Quantum Electron-Exchange Effects on the Buneman Instability in Quantum Plasmas

In plasmas, the two-stream or Buneman-type instability has received a considerable interest in investigating the physical characteristics of the plasma wave and properties of the plasma system since this phenomenon is one of the most effective heating mechanisms and is very common process in plasma physics [1–10]. It has been shown that two-stream instability happens whenever one plasma component has a relative velocity with respect to another component of the plasma [3]. It is also shown that the Buneman instability is the two-stream instability of the electron-ion plasma [1, 4]. In addition, the nonlinear effect associated with the two-stream instability has been of a great interest since this instability is connected with the plasma turbulence [5]. Moreover, the electron-beam instability has been found in the electron beam propagating through the corona in a solar radio burst [6]. In recent years, the physical characteristics and collision processes have been extensively investigated in dense plasmas such as semiclassical and quantum plasmas since the semiclassical and quantum plasmas have been found in various nanoscale objects in modern sciences and technologies, such as nanowires, quantum dots, and semiconductor devices [11–23]. It is shown that the plasma dielectric function and the effective interaction potential in quantum plasmas are quite different from those in classical plasmas due to the multiparticle correlation and quantum-mechanical effects [16]. Very recently, it is also found that the electron-exchange effect caused by the electron 1/2-spin plays a crucial role in the formation of the plasma response function as well as in the structure of the effective interaction potential in quantum plasmas [21–23]. In addition, it has been shown that the velocity associated with the electron-exchange effect alters the quantum recoil effect in degenerate quantum plasmas. Thus, it would be expected that the dispersion properties of the Buneman instability in dense quantum plasmas are quite different from those in classical plasmas since the influence of electron exchange and the Bohm potential alters the plasma dielectric function in quantum plasmas. Recently, the Buneman instability has been studied in the nonlinear regime and by particle in cell method in the absence of quantum effects [24, 25]. In addition, the low-frequency quantum Buneman instability has been investigated in the absence of electron-exchange effect [26]. However, the electron-exchange effect on the Buneman instability in quantum plasmas has not been investigated as yet. However, it would be expected that the electron-exchange effect caused by the quantum-mechanical electron-exchange and correlation potential modifies the energy transfer from the streaming of quantum plasma to the plasma oscillations. Thus, in this article, we investigate the influence of electron exchange on the Buneman instability for the quantum plasma system composed of the moving electron fluid relative to the stationary ion fluid since the instability would provide useful information on the resonant wave–particle interaction mechanism in quantum plasmas and also on the physical characteristics of the electron-exchange corrections. Hence, this work would provide useful information on the Buneman instability in quantum plasmas composed of the moving electron quantum plasma relative to the stationary ion plasma. The dispersion relation is obtained for the quantum plasma system composed of the moving quantum electron fluid relative to the ion fluid. We derive the analytic expressions of the wave frequency and the growth rate for the Buneman instability as functions of the electron-exchange parameter, de Broglie’s wave length, Debye’s length, and wave number. The variation in the growth rate of the Buneman instability in quantum plasmas due to the change in the electron-exchange effect and plasma parameters is also discussed.

Recently, it has been shown that the quantum hydrodynamic (QHD) formulation of the Schrödinger–Poisson system is extremely useful for understanding the physical characteristics and properties of quantum plasmas [19]. The QHD model [19, 20] in quantum plasmas for species α [α=e(electron), i(ion)] including the influence of exchange correlation, Bohm’s potential, and Fermi’s term would be represented by

(1)∂nα∂t + ∇⋅(nαvα) = 0, (1)

