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Dispersion Properties of the Propagation of Linear Waves

ABSTRACT

In electron-positron plasmas some of the plasma modes are decoupled because of the equal fee to mass percentage of both species. The dispersion properties of the propagation of linear waves in degenerate electron-positron magnetoplasma are looked into. Utilizing the quantum hydrodynamic equations with magnetic fields of the Wigner-Maxwell system, we have obtained a set of new dispersion relationships in which ions' motions are not considered. The general dielectric tensor comes from using the electron and positron densities and its own momentum respond to the quantum results credited to Bohm potential and the statistical effect of Femi temperature. It has been demonstrated the value of magnetic field and its own role with the quantum results in these plasmas which support the propagation of electromagnetic linear waves. Besides, the dispersion relationships in case of parallel and perpendicular methods are investigated for different positron-electron density ratios.

Keywords: Quantum Plasma; Dispersion relation ; Electron -Positron

1- INTRODUCTION

Electron-positron (e-p) plasmas are located in the first world, in astrophysical items (e. g. , pulsars, ultra nova remnants, and energetic galactic nuclei, in О -ray bursts, with the guts of the Milky Way galaxy [1].

In such physical systems, the e-p pairs can be created by collisions between allergens that are accelerated by electromagnetic and electrostatic waves and/or by gravitational pushes. Intense laser-plasma relationship tests have reported the development of MeV electrons and conclusive proof positron creation via electron collisions. Positrons have also been created in post disruption plasmas in large tokamaks through collisions between MeV electrons and thermal contaminants. The progress in the creation of positron plasmas of days gone by two decades makes it possible to consider laboratory experiments on e-p plasmas [2].

The previous theoretical studies on linear waves in electron-positron plasmas have essentially focused on the relativistic regime relevant to astrophysical contexts [3]. This is largely because of the fact that the creation of these electron-positron pairs requires high-energy techniques. In lab plasmas non-relativistic electron-positron plasmas can be created by using two different techniques. In one plan, a relativistic electron beam when impinges on high Z-target produces positrons by the bucket load. The relativistic pair of electrons and positrons is then trapped in a magnetic mirror and cools down rapidly by radiation, thus producing non-relativistic pair plasmas. In another program positrons can be gathered from a radioactive source. Such non-relativistic electron-positron plasmas have been produced in the lab by many experts.

This has given an impetus to numerous theoretical works on non-relativistic electron-positron plasmas. Stewart and Laing [4] analyzed the dispersion properties of linear waves in equal-mass plasmas and found that because of the special symmetry of such plasmas, well known phenomena such as Faraday rotation and whistler wave modes fade away. Iwamoto [5] analyzed the collective settings in non-relativistic electron-positron plasmas using the kinetic strategy. He found that the dispersion relationships for longitudinal settings in electron-positron plasma for both unmagnetized and magnetized electron-positron plasmas were similar to the modes in one-component electron or electron-ion plasmas. The transverse modes for the unmagnetized circumstance were also found to be similar. However, the transverse modes in the presence of your magnetic field were found to be different from those in electron-ion plasmas. Studies of wave propagation in electron-positron plasmas continue steadily to highlight the role performed by the identical mass of electrons and positrons. For instance, the low rate of recurrence ion acoustic influx, a feature of electron-ion plasmas due to significantly different people of electrons and ions, has no counterpart in electron-positron plasma. Shukla et al [6] produced a fresh dispersion connection for low-frequency electrostatic waves in firmly magnetized non-uniform electron-positron plasma. They revealed that the dispersion connection admits a fresh strictly growing instability in the existence of equilibrium denseness and magnetic field inhomogeneties. Linear electrostatic waves in a magnetized four-component, two-temperature electron-positron plasma are looked into by Lazarus et al in Ref. [7]. They have got produced a linear dispersion relation for electrostatic waves for the model and analyzed for different wave settings. Dispersion characteristics of the methods at different propagation sides are examined numerically.

