# Quantum Character of Electromagnetic Langmuir Oscillations in Conventional Electron-Ion Plasma

Authors: Boris Alexandrovich Veklenko
Journal: International Journal of Optics
 (2012)

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## Abstract

It is shown that the low-temperature plasma near-thermodynamic equilibrium cannot be classical because of a quantum nature of the longitudinal electromagnetic field and electron interaction with Rayleigh-Jeans distribution of Langmuir waves. The theory requires introduction of a dimensionless quantum charge whose value is greater than unity leading to a liquid-like behavior of the plasma.


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## Body

## 1. Introduction

Plasma is called classical and the low-temperature one, if electrons and ions in it are governed by equations of classical electrodynamics and obey Maxwell distribution at thermodynamic equilibrium. The latter condition implies that the de-Brogle temperature wave lengthλB≪1/n1/3  is much less than the mean distance between the particles in plasma. Here, n is the concentration of particles. The inequality written above does not mean that the electromagnetic field in plasma could be described also by classical physics methods. The latter is widely used as a traditional approach [1–5]. Describing the electromagnetic field with the help of the strength E(r,t), we suppose that it is in a quantum coherent state [6] and the occupation numbers are much greater than unity. Although the latter is meaningful, the former has no grounds.  Similarly, one could suppose that the field in plasma is in Fock quantum state with the mean value 〈E^(r,t)〉=0. This is more preferable since there are the Fock states, which are generated by a thermal source. In any case, a classical electromagnetic field description of the plasma should be justified. The procedure of field quantization in plasma leads to the appearance of the quantum of Langmuir oscillations energy ħΩ with frequency Ω=e2n/m. We use the rationalized Gauss unit system. Here, e is electron charge, m is its mass, and ħ is Planck constant. The quanta ħΩ do not play any role only if the classical energy of two-particle interaction at the distance of Debye radius rD=T/e2n, where T is the temperature, satisfying the inequality:
(1)e24πrD≫ħΩ.After substitution the explicit expressions forrD and Ω, we get
(2)e24πħve≫1,
where ve=T/m is the mean electron velocity in plasma. For T~104K0, it occurs that ve~107cm/s and e2/4πħve~10.  At first glance, one could think that the inequality (2) is satisfied and the quantization procedure could be omitted. But, surprisingly, a dimensionless charge (2) appears in the theory, with the value greater than one. The analogous dimensionless charge e2/ħc=1/137 is known in quantum electrodynamics which uses transversal electromagnetic waves as fine structure constant or Sommerfeld constant. Its small value permits there to use a perturbation technique. Basically, investigating the plasma, we deal with the same electrodynamics but dealing with the longitudinal waves. In this theory, the parameter denoting the vacuum light velocity c is absent. It is replaced now by ve≪c. That is why the dimensionless charge (2) occurs being great and there appears a question about the applicability of the perturbation technique, which is the base of modern theory of plasma. We state that, in our case, the interaction constant is big and plasma has liquid-like but not a gaze structure. Moreover, the value of dimensionless charge (2) arises at ħ→0 and the limiting transition to the conventional classical theory of plasma is not evident any more.To resolve the problem, we develop a quantum field theory of plasma and investigate the conditions of the applicability of the classical theory.