(2)mαnα(∂vα∂t + vα ⋅ ∇vα) = − nαqα∇φ − ∇Pα + ℏ24mα∇(∇2nα) − nα∇Vα,xc, (2)

where m_α, n_α(=n_α0 + n_α1), n_α0, n_α1, v_α, q_α, φ, Pα( = mαv″Fα2nα13/3nα03), and ℏ are the mass, total number density, unperturbed number density, perturbed number density, velocity, charge, electric potential, pressure, and rationalised Planck’s constant, respectively; v″Fα2( ≡ v′Fα2 + ℏ2k2/4mα2) = (3/5)vFα2[1 + (5π2/12)(Tα/TFα)2](1 + ℏ2k2/4mα2v′Fα2) is the square of the effective Fermi velocity, v′Fα2 ≡ (3/5)vFα2[1 + (5π2/12)(Tα/TFα)2] [27, 28], k is the wave number, v_Fα[=(2k_BT_Fα/m_α)^1/2] is the Fermi velocity, k_B is the Boltzmann constant; T_α is the thermal temperature [21], and TFα[ = ℏ2(3π2n0α2)2/3/(2mα)] is the Fermi temperature of species α. In the momentum equation (2), the ∇(∇²n_α) term represents the Bohm potential term due to the quantum-diffraction effect, and the additional potential Vα,xc[ = 0.985(e2/ε)nα1/3(1 + (0.034/a0αnα1/3)ln(1 + 18.37a0αnα1/3))] stands for the quantum-mechanical electron-exchange and correlation potential [20], where e is the magnitude of electron charge, ε is the relative dielectric constant, and a0α( = εℏ2/mαqα2) is the effective Bohr radius. For the streaming quantum plasma with velocity v₀, the plasma susceptibility χ_α(ω, k) including the influence of exchange and correlation potential, Bohm’s potential, and Fermi’s term is then obtained by the linearisation of (1) and (2) and the Fourier transformation ∂/∂t=−iω, ∇=ik:

(3)χα(ω, k) = − ωpα2(ω − v0 ⋅ k)2 11 − k2(ω − v0 ⋅ k)2(v″Fα2 + vα,xc2), (3)

where ω is the wave frequency, k is the wave vector, ωpα[ = (4πnαqα2/mα)1/2] is the plasma frequency, and v_α,xc is the exchange velocity associated with the influence of electron exchange associated with the quantum-mechanical electron-exchange and correlation potential V_α,xc since the Doppler-shifted wave frequency ω^* for the streaming electron plasma would be given by ω^*=ω−v₀·k. For the stationary cold classical ion plasma, the ion plasma susceptibility [5] is given by χi(ω, k) = − ωpi2/ω2, where ω_pi[=(4πn_ie²/m_i)^1/2] is the ion plasma frequency, n_i and m_i are the density and mass of the ion, respectively. Hence, the plasma dielectric function ε_p(ω, k) for the quantum plasma system composed of the moving electron quantum plasma relative to the stationary classical ion plasma is then found to be

(4)εp(ω, k) = 1 − ωpi2ω2 − ωpe2(ω − v0 ⋅ k)2 − k2(ve,xc2 + v′Fe2 + He2k2), (4)

where He2 ≡ ℏ2/(4me2) when ω >> kv_Fα. In obtaining (4), the relation between the perturbed number density n_α1 and the unperturbed density n_α0 is given by nα1/nα0 = (qαφ/mα)(k2/((ω − v0 ⋅ k)2 − (v″Fα2 + vα,xc2)k2)). In addition, the perturbed number density n_α1 is represented by n_α1=−χ_αk²φ/4πq_α. Very recently, an excellent investigation [29] on the screening in dense plasmas has shown that the electron-exchange and correlation potential alters the response function obtained by the orbital-free density functional theory (OF-DFT) and also shown the exact connection between the susceptibility and the local-field correction obtained by the exchange and correlation contribution. In addition, it is shown that the QHD model is consistent with the DFT formulation and the local-field correction is equivalent to the exchange and correlation potential. Hence, the expression of the plasma dielectric function ε_p(ω, k) (4) would be reliable to investigate the influence of electron exchange on the Buneman instability in quantum plasmas. Using (4), the dispersion relation for the electrostatic mode in quantum plasmas with a moving electron quantum plasma can be represented in the following non-dimensional form:

