Ion energy spectrum in expansion of plasmas with nonextensive electrons

Accepted Manuscript Ion Energy Spectrum in Expansion of Plasmas with Nonextensive Electrons B. Azarvand-Hassanfard, A. Esfandyari-Kalejahi, M. Akbari-Moghanjoughi PII: DOI: Reference:

S2211-3797(17)30990-7 https://doi.org/10.1016/j.rinp.2017.10.054 RINP 1032

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Please cite this article as: Azarvand-Hassanfard, B., Esfandyari-Kalejahi, A., Akbari-Moghanjoughi, M., Ion Energy Spectrum in Expansion of Plasmas with Nonextensive Electrons, Results in Physics (2017), doi: https://doi.org/ 10.1016/j.rinp.2017.10.054

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Ion Energy Spectrum in Expansion of Plasmas with Nonextensive Electrons B. Azarvand-Hassanfard,1, 2 A. Esfandyari-Kalejahi,1, 3 and M. Akbari-Moghanjoughi1 1

Azarbaijan Shahid Madani University, Faculty of Sciences, Department of Physics, 51745-406 Tabriz, Iran

2

Department of Physics, Payame Noor University (PNU), 19395-3697, Iran 3

Research Institute for Applied Physics and Astronomy, University of Tabriz, 51665-163, Tabriz, Iran (Dated: 21 April 2016)

Abstract The collisionless expansion of plasma with cold ions and nonextensive distributed electrons into vacuum is investigated using a self-similar hydrodynamic model in both planar and nonplanar geometries. The speed, size and the energy spectrum of ions in expanding plasma is studied in terms of nonextensivity and geometry parameters. It is found that superthermal electrons with nonextensive distribution accelerate ions to higher speeds and creating a high energy tail in ion-energy spectrum compared to that with Maxwellian electrons. The later may be useful in describing the high energy ions created in irregular space plasma expansion or strong intense laser-plasma accelerated expansion. A generalized Fokker-Planck like diffusion equation is obtained for plasmas with q-nonextensive electrons applicable to different geometry of expansion. From numerical investigation, it is observed that the rates of diffusive expansion for nonplanar geometries are different from planar ones. The anomalous character of diffusive plasma expansion is studied and concluded that three distinct regimes depending on the q-parameter value exists in all geometries. PACS numbers: 52.30.-q,71.10.Ca, 05.30.-d Keywords:

Plasma expansion. Nonextensive electrons. Ion energy spectrum. Self-similar. Nonplanar

geometries

1

I.

INTRODUCTION

After decades of research on plasma expansion and related fields,there still remain fundamental questions about many aspects of the expansion problem concerning those of both astrophysical and laboratory interest [1–4]. One of the important phenomenon associated with plasma expansion is regarding the laser-matter interaction and production of energetic ions in this process [5–9]. When a laser pulse strikes at thin solid target it produces layers of plasma at the front surface of the target in a number of different physical mechanisms. The hot inertialess electrons are detached from the target and leave the surface at the rear side. However, the reverse field produced by positive ions at the surface makes ions to be dragged along the electron cloud. That is, the instantaneous creation of electric fields due to the charge separation accelerates the ions to higher energies in order to maintain the quasi-neutrality in the plasma. The magnitude of this field is directly dependent on the energy-density distribution of the hot electrons motivated by the strong laser field and possibly a function of many other environmental parameters [10–15]. Hydrodynamic method is one of the most convenient approaches to the problem of plasma expansion. Great majority of works [16–19] on plasma expansion has been done in the framework of the hydrodynamic model. In the presence of free boundary condition, associated with the expansion of plasma, this approach has been used with solutions of self-similar type based on the quasi-neutral behavior [7, 20–23]. Because of a wide range of laser output characteristics used in plasma expansion experiments, 1010 ≤ Il λ2l ≤ 1019 , with Il (W/cm2 ) being the intensity and λl (µm) the wavelength, there can be a fundamental variety of expansion mechanisms. In low intensity pulse experiment bulk heating and collisional ionization by thermal electrons are dominant effects in the expansion process,which are best described under the assumption of Maxwellian electron distribution. However in the short-pulse intense laser experiments, acceleration of electrons by plasma expansion may be quite complex phenomenon leading to the generation of collisionless superthermal electrons with non-Maxwellian energy distribution. Liu et al. in 1994 [24] showed a direct indication for existence of a non-Maxwellian velocity distribution in plasma experiment. Pearlman et al. in 1978 [25] interpreted the ion velocity experiment in plasma expansion based on non-Maxwellian electron distribution function. Other studies [26, 27] reveal that the energetic electrons with non-Maxwellian distributions have a profound effect 2

