Subsonic Potentials in Ultradense Plasmas

1 Introduction

Ultradense plasmas have attracted plasma physicists because of their plentiful existence in various environments, such as the interior of compact astrophysical objects like white dwarf, neutron stars and Jovian planets (Jupiter and Saturn), as well as in solid-state configuration. There are many experimental phenomena, for example, the superluminal mode, which can be explained only by taking the relativistic effects [1], [2], [3], [4], [5]. The dense astrophysical objects are super compressed in such a way that Fermi levels are completely filled and electrons have no freedom to change their energy levels. The comprehensive study of degenerate plasmas in the dense astrophysical objects tends to unite relativistic and quantum mechanics [6], [7], [8], [9], [10]. Few examples of dense plasma systems are white dwarfs and neutron stars, which have the number density of order 10³⁰ cm⁻³ or even more. The corresponding Fermi energy is comparable with the electron’s rest mass energy [11], [12], [13]. The number density of plasma species increases even beyond 10³⁶ cm⁻³ when probed from the outer layer to the interior of neutron stars, whereas the energy exceeds the rest mass energy; more intrinsically, the relativistic effects appear [14]. The plasma scientists used a one-dimensional quantum hydrodynamic (QHD) model to study the influence of relativistic effects on the modulation instability of linear and non-linear waves in e-p-i plasmas [15]. There are many situations where quantum mechanics joins with the relativistic mechanics, such as the mechanisms of laser-matter interaction experiments and the evolution of the universe. Many researchers presented the relativistic and weak relativistic effect on the ion-acoustic shock waves in quantum plasmas [16], [17]. The instability comparison of three regions, namely, a Langmuir-type mode, a low-frequency ion acoustic mode and an ion-beam-driven mode, was studied by Elkamash et al. [18] in the relativistic plasma regime. Mendonca [19] described the kinetic model of relativistic quantum plasmas equivalent to that of Klein-Gordon and elaborated the condition of Landau damping. The Friedel oscillatory behavior associated with the static screening of test charge in a two-component relativistic quantum plasma across the external magnetic field was presented by Sivak [20].

Despite such quality work covering different aspects of relativistic quantum plasmas, there is much room for substantial work in the field of wake potential. The objective of this work is to fill the void of dynamic potential associated with the test charge in relativistic quantum plasmas. As far as the dynamic potential is concerned, when the speed of the test charge is comparable with the phase speed of the plasma mode, a wake field is formed behind it. However, it is proven experimentally that the dynamic potential may exist in either subsonic or supersonic scheme [21]. As a wake potential application, we can describe the fluid crystal formation and the origin of attractive force among the same polarity species. Nambu and Akama [22] were the pioneers who gave the concept of wake field. For decades, the researchers have shown their interest in this field whereas the experimentalists have demonstrated the formation of microscopic crystals and the particulate coagulation based on wake potential evolved behind the test particle [23], [24], [25]. Kroll et al. [26] reported the experimental work describing the role of wake fields on the three-dimensional particle arrangement in finite dust clouds. Khan et al. [27] studied wake potential in a magnetized semiconductor quantum plasma. The amount of wake potentials is important; it can be understood from the fact that theoretical and experimental studies have been extended to the computer simulation [21], [28], [29]. In this paper, we have studied the electrostatic wake patterns around a moving test charge by the dispersion equation of the acoustic mode obtained from the QHD model [30], [31] and the Poisson equations in a magnetized plasma B0z^. The format of the paper is as follows: in Section 2, the dielectric response function is obtained by using the quantum hydrodynamic fluid equations and the Poisson equation. The modified dielectric function is used for the derivation of the wake potential. The numerical analysis of the wake potential and the summary of results are given in Section 3.

2 Relativistic Dielectric Function

We consider a homogeneous, collisionless magnetized quantum plasma composed of electrons and ions in the presence of static ambient magnetic field B0z^. In this system, we consider a test charge particle q_t with velocity v_t parallel to the z-axis, which excites the low-frequency hybrid mode that nearly approaches the ion acoustic time scales. At equilibrium, the charge neutrality condition reads as ∑jqjnj0=0, where q_j is the charge of the jth species, j stands for electrons and ions and n_j0 is the density at equilibrium. The governing equations for the thermal ions in the presence of ambient magnetic field are as follows:

Here, B₀ is the ambient magnetic field, v_i is the total velocity of ions and Pi=γKBTini is the statistical thermal pressure for n_i, the total ion number density such that ni=ni0+ni1, n_{i ⁢ 1} and n_i0 are the perturbed and equilibrium number densities of ions, respectively, T_i is the thermal temperature of ions and K_B is the Boltzmann constant. After applying the Fourier transform, the linearized (1) becomes

