- Email: [email protected]
Obliquely propagating ion-acoustic solitary and shock waves in magnetized quantum degenerate multi-ions plasma in the presence of trapped electrons
Obliquely propagating ion-acoustic solitary and shock waves in magnetized quantum degenerate multi-ions plasma in the presence of trapped electrons
Journal Pre-proof
Obliquely propagating ion-acoustic solitary and shock waves in magnetized quantum degenerate multi-ions plasma in the presence of trapped electrons M.A. El-Borie, M. Abd-Elzaher, A. Atteya PII: DOI: Reference:
S0577-9073(19)30941-4 https://doi.org/10.1016/j.cjph.2019.10.004 CJPH 964
To appear in:
Chinese Journal of Physics
Received date: Revised date: Accepted date:
19 January 2019 2 October 2019 2 October 2019
Please cite this article as: M.A. El-Borie, M. Abd-Elzaher, A. Atteya, Obliquely propagating ion-acoustic solitary and shock waves in magnetized quantum degenerate multi-ions plasma in the presence of trapped electrons, Chinese Journal of Physics (2019), doi: https://doi.org/10.1016/j.cjph.2019.10.004
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.
Highlights
• Obliquely IAWS have been investigated. • KdV and Burger equations were derived. • Trapping effect is discussed. • Electron degenerate temperature effect is discussed. • Solitary and Shock waves are plotted.
1
Obliquely propagating ion-acoustic solitary and shock waves in magnetized quantum degenerate multi-ions plasma in the presence of trapped electrons M. A. El-Borie∗ Department of Physics, Faculty of Science, Alexandria University, Alexandria, P.O. 21511, Egypt, M. Abd-Elzaher† Department of Basic and Applied Sciences, Faculty of Engineering, Arab Academy for Science, Technology and Maritime Transport, Alexandria, Egypt. A. Atteya Department of Physics, Faculty of Science, Alexandria University, Alexandria, P.O. 21511, Egypt.‡
Abstract The properties of obliquely propagating ion-acoustic waves have been investigated in multi-ions magnetized plasma comprising of inertial, positively and negatively charged ion fluids, trapped electrons, and negatively charged stationary heavy ions. The propagation of the waves is oblique to the ambient magnetic field which is along the z-direction. Only fast type of modes exists in the linear regime. The reductive perturbation method was adopted to derive the Korteweg– de Vries (KdV) and Burger equations, as well as the solitary and shock wave solutions of the evolved equations, have been used to analyze the properties of the small but finite amplitude waves. The effects of the constituent plasma parameters, namely, the trapping effect of electrons, the electron degenerate temperature and the viscosity coefficient on the dynamics of the small amplitude solitary and shock waves have been examined. The influence of the magnetic field and the obliquity parameter on the propagation characteristics of ion-acoustic waves are discussed.
∗
Electronic address: [email protected]
†
Electronic address: [email protected]
1
‡
Electronic address: [email protected]
2
Keywords: Quantum plasma, Plasma waves, Multi-ions plasma, Solitary waves, Shock waves, Obliqness, Trapped electrons.
I.
INTRODUCTION
Great current interest has been given to the field of quantum plasma which has a long diverse tradition [1 − 6]. These interests are attributed to its applications in modern technology (metallic nanoparticles, nanotubes, quantum wells, and quantum dots, spintronics, nanoplasmonic devices, etc.). The ultrafast spectroscopy techniques have recent advancements that enabled us to monitor the confinement of electron gas in metallic plasmas which are considered in the femtosecond range [5]. Accordingly, relativistic degenerate plasmas have received great attention [6 − 8], where particle velocities become comparable to the speed of light. Astrophysical compact objects such as neutron stars, quasars, white dwarfs, black holes, pulsars are relativistic degenerate plasmas examples. The basic constituents of the white dwarfs are mainly carbon, oxygen, helium with hydrogen gas. It is assumed that the whole star is composed of these three elements with the heaviest at the core and the lightest at the crust [9]. It is well known from the nuclear structure studies that the isotopes of Mg, Ne, O, C, He, etc. found in white dwarfs are bosons [10]. The heavier element like iron is present in relatively massive white dwarfs. The heavy nuclei are mainly formed in the interior of these massive stars. When these stars under certain conditions Shrink to very high densities, they become degenerate and heavy elements begin to form. When an explosion occurs, a part of these heavy elements distributes into the surrounding space and leaves some stellar remnants in the aspects of the white dwarf [11]. The electron number density has very high value in this degenerate compact objects [2 − 4]. The equation of state for degenerate electrons and ions for these compact objects is mathematically explained by Chandrasekhar [12] for two cases, the non-relativistic and ultra-relativistic cases. The ion pressure for the non-relativistic case can be given by Pi = Ki nαi
(1)
where ni is the number density of ions and 3 π 13 π} 3 5 ' Λc }c, α = , Ki = 3 5 3 m 5 3
(2)
(in which Λc = π}/mc = 1.2 × 10−10 cm, and } is the Planck constant divided by 2π). Nowadays, the pressure laws (1) and (2) were used [13] to investigate the linear and nonlinear properties of electromagnetic and electrostatic waves, by using the non-relativistic quantum hydrodynamic[1] and quantum-magnetohydrodynamic [13] models. The range of quantization is achieved by bringing the Landau quantization of the electron motion and the magnetic field into account [14] and the condition KB T << ~Ωce (Ωce is the electron cyclotron frequency) has to be fulfilled. However, the Fermi ground state energies stay constant as the Landau energy-levels pair cancel out each other [14]. The particle trapping is a phenomenon that affects the plasma waves. The particle confinement in a certain region of phase space by the wave potential is the mean of trapping. Oscillations are performed by these trapped particles. The particle trapping has been observed in space plasmas and also in laboratory experiments [15 − 18]. The particle trapping cannot be understood by using the linear theory of waves, it is demonstrated that particle trapping is basically a nonlinear phenomenon. The usual concept of linear wave theory based on the van Kampen or Landau approach using the Vlasov theory breaks down entirely for the wave propagation with phase speeds is close to the thermal velocity in the presence of resonant particles. Modification of the wave propagation characteristics in collisionless plasmas may significantly be achieved by these nonisothermal particles. In the case of collisionless plasmas, the wave propagation manners are significantly altered by these nonisothermal particles. The first analytical method to construct equilibrium electrostatic structures including trapping particle was provided by Bernstein et al. [19]. However, the pseudo-potential method to construct equilibrium solutions was developed by Schamel [20 − 24]. As a consequence of this research, Schamel’s work is considered as a refinement in the phase space theory of vortices or holes. The large as well as small amplitudes electrostatic waves are affected by the electron trapping effect [25]. From this, the presence of trapped particles makes the lowest order excited plasmas distributions that satisfying the linearized Vlasov equation to loses its basis. Accordingly, due to the presence of trapped particles, an increased spectrum of waves takes place. The trapped electrons have lesser kinetic energy than the wave potential energy. These trapped electrons forth and bounce back in the potential well, which satisfies Schamel-like one-dimensional distribution [26, 27]. Many authors[25, 28−30] focused on the particles trapping in nonlinear solitary waves and showing that the basic properties of the ion-acoustic waves (IAWs) are significantly modified 4
by the effect of an external magnetic field. Anisotropic pressure is achieved in plasma as a result of applying a strong magnetic field [31]. Under the adiabatic condition, the anisotropic pressure has parallel and perpendicular components due to variations in the plasma density during the wave propagation, and the wave potential in plasmas leads to trapping effects [31]. Electron acoustic solitary waves in a magnetized plasma were investigated through the effect of the obliqueness and the presence of trapped hot electrons [28]. The vortices formation was investigated also due to the effect of trapping in the classical plasma [29], the trapping effect on the Sagdeev potential was also investigated. Obliquely propagating arbitrary amplitude IASWs have been studied in a magnetized electron-ion plasma [30]. The electrostatic waves were presented to be controlled by the nonlinearity of trapping for arbitrary amplitudes, where the infinitesimal amplitude limits are included by these arbitrary amplitudes [25]. Numerous observations related to space and laboratory plasmas have been revealed [32 − 35], which emphasize particle trapping in dense astrophysical plasmas. Investigation of the effect of trapping as a microscopic phenomenon in an inhomogeneous degenerate plasma is executed in the presence of a quantizing magnetic field [36]. The results were analyzed numerically for different values of the quantizing magnetic field and for different astrophysical plasmas. In the present paper, we consider the influence of trapping on the propagation of twodimensional ion-acoustic and shock waves in the quantum degenerate multi-ion plasma system as a result of the presence of Landau quantization and the inertialess electrons FermiDirac distribution function. Therefore, Korteweg–de Vries (KdV) and Burger equations have been derived and then the formation of solitary and shock waves has been studied. The layout of the present manuscript is as follows: the model and the basic equations with the carrying out of the nonlinear analysis are described in Sec. II. The numerical results are presented and discussed in Sec. III. The conclusion of our findings is given in Sec. IV.
II.
THE MODEL
Multi-ion collisionless, magnetized plasma system composed of trapped distributed electrons with inertial negatively and positively charged ions and negatively charged immobile heavy elements are considered in this study. The magnetic field B0 is along the z-direction, i.e., B0 = zˆB0 . The nonlinear propagation of finite-amplitude IA waves in such a multi-ion 5
plasma system, in case of planar geometry, is governed by ∂n+ ∂t ∂n− ∂t
+ ∇(n+ u+ ) = 0,
+ ∇(n− u− ) = 0, − 31 ∂u+ 5 ∇n+ + γ∇2 u+ + Ωc (u+ × zˆ) , + (u .∇)u = −∇φ − Kn + + + ∂t 3 1 − ∂u− + (u− .∇)u− = β∇φ − 53 Kβn− 3 ∇n− + γ∇2 u− − βΩc (u− × zˆ) , ∂t ∇2 φ = δne + σn− − n+ + Zh µ.
