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

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.

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 length 𝜆_𝐵≪1/𝑛^1/3 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 𝑟_𝐷=𝑇/𝑒²𝑛, where 𝑇 is the temperature, satisfying the inequality: 𝑒²4𝜋𝑟_𝐷≫ħΩ. (1)

After substitution the explicit expressions for 𝑟_𝐷 and Ω, we get 𝑒²4𝜋ħ𝑣_𝑒≫1, (2) where 𝑣_𝑒=𝑇/𝑚 is the mean electron velocity in plasma. For 𝑇∼10⁴𝐾⁰, it occurs that 𝑣_𝑒∼10⁷cm/s and 𝑒²/4𝜋ħ𝑣_𝑒∼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 𝑣_𝑒≪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 field 𝜓̌(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: 𝜓̌(𝑥);𝜓⁺(𝑥̌′)_𝑡=𝑡′=𝛿𝐫−𝐫^′,𝑥={𝐫,𝑡}. (3)

The field operator 𝜓̌(𝑥) is described by the Schrodinger equation: 𝜕̌𝑖ħ𝜓(𝑥)=1𝜕𝑡𝑝2𝑚²−𝑒𝑐𝐀̌(𝑥)²𝜓̌(𝑥),𝑝^𝜈=−𝑖ħ∇^𝜈. (4)

The electromagnetic field operator 𝐀̌(𝑥) obeys the Maxwell equation: ∇²𝐴̌^𝜈(𝑥)−1𝑐²𝜕²𝜕𝑡²𝐴̌^𝜈(𝑥)−∇^𝜈∇^𝜈1𝐴̌^𝜈1(𝑥)=−1𝑐𝑗^𝜈̌(𝑥), (5) and the conventional commutation relation: 𝐴̌^𝜈(𝑥);𝜕𝐴̌𝜕𝑡′^𝜈′(𝑥′)_𝑡=𝑡′=𝑖ħ𝑐²𝛿_𝜈𝜈′𝛿𝐫−𝐫′. (6)

We assume Einstein's summation rule over repeating indices 𝜈. The current density 𝑗^𝜈̌(𝑥) is defined by expression: 𝑗^𝜈̌(𝑥)=𝑒𝜓̌2𝑚⁺(𝑥)𝑝^𝜈𝜓̌(𝑥)𝑒+𝐻.𝑐−²𝜓̌2𝑚𝑐⁺(𝑥)𝐴̌^𝜈(𝑥)𝜓̌(𝑥). (7)

If the classical current 𝑗^𝜈_cl(𝑥) exists in the system, it is necessary to add it to 𝑗^𝜈̌(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 Φ̌(𝑥) by definition: 𝐄̌(𝑥)=−1𝑐𝜕̌𝐀(𝑥)Φ̌(𝑥),𝜕𝑡=−∇ (8) or after Fourier transformation: Φ̌(𝐤,𝜔)=−𝜔𝑐𝑘²𝑘^𝜈𝐴̌^𝜈(𝐤,𝜔). (9)

Further, it is convenient to exclude the vector potential 𝐀̌(𝑥) from (4) in order to deal with the linear term Φ̌(𝑥). Such a problem can be solved by using the auxiliary operator: 𝑋̌(𝑥)=−1𝑐𝐴̌^𝜈(𝑥)𝑑𝑙^𝜈. (10)

The integration is performed over any curve connecting points 𝐫 and 𝐫₀, where 𝐫₀ is any arbitrary point. By changing the variable: 𝜓̌(𝑥)=𝑒^{̌𝑖(𝑒/ħ)𝑋(𝑥)}𝜓̌(𝑥), (11) one can write (4) in the form: 𝜕𝑖ħ𝜓̌(𝑥)=𝑝𝜕𝑡²𝜓̌(𝑥)Φ̌(𝑥)𝜓̌(𝑥),2𝑚+𝑒 (12) and write 𝑗^𝜈̌(𝑥)=𝑒𝜓̌2𝑚⁺(𝑥)𝑝^𝜈𝜓̌(𝑥)+𝐻.𝑐. (13) instead of (7).

