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Physical review. B, Condensed matter

Correlation functions in quantum hard-sphere fluids at short times or high frequencies, and high-energy neutron scattering

T. R. Kirkpatrick

Published Aug 1, 1984• 23 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.30.1266

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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 properties of quantum fluids in Fourier space, as the system response functions to weak external fields, are analyzed taking the quantum hard-sphere fluid as a probe. This serves to clarify the physical meaning of the different radial correlation functions that can be defined in a path-integral quantum fluid, since these functions are the r-space counterparts of the response functions. The basic feature of the external field relevant to this discussion is connected with its localizing/nonlocalizing effect on the quantum particles composing the fluid (i.e., a localizing field causes the collapse of the particle thermal packet). Fields that localize the quantum particles reveal the so-called instantaneous quantities (e.g., the conventional static structure factor), which are related with the diagonal elements of the density matrix. Fields that do not localize the quantum particles show the so-called linear response quantities, which are related to the diagonal and the off-diagonal density matrix elements. To perform this study the path-integral formalism is considered from the functional analysis approach. Given that the Gaussian Feynman–Hibbs effective potential picture is known to represent well many structural features of the quantum hard-sphere fluid, the parallel study of the response functions within this picture is also presented. In particular, the latter picture provides an accurate Ornstein–Zernike scheme that can be used for numerical calculations of response functions over a wide range of conditions, and also gives fine estimates for quantities difficult to compute with the path integral. Results for the quantum hard-sphere fluid obtained within the latter scheme are reported, tests of consistency are given, and the possibility of approximating the instantaneous response function by means of the coherent part of the linear response function is assessed.
A many-body theory is presented here for the Van Hove pair (or density) correlation function, in a quantum fluid obeying Boltzmann statistics, of which the Fourier transform $S(q,\ensuremath{\omega})$ determines the differential cross section for inelastic coherent scattering of slow neutrons. The theoretical formalism of two-particle thermodynamic Green's functions is based on Baym and Kadanoff's conserving TTT-matrix approximation, developed for strong short-range (intermolecular) potentials. The only other approximations made are: (i) the smallness of the activity or fugacity $z\ensuremath{\ll}1$, which implies Boltzmann statistics for the single-particle states; and (ii) the TTT matrices are replaced by elastic scattering amplitudes. Our results for $S(q,\ensuremath{\omega})$ developed to order z2{z}^{2}z2 are found to consist of Boltzmann averages of Lorentzian distributions in the energy transfer $\ensuremath{\hbar}\ensuremath{\omega}$ (except for a three-particle scattering term which is developed in the $\ensuremath{\delta}$-function limit of the Lorentzians). Our results are general enough to satisfy detailed balancing at all stages and are found to generalize Van Leeuwen and Yip's results for the classical case. The only unknown in our results is the intermolecular scattering amplitude or phase shifts. Our results seem, however, to break down in the hydrodynamic limit $\ensuremath{\omega}$ and $q\ensuremath{\rightarrow}0$ since each term in our expansion diverges in this limit.
The thermodynamics of a homogeneous dilute Bose gas with an arbitrarily strong repulsion between particles is investigated on the basis of the exact relation connecting the pair correlation function with the in-medium pair wave functions and occupation numbers. It is shown that the effective-interaction scheme, which is reduced to the Bogoliubov model with the effective pairwise potential, is not acceptable for investigating the short-range particle correlations in a dilute strongly interacting Bose gas. In contrast to this scheme, our model is thermodynamically consistent and free of the ultraviolet divergences due to accurate treatment of the short-range boson correlations. An equation for the in-medium scattering amplitude is derived that makes it possible to find the in-medium renormalization for the pair wave functions at short boson separations. Low-density expansions for the main thermodynamic quantities are reinvestigated on the basis of this equation. In addition, the expansions are found for the interaction and kinetic energies per particle. It is demonstrated that for a many-boson system of hard spheres the interaction energy is equal to zero for any boson density. The exact relationship between the chemical potential and in-medium pair wave functions is also established.
The formalism based on Correlated Basis Functions (CBF) and the cluster-expansion technique has been recently employed to derive an effective interaction from a realistic nuclear Hamiltonian. One of the main objectives of the work described in this Thesis is establishing the accuracy of this novel approach--that allows to combine the flexibility of perturbation theory in the basis of eigenstates of the noninteracting system with a realistic description of short-range correlations in coordinate space--by focusing on the hard-sphere fermion system. As a first application of the formalism, the quasiparticle properties of hard spheres of degeneracy four have been determined from the two-point Green's function. The calculation has been performed carrying out a perturbative expansion of the self-energy, up to the second order in the CBF effective interaction. The main results of this study are the momentum distributions, the quasiparticle spectra and their description in terms of effective mass. The investigation of the hard-sphere fermion fluid has been extended to study the shear viscosity and thermal conductivity coefficients of the system with degeneracy two, that can be regarded as a model of pure neutron matter. The resulting transport coefficients show a strong sensitivity to the quasiparticle effective mass, reflecting the effect of second order contributions to the self-energy which are not taken into account in nuclear matter studies available in the literature. The difference between first and second order results is likely to play an important role in astrophysical applications and needs to be carefully investigated extending our analysis to nuclear matter.
The dynamic density response function, structure factor, and spectral function of a Luttinger liquid with Coulomb interaction are studied with the emphasis on the short-range electron correlations. The Coulomb interaction changes dramatically the density response function as compared to the case of the short-ranged interaction. The coordinate dependence of the density response is smoothing with time, and the oscillatory structure appears. However, the spectral functions remain qualitatively the same. The dynamic structure factor contains the $\ensuremath{\delta}$ peak in the long-wave region, corresponding to one-boson excitations. In addition, the multi-boson-excitations band exists in the wave-number region near to 2kF,{2k}_{F},2kF​, where kF{k}_{F}kF​ is the Fermi wave number. We develop a method to analyze the asymptotic behavior of the spectral functions near to the edges of the multi-boson-excitations band at zero temperature. The dynamic structure factor diverges at the edges of this band, while the dielectric function goes to zero there. Hence, a new collective charge mode appears in the one-dimensional electron liquid as a consequence of short-range dynamic correlations. The mode energy goes to zero at 2kF,{2k}_{F},2kF​, which means that the 2kF{2k}_{F}2kF​ mode is soft.