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# Zeitschrift für Naturforschung A

### A Journal of Physical Sciences

Editor-in-Chief: Holthaus, Martin

Editorial Board: Fetecau, Corina / Kiefer, Claus

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# Dense Electron-Positron Pair Plasma Expansion

• Corresponding author
• Theoretical Physics Laboratory, Faculty of Physics, B.P. 32 Bab-Ezzouar, 16079 Algiers, Algeria
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Published Online: 2015-08-26 | DOI: https://doi.org/10.1515/zna-2015-0244

## Abstract

The expansion of an electron-positron plasma is studied based on quantum hydrodynamical equations for two fluids. The quasi-neutral expansion, depicted through the quantum screening distance, is investigated numerically when the annealing processes is very slow. It was found that the pair plasma behaves as a single fluid with a front expansion velocity that depends on the density and degenerate parameters. Faster expansion results from the existence of exchange-correlation potential, which is enhanced in high-density plasma. The present investigation may be useful in understanding the expansion of a dense plasma produced by the interaction between high-energy laser and solid targets.

Keywords: Dense Matter; Expansion; Quantum Effects; Self-Similar

## 1 Introduction

Plasma expansion has been interpreted, to some extent, in analogy to expansions of hot neutral gases. The latter are driven by thermal pressure deposition of a heating source, such as thermonuclear reactions in astrophysical plasmas or laser pulse in laboratory plasmas. When the charge separation effect at the expanding front becomes more important, then the expansion is mainly driven by the ambipolar electrostatic potential [1]. Plasma expansion is associated with laser ablation of a metallic target, where the plasma plume expands far away from the target to distances larger than the dense gas thickness, depending on the laser intensity and nature of the target materials. The plasma expansion can also be observed in microdischarge [2], in the heliosphere when fast moving dust particles hit a solid target in space [3], and in astrophysics such as in solar wind [4] and comet tails [5].

Pair plasmas are common in high-energy astrophysical environments [6] and in blazar jets [7], and also can be used as a diagnostic tool through detection of radiation from positrons to understand the properties of the medium they propagate through [8]. However, advances in positron trapping techniques have been used to produce experimentally electron-positron plasma by transmitting an electron beam trough a positron plasma [9]. Based on a one-dimensional model, Fillion-Gourdeau et al. proposed a new mechanism for the production of electron-positron pairs from the interaction of a laser field and a fully ionized diatomic molecule. The production is achieved through a resonantly enhanced pair, which is analogous to the resonantly enhanced ionization in molecular physics [10]. Under conditions of thermonuclear burn in inertial confinement fusion, the formation of electron-positron pairs is achieved by a photon-photon pair creation process, where the highest steady-state positron number density predicted is about 1023 cm−3 [11]. Recently, X-ray laser of kiloelectronvolt (keV) photon energy has been used in a prototype atom and showed that sequential single-photon absorption dominates. This photoabsorption mechanism at high X-ray intensity is a promising method to produce plasma with degenerate electrons using multiple photons absorption [12]. Numerical simulations demonstrated the possibility of producing a pure dense electron-positron plasma by laser–solid interactions [13].

Dense plasma is of extremely high density, hence, requiring a quantum mechanical analysis for such a system. The latter rises when thin metal films are irradiated with femtosecond laser pulses [14] and in interactions of ultra-strong attosecond pulses through a laser-plasma at extremely short scales. In such a plasma, quantum mechanical effects are involved through quantum statistical pressure, quantum Bohm force involving electron tunnelling, forces due to electron-exchange, and electron correlations’ effects, as well as quantum screening in electron-positron system [15]. In laboratory experiments, making charge neutral electron-positron pair plasmas is a great challenge that is subject to the following constraints[16]:

• Confinement time must be longer than the plasma timescale of interest. On Earth, pair electron-positron plasmas are produced at low energies (sub-MeV).

• Debye length must be smaller than the plasma size.