We omit tildes below. If Φ̌(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<𝑘_𝑐=Ω/𝑣_𝑒. It means in connection with commutations relations (6) that 𝐴̌^𝜈(𝑥)=_𝐤𝜆ħ𝑐²(𝜆)𝑒2𝑉𝜔^𝜈𝐤𝜆𝛼_𝐤𝜆𝑒^{𝑖𝐤𝐫−𝑖𝜔(𝜆)𝑡}+𝛼⁺_𝐤𝜆𝑒^{−𝑖𝐤𝐫+𝑖𝜔(𝜆)𝑡},𝜔(𝜆)=𝜛(𝑘)𝑘if𝜆=1,2,Ω𝜗_𝑐−𝑘if𝜆=3. (14)

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(𝑘_𝑐−𝑘) is the Heaviside step function. The longitudinal part of vector potential satisfies the equation: −𝜕²𝑐²𝜕𝑡²𝐴̌^𝜈(𝑥)−Ω²𝑐²𝐴̌^𝜈(𝑥)=0at𝑘<𝑘_𝑐. (15)

Now, from (9), one gets Φ̌(𝑥)=−_𝐤ħΩ2𝑉𝑘²𝛼_𝐤3𝑒^{𝑖𝐤𝐫−𝑖Ω𝑡}+𝛼⁺_𝐤3𝑒^{−𝑖𝐤𝐫+𝑖Ω𝑡}𝜗𝑘_𝑐=Φ̌−𝑘₍₊₎(𝑥)+Φ̌₍₋₎(𝑥). (16)

The function 𝜛(𝑘) deals with transversal waves fall out of calculations. We shall use below some auxiliary constructions. strait calculation yields Φ̌₍₋₎(𝑥)Φ̌₍₊₎𝑥′=_𝐤ħΩ2𝑘²𝑒^{𝑖𝐤(𝐫−𝐫′)−𝑖Ω(𝑡−𝑡′)}𝜗𝑘_𝑐−𝑘𝑒^ħΩ/𝑇,Φ̌−1₍₊₎(𝑥)Φ̌₍₋₎𝑥′=_𝐤ħΩ2𝑘²𝑒^{𝑖𝐤(𝐫−𝐫′)−𝑖Ω(𝑡−𝑡′)}×11+𝑒^ħΩ/𝑇𝜗𝑘−1^𝑐−𝑘. (17)

The averaging in formulae (17) performs both in quantum and statistical senses over Gibbs distribution, and if ħΩ≪𝑇, then (Rayleigh-Jeans approximation) Φ̌₍₋₎(𝑥)Φ̌₍₊₎(𝑥)=Φ̌₍₊₎(𝑥)Φ̌₍₋₎(𝑥)=𝑇4𝜋²Ω𝑣_𝑒. (18)

3. Green Functions 𝐺_𝑟(𝑥,𝑥^′) and 𝐺_𝑎(𝑥,𝑥^′)

For investigation of the equation system (5)–(16) it is convenient to use the method of Green functions. Let by definition 𝐺_𝑟𝑥,𝑥^′=1𝜓̌(𝑥);𝜓̌𝑖ħ⁺𝑥^′𝜗𝑡−𝑡^′,𝐺_𝑎𝑥,𝑥^′1=−𝜓̌(𝑥);𝜓̌𝑖ħ⁺𝑥^′𝜗𝑡^′−𝑡. (19)

Using (12), we get 𝜕𝑖ħ𝐺𝜕𝑡^𝑟𝑥,𝑥^′+𝑝=𝛿𝑥,𝑥′²𝐺2𝑚^𝑟−𝑥,𝑥′𝑖𝑒ħΦ̌(𝑥)𝜓̌(𝑥);𝜓⁺̌𝑥′𝜗𝑡−𝑡′. (20)

