A Summary of the Introduction and Importance of Quantum Plasmas

Abstract

In this chapter, we will discuss the quantum plasmas that have their applications mainly in miniaturized semiconductors, optical fibers, waveguides, nanoplasmonics, and astrophysical systems. Quantum plasmas are the least explored field owing to the astronomical applications of classical plasmas. In this chapter we will discuss how quantum plasmas can be studied and which system of equations will be easier to follow. We will discuss the easiest method possible and more popular way to explore the quantum plasmas for the sake of understanding of a new reader to this subject.

1. Introduction

Plasma physics examines a system of charged particles that communicate with one another through the application of electromagnetic forces. At the start of the 20th century, researchers' interest in the physics of gas discharges sparked the beginning of the field of study known as plasma physics. Plasmas have been the subject of a significant amount of research, both experimental and theoretical, ever since the end of World War II. This is primarily because of the potential applications of nuclear fusion for both military (the hydrogen bomb) and civilian purposes (the production of energy through controlled thermonuclear fusion). It should not come as a surprise that plasma physics was created simultaneously by both astrophysicists and geophysicists given the widespread belief that around 90 percent of all matter and energy in the observable universe exists in the form of plasma. Plasmas can be observed in a variety of locations, including the atmosphere of the Sun, the magnetosphere of the earth, as well as the interplanetary and interstellar media.

Plasma physics is generally considered to be an entirely classical subject of study. The last ten years have seen a recent uptick in people’s interest in plasma systems, which are places where quantum effects play a key role.

Maxwell-Boltzmann statistics are frequently observed in space plasmas and laboratory plasmas because their densities are low even at temperatures that are rather high. At these temperatures, the de Broglie wavelength is not considerably impacted for electrons and ions.

When compared to the interparticle separation that occurs in plasmas, the thermal de Broglie wavelength of ions and electrons is easily observable in dense plasmas that are kept at low temperatures. Under these conditions, the quantum effect becomes more significant than the classical one. This is because of the significant overlap of matter waves that are connected with particles. Ions, degenerate light particles (such electrons and positrons), and holes are said to be in a dense state in quantum plasmas. Due to the large overlap of matter waves associated with particles in such circumstances, the quantum effect becomes more significant than the classical one. Quantum plasmas are described as having ions, degenerate light particles (such as electrons and positrons), and holes in a dense state. HEMT transistors, resonant tunneling diodes, and metallic nanostructures are examples of quantum semiconductor devices that exhibit plasmas with fermionic properties. Astrophysical systems also exhibit these properties (such as neutron stars, and white dwarfs). Quantum diodes, biophotonics, and ultra-small electrical devices can all be seen to share the same features.

2. Comparison of various methodologies for quantum plasmas

Over the past few decades, there has been an extraordinary amount of focus placed on the electrical properties of nanoscale objects [1, 2, 3, 4], particularly when such qualities are triggered into action by electromagnetic radiation. This field of research encompasses a wide range of nano-objects, such as semiconductor quantum dots, carbon nanotubes, metallic films, and nanoparticles; novel substances, such as graphene; and meta-materials, the composition of which can be tailored to display certain electromagnetic properties. Impressive potential applications include efficient high-performance computing (efficient storage and transfer of information), nanoplasmonics (optical filters, waveguides) [5, 6], and even in medical sciences (biomedical diagnostics and sensors) [7, 8]. These applications are all possible thanks to the properties of graphene. The electronic responses of such systems, which are frequently in the nonlinear and out-of-equilibrium zones, can be compared to those of a one-component quantum plasma and can be handled at various degrees of approximation. This comparison is possible because the electronic responses of such systems are frequent in these zones.

In order to account for quantum effects, the currently accepted classical theory for plasma needs to undergo some significant revisions. To properly account for this change, a suitable mathematical formalism is required. Several different research approaches have been established for the study of quantum plasmas. Techniques such as time-dependent density functional theory (TDDFT) and time-dependent Hartree–Fock (TDHF) are examples of such techniques. Although TDDFT and TDHF are wavefunction-based methods, they can be restated in the phase-space language that is common among plasma physicists by utilizing Wigner functions. This will allow the methods to be more easily understood.