(3)
where n+ , n− , and ne are the total number densities of inertial positive ions, inertial negative ions, and trapped electrons, respectively. ui is the plasma species fluid speed normalized by Cim = (me c2 /Zi mi )
1/2
with c being the light speed, mi and me are the rest mass of
plasma ion and electron species, respectively. Ωc = (Z+ eB0 /m+ ) /ωp+ is the positive ion cyclotron frequency, where e is the electron charge with ωp+ is the positive ion plasma 1/2 frequency ωp+ = 4πZ+2 n+0 e2 /m+ . φ is the electrostatic wave potential normalized
by me c2 /e . β = Z− m+ /Z+ m− represents the masses ratio of the positive ions and the negative ions multiplied by the negative to positive charge per ion ratio. δ = ne0 /Z+ n+0
is the electron and positive ion number density ratio multiplied by positive ion charge. 2
3 Ki /me c2 . γ is the viscosity coefficient and σ = Z− n−0 /Z+ n+0 is the negative K = n+0
and positive ion number density ratio multiplied by their ion charge and µ = nh0 /Z+ n+0 is the heavy element and positive ion number density ratio multiplied by Z+ , the number of electrons ejected from the heavy element is Zh . Space (x, y & z ) and time (t) variables 1/2 are normalized by λD = kB T+ /4πZ+2 n+0 e2 and the inverse of the positive ion plasma −1 frequency, ωp+ , respectively. The trapped electrons number density for partially degenerate
plasma is expressed by 1 3 3 ηT 2 − 32 − 21 2 2 2 ne = η (1 + φ) + (1 + φ − η) − (1 + φ) + T (1 + φ − η) . 2 2
(4)
The effect of the quantizing magnetic field appears through the parameter η = ~Ωce /EF e 3/2
where EF e (Fermi energy of the electron) = ~2 (3π 2 ne0 ) √ normalized by T = πTe / 2EF e .
A.
/2me . The temperature T
is
Derivation of the KdV Equation
The obliquely propagating small finite-amplitude IAWs dynamics in a multi-ion magnetized quantum plasma with trapping electrons has been investigated by employing the 6
reductive perturbation method to drive the KdV equation. the stretched coordinates are introduced as
1 ζ = 2 (Lx x + Ly y + Lz z − Vp t) , 3 τ = 2 t,
(5)
where is a smallness parameter for measuring the perturbation amplitude, Vp is the perturbation mode phase speed that is normalized by the IA speed. The directional cosines of the wave vector k are lx , ly , and lz along x, y, and, z -axes, respectively. Therefore lx2 + ly2 + lz2 = 1. The dependent variables ni,e , ui , and φ may be expanded in a power series of as; (1) (2) (3) ni = 1 + ni + 2 ni + 3 ni + ... , (1) (2) (3) ne = 1 + ne + 2 ne + 3 ne + ... , 3 5 (1) (2) (3) uix,y = 2 uix,y + 2 uix,y + 2 uix,y + ... , (1) 2 (2) 3 (3) uiz = uiz + uiz + uiz + ... , (1) 2 (2) 3 (3) φ = φ + φ + φ + ... .
(6)
To develop equations in various powers of , we substitute Eqs. (4 − 6) in Eq. (3). The
Poisson equation 0 coefficient gives the current plasma model neutrality condition, δ + σ − 1 + Zh µ = 0. The next order in gives the following relations (1) 3Vp Lz φ(1) , n+ 3Vp2 −5KL2z (1) (1) p Lz (1) u−z = − 3V 3βV , n− 2 −5KβL2 φ p z √ 3η (1+T 2 ) (1) ne = 3 21−η + 4 (1)
u+z =
=
3L2z (1) 3Vp2 −5KL2z 3βL2z 3Vp2 −5KβL2z
φ ,
=− −
T2 2(1−η)3/2
φ(1) ,
φ(1) .
(7)
The perturbed quantities are accompanied in Poisson’s equation. The expression of the IA waves phase speed in this magnetized plasma model is derived as; s p M + M 2 − 36δβF S Vp = Lz , (8) 18δF √ 3η (1+T 2 ) T2 where F = 3 21−η + − , M = 15K (1 + β) δF + 9 (1 + σβ) and S = 4 2(1−η)3/2
(15K (1 + σ) + 25K 2 δF ). From the equation, (8), we conclude that the phase velocity
depends on the ratio of masses and charges of negative and positive ions through β, the obliqueness through the change of Lz , the trapping parameter, η, and the negative ions and electron number densities through σ and δ, respectively. The x and y components of the 7
lowest order momentum equations can be expressed as 3Vp2 Ly
(1) ∂ (1) φ , u+y = − Ω 3V 2 −5KL2 ∂ζ c( p z) 3Vp2 Ly (1) (1) ∂ (1) u−x = − Ω 3V 2 −5KβL φ , u−y 2 ∂ζ (1) u+x
c
(
p
z
)
= =
3Vp2 Lx
∂ (1) φ , Ωc (3Vp2 −5KL2z ) ∂ζ 2 3Vp Lx ∂ (1) φ . Ωc (3Vp2 −5KβL2z ) ∂ζ
(9)
By applying these equations in the next higher order of the momentum, the following relations are considered 3Vp3 Lx
(2) u+x (2) u−x
=
(2) ∂ 2 (1) = Ω2 3V 2 −5KL2 ∂ζ , u+y 2φ ( ) c p z 3V 3 Lx (2) ∂ 2 (1) − βΩ2 3V 2p−5KβL φ , u−y 2 ∂ζ 2 c
(
z
p
)
=
3Vp3 Ly Ω2c (3Vp2 −5KL2z ) 3Vp3 Ly
= − βΩ2
∂ 2 (1) φ , ∂ζ 2
(
2 2 c 3Vp −5KβLz
)
∂ 2 (1) φ . ∂ζ 2
(10)
On substitution from the above equations in the next higher-order continuity equation and the z-component of the momentum equation, trapped electrons and Poisson’s equations the following relations are obtained; (2)
∂n+ ∂ζ
∂ (2) n ∂ζ e
=
√3 4 1−η
=
18Vp L2z
9Vp4 (1−L2z )
∂ (1) φ ∂τ
∂ 3 (1) φ ∂ζ 3
+ 2 2 2 2 Ωc (3Vp −5KL2z ) ) 2 ∂ (1) ∂ (2) z + φ(1) ∂ζ φ + 3V 23L φ , ( p −5KL2z ) ∂ζ ( ) (2) 2 9Vp4 (1−L2z ) ∂n− p Lz ∂ (1) ∂ 3 (1) = − 18βV − 2 ∂τ φ 2 3φ ∂ζ βΩ2c (3Vp2 −5KβL2z ) ∂ζ (3Vp2 −5KβL2z ) 2 2 4 3 6 2 81β Vp Lz −15Kβ Lz (1) ∂ (1) ∂ (2) z φ ∂ζ φ − 3V 23βL φ , + 3 2 p −5KβLz ∂ζ (3Vp2 −5KβLz2 ) √ 2 3T 2 ∂ (1) φ(1) ∂ζ φ + 3 21−η + 34 η (T 2 + 1) − − 15ηT8 +3η + 4(1−η) 5/2 (
3Vp2 −5KL2z 81Vp2 L4z −15KL6z 3 3Vp2 −5KL2z
T2 2(1−η)3/2
∂ 2 (1) (2) (2) φ = δn(2) e + σn− − n+ . ∂ζ 2
∂ ∂ζ
(2) φ ,
(11)
(12)
Substitution of Eq. (11) into Eq. (12) led to the following KdV equation for the obliquely propagating IAWs in a multi-ion magnetized plasma with trapped electrons, ∂φ(1) ∂ ∂3 + Aφ(1) φ(1) + B 3 φ(1) = 0, ∂τ ∂ζ ∂ζ
(13)
where A=
81Vp2 L4z −15KL6z
81σβ 2 Vp2 L4z −15Kσβ 3 L6z
√3 4 1−η
15ηT 2 +3η 8
− −δ − + 3 2 2 (3Vp −5KβLz ) 9(1−L2 )σV 4 9(1−L2 )V 4 B = 1 + 2 2 z p2 2 + 2 2 z p 2 2 Q, Ω 3V −5KLz ) βΩc (3Vp −5KβLz ) c( p −1 18Vp L2z 18σβVp L2z Q= . 2 + 2 (3Vp2 −5KL2z ) (3Vp2 −5KβL2z ) 3
(3Vp2 −5KL2z )
8
3T 2 4(1−η)5/2
Q,
(14)
The KdV solitary wave solution is obtained by considering a frame ξ = ζ − U0 τ (moving with speed U0 ) and the solution is (1)
φ
2
= φm sech
ξ ∆
,
(15)
where the solitary wave amplitude φm = 3U0 /A and its width ∆ =
p 4B/U0 . From Eq.(15),
small amplitude humped shape solitons with φ(1) > 0 and with φ(1) < 0 dip shape solitons are indicated.
B.
Derivation of the Burger Equation
To derive Burger equation, the stretched coordinates will be ζ = (Lx x + Ly y + Lz z − Vp t) , τ = 2 t.
(16)
The dependent variables ni,e , ui , and φ are expanded in a power series of as in Eq. (1)
(6). (1)
By using Eqs. (1)
(1)
(1)
(4, 6 and 16) in Eqs.
