Radial Distribution Function for Hard-Sphere Fermions at Zero and Finite Temperatures. J unzo CHIHARA. Japan Atomic Energy Research Institute. Tokai-mura ...
777 Progress of Theoretical Physics, Vol. 58, No. 3, September 1977
Radial Distributio n Function for Hard-Spher e Fermions at Zero and Finite Temperatur es J unzo CHIHARA
Japan Atomic Energy Research Institute Tokai-mura, Ibaraki 319-11
A quanta! version of the Percus-Yevick (PY) equation proposed previously is numerically solved for hard-sphere Fermions, as a model for liquid 'He, for a wide range of temperatures including two limits: zero and infinity. In the first place the density distributions n (rlv) around a fixed atom in the fluid, for which the integral equation is set up, are calculated at various temperatures, and the static density-density response functions XQ are obtained from the Fourier transforms of n(rlv) -no, no being the average density of the system. Next, the structure factors S(Q) are computed by the use of the sum rule, that is, by the integration, over frequency w, of the dynamic structure factors S(Q, w), which are obtained in a generalized Hartree approximation in conjunction with the quanta! direct correlation function determined by the quanta! PY equation. Finally, the radial distribution functions g (r) at corresponding temperatures are obtained by their inverse Fourier transforms. Thus, the quanta! PY equation is shown to give the reasonable results for n(rlv), XQ, g(r) and S(Q) of hard-sphere Fermions for a complete range of temperatures in a unified manner.
§ I.
Introduction
The a1m of the present investigation is to show that a quantal verswn of the Percus-Y evick equation derived previously!) can provide the radial distribution function (RDF) of hard--sphere Fermions for all range of temperatures including absolute zero and infinity (i.e., classical limit). For the ground state of liquid 4 He (Boson) there have been many works 2l based on the method employing a Jastrow-type trial wave function; this procedure has been also applied to liquid 3 He (Fermion) by several investigators,') although not so many as to liquid 4 He, on making a further approximation developed by Wu and Feenberg4l to take account of the antisymmetriza tion effect. However, this method is only effective at zero temperature, and cannot be applied to the liquid at nonzero temperatures. On the other hand, in classical fluids many integral equations for the RDF have been proposed and provided successful results for a wide variety of interatomic potentials. In this approach, however, quantum effects can be introduced only as a correction, described by an expansion in po·wers of It, to the classical result, so that it is difficult to treat large quantum effects ·which may appear at extremely low temperatures or at very high densities.") Since there is, until now, no method to calculate, by taking full account of quantum effects, significant RDF of a fluid at arbitrary temperatures from absolute
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(Received April 4, 1977)
778
J. Chihara
zero to high temperatures in a systematic manner, it is of interest for this purpose to soh·e our quantal PY equation, which reduces exactly to the classical PY equation in the high temperature limit to see the quantum effect on the RDF as a function of temperature and density. Previously, we have soh·ed one of integral equations for the RDF of an electron gas 61 derived by extending Percus' functional expansion method so as to be applicable to quantum system; 11 this method also provides a quanta! PY equation under consideration. Since an electron gas is considered as a typical system interacting via a long-range force and liquid helium as to be one with interacting via a short-range force, the application of such two integral equations to these systems may be taken as to provide a good test of our
§ 2.
Quantal Percus- Yevick equation
At first it should be noted that a quanta! PY equation has been derived not for the RDF g (r), but for the one-particle density distribution n (r [v) in a non-uniform fluid caused by imposition of an external potential U(r) equal to the interatomic potential v (r). This equation is \vritten in terms of the quanta! direct correlation function C(r) as follows :11
n0 B·C(r)=-n(r[v){ n° (r[v) n0
1}.
_
(2 ·1)
In this expressiOn n° (r Iv) represents the density distribution of a noninteracting Fermions with mass m m the presence of the external field v (r), namely, JZ 0
(r[v) = 'L;(exp{/1(-s~c- /I.)} -77] - 1 [1F~o (r) [2 ,
(2· 2)
I