on the expansion density and ion acceleration profile. In addition to the laser-produced plasma, superthermal electrons with nonMaxwellian high energy tail distribution are also observed in space plasma [28, 29]. In recent years several models for instance with kappa distribution [30], steplike electron energy distribution [31], a super-Gaussian distribution [32] and a truncated Maxwellian distribution [33] have been proposed in order to describe the plasma expansion behavior in the presence of non-Maxwellian electrons. The nonextensive distribution function on the other hand is a new statistical approach, based on Tsallis assumption [34–36] which has been employed to explain different phenomena in various plasma systems including those subject to non-Maxwellian electron distribution. In 1988, [37] proposed q-parameterized nonextensive entropy as a generalized form of the most celebrated Boltzmann-Gibbs entropy that is well-recognized for systems involving the spatial or temporal long-range interactions or memory effects. Many plasma systems in the astrophysical and laboratorial systems as a result of the spatial and temporal long-range behavior have since been successfully described by non-extensive statistical mechanics, for instance, plasma oscillations [38], Langmuir oscillations [39], Landau damping and shock structures in plasmas [40, 41], aspects of space plasmas [42, 43]. The generalized Tsallis statistics with excess superthermal or subthermal electrons in the velocity distribution provides a useful platform most suitable for fluids like plasmas. The Tsallis’s nonextensive distribution has also been studied [44] in connection with the kappa-distribution, first introduced by [45] in order for generalized modeling of different plasma states [46, 47]. The degree of nonextensivity of a system is specified by deviation of q-parameter from unity a value also describing the usual Maxwell-Boltzmann limit. Hence distinct regimes of this parameter q < 1, q = 1 and q > 1 are called the super-extensive (with super-additive entropy properties), extensive (additive) and sub-extensive (sub-additive) regimes, respectively. Therefore, the effects of superthermal charges with enhanced high energy tail distribution in the plasma expansion process refers to the class of nonextensive entropy with q-parameters smaller than unity. Recently the self similar and diffusive of plasma expansion with nonextensive distributed electrons has been studied in the framework of hydrodynamics model [48] in the planar geometry. However, it would be more realistic in laboratory and space plasmas to consider nonplanar expansion rather than planar one, since there are cases of particular interest in which the assumption of planar geometry breaks down. For instance the particular shape 3

of a target requires working in the planar geometries such as cylindrical or spherical ones. Supernova explosions, implosion of fusion targets or exploding wire or spherical targets in inertial confinement fusion experiments are few examples of nonplanar geometry expansion. Murakami et al. [49] showed that the reproduction of the experimental behavior of plasma expansion and resultant ion energy spectrum from those of the planar expansion model which is usually employed in analytical studies in literature is inappropriate. The layout of this article is as follows. In Sec. II, we describe the nonlinear equations governing the one-dimensional collisionless electron-ion plasma applicable to all geometries namely planar, cylindrical and spherical. Using the appropriate self-similar change of variables the nonlinear equations are transformed into the linear ones. Then we solve the linear equations and study the important quantities such as density, size of plasma expansion and electron-ion energy spectrum and their dependence on the q-parameter in different expansion geometries. In Sec. III we study the ambipolar diffusion of nonextensive plasmas and derive the Fokker-Planck equation for generalized geometry and the results of numerical simulations are discussed. Moreover by using the Fokker-Planck equation we study the anomalous character of diffusive plasma expansion. In Sec. IV we conclude the main findings of this research.

II.

THE HYDRODYNAMIC MODEL OF PLASMA EXPANSION A.