(1)
(3), the same expressions for
(1)
ne , n+ , n− , u+z , u−z , u+x,y , u−x,y and Vp are obtained. Following the same procedures, the next higher-order series of of continuity, momentum and Poisson’s equations with the aid of Eqs. (7 − 9) give the following relations (2)
∂n+ ∂ζ
=
(2)
∂n− ∂ζ
=−
∂ (2) n ∂ζ e
18Vp L2z
2
(3Vp2 −5KL2z )
18βVp L2z 2 3Vp2 −5KβL2z
∂φ(1) ∂τ
∂φ(1) ∂τ
+
+
81Vp2 L4z −15KL6z 3
(3Vp2 −5KL2z )
81β 2 Vp2 L4z −15Kβ 3 L6z 3
( ) ( ) 15ηT 2 +3η 3 3T 2 ∂ √ = 8 1−η − + 8(1−η)5/2 ∂ζ 16 3Vp2 −5KβL2z
9Vp L4z
(1)
φ(1) ∂φ∂ζ − γ ∂φ(1)
2
(3Vp2 −5KL2z )
4
∂ 2 (1) φ ∂ζ 2
+
3L2z 3Vp2 −5KL2z
2
∂ (2) φ , ∂ζ 2
∂ (2) ∂ (1) z + γ 29βVp Lz 2 2 ∂ζ − 3V 23βL φ , 2φ 2 p −5KβLz ∂ζ 3Vp −5KβLz ) ( √ 2 2 3 1−η 3 T ∂ (2) (1) 2 φ + + η (T + 1) − φ . 3/2 2 4 ∂ζ 2(1−η) (17)
φ(1)
∂ζ
Furthermore, the next higher order series of of Poisson’s equation gives: (2)
(2)
δn(2) e + σn− − n+ = 0.
(18)
By differentiation of Eq. (18) w.r.t ζ δ
∂ (2) ∂ (2) ∂ (2) ne + σ n− − n+ = 0. ∂ζ ∂ζ ∂ζ
By substitution of Eq. (17) into Eq. (19) the following Burger’s equation is obtained 9
(19)
∂φ(1) ∂ ∂2 + Aφ(1) φ(1) + C 2 φ(1) = 0, ∂τ ∂ζ ∂ζ where C=−
γL2z . 2
(20)
(21)
By using tanh method [37, 38], the shock wave stationary solution of Eq. (21) is obtained by transforming the independent variables ζ and τ to ξ = ζ − U0 τ , where U0 is the normalized speed of the shock wave in the moving reference frame. Thus, one can express the stationary solution as (1)
φ
ξ = φ◦ 1 − tanh ∆
(22)
with the amplitude and width of the shock wave equal φ◦ = U0 /A and ∆ = 2C/U0 , respectively. The electric field, E (1) = −∇φ(1) , associated with this analytical solution, Eq. (22), can be derived as E
III.
(1)
U2 = − 0 sech2 2AC
ξ ∆
(23)
RESULTS AND DISCUSSION
We have considered a degenerate multi-ion collisionless and dense magneto-plasma system by taking into account the effects of the finite degenerate temperature and the electron trapping in the presence of a quantizing magnetic field. The positive and negative ions following the nonrelativistic equation of state, Eq. (3). The small-amplitude reductive perturbative method was employed to derived the KdV and Burger equations, as well as their solitary and shock wave solutions are obtained for a partially degenerate quantum plasma, respectively. The analyzation of these solutions is also performed. The results obtained from these investigations can be summarized as follows: 1: The plasma model under consideration supports only the cyclotron wave propagation (as fast modes). The variation of the phase speed Vp with various plasma parameters is demonstrated in Fig. 1. It has been shown that the phase speed increases (decreases) as β, σ, and δ (T ) increases, as shown in Figs. 1a and 1b. It means that larger phase velocity is attained when the negative ions density decreases and its mass increases. Also as the 10
degeneracy of the plasma system decreases and as the electron density increases lead to an increase of phase velocity. Solitary waves can be generated also with larger phase velocity as the obliquity angle decreases and electrons become less trapped as depicted in Fig 1c. A balance between the nonlinear and the dispersion terms forms solitary structures in the considered plasma model. The nonlinear term, A, depends on the various plasma parameters, e.g., the inertial ions number density and the inertialess electrons via β, σ, and δ, trapping parameter, η and degenerate temperature through T . Therefore, it is important to analyze A in terms of the plasma compositional parameters. 2: Figure 2 illustrates the variation of the nonlinear term A with Lz for different values of T, η, and δ. The nonlinear term is seen to be positive, it increases with increasing of T, η, and Lz and for δ > 0.05, A acquires smaller value when δ increases. 3: The dispersive term, B, depends on the positive ion cyclotron frequency, Ωc . In addition to the same plasma parameters similar to the nonlinear term, A. We depict the variation of the dispersion term B with Lz for different values of T, η and Ωc in Fig. 3. The dispersive term is seen to be positive and it decreases with increasing of Lz , T, η, and Ωc . 4: The considered plasma system supports positive potential IASWs with the amplitude is influenced by the previously mentioned plasma parameters. Figure 4 depicts the amplitude decreases as the trapping and degenerate temperature coefficients increase. 5: The KdV solitary waves are depicted in Fig. 5. The positive potential KdV or compressive solitons existed, their amplitude and width are affected by T and Lz , as shown in Figs. 5a and 5b. As the positive ion cyclotron frequency, Ωc , increases the soliton width decreases, but the soliton amplitude of the produced soliton remains unchanged, as illustrated in Fig. 5c. We now trace the effect of the viscosity coefficient and the obliqueness on the formed shock waves since the dissipation term, C, depends on both the ion fluid’s kinematic viscosity and obliqueness. It is easier for the nonlinear term, A to balance with the dissipation term, C in a non-relativistic plasma to form a stronger shock profile[39] 6: It is observed that only compressive shocks are formed in this multi-ion plasma system. Figure 6 demonstrates the effect of kinematic viscosity, γ, and obliqueness, via Lz , on the shock wave profile. The shock wave width increases as γ and Lz increase. The amplitude decreases as Lz increases, while it remains constant with the increase of γ since it is independent on the variation of kinematic viscosity. 11
7: The behavior of the electric field associated with the solution (22) against ξ is presented graphically in Fig. 7. The viscosity coefficient and the obliqueness affect crucially on this associated electric field. The stronger associated electric field wave (larger in amplitude and narrower in width) has been observed for this positive electric field values with the decreasing of the viscosity coefficient and the obliqueness.
IV.
CONCLUSION
In conclusion, the obtained results reveal that the reductive perturbation method is valid only for small but finite amplitude IA solitary structures. It is noted that the results of theoretical analysis, the effects of obliquely quantizing magnetic field (η), ratio of electron number density and positive ion number density multiplied by positive ion charge (δ), mass for positive to negative ions times charge for negative to positive ions (β), kinematic viscosity coefficient (γ) and finite degeneracy temperature (T ) play an important role in the variation of width, amplitude, and polarity of the KdV solitons and shock waves (see Figs. 1 − 7). We can conclude that as the negative ions density and the mass-to-charge for positive to negative ions ratio, β, the electron number density via δ and obliqueness, via Lz (the degeneracy temperature, T and the trapped electrons, η) increase (decrease) the phase speed, Vp increases (see Fig. 1). The parametric regimes give positive nonlinear and dispersion terms (see Figs. 2 and 3). Accordingly, positive solitary waves with positive amplitudes are formed (see Figs. 4 and 5). The viscosity coefficient, γ, has positive effects on the width of the compressive shock waves (see Fig. 6a). It is significant to reveal here that the amplitude (width) of positive shock wave decreases (increases) due to the increase of Lz ; where it falls (raises) more rapidly (see Figs. 6b). It is observed that the increasing of Lz and γ have a negative effect on the positive electric field wave (see Fig. 7). Finally, it has been concluded that the system under consideration admits only compressive solitary and shock structures for partially degenerate plasma in the presence of a quantizing magnetic field. The present results maybe helpful in understanding the shock dynamics and the solitary structures in dense quantum plasmas such as those that exist in white dwarfs due to the existence of matters under extreme conditions and also in experiments of ultra-high-density plasmas where degeneracy effects become important.
12
V.