(5)Mω¯2 + 1(ω¯ − λ¯v¯0 k¯)2 − k¯2λ¯2(β2 + v′¯Fe2 + 18k¯2) = 1, (5)

where M( ≡ m_e/m_i) is the electron and ion mass ratio, ω̅( ≡ ω/ω_pe) is the scaled frequency, v̅₀( ≡ v₀/v_e) is the scaled streaming velocity, v_e[=(2k_BT_e/m_e)^1/2] is the electron thermal velocity, T_e is the electron thermal temperature, k̅( ≡ kλ_e) is the scaled wave number, λ_e[=ℏ /(m_ek_BT_e)^1/2] is the electron de Broglie wave length, λ̅ ≡ λ_De/λ_e, λ_De(=v_e/ω_pe) is the Debye length, v′¯Fe ≡ v′Fe/ve, and β( ≡ v_e,xc/v_e) is the electron-exchange parameter. In this normalisation process, we have used the electron thermal velocity v_e and the electron de Broglie wave length λ_e since the thermal temperature is given in the effective Fermi velocity v′Fα = (3/5)1/2[1 + (5π2/12)(Tα/TFα)2]1/2vFα [27, 28]. In classical electron-ion plasmas, it is known that if v₀=0, the complex roots disappear since the dispersion relation [30] is given by 1 = ωpi2/ω2 + ωpe2/(ω − v0 ⋅ k)2. However, in quantum plasmas as it is seen in (5), the complex roots exist due to the electron-exchange correction even for the non-streaming case. From (5), we can find that when the local minimum of the left hand side is below than unity, the fourth-order algebraic equation has no complex solutions, i.e., four real solutions. However, the Buneman instability, i.e., unstable mode, would occur when the local minimum of the left hand side of (5) is greater than unity. As a result of the Buneman instability in quantum plasmas, the energy excess by the streaming of electron quantum plasma would be transferred to plasma oscillations so that the electron streaming would be decelerated due to the energy transfer from the electron beam to the plasma oscillations. Hence, the constraint condition for the Buneman instability in quantum plasmas would be then given by

(6)Mω¯min2 + 1(ω¯min − λ¯v¯0k¯)2 − k¯2λ¯2(v′¯Fe2 + 18k¯2 + β2) > 1, (6)

where ω̅min( ≡ ω_min/ω_pe) is the scaled local minimum frequency. Then, the dispersion relation can be simplified as follows:

(7)ω¯4 − A(k¯, v¯0, λ¯)ω¯3+ [M + B′(k¯, v¯0, λ¯, β) + 1]ω¯2+ 2A(k¯, v¯0, λ¯)ω¯ + MB′(k¯, v¯0, λ¯, β) = 0, (7)

where A(k̅, v̅₀, λ̅) ≡ 2k̅λ̅v̅₀ and B′(k¯, v¯0, λ¯, β) ≡ k¯2λ¯2(v¯′Fe2 + k¯2/8 + β2 − v¯02). After some mathematical manipulations, the complex solution ω̅_BI(k̅, v̅₀, λ̅, β, M) corresponding to the unstable growing mode, i.e., the positive imaginary root: Imω̅_BI>0, when v′¯Fe = 1, for the Buneman streaming instability including the influence of electron exchange in quantum plasmas is then obtained by the following closed form:

(8)ω¯BI(k¯, v¯0, λ¯, β, M) = (ωR + iγGR)/ωpe = (1/2)[F + (21/3/3)G/(H + (− 4G3 + H2)1/2)1/3 + (2− 1/3/3)(H + (− 4G3 + H2)1/2)1/3]1/2 + (1/2)[I − (21/3/3)G/(H + (− 4G3 + H2)1/2)1/3 − (2− 1/3/3)(H + (− 4G3 + H2)1/2)1/3 − (1/4)J/(F + (21/3/3)G/(H + (− 4G3 + H2)1/2)1/3)1/2 + (2− 1/3/3)(H + (− 4G3 + H2)1/2)1/3]1/2, (8)

