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Physical Review Letters

Determination of the Momentum Distribution from the Dynamic Structure Factor in Quantum Liquids

Joseph Jacob Weinstein, John W. Negele

Published Oct 4, 1982• 80 citations
Publisher

Abstract

Scaling of the dynamic structure factor at high momentum transfer is established by using time-ordered perturbation theory. Whereas the scaling function is the longitudinal momentum distribution for systems interacting with smooth two-body potentials, additional terms contribute in the case of strong short-range repulsion. Illustrative results are presented for a hard-sphere Bose gas and the relevance to electron scattering from nuclei and neutron scattering from liquid helium is discussed.

DOI
10.1103/physrevlett.49.1016

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Scaling of the dynamic structure factor at high momentum transfer is established by using time-ordered perturbation theory. Whereas the scaling function is the longitudinal momentum distribution for systems interacting with smooth two-body potentials, additional terms contribute in the case of strong short-range repulsion. Illustrative results are presented for a hard-sphere Bose gas and the relevance to electron scattering from nuclei and neutron scattering from liquid helium is discussed.
The short-time or high-frequency expansions of several dynamical correlation functions in a quantum hard-sphere fluid of spinless bosons are computed. These expansions are shown to be of a different form than those obtained for either a fluid of quantum particles interacting through a smooth potential or for a classical fluid of hard spheres. Assuming that the same dynamical contributions that are important in the short-time expansions are also responsible for dominant final-state interactions in high-energy neutron scattering, we also approximately compute the shape of the dynamic structure factor near the free-particle frequency.
The dynamic structure factor of a trapped Bose-Einstein condensed gas is investigated at zero temperature in the framework of Bogoliubov theory. Different values of momentum transfer are considered, ranging from the phonon to the single-particle regime. Various approximate schemes are discussed, including the local density approximation, where the system is locally described as a uniform gas, and the impulse approximation, in which the response is fixed by the momentum distribution of the condensate. A comprehensive approach, based on the eikonal expansion, is presented. The predictions of theory are successfully compared with the results of recent two-photon Bragg scattering experiments, at both low and high momentum transfer. Relevant features of the dynamic structure factor are also discussed using the formalism of sum rules and the concept of scaling. Particular emphasis is given to the regime of high momentum transfer, in which the dynamic structure factor is sensitive to the behavior of the order parameter in momentum space, and some instructive examples showing the consequence of long-range coherence are presented.
With the pair Hamiltonian model as the starting point, perturbation theory calculations are performed for the radial distribution function and the structure factor through second order in the two-body interaction potential for a many-boson system below the Bose-Einstein transition temperature. For the special case of a dilute hard-sphere Bose gas at T=0° K, the structure factor is calculated in the low-momentum limit keeping terms through second order in the hard-sphere gas parameter (n a3)1/2, where n is the gas density and a is the hard-sphere diameter. The resulting expression for the structure factor is shown to satisfy the Feynman-Bijl relation explicitly.
The momentum distribution n(k)n(k)n(k) and condensate fraction N0N\frac{{N}_{0}}{N}NN0​​ for liquid helium and for a charged Bose gas are evaluated in perturbation theory through second order in 1N\frac{1}{\sqrt{N}}N​1​. For liquid helium at normal vapor pressure, to this order in perturbation theory, we obtain unphysical, negative values for n(k)n(k)n(k). This result will be discussed in detail.
The impulse approximation is used to describe the inelastic cross section for high-energy neutron scattering from superfluid helium in terms of a ground-state momentum distribution of the helium atoms as given by existing variational calculations. The predicted shape of the cross section is compared with recent experimental data and with a previous theory which approximated the noncondensate contribution to the dynamic structure factor by a single-Gaussian function. In contrast to conclusions obtained using the single-Gaussian approximation, our predicted inelastic cross section omitting a condensate contribution provides an acceptable fit to the experimental data. This fit is worsened by introducing a contribution from a condensate fraction in the amount consistent with the variational calculations, namely 11%. If the condensate fraction is arbitrarily reduced from this value, it is found that a much smaller value, with an upper limit of about 3%, is consistent with the available experimental data. It appears that unique assignments of condensate-fraction contributions to inelastic neutron scattering, for the range of neutron energies presently available, will require an improved theory of final-state effects on the shape of the condensate contribution.
The spectrum of noncondensate excitations in neutron scattering on bosons is obtained in the framework of the Bogoliubov models both for liquid 4He and a dilute gas. The problem is solved using a path-integral representation of the partition function of the system. We describe the influence of scattering of neutrons on a Bose condensate in a stationary (time-independent) picture in the Gibbs equilibrium ensemble. This influence is a stationary boson response, and it depends on the initial neutron momentum k, transfer momentum p, and the neutron-boson interaction λ, which is related to the scattering length. The contribution of the neutrons to the initial Bogoliubov spectrum is found to be important for “quasi-elastic” scattering on the noncondensate, while the contribution of deep inelastic scattering is small; no contribution from elastic scattering on the Bose condensate is found. In the case of liquid Helium, the response is unlikely to be observable for all values p. On the other hand, for a gas one may expect a visible effect, in particular for a small momentum transfer p and a small density of the Bose condensate ϱ.