# Quantum Treatment of Kinetic Alfvén Waves Instability in a Dusty Plasma: Magnetized Ions

Authors: N. Rubab; G. Jaffer
Journal: Advances in High Energy Physics
 (2016)

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## Abstract

Kinetic Alfvén wave instability is examined rigorously in a uniform nondegenerate quantum dusty plasma. A linear dispersion relation of kinetic Alfvén wave in inertial regime is derived by incorporating Bohm potential in the linearized Vlasov model. It is found that the quantum correctionCQ appears due to the insertion of Bohm potential in Vlasov model and causes the suppression in the Alfvén wave frequency and the growth rates of instability. A number of analytical expressions for various modes of propagation are derived. It is also found that the system parameters, that is, streaming velocity, dust charge, number density, and quantum correction, significantly influence the dispersion relation and the growth rate of instability.


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## Body

## 1. Introduction

Theoretical and laboratory studies on current-driven electromagnetic instabilities have been a popular subject in plasma research [1, 2]. Electromagnetic and electrostatic plasma instabilities driven by the parallel or cross-field currents give rise to narrow banded spontaneous emissions and have received a lot of attention in the past few decades. It has been speculated that low frequency electromagnetic waves, that is, Alfvén waves, are the most important electromagnetic waves which take part in various dynamical processes occurring in the interior of the sun [3–5]. Recently, kinetic Alfvén waves (KAWs) have been modified via quantum effects associated with electrons; for example, Hussain et al. [6] made an attempt to study spin and nonrelativistic quantum effects on KAWs within the frame work of kinetic theory of Alfvén waves and their analysis suggested that the spin quantum effects suppress the Alfvén wave frequency in a hot and magnetized plasma.Usually, for quantum effects to be significant in plasma, the temperature to density ratio is considered to be quite small. The quantum effects are expected to be significant on very small scales, where debye lengthλD=kT/4πne2 and gyroradius ρj should be smaller or of the order of the de Broglie wavelength λDB=ħ/2πmjkT. At such small scales the quantum kinetic theory is better suited for the proper treatment of these microphenomena. To study various aspect of quantum plasma, numerous efforts have been made whose applications are ranges from plasmonic device applications to dense astrophysical plasmas [7–16].Recently, various researches have been performed to study the collective effects in a quantum plasma. In these investigations, the researchers have focused on the linear dispersion effects of several low frequency modes and instabilities as a benchmark problem in quantum plasma. It has been treated by means of Bohm diffraction or quantum statistical effects that include the Fermi pressure [17]. Various fluid plasma models have been developed by introducing the quantum effects through quantum corrections. This can be done either through the quantum force produced by density fluctuations originating from the Bohm potential or through the spin of particles which includes the magnetic dipole force and magnetization energy. At large scales, plasma is treated as an electrically conducting fluid and can be described by magnetohydrodynamics, while the situation has to be interpreted within the frame work of kinetic theory when microscopic scales are involved. The proper treatment of such phenomena requires the development of kinetic theory based on quantum effects [18]. The presence of dust particles is known to significantly alter the electrostatic and electromagnetic modes [19–23]. This is due to the fact that their presence gives rise to some novel modes associated with dust particle dynamics and finite divergence of the cross-field plasma current density. More recently, Mahdavi and Azadboni investigated the nondegenerate quantum effects on Weibel instability by considering its application in the absorption layer of fuel pallet. In their study, the thermal energy was taken larger than the potential and Fermi energy by taking the quantum effects negligibly small and showed that nondegenerate quantum effects and density gradients tend to stabilize the Weibel instability [24].In this paper, we have made an effort to examine the role of quantum diffraction associated with electrons in a dense, hot, and magnetized plasma and its effect on low frequency modes. We discuss the characteristics of short wavelength dust kinetic Alfvén waves by emphasizing the quantum diffraction effects arising through Bohm potential of electrons, which is obviously dominant over the ions and the dust species due to large mass difference. The dispersion relation of quantum DKAW (QDKAW) is derived by incorporating Bohm quantum potential into the linearized Vlasov equation. Also, the coupling of electrostatic quantum dust acoustic mode (QDAW) with kinetic Alfvén wave (KAW) and the theoretical aspect of KAW instability in an inertial regime is discussed.The manuscript is organized as follows: basic assumptions leading to the general dispersion relation are presented in Sections2 and 3 while results and conclusions are discussed in Section 4.


