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Journal of Mathematical Physics

Excitation Spectrum of an Impurity in the Hard-Sphere Bose Gas

Toshio Soda

Published Jan 1, 1964• 2 citations
Publisher

Abstract

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.

DOI
10.1063/1.1704060

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•Sep 26, 1960•0 citations
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.
The dynamical properties of an impurity in the hard-sphere Bose gas are investigated by extending the pseudopotential method and Hamiltonian of the author's previous work. The interaction potential between two impurities is calculated up to one quasiparticle exchange and is shown to have a Yukawa form of attraction with a range of (16πρa)−1/2 in addition to the hard-core interaction at the origin. (ρ and a are the density of the boson gas and hard-core diameter of the boson, respectively.) Necessary conditions for impurities to have a bound state are also obtained. It is found that the mass of the impurity must be greater than approximately nine times the boson mass, and the density of the boson gas must lie in some range with a maximum and minimum. Then the differential cross section for Compton-type scattering and one quasiparticle production on an impurity are calculated. The Čerenkov radiation of sound quanta is studied when the impurity is traveling very fast in the hard-sphere Bose gas. It is found that there is no radiation unless the velocity v of the impurity is faster than [k2 + (16πρa)]1/2 and the radiation has a maximum for the angle θ0 such that cosθ0 = (k2 + 16πρa)½/v. The cross section for Čerenkov radiation is also calculated. (k is the momentum of the quasiparticle emitted in the Čerenkov radiation.)
The problem of the low-lying spectrum of a system of many bosons and one impurity particle interacting by repulsive, central, two-body forces is treated by first eliminating the dynamical variables of the impurity in terms of an added effective boson-boson interaction, and then treating the boson system by an extension of Bogolubov's method. In this way various physical properties of the clothed impurity particle, including its dispersion relation and effective mass, are obtained; singularities associated with the onset of acoustical wave drag as the impurity speed reaches the speed of sound in the boson medium are predicted. The parameters measuring the smallness of the corrections to the above results are shown to be the dimensionless boson-boson interaction, the dimensionless boson-impurity interaction, and the boson-impurity mass ratio. The treatment is then adapted to the case of hard spheres at low density by employing the Huang-Yang-Lee pseudopotential method; the results are valid if the dimensionless boson density and the boson-impurity mass ratio are small. A few remarks are made concerning the connection of the models treated with experimentally accessible results.
A many-body pseudopotential for a hard-sphere system which is mathematically equivalent to the boundary conditions imposed on the wave functions is obtained as a generalization of Huang and Yang's binary pseudopotential, and the lowest order term of the ground state energy resulting from the triple collision is calculated for the Bose system. Instead of the direct perturbational treatment of the pseudopotential, the use of the ρk-representation for the Bose system is proposed and the ground state energy is calculated in this representation. It is shown that the energy thus calculated is of the form of an expansion in powers of (ρa3)1/2, where ρ is the average density and a the hard-sphere diameter, which is agreement with Lee and Yang's conclusion drawn from the binary collision expansion method. Some discussions associated with the roton spectrum of liquid He4 are given.
We investigate the properties of an impurity immersed in a dilute Bose gas at zero temperature using quantum Monte Carlo methods. The interactions between bosons are modeled by a hard-sphere potential with scattering length aaa, whereas the interactions between the impurity and the bosons are modeled by a short-range, square-well potential where both the sign and the strength of the scattering length bbb can be varied by adjusting the well depth. We characterize the attractive and the repulsive polaron branch by calculating the binding energy and the effective mass of the impurity. Furthermore, we investigate the structural properties of the bath, such as the impurity-boson contact parameter and the change of the density profile around the impurity. At the unitary limit of the impurity-boson interaction, we find that the effective mass of the impurity remains smaller than twice its bare mass, while the binding energy scales with ${\ensuremath{\hbar}}^{2}{n}^{2/3}/m$, where nnn is the density of the bath and mmm is the common mass of the impurity and the bosons in the bath. The implications for the phase diagram of binary Bose-Bose mixtures at low concentrations are also discussed.
Using the energy-momentum relation derived by Brueckner and Sawada, the low-temperature thermodynamical properties of a Bose gas with hard-sphere interaction are investigated. The two cases, (1) when this energy spectrum is similar to that observed in liquid helium II and (2) when it becomes similar to that derived by Lee, Huang, and Yang, are investigated separately. In the first case (where the spectrum exhibits a nonmonotonic character), the analytical derivation of the expressions for the various thermodynamical properties is rendered possible by dividing the momentum range into three suitable intervals such that within each interval the epsilon -p relation can be approximated to a simple algebraic one. The numerical magnitudes of the various parameters characterizing the algebraic relations are calculated for the case of a gas with density equal to that of liquid helium(II) and for three plausible values of the hard-sphere diameter. The calculated results agree qualitatively with those obtained from the experimental study of this liquid. For the second case, expressions are derived for the various macroscopic properties of the assembly in terms of suitable mathematical functions and then are written down in the form of series expansions for the two particular cases of interest. (auth)