Using (12) again, we have for the right term in (20) 𝜕̌Φ(𝑥)𝜓̌(𝑥);𝜓̌𝜕𝑡⁺𝜗=𝜕Φ̌(𝑥)𝑥′𝑡−𝑡′𝜓̌(𝑥);𝜓̌𝜕𝑡⁺𝜗+1𝑥′𝑡−𝑡′Φ̌(𝑥)𝑝𝑖ħ²𝜓̌(𝑥);𝜓̌2𝑚⁺𝜗+𝑒𝑥′𝑡−𝑡′Φ̌(𝑥)Φ̌(𝑥)𝜓̌(𝑥);𝜓̌𝑖ħ⁺𝑥′𝜗𝑡−𝑡′. (21)

In order to get the closed equation system, we use the following conventional procedure of breaking up the correlators in (21): Φ̌(𝑥)Φ̌(𝑥)𝜓̌(𝑥);𝜓⁺̌𝜗=Φ̌(𝑥)Φ̌(𝑥𝜓̌(𝑥);𝜓̌𝑥′𝑡−𝑡′)⁺.𝑥′×𝜗𝑡−𝑡′ (22)

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 Φ̌𝑝²𝜓̌≈𝑝²Φ̌𝜓̌. Now, (21)-(22) could be easily solved using Fourier transformation: Φ̌₍₊₎𝜓̌;𝜓̌⁺_𝐩𝐸=𝑒Φ̌₍₊₎(𝑥)Φ̌₍₋₎(𝑥)(−𝑖/ħ)𝐸+𝑖Ω+𝑖𝑝²/2𝑚ħ×𝐺_𝑟𝐩ħ,𝐸ħ,Φ̌₍₋₎𝜓̌;𝜓̌⁺_𝐩𝐸=𝑒Φ̌₍₋₎(𝑥)Φ̌₍₊₎(𝑥)(−𝑖/ħ)𝐸−𝑖Ω+𝑖𝑝²/2𝑚ħ×𝐺_𝑟𝐩ħ,𝐸ħ. (23)

Finally, from (20), one gets 𝐺_𝑟𝐩ħ,𝐸ħ=𝑝𝐸+𝑖0−²2𝑚−2𝑒²Φ̌₍₊₎(𝑥)Φ̌₍₋₎(𝑥×𝑝)𝐸−²/2𝑚𝐸−𝑝²/2𝑚+𝑖0²−ħ²Ω²⁻¹. (24)

In accordance with (16), we took into account that 𝜕̌Φ𝜕𝑡⁽⁺⁾(𝑥)=−𝑖ΩΦ̌₍₊₎(𝑥),𝜕̌Φ𝜕𝑡⁽⁻⁾(𝑥)=𝑖ΩΦ̌₍₋₎(𝑥). (25)

The function 𝐺_𝑟(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 𝐺_𝑟𝐩ħ,𝐸ħ=𝐸−𝑝²/2𝑚²−ħ²Ω²𝐸−𝐸⁽⁰⁾(𝐩)𝐸−𝐸⁽¹⁾(𝐩)𝐸−𝐸⁽²⁾(𝐩), (26) where 𝐸⁽⁰⁾(𝐩)=𝑝²𝐸2𝑚⁽¹⁾,𝐸⁽¹⁾(𝐩)=𝑝²√2𝑚+ħΩ𝐸1+𝑍,₍₂₎(𝐩)=𝑝²√2𝑚−ħΩ11+𝑍,𝑍=2𝜋²𝑒^∗2𝑇=1ħΩ(2𝜋)³𝑟_𝐷𝑟_𝐵,𝑒^∗2=𝑒²ħ𝑣_𝑒, (27) and 𝑟_𝐵=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: 𝐺_𝑟𝐩ħ,𝐸ħ=111+𝑍𝐸−𝑝²+𝑍/2𝑚+𝑖02()11+𝑍𝐸−𝑝²/2𝑚−ħΩ√+𝑍1+𝑍+𝑖02()11+𝑍𝐸−𝑝²/2𝑚+ħΩ√.1+𝑍+𝑖0 (28)

The limit 𝑍→0 corresponds to free electrons without field Φ̌. 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 𝐺_𝑟𝐩ħ,𝐸ħ=121𝐸−𝑝²/2𝑚−ħΩ√+1𝑍+𝑖021𝐸−𝑝²/2𝑚+ħΩ√,𝑍+𝑖0 (29) and Plank's constant disappeared since using explicit form (27) for 𝑍, we get ħΩ√𝑍=𝑒²2𝜋²𝑟_𝐷𝑇,𝑍≫1. (30)

