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Hasan M M, Hossen M A, Rafat A, Mamun A A. Effect of Bohm quantum potential in the propagation of ion–acoustic waves in degenerate plasmas. Chinese Physics B, 2016, 25(10): 105203  
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Effect of Bohm quantum potential in the propagation of ion–acoustic waves in degenerate plasmas
Hasan M M, Hossen M A, Rafat A†, , Mamun A A
Department of Physics, Jahangirnagar University, Savar, Dhaka-1342, Bangladesh

 

† Corresponding author. E-mail: mehedi.ju.plasma@gmail.com

Abstract
Abstract

A theoretical investigation has been carried out on the propagation of the ion–acoustic (IA) waves in a relativistic degenerate plasma containing relativistic degenerate electron and positron fluids in the presence of inertial non-relativistic light ion fluid. The Korteweg-de Vries (K-dV), modified K-dV (mK-dV), and mixed mK-dV (mmK-dV) equations are derived by adopting the reductive perturbation method. In order to analyze the basic features (phase speed, amplitude, width, etc.) of the IA solitary waves (SWs), the SWs solutions of the K-dV, mK-dV, and mmK-dV are numerically analyzed. It is found that the degenerate pressure, inclusion of the new phenomena like the Fermi temperatures and quantum mechanical effects (arising due to the quantum diffraction) of both electrons and positrons, number densities, etc., of the plasma species remarkably change the basic characteristics of the IA SWs which are found to be formed either with positive or negative potential. The implication of our results in explaining different nonlinear phenomena in astrophysical compact objects, e.g., white dwarfs, neutron stars, etc., and laboratory plasmas like intense laser–solid matter interaction experiments, etc., are mentioned.

PACS: 52.27.Ny;52.35.Fp;52.35.Mw
Keyword:Bohm quantum potential;degenerate relativistic plasma;electron and positron quantum effect
1. Introduction

Nowadays, the quantum effects associated with the plasma particles have become very interesting due to their ubiquitous presence in different astrophysical as well as different space dense plasma environments.[1,2] Since dense electron–positron–ion (EPI) plasmas may exist in astrophysical environments, e.g., white dwarfs, neutron stars, active galactic nuclei, etc.[1–4] as well as in laboratory plasma viz intense laser–solid matter interaction experiments,[5–7] it is of practical interest to investigate the properties of the IA waves in such kinds of plasmas. In the case of dense EPI plasmas, the quantum effect plays a significant role in governing the dynamics of charged particles.[8,9] On the other hand, it is found that quantum plasmas occur abundantly in dense astrophysical environments such as white dwarf and magnetars.[10]

The equation of state for degenerate plasma occurring in the dense astrophysical compact objects (e.g., white dwarfs, neutron stars, etc.) was explained by Chandrasekhar.[11,12] The degenerate pressure exerted by plasma fluids can be given as[4,13]

where s is the plasma species (s = i for ions; s = e for electrons; and s = p for positrons),

for the non-relativistic limit (Λc = πħ/msc = 1.2 × 10−10 cm, and ħ = h/2π);

for the ultra-relativistic limit.

Recently, a number of authors have studied the nonlinear propagation of the IA waves in EPI plasmas either considering non-degenerate plasma[14–19] or degenerate plasmas.[13,20–25] In a degenerate dense plasma, Mamun and Shukla[13] studied the SWs, and implied their results in white dwarfs. Rahman et al.[22] investigated an EPI plasma, and showed that the amplitude, width and phase velocity of the IA waves are significantly modified due to the positron concentration as well as the relativistic plasma parameters. In degenerate plasma, the thermal de Broglie wavelength of the plasma species becomes comparable to the dimension of the system due to its high density. As a consequence, the quantum mechanical (such as tunneling associated with the Bohm potential) along with statistical (Fermi–Dirac pressure) effect come into play in such kinds of dense plasma. So, the quantum effects of the plasma particles viz electrons, positrons or ions must be taken into account to clearly describe the dynamics of these plasma systems. As far as we know, none of the earlier works on degenerate plasma considered the quantum mechanical effects (due to Bohm potential) of the plasma particles.