In the present work, we investigate the expansion of a pure neutral electron-positron plasma using quantum hydrodynamical equations. We studied the behavior of plasma properties such as density, velocity, and Fermi temperature during the expansion of a dense matter arising from the interaction of high-intensity laser with a solid target.

## 2 Modelling

The expansion of an electron-positron pair plasma is studied using a one-dimensional two fluids model defined by the following equations

$∂ne, p∂t + ∂∂x(ne, pve, p) = 0, (1)$(1)

$∂ve, p∂t + ve, p∂ve, p∂x = ±em∂ϕ∂x − ∂Vxc∂x − 1mne, p∂pe, p∂x, (2)$(2)

where n, v, and ϕ stand for density, velocity, and the electrostatic potential, respectively, while the subscripts e and p stand for electron (e) and positron (p), respectively. Symbols m and e are the electron mass and the elementary charge, respectively. The second term on the right hand side of (2), representing the exchange-correlation potential, arises from the interaction between electron (positron) which is considered as an ideal Thomas–Fermi gas [17],

$Vxc = 0.985e2ϵne, p1/3[1+ 0.034aB∗ne, p1/3ln(1 + 18.37aB∗ne, p1/3)], (3)$(3)

where $aB∗ = ϵℏ2/me2$ is the Bohr radius, ε the effective dielectric permeability, and ℏ the Planck constant divided by 2π.

The pressure gradient term of (2) has the same form as the one corresponding to classical plasma. However, the more electrons are accommodated in a fixed region of space, the more their wave functions would significantly overlap, implying the enhancement of Pauli pressure due to the exclusion principle [18]. Thus, electrons at high densities are degenerate and Fermi–Dirac statistics need to be used to obtain the electron pressure [19]

$pe,p=Kne,pγe,p, (4)$(4)

where

$γe, p = 53 and K = 53(π3)1/3πℏ2me. (5)$(5)

Quantum effects are included in the -dependent coefficient K. The leading terms in (2) are of the same magnitude as the degenerate pressure. Note that Bohm potential is neglected, because it would have a significant contribution in plasma applications with short wavelength, i.e. in semiconductor plasmas [20]. It is worth mentioning that relativistic corrections for mass and velocity are neglected because the electron temperature is <50 eV, such as for burning thermonuclear plasmas in inertial confinement fusion.

The set of (1–5) governing the expansion of dense, pure electron-positron plasma is cast in a tractable form using a self-similar transformation. The time and the space coordinates then are combined into one variable ξ=x/t which has the dimension of velocity. Using the following normalization,

$ne, p = Ne, pno, ϕ = meΦ, pe, p = mnoPe, p, (6)$(6)

leads to a set of differential equations that depend only on one variable ξ,

$(Ve, p − ξ)d Ne, pd ξ + Ne, pdVe, pd ξ = 0, (7)$(7)

$(Ve, p − ξ)dVe, pdξ ± dΦdξ +[δe, pNe, p(γe, p − 2) + αNe, p−2/3 + βνNe, p−2/3(1 + νN1/3)]∂N∂ξ = 0, (8)$(8)

where no being the initial plasma density. The constants are α=0.985e2 $no1/3/(3ϵm),$ β=0.034×18.376α, $δe, p = γe, pKno(γe, p − 1)/m,$ and $ν = 18.37aB∗no1/3.$ For non-relativistic dense plasma the dielectric permeability corresponds to $ϵ = 1 − ωL2/ω2,$ ω is the wave frequency and ωL is the plasma frequency [21]. As the time scale associated to electron-positron plasma expansion leads to ω>ωL we have assumed ϵ ~ 1.

During the expansion, one of the most crucial of all plasma properties is the Debye screening; the charge separation in a plasma can exist only on scales smaller than the Debye length. For a larger scale the plasma is quasi-neutral. However, in quantum plasma the analog of the Debye length is the quantum screening distance, or Thomas-Fermi length λF. For the characteristic spatial scale of expanding plasma much larger than the Thomas-Fermi we have Ne=Np=N. To get rid of the electrostatic force in (8), we combine the relevant equations for electrons and positrons but necessarily require that Ve=Vp=V. It turns out that the pair electron-positron plasma behaves as a single fluid of density N and fluid velocity V.

