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

$$\frac{\partial {n}_{e\mathrm{,}\text{\hspace{0.17em}}p}}{\partial t}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{\partial}{\partial x}\mathrm{(}{n}_{e\mathrm{,}\text{\hspace{0.17em}}p}{v}_{e\mathrm{,}\text{\hspace{0.17em}}p}\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\mathrm{0,}\text{\hspace{1em}(1)}$$(1)

$$\frac{\partial {v}_{e\mathrm{,}\text{\hspace{0.17em}}p}}{\partial t}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{v}_{e\mathrm{,}\text{\hspace{0.17em}}p}\frac{\partial {v}_{e\mathrm{,}\text{\hspace{0.17em}}p}}{\partial x}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\pm \frac{e}{m}\frac{\partial \varphi}{\partial x}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{\partial {V}_{xc}}{\partial x}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{1}{m{n}_{e\mathrm{,}\text{\hspace{0.17em}}p}}\frac{\partial {p}_{e\mathrm{,}\text{\hspace{0.17em}}p}}{\partial x}\mathrm{,}\text{\hspace{1em}(2)}$$(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],

$${V}_{xc}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}0.985\frac{{e}^{2}}{\u03f5}{n}_{e\mathrm{,}\text{\hspace{0.17em}}p}^{1/3}\left[1+\text{\hspace{0.17em}}\frac{0.034}{{a}_{B}^{\ast}{n}_{e\mathrm{,}\text{\hspace{0.17em}}p}^{1/3}}\text{ln}\mathrm{(}1\text{\hspace{0.17em}}+\text{\hspace{0.17em}}18.37{a}_{B}^{\ast}{n}_{e\mathrm{,}\text{\hspace{0.17em}}p}^{1/3}\mathrm{)}\right]\mathrm{,}\text{\hspace{1em}(3)}$$(3)

where $${a}_{B}^{\ast}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\u03f5{\hslash}^{2}\mathrm{/}m{e}^{2}$$ is the Bohr radius, ε the effective dielectric permeability, and ℏ the Planck constant divided by 2*π*.

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]

$${p}_{e\mathrm{,}p}\mathrm{=}K{n}_{e\mathrm{,}p}^{{\gamma}_{e\mathrm{,}p}}\mathrm{,}\text{\hspace{1em}(4)}$$(4)

where

$${\gamma}_{e\mathrm{,}\text{\hspace{0.17em}}p}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{5}{3}\text{\hspace{0.17em}and\hspace{0.17em}}K\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{5}{3}{\mathrm{(}\frac{\pi}{3}\mathrm{)}}^{1/3}\frac{\pi {\hslash}^{2}}{{m}_{e}}\mathrm{.}\text{\hspace{1em}(5)}$$(5)

Quantum effects are included in the *ℏ*-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,

$${n}_{e\mathrm{,}\text{\hspace{0.17em}}p}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{N}_{e\mathrm{,}\text{\hspace{0.17em}}p}{n}_{o}\mathrm{,}\text{\hspace{0.17em}}\varphi \text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{m}{e}\Phi \mathrm{,}\text{\hspace{0.17em}}{p}_{e\mathrm{,}\text{\hspace{0.17em}}p}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}m{n}_{o}{P}_{e\mathrm{,}\text{\hspace{0.17em}}p}\mathrm{,}\text{\hspace{1em}(6)}$$(6)

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

$$\mathrm{(}{V}_{e\mathrm{,}\text{\hspace{0.17em}}p}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\xi \mathrm{)}\frac{d\text{\hspace{0.05em}}{N}_{e\mathrm{,}\text{\hspace{0.17em}}p}}{d\text{\hspace{0.05em}}\xi}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{N}_{e\mathrm{,}\text{\hspace{0.17em}}p}\frac{d{V}_{e\mathrm{,}\text{\hspace{0.17em}}p}}{d\text{\hspace{0.05em}}\xi}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\mathrm{0,}\text{\hspace{1em}(7)}$$(7)

$$\begin{array}{l}\mathrm{(}{V}_{e\mathrm{,}\text{\hspace{0.17em}}p}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\xi \mathrm{)}\frac{\text{d}{V}_{e\mathrm{,}\text{\hspace{0.17em}}p}}{\text{d}\xi}\text{\hspace{0.17em}}\pm \text{\hspace{0.17em}}\frac{\text{d}\Phi}{\text{d}\xi}\text{\hspace{0.17em}}+\\ \left[{\delta}_{e\mathrm{,}\text{\hspace{0.17em}}p}{N}_{e\mathrm{,}\text{\hspace{0.17em}}p}^{\mathrm{(}{\gamma}_{e\mathrm{,}\text{\hspace{0.17em}}p}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}2\mathrm{)}\mathrm{}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\alpha {N}_{e\mathrm{,}\text{\hspace{0.17em}}p}^{-2/3}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{\beta \nu {N}_{e\mathrm{,}\text{\hspace{0.17em}}p}^{-2/3}}{\mathrm{(}1\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\nu {N}^{1/3}\mathrm{)}}\right]\frac{\partial N}{\partial \xi}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\mathrm{0,}\end{array}\text{\hspace{1em}(8)}$$(8)

where *n*_{o} being the initial plasma density. The constants are *α*=0.985*e*^{2} $${n}_{o}^{1/3}/\mathrm{(}3\u03f5m\mathrm{)},$$ *β*=0.034×18.376*α*, $${\delta}_{e\mathrm{,}\text{\hspace{0.17em}}p}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\gamma}_{e\mathrm{,}\text{\hspace{0.17em}}p}K{n}_{o}^{\mathrm{(}{\gamma}_{e\mathrm{,}\text{\hspace{0.17em}}p}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1\mathrm{)}\mathrm{}}\mathrm{/}m,$$ and $$\nu \text{\hspace{0.17em}}=\text{\hspace{0.17em}}18.37{a}_{B}^{\ast}{n}_{o}^{1/3}.$$ For non-relativistic dense plasma the dielectric permeability corresponds to $$\u03f5\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{\omega}_{L}^{2}\mathrm{/}{\omega}^{2},$$ *ω* is the wave frequency and *ω*_{L} is the plasma frequency [21]. As the time scale associated to electron-positron plasma expansion leads to *ω*>*ω*_{L} 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 *λ*_{F}. For the characteristic spatial scale of expanding plasma much larger than the Thomas-Fermi we have *N*_{e}=*N*_{p}=*N*. To get rid of the electrostatic force in (8), we combine the relevant equations for electrons and positrons but necessarily require that *V*_{e}=*V*_{p}=*V*. It turns out that the pair electron-positron plasma behaves as a single fluid of density *N* and fluid velocity *V*.

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