A number of works[26–32] have been done considering the quantum effects of plasma particles in different plasma medium. The behavior of high-densities and low-temperatures quantum plasmas was first studied by Pines.[33] A few years ago, Haas et al.[29] found in their investigation that the Bohm potentials associated with the plasma particles observably modify the basic features of the nonlinear IA waves. In a quantum electron-ion (EI) plasma, Bhowmik et al.[30] investigated the effects of the quantum diffraction parameter (H), and the equilibrium density ratio of the plasma species in modifying the electron–acoustic waves. While analyzing the IA waves in an EPI plasma, Ali et al.[34] found that the nonlinear properties of the IA waves are significantly affected by the inclusion of the quantum corrections of the corresponding plasma species. As stated earlier, in dense as well as degenerate plasma shows a crucial dependency on quantum effects of the plasma species, none of the earlier works have considered this case.

In our present communication, we have considered a degenerate EPI plasma (composed of inertial non-relativistic light ions, and degenerate electrons as well as positrons) considering the quantum mechanical (such as tunneling associated with the Bohm potential) effects for both electrons and positrons. The K-dV, mK-dV, and mmK-dV equations are derived, and their solutions are analyzed to study the basic properties of the IA waves propagating in such kinds of EPI plasma which abundantly occurs in different astrophysical situations (viz white dwarfs, neutron stars, active galactic nuclei, etc.[1–4]) and laboratory plasmas like intense laser-solid matter interaction experiments.[5–7]

The article is organized in the following way. The normalized governing equations are presented in Section 2. The K-dV, mK-dV, and mmK-dV equations for the SWs are derived on Section 3 (Subsections 3.1–3.3), respectively, and the SWs solution of the mmK-dV equation are derived in Subsection 3.4. The brief results and some discussions are given in Section 4. Finally, this paper ends with a summary in Section 5.

2. Theoretical model

The propagation of the IA waves in a degenerate dense EPI plasma system containing degenerate electron and positron fluids (both non-relativistic and ultra-relativistic), and an inertial non-relativistic light ion fluid has been considered. At equilibrium, the quasi-neutrality condition can be expressed as np0 + ni0 = ne0, where np0, ni0, and ne0 are the equilibrium number densities of positrons, ions, and electrons, respectively. The dynamics of the electrostatic IA waves propagating in such a plasma system is governed by the following set of normalized equations

where ns is the number densities of plasma species (s) normalized by their equilibrium value ns0; ui is the ion fluid speed normalized by the IA wave speed ci = (kBTFe/mi)1/2 (with mi being the ion rest mass, kB being the Boltzmann constant, and TFe being the Fermi temperature of the electron gas); ϕ is the electrostatic wave potential normalized by kBTFe/e (with e being the magnitude of an electron charge); η = TFp/TFe (with TFp being the Fermi temperature of positron); μ = ne0/ni0, σ = np0/ni0. The non-dimensional quantum parameter for electron (positron) which is proportional to quantum diffraction, is β = ni0He2/2ne0 (λ = ni0Hp2/2np0), where He = ħωpe/kBTFe (Hp = ħωpp/kBTFp). We note that the last terms in Eqs. (6) and (7) arise due to the quantum Bohm potential or tunneling effect.[32] The time variable (t) is normalized by ωpm−1 = (mi/4π e2ni0)1/2, the space variable (x) is normalized by λDi = (kBTFe/4π e2ni0)1/2. We have also defined as and , where j = e (p) for electrons (positrons).

