Local-Field Correction Function Of The Fermi System With A Short-Range Interaction

A model of fermion liquid with a short-range interaction potential which was simulated by the Yukawa repulsion potential was studied by the reference system approach. The local-eld correction function of such model and its dependences on potential parameters have been studied. The diierence between the local-eld correction function in the case of the electron liquid and the model under consideration has been shown. The energy of the ground state and the stability region of such model have been calculated.


Introduction
The local-eld conception holds a prior place in the modern electron liquid theory and the local-eld correction function is believed to be the fundamental characteristic of any many-particle system with a local interaction potential.The local-eld correction function was studied in detail for the electron liquid case 1,2].Up to the present, the investigation of this function for the case of a Coulomb repulsion potential cannot be considered to be copmleted.We do not know any investigation of the local-eld correction function of many-particle system with a non-Coulomb interaction potential.
Being a functional over the interaction potential, the local-eld correction function must possess the features, which are inherent to a particular physical system.But the local-eld correction function of a degenerate Fermi system also must possess common features caused by Fermi surface.In this paper we want to investigate in uence of the interaction potential on the local-eld correction function and other model characteristics.We shall use the reference system approach for investigation a model with a non-Coulomb interaction potential between particles.
We consider the degenerate Fermi system model of N fermions with charge Ae, which are located in a volume V with compensated background charge ?N V Ae.We describe the interaction between particles and between particles and background charge by the Yukawa repulsion potential V (r ij ) = A 2 e 2 (r ij ) ?1 exp ?r ij =a 0 ]: Here e is the charge of the electron, a 0 is the Bohr radius.Let us consider the thermodynamical limit N; V !1; N=V = const.The suggested potential has two parameters, which control the coupling strength and the interaction range.A 2 is the interaction constant and de nes the range of the potential.Such choice of the potential was caused, rstly, by our wish to have a look on the features of the nature of the electron liquid model with a general position, and secondly by possible use for the description of nuclear matter.The third parameter of the model is the particle mass m.A dimensionless coupling parameter of this model, de ned as the ratio of the average potential energy of the particle to its kinetic energy, has the form f( ) = ( ) ?2 f1 ? 1 + ] exp (? )g: Here " F = ~2k 2 F 2m is the energy of a free particle on the Fermi surface, m = m=m 0 (m 0 is the mass of the electron), g id 2 (r) is the pair distribution function of the ideal fermion system (without interaction) 3], R F = k ?1 F is the correlation length of the degenerate ideal fermion system, r s = (3V=4 N) 1=3 a ?1 0 is the Wigner parameter, = r s = , where = (9 =4) 1=3 .Taking into account the asymptotic behavior of f( ), one can see that the coupling parameter changes from 10A 2 r s ?4 (at !0) to 10A 2 r s ?4 ( ) ?2 (at 1).The electron liquid model is a limiting case of the model under consideration in the limit A ! 1; !0. Its coupling parameter r s has statistical meaning and is to a certain extent formal.Varying the parameters A; ; m , we can change the coupling parameter in a wide region.
For example, the weakly nonideal limit can be obtained in the di erent pathways 1) r s !0; A; ; m = const; 2) A ! 0; r s ; ; m = const; (1.3) 2) !0; A; r s ; m = const: The principal aim of our paper is to investigate the in uence of the interaction potential on the local-eld correction function and other model characteristics.
The local-eld approximation in the electron liquid theory generalizes expressions, which occur in the random phase approximation, to systems with arbitrary value of the coupling parameter r s .In general, the dynamic local-eld correction function G(x) can be de ned by the generalization of two-particle correlation function RPA 2 (x; ?x), which de nes the pair distribution function g 2 (r) = 1 + N(N ?1)] ?1 X X q6 =0 2 (x; ?x) exp (iqr); (1.4) where is the reciprocal temperature, x = (q; ); q is the wave vector, is the Bose-Matsubara frequency 4], r is the distance between two particles, and N is the numberol particles.
2.  The function G 1 (x) depends only on the parameter in the limit A ! 0, at r s ; = const.Let us use the representation (2.7) from 8] for ~ 0 4;1 (x; ?x; x 1 ; ?x 1 ) to investigate the asymptotic behavior of G 1 (x).We can obtain the following asymptotic for G id (x) in the short-wavelength region: G id (q; =q) = 1  2 q 2 + ( )] q ?2 (2.4) Local-eld correction function of the fermi system : : : 131 and G id (q; 0) = C 0 ( ) + C 2 ( ) q 2 + ; (2.6) G id (q; =q) = B 0 ( ) + B 2 ( ) q 2 + ; q ; C 0 ( ) = 1 4 ( ) As one can see from the asymptotes (2.3) -(2.7) the behavior of the localeld correction function is de ned by the e ective range of the interaction between particles R 0 = ?1 a 0 in the long-wavelength region.The local-eld correction function for any short-range potential has a nite value at q = 0, which rises, when the parameter rises.The asymptote ( )q 2 + occurs only in the Coulomb potential limit(R 0 = 1; C 0 ( ) = 0).In this way, the local-eld correction function of the system with the short-range interaction potential is more important than the one of the Coulomb system.Approximations G id (x) and G EL id (x) correspond to the result of 9], in which, for the rst time, the dynamic local-eld correction function of the electron-liquid model G EL id (q; !) in terms of (q; !), where ! is the Heisenberg frequency, has been investigated.As a function of ! the expression G EL id (q; !) has strong singularities.In the terms of (q; ) the function G id (x) has no singularities.This is convenient for its further use.
The results of the local-eld correction function calculation in the approximation (2.2), (2.8), (2.9) are depicted in gures 3, 4. When the parameter A changes, G(x) changes weakly in the region 0 6 jqj 6 2k F , but a strong dependence is noticed in the region jqj > 2k F .The increase of the localeld correction function with increasing A is similar to the dependence of the electron liquid local-eld correction function on the parameter r s 7].The dependence of G(x) on the parameter at A; r s ; = const is presented in gure 3.As one can see G(x) weakly depends on the parameter in the short-wavelength region.The range R 0 has a strong in uence on the localeld correction function in the region of small and medium wave vectors.
The dependence of G(x) on the parameter r s at = const is depicted in gure 4. The behavior of G(x) is very similar to the electron liquid local eld correction function in the region q > 2k F .Comparing G EL (x) with G(x) of our model, we notice that in the region q 6 2k F , G EL (x) and G(x) are very di erent.G(x) has the following asymptotic behavior: G(x) ! ( ; r s ; ) q 2 + ( ) 2 ] + for q 1; G 1 (A; r s ; ) + for q 1: (2.10) 3. 4.