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⁺̌(𝑥)𝜓̌(𝑥′)⟩ of electron gas with the help of Green functions 𝐺_𝑟,𝑎 and fluctuation-dissipation theorem [9]: 𝜓⁺̌(𝑥)𝜓̌𝑥′=𝑖𝑉_𝐩1𝑒^{(𝐸−𝜇/𝑇)}×𝐺−1_𝑟𝐩ħ,𝐸ħ−𝐺_𝑎𝐩ħ,𝐸ħ×𝑒^{(𝑖/ħ)𝐩(𝐫−𝐫′)−(𝑖/ħ)𝐸(𝑡−𝑡′)}𝑑𝐸.2𝜋 (31)

For nondegenerate gas, 𝑒^{(𝐸−𝜇)/𝑇}≫1, (32)𝜓⁺̌(𝑥)𝜓̌(𝑥=𝑖)𝑉_𝐩𝑒^{−(𝐸−𝜇)/𝑇}𝐺_𝑟𝐩ħ,𝐸ħ−𝐺_𝑎𝐩ħ,𝐸ħ𝑑𝐸.2𝜋 (33)

One gets from (28) that 𝐺_𝑟𝐩ħ,𝐸ħ−𝐺_𝑎𝐩ħ,𝐸ħ=−2𝜋𝑖𝛿1+𝑍𝐸−𝐸⁽⁰⁾𝑍−𝑖𝜋𝛿1+𝑍𝐸−𝐸⁽¹⁾𝑍−𝑖𝜋𝛿1+𝑍𝐸−𝐸⁽²⁾. (34)

Since 𝑛=𝜓⁺̌(𝑥)𝜓̌(𝑥), (35) one has the equation for chemical potential which is easily solved 𝑒^{−(𝜇−𝜇0)/𝑇}=1+𝑍1+𝑍√1+𝑍𝑐ℎħΩ1+𝑍𝑇, (36) where 𝜇₀1=−𝑇ln𝑛𝑚𝑇2𝜋ħ^23/2 (37) 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 ħΩ√𝑍=𝑇𝑒²/2𝜋²𝑟_𝐷, that is, 𝑒^{−(𝜇−𝜇0)/𝑇}√=𝑐ℎħΩ𝑍𝑇=𝑐ℎ𝑒²2𝜋²𝑟_𝐷𝑇𝑒=1−²4𝜋²𝑟_𝐷𝑇𝑒if²𝑟_𝐷𝑇≪1,𝑍≫1, (38) does not depend on Planck constant. So only for 𝑍∼1 this difference depends on ħ. We remark that in plasma always 𝑒²/𝑟_𝐷𝑇≪1. For typical values of parameters, 𝑛∼10¹⁹1/cm³ and 𝑇∼10⁴𝐾⁰, it turns out that 𝑍∼50, 𝑇/ħΩ∼10, ħΩ√1+𝑍/𝑇∼1. If the concentration 𝑛 diminishes then 𝑍 increases but the ratio ħΩ√1+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: 𝑥1−𝑥²=𝑛𝑚𝑇2𝜋ħ^23/2𝑒^−𝐼/𝑇1+𝑍1+𝑍√1+𝑍𝑐ℎħΩ1+𝑍𝑇, (39) 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 for Longitudinal Waves

Let the classical current 𝑗^𝜈_cl(𝑥)   generating the classical potential 𝜑_cl(𝑥) exists in plasma. Now, (12) takes the view: 𝜕𝑖ħ𝜓̌(𝑥)=𝑝𝜕𝑡²𝜓̌(𝑥)Φ̌(𝑥)2𝑚+𝑒+𝜑_cl(𝑥)𝜓̌(𝑥). (40)