Both the wavefunction and the phase space approaches need a significant amount of time and memory to employ, which is especially problematic in systems that contain hundreds of electrons.

Fluid models, which may be generated using the applicable kinetic equations (Wigner for a totally quantum approach or Vlasov for semiclassical modelling), offer the possibility of a cheaper alternative [9]. These models can be constructed by employing velocity moments of the distribution function. The fluid technique should be accurate enough to provide important insights into the underlying physical mechanisms while at the same time remaining affordable in terms of processing cost, despite the fact that some information will inevitably be lost throughout the course of this process. This is because some information cannot be recovered once it has been lost. Similar to their classical analogues, the validity of quantum fluid models is limited to long wavelengths in relation to the interparticle distance [10]. This is because quantum fluids behave differently from classical fluids. They are, nevertheless, capable of handling large excitations, Coulomb exchange (an effect associated to the Pauli exclusion principle), electron-electron correlations, and quantum processes such as tunnelling.

In the fluid approach, the dynamics of the electrons are modelled by a series of hydrodynamic equations (continuity, momentum balance, and energy balance), which incorporate quantum effects through the use of the so-called Bohm potential. When compared to Wigner or TDDFT simulations, one can anticipate a significant reduction in the amount of time required for computing. This is because TDDFT methods are required to solve N minus one Schrodinger-like equations, whereas the phase space approach doubles the number of independent variables (positions and velocities).

In the past, the field of condensed-matter physics made use of hydrodynamic models, primarily for semiconductors [11] and, to a lesser extent, for metal clusters [12, 13]. In more recent times, the concept of the quantum fluid has been expanded to include still another significant quality of the electron, which is referred to as its spin [14]. The resulting fluid equations are much more involved than their spinless counterparts, but they may find useful applications in the emerging field of spintronics [15] in condensed-matter physics and in the study of highly polarised electron beams [16] in plasma physics. Both of these fields are related to the study of condensed matter physics. Further applications of quantum fluid theory include dense plasmas, which are created when solid targets interact with intense laser beams [17], warm dense matter experiments [18], quantum nanoplasmonics [19, 20, 21, 22], and compact astrophysical objects, such as white dwarf stars [23].

3. Fluid approach for quantum plasmas

Quantum mechanical effects start playing a significant role when the Wigner-Seitz radius (average interparticle distance) $a={\left(\frac{3}{4\mathit{\pi n}}\right)}^{1/3}$ is comparable to or smaller than the thermal de Broglie wavelength ${\lambda }_B=\frac{\hslash }{m{V}_T}$ , where m is the mass of the quantum particles (e.g., degenerate electrons, degenerate positrons, degenerate holes), $V_T={\left(\frac{k_{B}T}{m}\right)}^{1/2}$ is the thermal speed of the quantum particles, T is the temperature, m is the mass, and $k_B$ is the Boltzmann constant, i.e., when

$$n{\lambda }_B^3\ge 1,$$E1

or, equivalently, when the temperature T is comparable to or lower than the Fermi temperature $T_F=\frac{E_F}{k_B}$ , where the Fermi energy is

$$E_F=\frac{{\hslash }^2}{2m}{\left(3{\pi }^2\right)}^{2/3}{n}^{2/3}$$E2

The relevant degeneracy parameter for the quantum plasma is

$$\frac{T_F}{T}=\frac{1}{2}{\left(3{\pi }^2\right)}^{2/3}{\left(n{\lambda }_B^3\right)}^{2/3}\ge 1$$E3

For typical metallic densities of free electrons, $n\sim 5\times {10}^{22}$ cm⁻³, we have $T_F\sim 6\times {10}^4$ K, which should be compared with the usual temperature T.

