2. DESCRIPTION OF THE MODEL

The present study of plasma expansion is deduced from high-intensity laser-matter interactions (I > 10¹⁸ W/cm²). The most interesting and efficient one for applications is the electrostatic acceleration at the rear-side of a thin dense target. In this case, electrons heated by the laser at the front surface of the target propagate through the solid material and form a space charge cloud in vacuum at the rear surface. The static electric field induced by this electron cloud is strong enough to fully ionize the material and accelerate ions perpendicularly to the rear surface up to multi-MeV energies. Depending on the intensity temperatures of around 1 MeV are obtained.

At first the stationary situation of a plasma slab of fixed ions and a hot electron population is studied. The metal target is supposed unlimited and is a homogeneously filled negative half space. At time t = 0, the target surface involved in the experiments is located at x = 0, and the target is in x < 0 region. Laser radiation is switched on in the -x direction and the plasma starts expanding. Due to the expansion, the plasma density will decrease and hence a rarefaction wave will propagate in the -x direction (Cheng et al., Reference Cheng, Perrie, Wub, Tao, Edwardson, Dearden and Watkins2009).

Ions are cold and initially at rest with density n _i = n _i0, (n _i0 ≈ 10¹⁹cm ⁻³) for x < 0 and n _i = 0 for x > 0 with a sharp boundary (see Fig. 1). At t > 0, plasma begins to expand into vacuum (Mora, Reference Mora2003).

Fig. 1. (Color online). Initial conditions of densities before plasma expansion, x = 0 is the semi-infinite border of the target.

3. BASIC EQUATIONS

3.1. Electron Function Distribution

The purpose of the present work is to study one-dimensional, non-relativistic, collisionless expansion into a vacuum of semi-infinite plasma in the presence of both trapped and non-thermal electrons and their influence on the ion dynamics. The approximation of a one-dimensional description of the expansion is justified as long as lateral heat conduction can be neglected.

We suppose the existence of a population of high energetic electrons following Cairns DF:

(1)$${\,f_e}\left({{{\rm \nu}_e}} \right)= \displaystyle{{{n_{e0}}} \over {\sqrt {2{\rm \pi} {\rm \nu}_{eth}^2 } }}\displaystyle{{\left({1 + {\rm \alpha} {\rm \nu}_e^4 /{\rm \nu}_{eth}^4 } \right)} \over {\left({3{\rm \alpha} + 1} \right)}}\exp \left({ - {\rm \nu}_e^2 /2{\rm \nu}_{eth}^2 } \right).$$

Where n _e0 is the unperturbed electronic density, ${{\rm \nu}_{eth}} = \sqrt {{T_e}/{m_e}} $ is the average electron thermal velocity, T _e is the electron temperature in the absence of non-thermal effects. α is an arbitrary non-thermal parameter that measures the deviation from the Maxwellian distribution function and defines the shape of the distribution. It indicates the population of the fast electrons. It is clear that Eq. (1) expresses the isothermally distributed electrons when α = 0. The shape of the function distribution is given for different values of α in Figure 2 showing the energetic tails of the distributions relatively to the Maxwell distribution.

Fig. 2. Normalized Cairns electron distribution function as function of the parameter α.

Propagation of high frequency waves in plasma can generate the longitudinal (potential) waves in which some of the plasma particles as electrons can be trapped and interact strongly with the wave during its evolution in the plasma (Hakimi Pajouh & Abbasi, Reference Hakimi Pajouh and Abbasi2002; Shorokhov, Reference Shorokhov2006).

We assume the existence of potential wells in the plasma U(x) = −eφ(x) large enough to trap a certain proportion of electrons following Gurevich distribution. It is in fact the study of the effect of trapping electrons on the plasma expansion, in the case of the presence of energetic electrons. The distribution of electrons in the presence of non-zero potential is expressed mathematically by replacing ν_e² / ν_eth² by ν_e² / ν_eth² − 2φ in the Cairns distribution to calculate the total density of free and trapped electrons.