REFERENCES
[1] Afshin Moradi, Phys. Scr. 90 (2015) 085601. [2] S. K. El-Labany, W. F. El-Taibany, A. E. El-Samahy, A. M. Hafez, A. Atteya, Braz. J. Phys. 45 (2015) 409. [3] S. K. El-Labany, W. F. El-Taibany, A. E. El-Samahy, A. M. Hafez, A. Atteya, IEEE Trans. Plasma Sci. 44 (2016) 842. [4] A. Atteya, E. E. Behery, W. F. El-Taibany, Eur. Phys. J. Plus, 132 (2017) 109. [5] M. Iqbal Shaukat, Phys. Plasmas. 24 (2017) 102301. [6] N. Roy, S. Tasnim, A. A. Mamun, Phys. Plasmas. 19 (2012) 033705. [7] L. Nahar, M. S. Zobaer, N. Roy, A. A.Mamun, Phys. Plasmas. 20 (2013) 022304. [8] Myoung-Jae Lee, Young-Dae Jung, Phys. Scr. 93 (2018) 025601. [9] J. M. Aparicio, G. Fontaine, White Dwarfs. Proceedings of the 10 th European Workshop on White Dwarfs, Kluwer Academic Publishers, London (1996). [10] M. R. Hossen, A. A. Mamun, Plasma Science and Technology 17 (2015) 3. [11] G. B.Van Albada, J. Astrophys. 105 (1947) 393. [12] S. Chandrasekhar, Mon. Not. R. Astron. Soc. 170 (1935) 405. [13] G. Brodin, M. Marklund, New J. Phys. 9 (2007) 277. [14] A. Y. Potekhin, Astron. Astrophys. 306 (1996) 999. [15] A. V. Gurevich, Sov. Phys. JETP 26 (1968) 575. [16] R. L. Morse, C. W. Nielson, Phys. Rev. Lett. 23 (1969) 1087. [17] H. L. Berk, C. E. Nielsen, K. V. Roberts, Phys. Fluids 13 (1970) 980. [18] M. Kako, T. Taniuti, T. Watanabe, J. Phys. Soc. Jpn. 31 (1971) 1820. [19] I. B. Bernstein, J. M. Greene, M. D. Kruskal, Phys. Rev. 108 (1957) 546. [20] H. Schamel, J. Plasma Phys. 14 (1972) 905. [21] H. Schamel, J. Plasma Phys. 9 (1973) 377. [22] H. Schamel, J. Plasma Phys. 13 (1975) 139. [23] H. Schamel, Phys. Rep. 140 (1986) 161. [24] H. Schamel, Phys. Plasmas 7 (2000) 4831. [25] H. Schamel, Phys. Plasmas 19 (2012) 020501. [26] S. A. Shan, S. A. El-Tantawy, Phys. Plasmas 23 (2016) 072112. [27] M. Tribeche, L. Djebarni, H. Schamel, Phys. Lett. A 376 (2012) 3164. 13
[28] A. A. Mamun, P. K. Shukla, L. Stenflo, Phys. Plasmas 9 (2002) 1474. [29] H. Siddiqui, H. A. Shah, N. L. Tsintsadze, J Fusion Energy 27 (2008) 216. [30] S. Sultana, I. Kourakis, N. S. Saini, M. A. Hellberg, Phys. Plasmas 17 (2010) 032310. [31] M. Irfan, S. Ali, Arshad M. Mirza, Phys. Plasmas 24 (2017) 052108. [32] P. Sharma, G. W. Hammett, E. Quataert, J. M. Stone, Astrophys. J. 637 (2006) 952. [33] P. Sharma, E. Quataert, G. W. Hammett, J. M. Stone, Astrophys. J. 667 (2007) 714. [34] S. K. El-Labany, W. F. El-Taibany, A. Atteya, Physics Letters A 382 (2018) 412. [35] E. E. Scime, P. A. Keiter, M. M. Balkey, R. F. Boivin, J. L. Kline, M. Blackburn, S. P. Gary, Phys. Plasmas 7 (2000) 2157. [36] S Arshad, H A Shah, M. N. S. Qureshi, Phys. Scr. 89 (2014) 075602. [37] S. K. El-Labany, W. F. El-Taibany, A. E. El-Samahy, A. M. Hafez, A. Atteya, Astrophys. Space Sci. 354 (2014) 385. [38]M. K.Deka, A. N. Dev, Annals of Physics 395 (2018) 45. [39] S. Islam, S. Sultana, A. A. Mamun, Phys. Plasmas 24 (2017) 092308.
14
Figure captions Fig. 1: (color online) The variation of the phase speed Vp with a) σ and β for Lz = 0.94, T = 0.4, η = 0.5 and δ = 0.3, b) T and δ for Lz = 0.94, η = 0.5, σ = 0.2 and β = 0.03 c) Lz and η for T = 0.6, σ = 0.3, δ = 0.3 and β = 0.03. Fig. 2: (color online) The variation of the nonlinear term A, represented by equation (14) with Lz for different values of a)T with η = 0.4, β = 0.01, σ = 0.1, δ = 0.3 and Ωc = 0.5, b) η with T = 0.4, β = 0.03, σ = 0.3, δ = 0.3 and Ωc = 0.5 c) δ with T = 0.7, η = 0.4, β = 0.03, σ = 0.3 and Ωc = 0.5. Fig. 3: (color online) The variation of the dispersive term B, represented by equation (14) with Lz for different values of a)T with η = 0.4, β = 0.01, σ = 0.1, δ = 0.3 and Ωc = 0.5, b) η with T = 0.4, β = 0.03, σ = 0.3, δ = 0.3 and Ωc = 0.5 c) Ωc with T = 0.4, η = 0.4, β = 0.03, σ = 0.3 and δ = 0.2. Fig. 4: (color online) The evolution of φm of IASWs with Lz for different values a) η with T = 0.4, β = 0.03, σ = 0.3, δ = 0.3, Ωc = 0.5 and U0 = 0.05 b)T with η = 0.4, β = 0.01, σ = 0.1, δ = 0.3, Ωc = 0.5 and U0 = 0.05. Fig. 5: (color online) The evolution of φ(1) of rarefactive IASWs that represented by equation (15) with ξ at U0 = 0.15 for different values of a) T with Lz = 0.94, η = 0.4, β = 0.03, σ = 0.2, δ = 0.3 and Ωc = 0.5, b) Lz with T = 0.5, η = 0.4, β = 0.03, σ = 0.3, δ = 0.2 and Ωc = 0.5 and c) Ωc with T = 0.5, Lz = 0.94, η = 0.4, β = 0.03, σ = 0.3 and δ = 0.2. Fig. 6: (color online) The evolution of φ(1) of rarefactive IA shock waves that represented by equation (22) with ξ at U0 = 0.05 for different values of a) γ with T = 0.6, η = 0.4, β = 0.03, σ = 0.3, δ = 0.2 and Lz = 0.94, b) Lz with T = 0.5, η = 0.4, β = 0.03, σ = 0.3, δ = 0.2 and γ = 0.4. Fig. 7: (color online) The evolution of the associated electric field E (1) of rarefactive IA shock wave that represented in Fig. 6 for different values of a) γ with the same parameters as Fig. 6a and b) Lz with the same parameters as Fig. 6b.
15
igure
0.645
Σ=0.1 Σ=0.2 Σ=0.3 Σ=0.4
0.640
Vp
0.635 0.630 0.625 0.620 0.615 0.00
0.05
0.10
0.15
0.20
0.25
0.30
Β
Fig. 1 a 0.7 0.6 0.5
T=0.1 T=0.2 T=0.3 T=0.4
Vp
0.4 0.3 0.2 0.1 0.0
0.0
0.1
0.2
∆
Fig. 1 b
0.3
0.4
0.62 0.60
Vp
0.58 0.56
lz=0.86 lz=0.9 lz=0.94 lz=0.98
0.54 0.52 0.50 0.1
0.2
0.3
0.4
Η Fig. 1 c
0.5
0.6
igure
2.20 2.15 2.10
A
2.05
T=0.1 T=0.2 T=0.3 T=0.4
2.00 1.95 1.90 1.85 0.86
0.88
0.90
0.92
0.94
0.96
0.98
1.00
0.94
0.96
0.98
1.00
lz Fig. 2 a 2.25
Η=0.3 Η=0.4 Η=0.5 Η=0.6
2.20 2.15
A
2.10 2.05 2.00 1.95 1.90 0.86
0.88
0.90
0.92
lz Fig. 2 b
∆=0.05 ∆=0.1 ∆=0.15 ∆=0.2
5.0 4.5
A
4.0 3.5 3.0 2.5 0.86
0.88
0.90
0.92
0.94
lz Fig. 2 c
0.96
0.98
1.00
igure
1.6
T=0.1 T=0.2 T=0.3 T=0.4
1.4 1.2
B
1.0 0.8 0.6 0.4 0.2 0.86
0.88
0.90
0.92
0.94
0.96
0.98
1.00
lz Fig. 3 a 1.4
Η=0.3 Η=0.4 Η=0.5 Η=0.6
1.2
B
1.0 0.8 0.6 0.4 0.2 0.86
0.88
0.90
0.92
0.94
lz Fig. 3 b
0.96
0.98
1.00
2.0
Wc=0.3 Wc=0.4 Wc=0.5
B
1.5
1.0
0.5
0.86
0.88
0.90
0.92
0.94
lz Fig. 3 c
0.96
0.98
1.00
igure
Η=0.3 Η=0.4 Η=0.5 Η=0.6
0.078 0.076
Φm
0.074 0.072 0.070 0.068 0.066 0.86
0.88
0.90
0.92
0.94
0.96
0.98
1.00
lz Fig. 4 a 0.080
T=0.1 T=0.2 T=0.3 T=0.4
0.078
Φm
0.076 0.074 0.072 0.070 0.068 0.86
0.88
0.90
0.92
0.94
lz Fig. 4 b
0.96
0.98
1.00
0.20
ΦH1L
0.15
T=0.1 T=0.3 T=0.5
0.10
0.05
0.00 -15
-10
-5
0
5
10
15
Ξ
Fig. 5 a
ΦH1L
0.15
lz=0.86 lz=0.9 lz=0.94 lz=0.98
0.10
0.05
0.00 -15
-10
-5
0
Ξ
Fig. 5 b
5
10
15
0.15
Wc=0.3 Wc=0.4 Wc=0.5
ΦH1L
0.10
0.05
0.00 -15
-10
-5
0
Ξ
Fig. 5 c
5
10
15
igure
0.035 0.030
ΦH1L
0.025
Γ=0.1 Γ=0.3 Γ=0.5
0.020 0.015 0.010 0.005 0.000 -15
-10
-5
0
5
10
15
Ξ
Fig. 6 a 0.04
ΦH1L
0.03
lz=0.86 lz=0.9 lz=0.94 lz=0.98
0.02
0.01
0.00 -20
-10
0
Ξ
Fig. 6 b
10
20
0.010
Γ=0.1 Γ=0.3 Γ=0.4
0.008
E H1L
0.006
0.004
0.002
0.000 -15
-10
-5
0
5
10
15
Ξ
Fig. 7 a 0.0035 0.0030
lz=0.86 lz=0.9 lz=0.94 lz=0.98
0.0025
E H1L
0.0020 0.0015 0.0010 0.0005 -15
-10
-5
0
Ξ
Fig .7 b
5
10
15
Journal Pre-proof
Obliquely propagating ion-acoustic solitary and shock waves in magnetized quantum degenerate multi-ions plasma in the presence of trapped electrons M.A. El-Borie, M. Abd-Elzaher, A. Atteya PII: DOI: Reference:
S0577-9073(19)30941-4 https://doi.org/10.1016/j.cjph.2019.10.004 CJPH 964
To appear in:
Chinese Journal of Physics
Received date: Revised date: Accepted date:
19 January 2019 2 October 2019 2 October 2019
Please cite this article as: M.A. El-Borie, M. Abd-Elzaher, A. Atteya, Obliquely propagating ion-acoustic solitary and shock waves in magnetized quantum degenerate multi-ions plasma in the presence of trapped electrons, Chinese Journal of Physics (2019), doi: https://doi.org/10.1016/j.cjph.2019.10.004
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.