When the plasma particle temperature approaches $T_F$ , one can show, by using a density matrix formalism [24], that the equilibrium distribution function changes from the Maxwell-Boltzmann $\propto exp\left(-E/k_{B}T\right)$ to the Fermi-Dirac (FD) distribution function

$$f_{\mathit{FD}}=2{\left(m/2\pi \mathrm{\hslash }\right)}^3{\left[1+exp\left(\frac{E-\mu }{k_{B}T}\right)\right]}^{-1}$$E4

where in the nonrelativistic limit the energy is $E=\left(m/2\right)v^2=\left(m/2\right)\left(v_x^2+v_y^2+v_z^2\right)$. The chemical potential is denoted by$\mu$ . The parameter $\mu /k_{B}T$ is large and negative in the nondegenerate limit, and is large and positive in the completely degenerate limit.

It is useful to define the quantum coupling parameters for electron-electron and ion-ion interactions. The electron-electron Coulomb coupling parameter is defined as the ratio of the electrostatic interaction energy $E_{\mathrm{int}}=e^2/a_e$ between electrons and the electron Fermi energy $E_{\mathit{Fe}}=k_B{T}_{\mathit{Fe}}$, where e is the magnitude of the electron charge and $a_e={\left(3/4\pi n_e\right)}^{1/3}$ is the mean interelectron distance. We have

$$T_e=\frac{E_{\mathit{int}}}{E_{\mathit{Fe}}}\approx 0.3{\left(\frac{1}{n_e{\lambda }_{\mathit{Fe}}^3}\right)}^{2/3}\approx 0.3{\left(\frac{\mathrm{\hslash }{\omega }_{\mathit{pe}}}{k_B{T}_{\mathit{Fe}}}\right)}^2$$E5

where ${\lambda }_{\mathit{Fe}}=V_{\mathit{Fe}}/{\omega }_{\mathit{pe}}$, is the electron Fermi speed, and ${\omega }_{\mathit{pe}}={\left(4{\pi }^{2}n{e}^2/m_e\right)}^{1/2}$ the electron plasma frequency. Furthermore, the ion-ion Coulomb coupling parameter is${\Gamma }_i=Z_i^2{e}^2/a_i{k}_B{T}_i$, where $Z_i$ is the ion charge state, $a_i={\left(3/4\pi n_i\right)}^{1/3}$ is the mean inter-ion distance, and $T_i$ is the ion temperature. Since $T_e$ for metallic plasmas could be larger than unity, it is of interest to enquire the role of interparticle collisions on collective processes in a quantum plasma. It turns that the Pauli blocking reduces the collision rate for most practical purposes [25, 26]. Because of Pauli blocking, only electrons with a shell of thickness $k_{B}T$ about the Fermi surface suffer collisions. For these electrons, the electron-electron collision frequency is proportional to $k_{B}T=\hslash$. The average collision frequency among all electrons turns out to be [25]

$$v_{\mathit{ee}}=\frac{k_B{T}^2}{\hslash T_{\mathit{Fe}}}$$E6

Typically, $v_{\mathit{ee}}⟨⟨{\omega }_{\mathit{pe}}$ when $T⟨T_{\mathit{Fe}}$, which is relevant for metallic electrons. On the other hand, the typical time scale for electron-ion (lattice) collisions is ${\tau }_{\mathit{ei}}\cong 10\mathit{fs}$ , which is 1 order of magnitude greater than the electron plasma period. Accordingly, a collisionless quantum plasma regime is relevant for phenomena appearing on the time scale of the order of a femtosecond in a metallic plasma.