In this work, The dispersion properties of the propagation of linear waves in degenerate electron-positron magnetoplasma are looked into. Utilizing the quantum hydrodynamic equations with magnetic areas of the Wigner-Maxwell system, we've obtained a couple of new dispersion relationships where ions' motions aren't considered. The general dielectric tensor comes from using the electron and positron densities and its momentum respond to the quantum results anticipated to Bohm probable and the statistical effect of Femi heat.

2- MODELING EQUATIONS

We consider quantum plasma made up of electrons and positrons whose qualifications stationary ions. The plasma is immersed within an exterior magnetic field. The quasi-neutrality condition reads as. From model, the dynamics of the debris are governed by the following continuity formula and the momentum formula:

(1)

(2)

Here and are the number density, the speed and the mass of particle respectively () and is the plank frequent divided by. Let electrons and positrons follow the following pressure regulation:

Where, is the Fermi thermal velocity, is the particle Fermi heat, is the Boltzmann's frequent and is the equilibrium particle amount density. We have included both quantum statistical results through Fermi temperatures and the quantum diffraction in the -dependent. If we placed equal to zero and identical the temp of electrons and positrons, we have the classical hydrodynamic formula. Assuming that the plasma is isothermal, the Fermi rates of speed for different debris may be similar.

Using the perturbation approach, assume the number representing (n, u, B, E) has the pursuing form where is the unperturbed value and is a tiny perturbation. Assuming the equilibrium electric field is zero and linearizing the continuity and the momentum equations, we have:

(3) (4)

Multiplying equation (4) by and Simplifying, we can obtain the following formula:

(5)

where,

,

, and

Assuming, , then your three components of the fluid velocity can be written as:

(6a) (6b)

(6c)

Where,

and

The current density and the dielectric permeability of the medium receive:

(7)

(8)

where is the machine tensor. So, we can obtain the dielectric tensor the following:

(9)

Where,

Then, corresponding to equations (8), (9) The propagation of different electromagnetic linear waves in quantum plasma can be obtained from the following general dispersion relation:

(10)

Where, is the plasma frequency and.

3- DISCUSSION

In this section, we concentrate our attention on the discussion of some different modes in two circumstances that the influx vector parallel and perpendicular to the magnetic field.

(3. I) Parallel modes

So, this case causes, with

.

Therefore the general dispersion connection (10) becomes:

(11)

This provides two dispersion relationships. The first one () investigates the dispersion of electrostatic quantum waves included the quantum results as follows

(12)

By neglecting the quantum results, equation (11) describes the following well-known classical modes

The second dispersion equation gives:

(13)

Equation (13) is similar to the dispersion of left and right waves (L- and R- modes). Due to the symmetry between the positively and adversely charged allergens, the dispersion relation for the right circularly polarized influx is identical left circularly polarized influx. It has been mentioned that no quantum effects on these methods. For unmagnetized plasma, the dispersion relationship becomes:

(14)

(3. II) Perpendicular mode

In this case, we have

So, the overall dispersion connection (10) becomes:

(15)

Where it has the following new elements

, ,

, ,

, ,

,

In the case of unmagnetized plasma, we have the following two dispersion equations:

(16)

and

(17)

The formula (16) is the popular dispersion connection which investigates the propagation of electromagnetic waves in traditional unmagnetized plasma. The damping is absent because the phase velocity of the influx obtained from this equation is definitely higher than the velocity of light, so that no allergens can be resonant with the wave. This results is analogous to the one-component electron plasma [5]. While the other connection (17) signifies the dispersion of the waves in electron-positron plasma under the quantum results.

4- NUMERICAL Research AND RESULTS

In this section, we will investigate the aforementioned dispersion relationships numerically. Introducing the normalized volumes, , , , and the plasmonic coupling () which describes the ratio of plasmonic energy density to the electron Fermi energy density, we rewrite some of the dispersion relationships in both of parallel and perpendicular settings.

