Effects of Spin Quantum Force in Magnetized Quantum Plasma

Transcription

1 Commun. Theor. Phys. 56 ( Vol. 56, No. 4, October 15, 011 Effects of Spin Quantum Force in Magnetized Quantum Plasma YANG Xiu-Feng ( ô, JIANG Hong (, QI Xue-Hong (, and DUAN Wen-Shan ( College of Physics and Electronic Engineering, Northwest Normal University, Lanzhou , China (Received November 8, 010; revised manuscript received March 31, 011 Abstract Starting from the governing equations for a quantum magnetoplasma including the electron spin 1/ effects and quantum Bohm potential, we derive Korteweg-de Vries (KdV equation of the system of quantum magnetohydrodynamics (QMHD. The amplitude and width of magnetosonic soliton with different parameters in the system are studied. It is found that the normalized Zeeman energy E plays a crucial role, for E 1 the amplitude r mξ and the width w ξ of solitary wave all decrease as E increases. That is, the introduction of spin quantum force modifies the shape of solitary magnetosonic waves and makes them more narrower and shallower. PACS numbers: 5.7.Gr, g, 5.35.Bj Key words: quantum magnetosonic soliton, reductive perturbation method, KdV equation 1 Introduction In recent years, the plasmas physics has been widely studied and different research has been presented, such as laser plasma, [1 ] dusty plasma, and dust lattice, [3 8] quantum plasmas, [9 10] magnetoplasma [11 1] and so on. The concept of a magnetoplasma has attracted a great deal of interest ever since it was first introduced by Alfvén, [13] who showed the existence of waves in magnetized plasmas. Since then, the study of magnetoplasma has developed rapidly because of their relevance and diverse applications in the laboratory environment [14 17] as well as in the astrophysical environment. [18 ] Today, the magnetoplasma has been a mature research topic, with a wide range of applications. Meanwhile, so called quantum plasmas in which the quantum properties of the plasma particles are taken into account, have received attention. [17 18,3] It has important applications in metallic nanostructures, semiconductor physics, and astrophysical physics. Quantum effects can be measured by the thermal de Broglie wavelength of the particles composing the plasma λ B = /mv T, the quantum mechanical effects become important when λ B > λ D, where λ D is the de Broglie wavelength. For λ B < λ D, the de Broglie wavelength is so small that the plasma particles are treated classically and particles can be considered as pointlike. Especially, in recent years, there has been a great deal of interest in investigating collective quantum effects in plasmas. [4 30] That means particle motions that depend not only on local conditions, but also on the positions and velocities of all other particles in the plasma, the collective motion of quantum particles in magnetic fields thus gives a natural extension to the classical theory of magnetohydrodynamic (MHD in terms of so called quantum magnetoplasma. [31] Many of studies of quantum plasma have shown that quantum collective effects can have interesting consequences both in laboratory and astrophysical environment and motivated by recent experimental progress and techniques. [3 35] Indeed, from the experimental perspective, a certain interest has been directed towards the relation of spin properties to the classical theory of motion. However, in strong magnetic fields, single-electron effects that depend on the electron spin properties, such as Landau quantization, will be important. [36] In the past few years, the investigations of the nonlinear waves propagation in a complex plasmas have attracted much attention. [37 4] It is thus not surprising that collective spin effects can influence the wave propagation in a strongly magnetized quantum plasma. [18,3,43 44] In the ultracold plasmas, the thermal energy of the particles can be very small compared to the Zeeman energy of the particles in a magnetic fields. Thus, collective spin properties of quantum plasmas may be possible to detect in the near future. [36] From the definition of λ B, it is