This equation can be rewritten in integral form: 𝜓̌(𝑥)=𝜓⁰̌(𝑥)+𝑒𝜓̌(𝑥);𝜓̌𝑖ħ⁺𝑥^′𝜑_cl𝑥^′𝜓̌𝑥^′𝑑𝑥^′, (41) where 𝜓⁰̌(𝑥) is the solution of (40) at 𝜑_cl(𝑥)=0. We use approximation: 𝑒𝜓̌(𝑥);𝜓̌𝑖ħ⁺𝑥^′𝜗𝑡−𝑡^′≈𝑒𝜓̌(𝑥);𝜓̌𝑖ħ⁺𝑥^′×𝜗𝑡−𝑡^′=𝐺_𝑟𝑥;𝑥^′. (42)

The electron density at first approximation over the potential 𝜑_cl(𝑥) is 𝑛(𝑥)=𝜓⁺̌(𝑥)𝜓̌(𝑥)=𝜓̌⁰⁺(𝑥)𝜓⁰̌(𝑥)+𝛿𝑛(𝑥), (43) where 𝛿𝑛(𝑥)=𝑒𝐺^∗_𝑟𝑥,𝑥₁𝜑_cl𝑥₁𝜓̌⁰⁺𝑥₁𝜓⁰̌(𝑥)𝑑𝑥₁+𝑐.𝑐. (44)

According to (33) and (34), one gets 𝜓̌⁰⁺𝑥₁𝜓⁰̌(𝑥)=1𝑉_𝐩𝑒^{(𝑖/ħ)𝐩(𝐫−𝐫1)−(𝑖/ħ)𝐸(𝑡−𝑡1)}𝑒^{−(𝐸−𝜇)/𝑇}×1𝛿1+𝑍𝐸−𝐸⁽⁰⁾+𝑍2()𝛿1+𝑍𝐸−𝐸⁽¹⁾+𝑍2()𝛿1+𝑍𝐸−𝐸⁽²⁾𝑑𝐸. (45)

Now, the increment of the electron concentration in plasma 𝛿𝑛(𝑥) as a function of 𝑥 due to their interaction with field 𝜑_cl(𝑥) can be written as (𝑥)=𝑒𝛿𝑛𝑉²_{𝐩𝐩1𝐤}𝛿𝐩₁ħ−𝐩ħ+𝐤𝑒^{𝑖𝐤𝐫−𝑖𝜔𝑡+(𝜇/𝑇)}×1𝐺1+𝑍^𝑟𝐩ħ,𝐸⁽⁰⁾𝐩₁+ħ𝜔ħ−𝐸exp⁽⁰⁾𝐩₁𝑇+𝑍2()𝐺1+𝑍^𝑟𝐩ħ,𝐸⁽¹⁾𝐩₁+ħ𝜔ħ−𝐸×exp⁽¹⁾𝐩₁𝑇+𝑍2(1+𝑍)×𝐺_𝑟𝐩ħ,𝐸⁽²⁾𝐩₁+ħ𝜔ħ−𝐸exp⁽²⁾𝐩₁𝑇×𝜑_cl(𝐤,𝜔)𝑑𝜔2𝜋+𝑐.𝑐, (46) where 𝜑_cl(𝐤,𝜔)=𝑒^{𝑖𝐤𝐫−𝑖𝜔𝑡}𝜑_cl(𝐫,𝑡)𝑑𝐫𝑑𝑡. (47)

The plasma dielectric permittivity can be found from (5) and equation of discontinuity: 𝜔𝑒𝛿𝑛(𝐤,𝜔)−𝑘^𝜈𝑗^𝜈_cl(𝐤,𝜔)=0. (48)

On the other hand, from the mean value equality −𝜕𝐀/𝑐𝜕𝑡=−∇𝜑_cl, it follows that 𝜔𝑘^𝜈𝐴^𝜈=−𝑐𝑘²𝜑_cl. After Fourier transformation of (5), multiplying the result by 𝑘^𝜈 and taking into account the presence of classical current and longitudinal waves, one gets 𝜔²𝑐²𝑘^𝜈𝐴^𝜈()𝜔𝐤,𝜔=−𝑐()−1𝑒𝛿𝑛𝐤,𝜔𝑐𝑘^𝜈𝑗^𝜈_cl(𝐤,𝜔). (49)