Highlights
• Obliquely IAWS have been investigated. • KdV and Burger equations were derived. • Trapping effect is discussed. • Electron degenerate temperature effect is discussed. • Solitary and Shock waves are plotted.
1
Obliquely propagating ion-acoustic solitary and shock waves in magnetized quantum degenerate multi-ions plasma in the presence of trapped electrons M. A. El-Borie∗ Department of Physics, Faculty of Science, Alexandria University, Alexandria, P.O. 21511, Egypt, M. Abd-Elzaher† Department of Basic and Applied Sciences, Faculty of Engineering, Arab Academy for Science, Technology and Maritime Transport, Alexandria, Egypt. A. Atteya Department of Physics, Faculty of Science, Alexandria University, Alexandria, P.O. 21511, Egypt.‡
Abstract The properties of obliquely propagating ion-acoustic waves have been investigated in multi-ions magnetized plasma comprising of inertial, positively and negatively charged ion fluids, trapped electrons, and negatively charged stationary heavy ions. The propagation of the waves is oblique to the ambient magnetic field which is along the z-direction. Only fast type of modes exists in the linear regime. The reductive perturbation method was adopted to derive the Korteweg– de Vries (KdV) and Burger equations, as well as the solitary and shock wave solutions of the evolved equations, have been used to analyze the properties of the small but finite amplitude waves. The effects of the constituent plasma parameters, namely, the trapping effect of electrons, the electron degenerate temperature and the viscosity coefficient on the dynamics of the small amplitude solitary and shock waves have been examined. The influence of the magnetic field and the obliquity parameter on the propagation characteristics of ion-acoustic waves are discussed.
∗
Electronic address: [email protected]
†
Electronic address: [email protected]
1
‡
Electronic address: [email protected]
2
Keywords: Quantum plasma, Plasma waves, Multi-ions plasma, Solitary waves, Shock waves, Obliqness, Trapped electrons.
I.
INTRODUCTION
Great current interest has been given to the field of quantum plasma which has a long diverse tradition [1 − 6]. These interests are attributed to its applications in modern technology (metallic nanoparticles, nanotubes, quantum wells, and quantum dots, spintronics, nanoplasmonic devices, etc.). The ultrafast spectroscopy techniques have recent advancements that enabled us to monitor the confinement of electron gas in metallic plasmas which are considered in the femtosecond range [5]. Accordingly, relativistic degenerate plasmas have received great attention [6 − 8], where particle velocities become comparable to the speed of light. Astrophysical compact objects such as neutron stars, quasars, white dwarfs, black holes, pulsars are relativistic degenerate plasmas examples. The basic constituents of the white dwarfs are mainly carbon, oxygen, helium with hydrogen gas. It is assumed that the whole star is composed of these three elements with the heaviest at the core and the lightest at the crust [9]. It is well known from the nuclear structure studies that the isotopes of Mg, Ne, O, C, He, etc. found in white dwarfs are bosons [10]. The heavier element like iron is present in relatively massive white dwarfs. The heavy nuclei are mainly formed in the interior of these massive stars. When these stars under certain conditions Shrink to very high densities, they become degenerate and heavy elements begin to form. When an explosion occurs, a part of these heavy elements distributes into the surrounding space and leaves some stellar remnants in the aspects of the white dwarf [11]. The electron number density has very high value in this degenerate compact objects [2 − 4]. The equation of state for degenerate electrons and ions for these compact objects is mathematically explained by Chandrasekhar [12] for two cases, the non-relativistic and ultra-relativistic cases. The ion pressure for the non-relativistic case can be given by Pi = Ki nαi
(1)
where ni is the number density of ions and 3 π 13 π} 3 5 ' Λc }c, α = , Ki = 3 5 3 m 5 3
(2)
(in which Λc = π}/mc = 1.2 × 10−10 cm, and } is the Planck constant divided by 2π). Nowadays, the pressure laws (1) and (2) were used [13] to investigate the linear and nonlinear properties of electromagnetic and electrostatic waves, by using the non-relativistic quantum hydrodynamic[1] and quantum-magnetohydrodynamic [13] models. The range of quantization is achieved by bringing the Landau quantization of the electron motion and the magnetic field into account [14] and the condition KB T << ~Ωce (Ωce is the electron cyclotron frequency) has to be fulfilled. However, the Fermi ground state energies stay constant as the Landau energy-levels pair cancel out each other [14]. The particle trapping is a phenomenon that affects the plasma waves. The particle confinement in a certain region of phase space by the wave potential is the mean of trapping. Oscillations are performed by these trapped particles. The particle trapping has been observed in space plasmas and also in laboratory experiments [15 − 18]. The particle trapping cannot be understood by using the linear theory of waves, it is demonstrated that particle trapping is basically a nonlinear phenomenon. The usual concept of linear wave theory based on the van Kampen or Landau approach using the Vlasov theory breaks down entirely for the wave propagation with phase speeds is close to the thermal velocity in the presence of resonant particles. Modification of the wave propagation characteristics in collisionless plasmas may significantly be achieved by these nonisothermal particles. In the case of collisionless plasmas, the wave propagation manners are significantly altered by these nonisothermal particles. The first analytical method to construct equilibrium electrostatic structures including trapping particle was provided by Bernstein et al. [19]. However, the pseudo-potential method to construct equilibrium solutions was developed by Schamel [20 − 24]. As a consequence of this research, Schamel’s work is considered as a refinement in the phase space theory of vortices or holes. The large as well as small amplitudes electrostatic waves are affected by the electron trapping effect [25]. From this, the presence of trapped particles makes the lowest order excited plasmas distributions that satisfying the linearized Vlasov equation to loses its basis. Accordingly, due to the presence of trapped particles, an increased spectrum of waves takes place. The trapped electrons have lesser kinetic energy than the wave potential energy. These trapped electrons forth and bounce back in the potential well, which satisfies Schamel-like one-dimensional distribution [26, 27]. Many authors[25, 28−30] focused on the particles trapping in nonlinear solitary waves and showing that the basic properties of the ion-acoustic waves (IAWs) are significantly modified 4
by the effect of an external magnetic field. Anisotropic pressure is achieved in plasma as a result of applying a strong magnetic field [31]. Under the adiabatic condition, the anisotropic pressure has parallel and perpendicular components due to variations in the plasma density during the wave propagation, and the wave potential in plasmas leads to trapping effects [31]. Electron acoustic solitary waves in a magnetized plasma were investigated through the effect of the obliqueness and the presence of trapped hot electrons [28]. The vortices formation was investigated also due to the effect of trapping in the classical plasma [29], the trapping effect on the Sagdeev potential was also investigated. Obliquely propagating arbitrary amplitude IASWs have been studied in a magnetized electron-ion plasma [30]. The electrostatic waves were presented to be controlled by the nonlinearity of trapping for arbitrary amplitudes, where the infinitesimal amplitude limits are included by these arbitrary amplitudes [25]. Numerous observations related to space and laboratory plasmas have been revealed [32 − 35], which emphasize particle trapping in dense astrophysical plasmas. Investigation of the effect of trapping as a microscopic phenomenon in an inhomogeneous degenerate plasma is executed in the presence of a quantizing magnetic field [36]. The results were analyzed numerically for different values of the quantizing magnetic field and for different astrophysical plasmas. In the present paper, we consider the influence of trapping on the propagation of twodimensional ion-acoustic and shock waves in the quantum degenerate multi-ion plasma system as a result of the presence of Landau quantization and the inertialess electrons FermiDirac distribution function. Therefore, Korteweg–de Vries (KdV) and Burger equations have been derived and then the formation of solitary and shock waves has been studied. The layout of the present manuscript is as follows: the model and the basic equations with the carrying out of the nonlinear analysis are described in Sec. II. The numerical results are presented and discussed in Sec. III. The conclusion of our findings is given in Sec. IV.
II.
THE MODEL
Multi-ion collisionless, magnetized plasma system composed of trapped distributed electrons with inertial negatively and positively charged ions and negatively charged immobile heavy elements are considered in this study. The magnetic field B0 is along the z-direction, i.e., B0 = zˆB0 . The nonlinear propagation of finite-amplitude IA waves in such a multi-ion 5
plasma system, in case of planar geometry, is governed by ∂n+ ∂t ∂n− ∂t
+ ∇(n+ u+ ) = 0,
+ ∇(n− u− ) = 0, − 31 ∂u+ 5 ∇n+ + γ∇2 u+ + Ωc (u+ × zˆ) , + (u .∇)u = −∇φ − Kn + + + ∂t 3 1 − ∂u− + (u− .∇)u− = β∇φ − 53 Kβn− 3 ∇n− + γ∇2 u− − βΩc (u− × zˆ) , ∂t ∇2 φ = δne + σn− − n+ + Zh µ.