## 2. Basic Assumptions

We consider an electromagnetic kinetic Alfvén wave in a collisionless electron-ion quantum dusty plasma. The strongly magnetized ions are considered to be Maxwellian and a beam of nondegenerate electrons drifting across an external magnetic field(B0∥z^) with constant ion drift velocity, V0x^, that is, V0⊥B0, while the dust component is cold and unmagnetized. The plasma beta βi is assumed to be very small, that is, βi≪1, where βi=4πni0Tt/B02. An electromagnetic wave with a wave vector k lies in xz plane, k=k⊥x^+k∥z^, making angle θ with x-plane. We have adopted two potential representations which are used in a low beta plasma to express the electromagnetic perturbations to describe the electric field E, that is, E⊥=-∇⊥φ and E∥=-∇∥ψ, where φ≠ψ.The linearized Poisson equation is(1)-k⊥2φ+k∥2ψ=4πene1-ni1-Zd0nd1,where ne1, ni1, and nd1 are electron, ion, and dust number densities, respectively. In (1) Zd0=-Qd0/e (with Zd0 as the number of electron charge on a grain) is the equilibrium charge on an average dust grain and e is the electron charge. For electromagnetic waves, we may ignore the dust charge fluctuation effects, that is, Zd1=0, [25, 26], as it causes damping (non-Landau type) which is relevant for electrostatic waves. Now, on combining Ampere’s and Faraday’s law, we get [27](2)c2k⊥2k∥φ-ψ=4πωJi1∥+Jd1∥,where Jj1∥ represent the field aligned current densities for plasma species. Since the electrons are streaming perpendicular to the field direction, therefore, Je1∥=0. The linearized Vlasov equation for quantum plasma after incorporating Bohm quantum potential in the direction of field will obtain the following:(3)∂fe1∂t+v∥∂fe1∂z+qemeEz+ħ24me2n0∇∇2ne1∂fe0∂v∥=0.The perturbed distribution function of streaming electrons is given by the aid of Vlasov equation and the zeroth order distribution function fe0, which is the usual distribution function and is assumed to obey a Maxwellian or a Fermi Dirac distribution.(4)fe0=Aeexp⁡v∥2+v⊥-V02vte2,where (5)Ae=ne0me2πTe3/2.Since the ions are hot and magnetized with finite Larmor radius effects, therefore, we can solve Vlasov equation in terms of guiding center coordinates by ignoring quantum effects and obtain the perturbed distribution for any electromagnetic wave, [28, 29],(6)fi1=ni0eTi∑l∑nk∥v∥ψ+nΩciφω-nΩci-k∥v∥exp⁡in-lθJnk⊥v⊥ΩciJlk⊥v⊥Ωcifi0,where fi0 is the equilibrium Maxwellian distribution function for electrons, Ωci is the electron cyclotron frequency, and Jn is the Bessel function of first kind of order n.