Governing Equations

We would like to study the one-dimensional plasma expansion using the self-similar approach for different geometries. The ion fluid is assumed to be cold and initially at rest whereas the electrons follow the generalized nonadditive entropy principle. Nonextensive R electron number density ne = f (v)dv with velocity distribution f (v), reads ne (φ) = ne0



eφ 1 + (q − 1) kB Te

(q+1)  2(q−1)

,

(1)

where ne0 is the unperturbed electron number density, φ is the electrostatic potential and q is the nonextensive parameter that characterizes the degree of nonextensivity, with q = 1, q < 1 and (q > 1) corresponding to standard Maxwellian, superextensive with a large num4

ber of superthermal particles and subextensive with a large number of low-speed particles distributed in the system, respectively. The plasma expansion into vacuum is described in the framework of a fluid model and characterized in the generalized (planar, cylindrical, spherical) geometry by the following set of equations: 1 ∂ ∂ρ + α−1 (rα−1 ρv) = 0, (2) ∂t r ∂r ∂v ∂v Ze ∂φ +v =− , (3) ∂t ∂r mi ∂r   1 ∂ α−1 ∂φ r = 4πe(ne − Zni ), (4) rα−1 ∂r ∂r in which α = 1, 2 and 3 correspond, respectively, to the planar, cylindrical, and spherical geometries, v is the ion velocity, mi is the ion mass, e is the electron charge, Z is the ion charge state and ρ(r, t) = mi ni is the mass density of the plasma. Equations (2)-(4) constitute required information to fully describe the ion acceleration in the plasma expansion.

B.

Self-Similar Solution

We assume a uniform collisionless plasma consisting of electrons and ions that are restricted to the half space, r < 0 at the initial time. For the time t > 0, the initial large electric field at the boundary, accelerates the ions in to vacuum at the r > 0 direction. If the characteristic length scale for plasma density variations is generally large compared with the Debye length, we can assume quasi-neutrality during the expansion i.e. ne ≈ Zni and may look for a self-similar solution. Eqs. (1) and (3) together with the quasi-neutrality condition may be combined to give rise to q−3

∂v 2c2s ρ q+1 ∂ρ ∂v +v =− ∂t ∂r q + 1 ρ 2(q−1) ∂r q+1

(5)

0

where c2s = ZkB Te /mi and ρ0 ≡ ρ(0, 0) are the sound speed and the initial mass-density at the center of expansion. The above system of equations admit the following self-similar solution (Murakami 2005) ˙ v = Rξ, ξ = r/R,

(6)

ρ = ρ0 (R/R0 )−α G(ξ),

(7)

in which the overdot denotes the time derivative, R(t) is the time-dependent characteristic plasma expansion size, ξ is the similarity coordinate, R0 is the initial plasma size and G(ξ) 5

FIG. 1: Variation of G-parameter in self-similar plasma expansion for different values of the nonextesivity parameter q.

is a density function with a normalized boundary condition with, G(0) = 1. By substituting ˙ −1 (d/dξ) and Eqs.(6) and (7) in Eq.(5) and making use of derivative rules, ∂/∂t = −ξ RR ∂/∂r = R−1 (d/dξ), we arrive at a separable ordinary differential equation 

R R0

 α(q−1) (q+1)

(q−3)

2 (q+1) dG ¨ = −2cs × G RR = ℓ0 , q+1 ξ dξ

(8)

in which ℓ0 > 0 is a separation constant (here it is set to ℓ0 = 2 for simplicity). The solution to the above equations for G(ξ) in the generalized statistic frame is 

2(q − 1) 2 G(ξ) = 1 − ξ (q + 1)

(q+1)  2(q−1)

,

(9)

As it is observed by inspection of Eq. (9) the power-law dependence of self-similar variable is predicted for arbitrary value of q. Figure 1 shows the variation of the normalized ion density G-parameter for different values of the nonextensivity parameter q. More over, the electrostatic potential φ and the corresponding electrostatic field E = −∇r φ is obtained using Eqs. (1) and (9) as follows φ=−

2Te ξ 2 , e (q + 1)

E = −∇r φ = 6

ξ 4Te . e (q + 1)R

(10) (11)

FIG. 2: The variation of normalized potential with ξ and q.

Figure 2, on the other hand, shows the variation of normalized electrostatic potential with the self-similar variable ξ for different nonextensive parameter values, i.e., q = 0.5, 1, 2. It is seen from this figure that the electrostatic potential increases for values of q < 1 which elevated number of superthermal electrons in the expanding quasi-neutral plasma. The later effect leads to an enhanced ion acceleration of ions which electrostatically couple to these electrons. It is observed that the density parameter G(ξ) and consequently, φ and E depend on the nonextensivity parameter q while they are not influenced by the geometry parameter α. From Eq. (8), one obtains ˙2

R =

R˙ 02

1 + α



4c2s q−1

h

1 − (R/R0 )−

2α(q−1) (q+1)

i

.

