Quantum ion acoustic solitary waves in electron–ion plasmas: A Sagdeev potential approach

Physics Letters A 372 (2008) 3467–3470 www.elsevier.com/locate/pla

Quantum ion acoustic solitary waves in electron–ion plasmas: A Sagdeev potential approach S. Mahmood ∗ , A. Mushtaq Theoretical Plasma Physics Division, PINSTECH, P.O. Nilore, Islamabad, Pakistan Received 31 October 2007; accepted 4 February 2008 Available online 9 February 2008 Communicated by F. Porcelli

Abstract Linear and nonlinear ion acoustic waves are studied in unmagnetized electron–ion quantum plasmas. Sagdeev potential approach is employed to describe the nonlinear quantum ion acoustic waves. It is found that density dips structures are formed in the subsonic region in a electron–ion quantum plasma case. The amplitude of the nonlinear structures remains constant and the width is broadened with the increase in the quantization of the system. However, the nonlinear wave amplitude is reduced with the increase in the wave Mach number. The numerical results are also presented. © 2008 Elsevier B.V. All rights reserved. PACS: 52.27.-h; 52.35.Fp; 52.35.Sb Keywords: Ion acoustic solitary waves; Quantum plasmas

1. Introduction The study of quantum plasmas has gain importance due to its applications in ultra small electronics [1], dense astrophysical environments [2] and laser plasmas [3]. The quantum effect becomes important in plasmas, when de Broglie wavelength associated with the particles is comparable to dimension of the system. Recently Haas et al. [4] have described the quantum hydrodynamic model (QHD) for quantum ion acoustic wave in electron–ion (ei) plasmas. The QHD model consists of a set of equations describing the transport of charge, momentum and energy in a charge particle system interacting through a self electrostatic potential. The QHD model generalizes the fluid model for plasmas with the inclusion of quantum correction term also known as Bohm potential. The Bohm potential term appropriately describes negative differential resistance in resonant tunneling diodes. Negative differential resistance is based on resonant tunneling which is a quantum phenomenon and it * Corresponding author. Fax: +92 51 9290275.

E-mail address: [email protected] (S. Mahmood). 0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2008.02.003

does not occur in classical transport model. Several authors [5– 8] have investigated different aspects of collective interactions in electron–ion (ei) quantum plasmas. The study of ion acoustic wave (IAW) in quantum ei plasmas is different from classical plasmas because in former case the electron equilibrium is described by Fermi–Dirac distribution rather than to be Maxwellian–Boltzmann distribution as described in later case. The equation of state for electrons is described from the Fermi-gas model at approximately zero temperature. The study of linear and nonlinear quantum ion acoustic waves have been recently investigated in unmagnetized ei plasmas [4]. They found that under the quasi-neutrality condition, the wave dispersion effect in a quantum plasma appears due to the inclusion of quantum diffraction phenomenon. The authors have also derived a deformed KdV (Korteweg– de Vries) equation for a quantum ion acoustic wave (QIAW) in the small amplitude limit using the reductive perturbation technique. The deformed KdV equation is destroyed at H = 2 (where H is a quantum parameter defined later in the Letter) due to the loss of dispersive term in the one dimensional QHD model. The authors have also described the arbitrary amplitude QIAW with Hamiltonian formulation and obtained the “energy

