Longitudinal dielectric permeability into quantum degenerate plasma with frequency of collisions proportional to the module of a wave vector
aa r X i v : . [ phy s i c s . p l a s m - ph ] F e b Longitudinal dielectric permeability into quantum degenerateplasma with frequency of collisions proportional to the module ofa wave vectorA. V. Latyshev and A. A. Yushkanov Faculty of Physics and Mathematics,Moscow State Regional University, 105005,Moscow, Radio str., 10–A
Abstract
Formulas for the longitudinal dielectric permeability in quantum degeneratecollisional plasma with the frequency of collisions proportional to the module ofthe wave vector, in approach Мермина, are received. Equation of Shr¨odinger—Boltzmann with integral of collisions relaxation type in Mermin’s approach is applied.It is spent numerical and graphic comparison of the real and imaginary partsof dielectric function of non-degenerate and maxwellian collisional quantum plasmawith a constant and a variable frequencies of collisions. It is shown, that the longitu-dinal dielectric function weakly depends on a wave vector.
Key words:
Mermin, quantum collisional plasma, conductance, degenerate plas-ma.PACS numbers: 03.65.-w Quantum mechanics, 05.20.Dd Kinetic theory, 52.25.DgPlasma kinetic equations.
1. Introduction
In Klimontovich and Silin’s work [1] expression for longitudinal and trans-verse dielectric permeability of quantum collisionless plasmas has been recei-ved.Then in Lindhard’s work [2] expressions has been received also for thesame characteristics of quantum collisionless plasma.By Kliewer and Fuchs [3] it has been shown, that direct generalisation offormulas of Lindhard on a case of collisionless plasmas, is incorrectly. Thislack for the longitudinal dielectric permeability has been eliminated in work avlatyshev @ mail.ru yushkanov @ inbox.ru of Mermin [4] for collisional plasmas. In this work of Mermin [4] on the basisof the analysis of a nonequilibrium matrix density in τ -approach expressionfor longitudinal dielectric permeability of quantum collisional plasmas in caseof constant frequency of collisions of particles of plasma has been announced.For collisional plasmas correct formulas longitudinal and transverse electricconductivity and dielectric permeability are received accordingly in works[5] and [6]. In these works kinetic Wigner—Vlasov—Boltzmann equation inrelaxation approximation in coordinate space was used.In work [7] the formula for the transverse electric conductivity of quantumcollisional plasmas with use of the kinetic Shr¨odinger—Boltzmann equationin Mermin’s approach (in space of momentum) has been deduced.In work [8] the formula for the longitudinal dielectric permeability ofquantum collisional plasmas with use of the kinetic Shr¨odinger—Boltzmannequation in approach of Mermin (in space of momentum) with any variablefrequency of collisions depending from wave vector has been deduced.In the present work on the basis of results from our previous work [8]formulas for longitudinal dielectric permeability in quantum degenerate colli-sional plasma with frequency of collisions, proportional to the module of awave vector are received. The modelling is thus used Shr¨odinger—Boltzmannequation in relaxation approximation.In our work [9] formulas for longitudinal and transverse electric conductivityin the classical collisional gaseous (maxwellian) plasma with frequency ofcollisions of plasma particles proportional to the module particles velocityhave been deduced.Research of skin-effect in classical collisional gas plasma with frequency ofcollisions proportional to the module particles velocity has been carried outin work [10].Let’s notice, that interest to research of the phenomena in quantum plasmagrows in last years [11] – [24].
