Quantum Space Charge Waves in a Waveguide Filled with Fermi-Dirac Plasmas Including Relativistic Wake Field and Quantum Statistical Pressure Effects

1 Introduction

The wave scattering and propagation in plasmas have been of a great interest since the interaction between waves and plasma system is ubiquitous and also provides useful spectroscopic information in many areas of physics such as astrophysics, atomic physics, plasma physics, and space sciences [1], [2], [3], [4], [5]. In addition, the propagation of surface plasma waves has been comprehensively prospected on plasma-vacuum or plasma-dielectric interfaces of semi-bounded or bounded plasma systems as applications in various physical sciences such as materials science, nanotechnology, plasma spectroscopy, and fusion science [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]. In conventional weakly coupled classical plasmas, the Yukawa-type screened Debye–Hückel model illustrated by the linearization of Poisson’s equation with the Maxwell–Boltzmann distribution function has been widely applied to express effective screened interaction potentials since the average value of the kinetic energy of a plasma particle is typically greater than the interaction energy between particles in plasmas [16]. However, it is shown that the Debye–Hückel model in weakly coupled plasmas is quite different from the screened pseudopotential model in dense strongly coupled plasmas due to the non-ideal collective correlations and quantum-mechanical effects in dense semi-classical and quantum plasmas [17]. In recent years, the physical and chemical characteristics of dense quantum plasmas have been investigated since the quantum plasmas can be found in many areas such as nano-wires, laser produced strongly coupled plasmas, quantum dot, quantum well, semiconductor devices as well as astrophysical environments in atmospheres of compact objects [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29]. Very recently, a theoretical investigation [30] on the propagation of positron-acoustic shock waves in an unmagnetised, collisionless, and dense plasma is carried out by employing the reductive perturbation method. A theoretical investigation [31] is also carried out for the properties of ion-acoustic solitary waves and double layers in a four-component magnetised degenerate quantum plasma system by the reductive perturbation method. In addition, a theoretical investigation [32] on the non-linear propagation of non-planar electrostatic modified ion-acoustic shock structures is carried out in an unmagnetised, collisionless four-component degenerate plasma system. Recently, the non-linear propagation of cylindrical and spherical modified ion-acoustic waves is investigated in an unmagnetised, collisionless, relativistic, degenerate multi-species plasma [33]. Moreover, a theoretical investigation [34] is carried out to investigate the existence and basic features of the ion acoustic shock structures in an unmagnetised, collisionless dense plasma system composed of degenerate electron and ion fluids, and charged heavy ions. In dense quantum and semi-classical plasmas, it is known that the quantum-mechanical effects including the Bohm potential, quantum pressure, and electron-exchange terms play important roles in the formation of effective interaction potential [24], [25]. Moreover, very recently, it is shown that the influence of quantum statistical degeneracy pressure plays crucial role in the formation of the plasma dielectric function in degenerate Fermi-Dirac quantum plasmas [28]. However, the effects of quantum statistical degeneracy pressure on the propagation and the stability of surface waves in a cylindrically bounded degenerate quantum plasma have not been investigated as yet. It would be then expected that the conditions for the propagation and instability modes of quantum surface waves in Fermi-Dirac quantum plasma waveguides are quite different from those in conventional plasma waveguides due to the influence of the quantum statistical degeneracy pressure. Therefore, in this paper, we investigate the influence of quantum statistical degeneracy pressure on the propagation of the quantum space charge wave in a cylindrically bounded plasma waveguide filled with relativistically degenerate quantum Fermi-Dirac plasmas and the relativistic ion wake field. Hence, we can expect that the propagation of quantum plasma waves on the plasma-vacuum interface would provide useful information on the spectral information of quantum surface waves as well as the geometrical effects of the configuration of quantum plasma systems. We obtain the analytic presentation of the dispersion relation of quantum space charge waves in a plasma waveguide filled with relativistically degenerate quantum Fermi-Dirac plasmas since the dispersion properties of the space charge waves are useful for understanding the physical properties of quantum Fermi-Dirac plasmas and the physical characteristics of the wave-particle interaction due to the relativistic ion wake field. We also discuss the variation of the dispersion properties of the quantum space charge wave due to the quantum and geometric effects in a degenerate quantum Fermi-Dirac plasma waveguide.