## 3. Basic Theory and Dispersion Relation

The perturbed number density for electrons including nondegenerate quantum effects has been recast by the aid of Vlasov equation and is determined by the relationne1=∫fe1dv as(7)ne1=2ene0φmevte21+ηZη1+CQ1+ηZη,where CQ=ħ2k3/4me2vte2k∥, which shows the quantum correction in the number density of electrons and Z(η) is the plasma dispersion function for cross-field streaming nondegenerate electrons [30] with argument η=(ω-k⊥V0)/k⊥vte.By using (6), the number density of hot and magnetized ions is(8)ni1=-ni0emi1k∥vti2∑nk∥vtiψ1+ξinZξin+nΩciφZξinInbie-bi,where In is the modified Bessel function with argument bi=k⊥2vti2/2Ωci2 and Z(ξin) is the usual dispersion function for a Maxwellian plasma with ξin=(ω-nΩci)/k∥vti.The perturbed number density of cold and unmagnetized dust grains by using hydrodynamic model is as follows:(9)nd1=nd0Qd0mdω2k⊥2φ+k∥2ψ.Since the nondegenerate electrons are streaming along thex-direction, therefore the longitudinal component of current density perturbation is taken to be zero; that is, Je1∥=0. The parallel components of current density for ions and dust species are (10)Ji1∥=-ni0e2Tik∥∑n1+ξinZξink∥vtiξinψ+nΩciφInbie-bi,Jd1∥=nd0Qd02mdωk∥ψ.Using the explicit expressions ofne1, ni1, and nd1 in (1) and Ji1∥, Je1∥, and Jd1∥ in (2) allows obtaining the following system of equations:(11)Aφ+Bψ=0,Cφ+Dψ=0,where(12)A=k⊥2+2ωpe2vte21+ηZη1+CQ1+ηZη+nωcik∥ZξinInbie-bi+k⊥2ωpd2ω2,B=k∥2+2ωpi2vti21+ξinZξinIne-bi-k∥2ωpd2ω2,C=c2k∥k⊥2+2ωpi2vti2nωΩcik∥1+ξinZξinIne-bi,D=k∥k⊥2c2+ωpd2-ω1λDi2∑nvti2ξinZ′ξinIne-bi.And by eliminatingφ and ψ from the homogeneous equation (11), the dispersion relation reduces to(13)AD-BC=0,which, after substituting the values of A, B, C, and D from (12) into (13), and some simple algebra, is given by(14)1+2ωpi2k∥2vti2∑n1+ωpd2k⊥2c2nΩcik∥vtiZξinIne-bi+1+ξinZξinIne-bi+2ωpe2k∥2vte21-ωpd2k⊥2c21+ηZη1+CQ1+ηZη+1+ωpd2k⊥2c2k⊥2k∥21-ωpd2ω2=0and represents the general dispersion relation of KAW streaming instabilities in a quantum dusty plasma. In the limit ωpd2/k⊥2c2≪1, we get the dispersion relation of modified two stream instabilities in a dusty plasma [31] and in a dust-free plasma the instability results are as in classical plasma [32].When the free energy associated with streaming is increased, that is,V0≫vte, the maximum growth rates also continue to increase and the dispersion equation for all plasma components becomes nonresonant (ξin,e>1). The unstable mode in a nonresonant regime is typically known as fluid instability. In the cold plasma limit,(15)A=k⊥2fiω2ω2-ωdlh2-ω2ωpe2/fiω-k⊥V02-CQk⊥2vte2,B=k⊥2fi-k∥2ωpd2ω2,C=c2k∥k⊥2,D=-k∥k⊥2c2+ωpd2,where ωdlh2=(ωpd2Ωci2)/ωpi2 and fi=ωpi2/Ωci2.Thus, from the dispersion relation (13), we obtain a biquadratic equation for ω in terms of various plasma parameters:(16)aω4+bω3+cω2+dω+e=0,where(17)a=α+k⊥2c2,b=-2k⊥3V0c2-2k⊥V0α-2k∥2VAi2k⊥V0,c=-ωdlh2-k∥2VAi2CQ-CQ-k⊥2V02-ωpe2fiα-k∥2VAi2ωpd2-CQk⊥2c2-k⊥4c2V02,d=2k⊥V0ωdlh2α+2k∥2VAi2k⊥V0ωpd2,e=CQ-k⊥2V02ωdlh2α-k∥2k⊥2V0VAi2ωpd2,where α=k⊥2c2+ωpd2.In the absence of streaming electrons the solution of (16) is given by (18)ω2=ωdlh2+2CQ+k∥2VAi21+k⊥2λd2,which is quantum corrected low frequency shear Alfvén wave in a dusty plasma, where VAi=B0/4πni0mi is the Alfvén speed and λd=c/ωpd is the dust skin depth. In the limit CQ=0, we obtain the dispersion relation of KAW in inertial regime; that is, ω2=ωdlh2+k∥2VAi2/1+k⊥2λd2. When CQ=0, k⊥2λd2≪1, we get the dust-modified dispersion relation of shear Alfvén waves, that is, ω2=ωdlh2+k∥2VAi2, which is a natural mode of any dusty magnetoplasma and ωdlh2 provides constraint for the electromagnetic wave propagation. Our analysis also indicates that cross-field streaming effects are not coupled with perpendicular wavenumber but appear as an additional term in the dispersion relation. The reason may be the absence of Je∥ due to streaming in cross-field direction and the Bohm potential induced Vlasov model. It is therefore tempting to introduce quantum correction which may significantly modify the classical modes and related instabilities.In order to observe the effect of dust grains on the wave in a low beta plasma, the limiting case ofk⊥2λd2≫1 can be obtained from (18) in the form(19)ω2=ωdlh2+ΩiΩdnd0Zdni0k∥2k⊥2.The second term on R.H.S can be recognized as the convective cell frequency which strongly depends on dust parameters. It can be recognized that dust inertia may play a major role in the wave dynamics, while the stationary dust excludes this mode. The heavy dust grains can be an important factor in diminishing the effect of wave frequency and unstable regions of propagation.