1. Longitudinal dielectric function of quantum collisional plasmawith variable collisional frequency
In work [5] longitudinal dielectric function of the quantum collisional plasmas with frequency of collisions, proportional to the module of a wavevector has been received ε l ( q , ω, ν ) = 1 + 4 πe q h B ( q , ω + i ¯ ν )++ ib ¯ ν ( q , ω + i ¯ ν ) b ( q , − b ( q , ω + i ¯ ν ) ωb ( q ,
0) + ib ω, ¯ ν ( q , ω + i ¯ ν ) i . (1 . In the formula (1.1) e is the electron charge, q is the wave vector, ω is thefrequency of oscillations of an electromagnetic field, ν ( k ) is the frequency ofcollisions of particles of plasma, ¯ ν = ¯ ν ( k , q ) = ¯ ν ( k + q , k − q ν (cid:0) k + q (cid:1) + ν (cid:0) k − q (cid:1) , (1 . B ( q , ω + i ¯ ν ) = Z d k π (cid:16) f k + q / − f k − q / (cid:17) Ξ( ω + i ¯ ν ( k + q / , k − q / , (1 . b ( q , ω + i ¯ ν ) = Z d k π (cid:16) f k + q / − f k − q / (cid:17) Ξ( ω + i ¯ ν ( k + q / , k − q / ×× ¯ ν ( k + q / , k − q / ω + i ¯ ν ( k + q / , k − q / , (1 . b ( q ,
0) = Z d k π (cid:16) f k + q / − f k − q / (cid:17) Ξ(0) ¯ ν ( k + q / , k − q / ω + i ¯ ν ( k + q / , k − q / , (1 . b ¯ ν ( q , ω + i ¯ ν ) = Z d k π (cid:16) f k + q / − f k − q / (cid:17) Ξ( ω + i ¯ ν ( k + q / , k − q / ×× ¯ ν ( k + q / , k − q / , (1 . b ω, ¯ ν ( q , ω + i ¯ ν ) = Z d k π (cid:16) f k + q / − f k − q / (cid:17) Ξ( ω + i ¯ ν ( k + q / , k − q / × × ¯ ν ( k + q / , k − q / ω + i ¯ ν ( k + q / , k − q / , (1 . In integrals (1.3) – (1.7) the following designations are accepted Ξ( ω + i ¯ ν ( k + q / , k − q / E k − q / − E k + q / + ~ [ ω + i ¯ ν ( k + q / , k − q / ,f k = 11 + exp (cid:16) E k − µk B T (cid:17) , E k ± q / = ~ m (cid:16) k ± q (cid:17) . Here m is the electron mass, k B is the Boltzmann constant, µ is thechemical potential of molecules of gas, ~ ie the Planck’s constant.Let’s show, that at ν ( k ) = ν = const , i.e. at a constant collisionalfrequency the formula (1.1) passes in the known Mermin’s formula [4] ε Mermin l = 1 + 4 πe q ( ω + iν ) B ( q , ω + iν ) B ( q , ωB ( q ,
0) + iνB ( q , ω + iν ) . (1 . In (1.8) the following designations are used B ( q , ω + iν ) = Z d k π ( f k + q / − f k − q / )Ξ( ω + iν ) , (1 . B ( q ,
0) = Z d k π ( f k + q / − f k − q / )Ξ(0) , Ξ( ω + iν ) = f k + q / − f k − q / E k − q / − E k + q / + ~ ( ω + iν ) . Let’s notice, that at ν ( k ) ≡ ν , ¯ ν ( k , q ) ≡ ν , and we receive followingequalities B ( q , ω + i ¯ ν ) ≡ B ( q , ω + iν ) ,b ( q , ω + i ¯ ν ) = νω + iν B ( q , ω + iν ) ,b ( q ,
0) = νω + iν B ( q , , b ¯ ν ( q , ω + i ¯ ν ) = νB ( q , ω + iν ) ,b ω, ¯ ν ( q , ω + i ¯ ν ) = ν ω + iν B ( q , ω + iν ) . It is as a result received, that ε l ( q , ω, ν ) = 1 + 4 πe q B ( q , ω + iν ) h iν B ( q , − B ( q , ω + iν ) ωB ( q ,
0) + iνB ( q , ω + iν ) i ≡ ε Mermin l ( q , ω, ν ) . Each of integrals (1.3) – (1.7) we will break into a difference of twointegrals. In each of two integrals it is realizable the obvious linear replacementof variables. It is as a result received, that B ( q , ω + i ¯ ν ) = Z d k π f k h Ξ( ω + i ¯ ν ( k , k − q )) − Ξ( ω + i ¯ ν ( k + q , k )) i , (1 . b ( q , ω + i ¯ ν ) = Z d k π f k h ¯ ν ( k , k − q ) ω + i ¯ ν ( k , k − q ) Ξ( ω + i ¯ ν ( k , k − q )) −− ¯ ν ( k + q , k ) ω + i ¯ ν ( k + q , k )) Ξ( ω + i ¯ ν ( k + q , k )) i , (1 . b ( q ,
0) = Z d k π f k h ¯ ν ( k , k − q ) ω + i ¯ ν ( k , k − q )( E k − q − E k ) −− ¯ ν ( k + q , k ) ω + i ¯ ν ( k + q , k ))( E k − E k + q ) i , (1 . b ¯ ν ( q , ω + i ¯ ν ) = Z d k π f k h ¯ ν ( k , k − q )Ξ( ω + i ¯ ν ( k , k − q )) −− ¯ ν ( k + q , k )Ξ( ω + i ¯ ν ( k + q , k )) i , (1 . b ω, ¯ ν ( q , ω + i ¯ ν ) = Z d k π f k h ¯ ν ( k , k − q ) ω + i ¯ ν ( k , k − q ) Ξ( ω + i ¯ ν ( k , k − q )) −− ¯ ν ( k + q , k ) ω + i ¯ ν ( k + q , k ) Ξ( ω + i ¯ ν ( k + q , k )) i . (1 . In integrals (1.10) – (1.14) following designations are accepted ¯ ν ( k , k − q ) = ν ( k ) + ν ( k − q )2 , ¯ ν ( k + q , k ) = ν ( k + q ) + ν ( k )2 , Ξ( ω + i ¯ ν ( k , k − q ) = 1 E k − q − E k + ~ [ ω + i ¯ ν ( k , k − q )] , Ξ( ω + i ¯ ν ( k + q , k ) = 1 E k − E k + q + ~ [ ω + i ¯ ν ( k + q , k )] .