After comparing this expression with the definition of dielectric permittivity 𝜀^𝑙(𝐤,𝜔): 𝜔²𝑐²𝜀^𝑙(𝐤,𝜔)𝑘^𝜈𝐴^𝜈()1𝐤,𝜔=−𝑐𝑘^𝜈𝑗^𝜈_cl(𝐤,𝜔), (50) one gets 𝜀^𝑙()𝑐𝐤,𝜔=1+𝑒𝜔𝛿𝑛(𝐤,𝜔)𝑘^𝜈𝐴^𝜈().𝐤,𝜔 (51)

Now, 𝜀^𝑙()𝑒𝐤,𝜔=1−²𝑘²𝑉_𝐩1𝐺1+𝑍^𝑟𝐩+ħ𝐤ħ,𝐸⁽⁰⁾+ħ𝜔ħ𝑒^{−𝐸(0)/𝑇}+𝑍2()𝐺1+𝑍^𝑟𝐩+ħ𝐤ħ,𝐸⁽¹⁾+ħ𝜔ħ𝑒^{−𝐸(1)/𝑇}+𝑍2()𝐺1+𝑍^𝑟𝐩+ħ𝐤ħ,𝐸⁽²⁾+ħ𝜔ħ𝑒^{−𝐸(2)/𝑇}𝑒^𝜇/𝑇+H.c. (52)

H.c. means the complex conjugate values and simultaneously changing 𝐤→−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<𝑘_𝑐=Ω/𝑣_𝑒 the inequalities ħ²𝑘²/2𝑚≪ħΩ√1+𝑍 and ħ𝑘≪𝑝 take place: 𝜀^𝑙()𝑒𝐤,𝜔=1−²𝑘²𝑉_𝐩𝑒^{−(𝐸(0)−𝜇)/𝑇}×𝑘²𝑚(𝜔−(𝐩𝐤/𝑚)+𝑖0)²×1(1+𝑍)²+12𝑍1+𝑍²√𝑐ℎħΩ1+𝑍𝑇+2𝑍ħ(1+𝑍)²×Ω√1+𝑍𝑠ℎħΩ√1+𝑍/𝑇+ħ𝑘²/𝑚(𝜔−(𝐩𝐤/𝑚)+𝑖0)²−Ω²()+21+𝑍ħ𝑍1+𝑍²×Ω√1+𝑍𝑠ℎħΩ√1+𝑍/𝑇(𝜔−(𝐩𝐤/𝑚)+𝑖0)²−4Ω²().1+𝑍 (53)

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∼Ω√1+𝑍 and 𝜔∼2Ω√1+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): Im𝜀^𝑙()𝑒𝐤,𝜔=𝜋²𝑚𝑛𝜔𝑘³𝑇√𝑒2𝜋𝑚𝑇^{(𝜇−𝜇0)/𝑇}×1(1+𝑍)²+12𝑍1+𝑍²√𝑐ℎħΩ1+𝑍𝑇×𝑒^{−(𝜔/𝑘)2(𝑚/2𝑇)}+𝑍(1+𝑍)²𝑇ħ𝜔×𝑒^{√−((𝜔−Ω1+𝑍)/𝑘)2(𝑚/2𝑇)}×Ω√1+𝑍𝑠ℎħΩ√1+𝑍/𝑇+ħ𝑘²/𝑚Ω√+𝑍1+𝑍1+𝑍²𝑇𝑒2ħ𝜔^{√−((𝜔−2Ω1+𝑍)/𝑘)2(𝑚/2𝑇)}√×𝑠ℎħΩ1+𝑍𝑇. (54)

From this equality at 𝑍→0, as expected, one finds the Landau attenuation. As the parameter 𝑍 goes up, the plasma starts to absorb at frequencies 𝜔∼Ω√1+𝑍. Then, this effect disappears. And if 𝑍≫1, Im𝜀^𝑙()𝑒𝐤,𝜔=𝜋²𝑚𝑛𝜔𝑘³𝑇√𝑒2𝜋𝑚𝑇^{(𝜇−𝜇0)/𝑇}×12𝑒^{−(𝜔/𝑘)2(𝑚/2𝑇)}√𝑐ℎħΩ𝑍𝑇+𝑇𝑒2ħ𝜔^{√−((𝜔−2Ω𝑍)/𝑘)2(𝑚/2𝑇)}√×𝑠ℎħΩ𝑍𝑇. (55)