Ground state energy of the model
As is known, the local-eld correction function de nes the integral and local characteristics of the system with a local two-particle interaction.In accordance with (1.5), the ground state energy expression has the form V q V ~ 0 2 (x; ?x) 1 ?G (x)] ) ?1 ;

Conclusion
The local-eld approximation is one of the most signi cant achievements in the electron liquid theory during the last decades and the local-eld correction function is one of many universal characteristics of the model.The generalized Fermi-system model with the interaction described by the Yukawa repulsion potential has been proposed.On the basis of this model the dependence of the local-eld correction function on the interaction range of the potential R 0 has been investigated.As one can see from g. 2 -4, the local eld correction function in such a model has a behavior di erent from the one for the electron liquid model.Especially the region 0 < q < 2k F is important, where the Fourier-transform of the potential V q is large.On the basis of the computation we may assert that the local eld correction function in the model with a short-range interaction potential is always larger than the model with the Coulomb potential in the region of small and medium values of the wave vector q.This fact shows the relative importance of the short-range part of the interaction potential for the systems with a nite value of the interaction range.As is known, the electron liquid model is stable, has negative total energy in the region r s > 2:0, and has the equilibrium density at r 0 s = 4:1825 : : : which corresponds to the minimum of the total energy (-0,15533 Ry per electron).As one can see from g. 5 the total energy of the generalized model in the ground state is smaller than that for the electron-liquid model for any value of the parameter at the same value of the parameter r s .Every value of r s in the generalized model has been proved to correspond to a critical range of interaction R c (r s ) = a 0 ?1 c (r s ).If the potential range is less than R c , the correlation e ects which have the range R F = k ?1 F cannot provide the stability of the model and a negative value of the total energy.Qualitative conclusions concerning the in uence of the range of the interaction potential between particles on the character of the local-eld correction function and the stability are valid not only for a model with the Yukawa potential but for any Fermi system with an arbitrary short-range potential.
The consideration of the represented model, rstly, has heuristic importance, and secondly, the model with a short-range interaction potential can be used as a reference system for the description of neutral fermion systems at 6 = 0.In this case the term V (q = 0) must be added to the expression of the total energy (3.2), and its coupling parameter will have the form 0 = 3 5 " F ?1 N V Z dr 0 V (r) g id 2 (r 0 ) = 10A 2 4 r s m f 0 ( ); f 0 ( ) = f2( ) ?2 ?(1 + ) exp (? )g:

Figure 2 .
Static local-eld correction function G id (q; 0) (2.2) in for di erent values of the parameter .

Figure 6 .
Figure 6.Stability and existence region of the model under consideration as a function of the parameters r s and at A = 1.Curve 1 is the solution of (3.6),curve 2 is the solution of (3.7).