4. Various systems of equations

4.1 The QHD equation

The nonrelativistic QHD equations [27] have been developed in condensed matter physics [28] and in plasma physics [9, 25]. The nonrelativistic QHD equations are composed of the electron continuity equation

$$\frac{\partial n_e}{\partial t}+\nabla .\left(u_e{n}_e\right)=0$$E7

the electron momentum equation

$$\mathit{me}\left(\frac{\partial u_e}{\partial t}+u_p.\nabla u_e\right)=e\nabla \phi -\frac{1}{n_e}\nabla p_e+F_Q=0$$E8

and Poisson’s equation

$${\nabla }^2\phi =4\mathit{\pi e}\left(n_e-Z_i{n}_i\right)$$E9

In a quantum plasma with nonrelativistic degenerate electrons, the quantum statistical pressure in the zero electron temperature limit can be modeled as [9, 29]

$$p_e=\frac{m_e{V}_{\mathit{Fe}}^2{n}_0}{3}{\left(\frac{n_e}{n_0}\right)}^{\left(D+2\right)/D}$$E10

where D is the number of space dimension of the system and $V_{\mathit{Fe}}=\left(\hslash /m_e\right){\left(3{\pi }^2{n}_e\right)}^{1/3}$ is the electron Fermi speed.

4.2 The NLS-Poisson equation

For investigating the nonlinear properties of dense quantum plasmas, it is also possible to work with a NLS equation. Hence, by introducing the wave function

$$\psi \left(r,t\right)=\sqrt{n_e\left(r,t\right)}exp\left(\iota \frac{S_e\left(r,t\right)}{\mathrm{\hslash }}\right)$$E11

where $S_e$ is defined according to $m_e{u}_e=\nabla S_e$ and $n_e={\left|\psi \right|}^2$ , it can be shown that continuity equation and equation for momentum can be cast into a NLS equation [9, 25]

$$\iota \mathrm{\hslash }\frac{\partial \psi }{\partial t}+\frac{{\mathrm{\hslash }}^2}{2m_e}{\nabla }^2\psi +\mathit{e\phi \psi }-\frac{\mathit{me}{V}_{\mathit{Fe}}^2}{2n_0^2}\left|\psi \right|{}^{4/D}\psi =0$$E12

where the electrostatic field $\phi$ arising from the charge distribution of N electrons is determined from Poisson’s equation

$${\nabla }^2\phi =4\mathit{\pi e}\left({\left|\psi \right|}^2-Z_i{n}_i\right)$$E13

We note that the third and fourth terms on the left-hand side of Eq. (12) represent the nonlinearities associated with the nonlinear coupling between the electrostatic potential and the electron wave function and the nonlinear quantum statistical pressure, respectively.

5. Numerical mode analysis method for investigation of two stream instabilities

Let us study the basic fluid equation for quantum plasmas to investigate the Two Stream instabilities using Normal Mode Analysis Method (NMA) which is one of the basic and simplest way to understand the instabilities in plasmas. For the sake of understanding and simplification let us rewrite the Fluid equations again, which includes continuity equations and momentum equations for a plasma model which is unmagnetized and possess non degenerate ions and moves with non-relativistic speed.

5.1 Basic equations

In an unmagnetized, non-relativistic collision-less dense plasma, the QHD equation for a degenerate electron under the effect of an electrostatics potential due to non-degenerate ions [30, 31] that move at a non-relativistic speed is self-possessed of a continuity equation and a momentum equation for a degenerate electron. This is the case because non-degenerate ions move at a non-relativistic electrons

$$\frac{\partial n_e}{\partial t}+\overrightarrow{\nabla }·\left(n_e{\overrightarrow{u}}_e\right)=0$$E14

$$m_e{n}_e\left[\frac{\partial \overrightarrow{u_e}}{\partial t}+\left({\overrightarrow{u}}_e·\overrightarrow{\nabla }\right){\overrightarrow{u}}_e\right]-e{n}_e\overrightarrow{\nabla }\phi +\overrightarrow{\nabla }{P}_e-n_e\overrightarrow{F_B}=0$$E15

Where

${\overrightarrow{F}}_B=F_{\mathit{Bx}}\stackrel{\wedge }{i}+F_{\mathit{Bz}}\stackrel{\wedge }{k}=\frac{{\mathrm{\hslash }}^2\overrightarrow{\nabla }}{2m_e}\left(\frac{{\nabla }^2\sqrt{n_e}}{\sqrt{n_e}}\right)$ is the force arising due to Bohm potential, which is self – consistent potential depends upon the amplitude of wave function of matter wave associated with degenerate electrons.

