Shock wave structures in a dissipative quantum ... - APS Link Manager

PHYSICAL REVIEW E 87, 033102 (2013)

Shock wave structures in a dissipative quantum plasma Samiran Ghosh1,* and Nikhil Chakrabarti2,† 1

Department of Applied Mathematics, University of Calcutta 92, Acharya Prafulla Chandra Road, Kolkata-700 009, India 2 Saha Institute of Nuclear Physics, 1/AF, Bidhannagar, Kolkata-700 064, India (Received 31 December 2012; published 12 March 2013) Weak nonlinear low-frequency modes in a two-component electron-ion dissipative quantum plasma are investigated. The dissipation arises due to the electron-ion collision. It is found that a standard perturbative approach leads to a new nonlinear equation for the dynamics of the finite amplitude wave. The quantum diffraction effect in the presence of collisions introduces a dissipative effect. The numerical solutions of this novel equation predict the existence of weak shock structures. DOI: 10.1103/PhysRevE.87.033102

PACS number(s): 52.35.Mw, 52.35.Tc, 67.10.Jn, 52.35.Fp

I. INTRODUCTION

Quantum effects in plasmas become important when the Fermi energy of the plasma species exceeds its thermal energy. Recently, quantum plasmas have been attracting significant attention because of their novelty as well as technological applications in intense laser-solid interaction experiments, plasmonic devices, and also in astrophysical objects like neutron stars and white dwarfs [1–10]. In metallic nanoscale structures the quantum diffraction effect of charge carriers has a significant influence in the collective processes in plasmas. Recent spectral measurements of x-ray Thomson scattering [1,11,12] experimentally verify the signature of the quantum diffraction effect on the dispersive character of electrostatic modes in dense degenerate (quantum) plasmas. These investigations indicate the possibility of the observation of degenerate quantum plasma in laboratory. To investigate the collective processes in quantum degenerate gases in plasmas, a quantum hydrodynamical model (QHM) [13–18] has been developed. This QHM has already been used to investigate the collective phenomena of lowfrequency modes [19–21] in quantum plasma. Even, this QHM will explain the closely related problems such as the nonlinear localized structures in self-gravitating Bose-Einstein condensates [22]. All these investigations are in collisionless limit. However, in atomic clusters irradiated by femtosecond laser pules, high-density and high-temperature plasmas are created. In such plasmas quantum effects are operative and the plasma heating is caused mainly due to the electron-ion collisions [23–26]. Thus in intense laser-solid interaction experiments electron-ion collisions also play an important role in the collective processes in quantum plasma. Therefore, it is pertinent to investigate the influence of collisions on collective modes in quantum plasmas. Despite the above-mentioned physical facts, very few investigations have been done in collisional quantum plasma [27–31]. Therefore, here, we investigate the dynamics of the weakly nonlinear low-frequency modes in two-component dissipative quantum plasma. The aim of this present work is to investigate the effects of the high electron-ion collisional rate on the formation of nonlinear localized structures of

* †

sran− [email protected] [email protected]

1539-3755/2013/87(3)/033102(5)

low-frequency weakly nonlinear waves in one-dimensional (1D) quantum plasmas. The dissipation arises due to the direct collisions between the electrons and the ions. The electrons are degenerate and their dynamics are described by the QHM. Due to the large inertia of ions, the quantum effects on ions are negligibly small and as a result the ion dynamics are modelled by the usual hydrodynamic equations. The electronion collisions are included in the momentum-conservation equation phenomenologically as a simple relaxation term [18]. The well-known reductive perturbation technique (RPT) is used to study the dynamics of the weakly nonlinear modes. To investigate the effects of collisions (dissipation) on the nonlinear dynamics, it is assumed that the electron-ion collision frequency is much larger than the (low-frequency) ion oscillation frequency (νei ωpi ). It is found that the dynamics of the wave is governed by a new nonlinear evolution equation. In the collisionless limit, this equation reduces to the usual deformed Korteweg–de Vries (KdV) equation for the low-frequency modes in quantum plasma [19]. On the other hand, in the classical limit, the derived equation reduces to the well-known Kakutani and Kawahara equation [32] for two-component electron-ion plasma. The numerical solution of this novel equation reveals the existence of the weak shock due to the collision induced dissipation. The manuscript is organized in the following manner. Physical assumptions and the basic equations to describe the model are discussed in Sec. II. The nonlinear evolution equation that governed the dynamics of the weakly nonlinear waves is derived in Sec. III. The numerical solutions of the nonlinear equation are discussed in Sec. IV. Finally, a brief summery of the results and its possible applications are discussed in Sec. V.