(3)
where n+ , n− , and ne are the total number densities of inertial positive ions, inertial negative ions, and trapped electrons, respectively. ui is the plasma species fluid speed normalized by Cim = (me c2 /Zi mi )
1/2
with c being the light speed, mi and me are the rest mass of
plasma ion and electron species, respectively. Ωc = (Z+ eB0 /m+ ) /ωp+ is the positive ion cyclotron frequency, where e is the electron charge with ωp+ is the positive ion plasma 1/2 frequency ωp+ = 4πZ+2 n+0 e2 /m+ . φ is the electrostatic wave potential normalized
by me c2 /e . β = Z− m+ /Z+ m− represents the masses ratio of the positive ions and the negative ions multiplied by the negative to positive charge per ion ratio. δ = ne0 /Z+ n+0
is the electron and positive ion number density ratio multiplied by positive ion charge. 2
3 Ki /me c2 . γ is the viscosity coefficient and σ = Z− n−0 /Z+ n+0 is the negative K = n+0
and positive ion number density ratio multiplied by their ion charge and µ = nh0 /Z+ n+0 is the heavy element and positive ion number density ratio multiplied by Z+ , the number of electrons ejected from the heavy element is Zh . Space (x, y & z ) and time (t) variables 1/2 are normalized by λD = kB T+ /4πZ+2 n+0 e2 and the inverse of the positive ion plasma −1 frequency, ωp+ , respectively. The trapped electrons number density for partially degenerate
plasma is expressed by 1 3 3 ηT 2 − 32 − 21 2 2 2 ne = η (1 + φ) + (1 + φ − η) − (1 + φ) + T (1 + φ − η) . 2 2
(4)
The effect of the quantizing magnetic field appears through the parameter η = ~Ωce /EF e 3/2
where EF e (Fermi energy of the electron) = ~2 (3π 2 ne0 ) √ normalized by T = πTe / 2EF e .
A.
/2me . The temperature T
is
Derivation of the KdV Equation
The obliquely propagating small finite-amplitude IAWs dynamics in a multi-ion magnetized quantum plasma with trapping electrons has been investigated by employing the 6
reductive perturbation method to drive the KdV equation. the stretched coordinates are introduced as
1 ζ = 2 (Lx x + Ly y + Lz z − Vp t) , 3 τ = 2 t,
(5)
where is a smallness parameter for measuring the perturbation amplitude, Vp is the perturbation mode phase speed that is normalized by the IA speed. The directional cosines of the wave vector k are lx , ly , and lz along x, y, and, z -axes, respectively. Therefore lx2 + ly2 + lz2 = 1. The dependent variables ni,e , ui , and φ may be expanded in a power series of as; (1) (2) (3) ni = 1 + ni + 2 ni + 3 ni + ... , (1) (2) (3) ne = 1 + ne + 2 ne + 3 ne + ... , 3 5 (1) (2) (3) uix,y = 2 uix,y + 2 uix,y + 2 uix,y + ... , (1) 2 (2) 3 (3) uiz = uiz + uiz + uiz + ... , (1) 2 (2) 3 (3) φ = φ + φ + φ + ... .
(6)
To develop equations in various powers of , we substitute Eqs. (4 − 6) in Eq. (3). The
Poisson equation 0 coefficient gives the current plasma model neutrality condition, δ + σ − 1 + Zh µ = 0. The next order in gives the following relations (1) 3Vp Lz φ(1) , n+ 3Vp2 −5KL2z (1) (1) p Lz (1) u−z = − 3V 3βV , n− 2 −5KβL2 φ p z √ 3η (1+T 2 ) (1) ne = 3 21−η + 4 (1)
u+z =
=
3L2z (1) 3Vp2 −5KL2z 3βL2z 3Vp2 −5KβL2z
φ ,
=− −
T2 2(1−η)3/2
φ(1) ,
φ(1) .
(7)
The perturbed quantities are accompanied in Poisson’s equation. The expression of the IA waves phase speed in this magnetized plasma model is derived as; s p M + M 2 − 36δβF S Vp = Lz , (8) 18δF √ 3η (1+T 2 ) T2 where F = 3 21−η + − , M = 15K (1 + β) δF + 9 (1 + σβ) and S = 4 2(1−η)3/2
(15K (1 + σ) + 25K 2 δF ). From the equation, (8), we conclude that the phase velocity
depends on the ratio of masses and charges of negative and positive ions through β, the obliqueness through the change of Lz , the trapping parameter, η, and the negative ions and electron number densities through σ and δ, respectively. The x and y components of the 7
lowest order momentum equations can be expressed as 3Vp2 Ly
(1) ∂ (1) φ , u+y = − Ω 3V 2 −5KL2 ∂ζ c( p z) 3Vp2 Ly (1) (1) ∂ (1) u−x = − Ω 3V 2 −5KβL φ , u−y 2 ∂ζ (1) u+x
c
(
p
z
)
= =
3Vp2 Lx
∂ (1) φ , Ωc (3Vp2 −5KL2z ) ∂ζ 2 3Vp Lx ∂ (1) φ . Ωc (3Vp2 −5KβL2z ) ∂ζ
(9)
By applying these equations in the next higher order of the momentum, the following relations are considered 3Vp3 Lx
(2) u+x (2) u−x
=
(2) ∂ 2 (1) = Ω2 3V 2 −5KL2 ∂ζ , u+y 2φ ( ) c p z 3V 3 Lx (2) ∂ 2 (1) − βΩ2 3V 2p−5KβL φ , u−y 2 ∂ζ 2 c
(
z
p
)
=
3Vp3 Ly Ω2c (3Vp2 −5KL2z ) 3Vp3 Ly
= − βΩ2
∂ 2 (1) φ , ∂ζ 2
(
2 2 c 3Vp −5KβLz
)
∂ 2 (1) φ . ∂ζ 2
(10)
On substitution from the above equations in the next higher-order continuity equation and the z-component of the momentum equation, trapped electrons and Poisson’s equations the following relations are obtained; (2)
∂n+ ∂ζ
∂ (2) n ∂ζ e
=
√3 4 1−η
=
18Vp L2z
9Vp4 (1−L2z )
∂ (1) φ ∂τ
∂ 3 (1) φ ∂ζ 3
+ 2 2 2 2 Ωc (3Vp −5KL2z ) ) 2 ∂ (1) ∂ (2) z + φ(1) ∂ζ φ + 3V 23L φ , ( p −5KL2z ) ∂ζ ( ) (2) 2 9Vp4 (1−L2z ) ∂n− p Lz ∂ (1) ∂ 3 (1) = − 18βV − 2 ∂τ φ 2 3φ ∂ζ βΩ2c (3Vp2 −5KβL2z ) ∂ζ (3Vp2 −5KβL2z ) 2 2 4 3 6 2 81β Vp Lz −15Kβ Lz (1) ∂ (1) ∂ (2) z φ ∂ζ φ − 3V 23βL φ , + 3 2 p −5KβLz ∂ζ (3Vp2 −5KβLz2 ) √ 2 3T 2 ∂ (1) φ(1) ∂ζ φ + 3 21−η + 34 η (T 2 + 1) − − 15ηT8 +3η + 4(1−η) 5/2 (
3Vp2 −5KL2z 81Vp2 L4z −15KL6z 3 3Vp2 −5KL2z
T2 2(1−η)3/2
∂ 2 (1) (2) (2) φ = δn(2) e + σn− − n+ . ∂ζ 2
∂ ∂ζ
(2) φ ,
(11)
(12)
Substitution of Eq. (11) into Eq. (12) led to the following KdV equation for the obliquely propagating IAWs in a multi-ion magnetized plasma with trapped electrons, ∂φ(1) ∂ ∂3 + Aφ(1) φ(1) + B 3 φ(1) = 0, ∂τ ∂ζ ∂ζ
(13)
where A=
81Vp2 L4z −15KL6z
81σβ 2 Vp2 L4z −15Kσβ 3 L6z
√3 4 1−η
15ηT 2 +3η 8
− −δ − + 3 2 2 (3Vp −5KβLz ) 9(1−L2 )σV 4 9(1−L2 )V 4 B = 1 + 2 2 z p2 2 + 2 2 z p 2 2 Q, Ω 3V −5KLz ) βΩc (3Vp −5KβLz ) c( p −1 18Vp L2z 18σβVp L2z Q= . 2 + 2 (3Vp2 −5KL2z ) (3Vp2 −5KβL2z ) 3
(3Vp2 −5KL2z )
8
3T 2 4(1−η)5/2
Q,
(14)
The KdV solitary wave solution is obtained by considering a frame ξ = ζ − U0 τ (moving with speed U0 ) and the solution is (1)
φ
2
= φm sech
ξ ∆
,
(15)
where the solitary wave amplitude φm = 3U0 /A and its width ∆ =
p 4B/U0 . From Eq.(15),
small amplitude humped shape solitons with φ(1) > 0 and with φ(1) < 0 dip shape solitons are indicated.
B.
Derivation of the Burger Equation
To derive Burger equation, the stretched coordinates will be ζ = (Lx x + Ly y + Lz z − Vp t) , τ = 2 t.
(16)
The dependent variables ni,e , ui , and φ are expanded in a power series of as in Eq. (1)
(6). (1)
By using Eqs. (1)
(1)
(1)
(4, 6 and 16) in Eqs.