## 3 Quantum Effects

In terms of N and V and from (7) and (8) the front velocity is

$V = ξ ± C = ξ ± δN(γe − 1) + αN1/3 + βνN1/3(1 + νN1/3) (9)$(9)

where δ=δe=δp. The plus sign is appropriate for an expansion occurring in the direction x>0. The expansion is connected to the propagation of a rarefaction wave which propagates in the opposite direction of the expansion. The expanding front moves with an acoustic speed. In the present plasma mix, and for a single-temperature electron-positron plasma, the electron-acoustic mode cannot exist. So the time scale is associated to the electron plasma mode (Langmuir wave). The front speed can be estimated from $c ~ T/m$ which is in the range of 108 cm/s in the present case. To find the evolution of density, velocity, and the normalized front speed C during the pair-plasma expansion, we solve numerically (7) and (8). The pure electron-positron plasma density is about 1020 cm−3. Such densities can be achieved when a 10 PW laser strikes a solid target [13], it is in the vicinity of a classical plasma. Quantum effects are more significant with higher densities. However, available current generation laser facilities of intensity ~1021 W/cm2 provide electron density about ~1022 cm−3. This density is expected to reach the value ~1024 cm−3 with laser of intensity ~1024 W/cm2 [22].

The expansion is studied at t=t0 which corresponds to the instant after which the laser pulse is switched off (the expansion is adiabatic). The target thickness is about 1 μm, the expanding front is followed at a distance several times the target thickness. In the present work it is assumed that the pair equilibrium (when pair production balances pair annihilation) is achieved. For non-relativistic plasma, we must have θ=kBT/mc2<1 [23]. The annihilation time τan is estimated using: τan=θ/[3(1 + 6θ)2 · 10−14n] [24]. When pair density is ~1020 cm−3, this gives τan ~ 10−8 s. Reducing the density can lead to a lifetime of the plasma, if limited by electron-positron annihilation processes, in the range of 105 s for n ~ 1013 cm−3 [25], but this is not a quantum plasma. Annihilation process, even at the highest initial density, is relatively slow. Kinetic processes in the expanding plasma further reduce the annihilation rate giving rise to streaming electrons and positrons [26].

To understand the mechanism that drives the expansion of a dense, pure electron-positron plasma, in Figure 1a we plotted the density vs. the self-similar variable. We found the same profile as in common plasma expansion into vacuum, that is, a density depletion associated with a rarefaction wave propagating in the opposite direction of the expanding front. The density depletion results from the combination of the degenerate pressure and the exchange-correlation potentials. We note that the self-similar variable limit corresponding to N ~ 0, is about ξL ~ 4×108 cm/s, when the initial density is 1020 cm−3. Increasing the plasma initial density turns out to enhance the self-similar limit. From Figure 1b, we can conclude that the exchange-correlations potential contribution to drive the expansion is of the same magnitude as the thermal pressure term. The higher the initial density is, the more important is the exchange-correlations contribution. For example, with no=1020 cm−3, ξL ~ 3.5×108 cm/s when the expansion is driven only by thermal pressure, while it corresponds to ξL ~ 14×108 cm/s under the combination of the exchange correlation term and the thermal pressure.

Figure 1:

Normalized ion density vs. the self-similar variable ξ with initial density no=1019 (···), 1020 (—), 1021 (– – –) cm−3; (a) expansion driven by thermal pressure and exchange-correlations potential; (b) plasma expansion driven only by thermal pressure.

The time scale of the expansion can be found if one assumes that the plasma can reach a position several times the size of the target interacting with the laser (~μm). The time is ~ps, well below the electron-positron pair plasma lifetime. As the exchange-correlations potential rises due to the electrostatic interaction of the electrons, higher density means stronger interaction to prevent the overlap of electron wave functions. This interaction can be depicted through the enhancement of the self-similar variable limit and the plasma acceleration, as shown in the velocity plots (Fig. 2). The acceleration is typically significant with higher initial density (dashed curves). The velocity sketched in these figures corresponds to the fluid velocity. It increases almost linearly, as it is the case in free expansion into vacuum [27] with an infinite limit. Physically this is not correct because the velocity loses its meaning when N ~ 0. However, the latter criterion did not allow us to determine the limit of the self-similar variable corresponding to the end of the expansion.

