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Physical Review

Simplified Model of Liquid Helium

K. Sawada, Ramabhadra Vasudevan

Published Oct 15, 1961• 13 citations
Publisher

Abstract

A simplified version of a Bose gas with negative scattering length is studied with a model Hamiltonian similar to that used in the case of repulsive interactions. A new transformation is used which takes into account the attractive interactions in the system, and the ground-state energy is calculated from experimental values of the sound velocity in the system. Real values for the phonon energy spectrum are obtained, and the ratio of the effective mass to ordinary mass is deduced using experimental values of interparticle separation and core radius. A justification for using this modified Hamiltonian is attempted in the Appendix.

DOI
10.1103/physrev.124.300

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The excitation spectrum for a Bose gas with repulsive and attractive interactions

Gerald V. Brown

The correct qualitative features of the helium II elementary excitation spectrum a r e derived microscopically for a realistic interatomic potential.The strong repulsive core is included by using a reaction matrix in the Hamiltonian.The attractive well i s success fully included by assuming a generalized Bose-Einstein condensation.The pair Hamilton ian is diagonalized by the thermodynamically equivalent Hamiltonian method.Numerical solutions yield spectra with phonon and roton regions.The spectrum energies a r e too high for the Yntema-Schneider potential.Another potential, constructed to fit virial co efficient data classically, gives better results.Spectra a r e presented for a series of at tractive well strengths.

•Oct 1, 1968•0 citations
A simplified version of a Bose gas with negative scattering length is studied with a model Hamiltonian similar to that used in the case of repulsive interactions. A new transformation is used which takes into account the attractive interactions in the system, and the ground-state energy is calculated from experimental values of the sound velocity in the system. Real values for the phonon energy spectrum are obtained, and the ratio of the effective mass to ordinary mass is deduced using experimental values of interparticle separation and core radius. A justification for using this modified Hamiltonian is attempted in the Appendix.
The correct qualitative features of the helium II elementary excitation spectrum are derived microscopically for a realistic interparticle potential using a form of pair Hamiltonian. The strong repulsive core is included by using reaction matrix elements in the Hamiltonian, and the attractive well is successfully included by assuming a generalized Bose-Einstein condensation. The pair Hamiltonian is diagonalized by the thermodynamically equivalent Hamiltonian method of Wentzel. Numerical solution of the equations yields spectra with phonon and roton regions. Since the energies in the spectrum are too high when the Yntema-Schneider potential is used, another potential is constructed which still fits the second-virial-coefficient data and gives better results. Spectra calculated for a series of attractive well strengths show the effect of weaker wells.
The properties of a Bose-Einstein gas with repulsive interparticle interactions are determined at low temperature. It is shown that the contribution to the energy arising from particle excitation with small momentum transfers computed in conventional perturbation theory is divergent, but that this difficulty can be avoided by an alternative procedure. An exact method is developed for dealing with that part of the Hamiltonian which gives rise to the perturbation divergence and an exact solution is obtained. The treatment of the region of large momentum transfers is then carried out by introducing the concept of the scattering length of the interaction. The energy is given by the well-known result of the optical theorem corrected by a series in powers of ${(\ensuremath{\rho}{a}^{3})}^{\frac{1}{2}}$, where $\ensuremath{\rho}$ is the density and aaa the scattering length. Examination of the processes contributing to the energy shows that 95.8% of the correction of order ${(\ensuremath{\rho}{a}^{3})}^{\frac{1}{2}}$ results from a simple alteration of the perturbation energy denominators which takes into account the interaction energy of excited pairs with the unexcited pairs of the medium. The remaining 4.2% arises from multiple particle excitation. These methods also predict the existence of phonon excitation with phonon energies that are linear in the phonon momentum for small excitation and approach q22M\frac{{q}^{2}}{2M}2Mq2​ for large momentum. The detailed consideration of this spectrum forms part of the following paper.The results are compared with the theory of fermion systems with strong interactions, where again the largest correction to the first-order "optical" energy arises from alteration of the two-particle propagator which is required to take into account interactions with the many unexcited particles of the medium. The propagator corrections in the fermion and boson systems are similar except for a characteristic difference arising from the statistics. The theory for bosons is shown to be essentially identical with that given by Brueckner and Levinson in their general formulation of the theory of many-body systems with strong interactions.Systems with attractive but saturating interactions are also considered and some new difficulties are shown to arise which cannot be treated by a straightforward application of the methods of this paper.
An electron in helium vapor at 4\ifmmode^\circ\else\textdegree\fi{}K is characterized by its sss-wave scattering length off helium atoms. This scattering length is small compared with the average interparticle spacing in the helium vapor. Consequently, we have taken as a model a particle interacting via hard-core repulsion with an ideal gas. This establishes a connection with the hard-sphere Bose problem of more general interest. The model described above is simpler (a) because it has only Boltzmann statistics and (b) because the electron is very light compared with the helium atoms. For this model, we have calculated the interaction free energy of the electron assuming that it is in statistical equilibrium with the helium gas. In the sss-wave approximation, it is shown that this interaction free energy is rigorously $2\ensuremath{\pi}\ensuremath{\rho}a(\frac{{\ensuremath{\hbar}}^{2}}{m})$ due to single scattering, all higher order multiple-scattering effects being zero. Here, $\ensuremath{\rho}$ is the average density of helium atoms, ddd is the scattering length, and mmm, is the electron mass. Since the ppp-wave approximation contributes a term of order $\ensuremath{\rho}{a}^{3}$, it is evident that the term $2\ensuremath{\pi}\ensuremath{\rho}a(\frac{{\ensuremath{\hbar}}^{2}}{m})$ is good to higher densities than might previously have been supposed. This provides partial justification for the "bubble" model of the electron mobility since the term $2\ensuremath{\pi}\ensuremath{\rho}a(\frac{{\ensuremath{\hbar}}^{2}}{m})$ is certainly good up to densities at which the free energy of the "bubble" configuration becomes smaller.
An NNN-body Bose-Einstein system of particles with long-range attraction and hard-sphere repulsion between particles is considered. It is shown that if the constants of the interaction have values within a certain range it is possible to calculate the ground-state energy of the system as a function of $\frac{\ensuremath{\Omega}}{N}$, where $\ensuremath{\Omega}$ is the volume of the box containing the system, in the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\Omega}\ensuremath{\rightarrow}\ensuremath{\infty}$, with $\frac{N}{\ensuremath{\Omega}}=\ensuremath{\rho}$ fixed. The results show that the system can possess an NNN-body bound state, which has an equilibrium density and negative energy, and that the interactions can be saturating. Excited states are also considered. It is shown that low-lying excitations consist purely of phonons, whose velocity agrees with that computed from the macroscopic compressibility, furnished by the ground-state energy. The formula for the general excited energy levels suggests that thermodynamically the system may have a "gas" phase and two "liquid" phases, the transition between the two "liquid" phases being the analog of the Bose-Einstein condensation of the ideal gas. Thermodynamic considerations are, however, not contained in this paper.
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)
The correct qualitative features of the helium II elementary excitation spectrum a r e derived microscopically for a realistic interatomic potential.The strong repulsive core is included by using a reaction matrix in the Hamiltonian.The attractive well i s success fully included by assuming a generalized Bose-Einstein condensation.The pair Hamilton ian is diagonalized by the thermodynamically equivalent Hamiltonian method.Numerical solutions yield spectra with phonon and roton regions.The spectrum energies a r e too high for the Yntema-Schneider potential.Another potential, constructed to fit virial co efficient data classically, gives better results.Spectra a r e presented for a series of at tractive well strengths.