2. Longitudinal dielectric function of the quantum collisionaldegenerate plasmas with frequency of collisions, proportional tothe module of a wave vector
Let’s consider the frequency of collisions proportional to the momentummodule, or, that all the same, to the module of a wave vector ν ( k ) = ν | k | . Then ¯ ν ( k , k ) = ν ( k ) + ν ( k )2 = ν (cid:16) | k | + | k | (cid:17) and ¯ ν ( k , q ) = ¯ ν (cid:16) k + q , k − q (cid:17) = ν (cid:16)(cid:12)(cid:12)(cid:12) k + q (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) k − q (cid:12)(cid:12)(cid:12)(cid:17) . The quantity ν we take in the form ν = νk F , where k F is the Fermi wavenumber, k F = mv F ~ , ~ is the Planck’s constant, v F is the Fermi electronvelocity. Now we have ν ( k ) = νk F | k | . (2 . Let’s notice, that on Fermi’s surface, i.e. at k = k F : ν ( k F ) = ν . So, furtherin formulas (1.1) – (1.7) frequency collisions according to (2.1) it is equal ¯ ν ( k , q ) = ν k F (cid:16)(cid:12)(cid:12)(cid:12) k + q (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) k − q (cid:12)(cid:12)(cid:12)(cid:17) . (2 . Instead of a vector k we will enter the new dimensionless wave vector ofintegration K = k k F , d k = k F d K. Let’s enter also new wave vector Q = q k F . At the specified replacement of variables we have f k = Θ( E F − E k ) = Θ( E F − ~ k m ) = Θ( E F − ~ k F m K ) == Θ( E F − E F K ) = Θ(1 − K ) = f K . Here Θ( x ) is the Heaviside function, Θ( x ) = ( , x > , , x < . According to the specified replacement of variables further it is received ¯ ν ( k , k − q ) = ν (cid:16) | K | + | K − Q | (cid:17) , ¯ ν ( k + q , k ) = ν (cid:16) | K + Q | + | K | (cid:17) , E k − q − E k + ~ [ ω + i ¯ ν ( k , k − q )] == ~ m h ( k − q ) − k i + ~ [ ω + i ¯ ν ( k , k − q )] == − E F Q (cid:16) K x − Q (cid:17) + ~ [ ω + i ¯ ν ( k , k − q )] == − E F Q (cid:16) K x − Q − z − Q (cid:17) . Here Q = Q (1 , , , z − = x + iyρ − , x = ωk F v F , y = νk F v F ,ρ − = 12 (cid:16) | K | + | K − Q | (cid:17) == 12 hq K x + K y + K z + q ( K x − Q ) + K y + K z i . Similarly we receive, that E k − E k + q + ~ [ ω + i ¯ ν ( k , k − q )] == − E F Q (cid:16) K x + Q − z + Q (cid:17) , z + = x + iyρ + , ρ + = 12 (cid:16) | K | + | K + Q | (cid:17) == 12 hq K x + K y + K z + q ( K x + Q ) + K y + K z i . Let’s pass to new variables in integrals (1.10) – (1.14). We receive followingequalities. For integral (1.10) it is had B ( q , ω + i ¯ ν ) = − k F π E F Q B ( Q, z ± ) , where B ( Q, z ± ) = Z f K h K x − Q/ − z − /Q − K x + Q/ − z + /Q i d K. For integral (1.11) it is received b ( q , ω + i ¯ ν ) = − yk F π E F Q b ( Q, z ± ) , where b ( Q, z ± ) = Z f K h ρ − z − ( K x − Q/ − z − /Q ) − ρ + z + ( K x + Q/ − z + /Q ) i d K. For integral (1.12) it is received b ( q ,