## 2. Principle Equations

Let us consider the plasma in thermodynamic equilibrium in which the ions are uniformly distributed over the space forming a certain background, which, along with electrons, is neutral. We use the Heisenberg representation to describe the quantum electron fieldψˇ(r,t), where r is the spatial coordinate and t is an arbitrary moment of time. Spin effects are omitted. Since the plasma electrons obey Maxwell distribution by definition, the results do not depend on the kind of statistics. Thus, we can use Bose-Einstein  commutation relations:
(3)[ψˇ(x);ψ+(x′)ˇ]t=t′=δ(r-r′),x={r,t}.The field operatorψˇ(x) is described by the Schrodinger equation:
(4)iħ∂ψ(x)ˇ∂t=12m(p^2-ecAˇ(x))2ψˇ(x),p^ν=-iħ∇ν.The electromagnetic field operatorAˇ(x) obeys the Maxwell equation:
(5)∇2Aˇν(x)-1c2∂2∂t2Aˇν(x)-∇ν∇ν1Aˇν1(x)=-1cjˇν(x),
and the conventional commutation relation:
(6)[Aˇν(x);∂∂t′Aˇν′(x′)]t=t′=iħc2δνν′δ(r-r′).We assume Einstein's summation rule over repeating indicesν. The current density jˇν(x) is defined by expression:
(7)jˇν(x)=e2mψ+ˇ(x)p^νψˇ(x)+H.c-e22mcψ+ˇ(x)Aˇν(x)ψˇ(x).If the classical currentjclν(x) exists in the system, it is necessary to add it to jˇν(x). The system of (4), (5), (7) is written in gauge condition with a zero scalar potential. It permits us to write the quantum Maxwell equations in a simple form (5)-(6). We are interested in the longitudinal electromagnetic field which is the potential field. That is why instead of zero scalar potential we introduce the pseudo-potential Φˇ(x) by definition:
(8)Eˇ(x)=-1c∂∂tAˇ(x)=-∇Φˇ(x),
or after Fourier transformation:
(9)Φˇ(k,ω)=-ωck2kνAˇν(k,ω).Further, it is convenient to exclude the vector potentialAˇ(x) from (4) in order to deal with the linear term Φˇ(x). Such a problem can be solved by using the auxiliary operator:
(10)Xˇ(x)=-1c∫Aˇν(x)dlν.The integration is performed over any curve connecting pointsr and r0, where r0 is any arbitrary point.  By changing the variable:
(11)ψˇ(x)=ei(e/ħ)Xˇ(x)ψ~ˇ(x),
one can  write (4) in the form:
(12)iħ∂∂tψ~ˇ(x)=p^22mψ~ˇ(x)+eΦˇ(x)ψ~ˇ(x),
and write
(13)jˇν(x)=e2mψ~+ˇ(x)p^νψ~ˇ(x)+H.c.
instead of (7).We omit tildes below.  IfΦˇ(x) is a classical function, we come to the conventional description of the classical plasma (see [7]). Below, we investigate how quantum longitudinal plasma waves affect the electrons. In a classical plasma, there are undamped Langmuir oscillations [1, 4] at the wave numbers k<kc=Ω/ve. It means in connection with commutations relations (6) that
(14)Aˇν(x)=∑kλħc22Vω(λ)ekλν(α^kλeikr-iω(λ)t+α^kλ+e-ikr+iω(λ)t),ω(λ)={ϖ(k)ifλ=1,2,Ωϑ(kc-k)ifλ=3.Here,α^kλ and α^kλ+ are the annihilation and the creation Langmuir’s wave operators with vector k and polarization index λ. For the transversal waves λ=1,2, for the longitudinal one, λ=3, ϑ(kc-k) is the Heaviside step function. The longitudinal part of vector potential satisfies the equation:
(15)-∂2c2∂t2Aˇν(x)-Ω2c2Aˇν(x)=0atk<kc.Now, from (9), one gets
(16)Φˇ(x)=-∑kħΩ2Vk2(α^k3eikr-iΩt+α^k3+e-ikr+iΩt)ϑ(kc-k)=Φ(+)ˇ(x)+Φ(-)ˇ(x).The functionϖ(k) deals with transversal waves fall out of calculations. We shall use below some auxiliary constructions. strait calculation yields
(17)〈Φ(-)ˇ(x)Φ(+)ˇ(x′)〉=∑kħΩ2k2eik(r-r′)-iΩ(t-t′)ϑ(kc-k)eħΩ/T-1,〈Φ(+)ˇ(x)Φ(-)ˇ(x′)〉=∑kħΩ2k2eik(r-r′)-iΩ(t-t′)×[1+1eħΩ/T-1]ϑ(kc-k).The averaging in formulae (17) performs both in quantum and statistical senses over Gibbs distribution, and if ħΩ≪T, then (Rayleigh-Jeans approximation)
(18)〈Φ(-)ˇ(x)Φ(+)ˇ(x)〉=〈Φ(+)ˇ(x)Φ(-)ˇ(x)〉=T4π2Ωve.