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integral” equation by including the Poison equation in their model. However, they have not described the solitary wave solution in ei quantum plasmas but presented the oscillatory wave solutions and discussed their instability of the fully nonlinear system. Haas et al. [4] have also predicted the existence of solitary density dips structures of Nonlinear Schrödinger (NLS) equation under the assumption of quasi-neutrality condition in ei quantum plasmas. Recently Wei and Wang [9] have done numerical and analytical study of QIAW in a single walled carbon nanotubes using QHD model. Their numerical results show that the frequency of QIAW strongly depends on the nanotube radius in the long wavelength case. The amplitude modulation of three dimensional (3D) ion acoustic waves in a magnetized ei quantum plasma have also been investigated [10]. Similarly three dimensional ion acoustic solitary wave in unmagnetized electron–positron–ion (epi) quantum plasmas has been carried out by Mushtaq and Khan [11] in the low amplitude limit. Therefore it seems important to study exact ion acoustic solitary wave using one dimensional QHD model with Sagdeev potential approach in unmagnetized ei quantum plasmas. The Sagdeev potential approach has already been used by many authors to find the exact solitary wave solutions in classical plasmas [12–14]. However, this important study has not been worked out in an ei quantum plasma case according to the authors best knowledge. The Letter is organized in the following way: In the next section, the set of nonlinear equations of quantum ion acoustic wave in unmagnetized ei plasmas and the dispersion relation have been presented. In Section 3, the “energy integral” equation has been derived using Sagdeev potential approach. Some of the possible numerical solutions have been presented in Section 4. In the final section, the conclusion has been presented. 2. Set of equations We are considering homogeneous and unmagnetized ei quantum plasmas. In order to study the ion acoustic wave in a quantum plasma, we assume the electrons to be inertialess and the ions are taken to be dynamic. The phase velocity of the wave is taken to be vF i  ωk  vF e (where vF i and vF e are the Fermi velocities of ion and electron respectively). Therefore ion pressure effects due to ion Fermi temperature are ignored. By employing the one dimensional quantum hydrodynamic (QHD) model, we can write the dynamic equations for individual particles as follows: The ion continuity equation can be written as ∂ ∂ni + (ni vi ) = 0, ∂t ∂x

(1)

and the ion momentum equation is given as ∂vi ∂vi e ∂ϕ h¯ 2 ∂ + vi =− + ∂t ∂x mi ∂x 2m2i ∂x



∂2 √  ni ∂x 2

√ ni

.

(2)

The equation of motion for inertialess electrons is described as 1 ∂pe h¯ 2 ∂ e ∂ϕ − + 0= me ∂x me ne ∂x 2m2e ∂x



∂2 √  ne ∂x 2

√

ne

.

(3)

The Poisson equation gives ∂ 2ϕ = −4πe(ni − ne ). ∂x 2 In equilibrium we have ni0 = ne0 = n0 ,

(4)

(5)

where electrostatic field is represented as E = −∇ϕ (here ϕ is the electrostatic potential), ne , me , −e are the perturbed electron density, mass and charge respectively, while ni , mi , e, vi are the perturbed density, mass, charge and perturbed velocity of the ion in ei quantum plasmas. Here n0 is the equilibrium density for both the electrons and ions. In Eqs. (1) and (2), two different quantum effects, i.e., the quantum diffraction and quantum statistics are included. The quantum diffraction effects appear due to wave nature of the particles in a quantum plasma which is taken into account by the terms proportional to h¯ 2 also known as Bohm potentials. However, quantum pressure term is obtained by using the quantum statistics which takes into account fermionic nature of electrons. For electron pressure, we assume that the electrons obey the equation of state pertaining to one dimensional Fermi gas, which is given as follows: pe =

me vF2 e 3n20

n3e .

(6)

The electron Fermi velocity (vF e ) is related to Fermi temperature of electrons (TF e ) by 12 me vF2 e = kB TF e , where kB is the Boltzmann constant. The electron equation of state is the same as in ordinary metals, metal clusters and nanoparticles for which the electron Fermi temperature is generally much higher than the room temperature. On solving Eqs. (1)–(6), the dispersion relation of QIAW is obtained by assuming all the perturbations to be linear, i.e., ω2 =

cs2 k 2 (1 + 14 H 2 λ2F e k 2 ) 1 + λ2F e k 2 (1 + 14 H 2 λ2F e k 2 )