In other words, the plasma absorbs at 𝑍≫1 not only at low frequencies, but at high quantum frequencies 𝜔∼2Ω√𝑍=2𝑇𝑒²/2𝜋²𝑟_𝐷/ħ as well. At 𝑍≫1 and low frequencies, Im𝜀^𝑙(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∼2Ω√𝑍∼1/ħ is, respectively, larger than intensity of low frequencies absorption.

Since Im𝜀^𝑙()=1𝐤,𝜔𝜔𝜎^𝑙(𝐤,𝜔), (56) where 𝜎^𝑙(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 at 𝑍=0, (52) shows the conventional results, and in particular the Vlasov dispersion formula [2]: 𝜔²=Ω²1+3𝑘²𝑟²_𝐷,𝑘²𝑟²_𝐷≪1. (57)

We remind that really Z>>1. If 𝑍 grows, but still it is small, the conventional results of Vlasov theory change considerably. Let for the beginning 𝑘²𝑟²_𝐷≪𝑍≪1. The explicit form of equation Re 𝜀^𝐿(𝐤,𝜔)=0 describing oscillation spectrum of the system in linear approximation over 𝑍 and by 𝑘𝑣_𝑒𝜔≪|𝜔²−Ω²(1+𝑍)| has the form: 𝐵1−¹(𝑘,𝜔,𝑍)𝜔²−𝐵₂(𝑘,𝜔,𝑍)𝜔²−Ω²(1+𝑍)=0, (58) where 𝐵₁(𝑘,𝜔,𝑍)=Ω²1+3𝑇𝑘²𝜔²𝑚×(1−𝑍)𝑒^{(𝜇−𝜇0)/𝑇},1≫3𝑇𝑘²𝜔²𝑚,𝐵₂(𝑘,𝜔,𝑍)=Ω²2𝑍𝑘²𝑟²_𝐷1+𝑇𝑘²𝑚(𝜔−Ω)²×𝑇𝑠ℎ(ħΩ/𝑇)ħΩ,1≫𝑇𝑘²𝑚(𝜔−Ω)², (59)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 𝜔²₁=Ω²2𝑍𝑘²𝑟²_𝐷𝑇𝑠ℎ(ħΩ/𝑇)ħΩ,𝑘²𝑟²_𝐷≪𝑍≪1. (60)

So, taking into account inequality 𝑍≠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 now 𝑍≪𝑘²𝑟²_𝐷≪1. The equation describing the oscillation spectrum has the form: 𝐵1−¹(𝑘,𝜔,𝑍)𝜔²−𝑒²𝑘²ħ𝑉_𝐩𝑒^{−(𝐸(0)−𝜇0)/𝑇}2𝑍Ω(𝜔−(𝐩𝐤/𝑚))²−Ω²×𝑠ℎħΩ𝑇=0. (61)

In this equation, the last term can be taken into account using perturbation technique. It yields 𝜔²₂=Ω²1+3𝑘²𝑟²_𝐷(1−2𝑍)𝑒^{(𝜇−𝜇0)/𝑇}×1+2𝑍𝑘²𝑟²_𝐷Ω²√()2𝜋𝑚𝑇𝑇𝑠ℎħΩ/𝑇𝑃ħΩ×^∞_−∞exp−𝑝²_𝑧/2𝑚𝑇𝑑𝑝_𝑧3Ω²𝑘²𝑟²_𝐷−2Ω𝑝_𝑧𝑍𝑘/𝑚−𝑖𝜋𝑘²𝑟²_𝐷√()2𝜋𝑇𝑠ℎħΩ/𝑇,ħΩ𝑍≪𝑘²𝑟²_𝐷≪1. (62)