## 3. Green FunctionsGr(x,x′) and Ga(x,x′)

For investigation of the equation system (5)–(16) it is convenient to use the method of Green functions. Let by definition
(19)Gr(x,x′)=1iħ〈[ψˇ(x);ψ+ˇ(x′)]〉ϑ(t-t′),Ga(x,x′)=-1iħ〈[ψˇ(x);ψ+ˇ(x′)]〉ϑ(t′-t).Using (12), we get
(20)iħ∂∂tGr(x,x′)=δ(x,x′)+p^22mGr(x,x′)-ieħ〈[Φˇ(x)ψˇ(x);ψ+ˇ(x′)]〉ϑ(t-t′).Using (12) again, we have for the right term in (20)
(21)∂∂t〈[Φˇ(x)ψˇ(x);ψ+ˇ(x′)]〉ϑ(t-t′)=〈[∂Φˇ(x)∂tψˇ(x);ψ+ˇ(x′)]〉ϑ(t-t′)+1iħ〈[Φˇ(x)p^22mψˇ(x);ψ+ˇ(x′)]〉ϑ(t-t′)+eiħ〈[Φˇ(x)Φˇ(x)ψˇ(x);ψ+ˇ(x′)]〉ϑ(t-t′).In order to get the closed equation system, we use the following conventional procedure of breaking up the correlators in (21):
(22)〈[Φˇ(x)Φˇ(x)ψˇ(x);ψ+ˇ(x′)]〉ϑ(t-t′)=〈Φˇ(x)Φˇ(x)〉〈[ψˇ(x);ψ+ˇ(x′)]〉×ϑ(t-t′).Such breaking is certainly correct if the fields do not interact or the interaction is sufficiently weak. In practice, the region of its validity is much greater. Its accuracy is greater than the accuracy of a conventional Hartree-Fock approximation when one supposes the correlator braking in the first (20) equation of system. But now, one breaks the correlators in the second equation (21) of the system. In some cases by such a way, one can develop a theory of phase transitions [8]. If ħΩ≪T, one can suppose that Φˇp^2ψˇ≈p^2Φˇψˇ. Now, (21)-(22) could be easily solved using Fourier transformation:
(23)〈[Φ(+)ˇψˇ;ψ+ˇ]〉pE=e〈Φ(+)ˇ(x)Φ(-)ˇ(x)〉(-i/ħ)E+iΩ+(ip2/2mħ)×Gr(pħ,Eħ),〈[Φ(-)ˇψˇ;ψ+ˇ]〉pE=e〈Φ(-)ˇ(x)Φ(+)ˇ(x)〉(-i/ħ)E-iΩ+(ip2/2mħ)×Gr(pħ,Eħ).Finally, from (20), one gets
(24)Gr(pħ,Eħ)=[E+i0-p22m-2e2〈Φ(+)ˇ(x)Φ(-)ˇ(x)〉×E-(p2/2m)(E-(p2/2m)+i0)2-ħ2Ω2]-1.In accordance with (16), we took into account that
(25)∂∂tΦ(+)ˇ(x)=-iΩΦ(+)ˇ(x),∂∂tΦ(-)ˇ(x)=iΩΦ(-)ˇ(x).The functionGr(p/ħ,E/ħ)  is analytical in upper half plane of complex E by definition due to Heaviside step function ϑ(t-t′). That is why the term i0 appears in (24).In accordance with (24), the interaction of plasma electrons with thermal quantum electromagnetic fields is described by a pole leading to qualitative deformation of their properties. Formula (24), in accordance with (18), can be rewritten  as
(26)Gr(pħ,Eħ)=(E-(p2/2m))2-ħ2Ω2(E-E(0)(p))(E-E(1)(p))(E-E(2)(p)),
where
(27)E(0)(p)=p22mE(1),E(1)(p)=p22m+ħΩ1+Z,E(2)(p)=p22m-ħΩ1+Z,Z=12π2(e*)2TħΩ=1(2π)3rDrB,(e*)2=e2ħve,
and rB=4πħ2/e2m is the Bohr radius of Hydrogen atom. So, the procedure of quantization of the longitudinal electromagnetic field leads to the emergence of the dimensionless charge e* introduced in the introduction qualitatively. Formula (26) can be rewritten in a form:
(28)Gr(pħ,Eħ)=11+Z1E-(p2/2m)+i0+Z2(1+Z)1E-(p2/2m)-ħΩ1+Z+i0+Z2(1+Z)1E-(p2/2m)+ħΩ1+Z+i0.The limitZ→0 corresponds to free electrons without field Φˇ. If Z≫1, the free electrons are replaced by two types of quasi-particles described by the two last terms of (28). It is the first result of present paper. Namely, these quasi particles describe electrodynamics, thermodynamic, and especially kinetic properties of plasma at Z≫1. Since such inequality is always satisfied for low-temperature plasma, the quantum parameter Z>1 should be present in all formulae. Nevertheless, one should stress that the used formalism and, in particular, the procedure of correlator breaking does not guarantee the accuracy of quantitative results at Z≫1. For this reason, the extrapolation of the results to the region Z≫1 is expected to lead as in theory of liquids to semiquantitative predictions. But at the same time, this way proves that theories which use limit Z→0 (Vlasov equation) are not correct.It follows from formula (28) that at Z≫1 there exists any “camouflage effect” that sometimes hides the Plank's constant ħ. For instance, in such region
(29)Gr(pħ,Eħ)=121E-(p2/2m)-ħΩZ+i0+121E-(p2/2m)+ħΩZ+i0,
and Plank's constant disappeared since using explicit form (27) for Z, we get
(30)ħΩZ=e22π2rDT,Z≫1.That is why, in this region, the erroneous picture of electron clouds as an ensemble of free electrons sometimes leads to the pseudo reasonable results following, in particular, from Vlasov equation [2].