Along with Poisson equation

$${\epsilon }_0{\nabla }^2\phi -\frac{e}{{\epsilon }_0}\left(n_e-Z_i{n}_i\right)=0$$E16

In unmagnetized quantum plasma, the dynamics of non-degenerate ions, is described by the QHD equation in non-relativistic region, which includes continuity equation and momentum equation for non-degenerate ions

$$\frac{\partial n_i}{\partial t}+\overrightarrow{\nabla }·\left(n_i{\overrightarrow{u}}_i\right)=0$$E17

$$\left(1+{\tau }_m\frac{\partial }{\partial t}\right)\left[\begin{array}{c}\left(\frac{\partial }{\partial t}+\overrightarrow{u}.\overrightarrow{\nabla }\right)\overrightarrow{u}+\frac{Z_{i}e}{m_i}\overrightarrow{\nabla }\phi +\frac{{\gamma }_i{k}_B{T}_i}{m_i{n}_i}\overrightarrow{\nabla }{n}_i\\ \end{array}\right]-\frac{\eta }{{\rho }_i}{\nabla }^2{u}_i-\frac{\left(\xi +\frac{\eta }{3}\right)}{{\rho }_i}\overrightarrow{\nabla }\left(\overrightarrow{\nabla }.\overrightarrow{u_i}\right)=0$$E18

Where ${\mathrm{Z}}_i$ is atomic number of ions, $\phi$ is electrostatics potential due to space charge distribution of electron-ion system., $n_e,n_i$ are number density of electrons and ions $m_{e,}{m}_i$ are Mass of electrons and ions, $u_e$, $u_i$ are velocity of electrons and ions, ${\mathrm{\rho }}_{\mathrm{i}}$ is mass density of ion, ${\mathrm{\gamma }}_{\mathrm{i}}$ is Adiabetic index of ion, ${\mathrm{\tau }}_{\mathrm{m}}$ relaxation time of ion η and ξ = 0.1 and 2.7 = Bulk ion visco-elastic coefficient for ions while the rest of physical quantities have their usual meaning.

Here momentum equation includes the quantum statistical pressure term for degenerate electron [32] which behaves like a Fermi gas at zero-electron temperature, which depends upon Fermi speed of degenerate Fermi electron gas and degree of space-dimension of electron [33, 34].

At zero-electron temperature, quantum statistical pressure is

$$P_e=\frac{m_e{\left(V_{\mathit{Fe}}\right)}^2{n}_o}{3}{\left(\frac{n_e}{n_o}\right)}^{\left(\frac{2}{f}+1\right)}$$E19

Where

$V_{\mathit{Fe}}=\frac{{\left(3{\pi }^2{\mathrm{\hslash }}^3{n}_e\right)}^{\frac{1}{3}}}{m_e}$ = Fermi speed of degenerate electron

f = degree of space-dimension of degenerate electron.

$n_o$ = number density of electron in equilibrium state.

5.2 Mathematical analysis

The continuity and momentum equations for electrons and ions, as well as Poisson’s equation, are solved using Normal Mode Analysis and linearization, where the physical quantities (density, velocity, and electrostatic potential) $n_i$ ,$n_e$, $u_{\mathit{ix}}$, $u_{\mathit{iy}}$, $u_{\mathit{iz}}$, $u_{\mathit{ex}}$, $u_{\mathit{iy}}$, $u_{\mathit{iz}}$ and $\mathrm{\phi }$ are expressed as $P=P_0+P_1$ where $P_0$ and $P_1$ represent the unperturbed and perturbed part respectively. Due to their tiny size, higher order perturbed terms, particularly those of the second order, are disregarded during linearization. After calculating the perturbed velocities for ions and electrons, $u_{i1}$ and $u_{e1}$, from momentum equations, these values are substituted into the expression for perturbed densities of ions and electrons obtained from continuity equations under the assumption that the unperturbed densities of ions and electrons are constant with respect to the electron oscillation period, such that-