II. PHYSICAL MODEL AND BASIC EQUATIONS

The plasma is fully ionized with quasineutrality condition ne0 = ni0 = n0 , where ni(e)0 is the equilibrium number density of ion (electron). The plasma is homogeneous, unmagnetized, and unbounded consisting of two-component electrons and ions. The electrons collide with the ions. Electrons are degenerate (χe 1), and due to large inertia (compared to electrons), the quantum effect on ions is negligibly small (χi 1). The degeneracy (dimensionless) parameter χs for

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SAMIRAN GHOSH AND NIKHIL CHAKRABARTI


a particle of species s is defined as [17] 2/3 εF 1 χs ≡ s = (3π 2 )2/3 n0 λ3Bs , Ts 2 where n0 is the equilibrium plasma density. The√other physical parameters εFs = h ¯ 2 (3π 2 n0 )2/3 /2ms , λBs = h ¯ / ms Ts , Ts , and ms are the Fermi energy in the ground state, thermal de Broglie wavelength, thermal energy, and mass of the particle species s, respectively. Also, both the species are nonrelativistic (εFs ms c2 ; c is the velocity of light). The present model is relevant for the ordering TFi Ti Te TFe , where TFs ≡ εFs is the Fermi temperature of the particle species s. The collisional effects in condensed matter and quantum liquids has been investigated kinetically as the limiting case of a nondegenerate plasma with the help of the Lindhard dielectric function [33,34]. Later investigations reveal [35,36] that, as a first approximation, the usual Bhatnagar-Gross-Krook–type collisional model [37] is the kinetic analog of the relaxation models in condensed-matter physics and quantum liquids. Therefore, as a first approximation, we consider Krook’s collisional model to incorporate the effects of the electron-ion collision in QHM [17]. Accordingly, if Ri = −mi ni νie (vi − ve ) and Re = me ne νei (ve − vi ) are the momentum losses of ion and electron fluids per unit volume due to electron-ion elastic collisions, where ni(e) is the ion (electron) unperturbed number density, vi(e) is the ion (electron) fluid velocity, and νie(ei) is the ion-electron (electron-ion) collision frequency, then Krook’s model suggests that Ri + Re = 0, which implies that ne me νei . νie = (1) mi ni On the basis of this collisional model, we consider the ion and electron momentum-conservation equations. The ion dynamics are given by the following ion momentum equation: ∂ ∂ mi ni + vi vi = eni E − mi ni νie (vi − ve ), (2) ∂t ∂x where E is the electric field. The pressure effects of ions are neglected because of their large masses (compared to electrons). Due to the same reason the quantum effects of ions are also negligibly small. The dynamics of the electrons are described by the QHM and therefore the electron momentum equation becomes [13] √ ∂ 1 ∂ 2 ne ∂ ¯2 ∂ neh ve = −ene E + + ve me ne √ ∂t ∂x 2me ∂x ne ∂x 2 ∂pe − − me ne νei (ve − vi ). (3) ∂x In the above the term containing h ¯ 2 arises due to the quantum diffraction (Bohm potential [38]) and pe is the electron pressure. For a 1D fluid equation, we assume that the electrons obey the equation of state pertaining to 1D zero-temperature Fermi pressure given by pe = me vF2 e n3e /3n20 [17]. To write equations in the dimensionless form, let us introduce the following dimensionless variables: x˜ = x/λFe , t˜ = ωpi t, n˜ i(e)= ni(e) /n0 , v˜i(e) = vi(e) /cs , E˜ = eλFe E/2TFe , where λFe = 2TFe /4π n0 e2 is the Fermi length, ωpi = 4π n0 e2 /mi is the ion plasma frequency, and cs = 2TFe /mi is the ion acoustic velocity in quantum plasma. Hereafter,

we will be using these new variables and remove all the tildes for simplicity of notations. In dimensionless form, ion momentum-conservation equation (2) with the help of Eq. (1) can be written as ne ∂ ∂ (vi − ve ), + vi vi = E − μ¯νei (4) ∂t ∂x ni where μ = me /mi and ν¯ ei = νei /ωpi . With these normalizations, the electron momentum equation (3) can be rewritten as μ