## 4. Chemical Potential of Electron Gas

On can find the correlation function〈ψ+ˇ(x)ψˇ(x′)〉 of electron gas with the help of Green functions Gr,a and fluctuation-dissipation theorem [9]:
(31)〈ψ+ˇ(x)ψˇ(x′)〉=iV∑p∫1e(E-μ/T)-1iV∑×[Gr(pħ,Eħ)-Ga(pħ,Eħ)]iV∑×e(i/ħ)p(r-r′)-(i/ħ)E(t-t′)dE2π.For nondegenerate gas,(32)e(E-μ)/T≫1,(33)〈ψ+ˇ(x)ψˇ(x)〉=iV∑p∫e-(E-μ)/T[Gr(pħ,Eħ)-Ga(pħ,Eħ)]dE2π.One gets from (28) that
(34)Gr(pħ,Eħ)-Ga(pħ,Eħ)=-2πi1+Zδ(E-E(0))-iπZ1+Zδ(E-E(1))-iπZ1+Zδ(E-E(2)).Since(35)n=〈ψ+ˇ(x)ψˇ(x)〉,
one has the equation for chemical potential which is easily solved
(36)e-(μ-μ0)/T=11+Z+Z1+ZchħΩ1+ZT,
where
(37)μ0=-Tln⁡1n(mT2πħ2)3/2
is the chemical potential of ideal electron gas. Formula (36) possesses a remarkable property. The difference μ-μ0 in both regions as Z≪1 and Z≫1 according to equality ħΩZ=Te2/2π2rD, that is,
(38)e-(μ-μ0)/T=chħΩZT=che22π2rDT=1-e24π2rDTife2rDT≪1,Z≫1,
does not depend on Planck constant. So only for Z~1 this difference depends on ħ. We remark that in plasma always e2/rDT≪1. For typical values of parameters, n~10191/cm3 and T~104K0,  it turns out that Z~50, T/ħΩ~10, ħΩ1+Z/T~1. If the concentration n diminishes then Z increases but the ratio ħΩ1+Z/T diminishes. So at higher concentration n the procedure of electromagnetic field quantization according to (36) leads to the violation of the Sacha formula:
(39)x1-x2=n(mT2πħ2)3/2e-I/T(11+Z+Z1+ZchħΩ1+ZT),
where I is the ionization potential. The corrector term unclosed in brackets depends on ħ only if Z is of the order of unity.