0) = − yk F π E F Q b ( Q, ± ) , where b ( Q, ± ) = Z f K h ρ − z − ( K x − Q/ − ρ + z + ( K x + Q/ i d K. For integral (1.13) it is received b ¯ ν ( q , ω + i ¯ ν ) = − yk F v F π E F Q b ¯ ν ( Q, z ± ) , where b ¯ ν ( Q, z ± ) = Z f K h ρ − K x − Q/ − z − /Q − ρ + K x + Q/ − z + /Q i d K. At last, for integral (1.14) it is similarly received b ω, ¯ ν ( q , ω + i ¯ ν ) = − y k F v F π E F Q b ω, ¯ ν ( Q, z ± ) , where b ( Q, z ± ) = Z f K h ρ − z − ( K x − Q/ − z − /Q ) − ρ +2 z + ( K x + Q/ − z + /Q ) i d K. Let’s substitute the received equalities in the formula (1.1). We receivethe expression for longitudinal dielectric function ε l ( Q, x, y ) = 1 − x p πQ h B ( Q, z ± ) + iyb ¯ ν ( Q, z ± ) b ( Q, − b ( Q, z ± ) xb ( Q,
0) + iyb ω, ¯ ν ( Q, z ± ) i . (2 . Here x p is the dimensionless plasma (Langmuir) frequency, x p = ω p k F v F , ω p = 4 π eNm ,ω p is the dimension plasma (Langmuir) frequency.Let’s notice, that in case of constant frequency of electron collisions thequantity ρ ± passes in unit. Then we have B ( Q, z ± ) = QB ( Q, z ) , b ( Q,
0) =
Qz B ( Q, ,b ¯ ν ( Q, z ± ) = QB ( Q, z ) , b ω, ¯ ν ( Q, z ± ) = Qz B ( Q, z ) , where B ( Q, z ) = Z f K d K ( K x − z/Q ) − ( Q/ . Substituting these equalities in (2.3), we receive expression of dielectricfunction for quantum degenerate collisional plasmas with constant frequencyof collisions ε l ( Q, x, y ) = 1 − x p πQ B ( Q, z ) h iy B ( Q, − B ( Q, z ) xB ( Q,
0) + iyB ( Q, z ) i . Let’s result the formula (2.3) in the calculation form. For this purpose inthe plane ( K y , K z ) we will pass to polar coordinates K y + K z = r , dK y dK z = rdrdϕ. Then ε l ( Q, x, y ) = 1 − − x p Q h D ( Q, z ± ) + iyd ¯ ν ( Q, z ± ) d ( Q, − d ( Q, z ± ) xd ( Q,
0) + iyd ω, ¯ ν ( Q, z ± ) i . (2 . Here D ( Q, z ± ) = Z − dK x √ − K x Z (cid:16) K x − Q/ − z − /Q − K x + Q/ − z + /Q (cid:17) rdr,ρ − = 12 (cid:16)p ( K x − Q ) + r + p K x + r (cid:17) ,ρ + = 12 (cid:16)p ( K x + Q ) + r + p K x + r (cid:17) . Besides, d ( Q, z ± ) == Z − dK x √ − K x Z (cid:16) ρ − z − ( K x − Q/ − z − /Q ) − ρ + z + ( K x + Q/ − z + /Q ) (cid:17) rdr,d ( Q,
0) == Z − dK x √ − K x Z h ρ − ( x + iyρ − )( K x − Q/ − ρ + ( x + iyρ + )( K x + Q/ i rdr,d ¯ ν ( Q, z ± ) = Z − dK x √ − K x Z (cid:16) ρ − K x − Q/ − z − /Q − ρ + K x + Q/ − z + /Q (cid:17) rdr, and, at last, d ω, ¯ ν ( Q, z ± ) == Z − dK x √ − K x Z (cid:16) ρ − z − ( K x − Q/ − z − /Q ) − ρ +2 z + ( K x + Q/ − z + /Q ) (cid:17) rdr. On Figs. 1-8 comparison of the real and imaginary parts of dielectricfunction depending on quantity of the dimensionless wave vector Q (Figs. 1-4)and depending on the dimensionless quantities of frequency of an electromag-netic field x (Figs. 5-8) is shown. Thus curves 1 and 2 correspond to valuesof dimensionless frequency collisions y = 0 . and y = 0 . . Everywhere morelow x p = 1 . On Figs. 9 and 10 comparison of relative deviation of real (curves 1) andimaginary parts (curves 2) of dielectric function from the present work (withfrequency of collisions, proportional to the module of a wave vector) withthe corresponding parametres of dielectric Mermin function (with constantfrequency collisions) at the same parametres, and quantity y = νk F v F = 0 . is the same. The last means, that on border Fermi’s surfaces quantity offrequency of collisions in both dielectric functions is the same. Curves 1 onFigs. 9 and 10 are defined by function O r ( Q, x, y ) = Re ε Mermin l ( Q, x, y ) − Re ε l ( Q, x, y )Re ε Mermin l ( Q, x, y ) , and curves 2 defined by function O i ( Q, x, y ) = Im ε Mermin l ( Q, x, y ) − Im ε l ( Q, x, y )Im ε Mermin l ( Q, x, y ) . On Figs. 11-14 comparison of the real and imaginary parts of dielectricfunction according to frequency of collisions proportional to the module ofa wave vector (curves 1) and constant frequency of collisions (curves 2) isshown.