$$\frac{\partial n_{i0}}{\partial t}=\frac{\partial n_{e0}}{\partial t}=0$$E20

Also unperturbed components of velocities of ions and electrons are considered constant in space-time, so that

$$\frac{\partial u_{\mathit{ix}0},u_{\mathit{iy}0},u_{\mathit{iz}0}}{\partial x,z}=\frac{\partial u_{\mathit{ex}0},u_{\mathit{ey}0,}{u}_{\mathit{ez}0}}{\partial x,z}=0$$E21

$$\frac{\partial u_{\mathit{ix}0},u_{\mathit{iy}0},u_{\mathit{iz}0}}{\partial t}=\frac{\partial u_{\mathit{ex}0},u_{\mathit{ey}0,}{u}_{\mathit{ez}0}}{\partial t}=0$$E22

After applying all this conditions, the density of electrons and ions are evaluated in term of perturbed potential ${\phi }_1$ and Using these values of np1 and ne1 in Poisson’s Eq. (18), we get the following equation in ${\phi }_1$

$${\phi }_1=\frac{{\phi }_N}{{\phi }_D}=\frac{{\phi }_{\mathit{NR}}+i{\phi }_{\mathit{NI}}}{{\phi }_{\mathit{DR}}+i{\phi }_{\mathit{DI}}}$$E23

Where

$${\phi }_N={\phi }_{\mathit{NR}}+i{\phi }_{\mathit{NI}}$$E24

and

$${\phi }_D={\phi }_{\mathit{DR}}+i{\phi }_{\mathit{DI}}$$E25

Real and imaginary parts of numerator and denominator are written as:

$${\phi }_{\mathit{NR}=}\left[\begin{array}{c}{\omega }^8{G}_{\mathrm{NR}1}+{\omega }^7{G}_{\mathrm{NR}2}+{\omega }^6{G}_{\mathrm{NR}3}+{\omega }^5{G}_{\mathrm{NR}4}+{\omega }^4{G}_{\mathrm{NR}5}\\ +{\omega }^3{G}_{\mathrm{NR}6}+{\omega }^2{G}_{\mathrm{NR}7}+{\omega }^1{G}_{\mathrm{NR}8}+{\omega }^0{G}_{\mathrm{NR}9}\end{array}\right]$$E26

$${\phi }_{\mathit{NI}}=\left[\begin{array}{c}{\omega }^8{G}_{\mathrm{NI}1}+{\omega }^7{G}_{\mathrm{NI}2}+{\omega }^6{G}_{\mathrm{NI}3}+{\omega }^5{G}_{\mathrm{NI}4}+{\omega }^4{G}_{\mathrm{NI}5}\\ +{\omega }^3{G}_{\mathrm{NI}6}+{\omega }^2{G}_{\mathrm{NI}7}+{\omega }^1{G}_{\mathrm{NI}8}+{\omega }^0{G}_{\mathrm{NI}9}\end{array}\right]$$E27

$${\phi }_{\mathit{DR}}=\left[\begin{array}{c}{\omega }^{10}{G}_{\mathrm{DR}1}+{\omega }^9{G}_{\mathrm{DR}2}+{\omega }^8{\mathit{FG}}_{\mathrm{DR}3}+{\omega }^7{G}_{\mathrm{DR}4}\\ +{\omega }^6{G}_{\mathrm{DR}4}+{\omega }^5{G}_{\mathrm{DR}5}+{\omega }^4{G}_{\mathrm{DR}6}+{\omega }^3{G}_{\mathrm{DR}7}\\ +{\omega }^2{G}_{\mathrm{DR}8}+{\omega }^1{G}_{\mathrm{DR}9}+{\omega }^0{G}_{\mathrm{DR}10}\end{array}\right]$$E28