3. Nonlinear equations for the SWs
3.1. Derivation of the K-dV equation

In order to examine the characteristics of the IA SWs propagating in our considered plasma system, we derive the K-dV equation employing the reductive perturbation method.[35] So, we first introduce the stretched coordinates[35] as

where Vp [= ω/k (ω is angular frequency and k is the wave number)] is the phase speed of the IA wave, and ɛ (0 < ɛ < 1) is a smallness parameter measuring the weakness of the dispersion. We then expand the variables ns, ui and ϕ, in a power series of ɛ as

Now, after substituting Eqs. (9)–(13) in Eqs. (4)–(8), the lowest order coefficients of ɛ can be written as

where , and . The equation (18) represents the dispersion relation for the IA-type electrostatic waves propagating in the degenerate EPI plasma under consideration. Again, substituting Eqs. (9)–(13) into Eqs. (4)–(8), and equating the coefficient of ɛ5/2 from Eqs. (4)–(7), and the coefficient of ɛ2 from Eq. (8), we obtain a set of equations that can be simplified as

Now, simplifying Eqs. (19)–(22) by using Eqs. (14)–(18), and combining each other, we finally obtain our desired equation in the form

Equation (23) is the well-known K-dV equation describing the dynamics of the IA SWs propagating in the degenerate EPI plasma, where the nonlinear coefficient (A) and the dispersion coefficient (B) are given as

Now, to investigate the properties of the IA SWs, we are interested in the SWs solution of the K-dV equation. To do so, we introduce another stretched coordinate, ζ = ξ − U0τ. After the coordinate transformation, the steady state (∂/∂τ = 0) solution of the SWs can be written as (by taking ϕ(1) = Φ)

where the amplitude Φm = 3U0/A, and the width . It is seen that the quantum parameters both β and λ are present in δ1, whereas Φm is totally independent of these parameters.

3.2. Derivation of the mK-dV equation

The K-dV equation [Eq. (23)] is the result of the second-order calculation in the smallness parameter ɛ, where the quadratic nature has been revealed by the nonlinear term Aϕ(1) ∂ϕ(1)/∂ξ. For plasmas with more than two species as like our system, however, there can arise cases where (A) vanishes at a particular value of a certain parameter μ, and equation (23) fails to describe the nonlinear evolution of the perturbation. So, a higher order calculation (i.e., derivation of the mK-dV equation) is needed to describe the dynamics of the system at the critical value of a certain parameter. So, at the critical value μ = μc, we derive the mK-dV equation by applying the following stretched coordinates[36]

By using Eqs. (27) and (28) in Eqs. (4)–(8), we found the same values of , , , , and Vp as those of the K-dV equation. To the next higher order of ɛ, we obtain a set of equations, which, after using the values of , , , ui(1), and Vp can be simplified as

where

To the next higher order of ɛ, we obtain a set of equations as follows:

Now, combining Eqs. (34)–(37) after simplifying by using Eqs. (29)–(32), we obtain the well-known mK-dV equation as follows:

where

where

and

Now, taking the same stretching as that employed in the K-dV solution, the stationary SWs solution of Eq. (38) can be directly given as (by taking ϕ(1) = Ψ)

where the amplitude , and the width . It is seen from δ2 that it depends on both β and λ; on the other hand, Ψm shows no dependency on these parameters.

3.3. Derivation of the mmK-dV equation

For μ in the vicinity of μc, however, (A) is small and can be of the order ɛ. Then, the term Aϕ(1) ∂φ(1)/∂ξ as a whole becomes of the same order as the terms in Eq. (38). Including this term in the mK-dV equation, we need to derive the mmK-dV. So, for μ around its critical value μc, A = A0 can be expressed as

where |μ − μc| is a small and dimensionless parameter, also known as the expansion parameter ɛ, i.e., |μ − μc| ≃ ɛ, and s = 1 for μ > μc and s = −1 for μ < μc. c1 is a constant depending on plasma parameter γ and which is given by

and ρ(2) can be expressed as

Now taking the coefficient of ɛ3 from Eq. (8), we get

Now we can find the value of ρ(3) from Eq. (45), where the values of , , and can be found from Eqs. (34)–(36). Therefore, combining these equations into Eq. (45), we can finally write the following equation as

where . It is called the mmK-dV equation[36] which describes the propagation of the IA waves in the EPI plasma system in the vicinity of the critical value of the plasma parameter μ.