## 5. Plasma Dielectric Permittivity forLongitudinal Waves

Let the classical currentjclν(x)   generating the classical potential φcl(x) exists in plasma. Now, (12) takes the view:
(40)iħ∂∂tψ~ˇ(x)=p^22mψ~ˇ(x)+e(Φˇ(x)+φcl⁡(x))ψ~ˇ(x).This equation can be rewritten in integral form:(41)ψˇ(x)=ψ0ˇ(x)+eiħ∫[ψˇ(x);ψ+ˇ(x′)]φcl⁡(x′)ψˇ(x′)dx′,
where ψ0ˇ(x) is the solution of (40) at φcl(x)=0. We use approximation:
(42)eiħ[ψˇ(x);ψ+ˇ(x′)]ϑ(t-t′)≈eiħ〈[ψˇ(x);ψ+ˇ(x′)]〉×ϑ(t-t′)=Gr(x;x′).The electron density at first approximation over the potentialφcl(x) is
(43)n(x)=〈ψ+ˇ(x)ψˇ(x)〉=〈ψ0+ˇ(x)ψ0ˇ(x)〉+δn(x),
where
(44)δn(x)=e∫Gr*(x,x1)φcl⁡(x1)〈ψ0+ˇ(x1)ψ0ˇ(x)〉dx1+c.c.According to (33) and (34), one gets
(45)〈ψ0+ˇ(x1)ψ0ˇ(x)〉=1V∑p∫e(i/ħ)p(r-r1)-(i/ħ)E(t-t1)e-(E-μ)/T×[11+Zδ(E-E(0))+Z2(1+Z)δ(E-E(1))+Z2(1+Z)δ(E-E(2))]dE.Now, the increment of the electron concentration in plasmaδn(x) as a function of x due to their interaction with field φcl(x) can be written as
(46)δn(x)=eV2∑pp1k∫δ(p1ħ-pħ+k)eikr-iωt+(μ/T)×[11+ZGr(pħ,E(0)(p1)+ħωħ)exp⁡(-E(0)(p1)T)+Z2(1+Z)Gr(pħ,E(1)(p1)+ħωħ)×exp⁡(-E(1)(p1)T)+Z2(1+Z)×Gr(pħ,E(2)(p1)+ħωħ)exp⁡(-E(2)(p1)T)]×φcl⁡(k,ω)dω2π+c.c,
where
(47)φcl⁡(k,ω)=∫eikr-iωtφcl⁡(r,t)drdt.The plasma dielectric permittivity can be found from (5) and equation of discontinuity:
(48)ωeδn(k,ω)-kνjcl⁡ν(k,ω)=0.On the other hand, from the mean value equality-∂A/c∂t=-∇φcl, it follows that ωkνAν=-ck2φcl. After Fourier transformation of (5), multiplying the result by kν and taking into account the presence of classical current and longitudinal waves, one gets
(49)ω2c2kνAν(k,ω)=-ωceδn(k,ω)-1ckνjcl⁡ν(k,ω).After comparing this expression with the definition of dielectric permittivityεl(k,ω):
(50)ω2c2εl(k,ω)kνAν(k,ω)=-1ckνjcl⁡ν(k,ω),
one gets
(51)εl(k,ω)=1+ecωδn(k,ω)kνAν(k,ω).Now,(52)εl(k,ω)=1-e2k2V∑p[11+ZGr(p+ħkħ,E(0)+ħωħ)e-E(0)/T+Z2(1+Z)Gr(p+ħkħ,E(1)+ħωħ)e-E(1)/T+Z2(1+Z)Gr(p+ħkħ,E(2)+ħωħ)e-E(2)/T]eμ/T+H.c.H.c. means the complex conjugate values and simultaneously changingk→-k and ω→-ω.The explicit form of the dielectric permittivity one can obtain from (52) and (28) if simultaneously one takes into account that for not too large k with respect to k<kc=Ω/ve the inequalities ħ2k2/2m≪ħΩ1+Z and ħk≪p take place:
(53)εl(k,ω)=1-e2k2V∑pe-(E(0)-μ)/T×{Ω1+Zsh(ħΩ1+Z/T)(ω-(pk/m)+i0)2-4Ω2(1+Z)k2m(ω-(pk/m)+i0)2×[1(1+Z)2+12(Z1+Z)2chħΩ1+ZT]+2Zħ(1+Z)2×Ω1+Zsh(ħΩ1+Z/T)+(ħk2/m)(ω-(pk/m)+i0)2-Ω2(1+Z)+2ħ(Z1+Z)2×Ω1+Zsh(ħΩ1+Z/T)(ω-(pk/m)+i0)2-4Ω2(1+Z)}.From this formula, it follows that there are three resonance regions in plasma:ω~0—the Vlasov-Landau region at small frequencies and the new resonance regions in vicinities of quantum frequencies ω~Ω1+Z and ω~2Ω1+Z. The existence of these regions manifests themselves in a characteristic way in the imaginary part of the dielectric permittivity. From (53), one gets (ω>0):
(54)Im⁡εl(k,ω)=πe2mnωk3T2πmTe(μ-μ0)/T×{[1(1+Z)2+12(Z1+Z)2chħΩ1+ZT]×e-(ω/k)2(m/2T)+Z(1+Z)2Tħω×e-((ω-Ω1+Z)/k)2(m/2T)×Ω1+Zsh(ħΩ1+Z/T)+(ħk2/m)Ω1+Z+(Z1+Z)2T2ħωe-((ω-2Ω1+Z)/k)2(m/2T)×shħΩ1+ZT}.From this equality atZ→0, as expected, one finds  the Landau attenuation. As the parameter Z goes up, the plasma starts to absorb at frequencies ω~Ω1+Z.  Then, this effect disappears. And if Z≫1,