(12)

As it is evident from Eq. (12) the rate of plasma expansion is function of nonextesivity parameter q and geometry parameter α. Integration of Eq. (12) leads to an expression for R(t) which is not a simple function of time and for particular case of q and α, the solution is in the form of hypergeometric function. Numerical solution of Eq. (12) for different values of q and initial condition R˙ 0 = 0 is shown in Fig. 3. This figure reveals that the normalized size of the plasma expansion is significantly affected by the q-parameter. Along with the decrease of q-parameter, the size of expansion increases further with time. Furthermore, it is confirmed that for the value of q → 1, i.e., for a Maxwellian electron 7

FIG. 3: Temporal profile of the normalized size of plasma expansion for different values of q under cylindrical geometry.

distribution, the isothermal Maxwellian limit is recovered [49].

C.

Energy Spectrum of Accelerated Ions

Based on the solution given above the ion energy spectrum can be derived as follows. The total conserved mass of the plasma M0 can be readily obtained by using Eqs. (7) and (9) as 2π α/2 M0 = ρ0 R0α Γ (α/2)

Z

∞

ξ

α−1

0



(q − 1) 2 1− ξ (q + 1)

q+1  2(q−1)

dξ.

(13)

The integration for all values of q-parameter can be analytically evaluated in terms of the hypergeometric functions. Also, the ion kinetic energy Ei may be derived using Eqs. (6), (7) and (13) in the form of hyper-geometric functions α/2 1 ˙ 2= π Ei = M0 (Rξ) ρ00 R0α R˙ 2 2 Γ (α/2)

Z

∞

ξ α+1 0



(q − 1) 2 ξ 1− (q + 1)

q+1  2(q−1)

dξ.

(14)

The total number of ions in the interval [ξ, ξ + dξ] is given as   q+1 2π α/2 2(q − 1) 2 2(q−1) α α−1 dN = ni0 R0 ξ ξ 1− dξ. Γ (α/2) q+1 Thus, the kinetic energy of an ion εi at the position ξ is 1  ˙ 2 εi = mi Rξ , 2 8

(15)

(16)

so that, dεi = mi R˙ 2 ξdξ.

(17)

Using Eq. (15) and Eq. (17), the self-similar distribution of ion kinetic energy is given, as   q+1 2(q − 1) 2 2(q−1) 2π α/2 ni0 R0α α−2 dN ξ 1− = ξ , (18) dεi Γ (α/2) mi R˙ 2 q+1 or equivalently, q+1   2(q−1)   α−2  2 2π α/2 ε 2(q − 1) ε dN α = ni0 R0 . (19) 1− dεi Γ (α/2) ε0 q+1 ε0 ˜ ≡ N/N0 with N0 = By using normalized quantities ε˜ ≡ ε/ε0 with ε0 = (mi /2)R˙ 2 and N  2π α/2 ni0 R0α Γ (α/2), the normalized ion energy spectrum is obtained as follow  q+1  ˜ dN 2(q − 1) 2(q−1) (α−2)/2 = ε˜ ε˜ 1− , d˜ ε q+1

(20)

where ε˜ is the energy of an ion at position ξ normalized to the energy of an ion at ξ = 1. Figure 4 shows the normalized ion energy distribution functions in term of ion kinetic energy given in Eq. (20) for different values of the q-nonextensivity parameter in the spherical geometry expansion. This figure confirms that the ion acceleration depends essentially on the electron distribution function because as the parameter q decreases below unity (the Mawellian limit) the enhanced presence of superthermal electrons cause further acceleration of ions to larger speed and higher energies at a given self similar position (e.g. see q = 0.8 in Fig. 4). It is also confirmed that, in the limiting case of q → 1, Eq. (20) reduces to the similar Maxwellian ion energy spectrum [49]: ˜ dN = ε˜(α−2)/2 exp(−˜ ε) d˜ ε

(21)

In Fig. 5 we show the structure of ion energy spectrum in different expansion geometries, namely, planar (α = 0), cylindrical (α = 1) and spherical (α = 2) ones. It is found that the highest energy distribution of ions occurs with q = 0.8 for the spherical expansion and the lowest occurs for the planar expansion.

III.

AMBIPOLAR DIFFUSIVE EXPANSION

Diffusive expansion of plasma particles play an important role in many physical situations such as electrons and holes in semiconductors [50–52]. In this section, we first obtain a 9

FIG. 4: Normalized ion energy spectrum corresponding to Eq. (20) for different values of qparameter in an spherical plasma expansion.