clear that quantum behavior will be reached much more easily for the electrons than for the ions, due to the large mass difference. In this paper, we study the effect of spin quantum force on solitary waves in an ultracold and magnetized quantum plasma. Starting from the governing equations for a quantum magnetoplasma including the electron spin 1/ effects and quantum Bohm potential, we use the reductive perturbation method, a Korteweg-de Vries (KdV equation of the system of QMHD is derived. The amplitude and width of magnetosonic soliton are also derived, we Supported by the National Natural Science Foundation of China under Grant No and the Natural Science Foundation of Northwest Normal University under Grant No. NWNU-KJCXGC Corresponding author, duanws@nwnu.edu.cn c 011 Chinese Physical Society and IOP Publishing Ltd

2 770 Communications in Theoretical Physics Vol. 56 study the influence of the spin quantum force on the amplitude r mξ and the width w ξ of magnetosonic soliton. The paper is organized as follows. In Sec., the governing equations are presented. The KdV equation and discussion are presented in Sec. 3, finally the conclusion is presented in Sec. 4. Governing Equations At room temperature and standard metallic densities, quantum effects can no longer be ignored, so that the electron gas constitutes a true quantum plasma, the quantum magnetohydrodynamics (QMHD model has to be employed to study quantum plasma system instead of the classical fluid model. [17,45] We consider a QMHD model to describe the dynamics of magnetosonic soliton in a two component quantum magnetoplasma consisting of electrons and ions. We begin by presenting the general governing equations for a quantum magnetoplasma in which the electron 1/ spin effect are included. We introduce the total mass density ρ (m e n e + m i n i, the center-ofmass fluid flow velocity V (m e n e v e + m i n i v i /ρ, and the current density j = en e v e + en i v i. Here m e (m i is the electron (ion mass, n e (n i is the electron (ion number density, v e (v i is the electron (ion fluid velocity, and e is the magnitude of the electron charge. From Refs. [18] and [3], we obtain the continuity equation ρ + (ρv = 0. (1 t Assuming quasi-neutrality, i.e., n e = n i, we next add the momentum conservation equations for the electrons and positrons to obtain ( ρ t + V V = j B P + F Q, ( where P is the scalar pressure in the center-of-mass frame, j = µ 1 0 (B µ 0M is the current produced by the free charges, where we have neglected the displacement current is j D = ǫ 0 t E. M = (µ B ρ/m i tanh (µ B B/k B T e ˆB is the plasma magnetization due to the electron spin, and the F Q is the total quantum force due to collective tunneling and spin alinment. Here we will make use of the following expression for the quantum force on the electrons/positrons [18,3] F Q = ρ ( 1 ρ ρ + µ ( Bρ µb B tanh B, (3 m e m i m i k B T e where the first term is the gradient of the so called Bohm potential, the second term comes from the spin, we call the spin quantum force. Here µ B = e /m e is the magnitude of the Bohr magneton, is Plank s constant divided by π. The coupling between the quantum plasmas species is mediated by the electromagnetic field, the total magnetic field include both the classical contribution (from currents j = q j n j V j and the spin sources. The generalized Faraday law takes the form B { t = V B j B ηj m [ e ( B ] B en e e µ 0 t F } Q, (4 eµ 0 n e n e en e where η is the plasma resistivity and we have omitted the anisotropic part of the pressure. 