where ω_R is the real part of the frequency; γ_GR is the growth rate; the solution parameters F(k̅, v̅₀, λ̅, β, M), G(k̅, v̅₀, λ̅, β, M), H(k̅, v̅₀, λ̅, β, M), I(k̅, v̅₀, λ̅, β, M), and J(k̅, v̅₀, λ̅, β, M) are, respectively, represented in terms of the parameters A(k̅, v̅₀, λ̅)(=2k̅λ̅v̅₀) and B(k¯, v¯0, λ¯, β)[ = k¯2λ¯2(1 + k¯2/8 + β2 − v¯02)] with the ratio M(=m_e/m_i), such as F ≡ −1 + A²/4 + (1−2B−2M)/3, G ≡ 6A² + 12MB + (1 + B + M)², H ≡ A²(108 + 27BM) + (18A²−72BM)(1 + B + M) + 2(1 + B + M)³, I ≡ −1 + A²/4−(1 + 4B + 4M)/3, and J ≡ −16A² + A³−4A(1 + B + M), since the perturbation varies as exp(−iωt + ik·r)=exp(−iω_Rt + ik·r)exp(γ_GRt). Hence, the scaled growth rate γ̅_GR(k̅, v̅₀, λ̅, β, M)( ≡ γ_RG/ω_pe) in units of the electron plasma frequency for the Buneman instability in quantum plasmas composed of the moving electron quantum plasma relative to the stationary ion plasma is then found to be

(9)γ¯GR(k¯, v¯0, λ¯, β, M) = {Im(1/2)[F + (21/3/3)G /(H + (− 4G3 + H2)1/2)1/3 +(2− 1/3/3)(H + (− 4G3 + H2)1/2)1/3]1/2 + (1/2)[I − (21/3/3)G/(H + (− 4G3 + H2)1/2)1/3 − (2− 1/3/3)(H + (− 4G3 + H2)1/2)1/3 − (1/4)J/(F + (21/3/3)G/(H + (− 4G3 + H2)1/2)1/3)1/2 + (2− 1/3/3)(H + (− 4G3 + H2)1/2)1/3]1/2}. (9)