FIG. 5: Ion energy spectrum with q = 0.8 for different expansion geometries α = 0, 1, 2.

generalized Fokker-Planck like equation for non-planar geometry with nonextensive electrons assuming density dependence of electron-ion collision frequency. Then we find numerical soulsion of this equasion on diffusion rate of plasma expansion. We also make some remarks on the anomalous character of diffusive plasma expansion. The continuity equation in a generalized geometry can be written as ∂ne ∂ni + ∇ · Je = 0, + ∇ · Ji = 0, ∂t ∂t 10

(22)

Ji = −Di ∇ni − µi ni Ed , Je = −De ∇ne − µe ne Ed ,

(23)

where Je (Ji ) is the electron(ion)current density and Ed , De (Di ) and µe (µi ) are the electric field, electron(ion) fluid diffusion constant, and electron(ion) mobility, respectively. The ion momentum equation, considering also collisions between electrons and ions, can be written as e ∂ui + (ui · ∇)ui + ∇(φ + φd ) + νei ui = 0, ∂t mi

(24)

where φd is the potential caused by the ambipolar diffusion and the collision frequency νei between electrons and ions, in terms of the number density, is given as [53, 54] √ 2  4π 2 Zi e 2 ln Λe , νei ≈ √ ni 1/2 3/2 4πε0 3 π me Te

(25)

For typical values of ln Λe = 15 [51] and ion charge state Zi = 1 one obtains νei ≈ 6 × 10−11 ni /(Te )3/2 ,

(26)

where, ni is in cubic meter and Te in electron-volts. Furthermore, in the steady state, when the charge does not locally build up, the flux of electrons and ions must be equal, i.e., Je = Ji = J and quasi-neutrality allows to assume that the electron and ion densities are approximately equal ni ≈ ne = n. Equation (24) can be solved for Ed in terms of ∇n and then by substituting Ed in electron’s or ions flux equations 23, we obtain expressions for the combined ambipolar diffusion coefficient as Da =

µ i De + µ e Di . µe + µi

(27)

Then use the normalized parameters where the ion fluid velocity ui is normalized to the ion p acoustic speed cs0 = ZkB Te /mi , ion density ni is normalized to ion equilibrium density ni0 ,

−1 the time t and space r variables are normalized to inverse of ion plasma frequency ωpi0 = p mi ε0 /n0 e2 and the ion Debye length λs0 = cs0 /ωpi0 respectively, the electric potential

φ and ambipolar potential φd are normalized to kB Te /e and the collision frequency νei is normalized to ωpi0 .The following momentum and continuity equations can be obtained " q−3 # ∂u 2n q+1 1 − Di /De 1 + (u.∇)u + νu + +( ) ∇n = 0, (28) ∂t q+1 1 + µi /µe n ∂n − Da ∇2 n = 0, ∂t 11

(29)

With typical laboratory values n0 ≈ 1016 (m−3 ) and Te = 1(ev) and using a normalized

collision frequency ν = νei /ωpi0 we find ν ≃ b0 n (b0 ≃ 4.3 × 102 ). Note that the assumptions Di ≪ De and µi ≪ µe are due to large mass-ratio mi /me ≥ 1. The simplified hydrodynamic equations for generalized geometry parameter, α, is given below # " q−3 ∂u 1 ∂n 2n q+1 ∂u +u + (b0 n) u + + = 0, ∂t ∂r q+1 n ∂r ∂n 1 ∂ + α−1 (rα−1 nu) = 0. ∂t r ∂r

(30)

(31)

In the steady-state expansion by ignoring the convective derivative (du/dt = 0) in Eq. (30) we arrive at the following relation " q−3 # 1 2n q+1 1 ∂n nu = − + b0 q + 1 n ∂r

(32)

which is used together with Eq. (31) to obtain the following generalized Fokker-Planck-like diffusion equation in α-geometry ∂n ∂t

= h

+ b10 + b10

r −1 (α−1) b0



q−3

2n q+1 q+1

−4 2(q−3) q+1 2n  (q+1) q−3

2n q+1 q+1

−n 

+ n−1



∂n +n ∂r i
∂n 2 −2 −1

∂r

(33)