∂ ∂ + ve ∂t ∂x

ve

√ H2 ∂ 1 ∂ 2 ne = −E + √ 2 ∂x ne ∂x 2 ∂ne − δ(ve − vi ), − ne ∂x

where δ = μ¯νei and H = h ¯ ωpe /2TFe is the dimensionless quantum parameter that arises due to the electron-diffraction effect. For the low-frequency (ω ωpe ) modes, we neglect the electron inertia, i.e., we neglect the term O(me /mi ). Also, for large electron-ion collisions compared to ion plasma frequency, where νei ωpi ≡ ν¯ ei 1 such that ν¯ ei 1/μ, we retain the term δ and finally get the following electron momentum equation: √ H2 ∂ 1 ∂ 2 ne ∂ne − δ(ve − vi ). (5) = −E + ne √ ∂x 2 ∂x ne ∂x 2 The above two equations (4) and (5) are coupled with the dimensionless ion continuity equation, ∂ni ∂ (ni vi ) = 0, + ∂t ∂x

(6)

and spatial and temporal electric-field variation, ∂E = ni − ne , ∂x

∂E = ne ve − ni vi . ∂t

(7)

The first equation is Poisson’s equation and the second is the x component of the ∇ × B (B is the magnetic field) equation. Here we are studying the electrostatic mode in a uniform magnetic field; therefore, the temporal evolution of the electric field can be obtained from the x component of the Maxwell equation ∂E/∂t stated above. This equation dictates that the conduction current is being balanced by the displacement current. In a fluid picture the total variation (i.e., in both space and time) is obtained combining this with Poisson’s equation ∂E/∂x [39]. With the help of Eq. (7), the momentum equations for ions [Eq. (4)] and electrons [Eq. (5)] become ∂ ∂ ∂E ∂E ni + vi vi = ni E + δ + vi (8) ∂t ∂x ∂t ∂x and n2e

√ 1 ∂ 2 ne ∂ne H2 ∂ = −ne E + ne √ ∂x 2 ∂x ne ∂x 2 ∂E ∂E −δ + vi . ∂t ∂x

(9)

Finally, the above equations (6),(7), (8), and (9) are the basic equations for the model described in this work.

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III. NONLINEAR EVOLUTION EQUATIONS

To extract the essential feature of weakly nonlinear localized structures, the reductive perturbative technique (RPT) is employed and the following stretched coordinate is introduced: √ ξ = (x − t), τ = 3/2 t, (10) where is a small nonzero parameter proportional to the amplitude of the perturbation. The (normalized) dynamical variables ni(e) , vi , and E are expanded about their equilibrium value in power series of in the following way: ⎛ (2) ⎞ ⎛ (1) ⎞ ⎞ ⎛ ⎞ ⎛ ni(e) ni(e) 1 ni(e) ⎟ (1) ⎟ + 2 ⎜ ⎝ vi ⎠ = ⎝ 0 ⎠ + ⎜ ⎠ ⎝ vi(2) ⎠ + · · · . ⎝ vi √ (1) √ (2) E 0 E E (11)

To incorporate the large electron-ion collisional rate and also for the perturbation consistent with that of (10) and (11), we consider the following scaling: ν δ = μ¯νei = √ . Note that the above scaling is also consistent with our assumption that ν¯ ei 1/μ, i.e., δ 1. With the new independent coordinates defined in Eq. (10), the perturbation expansion (11) and the above-mentioned scaling on δ, we transform the continuity equation [Eqs. (6)], momentum equations [Eqs. (8) and (9)], and Poisson’s equation [Eq. (7)] into a set of four equations in the form of a power series in . The resulting system can be written as

(1) ∂ni ∂ (1) ∂ (2) (1) (1) (1) (2) v − ni + v + ni vi − ni + O( 2 ) = 0, + ∂ξ i ∂τ ∂ξ i

(12)

(1) (1) (1) (1) ∂vi ∂vi(1) ∂ 1 (1) 2 ∂E (2) ∂ ∂E (2) (1) ∂vi (1) ∂E (1) (1) (2) − ni −E −ν −1 E − + + v +E − + vi ν − vi ∂ξ ∂ξ ∂τ ∂ξ 2 i ∂ξ ∂τ ∂ξ ∂ξ + O( 2 ) = 0,