(7)

where ω and k are the wave frequency and wave number respectively. The nondimensional quantum parameter is defined 2 1 hω 0e 2 as H = ( 2k¯B TpeF e ) (where ωpe = ( 4πn me ) is the electron plasma frequency). It is worth mentioning that H 2 is proportional to the rs (Wigner–Seitz radius in units of the Bohr radius) parameter of the electron gas, and it takes on values in the range 2–6 for metallic electrons [4]. The ion acoustic speed is defined 1 1 B TF e 2 s ) = ωcpe is the electron as cs = ( 2kBmTi F e ) 2 and λF e = ( 2k 4πn0 e2 Fermi wavelength in a quantum plasma. The quantum force acting on the ions is smaller due to the large ion mass as compared to electron and therefore it has been neglected. In long wavelength approximation, i.e., λ2F e k 2  1, the above dispersion relation of ion acoustic wave in quantum plas-

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mas can be written as   1 ω2 = cs2 k 2 1 + H 2 λ2F e k 2 . 4

(8)

From Eq. (8), it is evident that wave dispersion effects appear due to the inclusion of quantum effects in plasmas. It is well known that stable solitary structures are formed when the nonlinearity is balanced by the dispersive term in plasmas. Therefore it seems important to study the effects of quantum parameter H on arbitrary amplitude ion acoustic solitary wave in an unmagnetized ei quantum plasma. 3. Localized stationary solution In order to find the localized stationary solution, Eqs. (1)–(5) can be re-written in the normalized form as follows: ∂ ∂ni + (ni vi ) = 0, ∂t ∂x ∂vi ∂vi ∂Φ + vi =− , ∂t ∂x ∂x and ∂ne H 2 ∂ ∂Φ − ne + 0= ∂x ∂x 2 ∂x

(9) (10) 

∂2 √  ne ∂x 2

√ ne

(11)

.

(12)

t¯ = ωpi t , The normalized variables are defined as x¯ = eϕ ni vi ne n¯ i = n0 , n¯ e = n0 , v¯i = cs and Φ = 2kB TF e . The ion plasma x λF e ,

1

0e 2 frequency is defined as ωpi = ( 4πn mi ) for ei plasmas. For simplicity, we have ignored the bars on the normalized variables defined in the above normalized equations. For localized stationary solution, let us choose a transformed coordinate ξ in the moving frame such that ξ = x − Mt, where Mach number is defined as M = cVs and V is the velocity of the nonlinear structure moving with the frame. After integration, the electron equation of motion in the transformed frame becomes

(13)

where we have used the B.Cs. such that as ξ → |±∞|, then Φ → 0 and ne → 1. From ion continuity equation, we have   1 vi = M 1 − (14) ni and the ion momentum equation gives vi2 − 2Mvi = −2Φ

(16)

Now using quasi-neutrality condition, i.e., ni  ne = n and also √ substituting Z = n [4], then from Eqs. (13)–(15) we obtain     1 Z5 Z M 2 H 2 ∂ 2Z − − Z − = . (17) 2 ∂ξ 2 2 2 2 Z3 Now multiplying both sides by ( ∂Z ∂ξ ) of Eq. (17) and after integration once, we obtain the nonlinear differential equation in terms of density as follows   1 ∂n 2 + U (n) = 0 (18) 2 ∂ξ where the Sagdeev potential is defined as    1 8n 1  3 n − 1 − 1 + M 2 (n − 1) U (n) = − 2 4 H 12   M2 1 −1 . − 4 n

(19) 2

ni  ne = n.

H 2 ∂2 √ 1 n2 ne Φ =− + e − √ 2 2 2 ne ∂ξ 2

Using Eq. (14) in Eq. (15) we have   1 M2 1− 2 . Φ= 2 ni

We have used the B.Cs. such that as ξ → |±∞|, then ( ∂∂ξn2 ) →0,

Using quasi neutrality condition (with the assumption that dispersion of the wave is coming from the Bohm quantum diffraction effects) we have

2

3469

(15)

where we have used the B.Cs. such that as ξ → |±∞|, then Φ → 0, vi → 0 and ni → 1.

