Calculation Of The Collective Excitations Spectrum For The Electron Liquid Model In The Local-Field Approximation

Within the reference system approach a method of calculation of plasmons excitations spectrum for the electron liquid model is proposed. It was made by transition from Matsubara's frequency representation to Heisenberg's one and to the mixed representation for the many-particle dynamical correlation functions of the reference system (ideal degenerated electron system) and for the dynamical local eld correction. For the ground state of model the plasmon dispersion curves are investigated in the wide region of the coupling parameter r s .


Introduction
The local-eld correction function for the electron liquid model was investigated for the rst time in the papers 1 -3].It was done in the terms of reference system approach in the variables (q; ), where q is the wavevector and = 2 n ?1 is the Bose-Matsubara frequency 4].The local-eld correction G(q; ) in these variables has no any singularities at all which makes it very convenient for the calculations of di erent thermodynamic and structural characteristics 1, 2].However, the mentioned approach makes also possible the investigation of dynamical characteristics.One of them is the spectrum of collective excitations -plasmons != !(q)which, as usual, is obtained as the root of equation "(q; ~!) = 0; (0.1) where "(q; ~!) is the dielectric function in ordinary Heisenberg representation (see 5, 6]).
To simplify the calculations of G(q; !) it is very useful the frequency over which the summation must be maken to be left the Matsubara's one.

Dispersion coe cient of plasmons
We shall con ne the analytical investigation of the long-wave plasmon spectrum limit to the weakly nonideal systems approximation, using G 1 (q; ~!) and G RPA 1 (q; ~!) as a local eld correction.At the mentioned limit the solution of equation (1.2), as usual, we represent in the following form ! q = !p + 2 !F q 2 ; (3.1)   where !p = 4 Ne 2 =mV ] 1=2 is the so called plasmons frequency, !F = " F =~, and is the plasmons dispersion coe cient.Using formulae (1.5) and (2.1) one may obtain the following relation: Here RPA = 3 5 !F =! p is plasmons dispersion coe cient calculated in RPA.Dependence of the ratio = RPA on coupling parameter r s is the important characteristic being considered by many authors.There are many experimental papers where this characteristic has been studied (see 7] where = RPA for the metals Be; Mg; Ba; Li; Na; K was considered).As fas as (! p =2! F ) 2 = 4r s (3 ) ?1 and = (9 =4) 1=3 in the static approximation of local eld correction we obtain 2 (? ! 2 p 4! 2 F q 2 jr s ) ) 2 (0j0) = 1 4 : (3.3)This formally corresponds to the weakly nonideal system (r s !0) and the relationship RPA = 1 ? 59 r s (3.4) has been obtained in 8] for the rst time.Taking a formal limit 2 (1j0) = 3=20 which corresponds to great frequencies and weakly nonideal system we have a result

Calculation of plasmon spectrum
In contrast to G id (q; ), function G id (q; ~!) is not a smooth function of frequency because it has rst order poles.The numerical calculation of this function by its integral representation or especially as its analytical approximation makes up a pronounced mathematical di culties.
The dielectric function "(q; ~!) calculated in di erent approximations (RPA and G 1 (q; ~!)) as a function of variable !=! p at the given values of wavevector q = 0:4, q = 0:7 for the case r s = 1 is presented in gure 2. As it is visible the equation "(q; ~!) = 0 has two real roots at not great values of q.Curve "(q; ~!) calculated in G id (q; ~!) approximation has a complicated form and the function "(q; ~!) has a pole near to its greatest zero.The high { frequency (plasmon) branch of the spectrum calculated in RPA exists only at nongreat values of q.At the same time in G id (q; ~!) or G RPA id (q; ~!) approximation it exists when q takes great values too.For the rst time this result was established in paper 1], where the exchange correction to "(q; ~!) in variebles (q; ~!) immediately was investigated numerically.
In gure 3 the solution of equation (1.2) at r s = 3, r s = 10 in di erent approximations: RPA (curve 1) and G id (q; ~!) (curve 2), is shown.As one can see, in the long-wave region these di erent approximations give almost the same results, and in the great wave vectors region the solution of (1.2) in the RPA escapes at all.A family of solutions of equation (1.2) in the region 0; 5 6 r s 6 10 in G RPA 1 (q; ~!) approximation is shown in gure 4. It is my pleasant duty to express profound gratitude to Dr. M.V.Vavrukh whose collaboration, scienti c and moral support were of unquestionable signi cance in the process of writing the present paper.

Figure 4 .
Figure 4. which was obtained for the rst time in the paper 9].The relation-