∂2n ∂r 2

Equation 33 is numerically solved using the following initial and boundary conditions. At time t = 0 , the normalized plasma density is n(r, 0) = 1 in initial normalized size of plasma (−0.1 ≤ r ≤ 0.1) and n(r, 0) = 0 for other places. For other times, varition of density is negligible at the boundaries ( ∂n(r, t)/∂r|r=±∞ = 0 ) Let us discuss the effects of geometry and non-extensive q parameters on rate of diffusive expansion in Figs. 6 and 8. Figure 6, shows numerical solution of Eq. (33) for plasma expansion with different ranges of the nonextensive q-parameter in the cylindrical geometry. It is clearly observed that the rate of diffusive expansion increases with the decrease of q-parameter. The later is due to the fact that superthermal electrons in the case of q < 1 are the main ingredients in diffusion engine. Figure 7, demonstrates the diffusive expansion density variation for different expansion geometries with fixed value of q. It is evident that for the case of spherical geometry the 12

diffusion expansion process takes places faster compared to that of cylindrical and planar geometries. It is obvious that as the geometry parameter of system increases, the diffusive expansion amplitude also increases. The reason could be related to increasing the degree of freedom in expansion by increasing the geometry index in diffusion. Another interesting feature of diffusive expansion is revealed when one compares the mean-squared expansion distance with time, < r2 >∝ tγ . The diffusion exponent value γ = 1 is related to the normal diffusion, so that, when the exponent γ deviates from unity, the diffusion is referred to as anomalous. The case with γ < 1 corresponds to sub-diffusion and γ > 1 to super-diffusion. The anomalous character of the diffusion may be observed in a variety of systems with different range and types of correlations between species. Dyrud et al. [55] showed with numerical simulations that the anomalous diffusion of meteor trails enhances cross-field expansion and may explain the gradient-drift instability observations at the edges of meteor trails. The nonlinear Fokker-Planck equation with non-extensive species is observed to beautifully describe thise anomalous property, as discussed below. In Fig. 8, we show the plot of mean-squared expansion size given by Eq. (33) for different values of q in the cylindrical geometry. Comparing different curves in this plot, three distinct cases of diffusion according to q-values are observed namely normal diffusion for q = 1 (Maxwellian limit), sub-diffusion for q > 1 and super-diffusion for q < 1. It is seen that both of the sub-diffusion and super-diffusion are described in a nonextensive plasma with ( q > 1) and (q < 1), respectively. In fact, the nonextensive distributions can be suitable for describing the plasma systems containing the anomalous character of diffusion.

IV.

CONCLUSION

We have investigated the self-similar and diffusive expansion of plasma into vacuum using a hydrodynamic approach. The electron distribution function is modeled by a nonextensive entropy formalism. Behaviors of the density, velocity and size of plasma expansion are investigated in one dimension by emphasis on the effects of nonextensive parameter and nonplanar geometry. Furthermore, the analytical expressions for the energy spectrum of ions were derived. It was seen that, the existence of excessive high energy ions in the spectrum for case q < 1, which is due to existence of excess super-thermal electrons in velocity space, enhances the ion acceleration in comparison with Maxwellian electron case. 13

FIG. 6: Surfaces of ambipolar diffusive expansion plasma in Eq.(33) with q = 1.5, 1, 0.8. for cylindrical geometry α = 2

14

FIG. 7: Comparison of plasma ambipolar diffusive expansion surfaces for different geometries with nonextensivity parameter q = 3.

15

FIG. 8: Comparison of squared displacement of density depletion for different q in cylindrical geometry.

Our results reduce to the solution with Maxwellian electron at the extensive entropy limit q = 1. It was also found that, the population of the high energy ions in the spectrum maximizes for the spherical geometry compared to for cylindrical geometry, where as it is minimum for the planar geometry. Investigation of the plasma expansion structures in case of planar and non-planar geometries assuming nonextensive distribution for electrons makes our present work appropriate to reproducing the experimental behavior of plasma expansion and resultant ion energy spectrum situation in both laser-matter experiment and astrophysical situations. Additionally, the Fokker-Planck-like diffusion equation was derived in generalized geometry in terms of two key parameters, namely, the nonextensivity parameter q and the geometry index α taking in to account the density dependent electron-ion collision frequency. Numerical results revealed that the rate of diffusive expansion of plasma and the anomalous character of diffusive expansion corresponds to the deviations of the electron velocity distribution from Maxwellian distribution. Interestingly existence of excess superthermal electrons significantly increases the diffusive expansion rate. A further insight into the diffusion expansion was achieved by considering different expansion geometries. Finally, a more realistic approach to the plasma expansion problem should address the charge separation when the finite value of the Debye length is taken into account [9] which

16

is left for future works.

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