(1)
(3), the same expressions for
(1)
ne , n+ , n− , u+z , u−z , u+x,y , u−x,y and Vp are obtained. Following the same procedures, the next higher-order series of of continuity, momentum and Poisson’s equations with the aid of Eqs. (7 − 9) give the following relations (2)
∂n+ ∂ζ
=
(2)
∂n− ∂ζ
=−
∂ (2) n ∂ζ e
18Vp L2z
2
(3Vp2 −5KL2z )
18βVp L2z 2 3Vp2 −5KβL2z
∂φ(1) ∂τ
∂φ(1) ∂τ
+
+
81Vp2 L4z −15KL6z 3
(3Vp2 −5KL2z )
81β 2 Vp2 L4z −15Kβ 3 L6z 3
( ) ( ) 15ηT 2 +3η 3 3T 2 ∂ √ = 8 1−η − + 8(1−η)5/2 ∂ζ 16 3Vp2 −5KβL2z
9Vp L4z
(1)
φ(1) ∂φ∂ζ − γ ∂φ(1)
2
(3Vp2 −5KL2z )
4
∂ 2 (1) φ ∂ζ 2
+
3L2z 3Vp2 −5KL2z
2
∂ (2) φ , ∂ζ 2
∂ (2) ∂ (1) z + γ 29βVp Lz 2 2 ∂ζ − 3V 23βL φ , 2φ 2 p −5KβLz ∂ζ 3Vp −5KβLz ) ( √ 2 2 3 1−η 3 T ∂ (2) (1) 2 φ + + η (T + 1) − φ . 3/2 2 4 ∂ζ 2(1−η) (17)
φ(1)
∂ζ
Furthermore, the next higher order series of of Poisson’s equation gives: (2)
(2)
δn(2) e + σn− − n+ = 0.
(18)
By differentiation of Eq. (18) w.r.t ζ δ
∂ (2) ∂ (2) ∂ (2) ne + σ n− − n+ = 0. ∂ζ ∂ζ ∂ζ
By substitution of Eq. (17) into Eq. (19) the following Burger’s equation is obtained 9
(19)
∂φ(1) ∂ ∂2 + Aφ(1) φ(1) + C 2 φ(1) = 0, ∂τ ∂ζ ∂ζ where C=−
γL2z . 2
(20)
(21)
By using tanh method [37, 38], the shock wave stationary solution of Eq. (21) is obtained by transforming the independent variables ζ and τ to ξ = ζ − U0 τ , where U0 is the normalized speed of the shock wave in the moving reference frame. Thus, one can express the stationary solution as (1)
φ
ξ = φ◦ 1 − tanh ∆
(22)
with the amplitude and width of the shock wave equal φ◦ = U0 /A and ∆ = 2C/U0 , respectively. The electric field, E (1) = −∇φ(1) , associated with this analytical solution, Eq. (22), can be derived as E
III.
(1)
U2 = − 0 sech2 2AC
ξ ∆
(23)
RESULTS AND DISCUSSION
We have considered a degenerate multi-ion collisionless and dense magneto-plasma system by taking into account the effects of the finite degenerate temperature and the electron trapping in the presence of a quantizing magnetic field. The positive and negative ions following the nonrelativistic equation of state, Eq. (3). The small-amplitude reductive perturbative method was employed to derived the KdV and Burger equations, as well as their solitary and shock wave solutions are obtained for a partially degenerate quantum plasma, respectively. The analyzation of these solutions is also performed. The results obtained from these investigations can be summarized as follows: 1: The plasma model under consideration supports only the cyclotron wave propagation (as fast modes). The variation of the phase speed Vp with various plasma parameters is demonstrated in Fig. 1. It has been shown that the phase speed increases (decreases) as β, σ, and δ (T ) increases, as shown in Figs. 1a and 1b. It means that larger phase velocity is attained when the negative ions density decreases and its mass increases. Also as the 10
degeneracy of the plasma system decreases and as the electron density increases lead to an increase of phase velocity. Solitary waves can be generated also with larger phase velocity as the obliquity angle decreases and electrons become less trapped as depicted in Fig 1c. A balance between the nonlinear and the dispersion terms forms solitary structures in the considered plasma model. The nonlinear term, A, depends on the various plasma parameters, e.g., the inertial ions number density and the inertialess electrons via β, σ, and δ, trapping parameter, η and degenerate temperature through T . Therefore, it is important to analyze A in terms of the plasma compositional parameters. 2: Figure 2 illustrates the variation of the nonlinear term A with Lz for different values of T, η, and δ. The nonlinear term is seen to be positive, it increases with increasing of T, η, and Lz and for δ > 0.05, A acquires smaller value when δ increases. 3: The dispersive term, B, depends on the positive ion cyclotron frequency, Ωc . In addition to the same plasma parameters similar to the nonlinear term, A. We depict the variation of the dispersion term B with Lz for different values of T, η and Ωc in Fig. 3. The dispersive term is seen to be positive and it decreases with increasing of Lz , T, η, and Ωc . 4: The considered plasma system supports positive potential IASWs with the amplitude is influenced by the previously mentioned plasma parameters. Figure 4 depicts the amplitude decreases as the trapping and degenerate temperature coefficients increase. 5: The KdV solitary waves are depicted in Fig. 5. The positive potential KdV or compressive solitons existed, their amplitude and width are affected by T and Lz , as shown in Figs. 5a and 5b. As the positive ion cyclotron frequency, Ωc , increases the soliton width decreases, but the soliton amplitude of the produced soliton remains unchanged, as illustrated in Fig. 5c. We now trace the effect of the viscosity coefficient and the obliqueness on the formed shock waves since the dissipation term, C, depends on both the ion fluid’s kinematic viscosity and obliqueness. It is easier for the nonlinear term, A to balance with the dissipation term, C in a non-relativistic plasma to form a stronger shock profile[39] 6: It is observed that only compressive shocks are formed in this multi-ion plasma system. Figure 6 demonstrates the effect of kinematic viscosity, γ, and obliqueness, via Lz , on the shock wave profile. The shock wave width increases as γ and Lz increase. The amplitude decreases as Lz increases, while it remains constant with the increase of γ since it is independent on the variation of kinematic viscosity. 11
7: The behavior of the electric field associated with the solution (22) against ξ is presented graphically in Fig. 7. The viscosity coefficient and the obliqueness affect crucially on this associated electric field. The stronger associated electric field wave (larger in amplitude and narrower in width) has been observed for this positive electric field values with the decreasing of the viscosity coefficient and the obliqueness.
IV.
CONCLUSION
In conclusion, the obtained results reveal that the reductive perturbation method is valid only for small but finite amplitude IA solitary structures. It is noted that the results of theoretical analysis, the effects of obliquely quantizing magnetic field (η), ratio of electron number density and positive ion number density multiplied by positive ion charge (δ), mass for positive to negative ions times charge for negative to positive ions (β), kinematic viscosity coefficient (γ) and finite degeneracy temperature (T ) play an important role in the variation of width, amplitude, and polarity of the KdV solitons and shock waves (see Figs. 1 − 7). We can conclude that as the negative ions density and the mass-to-charge for positive to negative ions ratio, β, the electron number density via δ and obliqueness, via Lz (the degeneracy temperature, T and the trapped electrons, η) increase (decrease) the phase speed, Vp increases (see Fig. 1). The parametric regimes give positive nonlinear and dispersion terms (see Figs. 2 and 3). Accordingly, positive solitary waves with positive amplitudes are formed (see Figs. 4 and 5). The viscosity coefficient, γ, has positive effects on the width of the compressive shock waves (see Fig. 6a). It is significant to reveal here that the amplitude (width) of positive shock wave decreases (increases) due to the increase of Lz ; where it falls (raises) more rapidly (see Figs. 6b). It is observed that the increasing of Lz and γ have a negative effect on the positive electric field wave (see Fig. 7). Finally, it has been concluded that the system under consideration admits only compressive solitary and shock structures for partially degenerate plasma in the presence of a quantizing magnetic field. The present results maybe helpful in understanding the shock dynamics and the solitary structures in dense quantum plasmas such as those that exist in white dwarfs due to the existence of matters under extreme conditions and also in experiments of ultra-high-density plasmas where degeneracy effects become important.
12
V.
