The Quantum Effects Role on Weibel Instability Growth Rate in Dense Plasma

The Weibel instability is one of the basic plasma instabilities that plays an important role in stopping the hot electrons and energy deposition mechanism. In this paper, combined effect of the density gradient and quantum effects on Weibel instability growth rate is investigated.The results have shown that, by increasing the quantum parameter, for large wavelengths, theWeibel instability growth rate shrinks to zero. In the large wavelengths limit, the analysis shows that quantum effects and density gradient tend to stabilize the Weibel instability. The density perturbations have decreased the growth rate of Weibel instability in the near corona fuel, η > 0.1. In the small wavelengths limit, for the density gradient, η < 0.1, the tunneling quantum effects increase anisotropy in the phase space. The quantum tunneling effect leads to an unexpected increase in the Weibel instability growth rate.


Introduction
Quantum plasmas have attracted renowned attention in the recent years, due to, for example, the relevance of quantum effects in dense plasmas, very intense laser plasmas, and ultra-small semiconductor devices.The inclusion of quantum terms in the plasma fluid equations such as quantum diffraction effects, modified equations of state [1], and spin degrees of freedom leads to a variety of new physical phenomena [2].Quantum plasmas can be found in laser-solid density plasma interaction experiments, laser-based inertial fusion, astrophysical and cosmological environments, and quantum diodes.In laser-dense plasma interaction, when the beam propagates in the plasma, it will induce a return current to keep current neutralization of the beam-plasma system, resulting in the well-known three instabilities; the twostream, current filamentation, and the Weibel instabilities [3][4][5].The Weibel instability is one of the basic plasma instabilities that is driven by an anisotropic velocity distribution of plasma particles.Weibel instability arises from temperature anisotropy in the equilibrium distribution function and it is one of the fundamental instabilities of plasma physics.In more recent years, it has been the central concept in several instances, like fast ignition scenarios [6][7][8][9][10][11][12][13][14][15], particle acceleration, and magnetic field generation in astrophysical settings, collective non-Abelian Weibel instabilities in melting color glass condensates, covariant relativistic scenarios, electron-positron relativistic shocks, and laser heated plasmas.The Weibel instability plays an important role in stopping the hot electrons and energy deposition mechanism in the target.The energy deposition of a relativistic electron beam in a plasma can be managed through sufficiently steep plasma density gradients [16][17][18][19][20][21][22][23][24].At the very initial stage of heating of the dense plasma, the electron temperature at the critical layer may be as low as a few eV.The laser intensity is absorbed at a region with electron density of 0.02-1.0 of the critical density.In this region the quantum effects are not totally negligible.For this purpose, the kinetic Wigner-Maxwell model which is the quantum counterpart of the Vlasov-Maxwell system has been used.The aim of this contribution is to get detailed information about the influence of quantum effects on Weibel instability in density gradient of dense plasma.

Theoretical Model
We consider linear transverse waves in a dense plasma composed of electrons and immobile ions, with ( ⃗  ⋅ ⃗  = 0), where ⃗  is the wave vector and ⃗  is the wave electric field.The propagation direction is the -direction.The plasma will be heated only in the velocity dimension along the wave propagation direction, leading to a temperature anisotropy of the electron distribution that varies with the wave motion.Due to the density gradient of the target (the center is about 10 4 denser than the border) that is in the -direction, the beam-plasma interaction is collisionless near the relativistic electron beam-emitting region.In a dense laser created plasma, quantum effects must play an important role in the context of the Weibel instability.Following the standard procedure, one then obtains the general dispersion relation for the transverse waves of the Wigner-Maxwell system.The quantum dispersion relation for transverse waves is as given in where   = √4 2  0 /   is the plasma frequency and (, ) = [ 0 /  (, )] 2 is the density gradient in the direction.In general, in the absorption layer of fuel pellet, the thermal energy is much larger than both the potential energy and the Fermi energy, that is, the degeneracy parameter Θ =   /    is much smaller than unity.Here e is the electron charge,   is the Boltzmann constant, and   is the electron temperature.This means that quantum degeneracy effects are negligibly small.Therefore, the analysis is restricted to nondegenerate plasmas, so that quantum effects due to spin and statistics are beyond the scope of this work.Therefore, the distribution of the electrons is described by anisotropic Maxwell-Boltzmann.The electron distribution function is where  ‖ and  ⊥ are related to the velocity dispersion in the direction perpendicular and in parallel to the -axis, respectively,  is  ‖ / ⊥ , and V  is the mean velocity of the particles.In application of these results to implosion of fuel pellet, density is [25]   (, ) =  0 exp (   + 10) , where  is the density gradient scale length.The electron density and initial temperature of the plasma are taken to be  0 = 1 × 10 21 cm −3 and  0 = 500 keV, respectively.Considering spacetime dependence of the perturbations of the form exp (−), the dispersion equation for the Weibel transverse electromagnetic wave will obtain the following: where is the electron thermal speed, and  = ℏ/2  V ‖ is a characteristic parameter representing the quantum diffraction effect.With considering the plasma dispersion function as the dispersion relation will be corrected as follows: In limit large quantum effects,  ≫ 1, and thus  2 ≫ || 2 , the dispersion relation simplifies to Specifically, for large quantum effects, there will be a purely growing wave only if However, since the right-hand side of the last inequality is an increasing function of  and √  > 0, one can conclude that quantum effects play a stabilizing role.In limit small quantum effects,  ≪ 1, and, small wavelengths,  ≪ 1, the expansion of dispersion relation is as follows: And, in the plasma dispersion function, () ≅ √, the dispersion relation is given by The growth rate of the Weibel instability  ( =   +, where   and  are both real numbers) is determined by There can be instability ( > 0), provided that there is also sufficient temperature anisotropy: This condition implies that growing instability at the edge of fuel pellet needs extra anisotropy, due to quantum effects.But with the advance towards the center of the fuel pellet and decreasing the density gradient, required temperature anisotropy decreases.The unstable wave numbers of the Weibel instability are equal to where the allowable unstable wave numbers,  cut , occur for a smaller range for increasing quantum parameter, .In limit small quantum effects,  ≪ 1, and, large wavelengths,  ≫ 1, using the expansion () ≅ −1/ − 1/2 3 , the dispersion relation is given by Using definition thermal speed and , the dispersion relation reduces to For || ≪ , the growth rate of the Weibel instability is given by It is shown that purely growing waves are excited when One sees that for increasing quantum effects there is the need for extra temperature anisotropy.Due to wave-particle spreading and tunneling, quantum effects tend to enhance the dispersion of particles in phase space [26].Therefore, the effectively temperature anisotropy decreases, so that the temperature anisotropy has to be greater to produce the same instability results as in classical plasma.