$${\phi }_{\mathit{DI}}=\left[\begin{array}{c}{\omega }^{10}{G}_{\mathrm{DI}1}+{\omega }^9{G}_{\mathrm{DI}2}+{\omega }^8{G}_{\mathrm{DI}3}+{\omega }^7{G}_{\mathrm{DI}4}\\ +{\omega }^6{G}_{\mathrm{DI}3}+{\omega }^5{G}_{\mathrm{DI}4}+{\omega }^4{G}_{\mathrm{DI}5}+{\omega }^3{G}_{\mathrm{DI}6}\\ +{\omega }^2{G}_{\mathrm{DI}7}+{\omega }^1{G}_{\mathrm{DI}8}+{\omega }^0{G}_{\mathrm{DI}9}\end{array}\right]$$E29

Here, G’s are coefficients of different powers of $\omega$, which are functions of unperturbed quantities.

Eq. (29) relates the perturbed quantity ${\varphi }_1$ with the unperturbed quantities. Since the first-order quantity cannot be explicitly expressed in terms of zeroth-order quantities [35], we make the RHS of this equation indeterminate by putting numerator and denominator to be individually zero. Thus, we get two equations of order seven for frequency of oscillations. These equations are solved numerically for the values of growth rate (imaginary part of $\omega$ ) by giving typical values of $n_{e0}$, $n_{p0}$, $u_{x0}$, $u_{y0}$, $u_{z0}$, $v_{x0}$, $v_{y0}$, $v_{z0}$, and $T_e$. The real roots of these equations give rise to the propagating waves, which are not of our concern here. The positive imaginary part of the complex root gives the instability, whereas its negative part gives the damped wave. It means the oscillations are unstable when Im(ω) < 0.

Here Figure 1, presents the variation of normalized growth rate of the instabilities occurring into the plasma. From above mentioned polynomial equation in terms of $\omega$, several equations will have complex conjugate roots and same may not have. The growth rate can be analyzed with variation in other typical parameters like ion density, angle of propagation in case of magnetized plasmas, with ion temperature in case of warm plasmas and also with respect to variation in magnetic field in magnetized plasmas.

Figure 1.

Variation of growth rate Γ with Propagation Vector (k in m⁻¹) When θ = 45°, ${\phi }_0$ = 0.05, $n_{e0}$ = 5 × 1017 m⁻³, $m_e$ = 9.1 $\times {10}^{-31}\mathrm{kg},m_i$ = 1.67 × ${10}^{-27}$ kg, $U_{\mathit{ix}0}$ = $U_{\mathit{iy}0}$ = $U_{\mathit{iz}0}$ = 0.25Cs, $U_{\mathit{ex}0}$ = $U_{\mathit{ey}0}=U_{\mathit{ez}0}=$5 Cs, $T_e=0.25\mathit{eV},T_i=0\text{and}{Z}_i$ = 1.

6. Future perspectives

In the past many derived the dielectric constant for the high-frequency (in comparison with the ion plasma frequency) ES waves [36, 37] and the refractive index for EM waves [38] by using a quantum kinetic theory based on the Wigner and Poisson-Maxwell equations in an unmagnetized quantum plasmas. But no one has found the same in magnetized plasma. In 2D electrons, at very high magnetic field, electron density is quantized which is associated with quantized Hall resistance. After solving this problem we will be able to understand the features of quantum oscillations of electrons and possible formation of bound states of electrons in the presence an external magnetic field.

In the past researchers have given dielectric constant without quantum statistical pressure to investigate the screening and wake potentials around a test charge in an electron ion quantum plasma. The wake potential behind a test charge arises due to collective interactions between a test charge and the ion oscillation. The screening potential can be used to study the doubly excited resonance states of He and H atoms embedded in a quantum plasma and lattice waves in 2D hexagonal quantum plasma crystals. In this problem we will deduce potentials around a moving test charge in a quantum plasma with inclusion of statistical pressure term.