We write the density of free non-thermal electrons, with a total positive energy ε > 0, and density of trapped electrons, with a total negative energy ε < 0, as follows (Landau & Lifshitz, Reference Landau and Lifshitz1981; Bennaceur-Doumaz et al., Reference Bennaceur-Doumaz, Bara and Djebli2011):

(2)$${n_e} = 2\int_{{{\rm \nu}_1}}^\infty {{\,f_e}\left({{{\rm \nu}_e}} \right)d{{\rm \nu}_e}} + 2\int_0^{{{\rm \nu}_1}} {{\,f_e}\left(0 \right)d{{\rm \nu}_e}}.$$

With ν₁ = (2|U| / m _e)^1/2, |U| = |eφ| and ε = m _eν_e² / 2 + U(x). The factors 2 take account of particles with ν₁ > 0 and ν_e < 0.

Using Cairns distribution (1) with the effects of potential in Eq. (2) and after integration by parts of the resulting equation, we find the normalized density as:

(3)$$\eqalignno{&{\tilde n_e} = {n_e}/{n_{e0}} = \displaystyle{2 \over {3{\rm \alpha} + 1}} \left({2{\rm \alpha} {\Phi ^2} - 2{\rm \alpha} \Phi + \displaystyle{{3{\rm \alpha} + 1} \over 2}} \right)\cr & \quad \left\{{{e^\Phi }\left({1 - erf\left({\sqrt {\left\vert \Phi \right\vert } } \right)} \right)+ 2\displaystyle{{\sqrt {\left\vert \Phi \right\vert } } \over {\sqrt {\rm \pi} }}} \right\}.}$$

Setting $b = 4{\rm \alpha} /\left({3{\rm \alpha} + 1} \right)$, the electron density normalized to its initial value is given as follows:

(4)$${\tilde n_e} = \left({b{\Phi ^2} - b\Phi + 1} \right)\left\{{{e^\Phi }\left({1 - erf\left({\sqrt {\left\vert \Phi \right\vert } } \right)} \right)+ 2\displaystyle{{\sqrt {\left\vert \Phi \right\vert } } \over {\sqrt {\rm \pi} }}} \right\}$$

|Φ| = |eφ| / T _e is the normalized electrostatic potential.

If we put b = 0, that is to say in the absence of non-thermal electrons, we find the case of trapping of Gurevich (Landau & Lifshitz, Reference Landau and Lifshitz1981).

(5)$${\tilde n_e} = {e^\Phi }\left({1 - erf\left({\sqrt {\left\vert \Phi \right\vert } } \right)} \right)+ 2\displaystyle{{\sqrt {\left\vert \Phi \right\vert } } \over {\sqrt {\rm \pi} }}.$$

3.2. Ion Motion Modelling

The plasma is considered as a non-dissipative and non-heat-conductive fluid and the ions with density n _i and velocity ν_i are described by the following Euler continuity and momentum fluid equations, respectively:

(6)$$\displaystyle{{\partial {n_i}} \over {\partial t}} + \displaystyle{{\partial \left({{n_i}{v_i}} \right)} \over {\partial x}} = 0\comma \;$$

(7)$$\displaystyle{{\partial {v_i}} \over {\partial t}} + {v_i}\displaystyle{{\partial {v_i}} \over {\partial x}} + \displaystyle{1 \over {{m_i}{n_i}}}\displaystyle{{\partial {P_i}} \over {\partial x}} + \displaystyle{e \over {{m_i}}}\displaystyle{{\partial {\rm \varphi} } \over {\partial x}} = 0\comma \;$$

m _i is the mass of the plasma particles. Eqs. (6) and (7) are closed by the ideal gas equation of state such that plasma pressure is given by P _i = n _iT _i, where T _i is the ion temperature.

The electron thermal conductivity in the hot expanding plasma is considered sufficiently high so that the plasma tends to be isothermal which means T _e and T _i are assumed to remain constant during the plasma expansion (Pearlman & Morse, Reference Pearlman and Morse1978; Sack & Schamel, Reference Sack and Schamel1987).

