Gravitational Drift Instability in Quantum Dusty Plasmas

1 Introduction

Dusty or complex plasmas contain three species: electron, ions, and tiny solid particles called dust. The dust particles collect charges due to many processes like ultraviolet photons and plasma currents [1], [2], [3]. The corresponding mechanism, for example, for the mutual interaction of particles and the particle-plasma interactions evolve the physics of complex plasmas. The presence of dusty plasmas are probed in the astrophysical objects, the medium between planets, the planetary rings, the Earth’s plasmas, the fusion reactors, and the comet tails [4], [5]. The plasma engineering in the manufacturing industry has developed an interest among the plasma community to search for the processing of dusty plasmas [6], waves in dusty plasmas [7], wave excitation in plasma crystals [8], [9], liquid and solid state dusty plasmas [10], [11], [12], etc. The dust particulates become highly charged because of the possession of electrons in dense plasmas, whereas in the atmosphere of space plasmas, it becomes either less negative or positive because of secondary electron emission and photoemission [13], [14]. The eaten up electrons in the dust particles affect the mobility of plasma species. Depending upon the time scales, the charge of dust becomes a dynamical variable that defines a new phenomenon in the dusty plasma systems [15], [16]. In addition, such plasma systems host different kinds of low-frequency wave instabilities, for example, dust ion-acoustic waves [17], dust-acoustic waves [18], and dust lattice waves [19], [20].

Self-gravity and gravitational drift play a fundamental role in both magnetized and unmagnetized plasmas. The presence of electric and magnetic fields across the plasmas at equilibrium tends to modify the usual Larmor gyration by the addition of a guiding center drift [21]. Thus, the E0×B0 drift takes charge particles away from the line of force, which is a serious problem in plasma confinement losses. The centrifugal force is the basis of gravitational instability that arises in the absence of weak electric field. Several researchers have accounted a huge amount of work to study the variety of instabilities resulting from different drifts in the complex plasma. For example, Salimullah et al. [22] studied the drift waves and associated instabilities due to density inhomogeneity in quantum complex plasmas. Ren et al. [23] pointed out the significant instability of electromagnetic drift waves in e, p, and i quantum plasmas. Misra [24] did a study on the amplitude modulation of drift waves arising from nonuniformity in quantum plasmas. It is noticed that a very small amount of work has so far been done to investigate the effects of the g0×B0 drift motion of the micron-sized charged dust grains in the dense plasma environment.

Quantum plasmas have been regarded as the active research area for plasma researchers for the past decades owing to the importance of explaining many phenomena regarding microelectronics, nanostructure materials, dense astrophysical environment, and laser-produced plasmas [25], [26], [27], [28], [29], [30], [31], [32]. The description of quantized complex plasma can be formulated by adopting several approaches. Quantum magnetohydrodynamics incorporates several forces acting on the plasma species like quantum effects of Fermi degenerate electrons, Bohm potential due to tunneling of quantized particles, and Lorentz force.

This work is motivated from Salahshoor and Niknam [33], where the effect of gravitational drift in a collisional magnetoplasma is studied in the classical ionospheric environment. Here, in the current study, the same working model is extended to the quantum hydrodynamic plasma system where quantum effects contribute through the statistical Fermi pressure and the tunneling potential. These characteristics are only associated with electrons existing in the dense environment like white dwarf. Moreover, [33] deals with the fixed parameter of gravity, i.e. g = 981 cm s⁻², with its application only in the Earth environment, whereas we have taken the generalized g depending upon the density of the plasma system for the dense astrophysical objects. The modified dispersion relation of electrostatic waves is derived depicting the significant contribution of quantum characteristics that was never probed yet before. The effects of strong magnetic field, weak electric field, and quantum effects through Fermi degenerate pressure and Bohm potential are considered. We demonstrated that the E0×B0 drift dynamics of the plasma species is the main source of instability progression; however, because of weak electric field g0×B0, the drift motion of the dust particulates is the source of generating unstable waves in the low-frequency limit. This paper is organized in four sections. In Section 2, the main equations of the multi-fluid model are solved to derive the dispersion relation of the electrostatic waves. In Section 3, the dispersion relation of the gravitational drift instability is analyzed, and two cases are discussed under various approximations. Finally, in Section 4, a numerical analysis of the electrostatic g0×B0 drift instability is depicted in graphical representation, and a summary of the results is presented.