(13) (2) ∂n(1) ∂ ∂n(1) ∂ne H 2 ∂ 3 n(1) e e e (1) (2) 4 + E − ν − 1 E (1) + + n(1) + 3E − e ∂ξ ∂ξ ∂ξ ∂ξ 2 ∂ξ 3 (1) ∂E (2) ∂E (1) ∂E (1) ∂E − + vi(1) − 2n(1) + O( 2 ) = 0, +ν e ∂τ ∂ξ ∂ξ ∂ξ ∂E (1) (2) (1) (2) + O( 2 ) = 0. n(1) − n + n − n − e e i i ∂ξ

These equations (13) and (15) are to be satisfied to all orders in . The zeroth-order terms subject to the boundary conditions (1) (1) → 0 as ξ → −∞ give the following n(1) i(e) , vi , and E relations: (1) (1) n(1) e = ni = vi ≡ n(ξ,τ )

and

∂n ∂ = ν − 1 E (1) , ∂ξ ∂ξ

(16)

(17)

where we define a new function n(ξ,τ ). After a little algebra, the first-order terms in together with the relations (16) and (17) yield the following nonlinear evolution equation: ∂n ∂ 4n ∂ ∂n ∂n ∂ 3n ∂n +n +β 3 +γ 4 =α +n , (18) ∂τ ∂ξ ∂ξ ∂ξ ∂ξ ∂τ ∂ξ where β=

(14)

(15)

In the collisionless limit ν → 0, we recover the usual deformed KdV equation for the low-frequency modes in quantum plasma, which has been discussed in detailed in [19]. Retaining the collisional effect in the classical limit, i.e., H → 0, we recover the Kakutani and Kawahara equation [32] for a two-component electron-ion collisional plasma. Therefore, the presence of both quantum and collisional effects in the above equation [Eq. (18)] suggests a new nonlinear evolution equation for low-frequency modes. The electron-ion collisional (compared to ion plasma frequency) effects are reflected for the terms α and γ in the above nonlinear equation [Eq. (18)]. The quantum diffraction (term ∝ H ) introduces the usual dispersive correction (in the expression for coefficient of dispersion β) [19]. Also, in the presence of electron-ion collision, quantum diffraction introduces a dissipative effect in the system that can be seen from the term γ . This can be easily seen if we analyze the linear version of Eq. (18). Assuming the Fourier mode solution n ∼ exp[i(kξ − ωt)], from linear version of Eq. (18), we have

2 H2 ν H 1 1− , α= , γ =ν . 2 4 2 8

ω= 033102-3

(γ αk 5 − βk 3 ) − ik 4 (βα + γ ) . k2α2 + 1

SAMIRAN GHOSH AND NIKHIL CHAKRABARTI

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FIG. 1.√(Color online) Solution of (18) in different time τ in the presence of electron-ion collision. The initial condition is n(ξ,0) = 3U sech 2 ( U/4βξ ). The parameters are U = 1, ν = 0.1 with (a) H = 0, (b) H = 1.2, and (c) H = 1.5.

(γ αk 5 − βk 3 ) , k2α2 + 1 k4ν H2 −k 4 (βα + γ ) ≡− . 1+ ωi = k2α2 + 1 4 + k2ν 2 4

ωr =

Since ωi < 0 the mode is damped and the expression of ωi suggests that this damping is due to the combined effect of quantum diffraction and the collisionality. Interestingly, in the fine-tuning case H = 2, the usual dispersive character (dispersive character of the deformed KdV equation [19]) of the above nonlinear equation (18) vanishes, viz. (β = 0), and the dynamics of the weakly nonlinear wave is governed by the following equation: ∂n ∂n ∂ 4n ∂n ν ∂ ∂n +n +ν 4 = +n . (19) ∂τ ∂ξ ∂ξ 2 ∂ξ ∂τ ∂ξ Moreover, for higher quantum diffraction effect H > 2, the derived nonlinear equation (18) possesses negative dispersive effect (β < 0). Therefore, there are a variety of nonlinear solutions one can look to for depending physical situations. IV. NUMERICAL ANALYSIS

