PROPERTIES OF ELECTRON
DEGENERATE
*1
Promi Halder,
1
Department of Physics, Jagannath
2Department of Physics,
ARTICLE INFO ABSTRACT
This research focuses on the
acoustic solitary waves in a degenerate quantum plasma (containing relativistic magnetized quantum electrons and light ions in presence of stationary heavy ions) have been theoretically investigated. The Korteweg
reductive perturbation method. Their stationary solutions are derived and analyzed analytically as well as numerically to study some new basic fea
commonly found to exist in degenerate quantum plasma. It is found that the basic properties (viz. amplitude, width, and phase speed, etc.) of the El acoustic waves are significantly modified by the effects of relativistic
plasma particles, and external magnetic field, etc. The results of this theoretical investigation may be useful for understanding the formation and fea
objects like white dwarfs, neutron stars, etc.
Copyright © 2018, Promi Halder et al. This is an open distribution, and reproduction in any medium, provided
INTRODUCTION
The physics of dense quantum plasmas has received an immense interest not only because of astrophysical compact objects (like white dwarfs, neutron stars, active galactic nuclei, etc.
Garcia-Berro, 2010; Mamun, 2010)), but also for their available application in the lab interaction experiment (Berezhiani, 1992; Murkhmd
are usually consist of extremely dense iron/oxygen/carbon and helium nuclei
degenerate plasma systems where the quantum mechanical effects (i.e., when the de Broglie thermal wavelength of the charged particles is equal or larger than the average inter
2016). The plasma in the interior of white dwarfs and in the crust of neutron stars are extremely dense and highly degenerate (Mamun, 2010; Chabrier, 2006; Lai, 2001; Harding
even more and the magnetic field strength can also be very large, i.e., high and the quantum-mechanical effects as
Rohm potential (causing the plasma particles tunneling) are expected to play an im adding the quantum statistical pressure term (the Fermi
to the fluid model generalized QHD model Hossain, 2011; Haas, 2003; Bhowmik, 2007; quantum effects of plasma particles in dif quantum plasmas was first studied by Pines
Bohm potentials associated with the plasma particles significantly modify the basic features of the nonlinear IA waves. Bhowmik
et al. (2007) investigated the effects of quantum diffraction parameter modifying the electron-acoustic (EA) waves in a quan
*Corresponding author:Promi Halder,
Department of Physics, Jagannath University, Dhaka
ISSN: 0975-833X
Article History:
Received 19th January, 2018
Received in revised form 08th February, 2018
Accepted 22nd March, 2018
Published online 30th April, 2018
Citation: Promi Halder, Mukta, K. N. and Mamun, A. A.
Quantum Plasma”, International Journal of Current Research
Key words:
Ion-acoustic Waves, Solitary waves, Relativistic Effect,
Degenerate Pressure, Magnetized Dense Objects.
RESEARCH ARTICLE
PROPERTIES OF ELECTRON-ION ACOUSTIC SOLITARY WAVES IN A MAGNETIZED
DEGENERATE QUANTUM PLASMA
Promi Halder,
2Mukta, K. N. and
2Mamun, A. A.
Department of Physics, Jagannath University, Dhaka-1100, Bangladesh
Department of Physics, Jahangirnagar University, Savar, Dhaka-1342, Bangladesh
ABSTRACT
This research focuses on the " Quality of service The nonlinear propagation of electron
acoustic solitary waves in a degenerate quantum plasma (containing relativistic magnetized quantum electrons and light ions in presence of stationary heavy ions) have been theoretically investigated. The weg-de Vries (K-dV) and modified K-dV (mK-dV) equations are derived by adopting the reductive perturbation method. Their stationary solutions are derived and analyzed analytically as well as numerically to study some new basic features of the El acoustic
commonly found to exist in degenerate quantum plasma. It is found that the basic properties (viz. amplitude, width, and phase speed, etc.) of the El acoustic waves are significantly modified by the effects of relativistic ally degenerate electrons and light ions, quantum pressure, number densities of plasma particles, and external magnetic field, etc. The results of this theoretical investigation may be useful for understanding the formation and features of the solitary structu
objects like white dwarfs, neutron stars, etc.
access article distributed under the Creative Commons Attribution the original work is properly cited.