4. SELF-SIMILAR THEORY AND NUMERICAL CALCULATION

In the presence of free boundary associated to plasma expansion, it is a hard task to solve hydrodynamic equations describing the expansion numerically. But under certain assumptions, these partial differential equations can be reduced to ordinary differential equations that greatly simplify the problem. This transformation is based on the assumption that we have a self-similar solution, i.e., every physical parameter distribution preserves its shape during expansion and there is no scaling parameter (Sack & Schamel, Reference Sack and Schamel1987; Zel'dovich & Raizer, Reference Zel'dovich and Raizer1966).

Self-similar solutions usually describe the asymptotic behavior of an unbounded problem and the time t and the space coordinate x appear only in the combination of (x/t). It means that the existence of self-similar variables implies the lack of characteristic lengths and times. Indeed, this is justified when we deal with the assumption of charge quasi-neutrality in the plasma. The Debye length then, loses its importance as a characteristic length in the plasma.

The one-dimensional self-similar solution of Eqs. (6) and (7) with quasi-neutrality assumption can be constructed by using the ansatz defined as ${\tilde n_i} = {n_i}/{n_{i0}}$, ${\tilde {\rm \nu}_i} = {{\rm \nu}_i}/{c_s}$, c _s is the ion sound velocity given by c ²_s = T _e/m _i and n _i0 is the initial density of the plasma. By using the dimensionless self-similar variable ξ = x/c _st, we obtain the following set of normalized ordinary equations:

(8)$$\left({{{\tilde {\rm \nu}}_i} - {\rm \xi} } \right)\displaystyle{{d{{\tilde n}_i}} \over {d{\rm \xi} }} + {\tilde n_i}\displaystyle{{d{{\tilde {\rm \nu}}_i}} \over {d{\rm \xi} }} = 0\comma \;$$

(9)$$\left({{{\tilde {\rm \nu}}_i} - {\rm \xi} } \right)\displaystyle{{d{{\tilde {\rm \nu}}_i}} \over {d{\rm \xi} }} + \displaystyle{{\rm \delta} \over {{{\tilde n}_i}}}\displaystyle{{d{{\tilde n}_i}} \over {d{\rm \xi} }} + \displaystyle{{d\Phi } \over {d{\rm \xi} }} = 0.$$

The ion motion is modelled by the cold ion approximation where δ = T _i/T _e≪1.

Differentiating Eq. (4) and using the condition of quasi-neutrality of charge, we obtain

(10)$$\displaystyle{{d{{\tilde n}_i}} \over {d{\rm \xi} }} = H\left(\Phi \right)\displaystyle{{d\Phi } \over {d{\rm \xi} }}.$$

Where $ H = 2\left({2b\Phi - b} \right)\left({\displaystyle{{\sqrt {\left\vert \Phi \right\vert } } \over {\sqrt {\rm \pi} }}} \right)+ \left({b{\Phi ^2} - b\Phi + 1 - b} \right){e^\Phi }$$erfc\left({\sqrt {\left\vert \Phi \right\vert } } \right)$. The equation of ion motion (9) becomes:

(11)$$\left({\displaystyle{{\rm \delta} \over {{{\tilde n}_i}}} + \displaystyle{1 \over H}} \right)\displaystyle{{d{{\tilde n}_i}} \over {d{\rm \xi} }} + \left({{{\tilde {\rm \nu}}_i} - {\rm \xi} } \right)\displaystyle{{d{{\tilde {\rm \nu}}_i}} \over {d{\rm \xi} }} = 0.$$

If we treat the entire derivative terms as independent variables and the resulting set of equations as algebraic ones, then the nontrivial solution to the system of Eqs. (5) and (8) requires that the determinant of their coefficients must vanish (Yu, Reference Yu and Luo1995), i.e., while choosing the positive solution corresponding to an expansion in the +x direction and a velocity increasing with increasing x:

(12)$${\tilde {\rm \nu}_i} = {\rm \xi} + \sqrt {{\rm \delta} + \displaystyle{{{{\tilde n}_i}} \over H}}.$$

Differentiating (12) and using Eqs. (8) and (9) we found the system of equations to solve for the density and electrostatic potential

(13)$$\displaystyle{{d{{\tilde n}_i}} \over {d{\rm \xi} }} = \displaystyle{{ - {{\tilde n}_i}\sqrt {{\rm \delta} + {{\tilde n}_i}/H} } \over {\left({{\rm \delta} + 1.5{{\tilde n}_i}/H - 0.5\tilde n_i^2 L/{H^3}} \right)}}\comma \;$$

(14)$$\displaystyle{{d\Phi } \over {d{\rm \xi} }} = \displaystyle{{ - {{\tilde n}_i}\sqrt {{\rm \delta} + {{\tilde n}_i}/H} } \over {H\left({{\rm \delta} + 1.5{{\tilde n}_i}/H - 0.5\tilde n_i^2 L/{H^3}} \right)}}\comma \;$$

where $L = {e^\Phi }erfc\left({\sqrt {\left\vert \Phi \right\vert } } \right)\left({b{\Phi ^2} - b\Phi + 1 - b} \right)+ 4b\left({\displaystyle{{\sqrt {\left\vert \Phi \right\vert } } \over {\sqrt {\rm \pi} }}} \right)- \left({\displaystyle{{b{\Phi ^2} - b\Phi + 1} \over {\sqrt {\rm \pi} \sqrt {\left\vert \Phi \right\vert } }}} \right)$.

The plasma under consideration is expanding into vacuum at t > 0. The initial time t = 0 in our case corresponds to an unperturbed plasma with initial parameters ${\tilde n_0} = 1$ and ${\tilde {\rm \nu}_0} = 0$ (Ivlev, Reference Ivlev and Fortov1999). As a consequence, we require that there should exist a point ξ₀ at t ≤ 0 for which the plasma is unperturbed and at rest, such that $\tilde {\rm \nu}\left({{{\rm \xi} _0}} \right)= 0$, $\tilde n\left({{{\rm \xi} _0}} \right)= 1$ and Φ(ξ₀) = Φ₀ (see Fig. 3).

Fig. 3. Plasma expansion at rear surface of target in the direction of the laser.

The system of Eqs. (13) and (14) is solved numerically with Runge-Kutta method. Densities, velocities, and plasma potential are deduced according to the variable ξ, and depend on the initial conditions of the plasma expansion.

5. INFLUENCE OF ELECTRON TRAPPING AND ENERGETIC ELECTRONS ON THE PLASMA EXPANSION

The plasma dynamics is governed by ion motion. To study self-similar expansion of the plasma, we present, in Figures 4–7, ion densities normalized to their initial values and ion velocities normalized to the sonic velocity as functions of the self-similar variable ξ for different values of the parameter b at a given initial potential Φ₀ (Figs. 4 and 6), and for different values of the initial potential well Φ₀ at a given value b (Figs. 5 and 7). Assuming that ions are modeled by cold approximation and are much colder than electrons, the temperature ratio δ is arbitrary taken equal to 0.01. It is important to note that the obtained profiles for the limit b = 0 in the density expression (4) are the same than those which are derived using the distribution of Gurevich.

Fig. 4. Ion densities normalized to their initial values as functions of ξ for different values of b.

Fig. 5. Ion densities normalized to their initial values as functions of ξ for different values of Φ₀.

Fig. 6. Ion velocities normalized to the sonic velocity as functions of ξ for different values of b.

Fig. 7. Ion velocities normalized to the sonic velocity as functions of ξ for different values of Φ₀.