(55)Im⁡εl(k,ω)=πe2mnωk3T2πmTe(μ-μ0)/T×[12e-(ω/k)2(m/2T)chħΩZT+T2ħωe-((ω-2ΩZ)/k)2(m/2T)×shħΩZT].In other words, the plasma absorbs atZ≫1 not only at low frequencies, but at high quantum frequencies ω~2ΩZ=2Te2/2π2rD/ħ as well. At Z≫1 and low frequencies, Imεl(k,ω) is a half of the imagine part of permittivity predicted by Landau theory [3] at Z=0. The intensity of absorption at the frequencies ω~2ΩZ~1/ħ is, respectively, larger than intensity of low frequencies absorption.Since(56)Im⁡εl(k,ω)=1ωσl(k,ω),
where σl(k,ω) is the conductivity of substance one states that the parameter Z plays an important role in kinetic properties of plasma.As for the real part of dielectric permittivity atZ=0, (52) shows the conventional results, and in particular the Vlasov dispersion formula [2]:
(57)ω2=Ω2(1+3k2rD2),k2rD2≪1.We remind that really Z>>1. IfZ grows, but still it is small, the conventional results of Vlasov theory change considerably. Let for the beginning k2rD2≪Z≪1. The explicit form of equation Re εL(k,ω)=0 describing oscillation spectrum of the system in linear approximation over Z and by kveω≪|ω2-Ω2(1+Z)| has the form:
(58)1-B1(k,ω,Z)ω2-B2(k,ω,Z)ω2-Ω2(1+Z)=0,
where
(59)B1(k,ω,Z)=Ω2(1+3Tk2ω2m)×(1-Z)e(μ-μ0)/T,1≫3Tk2ω2m,B2(k,ω,Z)=Ω22Zk2rD2(1+Tk2m(ω-Ω)2)×Tsh(ħΩ/T)ħΩ,1≫Tk2m(ω-Ω)2,B1(k,ω,Z) and B2(k,ω,Z) slightly depend on the frequency. In the present conditions, it occurs that B2(k,ω,Z)≫B1(k,ω,Z). That is why B1(k,ω,Z) can be taken into account by the perturbation technique. The only one solution of (58) satisfies inequalities (59). As a result, we have
(60)ω12=Ω22Zk2rD2Tsh(ħΩ/T)ħΩ,k2rD2≪Z≪1.So, taking into account inequalityZ≠0 at small k, we state that the Vlasov spectrum (57) changes significantly.The more important fact is the depending of the result through Z on Planck constant ħ. The larger is Z,  the larger is the region of validity of the formula (60).Let nowZ≪k2rD2≪1. The equation describing the oscillation spectrum has the form:
(61)1-B1(k,ω,Z)ω2-e2k2ħV∑pe-(E(0)-μ0)/T2ZΩ(ω-(pk/m))2-Ω2×shħΩT=0.In this equation, the last term can be taken into account using perturbation technique. It yields(62)ω22=Ω2(1+3k2rD2)(1-2Z)e(μ-μ0)/T×(1+2Zk2rD2Ω22πmTTsh(ħΩ/T)ħΩP×∫-∞∞exp⁡(-pz2/2mT)dpz3Ω2k2rD2-2Ω(pzk/m)-iπZk2rD22πTsh(ħΩ/T)ħΩ),Z≪k2rD2≪1.With respect to (60), it is another brunch of spectrum. The symbol P means the principal value of the integral. The expression (62) is written in the form, which allows comparison with Vlasov formula (57). It is evident that in the region under consideration the Vlasov formula is correct but the presence of parameter   Z≠0 means that all the oscillation modes through imagine part Imω22 are damped. Moreover, the dispersion condition (57) can hardly be implemented in practice. In real electron-ion low-temperature plasma condition is always Z≫1.Now, we consider the more important case whenZ≫1. The equation describing the oscillation spectrum reads
(63)1-B1(k,ω,Z)ω2-B3(k,ω,Z)ω2-4Ω2(1+Z)=0,
where
(64)B1(k,ω,Z)=Ω22(1+3Tk2mω2)e(μ-μ0)/T,1≫3Tk2mω2,B3(k,ω,Z)=2Ω2Zk2rD2e(μ-μ0)/TTth(ħΩZ/T)ħΩZ×(1+Tk2m(ω-2ΩZ)2),1≫Tk2m(ω-2ΩZ)2.Also, (63) describes two spectrum branches:
(65)ω3,42=12(4Ω2Z+B1+B3)±14(4Ω2Z-B1+B3)2+B1B2,
as a solution of biquadrate equation. In conditions under consideration, the following inequality is correct B3(k,ω,Z)≫B1(k,ω,Z). The only one result coinciding with inequalities (64) follows:
(66)ω32=4Ω2Z(12k2rD2e(μ-μ0)/TTth(ħΩZ/T)ħΩZ+1),k2rD2≪Z.This formula differs significantly from Vlasov’s formula (57). We state that for the real conditions Z≫1 the presence (63) of the second term dramatically changes the Vlasov result. This branch of spectrum depends principally on ħ since Z~1/ħ2.