Figure 2:

This is the same as Figure 1 but for velocity versus ξ.

The present model is valid only when quantum effects are significant; it is inappropriate to follow the expansion until N ~ 0 after a critical density the model becomes non-valid. In plasma physics and due to particles size, quantum effects are reflected in the individual aspect. However, there is no significant contribution on the collective behavior because of the long range of the electromagnetic forces. There are myriads of parameters, that give us the possibility to state whether quantum effects are relevant for a given plasma [28]. Among them is the mean particle distance d=(3n/4π)1/3 which must be comparable to or smaller than the de Broglie wavelength $λB = h/(2πmkBT).$ Another parameter is the degeneracy parameter $χ∗ = n ⋅ λB3,$ when χ* ≥ 1 Fermi-Dirac statistic is applied instead of Maxwell-Boltzmann statistic. Thus, two parameters play a crucial role in deciding whether quantum effects are important, as well as the density and the temperature. The latter, for quantum plasma, is defined through Fermi energy: EF=2 (3π2n)2/3/2m. The dimensionless parameter:

$χ = TfT = 12(3π2)2/3(nλB3)2/3 (10)$(10)

where Tf is the Fermi temperature (in energy unit) and T, the thermodynamic temperature, is related to the degeneracy parameter. Quantum effects become important when χ ≥ 1 [29]. From the density-temperature phase diagram (Fig. 4.1 of [28]), to have quantum effects with densities (n ≤ 1023 cm−3), the temperature must be low (~eV). This condition is shown in Figure 3 by the horizontal line. Now, we can find the self-similar domain relevant to the expansion of a fully degenerate pair plasma [0, ξl]. Two factors affect ξl, the plasma initial density no and the thermodynamic temperature of the plasma system. The higher the initial density the higher will be the temperature. For low density plasma the temperature must be low as it is the case in semi-conductor plasmas.

Figure 3:

The parameter χ versus the self-similar variable ξ. Horizontal line corresponds to ξ domain of the fully degenerate plasma (χ>1). The parameters labelling the curves are the initial plasma density and the thermodynamic temperature.

## 4 Conclusions

We have investigated the expansion of a fully degenerate plasma composed of pair particles i.e. electron and positron, resulting from laser-irradiated solids. Quantum effects are included through thermal pressure and exchange-correlations potential. Based on the self-similar approach the space coordinates and the time are combined into one variable having the dimension of velocity. We found that the expansion can occur on a time scale smaller than the pair plasma life time. The expansion is driven by the combination of the degenerate thermal pressure and the exchange-correlations potential which are found to be of the same magnitude, particularly close to the plasma source region. The end of the expansion did not correspond to the vanishing of the density, rather it depends on the domain of validity of the model. Quantum effects were found to be relevant in a given plasma for an appropriately chosen combination of initial density and thermodynamic temperature. It is important to note that the quasineutral assumption is valid on length scales corresponding to the Thomas-Fermi length rather than the Debye length. The present investigation is conducted for plasmas produced by a high-intensity laser which provides electron densities in the range from 1020 cm−3 to 1024 cm−3, for which quantum effects are significant.

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Corresponding author: Mourad Djebli, Theoretical Physics Laboratory, Faculty of Physics, B.P. 32 Bab-Ezzouar, 16079 Algiers, Algeria, E-mail: mdjebli@usthb.dz

Accepted: 2015-07-23

Published Online: 2015-08-26

Published in Print: 2015-10-01

Citation Information: Zeitschrift für Naturforschung A, Volume 70, Issue 10, Pages 875–880, ISSN (Online) 1865-7109, ISSN (Print) 0932-0784,

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