2 Analytical Model of the Dispersion Equation

We take homogeneous high-density quantum magnetoplasma [33], [34], [35] containing electrons, ions, and uniformly charged dust grains in the presence of a homogeneous static ambient magnetic field B0∥z^. Such plasmas exist in the interiors and environments of astrophysical compact objects, e.g. white dwarfs and neutron stars/magnetars, and supernovae [36], [37]. We confine ourselves to an orthogonal configuration in which g ⊥ B. The electrostatic wave is taken to be propagated perpendicular to these fields. Thus, a complex plasma system is considered in the gravitational field g0=g0x^ and in a weak electric field E0=E0x^. The linearized equations of the quantum magnetohydrodynamic model for the electrons, ions, and charged dust grains (j=e,i,d) in the presence of the ambient magnetic field B₀ are

where ℏ=h/2π and qj,mj,n0j,ϕ, and c are the charge, mass, equilibrium number density of the j^th species, total electrostatic potential, and the velocity of light in a vacuum, respectively. Here, qe=−e,qi=e, and qd=−zde, with e being the magnitude of electronic charge. Here, g0=(43πGρR) is the gravitational field strength where G is the universal gravitational constant, ρ is mass density, and R is the radius of the plasma system. In (1), we assume the plasma particles moving in a three-dimensional Fermi gas satisfying the pressure law, pj=mjvFj2n1j33n0j, where vFj2=65TFjmj{1+512π2(TjTFj)2} is the Fermi speed and k_B, TFj, T_j, and n1j are the Boltzmann constant, Fermi temperature, thermal temperature, and the number density with its equilibrium value n0j, respectively. Equation (1) contains a term with ℏ called Bohm or tunneling potential. The tunneling potential can significantly contribute in the comparison of Fermi pressure only for super cool dense plasmas where number density of plasmas is extraordinarily high; however, for the astrophysical objects where number density is of the order of 1027 cm⁻³, the Fermi term dominates over the tunneling potential. The thermal temperatures of ions and dust are small and are therefore ignored. At equilibrium, the charge quasi neutrality condition is satisfied, that is, n0i=n0e+(qd/e)n0d. The unperturbed velocity components of the particles can be derived from the zero-order equilibrium solution of the momentum equations as follows:

where ωcj=qjB0/mjc is the cyclotron frequency of the j^th species, qd=−zde is the dust charge, z_d is the number of electrons, and e is the magnitude of electronic charge. The drift motion of particles in the y direction is due to the cross products of external fields. For electrons and ions, the drift motion is a E0×B0 type, but for dust, the g×B0 drift contributes to the cross-field motion additionally. We assume that the perturbed quantities are proportional to exp(−iωt+iky), where ω and k are the frequency and the wavenumber of the mode under study. We suppose that the wavevector lies in the y direction, i.e. k=ky^.

Using (1)–(3) and after some straightforward calculations, the perturbed velocities and number densities of j^th species are

Here, we noted that v1jz=0,

Here, vFGj′2={KBTjmj(1+ℏ2k24kBTjmj)+ig0ky} and ω∗=ω−v0j⋅k. On comparing (10) with the standard expression of the perturbed number density of dielectric medium, i.e. n1j=−χjky2ϕ14πqj, we obtain the dielectric susceptibility for the j^th species as

where ωpj=4πn0jqj2mj is the j^th plasma frequency. We can use (11) to find the dielectric response function of uniform quantum dusty magnetized plasma under various possible conditions from

3 Gravitational Drift Instability

Now, we analyze the gravitational drift instability of micron-sized dust particles in a magnetized gravitating dense plasma in which quantum effects of electrons significantly affect the characteristics of the propagating modes. Here, we assumed such parameters of the plasma system in which dust grains are magnetized as well as affected by gravity. Dust grains with radius of 1 μm at a gas pressure smaller than 1 m Torr can be magnetized by a magnetic field strength of more than 2 T. Dust grains with a radius of 1 μm surely feel the gravity particularly at such a low-pressure regime. Hence, gravitational effects are only considered for heavier dust grains, while the quantum effects on ions and dust grains are neglected because of their heavier masses. Also, being insignificantly small, we can neglect the gravitational effects on electrons and ions. Under these considerations, the dielectric susceptibilities can be obtained from (11) as

where vGd′2=ig0ky. The dispersion relation in the homogeneous quantum dusty magnetoplasma is given as

Here, we ignore streaming of electrons and ions at equilibrium and made the approximation of low-frequency electrostatic wave propagation as, for electrons, (ω−v0e⋅k)2≪(ωce2,vFe′2ky2), for ions, (ω−v0i⋅k)2≪ωci2, and for unmagnetized dust, ωcd2≪(ω−v0d⋅k)2.