where “Im” extracts the imaginary part of the expression. Since it is shown that the plasma instability is responsible for the electromagnetic radiation [30], it would be expected that the radiation related to the Buneman instability (9) provides the useful spectral information on the physical characteristics and properties of the quantum plasma composed of the streaming electron plasma and the stationary ion plasma. Recently, the nonthermal effect on the Buneman instability due to the ion streaming has been investigated in Lorentzian dusty plasmas [31]. However, the influence of electron exchange on the Buneman instability has not been investigated as yet. Hence, (9) would be quite useful to investigate the electron-exchange effect on the Buneman instability in quantum plasmas. In dense, warm, semiclassical or quantum plasmas [18], the number density n and temperature T are shown to be about 10²⁰–10²⁴ cm⁻³ and 5×10⁴–10⁶ K, respectively. Additionally, it has been shown that the physical properties of the dense plasmas¹⁴ would be represented by the plasma coupling parameter Γ[=(Ze)²/ak_BT], degeneracy parameter θ(=k_BT/E_F), and density parameter r_s(=a/a₀), where a is the average distance between particles in plasmas. In order to explicitly investigate the physical characteristics and the electron-exchange effects on the Buneman instability in quantum plasmas, we analyse the dependence of the unstable root ω̅_BI(k̅, v̅₀, λ̅, β, M) on the exchange parameter β of the quantum plasma when v₀>v_e, i.e., cold quantum plasmas. Figures 1–3 represent the scaled growth rate γ̅_GR(=γ_RG/ω_pe) for the Buneman instability in quantum plasmas as a function of the scaled wave number k̅(=kλ_e) for various values of the exchange parameter β and the ratio λ̅(=λ_DFe/λ_e) of the Fermi–Debye length λ_DFe to the de Broglie wave length λ_e. In these figures, it is shown that the electron-exchange effect suppresses the growth rate of the Buneman instability in quantum plasmas. It would be expected that the influence of electron exchange prevents the transfer of the energy excess by the electron beam to the plasma oscillations due to the additional force −∇V_{α, xc} produced by the quantum-mechanical electron-exchange and correlation potential V_α,xc so that the electron-exchange effect reduces the energy transfer from the streaming of electron quantum plasma. Hence, we have found that the electron-exchange effect diminishes the growth rate of the Buneman instability. Hence, we have found that the growth rates γ̅_GR of the Buneman instability in quantum plasmas including the influence of electron exchange are always smaller than those neglecting the electron-exchange effect. As shown, it is also found that the magnitude of the growth rate γ̅_GR of the Buneman instability significantly decreases with an increase in the ratio λ̅. In addition, it is shown that the domain of the Buneman instability increases with an increase in the ratio of the Debye length to the de Broglie wave length. Thus, we expect that the instability mode is impossible in small wave number domains when the ratio λ̅ is quite large, i.e., low-density plasmas. Hence, we find that the density effect is quite sensitive to the instability condition in quantum plasmas. Moreover, the electron-exchange effect on the growth rate decreases with increasing wave number k̅ and, however, increases with an increase in the ratio λ̅. It would be then expected that the electron-exchange effect can only be figured out in the growth rate with small wave numbers. As it is shown in Figures 1–3, the growth rate γ̅_GR of the Buneman instability saturates with an increase in the wave number k̅. As in classical plasmas [32], it is also found that the growth rate γ_GR is small compared to the electron plasma frequency ω_pe in quantum plasmas since the Doppler-shifted frequency seen by streaming electrons is in the range of the ion plasma frequency ω_pi. Figure 4 shows the surface plot of the scaled growth rate γ̅_GR for the Buneman instability in quantum plasmas as a function of the exchange parameter β and the scaled wave number k̅. It is interesting to note that the electron-exchange effect reduces the domain of the wave number of the Buneman instability. Figure 5 represents the surface plot of the scaled growth rate γ̅_GR for the Buneman instability in quantum plasmas as a function of the ratio λ̅ and the scaled wave number k̅. As it is seen, the domain of the Buneman instability decreases with an increase in the de Broglie wave length λ_e and, however, increases with increasing Debye length λ_De. Hence, we find that the density effect on the growth rate γ̅_GR for the Buneman instability is quite significant in quantum plasmas. Very recently, the quantum-mechanical diffraction effect on the formation of the effective potential has been obtained in two component semiclassical plasma by using the dielectric response function method [33]. Hence, the investigation of the influence of quantum diffraction on the Buneman instability in quantum plasmas will be treated elsewhere. From these results, we conclude that the influence of electron exchange and plasma density plays a crucial role on the Buneman instability due to the streaming of the electron fluid in quantum plasmas. These results would provide useful information of the quantum electron-exchange effect and physical characteristics of the streaming instability in quantum plasmas.

Figure 1:

The scaled growth rate γ̅_GR for the Buneman instability in quantum plasmas as a function of the scaled wave number k̅ when λ̅=20 and v̅₀=80. The solid line represents the case of β=0. The dashed line represents the case of β=10. The dotted line represents the case of β=20.

Figure 2:

The scaled growth rate γ̅_GR for the Buneman instability in quantum plasmas as a function of the scaled wave number k̅ when λ̅=15 and v̅₀=80. The solid line represents the case of β=0. The dashed line represents the case of β=10. The dotted line represents the case of β=20.

Figure 3:

The scaled growth rate γ̅_GR for the Buneman instability in quantum plasmas as a function of the scaled wave number k̅ when λ̅=12 and v̅₀=80. The solid line represents the case of β=0. The dashed line represents the case of β=10. The dotted line represents the case of β=20.

Figure 4:

The surface plot of the scaled growth rate γ̅_GR for the Buneman instability in quantum plasmas as a function of the exchange parameter β and the scaled wave number k̅ when λ̅=12 and v̅₀=80.

Figure 5:

The surface plot of the scaled growth rate γ̅_GR for the Buneman instability in quantum plasmas as a function of the ratio λ̅ and the scaled wave number k̅ when λ̅=12 and v̅₀=80.