We are interested in finding the solution of Eq. (18) with its full generality. However, in the presence of dissipation (ν = 0), Eq. (18) is not an exactly integrable Hamiltonian system. Therefore, to investigate the effect of electron-ion collision on weakly nonlinear low-frequency waves, we solve the nonlinear equation (18) numerically with the help of a MATHEMATICAbased finite difference scheme for β > 0 (0 H < 2). Note here that, in the absence of collision (ν = 0), the nonlinear equation (18) is an exactly integrable Hamiltonian system and √ possesses a single soliton solution, n(ξ,τ ) = 3U sech 2 [ U/4β (ξ − U τ )], where U is the soliton velocity. As mentioned before for 0 H < 2, Eq. (18) possesses positive dispersion, whereas for H > 2, Eq. (18) possesses negative dispersion for the weakly nonlinear wave. One can obtain the nonlinear equation with positive dispersion by replacing n → −n and ξ → −ξ [40,41]. The sign of the dispersion only determines the direction of the propagation of the wave and therefore the investigations for 0 H < 2 to study the weak nonlinear wave can also be equally applied to a medium with negative dispersion [40,41].

For the time-dependent numerical solution, we use the soliton√ solution as the initial wave form: n(ξ,0) = 3U sech 2 ( U/4βξ ), ξ ∈ [−L,L], where L is the spatial length. √The boundary conditions are n(±L,τ ) = 3U sech 2 (± U/4βL) and nξ (−L,τ ) = 0 = nξ (L,τ ). To obtain adequate results for the computation, we take L = 15 and U = 1. This numerical estimation confirms the existence of weak shock structures, as shown in Fig. 1. The time-dependent solutions for H = 0, H = 1.2, and H = 1.5 are shown in Figs. 1(a), 1(b), and 1(c). For the same value of dissipation (ν = 0.1), these figures show the oscillatory nature of the shock (dispersive shock). In the absence of quantum diffraction [Fig. 1(a)], the shock is more dispersive than that of the presence of quantum diffraction [Figs. 1(b) and 1(c)]. This clearly demonstrates that quantum diffraction in the presence of collision enhances the dissipative effect. In the case with H = 2(⇒ β = 0), the above equation (18) reduces to the nonlinear equation (19), which is also a nonintegrable Hamiltonian system. Therefore, we also solve Eq. (19) numerically for H = 2. In this case, instead of the previous initial condition, we use n(ξ,0) = A sech [ξ ], ξ ∈ [−L,L] (where A is the amplitude) as the initial condition. The boundary conditions are n(±L,τ ) = A sech [±L] and nξ (−L,τ ) = 0 = nξ (L,τ ). To obtain adequate results for the computation, we take L = 10 and A = 1. Again, these numerical estimations confirm the existence of shock structures, as shown in Fig. 2. This figure shows the monotonic nature of the shock structure. The comparison between Fig. 1 and Fig. 2

1.0

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0.6 n

Calculating real (ωr ) and imaginary (ωi ) parts of the frequency ω, we have

0.4

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FIG. 2. (Color online) Solution of (19) in different time τ in the presence of electron-ion collision. The initial condition is n(ξ,τ ) = A sech [ξ ]. The parameters are H = 2, A = 1, and ν = 0.1.

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reveals that the quantum diffraction in the presence of collision introduces a dissipative effect (γ ).

In this paper, we investigate the propagation characteristics of weak nonlinear low-frequency modes in dissipative quantum plasma. The dissipation arises due to the electron-ion direct collisions. It is found that when the electron-ion collision time scale is smaller that the low-frequency mode oscillation time scale, the dynamics of the nonlinear wave is governed by a new nonlinear equation. Numerical solution of this novel equation reveals that electron-ion collision induced dissipation

leads to the formation of low-frequency weak shock structures in the quantum plasma. Time-dependent numerical solution clearly demonstrates that, in the presence of collision, the quantum effects enhance the dissipation. Note that, till now, we have not encountered any experimental observations. However, it would be very interesting to look at this situation in a laboratory. We hope that in the future the development of the coherent x-ray radiation sources [42] and new generation laser plasma experiment, e.g., keV free-electron lasers [43], will provide the experimental observation of this type of nonlinear (weak shock) structure in quantum plasmas.

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V. CONCLUSIONS

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