( ∂n ∂ξ ) → 0 and n → 1 in order to obtain Eq. (18). Eq. (18) is a well-known “energy integral” of an oscillating particle of a unit mass moving with velocity ( ∂n ∂ξ ) at position n in a potential well U (n). The conditions for the existence of localized solutions of Eq. (18) require that (i) U (1) = U (N ) = ∂U ∂n |n=1 = 0 and 2

(ii) ∂∂nU2 |n=1 < 0 (where the fixed point at n = 1 is unstable). Here N is the point where the curve crosses the n-axis and it represents the maximum amplitude of the solitary wave. From second condition, it can be seen that ion acoustic solitary structures are formed only in the subsonic region, i.e., M < 1. 4. Numerical solutions Some of the possible numerical solutions of Eq. (18) have been obtained for different values of quantum parameter H and Mach number M shown in Figs. 1–3. The Sagdeev potential curves have been plotted for different values of quantum parameter, i.e., H = 1 (solid curve), H = 3 (dashed curve) and H = 7 (dashed and dotted curve) for M = 0.6(< 1) shown in Fig. 1. It can be seen from the figure that the amplitude of the potential curve reduces with the increase in the value of the quantum parameter. However, the crossing point on the n-axis remains the same for all the nonlinear potential curves in a ei quantum plasma. The corresponding density dips structures of the Sagdeev potentials are shown in Fig. 2. It is evident from the figure that the amplitude of the nonlinear waves remain constant with the increase in the quantum parameter H . However, only the width of the nonlinear structure is broadened with the increase in the value of quantum parameter H . The nonlinear density dips structures for different values of Mach number, i.e., M = 0.8 (solid curve) and M = 0.6 (dotted and solid curve) at the same quantum parameter H have also been shown in Fig. 3.

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5. Conclusion

Fig. 1. The Sagdeev potential curves are plotted for different values of quantum parameters, i.e., H = 1 (solid curve), H = 3 (dotted curve) and H = 7 (dotted and dashed curve) for the same subsonic Mach number, i.e., M = 0.6(< 1).

Fig. 2. The corresponding density dip profiles are shown for the same parameters as given in Fig. 1.

To conclude, we have studied the arbitrary amplitude ion acoustic wave in unmagnetized ei quantum plasmas with Sagdeev potential approach. We have found that nonlinear ion acoustic wave density dips structures are formed in the subsonic region. It is found that only the width of the nonlinear ion acoustic wave structure is broadened and the wave amplitude remains the same with the increase in the value of quantum parameter H . However, the wave amplitude is increased with the decrease in the Mach number. The results obtained of nonlinear IAW in unmagnetized ei quantum plasmas are quite different from unmagnetized ei classical plasmas in which solitary density humps structures are formed in the supersonic region (M > 1) [12–14]. Haas et al. [4] have already studied thoroughly the linear and nonlinear ion acoustic waves in ei quantum plasmas but they have not studied the arbitrary amplitude single pulse solitary wave solutions. They have derived the deformed Korteweg–de Vries (KdV) equation in the low amplitude limit of quantum ion acoustic wave. The dispersive coefficient disappears at H = 2 in the deformed KdV equation and there is no soliton but only free streaming occurs which eventually produces a shock wave. However in the case of arbitrary amplitude quantum ion acoustic wave in unmagnetized ei plasmas, we have not found any disappearance of soliton solution at any value of quantum parameter H (where H > 0 in a quantum plasma). Our results are general and applicable to unmagnetized ei quantum plasmas, which can exit in micro and nanoscale electronic devices and in dense astrophysical plasma situations like in neutron star or in the core of white dwarf. References

Fig. 3. The decrease in the wave amplitude with the increase in the Mach number are shown for M = 0.8 (solid curve) and M = 0.6 (dashed and dotted curve) for the same quantum parameter H = 7.

It can be seen from the figure that the wave amplitude increases with the decrease in the Mach number.

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