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 when 𝑍≫1. The equation describing the oscillation spectrum reads 𝐵1−¹(𝑘,𝜔,𝑍)𝜔²−𝐵₃(𝑘,𝜔,𝑍)𝜔²−4Ω²(1+𝑍)=0, (63) where 𝐵₁()=Ω𝑘,𝜔,𝑍²21+3𝑇𝑘²𝑚𝜔²𝑒^{(𝜇−𝜇0)/𝑇},1≫3𝑇𝑘²𝑚𝜔²,𝐵₃(𝑘,𝜔,𝑍)=2Ω²𝑍𝑘²𝑟²_𝐷𝑒^{(𝜇−𝜇0)/𝑇}𝑇𝑡ℎħΩ√𝑍/𝑇ħΩ√𝑍×1+𝑇𝑘²𝑚𝜔−2Ω√𝑍²,1≫𝑇𝑘²𝑚𝜔−2Ω√𝑍². (64)

Also, (63) describes two spectrum branches: 𝜔²_3,4=124Ω²𝑍+𝐵₁+𝐵₃±144Ω²𝑍−𝐵₁+𝐵₃²+𝐵₁𝐵₂, (65) 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: 𝜔²₃=4Ω²𝑍12𝑘²𝑟²_𝐷𝑒^{(𝜇−𝜇0)/𝑇}𝑇𝑡ℎħΩ√𝑍/𝑇ħΩ√𝑍,𝑘+1₂𝑟²_𝐷≪√𝑍. (66)

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: 𝑘²𝜀^𝑙(𝑘,𝜔)𝜑_cl(𝐤,𝜔)=𝜌_cl(𝐤,𝜔), (67) where 𝜌_cl(𝐤,𝜔) is the density of external classical charge present in plasma and 𝜑_cl(𝐫,𝑡)=𝑒^{𝑖𝑘𝑟−𝑖𝜔𝑡}𝜌_cl(𝐤,𝜔)𝑘²𝜀^𝑙(𝐤,𝜔)𝑑𝐤𝑑𝜔(2𝜋)⁴. (68)

If the point charge 𝑄 is placed at point 𝐫=0 and does not depend on time, then 𝜌_cl(𝐤,𝜔)=2𝜋𝛿(𝜔)𝑄. Using (68) at 𝑍=0, one gets the conventional result: 𝜑_cl(𝐫)=𝑄𝑒4𝜋𝑟^{−𝑟/𝑟𝐷}. (69)

According to (53) at arbitrary 𝑍 and 𝑘→0, 𝜀^𝑙()1𝐤,0=1+𝑘²𝑟²_𝐷𝑒^{(𝜇−𝜇0)/𝑇}×1(1+𝑍)²+12𝑍1+𝑍²√𝑐ℎħΩ1+𝑍𝑇√+𝑍2𝑇1+𝑍ħΩ(1+𝑍)³√𝑠ℎħΩ1+𝑍𝑇+𝑍²𝑇√1+𝑍2ħΩ(1+𝑍)³√𝑠ℎħΩ1+𝑍𝑇+2𝑍(1+𝑍)³𝑒^{(𝜇−𝜇0)/𝑇}×𝑇1+ħΩ√𝑍1+𝑍1+√16𝑠ℎħΩ1+𝑍𝑇. (70)

The term multiplied by 1/𝑘²𝑟²_𝐷 describes the Debye radius renormalization. The last term takes part in radius 𝑟_𝐷 and charge 𝑄 renormalization. If 𝑍≫1, then using (36): 𝜀^𝑙()1𝐤,0=1+2𝑘²𝑟²_𝐷𝑇1+ħΩ√𝑍√𝑡ℎħΩ𝑍𝑇, (71) and as a consequence, we return for 𝑟→∞ to the formula (69) in which now instead of 𝑟_𝐷 the value 𝑟^∗_𝐷 is placed where 𝑟^∗_𝐷=2𝑟_𝐷𝑇1+ħΩ√𝑍√𝑡ℎħΩ𝑍𝑇⁻¹≈𝑟_𝐷11−12𝜋²𝑒²𝑟_𝐷𝑇𝑒since²𝑟_𝐷𝑇≪1. (72)

This formula does not depend on ħ. 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 parameter 𝑍 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.