5. Conclusions
In the present work the formula for longitudinal dielectric permeabilityinto quantum collisional degenerate plasma is deduced. Comparison of thereal and imaginary parts of dielectric function at various parametres is shown. R e l Q1 2
Fig. 1. Real part of dielectric function, x = 0 . . Curves , correspond to values of dimensionless collision frequency y = 0 . , . . Q I m l Fig. 2. Imaginary part of dielectric function, x = 0 . . Curves , correspond to values ofdimensionless collision frequency y = 0 . , . .3 R e l Q1 2
Fig. 3. Real part of dielectric function, x = 1 . Curves , correspond to values ofdimensionless collision frequency y = 0 . , . . I m l Q1 2
Fig. 4. Imaginary part of dielectric function, x = 1 . Curves , correspond to values ofdimensionless collision frequency y = 0 . , . .4 R e l x1 2 Fig. 5. Real part of dielectric function, Q = 0 . . Curves , correspond to values ofdimensionless collision frequency y = 0 . , . . I m l x1 2 Fig. 6. Imaginary part of dielectric function, Q = 0 . . Curves , correspond to values ofdimensionless collision frequency y = 0 . , . .5 R e l x 12 Fig. 7. Real part of dielectric function, Q = 1 . Curves , correspond to values ofdimensionless collision frequency y = 0 . , . . I m l x1 2 Fig. 8. Imaginary part of dielectric function, Q = 1 . Curves , correspond to values ofdimensionless collision frequency y = 0 . , . .6 O ( Q ) Q12
Fig. 9. Relative deviation of the real part of dielectric function, x = 1 . Curves , correspond to values of dimensionless collision frequency y = 0 . , . . O ( x ) x 1 2 Fig. 10. Relative deviation of the imaginary part of dielectric function, Q = 1 . Curves , correspond to values of dimensionless collision frequency y = 0 . , . .7 R e l x1 2 Fig. 11. Real part of dielectric function, Q = 1 , y = 0 . . Curves 1 and 2 correspondaccordingly to variable and constant collision frequency. I m l x12 Fig. 12. Imaginary part of dielectric function, Q = 1 , y = 0 . . Curves 1 and 2 correspondaccordingly to variable and constant collision frequency.8 R e l Q1 2
Fig. 13. Real part of dielectric function, x = 1 , y = 0 . . Curves 1 and 2 correspondaccordingly to variable and constant collision frequency. I m l Q12
Fig. 14. Imaginary part of dielectric function, x = 1 , y = 0 . . Curves 1 and 2 correspondaccordingly to variable and constant collision frequency.9 REFERENCES Klimontovich Y. and Silin V. P.
The Spectra of Systems of Interacting Particles//JETF (Journal Experimental Theoreticheskoi Fiziki), , 151 (1952).2. Lindhard J.
On the properties of a gas of charged particles// Kongelige DanskeVidenskabernes Selskab, Matematisk–Fysiske Meddelelser. V. 28, No. 8 (1954), 1–57.3.
Kliewer K. L., Fuchs R.
