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arXiv:cond-mat/9806179

Impurity effective mass in superfluid $^4$He

I. O. Vakarchuk, Volodymyr Babin

Published Jun 15, 1998• 0 citations
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Abstract

The system of bose liquid + impurity is considered. The energy spectrum as well as effective mass of impurity is calculated. For the case of the $ ^3He$ atom in superfluid $^4He$ numerical calculations are performed.

DOI
10.48550/arxiv.cond-mat/9806179

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Impurity effective mass in superfluid 4^44He
The system of bose liquid + impurity is considered. The energy spectrum as well as effective mass of impurity is calculated. For the case of the 3He ^3He3He atom in superfluid 4He^4He4He numerical calculations are performed.
We consider the movement of a 3He impurity atom in the Bose liquid. We suggest to describe the many-particle correlations between atoms of the Bose liquid, by using a deformed Heisenberg algebra. As generalized coordinates, we choose the collective variables that are the Fourier components of fluctuations of the density of Bose particles. The wave function of the investigated system in the zeroth approximation is the product of the wave function of the liquid helium-4 within deformed commutation relations between generalized coordinates and momenta and the plane wave of the impurity atom. We calculate the correction to the ground-state energy of the system “Bose liquid plus impurity” and the effective mass of the impurity atom 3He, by assuming that the boson-impurity interaction is a small perturbation.
A finite one-dimensional microscopic model of a superfluid is presented. The model consists of interacting Bose particles with an additional impurity particle confined to a ring. Both semiclassical and exact quantum calculations reveal dissipationless motion of impurity with increased effective mass due to its interaction with the excitations of Bose fluid. It is shown that both the excitation spectrum of Bose fluid and the excitation spectrum of impurity can be analyzed using the structure of the ground state of the system.
We calculate the binding energies of a 3He{}^{3}\mathrm{He}3He impurity and a fictitious 4He{}^{4}\mathrm{He}4He impurity to a superfluid 4He{}^{4}\mathrm{He}4He vortex. Within the accuracy of the calculation, there is no difference between the mass-3 and mass-4 binding energies, suggesting that the binding is due to the displacement of atoms from the core. Comparing the results between Onsager-Feynman ( ${\ensuremath{\Psi}}_{\mathrm{OF}}$) and ``shadow'' ( ${\ensuremath{\Psi}}_{D}$) variational functions, the energy is dependent on the density for ${\ensuremath{\Psi}}_{\mathrm{OF}}$, but independent for ${\ensuremath{\Psi}}_{D}$. The binding of 2.3 K for ${\ensuremath{\Psi}}_{D}$ at zero pressure is smaller than the experimental estimate. We also estimate the binding energies of charged ions.
The impurity excitation spectrum is calculated for the hard-sphere Bose gas with dilute impurity concentration, using the pseudopotential method of Lee, Huang, and Yang. The spectrum for the ideal case, in which the mass and interaction of impurity are the same with those of the bosons in the gas, is k2 in the unit of 2ma = h/ = 1 (ma being the mass of boson, and k the momentum of impurity), with an effective mass correction of −6445[(ρa3)12/π12]k2+O[(ρa3)k2,k4], where ρ is the density of the boson gas, and a is the hard-sphere diameter of the boson. The general impurity case is also calculated and found to have an impurity spectrum rk2, with a similar effective mass correction, where r is the mass ratio of the boson to the impurity.