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Physical Review A

Many-body clusters in a dense strongly interacting quantum liquid at O°K

RAMAMURTI RAJARAMAN

Published Jun 1, 1974• 0 citations
Publisher

Abstract

A beginning is made towards determining the energy of dense and strongly interacting quantum liquids at 0 \ifmmode^\circ\else\textdegree\fi{}K, starting all the way from the interparticle potential. Liquids ${\mathrm{He}}^{3}$ and ${\mathrm{He}}^{4}$ as well as neutron-star interiors approximate such a system. Since expansions do not converge either in powers of the interaction nor in the density, $n$-body clusters have to be calculated for all $n$ and to all orders in the potential. Working within a simplified model, a method is suggested for estimating $n$-body contributions to the binding energy. The method is far simpler than more-direct methods thus far employed, which can be used only for low values of $n$. The method is verified and illustrated by comparison with these existing calculations of two-, three-, and four-body energies. A discussion is also given of the steps remaining to be taken from where this paper leaves off. Possible "pairing" and "super" properties are outside the scope of this paper.

DOI
10.1103/physreva.9.2761

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A beginning is made towards determining the energy of dense and strongly interacting quantum liquids at 0 \ifmmode^\circ\else\textdegree\fi{}K, starting all the way from the interparticle potential. Liquids He3{\mathrm{He}}^{3}He3 and He4{\mathrm{He}}^{4}He4 as well as neutron-star interiors approximate such a system. Since expansions do not converge either in powers of the interaction nor in the density, nnn-body clusters have to be calculated for all nnn and to all orders in the potential. Working within a simplified model, a method is suggested for estimating nnn-body contributions to the binding energy. The method is far simpler than more-direct methods thus far employed, which can be used only for low values of nnn. The method is verified and illustrated by comparison with these existing calculations of two-, three-, and four-body energies. A discussion is also given of the steps remaining to be taken from where this paper leaves off. Possible "pairing" and "super" properties are outside the scope of this paper.
The variational method for calculating energy of quantum fluids, and its applications to the Bose liquid 4He^{4}\mathrm{He}4He, Fermi neutron gas, and liquid 3He^{3}\mathrm{He}3He are discussed. The correlation functions are parametrized by their healing distance, and can depend on the states occupied by the correlated particles in the model wave function. They are calculated by constrained variation of the lowest-order contributions. The healing distance has a prescribed value in lowest-order calculations, whereas it is sufficiently large in hopefully exact energy calculations. The many-body cluster contributions in Bose fluids are summed with successive approximations of an integral equation due to van Leeuwen et al. A simple diagrammatic cluster expansion is presented for Fermi liquids, and its direct diagrams are summed with the integral equation. The contribution of exchange diagrams is shown to decrease rapidly with the number of exchanges, and their sums are truncated after the energy has converged to within a few percent.
The variational method for calculating energy of quantum fluids, and its applications to the Bose liquid 4He, Fermi neutron gas, and liquid 3He are discussed. The correlation functions are parameterized by their healing distance, and can depend on the states occupied by the correlated particles in the model wave function. They are calculated by constrained variation of lowest order contributions. The healing distance has a prescribed value in lowest order calculations, whereas it is sufficiently large in hopefully exact energy calculations. The direct many-body cluster diagrams are summed with successive approximations of an integral equation. The contribution of exchange diagrams is shown to decrease rapidly with the number of exchanges, and their sums are truncated after the energy has converged to within a few percent.
An approximation method developed previously to deal with many particles in strong interaction is examined in further detail. It is shown that the series giving the interaction energy is a development in a sequence of linked or irreducible cluster terms each of which gives a contribution to the energy proportional to the total number of particles. Consequently the convergence of the expansion is independent of the total number of particles. The origin of this simple feature is illustrated by showing that a similar situation exists in the expansion of standard perturbation theory. The numerical convergence of the expansion is quantitatively discussed for the nuclear problem where it is shown that the correction arising from the first cluster term involving three particles is less than the leading term by a factor of about ${10}^{\ensuremath{-}4}$. The smallness of the correction is largely a result of the action of the exclusion principle.
In this work we investigate the structure and dynamics of small clusters of Helium atoms. We consider bound states of clusters having A = 2, 3, 4, 5, 6 atoms and continuum states in the three-atom system. Motivated by the fact that the He-He system has a very large scattering length a compared to the range r 0 of the He-He potential (r 0/a < 1/10), we propose the use of a soft-core interparticle potential. We use an attractive gaussian potential that reproduces the values of the dimer binding energy and the atomatom scattering length obtained with one of the widely used He-He interactions, the LM2M2 potential. In addition, we include a repulsive three-body force to reproduce the trimer binding energy. With this model, consisting in the sum of a two- and a three-body potential, we show the spectrum of the four, five, and sixparticle systems and phase-shifts and inelasticities in the three-atom system. Comparisons to calculations using realistic He-He potentials are given. In addition some universal relations are explored.
We derive coupled-cluster equations for three-body Hamiltonians. The equations for the one- and two-body cluster amplitudes are presented in a factorized form that leads to an efficient numerical implementation. We employ low-momentum two- and three-nucleon interactions and calculate the binding energy of 4He^{4}\mathrm{He}4He. The results show that the main contribution of the three-nucleon interaction stems from its density-dependent zero-, one-, and two-body terms that result from the normal ordering of the Hamiltonian in coupled-cluster theory. The residual three-body terms that remain after normal ordering can be neglected.
Variational calculations of the ground-state energy of hard-sphere, helium-4, Coulomb, and nuclear Yukawa fluids are carried out. The long-range part v(r>d)v(r>d)v(r>d) and a constant part $\ensuremath{\lambda}$ at r<dr<dr<d of the bare interaction are assumed to contribute to the average field. The correlation function f(r)f(r)f(r) is obtained by minimizing the two-body cluster contribution due to the remainder interaction, and the distance ddd is determined by minimizing the energy as calculated with the hierarchy of hypernetted-chain-type equations. The convergence of two- and three-particle distribution functions given by these equations is discussed. The method is shown to be exact in the low-density limit, and the comparison of the present results with the available exact (standard) results in the high-density limit (region) verifies its accuracy over the entire density range. It is shown that the variationally obtained value of ddd in the high-density Coulomb fluid can be understood with conventional meanfield theory. A solidification criterion is obtained by studying the energies in liquid and solid phases with the present f(r)f(r)f(r). This criterion explains the solidification of the Coulomb, hard-sphere, and helium systems, and the inability of the nuclear Yukawa system to solidify. The variational method in the form presented here could be very useful for studying complicated quantum systems.