In (2), vti=(γKBTimi)0.5 is the thermal speed of ions, ωci=eB0mic is the cyclotron frequency of ions in cgs units, c is the speed of light, k is the propagation vector of wave, ω is the angular frequency of wave and E=−(∇⊥+∇∥)ϕ is the perturbed electric field where ϕ is the perturbed electrostatic potential. Now, the linearized continuity equation is

Equation (3) derives the following equation after Fourier transform in reciprocal quantities:

The straightforward calculations after combining (2) and (4) give the perturbed velocity of ions in terms of ϕ, that is,

Here, subscripts ∥ and ⊥ represent the corresponding parallel and perpendicular components of quantities, gi=1−k2vti2ω2=1−k2Csf2ω2τi, Csf2=TFemi, τi=TiTFe, Csf2 is the square of sound speed, τ_i is the normalized temperature and the wavevector k2=k∥2+k⊥2. T_i and T_Fe are ion thermal and electron Fermi temperatures. On substituting (5) into (4), the perturbed number density for ions is

The dynamics of quantized electrons containing momentum and relativistic pressure is given by the following:

At a very high number density, the statistical pressure dominates over the Bohm potential because the quantum characteristic of the de Broglie length defined with the Fermi speed in non-thermal degenerate plasmas becomes very small. In (7), n_e is the total number density, p_e is the total momentum, v_e is the total velocity of relativistic electrons and the Fermi pressure of the electrons in relativistic regime P_Fe given in [32], [33] is defined as

where m_e is the rest mass of electrons. After linearizing for inertia-less electrons, (7) turns to

where μe=(3π2ne0)2/3ℏ2c2+me2c4 is the chemical potential for electrons and me∗ is the relativistic mass of electrons. The density-dependent Fermi momentum of the electron gets its form pFe=(3π2ne0)1/3ℏ, and the Fermi temperature is TFe=(3π2ne0)23ℏ22me in the energy units. Applying sinusoidal behavior ei(k.r−ωt) on (8), the perturbed number density of electrons in relativistic regime is

The Poisson equation in order to explore the electrostatic potential ϕ(x, t) of the electrostatic perturbation in space and time frame due to linearized charge species is

By substituting (6) and (9) into the Poisson equation, we obtain

We apply the Fourier transform and the neutrality condition ne0≃ni0 and perform simple calculations to get

Hence, the dispersion equation from (12) can be written as

Equation (13) is the dispersion relation of an electrostatic wave in the presence of uniform external magnetic field. The notation kDFe2 is substituted for 1/λDFe2, where λDFe=TFe4πe2ne0 is the effective Fermi length. Also applying the condition of ion acoustic wave k∥2≫k⊥2,

Here, ωk2=ωpi2(1+kDFe2μeTFek2(μe2−me2c4)+ωpi2 K⊥2k2ωci2) and Csf2 KDFe2=ωpi2. From the first part of (14), we will get static shielding, and from the second part, a modified wake potential is obtained. The modified potential of test charge in the relativistic quantum plasmas can be obtained by using the dielectric constant from (14) into the formula given by Krall and Trivelpiece [34] and Nambu and Salimullah [35],

where r=x−vtt, v_t is the velocity vector of the test particulate, and q_t is its charge. By using (14) into (15), we get the total potential ϕ=ϕc+ϕw, where ϕ_c is the coulomb potential and ϕ_w is the wake part. The dynamic wake potential can be solved by substituting the second part of (14) into (15),

where Γ=k⊥2(1+β)+kDFe2μeTFe(μe2−m0e2c4) and β=ωpi2ωci2. For the speed of test particle v_t parallel to the z-direction, k.vt=k∥vt and the cylindrical coordinates give k.r=k⊥ρcosθ+k∥ξ. The ω integration obeys the general integral property of ∫δ(x−x0)f(x)=f(x0), which can be applied on replacing simply ω by the k∥vt [27].

For the ion acoustic wave propagating at a small angle with the z-axis, the spatial scale approaches the acoustic scale so that k∥2≫k⊥2, which implies that k2≃k∥2. Similarly, θ integration is performed following [27], and hence,

where α=ωpi2vt2−Γ2. The subsonic wake (at small v_t) of the test particle restricts that α should be positive, which is only possible for ωpi2vt2>Γ2. The wake potential is associated with the motion of test particle; however, the lower speed of test particle corresponds to the shielding potential as well in addition to the wake. Henceforth, (14) may contain both the coulomb part and the wake potential defined at the poles k∥=±iΓ and k∥=±α. The singularities at k∥=±iΓ provide the static or coulomb part [30], whereas the poles k∥=±α provide oscillating wake potential. The wake potential gets its form: ϕ(ρ,ξ)=−∫qtπvt2ωpi2J0(k⊥ρ)k⊥2παsin(αξ)dk⊥. The long wavelength condition k⊥ρ≪1 implies that ρ ≃ 0 and J0(0)=1, and after some substitution of η=ωpi2vt2−kDFe2μeTFe(μe2−m0e2c4), χ=1+β,

After applying the limit (0, 1) [36], [37], we finally get

Equation (19) is the main result of our analytical studies, about the modified dynamic potential in quantum relativistic magnetoplasma.