2 Theory and Calculations

In this section, we discuss the quantum Fermi-Dirac plasmas including the generalised quantum statistical degeneracy pressure effects and the cylindrically bounded plasma waveguide filled with Fermi-Dirac plasmas and the relativistic ion flow. Based on the quantum hydrodynamic (QHD) analysis [28], the continuity equation and the quantum Euler equation for the j species of plasma particle in degenerate quantum Fermi-Dirac plasmas, i.e. j=e for electrons and j=i for ions are, respectfully, given by

where n_j(=n_j0+n_j1) is the number density, n_j0 is the unperturbed equilibrium density, n_j1 is the density perturbation, v_j(=v_j0+v_j1) is the velocity, v_j0 is the unperturbed velocity, v_j1 is the velocity perturbation, m_j is the mass of the plasma particle, ħ(=h/2π) is the rationalised Planck constant, and φ is the electric potential obtained by the Poisson equation:

In (2), the last term ∇(∇2nj/nj) is known as the Bohm potential expression that stands for the quantum tunneling and wave packet spreading in quantum plasmas. In dense quantum or semi-classical plasmas, it is shown that the Bohm potential term leads the propagation mode with the dispersion relation: ω(k)=ωpe(1+k4λq4)1/2, where λq(≡ℏ2/4me2ωpe2)1/4 is the quantum wave length [20]. The term containing P_G(=P_Cou+P_XC+P_Deg) in (2) represents the generalised electron quantum statistical degeneracy pressure effect [28], [29] including the Coulomb P_Cou(R₀), the spin-exchange P_XC(R₀), and the relativistic degeneracy P_Deg(R₀) pressure terms represented by

and

where R(R₀)[(≡p_Fe/m_ec)=R₀(n_e/n₀)^1/3] is the relativity parameter, p_Fe is the Fermi momentum, R₀[≡(n₀/n_c)^1/3] is the relativistic degeneracy parameter, n_c(≅5.9×10²⁹ cm⁻³) is the critical electron density, α_f[(=e²/ħc)≈1/137] is the fine structure constant, e is the elementary electron charge, c is the speed of light, Z is the ion charge number, and β(R0)≡R(R0)+1+R2(R0). If the direction of the relativistic ion wake field is parallel to the axial direction of a plasma waveguide, i.e. v_i0(=u_iẑ), the longitudinal component of the quantum-mechanical Akbari–Moghanjoughi (A–M) plasma dielectric function ε_AM(ω, k, R₀) [28] in quantum Fermi-Dirac plasmas including the generalised quantum statistical degeneracy pressure effects is then represented by

where k(=|k|=kz2+k⊥2) is the wave number, k_z(=k_∥) and k_⊥ are the parallel and perpendicular components of the wave vector, ω is the wave frequency, H[(=ħω_cp/2m_ec²)≅0.0279087] is the quantum parameter, ωcp(=4πnce2/me) is the critical plasma frequency, ωpj(=4πnj0qj2/mj) is the plasma frequency of species j, ΓL(ui)(=1/1−ui2/c2) is the Lorentz factor, and the parameter T(R₀) determined by the generalised quantum statistical degeneracy pressure effects is represented by

In order to describe the cylindrically confined degenerate quantum Fermi-Dirac plasma, we assume that the radius of the cylindrical wave guide is R and the sufficiently long along the z-axis in cylindrical polar coordinates (ρ, θ, z). In the cylindrical coordinate system, we can set the Laplacian ∇² as ∇2=∇⊥2+∂2/∂z2 in with the transverse Laplacian ∇⊥2 and the all perturbation quantities n_j1(r), v_j(r), and φ(r) as

where n̅_j1(ρ), v̅_j(ρ), and φ̅(ρ) are the perturbation quantities in the perpendicular x-y plane [10] since the field fluctuation is proportional to exp(−iωt). After some mathematical manipulations using (1), (2), (3), (7), (8), and (9) with the azimuthally symmetric condition, i.e. ∂/∂θ=0 and ξ=0, with the longitudinal dielectric function ε_zz=ε_A−M(k_z, ω, R₀), the equation for the potential is given by

where the parameter χ²(k_z, ω, R₀) is given by

Since (10) is Bessel’s equation of order 0 and has also a regular singular point at the origin ρ=0, the general solution of (10) is then found to be φ̅(ρ)=CJ₀(χρ)+DN₀(χρ), where J₀(χρ)(≈1−χ²ρ²/4+⋯) is the zeroth-order cylindrical Bessel function, N₀(χρ)[≈(π/2)(ln χρ/2+γ)+⋯] is the zeroth-order Neumann function with constant coefficients C and D, and γ(≅0.57721) stands for the Euler–Mascheroni constant. At the origin (ρ=0), we should have a finite value of the perturbation potential φ̅₁ so that the irregular solution must be vanished, i.e. D=0. Since the boundary condition at ρ=R leads to the vanishing of the perturbation potential at the surface of the waveguide, i.e. φ̅₁(R)=CJ₀(χR)=0, the parameter χ(k_z, ω, R₀) is then determined by the condition χ(k_z, ω, R₀)R=α_0n, where α_0n(=2.4048, 5.5201, 8.6537, 11.7915, …) are the n^th-roots of the zero-order cylindrical Bessel function of the first kind J₀(α_0n). Since the asymptotic expression [35] of the zero-order cylindrical Bessel function J₀(χρ) is represented by

the n^th-roots for χR≫1 are then found to be

3 Results and Discussion

In this section, we discuss the dispersion relation and the physical characteristics of quantum space charge waves. The dispersion relation D(ω, k_z) of space charge waves propagating on the surface of the plasma column composed of relativistically degenerate quantum Fermi-Dirac electrons and relativistic ion flows:

