wave functions. Because of identity of the nuclei, g and u states of interaction occur, the energy of the u state having a minimum at about 2·1a0, and a positive maximum of 0·29 and 0·26 eV (for triplet and singlet states respectively) at about 4a0; the g state is entirely repulsive. Comparison is made with experimental evidence for the binding energy of normal and metastable (triplet) atoms. An estimate is made of the second-order dipole-dipole interaction for large separations, and it seems certain that the first-order interaction dominates at least to 12a0. Methods of calculating the necessary integrals are discussed in an appendix.
Superfluidity in liquid 3 He, discovered in 1972, is due to pairing, as is the case for superconductivity in ordinary metals. However, the pair is in the 3 P state so that a new aspect of superfluidity appears with internal degrees of freedom. It offers quite an interesting real example of ‘spontaneous symmetry breaking’. This aspect will be emphasised in the following discussion. Fermi liquid 3 He A 3 He atom is a fermion with nuclear spin ½. Although 3 He gas liquefies at low temperatures due to the weak van der Waals force, it does not solidify even at T = 0 unless the pressure exceeds 33.5 atm (3.4 × 10 6 Pa) as shown in the phase diagram (Fig. 6.1 (a)). While Bose liquid 4 He becomes superfluid at about 2K, 3 He is Fermi-degenerate at less than T F ∼ 0.1 K and shows the properties of a normal Fermi gas. For example, the specific heat and the susceptibility are proportional to T at T < T F . It is only at an ultra-low temperature in the mK-region that the system undergoes a transition to the superfluid state. Figure 6.1(b) is the phase diagram showing the superfluid phases with variables T,P and external magnetic field H in the mK-region. It should be noted that there are two thermodynamically distinct phases called the A phase and the B phase even at H = 0, and that for H ≠ 0 the A 1 phase appears.
The bulk, ground state properties of atomic hydrogen, deuterium, and tritium systems are calculated assuming that all pair interactions occur via the 3Σ+u atomic triplet potential. The conditions required to obtain this system, including inhibition of recombination through the energetically favorable singlet interaction, are discussed. The internal energy, pressure, and compressibility are calculated over the volume range 40 cm3/mole ? V ⩽200 cm3/mole applying the Monte Carlo technique with a quantum mechanical variational wavefunction. The system studied consisted of 32 atoms in a box with periodic boundary conditions. Results show that atomic triplet hydrogen and deuterium remain gasous at 0 K; i.e., the internal energy is positive at all molar volumes considered. Under the same conditions, tritium forms a liquid with a binding energy of E0 ? −0.75 K per atom, at a volume of V ? 130 cm3/mole. The pair distribution function for these systems is also calculated. A brief discussion of the predicted superfluid behavior of 3Σ+u hydrogen and tritium is presented.
The three-body Schrdinger equation is approximately solved in the hyperspherical coordinates and the binding energies of the three-body weakly bound systems are calculated with the purpose to find if He-He-Ba trimers could exist. Using the special feature of the B-spline function like the flexible and highly localized properties, hypersphercial potentials are obtained by modifying the knots distribution of the B-spline basis of different weakly bound three-atom systems. Employing the best empirical interaction potentials between each pair of particles, we obtain that in the ground state binding energies of the weakly bound typical three-atom systems, the bindings of the molecules, 4He-4He-138Ba, 4He-3He-138Ba and 3He-3He-138Ba are possible. The binding energies of these systems are shown in the order of 1 Kelvin, each system could support only one bound state. These weakly bound molecules can exist only in a very cold environment. To get insight into the geometry of the molecules, the features of the channel functions associated with the hyperspherical potential curves of each system are investigated.
We investigate a two-component mixture of resonantly interacting Fermi gases as a function of the ratio $\ensuremath{\kappa}$ of the heavy to the light mass of the two species. The diffusion Monte Carlo method is used to calculate the ground-state energy and the pair correlation function starting from two different guiding wave functions, which describe, respectively, the superfluid and the normal state of the gas. Results show that the mixture is stable and superfluid for mass ratios smaller than the critical value ${\ensuremath{\kappa}}_{c}=13\ifmmode\pm\else\textpm\fi{}1$. For larger values of $\ensuremath{\kappa}$ simulations utilizing the wave function of the normal state are unstable towards cluster formation. The relevant cluster states driving the instability appear to be formed by one light particle and two or more heavy particles within distances on the order of the range of the interatomic potential. The small overlap between the wave function of the trimer bound state and the guiding wave function used to describe the superfluid state produces the unphysical stability of the superfluid gas above ${\ensuremath{\kappa}}_{c}$.