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Progress of Theoretical Physics

Long-Distance Behavior of the Pair Distribution Fnction of Hard-Sphere Fermi Gases

Hiroaki Hara, Tohru Morita

Published May 1, 1971• 0 citations
Publisher

Abstract

The long-distance behavior of the pair distribution function of a hard-sphere Fermi gas, ρ(2)(r) is investigated. It is shown that its distance dependence at zero temperature is the same as that of the ideal Fermi gas, though the amplitudes are different; namely ρ(2)(r)∼k0-2r-4[C1+C2 cos2k0r]. where k0 is the Fermi wave number and Ci's are constants. Corrections arise from the temperature dependence of the chemical potential and from the deformation of the Fermi distribution function fp. The temperature dependence of the terms due to these corrections is estimated to be of the order O(T2), while the distance behavior is found to be O(1/r3) and O(1/r2), respectively. Comments are given on higher order contributions. It is argued that the exchange and the direct contribution to the interaction parts of ρ(2)(r) have the same long-distance behavior for an electron gas as for a hard-sphere Fermi gas.

DOI
10.1143/ptp.45.1437

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Physical Review•May 17, 1965•6 citations
The long-distance behavior of the pair distribution function of a hard-sphere Fermi gas, ρ(2)(r) is investigated. It is shown that its distance dependence at zero temperature is the same as that of the ideal Fermi gas, though the amplitudes are different; namely ρ(2)(r)∼k0-2r-4[C1+C2 cos2k0r]. where k0 is the Fermi wave number and Ci's are constants. Corrections arise from the temperature dependence of the chemical potential and from the deformation of the Fermi distribution function fp. The temperature dependence of the terms due to these corrections is estimated to be of the order O(T2), while the distance behavior is found to be O(1/r3) and O(1/r2), respectively. Comments are given on higher order contributions. It is argued that the exchange and the direct contribution to the interaction parts of ρ(2)(r) have the same long-distance behavior for an electron gas as for a hard-sphere Fermi gas.
The behavior of the one-particle reduced density matrix ρ(1)(r) and the two-particle distribution function ρ(2)(r) of the ideal Fermi gas are investigated for various limiting cases. These cases are classified by the relative magnitude of the following parameters; the distance r, the thermal wave length λT and an “effective length” R0 = (6π2ρ/2S+1)-1/3, where ρ is the density and S is spin. At T = 0, R0-1 is reduced to the Fermi wave number k0 and the obtained long distance behaviors are well known; for instance, R0-2r-4[1+cos (2k0r)] for ρ(2)(r), whereas for T ≠0, this well-known behavior is shown to be valid only when r is smaller than λT and the asymptotic behavior is found to be λT-4r-2[1+cos (2k0r)]exp (-arλT), where a is a constant. This is a sum of terms which shows exponential decay and oscillatory exponential decay, respectively. An explicit form of the three-particle distribution function ρ(3)(r12, r23, r31) of the ideal Fermi gas is also given and its behaviors are discussed.
The singlet and pair distribution functions of a Fermi gas are evaluated for low temperatures. It is assumed that the particles interact with a hard-sphere potential and have an arbitrary spin, and the evaluation is made to first order in the hard-sphere diameter. The binary-kernel method developed by Siegert, Lee, and Yang is used and the cluster-expansion theory given recently by Isihara and Yee for the distribution functions is applied. The pair distribution function is determined by a dimensionless variable $\frac{r}{\ensuremath{\lambda}}$, with $\ensuremath{\lambda}$ being the thermal de Broglie wavelength. Thus, changing the distance corresponds to varying the temperature. Moreover, the pair distribution function depends on the density. Therefore, its asymptotic expression for $T\ensuremath{\rightarrow}0$ depends on the ways to approach the limit. If we keep the density finite and reduce the temperature, the gas will be degenerate when $\ensuremath{\lambda}$ is of order of ${n}^{\ensuremath{-}\frac{1}{3}}$, where nnn is the number density. The pair distribution function for this case is an oscillating function of the Fermi momentum. On the other hand, if we reduce the density first and then decrease the temperature, the pair distribution function decays as ${r}^{\ensuremath{-}4}$, and the free-particle diagrams give a contribution of the same decay.
The pair distribution function of a hard-sphere Bose gas is evaluated using a modified cluster expansion formula developed in the preceding papers of this series [H. Ichimura, Prog. Theor. Phys. 31 (1964), 538; ibid. 37 (1967), 484]. The contributions for the pair distribution function are expressed in terms of diagrams, and a partial summation of the chain-type diagrams is carried out. The temperature dependence of the resultant expression is investigated for low temperatures, at short and long distances. The result is compared with the expression for the pair distribution function which has been obtained for a weakly interacting Bose gas. The two- and three-particle distribution functions for a degenerate ideal Bose system are derived.
The pair distribution function for a quantum Bose gas is expressed as a power series in terms of the fugacity, the coefficients of which are temperature dependent. For the hard sphere case, these coefficients have been evaluated to the first order in a/λ, (a being the scattering length and λ the thermal wavelength) by using torons with two fixed points, or alternatively U cluster functions. The result gives the first order correction to the ideal gas formula of London and Placzek introduced by the interactions between the particles.
Using a method due to the author which correctly reproduces the scattering of two hard spheres for all relative momenta and all partial waves, we study the properties of a low-density hard-sphere boson gas at T=0T=0T=0\ifmmode^\circ\else\textdegree\fi{}K. The system is described by a Hamiltonian H\mathcal{H}H defined for all relative distances of the NNN particles. The terms of H\mathcal{H}H corresponding to the forward motion of a particle through the system lead to the notion of an effective mass m∗{m}^{*}m∗ for each particle in excess of its actual mass mmm, with m∗m\frac{{m}^{*}}{m}mm∗​ given by a monotonically increasing function of $\ensuremath{\rho}{a}^{3}$, where $\ensuremath{\rho}$ and aaa are the number of particles per unit volume and the hard-sphere diameter, respectively. The excitation spectrum is linear at low momenta, displays a roton-like behavior for momenta $p\ensuremath{\gtrsim}\frac{\ensuremath{\hbar}}{a}$, possesses further extrema at higher momenta, and asymptotically approaches p22m∗\frac{{p}^{2}}{2{m}^{*}}2m∗p2​. The pair distribution function vanishes in the low-density limit for particle separations $r\ensuremath{\le}a$.