## 6. Debye Screen Radius

Equation (50) can be cast in the form:
(67)k2εl(k,ω)φcl⁡(k,ω)=ρcl⁡(k,ω),
where ρcl(k,ω) is the density of external classical charge present in plasma and
(68)φcl⁡(r,t)=∫eikr-iωtρcl⁡(k,ω)k2εl(k,ω)dkdω(2π)4.If the point chargeQ is placed at point r=0 and does not depend on time, then ρcl(k,ω)=2πδ(ω)Q. Using (68) at  Z=0, one gets the conventional result:
(69)φcl⁡(r)=Q4πre-r/rD.According to (53) at arbitrary Z and k→0,
(70)εl(k,0)=1+1k2rD2e(μ-μ0)/T×[1(1+Z)2+12(Z1+Z)2chħΩ1+ZT+Z2T1+ZħΩ(1+Z)3shħΩ1+ZT+Z2T1+Z2ħΩ(1+Z)3shħΩ1+ZT]+2Z(1+Z)3e(μ-μ0)/T×[1+TħΩ1+Z(1+Z16)shħΩ1+ZT].The term multiplied by1/k2rD2 describes the Debye radius renormalization. The last term takes part in radius rD and charge Q renormalization. If  Z≫1, then using (36):
(71)εl(k,0)=1+12k2rD2(1+TħΩZthħΩZT),
and as a consequence, we return for r→∞ to the formula (69) in which now instead of rD the value rD* is placed where
(72)rD*=2rD(1+TħΩZthħΩZT)-1≈rD(1-112π2e2rDT)sincee2rDT≪1.This formula does not depend onħ. Such a fact once again manifests the “camouflage effect” explaining in some cases the pseudoreasonable results following from conventional classical theory of plasma.


## 7. Conclusion

The electromagnetic field is always quantized. The question is whether the quantum properties of the field are significant or not in a particular system.In the present work, we show that as quantization procedure of longitudinal (Langmuir) electromagnetic field and taking into account the electron interaction with Rayleigh-Jeans distribution of Langmuir waves at finite temperature change dramatically the structure of the electron cloud in a low-temperature plasma. The dimensionless quantum parameterZ is proportional to the charge squared whose value is much larger than the one appears in theory. It makes impossible to use a perturbation technique. We tried to construct the closed system of equations, the solution of which using perturbation technique (small Z) returns to the consequences of Vlasov equation.  In real conditions of the low-temperature electron-ion plasma, Z≫1 that is why, in any case, the plasma optical properties could not be investigated using Vlasov’s equation. The analysis of the obtained system of equations at large Z leads to the conclusion that the quantum properties of plasmas manifest themselves dramatically in kinetic and hydrodynamic properties.



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*Source: 648741-2012-07-08.xml*