Linear Properties of Electrostatic Wave in Two-Component Fermi Plasma
Shraddha Sahana, Spandita Mitra, Swarniv Chandra, Suman Pramanick
LLinear Properties of Electrostatic Wave inTwo-Component Fermi Plasma
Shraddha Sahana, Spandita Mitra, Swarniv Chandra and Suman Pramanick
Abstract —We study the two component Fermi plasma.Two components are electrons and ions. Using the Quantum-Hydrodynamic model (QHD), we study the linear propertiesof electrostatic wave. We derive the linear dispersion relationfor the system from dynamical governing equations for thesystem. We study the dependence of linear-dispersion relationon various parameters of the system.
PACS —52.35.Fp; 71.10.Ca
Index Terms —Dispersion relation; Electrostatic wave; Fermipressure; QHD model. N OMENCLATURE
EW-Electrostatic Wave; QHD: Quantum Hydrodynamics;DR- Dispersion RelationI. I
NTRODUCTION I n recent times, scientists believe that the universe consistsof 69 % of dark energy, 27 % dark matter, and 1 % normalmatter, and all these celestial objects we see in the nightsky are in the plasma state. Here, plasma (not the bloodplasma) is an ionized gas that consists of electrons and ions.Plasma is called “The Fourth State of Matter” [1]. Nowadays,Quantum plasma has become a major topic for research. Itsapplication is spreading from neutron stars, giant planets,dwarfs to laboratory plasmas, also in electron- hole plasma,electron gas etc. The basic condition for Quantum plasma islow temperature and sufficiently high density, unlike classicalplasma [2]. In extreme densities when the distance betweenparticles becomes as close as the quantum level, quantumtunneling plays a significant role. An electron feels a pressuredue to its quantum effect. The examples of these pressuresare Fermi pressure, relativistic pressure etc.[3]. In Quantumplasma field, many researchers (Haas et. al, 2003 [4]; Ali andShukla, 2006 [5]; Manfredi, 2005 [6] ) have used QuantumHydrodynamic Model (QHD). This QHD model treats plasmaas a fluid and uses fluid equations for investigating this [7].The electromagnetic fields in plasma have two parts- elec-trostatic and oscillatory. Waves are differentiated as electro-magnetic or electrostatic according to there is any oscillatory Shraddha Sahana is pursuing her B.Sc. degree from University of Calcutta,West Bengal 700073, India (E-mail: [email protected])Spandita Mitra is with Earth and Atmospheric Science, National Insti-tute of Technology, Rourkela, Odisha 769001, India (E-mail: [email protected])Swarniv Chandra is with the Physics Department, Government GeneralDegree College at Kushmandi, Dakshin Dinajpur 733121, India (E-mail:[email protected])Suman Pramanick is with The Department of Physics, Indian Instituteof Technology Kharagpur, Kharagpur, West Bengal 721302, India (E-mail:[email protected]) magnetic field or not. And based on these, we can say, EWsmust be longitudinal. So, If the magnetic field is zero thenthe wave is EW [1]. Now we will talk about dispersionproperties. We know that plasma is a dispersive medium.Dispersion relation (DR) describes the dispersive effects on theproperties of waves in plasma. It relates the frequency of wavewith its wave number. Whenever dispersion is present, wavevelocity cannot be defined separately, we need to define groupand phase velocity. The linear and non-linear properties ofelectron-acoustic solitary waves with three-component Fermiplasma has been studied in [8]. For three-component Fermiplasma the Rouge-wave formation and dynamical propertiesof electron-acoustic solitary waves has been done in [9].The paper is organized in the following structure: In Sec-[I], we have discussed about plasma waves and DR. In Sec-[II], we will set the governing equations using the QHD modeland will normalize them with normalization schemes. In Sec-[III], we will derive the dispersion relation using normalizedequations and standard reductive perturbation method. In Sec-[IV], we will discuss about plots of DR. And then, we willconclude the effects of dispersion relation on electrostaticwaves in Sec-[V]. II. B
ASIC E QUATIONS
Let us consider the propagation of electrostatic wavesin Quantum plasma consisting of electrons and ions. Also,assuming the plasma particles act like Fermi gas at zerotemperature, the pressure term will be- P j = m j V F j n j n j (1)where, j = e for electron j = i for ion m j = mass n j = Number density V F j = √ k B T Fj m j = Fermi speed T F j = Fermi temperature k B = Boltzmann’s constant.So, the set of QHD equations governing the dynamics ofQuantum plasma waves in a two-component plasma is listedbelow- ∂n e ∂t + ∂ ( n e u e ) ∂x = 0 (2) ∂n i ∂t + ∂ ( n i u i ) ∂x = 0 (3) (cid:18) ∂∂t + u i ∂∂x (cid:19) u i = 1 m i (cid:20) Q i ∂φ∂x + η i ∂ u i ∂x (cid:21) (4) a r X i v : . [ phy s i c s . p l a s m - ph ] J a n = 1 m e (cid:20) Q e ∂φ∂x − n e ∂P e ∂x + ¯ h m e ∂∂x (cid:20) √ n e ∂ √ n e ∂x (cid:21)(cid:21) (5) ∂ φ∂x = 4 π ( Q e n e − Q i n i ) (6)where, u j = Fluid velocity p j = Pressure q j = Charge j = e (For electron) j = i (For ion)Here, q e = − e , q i = + e ¯ h = ( h/ π ) where h is the Planck’s constantNow the normalization schemes for normalizing thegoverning equations which we have used are- x → xω pe /V F e , t → tω pe , φ → eφ/ k B T F e ,u j → u j /V F e , n j → n j /n j where, ω pe = (cid:115) πn e m e is the electron plasma wave frequency V F e = (cid:114) k B T F e m e is Fermi speed H = ¯ hω pe k B T F e