Since Maxwell’s equations for the perturbation electric E₁ and magnetic B₁ fields [2] are, respectively, obtained by ∇×E₁=(iω/c)B₁ and ∇×B₁=−(iω/c)ε·E₁, where ε is the plasma dielectric tensor, the dispersion relation for the space-charge electromagnetic wave in a magnetised plasma column, i.e. when a cylindrical plasma waveguide is immersed in an axial magnetic field, is then found to be

Here we seek a streaming instability with the high-frequency mode, ω≈ω_pe, where the wave frequency has the following form: ω(k_z)≅k_zu_i+iγ, where γ is the imaginary part of the wave frequency of the quantum space charge wave. Solving (14) for the imaginary part γ yields

where γ̅(=γ/ω_pe) is the scaled imaginary part of the wave frequency, k̅_z(=k_zc/ω_pe) is the scaled wave number, u̅_i≡u_i/c, and R̅(=Rω_pe/c) is the scaled radius of the waveguide. According to the double sign in (14), the wave frequency has the damping (ω̅_Damp=ω_Damp/ω_pe) and the unstable (ω̅_Unst=ω_Unst/ω_pe) modes:

and

As shown in (17) and (18), the denominator in square root term would be negative for high harmonics, i.e. 1(k¯zu¯i)2−H2R03k¯z4−T(R0)k¯z2−α0n2k¯z2R¯2<1, so that the streaming instability mode is only possible for low harmonic cases. When R→∞, the wave frequencies would be written as

which are the cases for bulk quantum plasmas. Equation 18 stands for the streaming resonant instability mode of the space charge wave. For a time interval Δt after the disruption of the wave oscillation owing to the exponential growing factor exp(γt) for the streaming resonant instability mode, the temporal characteristic function ζ_ch(t) can be defined as ζ_ch(t)=exp(−ik_zu_it) exp(γt) since the perturbation of the space charge wave is proportional to the exponential term: exp[i(k_zz−ωt)]. When the temporal characteristic function ζ_ch(t) is valid, the time derivative d|ζ_ch(t)|²/dt of the squared amplitude |ζ_ch(t)|² over times Δt large compared with the period of the wave oscillation, 2π/ω_R[=2π/(k_zu_i)], would be then represented by the following relation: d|ζ_ch(t)|²/dt≅2γ|ζ_ch(t)|²−μ_L|ζ_ch(t)|⁴+…, where ω_R is the real part of the wave frequency and μ_L is the Landau constant [36]. Hence, the temporal characteristic function based on the Landau expression [37] for the weakly non-linear equation due to the streaming resonant instability of the space charge wave is found to be |ζ_ch(t)|²≅e^2γt|ζ_ch(0)|²/[1+|ζ_ch(0)|²(μ_L/2γ)(e^2γt−1)]. The quantum plasmas can be found in various dense plasma environments in nanostructures, laser-plasma interactions, and astrophysical compact objects such as neutron stars and white dwarfs [21], [38]. In these dense quantum plasmas composed of electrons and ions, the ranges of the number density n and the temperature T are, respectfully, known to be about 10²⁰–10²⁴ cm⁻³ and 5×10⁴–10⁶ K. Recently, the ultra-relativistic two-stream instability is investigated in an unmagnetised plasma [38]. Therefore, the influence of quantum statistical degeneracy pressure on the two-stream instability in relativistically degenerate quantum Fermi-Dirac plasmas will be treated elsewhere. Very recently, the excellent investigations on the propagation of linear and non-linear heavy ion-acoustic waves [39] and the heavy ion-acoustic non-planar shocks and solitons [40] have been carried out in unmagnetised, collisionless, strongly coupled plasma system. It is shown that the influence of positively charged heavy ions have significantly modified the basic features of the wave propagation. It can be then expected that the influence of heavy ions would alter the instability domain of the wave number and the growth rates for the resonant beam instability. Hence, the detailed investigation of the influence of positively charged heavy ions on the propagation of the quantum space charge wave will be treated elsewhere. Recently, the excellent investigations on the propagation and modulational instability of plasma waves have been carried by Chandra and Ghosh [41], [42], [43], [44] using the QHD model with the perturbation technique in quantum Fermi plasmas. Hence, the investigation on the modulational instability of plasma waves in semi-bounded quantum plasmas will also be treated elsewhere.