## 4. Results

In the presented work, we have investigated the cross-field streaming instability and dispersion propertied of KAW due to nondegenerate electrons in a quantum dusty magnetoplasma. We have derived the KAW instability growth rates using various parameters close to dense astrophysical plasma, that is,ne=1026, ni=1.001×1026, Zd=103, nd0=0.3×1023, and B0=1013G [33]. The influence of quantum Bohm potential is found to act against the KAW instability, while classical effects reinforce the unstable regions, as depicted in Figure 1. The cross-field nondegenerate electron beam streaming with velocity V0 helps in the growth of unstable regions and a further increase inhibits the stabilization which maintains the wave amplitude as illustrated in Figure 2. It has also been observed that quantum parameter CQ is inclined to suppress the instability. In Figure 3, we can observe that the growth rates are enhanced with the increase in number density, followed by increase in cross-filed ion streaming velocity. Physically, when the dust concentration is increased, it also enhances the depletion of electrons due to attachment to the dust grain surface, which in turn increases the streaming velocity and ultimately is responsible for the excitation of an electromagnetic wave. The role of dust charge is also found to enhance the wave activity which can be seen in Figure 4. Our results indicate that KAWs in inertial regime are less affected by the Bohm potential effects and hence CQ appears to suppress the Alfvén wave frequency in a hot and magnetized nondegenerate quantum plasma which is evident from Figure 5.Figure 1
Contour plots betweenk~⊥ and CQ for V0=10, Zd=103, and nd=10-8ni.Figure 2
Contour plots betweenk~⊥ and V0 for CQ=10, Zd=103, and nd=10-8ni.Figure 3
Plots betweenk~ and nd for V0=10, CQ=10, and Zd=103.Figure 4
Effect ofZd on the dispersion characteristics.Figure 5
Effects ofCQ on the real part of the dispersion relation.To summarize, we have investigated the effects of quantum contribution on the growth and dispersion of low frequency kinetic Alfvén wave in inertial regime by using Vlasov-Maxwell equations. We have neglected the quantum effects of ions as they are heavier than electrons. It is found that quantum effects suppress the instability in a hot and magnetized plasma. The density fluctuations are not accompanied by pure electromagnetic nonstreaming dense plasma; therefore, applying quantum corrections makes sense to electromagnetic waves when density fluctuations are involved especially in the inertial turbulent range. We therefore have recast the perturbed nondegenerate electron density which has a strong dependence on quantum correction parameterCQ. Our analysis is based on linear approximation and we believe that we have contributed to the analysis of KAWI in a nondegenerate quantum plasma and there is a large set of nonlinear investigations which should be addressed. The present analysis could be applied to dense astrophysical streaming and ICF plasmas where quantum diffraction effects are nonnegligible and significant.



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