The curves are plotted using conditions at ξ₀ corresponding to zero initial velocity, instead of ξ = 0, showing that initial conditions depend strongly on physical parameters like temperatures, initial potentials, and non-thermal effects. Figure 4 shows that for a large enough initial potential well: Φ₀ = 10, and for different population ratios of energetic electrons, between 5 and 90%, two behaviors of the density are observed relatively to the intersection point ξ₀. A large enough initial potential is required to have electron trapping; the value Φ₀ = 10 is chosen arbitrary finding this value numerically suitable to have non-negligible trapping (Bennaceur-Doumaz et al., Reference Bennaceur-Doumaz, Bara and Djebli2011).

The two limits values of b have been chosen in order to handle both the case of very small number of non-thermal electrons (b = 0.05) and the case where almost all the electrons are energetic (b = 0.9). Near the source (ξ < 0), from the early stages of expansion, not only the effect of pressure is observed but also the potential well influences the plasma expansion: as b is decreasing, the expansion is slowing down. This effect is much more efficient beyond the point of intersection. In fact, for ξ > 0, for large values of b (for example b = 0.9) where the majority of electrons are energetic, the behavior of the density decreases very rapidly comparatively to that for small values of b (example b = 0.05) where the decrease in density is much slower. So, the presence of an important number of non-energetic trapped electrons in the plasma potential wells has the effect of slowing down the expansion, these electrons have not enough energy to go across the potential well and remain trapped and oscillating in it; whereas the phenomenon of presence of energetic electrons makes the influence of trapping effect on the self-similar expansion very weak.

On the other hand, we can make the following remarks concerning Figure 5 where density is represented for different potentials wells for a large value of b parameter. The figure shows that when the population of energetic electrons is dominant (b = 0.9), as the potential increases, the trapping effect on these particles has another role. It seems that the energetic particles trapped in the potential gained energy from the waves where they are in resonance with them, making the plasma expansion more accelerated. The case of smaller potential Φ₀ = 2 is approaching the case of Cairns function distribution with zero initial potential, without trapping.

In Figures 6 and 7, the represented ion velocities are approximately linear as functions of the self-similar variable ξ. In Figure 6, for Φ₀ = 10, for all values of b, the velocities are increasing until reaching the same value. We point out that the expansion velocity is increasing slowly when b is decreasing. We can see this phenomenon in the decrease of the expansion acceleration (Δν/Δξ) when b decreases: this is due to the influence of electronic trapping that is to slow down the expansion, especially in the presence of a minimum population of energetic electrons (example Δν/Δξ = 0.962_s for b = 0.9 and 0.773c _s for b = 0.05). The very slow electrons cannot exchange energy with the wave and cannot remain in phase with it because the wave exceeds them very quickly.

In Figure 7, the same phenomenon is observed for the behavior of velocity as a function of ξ for different values of the potential and for b = 0.9 where it is shown that the acceleration is less pronounced for little values of the potential. This is due to the acceleration of energetic electrons that are trapped by the potential. There is a population of electrons which can be in phase or resonance with the plasma wave, and then, can be trapped. The resonant interaction only takes place for those electrons that have a velocity close to the ion-acoustic velocity.

Figure 8 tracing the limit of self-similar ξ_Lim expansion as a function of b for a well of Φ₀ = 100 shows that the self-similar expansion is more extended and the plasma lifetime increases when the parameter b is decreasing (example: ξ_Lim = 1.111 for b = 0.9 and ξ_Lim = 1.654 for b = 0.05). The limit of self-similar expansion is nearly constant especially for very large electrostatic potentials, and the expansion is much slower for b = 0 than for b ≠ 0 even in the case of a very small number of energetic electrons (b = 0.05). For example, for Φ₀ = 10, if b = 0.05, ξ_Lim = 1.654 and if b = 0, ξ_Lim = 5.179, showing the slowing down of expansion even in the presence of a very small number of energetic electrons. The obtained self-similar solution shows that when the population of the energetic electrons is dominant, it is the phenomenon of acceleration due to these electrons which overrides the trapping in the expansion of plasma.

Fig. 8. Limit of the self-similar variable as function of b.