3 KdV Equation and Discussion Now, we assume that the magnetic field is along the z direction such that B = B(x, tẑ, while we have the velocity V = V (x, tˆx and the density ρ(x, t. We obtain the governing equations for this system as follows: [36] V t + V ( V B t + (Bv B λ = 0. = B B µ 0 ρ lnρ C s + c λ m e ( 1 ρ c ρ m i + µ B m i ρ [ ( µb B ] ρbtanh, (5 k B T e (6 Here λ = η/µ 0 is the magnetic diffusivity and c is the speed of light in vacuum, where we have introduced the Compton wavelength λ c = c/ω c = /m e c, ω c is Compton frequency, C s = [k B (T e + T i /m i ] 1/ is the sound speed, we also have introduced the Alfvén speed C A = (B0/µ 0 ρ 0 1/ and the temperature-normalized Zeeman energy E = µ B B 0 /k B T e, B 0 = B 0 ẑ is the external magnetic field. If the plasma resistivity is weak, we can obtain the frozen-in-field condition ρ = ρ 0 b, here b = B/B 0 is the magnetic field, and we may neglect the last term in the Eq. (6. Then Eqs. (5 and (6 form a closed system. We now introduce the normalizations: t = tω ci, x = x(ω pi /c, ω pi = (n i0 e /ǫ 0 m i 1/ is the frequency of the ion plasma. With this Eq. (5 and (6 can be rewritten into v t + ( v = b lnb c s + ω pe ( 1b b ω ce ω c +v {ln[cosh(eb]+eb tanh(eb}, (7 B b t + (bv = 0, (8 where v = V/C A, c s = C S /C A, v B = k B T e /m i C A,

3 No. 4 Communications in Theoretical Physics 771 ω pe is the electron plasma frequency, and ω ce = eb 0 /m e is the electron cyclotron frequency. For generality and the simplicity, we have droped the bars on the normalized coordinates. From the last term of Eq. (7, we can clearly see that for E = 0, it means that we have not considered the spin quantum force on this system, and E 0 the spin quantum force term have been considered. If we neglect the term of spin quantum force on the system, we can obtain the governing equations for this system as follows v t + ( v = b lnb c s + ω pe ( 1 b ω ce ω c b, (9 b t + (b v = 0, (10 where b is the magnetic field in the case of not considered the spin quantum force on this system, and v is the velocity. In order to more clearly understand the impact of spin quantum force on the solitary wave, next, we use the traditional perturbation method as previous used extensively to obtain a KdV equation. Firstly, we introduce the stretched coordinates as follows: ξ = ε(x u 0 t, τ = ε 3 t, (11 where u 0 is the phase velocity of waves, and ε is a small parameter characterizing the strength of nonlinearity. Secondly, the dependent variables are expended as follows: b = 1 + ε b 1 + ε 4 b +, υ = ε υ 1 + ε 4 υ + (1 Substituting Eqs. (11 and (1 into Eqs. (7 and (8 and collecting the same order of ε. Finally, we derive the famous Korteweg-de Vries (KdV equation b 1 τ + αb b 1 1 ξ + β 3 b 1 = 0, (13 ξ3 where the coefficients are α = 3 + c s vbe (6 5E s v B E (3 (5/3E, (14 ω pe β = ω ce ω c s vbe (3 (5/3E. (15 In Eq. (13, the second term stands for the nonlinear term, it is observed that the nonlinear term increases as α increases, the last is dispersion term, we can see that as β increases, the dispersion term increases. It is well known that the stationary solitary wave solution of the KdV equation (13 can be written as follows: b 1 = r mξ sech ( ξ + u 0ξ τ, (16 w ξ where r mξ = 3u 0ξ α, (17 w ξ = ( 4β u 0ξ 1/, (18 are the amplitude and the width of a spin soliton respectively. u 0ξ is a constant speed normalized by C A. Substituting Eqs. (14 and (15 into Eqs. (17 and (18, we immediately obtain the amplitude and the width of a spin soliton as follows: r mξ = 6u 0ξ s vbe (3 5E /3 3 + c s v B E (6 5E, (19 [ ww ξ = ω pe u 0ξ ω ce ω c s v BE (3 5E /3 ] 1/. (0 From Eqs. (19 and (0 we can clearly see that the amplitude r mξ and width w ξ of a soliton are related to the speed u 0ξ and the normalized Zeeman energy E. Without considering the spin quantum force, we obtain the amplitude and the width of a soliton as r mξ = 6u 0ξ s 3 + c, (1 s ( w ξ = ωpe 1/. ( u 0ξ ω ce ω c s Next, we use the numerical simulation method to discuss the spin quantum force parameters E on the amplitude and width of magnetosonic soliton, in our simulation we choose E = 0,, 5, [3] the results are shown as follows: Figure 1 shows how the spin quantum force affects the amplitude r mξ and the width w ξ of solitary wave for different values of E. From the Eq. (7 we can clearly see that the parameter E stands for the effect of spin quantum force on the system of quantum magnetohydrodynamics (QMHD. For E = 0, it means that we have not considered the spin quantum force on this system, and E 0 (E = and E = 5 the spin quantum force have been considered. As observed, in Fig. 1, those curves are remarkable