Figures 1–3 represent the scaled growth rates γ̅ for the resonant beam instability of the quantum space charge wave as functions of the degenerate parameter R₀ and the scaled wave number k̅_z for various values of the scaled beam velocity u_i. From these figures, it is shown that the domain of the degenerate parameter R₀ for the resonant beam instability significantly increases with an increase of the scaled beam velocity u̅_i. Hence, it would be expected that the influence of quantum degeneracy pressure on the resonant beam instability of the quantum space charge wave is more significant for higher ion beam cases. It is also found that the instability domain of the scaled wave number k̅_z increases with an increase of the degenerate parameter R₀. In addition, it is found that the scaled growth rates γ̅ for the resonant beam instability decreases with an increase of the degenerate parameter R₀. Figure 4 shows the scaled growth rate γ̅ for the resonant beam instability of the quantum space charge wave as a function of the scaled wave number k̅_z for various values the root α_0n of the zeroth order Bessel function J₀(α_0n)=0. As it is seen, the scaled growth rate γ̅ decreases with an increase of the order of the root α_0n of the Bessel function. Hence, it would be expected that the lowest harmonic mode provides the maximum value of the scaled growth rates γ̅ for a given degenerate parameter R₀ and the beam velocity u_i. Figure 5 represents the scaled growth rate γ̅ for the resonant beam instability of the quantum space charge wave as a function of the scaled beam velocity u̅_i and the scaled wave number k̅_z for a given degenerate parameter R₀. As shown in this figure, it is shown that the resonant beam instability cannot be generated in small values of the beam velocity u_i. It is also shown that the scaled growth rate γ̅ increases as an increase of the scaled wave number k̅_z. However, it is interesting to note that the instability domain of the scaled wave number k̅_z decreases with an increase of the beam velocity u_i. We now summarize results and issues as follows: The results in this article show that the domain of the degenerate parameter for the resonant beam instability significantly increases with an increase of the scaled beam velocity. It is also found that the instability domain of the wave number increases with an increase of the degenerate parameter. In addition, it is found that the growth rates for the resonant beam instability decreases with an increase of the degenerate parameter. Moreover, it is found that that the lowest harmonic mode provides the maximum value of the growth rates. Finally, it is found that the instability domain of the wave number decreases with an increase of the beam velocity. From this article, we have shown that the influence quantum statistical degeneracy pressure plays a significant role in the propagation and the stability of quantum space charge wave in a cylindrically bounded plasma waveguide filled with relativistically degenerate quantum Fermi-Dirac plasmas with the relativistic ion flow. It can be also expected that these results would be useful for understanding the characteristics of the quantum space charge wave as well as the various physical properties of relativistically degenerate quantum Fermi-Dirac plasmas.

Figure 1:

The surface plot of the scaled growth rate γ̅ of the quantum space charge wave in a plasma waveguide filled with degenerate quantum Fermi-Dirac plasmas with the relativistic ion flow as a function of the relativistic degeneracy parameter R₀ and the scaled wave number k̅_z for R̅=5, α₀₁=2.4048 (the first-root), and u̅_i=0.5.

Figure 2:

The surface plot of the scaled growth rate γ̅ of the quantum space charge wave in a plasma waveguide filled with degenerate quantum Fermi-Dirac plasmas with the relativistic ion flow as a function of the relativistic degeneracy parameter R₀ and the scaled wave number k̅_z for R̅=5, α₀₁=2.4048 (the first-root), and u̅_i=0.3.

Figure 3:

The surface plot of the scaled growth rate γ̅ of the quantum space charge wave in a plasma waveguide filled with degenerate quantum Fermi-Dirac plasmas with the relativistic ion flow as a function of the relativistic degeneracy parameter R₀ and the scaled wave number k̅_z for R̅=5, α₀₁=2.4048 (the first-root), and u̅_i=0.1.

Figure 4:

The scaled growth rate γ̅ for the resonant beam instability of quantum the space charge wave as a function of the scaled wave number k̅_z for various values the root α_0n of the zeroth order Bessel function J₀(α_0n)=0. The solid line represents the case of the first-root, i.e. α₀₁=2.4048. The dashed line represents the case of the second-root, i.e. α₀₂=5.5201.

Figure 5:

The surface plot of the scaled growth rate γ̅ of the quantum space charge wave in a plasma waveguide filled with degenerate quantum Fermi-Dirac plasmas with the relativistic ion flow as a function of the scaled beam velocity u̅_i and the scaled wave number k̅_z for R̅=5, α₀₁=2.4048 (the first-root), and R₀=0.5.