3.4. SWs solution of the mmK-dV equation

The stationary SWs solution of the mmK-dV equation [Eq. (46)] can be derived by the same transformation done in case of the solution of the K-dV and mK-dV equation. To do so, we introduce a transformation ζ = ξ − U0τ which allows us to write Eq. (46), under the steady state condition (∂/∂τ = 0), as (by taking ϕ(1) = Φ)

where the pseudo-potential V(Φ) is

we can write from Eq. (48) that

The conditions of Eqs. (49) and (50) imply that the SWs solution of Eq. (47) exists if

The latter can be solved as

where Φm = −sc1D/C and V0 = (sc1D)2/6C. Now, using Eqs. (48) and (53) in Eq. (47), we have

where κ = C/6B. The SWs solution of Eq. (47) or Eq. (54) is, therefore, directly given by

where Φm1,2 are given in Eq. (53), and the SWs width Δ is

4. Results and discussions

In this section, we numerically investigate the dependency of the nonlinear wave potential ϕ(1) on the densities of the degenerate electrons (ne0), positrons (np0), and ions (ni0); the Fermi temperatures of electrons (TFe) and positrons (TFp); and the quantum diffraction parameters of both electrons (β) and positrons (λ). For our purpose, we have numerically derived the SWs solutions from our derived K-dV, mK-dV, and mmK-dV equations, and analyzed their stationary SWs solutions based on some typical plasma parameters relevant to different astrophysical and laboratory plasma situations. We consider some typical plasma species density which is consistent with the relativistic degenerate astrophysical plasmas, e.g., ni0 = 1.1 × 1029 cm−3, ne0 = 9.1 × 1029 cm−3, and np0 = 1.5 × ne0 cm−3,[13,24,37] and the other quantum parameters e.g., λ = 0.1–0.9, β = 0.1–0.9, η = 0.2–0.6, and H = 0.2–0.9[27,32,38] related to the different EPI plasma environments existing in some published works. The values of the parameters may change depending on different plasma situations. The results, we have found from this investigation can be summarized as follows.

We have considered the non-relativistic case for the degenerate inertial light ion fluids in our model. As light ions may have a little/significant contribution to the degenerate pressure due to their light mass, in our present work we did not neglect this possibility.[20,39] It should be remarked here that we have neglected the quantum effect for ions since ions are heavier than electrons and positrons.[32] We have also neglected the electron-positron annihilation process by considering the conditions that , where ωpj is the plasma frequency of the j-th species, and Tann is the annihilation time. Many authors have neglected the annihilation process in the relativistic dense plasmas.[40–42]

5. Summary

We have numerically analyzed, and discussed the effects of relativity (both ultra-relativistic and non-relativistic cases), quantum parameters, Fermi temperatures, and number densities of the plasma species on the basic features (viz phase speed, polarity, amplitude, width, etc.) of the IA SWs propagating in a degenerate EPI plasma. It is found that: the degenerate pressure of electrons, positrons, and ions; the inclusion of the new phenomena such as Fermi temperatures and quantum effects of both electrons and positrons; the number densities of electrons, positrons, and ions, etc. remarkably change the basic characteristics of the IA SWs. Finally, it should be noted here that our lower order to higher order analysis for the SWs formulation, which includes not only at the critical values μ = μc of certain plasma parameters but also in the vicinity of the critical value, will be very much helpful in analyzing the IA SWs propagating in the different astrophysical compact objects viz. white dwarfs, neutron stars, etc.[1,2,4] and also in the laboratory plasmas, e.g., intense laser-solid matter interaction experiments.[5–7]

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