REFERENCES
[1] Afshin Moradi, Phys. Scr. 90 (2015) 085601. [2] S. K. El-Labany, W. F. El-Taibany, A. E. El-Samahy, A. M. Hafez, A. Atteya, Braz. J. Phys. 45 (2015) 409. [3] S. K. El-Labany, W. F. El-Taibany, A. E. El-Samahy, A. M. Hafez, A. Atteya, IEEE Trans. Plasma Sci. 44 (2016) 842. [4] A. Atteya, E. E. Behery, W. F. El-Taibany, Eur. Phys. J. Plus, 132 (2017) 109. [5] M. Iqbal Shaukat, Phys. Plasmas. 24 (2017) 102301. [6] N. Roy, S. Tasnim, A. A. Mamun, Phys. Plasmas. 19 (2012) 033705. [7] L. Nahar, M. S. Zobaer, N. Roy, A. A.Mamun, Phys. Plasmas. 20 (2013) 022304. [8] Myoung-Jae Lee, Young-Dae Jung, Phys. Scr. 93 (2018) 025601. [9] J. M. Aparicio, G. Fontaine, White Dwarfs. Proceedings of the 10 th European Workshop on White Dwarfs, Kluwer Academic Publishers, London (1996). [10] M. R. Hossen, A. A. Mamun, Plasma Science and Technology 17 (2015) 3. [11] G. B.Van Albada, J. Astrophys. 105 (1947) 393. [12] S. Chandrasekhar, Mon. Not. R. Astron. Soc. 170 (1935) 405. [13] G. Brodin, M. Marklund, New J. Phys. 9 (2007) 277. [14] A. Y. Potekhin, Astron. Astrophys. 306 (1996) 999. [15] A. V. Gurevich, Sov. Phys. JETP 26 (1968) 575. [16] R. L. Morse, C. W. Nielson, Phys. Rev. Lett. 23 (1969) 1087. [17] H. L. Berk, C. E. Nielsen, K. V. Roberts, Phys. Fluids 13 (1970) 980. [18] M. Kako, T. Taniuti, T. Watanabe, J. Phys. Soc. Jpn. 31 (1971) 1820. [19] I. B. Bernstein, J. M. Greene, M. D. Kruskal, Phys. Rev. 108 (1957) 546. [20] H. Schamel, J. Plasma Phys. 14 (1972) 905. [21] H. Schamel, J. Plasma Phys. 9 (1973) 377. [22] H. Schamel, J. Plasma Phys. 13 (1975) 139. [23] H. Schamel, Phys. Rep. 140 (1986) 161. [24] H. Schamel, Phys. Plasmas 7 (2000) 4831. [25] H. Schamel, Phys. Plasmas 19 (2012) 020501. [26] S. A. Shan, S. A. El-Tantawy, Phys. Plasmas 23 (2016) 072112. [27] M. Tribeche, L. Djebarni, H. Schamel, Phys. Lett. A 376 (2012) 3164. 13
[28] A. A. Mamun, P. K. Shukla, L. Stenflo, Phys. Plasmas 9 (2002) 1474. [29] H. Siddiqui, H. A. Shah, N. L. Tsintsadze, J Fusion Energy 27 (2008) 216. [30] S. Sultana, I. Kourakis, N. S. Saini, M. A. Hellberg, Phys. Plasmas 17 (2010) 032310. [31] M. Irfan, S. Ali, Arshad M. Mirza, Phys. Plasmas 24 (2017) 052108. [32] P. Sharma, G. W. Hammett, E. Quataert, J. M. Stone, Astrophys. J. 637 (2006) 952. [33] P. Sharma, E. Quataert, G. W. Hammett, J. M. Stone, Astrophys. J. 667 (2007) 714. [34] S. K. El-Labany, W. F. El-Taibany, A. Atteya, Physics Letters A 382 (2018) 412. [35] E. E. Scime, P. A. Keiter, M. M. Balkey, R. F. Boivin, J. L. Kline, M. Blackburn, S. P. Gary, Phys. Plasmas 7 (2000) 2157. [36] S Arshad, H A Shah, M. N. S. Qureshi, Phys. Scr. 89 (2014) 075602. [37] S. K. El-Labany, W. F. El-Taibany, A. E. El-Samahy, A. M. Hafez, A. Atteya, Astrophys. Space Sci. 354 (2014) 385. [38]M. K.Deka, A. N. Dev, Annals of Physics 395 (2018) 45. [39] S. Islam, S. Sultana, A. A. Mamun, Phys. Plasmas 24 (2017) 092308.
14
Figure captions Fig. 1: (color online) The variation of the phase speed Vp with a) σ and β for Lz = 0.94, T = 0.4, η = 0.5 and δ = 0.3, b) T and δ for Lz = 0.94, η = 0.5, σ = 0.2 and β = 0.03 c) Lz and η for T = 0.6, σ = 0.3, δ = 0.3 and β = 0.03. Fig. 2: (color online) The variation of the nonlinear term A, represented by equation (14) with Lz for different values of a)T with η = 0.4, β = 0.01, σ = 0.1, δ = 0.3 and Ωc = 0.5, b) η with T = 0.4, β = 0.03, σ = 0.3, δ = 0.3 and Ωc = 0.5 c) δ with T = 0.7, η = 0.4, β = 0.03, σ = 0.3 and Ωc = 0.5. Fig. 3: (color online) The variation of the dispersive term B, represented by equation (14) with Lz for different values of a)T with η = 0.4, β = 0.01, σ = 0.1, δ = 0.3 and Ωc = 0.5, b) η with T = 0.4, β = 0.03, σ = 0.3, δ = 0.3 and Ωc = 0.5 c) Ωc with T = 0.4, η = 0.4, β = 0.03, σ = 0.3 and δ = 0.2. Fig. 4: (color online) The evolution of φm of IASWs with Lz for different values a) η with T = 0.4, β = 0.03, σ = 0.3, δ = 0.3, Ωc = 0.5 and U0 = 0.05 b)T with η = 0.4, β = 0.01, σ = 0.1, δ = 0.3, Ωc = 0.5 and U0 = 0.05. Fig. 5: (color online) The evolution of φ(1) of rarefactive IASWs that represented by equation (15) with ξ at U0 = 0.15 for different values of a) T with Lz = 0.94, η = 0.4, β = 0.03, σ = 0.2, δ = 0.3 and Ωc = 0.5, b) Lz with T = 0.5, η = 0.4, β = 0.03, σ = 0.3, δ = 0.2 and Ωc = 0.5 and c) Ωc with T = 0.5, Lz = 0.94, η = 0.4, β = 0.03, σ = 0.3 and δ = 0.2. Fig. 6: (color online) The evolution of φ(1) of rarefactive IA shock waves that represented by equation (22) with ξ at U0 = 0.05 for different values of a) γ with T = 0.6, η = 0.4, β = 0.03, σ = 0.3, δ = 0.2 and Lz = 0.94, b) Lz with T = 0.5, η = 0.4, β = 0.03, σ = 0.3, δ = 0.2 and γ = 0.4. Fig. 7: (color online) The evolution of the associated electric field E (1) of rarefactive IA shock wave that represented in Fig. 6 for different values of a) γ with the same parameters as Fig. 6a and b) Lz with the same parameters as Fig. 6b.
15
igure
0.645
Σ=0.1 Σ=0.2 Σ=0.3 Σ=0.4
0.640
Vp
0.635 0.630 0.625 0.620 0.615 0.00
0.05
0.10
0.15
0.20
0.25
0.30
Β
Fig. 1 a 0.7 0.6 0.5
T=0.1 T=0.2 T=0.3 T=0.4
Vp
0.4 0.3 0.2 0.1 0.0
0.0
0.1
0.2
∆
Fig. 1 b
0.3
0.4
0.62 0.60
Vp
0.58 0.56
lz=0.86 lz=0.9 lz=0.94 lz=0.98
0.54 0.52 0.50 0.1
0.2
0.3
0.4
Η Fig. 1 c
0.5
0.6
igure
2.20 2.15 2.10
A
2.05
T=0.1 T=0.2 T=0.3 T=0.4
2.00 1.95 1.90 1.85 0.86
0.88
0.90
0.92
0.94
0.96
0.98
1.00
0.94
0.96
0.98
1.00
lz Fig. 2 a 2.25
Η=0.3 Η=0.4 Η=0.5 Η=0.6
2.20 2.15
A
2.10 2.05 2.00 1.95 1.90 0.86
0.88
0.90
0.92
lz Fig. 2 b
∆=0.05 ∆=0.1 ∆=0.15 ∆=0.2
5.0 4.5
A
4.0 3.5 3.0 2.5 0.86
0.88
0.90
0.92
0.94
lz Fig. 2 c
0.96
0.98
1.00
igure
1.6
T=0.1 T=0.2 T=0.3 T=0.4
1.4 1.2
B
1.0 0.8 0.6 0.4 0.2 0.86
0.88
0.90
0.92
0.94
0.96
0.98
1.00
lz Fig. 3 a 1.4
Η=0.3 Η=0.4 Η=0.5 Η=0.6
1.2
B
1.0 0.8 0.6 0.4 0.2 0.86
0.88
0.90
0.92
0.94
lz Fig. 3 b
0.96
0.98
1.00
2.0
Wc=0.3 Wc=0.4 Wc=0.5
B
1.5
1.0
0.5
0.86
0.88
0.90
0.92
0.94
lz Fig. 3 c
0.96
0.98
1.00
igure
Η=0.3 Η=0.4 Η=0.5 Η=0.6
0.078 0.076
Φm
0.074 0.072 0.070 0.068 0.066 0.86
0.88
0.90
0.92
0.94
0.96
0.98
1.00
lz Fig. 4 a 0.080
T=0.1 T=0.2 T=0.3 T=0.4
0.078
Φm
0.076 0.074 0.072 0.070 0.068 0.86
0.88
0.90
0.92
0.94
lz Fig. 4 b
0.96
0.98
1.00
0.20
ΦH1L
0.15
T=0.1 T=0.3 T=0.5
0.10
0.05
0.00 -15
-10
-5
0
5
10
15
Ξ
Fig. 5 a
ΦH1L
0.15
lz=0.86 lz=0.9 lz=0.94 lz=0.98
0.10
0.05
0.00 -15
-10
-5
0
Ξ
Fig. 5 b
5
10
15
0.15
Wc=0.3 Wc=0.4 Wc=0.5
ΦH1L
0.10
0.05
0.00 -15
-10
-5
0
Ξ
Fig. 5 c
5
10
15
igure
0.035 0.030
ΦH1L
0.025
Γ=0.1 Γ=0.3 Γ=0.5
0.020 0.015 0.010 0.005 0.000 -15
-10
-5
0
5
10
15
Ξ
Fig. 6 a 0.04
ΦH1L
0.03
lz=0.86 lz=0.9 lz=0.94 lz=0.98
0.02
0.01
0.00 -20
-10
0
Ξ
Fig. 6 b
10
20
0.010
Γ=0.1 Γ=0.3 Γ=0.4
0.008
E H1L
0.006
0.004
0.002
0.000 -15
-10
-5
0
5
10
15
Ξ
Fig. 7 a 0.0035 0.0030
lz=0.86 lz=0.9 lz=0.94 lz=0.98
0.0025
E H1L
0.0020 0.0015 0.0010 0.0005 -15
-10
-5
0
Ξ
Fig .7 b
5
10
15