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Nonlinear properties of ion acoustic solitary waves are studied in the case of dense magnetized plasmas. The degenerate electrons with relative density effects from their spin states in the same direction and from equally probable up and down spinning states are taken up separately. Quantum statistical as well as quantum tunneling effects for both types of electrons are taken. The ions have large inertia and are considered classically, whereas the electrons are degenerate. The collisions of ions and electrons with neutral atoms are considered. We derive the deformed Korteweg de–Vries (DKdV) equation for small amplitude electrostatic potential disturbances by employing the reductive perturbation technique. The Runge–Kutta method is applied to solve numerically the DKdV equation. The analytical solution of DKdV is also presented with time dependence. We discuss the profiles for velocity, amplitude, and time variations in solitons for the cases when all the electrons are spinning in the same direction and for the case when there is equal probability of electrons having spin up and spin down. We have found that the wave is unstable because of the collisions between neutral gas molecules and the charged plasmas particles in the presence of degenerate electrons.
The ion acoustic wave is the basic mode of plasma waves in collisionless as well as collisional plasmas. In collisionless plasmas, the pressure of electrons provides the restoring force, whereas the inertia comes from the mass of ions. The species couple with each other through electrostatic forces. In collisional plasmas, the collisions between different types of particles can also couple the dynamics of plasma species. This leads to more complicated dynamics of ions in the ion acoustic waves generated in collisional plasmas.[1,2]
Many authors have researched the ion acoustic waves propagating in collisional plasmas. Different aspects of ion acoustic waves in collisional plasmas were studied by Vranjes and Poedts.[3] They pointed out that the collisions with the neutral atoms modify the ion acoustic wave mode. This mode becomes evanescent at low density and then reappears at high density. The neutrals are dragged along electrons and ions at high density and the three fluids propagate together. The neutrals and plasma species are strongly coupled such that these two different fluids contribute towards the propagation of the wave together.
There are numerous laboratory applications of quantum plasmas. Some examples of their presence are quantum Coulomb crystals, inertial confinement fusion, as well as ultra small electronic devices. The dense plasma systems are also found in astrophysical objects such as white dwarfs, and even in more dense environments of neutron stars and pulsars.[4–9] The quantum effects in plasma appear when the Fermi temperature associated with the plasma number density exceeds the system temperature. The other condition is that the de Broglie wave length should be comparable to the inter-particle distance. Highly dense plasmas are created in atomic clusters which are irradiated by femtosecond laser pulses, where quantum effects appear and electron–ion collisions result in heating the plasmas.[10–12] Ghosh and Chakrabarti[13] studied the nonlinear electrostatic structures with electron–ion collisions and found that when the time scale of the electron–ion collisions is smaller than the time scale for the frequency mode oscillation of the nonlinear wave, the structures are described by the deformed Korteweg de–Vries (DKdV) equation.
The solitons with weak dissipation in quantum plasmas were investigated by Ghosh.[14] The collisions of neutral atoms with ions and electrons were taken into account. The author found that a solitary structure can get excited in the dissipative plasma system when the ion-plasma frequency is sufficiently high as compared to the ion-neutral collision frequency.
The damped solitary wave structures having degenerate attributes of ultra-relativisitic electrons, with nonrelativistic ions, were investigated by Ahmad et al.[15] The authors considered the neutral atom–electron and neutral atom–ion collisions using simple relaxation mechanism and concluded that the solitary wave is completely damped for τ → ∞.
In the above cited works, quantum statistical effects as well as quantum tunneling of fermions were taken into account in the presence of electron–neutrals and ion–neutrals collisions. The aim of this work is to include the spin effects of fermions in the presence of collisions. The spin effects play an important role in dense plasmas where they are used to measure more precisely the plasma number density as compared to the situations employed in the conventional techniques. Spin effects further contribute towards the measurement of magnetic field in association with plasma waves as well as in the resonance line of plasmas and its Doppler broadening width.[16,17]
The paper is arranged as follows. Section
We consider, in this work, the dense magnetized homogeneous plasmas comprised of ions and electrons. We take the electrons with spin-up and spin-down as separate species.[18] The collisions of neutral atoms with electrons and ions are also included. The equations that govern the dynamics of ions and degenerate electrons (in spin-up as well as spin-down states) in the presence of electric and magnetic fields are given as follows.
The ion continuity equation is
The equation for momentum conservation is
The electrons are degenerate and the effects of quantum statistics along with quantum tunneling and spin are also included in their dynamic equations. The momentum conservation equation for the spin-up electrons is
The spin-down electrons are considered separately and their momentum conservation equation is
We have scaled temporal coordinates by ion cyclotron frequency ωci while spatial coordinates by ion Larmor radius ρi. The densities of plasma species are scaled by their respective densities in equilibrium. The normalization of the electrostatic potential is done by eΦ/2kBTF, where TF denotes the Fermi temperature of electrons such that
For ions, the continuity equation, after normalization, is now
The x component of the spin-up electron momentum conservation equation is
The respective components of the spin-down electron momentum conservation equations are
The Poisson equation in these cases is normalized as
We assume the perturbation of the form expi(kxx + kyy + kzz − ωt), where ω is the frequency, we obtain the following dispersion relation:
In the absence of collision, we obtain the dispersion relation as that in Ref. [19].
In this section, we are going to derive the DKdV equation using the reductive perturbation technique.[20,21] The variables for space and time are stretched as
Perturbed quantities ni, vix, viy, viz, Φ, and μi,e are expanded in powers of ε in the following form:
The terms (ε5/2) in the ion continuity equation lead to
In the absence of dissipation, nonlinearity balances the dispersion, which is due to the quantum tunneling effects and solitons are retrieved. The dissipation in the system appears due to the collisions between neutrals and plasma species.
For numerical studies of Eq. (
Let ζ = μξ + U0τ, here α is the wave number and is considered as 1 and U0 is the velocity of the nonlinear structure, then equation (
We have solved Eq. (
We observe that by increasing the collisions between neutrals and plasma species, the amplitude of the oscillation increases, as shown in Figs.
If we ignore the collisions between plasma species and the neutrals, then C = 0 and we obtain KdV equation
Equation (
We also investigate the variations in time dependent velocity amplitude and the width of soliton profile. It is observed that for δ = 1, the degenerate electrons attain higher velocities initially and then decay with the passage of time. On the other hand, when the electrons are populated in both spin states, they attain smaller values of velocities. This fact is depicted in Fig.
We studied small amplitude damped ion acoustic solitary waves with three spatial component dense plasmas. We also derived the DKdV equation using the method for reductive perturbation. The numerical solution of the DKdV equation predicts that instability exists due to collisions between plasma species and neutrals. We have also observed that the time dependent speed and amplitude of the soliton profile decays sharply, when all the electrons are spinning in the same direction. The width enhances slowly for the situation when the degenerate electrons have equally populated both spin-up and spin-down states. Such results are relevant in astrophysical plasma environments where the spin of electrons has to be included, along with collisions, to describe highly dense systems. Our findings are also applicable to measure the plasma density and the width of Doppler broadening of resonance line of plasmas where the electron spin effects play an important role.[17,27]
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