Results and Discussions
As expected and displayed in Figure 1, for increasing quantum effects (larger values of  = ℏ/2  V ‖ ), the instability occurs with smaller maximal growth rates and unstable wave numbers shrink to zero.In Figure 2, the normalized growth rate, /  , is plotted as a function of the normalized wavenumber, /  , for different values of the density gradient, .Generally, as discussed by Schaefer-Rolffs and Tautz [23], the quantum effects without density gradient produce smaller Weibel instability growth rate and smaller ranges for  > 0.1, in the case of equilibria with distribution functions anisotropic in temperature.Considering density gradient in fuel pellet in limit large wavelengths, quantum effects lead to increasing growth rates and smaller ranges for unstable wave numbers (Figure 2(a)).Also, for the larger values of density gradient,  > 0.1, the maximum growth rate is quickly decreased and the instability occurs with smaller maximal wave numbers.As is shown in Figure 2(b), in limit small wavelengths for  < 0.1, quantum effects lead to an unexpected increase in the rate of growth.In interaction short-pulse laser-dense plasma of fuel, due to wave-particle spreading and tunneling, quantum effects tend to enhance the dispersion of particles in phase space.This corresponds to increases of temperature anisotropy and increases of Weibel instability rate.Figure 3 shows the growth rate of Weibel instability in terms of several values of density gradient, small wavelengths, and large wavelengths.For fixed quantum parameter, in transmission of relativistic electron beam from the side of the fusion pellet to its core and decreasing gradient density (the quantity  will have a variation range between zero and one, 0.0001 <  < 1), the thermal spread of the energetic electrons reduces the growth rate of Weibel instability.For  = 0.1, in limit small wavelengths,  ≪ 1, the instability growth rate and unstable wavenumber are

Conclusions
The Weibel instability is one of the basic plasma instabilities that is driven by an anisotropic velocity distribution of plasma particles.The study of this kind of instability is very important as it plays a very crucial role in heat deposition process from laser to dense plasma.In the interaction laser-dense plasma, a fast electron beam which makes its way to the target core generates at the critical density.The fast electron beam traveling through a very important density gradient, since the critical density where the laser deposits its energy, is smaller than the core density by four orders.Therefore, it is important to evaluate the effect of a density gradient upon the growth rate of Weibel instability.Calculations show that the Weibel instability growth rate is dependent on the density gradient of target and the quantum effects such as spin and tunneling, so increasing the quantum parameter and decreasing the density gradient in large wavelength will decrease the instability.The maximum growth rate of Weibel instability and wavenumber decreases by traveling the relativistic electron beam towards the target center.There is a critical scale length   = 0.1, under which the growth rate starts to increase as  decreases.For  <   , this could be attributed to sometimes quantum effects; for example, tunneling can give unexpected enhancement of Weibel instability growth rate and unstable wavenumber.

Figure 1 :
Figure 1: The normalized growth rate of the Weibel instability, /  , as a function of quantum parameter  = ℏ/2  V ‖ , for  = 0.1, and the large wavelengths.

Figure 2 :
Figure2: The growth rate of the Weibel instability, /  , as a function of wavenumber, /  , for the fixed density gradient, , (a) for the large wavelengths, (b) for the small wavelengths.

Figure 3 :
Figure 3: The normalized growth rate of the Weibel instability, /  , as a function of density gradient, , for the fixed quantum parameter  = ℏ/2  V ‖ .

Figure 4 :
Figure4: The growth rate of the Weibel instability, /  , as a function of wavenumber, /  , and quantum parameter, , for the fixed density gradient, , (a) for the small wavelengths, (b) for the large wavelengths.