Lindhard Dielectric Functions with a Finite ElectronLifetime// Phys. Rev. 1969. V. 181. No. 2. P. 552–558.4.
Mermin N. D.
Lindhard Dielectric Functions in the Relaxation–TimeApproximation. Phys. Rev. B. 1970. V. 1, No. 5. P. 2362–2363.5.
Latyshev A.V., Yushkanov A.A.
Longitudinal permettivity of a quantum degeneratecollisional plasma// Teor. and Mathem. Physics, (3): 1739–1749 (2011).6.
Latyshev A.V., Yushkanov A.A.
Transverse Electric Conductivity in CollisionalQuantum Plasma// Plasma Physics Reports, 2012, Vol. 38, No. 11, pp. 899–908.7.
Latyshev A.V., Yushkanov A.A.
Transverse electric conductivity in quantumcollisional plasma in Mermin approach// arXiv:1109.6554v1 [math-ph] 29 Sep 2011.8.
Latyshev A.V., Yushkanov A.A.
Longitudinal electric conductivity and dielectricpermeability in quantum plasma with variable frequency of collisions in Mermin’approach// arXiv:1212.5659v1 [physics.plasma-ph] 17 Jan 2013, 28 p.9.
Latyshev A.V., Yushkanov A.A.
Transverse and Longitudinal Permitivities of aGaseous Plasma with an Electron Collision Frequency Proportional to the ElectronVelocity. – Plasma Physics Report, 2007, Vol. 33, No. 8, pp. 696–702 (Fizika Plasmy,Vol. 33, No. 8, pp. 762–768, russian).10.
Latyshev A.V., Yushkanov A.A.
Skin Effect in a Gaseous Plasma with a CollisionFrequency Proportional to the Electron Velocity. – Plasma Physics Report. 2006. Vol.32. No. 11, pp. 943 – 948 (Fizika Plasmy, Vol. 32, No. 11, pp. 1021–1026, russian).11.
Manfredi G.
How to model quantum plasmas// arXiv: quant - ph/0505004.12.
Anderson D., Hall B., Lisak M., and Marklund M.
Statistical effects in themultistream model for quantum plasmas// Phys. Rev. E (2002), 046417.13. Andr´es P.,de, Monreal R., and Flores F.
Relaxation–time effects in the transversedielectric function and the electromagnetic properties of metallic surfaces and smallparticles// Phys. Rev. B . 1986. Vol. 34, No. 10, 7365–7366.014. Shukla P. K. and Eliasson B.
Nonlinear aspects of quantum plasma physics//Uspekhy Fiz. Nauk, (1) 2010;[V. 180. No. 1, 55-82 (2010) (in Russian)].15. Eliasson B. and Shukla P.K.
Dispersion properties of electrostatic oscillations inquantum plasmas// arXiv:0911.4594v1 [physics.plasm-ph] 24 Nov 2009, 9 pp.16.
Opher M., Morales G. J., Leboeuf J. N.
Krook collisional models of the kineticsusceptibility of plasmas// Phys. Rev. E. V.66, 016407, 2002.17.
Gelder van, A.P.
Quantum Corrections in the Theory of the Anomalous Skin Effect//Phys. Rev. 1969. Vol. 187. No. 3. P. 833–842.18.
Fuchs R., Kliewer K. L.
Surface plasmon in a semi–infinite free–electron gas// Phys.Rev. B. 1971. V. 3. No. 7. P. 2270–2278.19.
Fuchs R., Kliewer K. L.
Optical properties of an electron gas: further studies of anonlocal description// Phys. Rev. 1969. V. 185. No. 3. P. 905–913.20.
Dressel M., Gr¨uner G.
Electrodynamics of Solids. Optical Properties of Electrons inMatter. - Cambridge. Univ. Press. 2003. 487 p.21.
Wierling A.
Interpolation between local field corrections and the Drude model by ageneralized Mermin approach// arXiv:0812.3835v1 [physics.plasm-ph] 19 Dec 2008.22.
Brodin G., Marklund M., Manfredi G.
Quantum Plasma Effects in the ClassicalRegime// Phys. Rev. Letters. , (2008). P. 175001-1–175001-4.23.
Manfredi G. and Haas F.
Self-consistent fluid model for a quantum electron gas//Phys. Rev. B (2001), 075316.24. Reinholz H., R¨opke G.
Dielectric function beyond the random-phase approximation:Kinetic theory versus linear response theory// Phys. Rev.,