(4. I) Parallel modes

In the first, formula (12), () becomes:

(18)

Where, .

The dispersion relation (17) has two positive alternatives, Fig 1, for positron electron density ration with and. One of alternatives of the dispersion formula (19) can be investigated in Fig. (2) to review the parallel methods for different density ratios with in quantum plasma.

The solution of the normalized dispersion formula (17) has been also shown in 3D amount (3) for quantum unmagnetized plasma.

It is clear from the prior numbers that the dispersion relationships depend highly on the density ratio of positron to electron. As the positron density is increased to add up to the electron density, the period velocity has been increased. In the beginning, with very small positron density the wave consistency equals the electron plasma consistency and lowered with positron density increased.

Besides, in the Fig. (4), the dispersion connection of parallel methods is shown for different quantum ratios, in the case of positron-electron density percentage and identical velocities of these. It really is clear that the phase velocity of the setting is increased with the raises of plasmonic coupling ratio.

(4. II) Perpendicular mode

In the situation of perpendicular modes, equation (15) can be normalized and fixed numerically (here, ). Body (5) shows the dispersion curves of electromagnetic settings under the effect of different density ratios in traditional plasma.

Also, the other equation (16) can be solve numerically to give two real solutions. One of them is the same solution roughly of formula (15) (which is clear in Figure (6). The other solution of dispersion equation (16) is displayed in shape (7).

It is clear in the figures that the dispersion curves at count essentially on the positron-electron density ratio. As the positron density increases to similar electron density, the wave frequency is risen to be bigger than the plasma consistency.

On the dispersion curves (figures (5) and (6)), it has been observed the phase velocity of methods (+ve slope of the curves) lessens as density ratio raises. But, on the body (7), the stage velocities of the modes (-ve slope) will be the same with changes of the density percentage. They tend to zero with large wave number which means that these modes cannot propagate in plasmas.

Figure (8) investigates the dispersion relationships of the electromagnetic waves in electron-positron plasma under the quantum results. It is clear that, regarding classical plasma, the wave frequency reduces as wave number increases (the stage speed is negative). But, regarding quantum plasma (for small ratio ), the wave rate of recurrence deceases as influx number rises (the phase speed is negative). Then, the period velocity and group velocity tends to zero at certain wave number () is determined by the quantum ratio (). For high quantum proportion, the phase velocity begins to be +ve and raises again.

5-CONCLOUSION

In this work, The dispersion properties of the propagation of linear waves in degenerate electron-positron magnetoplasma are looked into utilizing the quantum hydrodynamic equations with magnetic fields of the Wigner-Maxwell system. The overall dielectric tensor is derived using the electron and positron densities and its own momentum respond to the quantum effects anticipated to Bohm potential and the statistical aftereffect of Femi temperature. We have obtained a couple of new dispersion relations in two conditions that the influx vector parallel or perpendicular to the magnetic field to research the linear propagation of different electromagnetic waves. It really is clear that the quantum results increase or reduce the phase speed of the methods will depend on the exterior magnetic field. Besides, it has shown that the dispersion curves at rely essentially on the positron-electron density proportion including the positron density is risen to equal electron density, the wave rate of recurrence of the settings is increased. .

Fig. (1). The dispersion connection (5. 19) has two positive solutions for positron electron density ration with and

Fig. (2) The dispersion relations of the modes for different density positron-electron ratios with and

Fig. (3). The dispersion relationships of the parallel methods along density ratioaxis with and

Fig. (4). The dispersion relations of different settings for different quantum effects with positron-electron density proportion and velocity proportion. .

, Fig. (5. 5). The dispersion relationships of electromagnetic methods for different ratios in classical plasma.

Fig. (6). The dispersion solutions of the equations (5. 17) and (5. 18) for different density ratios.

Fig. (7). The other dispersion alternatives of the formula (18) for different density ratios.

Fig. (8). 3D plotting for dispersion relationship for perpendicular modes in quantum unmagnetized plasma along quantum ratio axis with

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