Figure 1:

Relationship of the wake potential ϕ [StatV] vs. ξ [cm] with varying B₀ [G].

Figure 2:

Relationship of the wake potential ϕ [StatV] vs. ξ [cm] with varying v_t [cm/s].

Figure 3:

Relationship of the wake potential ϕ[StatV] vs. ξ [cm] with varying n_0e [cm⁻³].

3 Numerical Results and Conclusion

We know that the threshold of degenerate plasmas is defined with the comparison of the de Broglie length λB=ℏ/(mkBT)1/2 with the interparticle distance d=n−1/3, where ℏ,m,kB,T and n are the Planck constant, the mass of plasma particle, the Boltzmann constant, the plasma temperature and the plasma density, respectively. For quantum plasmas, the condition of nλB3=1 should be obeyed. The electron density in the space plasmas in the order of 10²⁵ cm⁻³ falls in the discipline of dense plasmas, whereas 10²⁸ cm⁻³ and onward ultradense satisfy the quantum condition. The number density of electrons in the Fermi sphere with Fermi radius in terms of momentum p_F = (3π²n)^(1/3) can be in the order of 1029(pFm0c)3 cm−3. In this case, when relativistic factor pFm0c approaches unity, the plasma is referred to as relativistic that, in turn, gives the density threshold in the order of 10²⁹ in relativistic ultradense plasmas, whereas ≤1 is the semirelativistic and ≫1 is the ultrarelativistic [12], [18], [33]. At pFm0c≃1, the analytical solution given in (19) is numerically analyzed for the given typical parameters in cgs system of units, n0e=5.9×1029 cm−3, B0=(1.0−1.2)×107 G, vt=3.3×108 cm/s and T_i = 300 K [9], [30], [33]. The dependence of ϕ(ρ=0,ξ) on magnetic field B₀, the velocity of the test particle v_t and the number density n_e0 are depicted in Figures 1–3 for particular parameters of astrophysical regions (white dwarfs). The effect of magnetic field is shown Figure 1, at B0=1.0×107 G (solid), B0=1.1×107 G (dashed) and B0=1.2×107 G (dotted), with all other variables kept the same. In Figure 1, the peak of the wake potential increases with the increasing magnetic field, which strengthens the concept proven by Nambu and Salimullah [35]. Physically, by increasing the magnetic field and keeping the density the same, the plasma species are kept bounded to confine the maximum energy; thus far, the amplitude of wake is increased. This behavior is analogous to the strong interaction between the plasma species that in turn enhances the potential amplitude, i.e. there is a strong attractive force among the particles of same polarity but at the same distance from the test particle. For the particular values given above, Fermi temperature is calculated as TFe=2.98114×109 K and Csf=5.07073×108 cm/s as well. Implying the explicit condition for the wake potential, the speed of the test particle has been chosen, which cannot be a big difference from the sound speed. Figure 2 shows the dependence of wake on the speed of the test particle at vt=3.3×108 cm/s (solid), vt=3.5×108 cm/s (dashed) and vt=3.7×108 cm/s (dotted). It manifests that by increasing the speed of the test particle, but less than the sound speed, the amplitude of the wake potential is decreased. In addition to the amplitude, there is a phase shift as well. This behavior is analogous to some weak force existing among the particles observed at some large distance. By contrast, when the speed of the particle is reduced, there is a strong force that exists among the particles that have large spacing, and wake potential is increased but not realized at a large distance. Figure 3 elaborates the behavior of wake potential by varying the number density for ne0=5.9×1029 cm−3 (solid), ne0=5.5×1029 cm−3 (dashed) and ne0=5.1×1029 cm−3 (dotted). Physically, increasing the density increases a strong attractive force among the particles, which enhances the existence of a higher potential at some small length scales, as shown in Figure 3. By decreasing the density, the amplitude of the wake potential is decreased, but its effects are realized at some large distance. In summary, we have studied the wake potential analytically and numerically for the relativistic quantum magnetoplasmas. The ions are non-relativistic and classical, whereas electrons are taken relativistically and quantized. It is observed that the wake potential is significantly affected by the external magnetic field B₀z, the speed of the test particle and the electron number density at the quantum scales. For relativistic degenerate system, the dynamic potential prevails only in the subsonic zone, which is the vital outcome of this paper. The wake potential has its application in the crystal formation in the plasmas even with the like charge polarity.