similar, it is also clearly seen that the amplitude r mξ and the width w ξ of solitary wave are all larger for E = 0 than E 0, it is also found that r mξ and w ξ decrease with increasing of E. In particular, the spin quantum force contribution to the soliton dynamics is enhanced when the Zeeman energy E is of the order of or greater than one. Traditional plasma physics has mainly focused on regimes characterized by high temperatures and low densities, for which quantum mechanical effects have virtually no impact. However, we mainly consider quantum plasma, which is high-density and the low temperature. In such cold plasmas, the thermal energy of the particles can be very small compared to the Zeeman energy of the particles in a magnetic fields. Thus, the Zeeman energy E plays an important role in this environment. It is clear from Fig. 1 for larger values of the Zeeman energy E the solitary wave became narrower and shallower. It means that the spin introduces a

4 77 Communications in Theoretical Physics Vol. 56 pressure-like term with negative sign in this system, which makes the solitary wave became narrower and shallower. that as E increases the nonlinear term increases and the dispersion term decreases in Kdv equation, for which the solitary waves become narrower and shallower. Fig. 1 The b 1 is plotted against ξ, keeping different spin quantum force parameters E, E = 0 (solid line, E = (dashed line, E = 5 (dotted line. Other parameter values are c s = 0.1, v B = 0., u 0ξ = 1.0, and ω pe/ ω ce ω c = 1. Fig. 3 The variation of the width ω ξ with the spin quantum force parameter E for different values of speed u 0ξ = 0., 0.5, 1.0, other parameter values are c s = 0.1, v B = 0., and ω pe/ ω ce ω c = 1. Fig. The variation of the amplitude r mξ with the spin quantum force parameter E for different values of speed u 0ξ = 0., 0.5, 1.0, other parameter values are c s = 0.1, v B = 0.. Figures and 3 show the variation of r mξ and w ξ as the parameter E for different values of speed u 0ξ = 0., 0.5, 1.0 and these curves are also remarkable similar. It is clear from Fig. that for E 1 the amplitude r mξ decreases as E increases, but the r mξ increases as u 0ξ increases for a given value E, which is similar to the Fig. 1. In the Fig. 3, we can clearly see that w ξ decreases as E and u 0ξ increase. From the Eqs. (14 and (15 we know that α increases as E increases, while β decreases as E increases. It indicates 4 Conclusion The work in this paper can be summarized as follows: The effects electron spin 1/ on the existence of magnetosonic solitary waves in an ultracold and magnetized quantum plasma are investigated by theoretical and numerical simulation. By using the reductive perturbation method, a KdV equation is derived. The amplitude r mξ and width w ξ of a soliton are also derived. The effects of spin quantum force parameter E and speed u 0ξ on the amplitude and width are studied in this paper. It is found that the normalized Zeeman energy E played a crucial role, for E 1 the amplitude r mξ and the width w ξ of solitary wave all decrease as E increases. It is also found that the speed u 0ξ is faster, the amplitude r mξ of solitary wave is larger but the width w ξ is smaller. Therefore, we conclude that the enhancement of the speed makes the solitary wave wider and deeper, and the introduction of spin quantum force modifies the shape of solitary magnetosonic waves and makes them more narrower and shallower. As a final statement, it is important to stress that our numerical simulations could give a great contribution to understanding the effect of spin in magnetized quantum plasma and to detect the collective spin properties of quantum plasmas in the near future. References [1] B.S. Xie, H.C. Wu, H.Y, Wang, N.Y. Wang, and M.Y. Yu, Phys. Plasmas 14 ( [] D.X. Tian, G.J. He, J.N. Han, and W.S. Duan, Commun. Theor. Phys. 51 ( [3] P.K. Shukla and V.P. Silin, Scr. 45 ( [4] P. Duan, J.Y. Liu, Y. Gong, Y. Zhang, Y. Liu, and X.G. Wang, Acta Phys. Sin. 56 ( [5] D.Z. Wang and H.T. Wu, Chin. Phys. 11 (

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