Equation (17) is the main result of this study, which elaborates the competition of the E0×B0 drift and the g0×B0 drift for the propagation of modified electrostatic wave. The standard expression of electrostatic wave is ω=csk, where c_s represents the acoustic speed of the wave due to multi-drifts. It is noted that the gravitational field drift does contribute significantly for only the massive dust species, whereas the electric field drift modifies the dynamics of electrons and ions in the quantum plasmas. The dispersion relation is a complex root, that is, ω=ωr+iγ. The ω_r is real part of the ω that describes the phase speed in the presence of two kind of drifts, while the γ reflects the damping or growth depending upon the sign, negative or positive.

Figure 1:

Real part of the wave frequency (ω_r) as a function of wavenumber (k) for different values of electron number density. Bold line for n0e=0.1×1027 cm⁻³, dotted for n0e=0.3×1027 cm⁻³, and dashed-dot for n0e=0.5×1027 cm⁻³.

Figure 2:

Imaginary part of the wave frequency (γ; growth rate of the drift wave instability) as a function of wavenumber (k) for different values of electron number density. Bold line for n0e=0.1×1027 cm⁻³, dotted for n0e=0.3×1027 cm⁻³, and dashed-dot for n0e=0.5×1027 cm⁻³.

Figure 3:

Real part of the wave frequency (ω_r) as a function of wavenumber (k) for different values of external magnetic field. Bold line for B0=1.2×105G, dotted for B0=1.25×105G, and dashed-dot for B0=1.3×105G.

Figure 4:

Imaginary part of the wave frequency (γ; growth rate of the drift wave instability) as a function of wavenumber (k) for different values of external magnetic field. Bold line for B0=1.2×105G, dotted for B0=1.25×105G, and dashed-dot for B0=1.3×105G.

4 Numerical Results and Conclusion

Equation (17) describes the analytical studies of gravitational drift instability in quantum dusty magnetoplasmas. The drift instability can be studied over a wide range of astrophysical objects by selecting an appropriate set of parameters in the model. It is observed that gravity plays a dominant role in the dynamics of micron-sized dust particles. For a graphical explanation of the dispersion relations of the dust gravitational drift wave in quantum plasmas, we have plotted ω as a function of k, (17) for the following typical parameters in interstellar and magnetospheric environments: for example, white dwarf for which ρ=1×106 g cm⁻³, R=7×108 cm, B0=105G, mi=mp, zd=n0i−n0en0d, qd=−zde esu, n0e=1027 cm⁻³, n0i=1.001×1027 cm⁻³, n0d=10−6×n0i cm⁻³, TFe=(3π2n0e)23ℏ2(2mekB), and mdmi=109 [38], [39], [40]. The standard values of m_e, m_p, Planck constant, Boltzmann constant, gravitational constant, electron charge, and ion charge are used in the cgs system. It can be seen from Figures 1–4 that the drift motion of the particles considerably alters the nature of the dispersive properties of the system, which results in an unstable wave with growth rate in the system. Thus, in turn, a drifting plasma species becomes the cause of the generation of free energy in the system that results in a wave instability. Figures 1 and 2 represent the real and imaginary plots of the wave frequency as a function of wavenumber, respectively. Figure 1, describes the phase speed at different values of electron number density. At the n0e=0.5×1027 cm⁻³, the phase speed reflects a very small increment for a small wavenumber; however, for values at higher k, phase speed increases sharply. At n0e=0.3×1027 cm⁻³ and n0e=0.1×1027 cm⁻³, the threshold frequency shifts to higher k. However, for the whole range of the wavevector spectrum, the phase speed increases with the same pattern. Figure 2 describes the growth rate of the drift wave instability. The growth rate increases as we increase k at n0e=0.5×1027 cm⁻³, and the maximum growth rate is achieved at k between 2000 and 2500 cm⁻¹. Physically with the increasing wavevector spectrum, the g0×B0 drift enhanced the E0×B0 drifts, and maximum growth is obtained. Beyond the maximum instability, growth rate decreases because of the weak resonance of the two drifts. Figures 3 and 4 describe the phase speed and growth rate of drift instabilities for a gradual increase in the magnetic field with a fixed value of number density B0=1.2×105G to B0=1.3×105G. Figure 3 shows the same trend of phase velocity of the drift wave as depicted in Figure 1. As we increase the magnetic field, the threshold value of the phase velocity of the wave decreases. Figure 4 shows that the growth rate linearly increases as we increase the wavenumber k. However, for a fixed value of k, as we increase the value of the external magnetic field, the growth rate also increases. The higher magnetic field drives the g0×B0 drift instability to the maximum limit. In summary, we studied quantum dusty magnetoplasmas under gravity by taking into account the effects of weak electric field and quantum mechanical effects arising from Bohm potential and Fermi degenerate pressure for plasma electrons only. We carried out the linearized analysis using the quantum hydrodynamic model and derived the susceptibilities of complex plasma species for the electrostatic wave propagating perpendicular to the external fields.