Using the normalization consideration, equation (2) to equa-tion (6) can be rewritten as- ∂n e ∂t + ∂ ( n e u e ) ∂x = 0 (7) ∂n i ∂t + ∂ ( n i u i ) ∂x = 0 (8) (cid:18) ∂∂t + u i ∂∂x (cid:19) u i = − µ ∂φ∂x + η i ∂ u i ∂x (9) ∂φ∂x − n e ∂n e ∂x + H ∂∂x (cid:20) √ n e ∂ √ n e ∂x (cid:21) (10) ∂ φ∂x = ( n c − n i ) (11)Here, µ = m e m i η = η i V F e ω pe H = ¯ hω pe k B T F e
H is a quantum parameter which is dimensionless and pro-portional to quantum diffraction. ¯ hω pe is the energy of initialoscillation of plasma waves and k B T F e is the Fermi energy. III. A
NALYTICAL S TUDIES
In order to study linear properties of plasma waves we usethe reductive perturbation expansion for the field quantities n e , n i , u e , u i and φ about equilibrium values listed below- n j u j φ = u φ + ε n (1) j u (1) j φ (1) + ε n (2) j u (2) j φ (2) + · · · (12)Now substituting the expansion (12) in equations (7) to(11),and then linearizing and assuming all these field quantitieschanges like e i ( kx − ωt ) , we got k which is wave numberand ω which is normalized wave frequency. So, the deriveddispersion relation is- ω = ku ± k (cid:115) µ (4 + H k ) k (4 + H k ) + 4 (13)Where, µ = m e m i , H = ¯ hω pe k B T Fe The equation (13) represents the dispersion relation ofelectrostatic waves in Fermi plasma. This quadratic equationhas two solutions- ω = ku + k (cid:115) µ (4 + H k ) k (4 + H k ) + 4 ω = ku − k (cid:115) µ (4 + H k ) k (4 + H k ) + 4 During the calculation of dispersion relation, we have assumedthe particles to be free from viscosity.IV. R
ESULTS AND D ISCUSSIONS
The linear dispersion relation of electrostatic wave in Fermiplasma with the help of one dimensional QHD model andstandard reductive perturbation method has been investigated.Keeping all the parameters within their range, we have plottedthe dispersion curve for different quantum diffraction parame-ters (H) and different streaming velocities u . And we have gotsimilar type of two dispersion curves that have been plottedbelow.The dispersion curve corresponds to electrostatic waves, wehave seen in both types of graphs keeping one parameterconstant and varying the other parameter, the curves showthe upward trend (We have plotted the dispersion curve withpositive ω vs k ).In fig (1) and (2), keeping the streaming velocity u constantand increasing the values of H we have plotted the graphs. Anincrease in H shows non-linearity in the graph. And after someregion, ω attains maximum value. In fig (3) and (4), keepingH constant and increasing the values of u we have seen theshift towards the upper direction in the wave frequency vswave number graph. ig. 1. 2D plot of dispersion relation for different Quantum diffractionparameter (H) with u = 0.5, µ = 1000.Fig. 2. 3D plot of dispersion relation for different Quantum diffractionparameter (H) with u = 0.5, µ = 1000. V. C
ONCLUSIONS
The Fermi-plasma consists of non-relativistic electrons andions. The dependency of wave frequency on Quantum diffrac-tion parameter and streaming velocity are studied thoroughly.It is shown that these two components H and u have animportant role in determining the linear properties of elec-trostatic waves. At high wave-number region, the dispersioncurve with constant H starts to split into different lines fordifferent streaming velocities u because at the high wave-number region the energy carrying capacity for different u is different. In the graph of dispersion, with increasing H,the quantum effect is increasing and so the non-linearity isincreasing as well. At high wave-number range, the frequencybecomes high, and for this reason, energy becomes high. Andbecause of the high energy, the curve becomes observablein the classical range. That’s why it is classically lineareverywhere. At a higher range of wave-number, ω attainsa maximum value. The newly gotten results will be usefulfor understanding the dispersion properties and obtaining thegroup velocities and phase velocities of the waves. And it willbe also helpful for studying the instabilities of the waves inFermi plasma. Fig. 3. Dispersion relation 2D plot for different streaming velocities u withH=2, µ =1000.Fig. 4. Dispersion relation 3D plot for different streaming velocities u withH=2, µ =1000. R EFERENCES[1] F. F. Chen et al. , Introduction to plasma physics and controlled fusion .Springer, 1984, vol. 1.[2] S. Pramanik, S. Mandal, and S. Chandra, “International journal of engi-neering sciences & management,”
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