The physics of dense quantum plasmas has received an immense interest not only because of astrophysical compact objects (like white dwarfs, neutron stars, active galactic nuclei, etc. (Woolsey
but also for their available application in the laboratory
Murkhmd, 2006). The degenerate compact objects e.g. white dwarfs, neutron stars, etc. are usually consist of extremely dense iron/oxygen/carbon and helium nuclei (Shiikla, 2011; Massey
degenerate plasma systems where the quantum mechanical effects (i.e., when the de Broglie thermal wavelength of the charged particles is equal or larger than the average inter-particle distance) play an important role in the plasma dynamics
The plasma in the interior of white dwarfs and in the crust of neutron stars are extremely dense and highly degenerate Harding, 2006) (plasma particles number densities can be of the order of 10
even more and the magnetic field strength can also be very large, i.e., B » WgG). For such plasmas, the Fermi tempera
mechanical effects associated with the quantum statistical pressure and the quantum force involving the Rohm potential (causing the plasma particles tunneling) are expected to play an important role (
tistical pressure term (the Fermi-Dirac distribution) and the quantum diffraction term (the Bohm potential) to the fluid model generalized QHD model (Manfredi, 2001; Manfredi, 2005) is obtained. A number of works
, 2007; Saeed-ur-Rehman, 2012; Hossen, 2015) have been done by con
quantum effects of plasma particles in different plasma medium. The behaviour of high densities and low temperatures ied by Pines (Pines, 1961). Haas et al. (Haas et al., 2003) found in t
plasma particles significantly modify the basic features of the nonlinear IA waves. Bhowmik tigated the effects of quantum diffraction parameter H, and the equilibrium density ratio of the plasma species in
acoustic (EA) waves in a quantum ET plasma.
Department of Physics, Jagannath University, Dhaka-1100, Bangladesh.
International Journal of Current Research Vol. 10, Issue, 04, pp.68243-68256, April, 2018
Promi Halder, Mukta, K. N. and Mamun, A. A. 2018. “Properties of Electron-Ion Acoustic Solitary Waves in a Magnetized Degenerate
Journal of Current Research, 10, (04), 68243-68256.
ION ACOUSTIC SOLITARY WAVES IN A MAGNETIZED
1100, Bangladesh
1342, Bangladesh
The nonlinear propagation of electron-ion (El) acoustic solitary waves in a degenerate quantum plasma (containing relativistic magnetized quantum electrons and light ions in presence of stationary heavy ions) have been theoretically investigated. The dV) equations are derived by adopting the reductive perturbation method. Their stationary solutions are derived and analyzed analytically as tures of the El acoustic solitary structures that are commonly found to exist in degenerate quantum plasma. It is found that the basic properties (viz. amplitude, width, and phase speed, etc.) of the El acoustic waves are significantly modified by the degenerate electrons and light ions, quantum pressure, number densities of plasma particles, and external magnetic field, etc. The results of this theoretical investigation may be tures of the solitary structures in astrophysical compact
License, which permits unrestricted use,
The physics of dense quantum plasmas has received an immense interest not only because of their omnipresence in many Woolsey et al., 2004; Shapiro, 1983; oratory i.e., intense laser-solid matter . The degenerate compact objects e.g. white dwarfs, neutron stars, etc. Massey, 1976) are ideal examples of degenerate plasma systems where the quantum mechanical effects (i.e., when the de Broglie thermal wavelength of the charged the plasma dynamics (Abdikian,
The plasma in the interior of white dwarfs and in the crust of neutron stars are extremely dense and highly degenerate ties can be of the order of 1030cm~3, or
For such plasmas, the Fermi temperature Tp is pressure and the quantum force involving the (Manfredi, 2005). Therefore, by e quantum diffraction term (the Bohm potential) A number of works (Rouhani, 2014; have been done by considering the ferent plasma medium. The behaviour of high densities and low temperatures found in their investigation that the plasma particles significantly modify the basic features of the nonlinear IA waves. Bhowmik and the equilibrium density ratio of the plasma species in INTERNATIONAL JOURNAL OF CURRENT RESEARCH
On the other hand, Ali et al. (2007) analyzed the IA waves in an EPI plasma, and found that the nonlinear properties of the IA waves are significantly affected by the inclusion of the quantum terms in the momentum equations of electrons and positrons. But none of them considered the effects of external magnetic field and the presence of the heavy ions. It is well known that (Miller, 1987; Plastino, 1993; Gervino, 2012) the presence of external magnetic field (which causes the obliqueness of the wave propagation) plays a vital role in modifying the basic features of the linear and nonlinear waves in space and astrophysical plasmas (Mahmood, 2008; Sultana, 2010; El-Tantawy, 2012; Shahmansouri, 2013; Alinejad, 2013; Ashraf et al., 2014), and that the most of astrophysical degenerate quantum plasma systems like white dwarfs and neutron stars usually contain degenerate electrons and light ions along with heavy ions (Koester, 1990). This means that the effects of heavy ions and magnetic field must be considered, specially for the study of the nonlinear phenomena in the degenerate astrophysical objects (Woolsey, 2004; Shapiro, 1983; Torres
et al., 2010; Mamun, 2010). Obliquely propagating electron-acoustic (EA) solitary waves (SWs) in a two electron population quantum rnag-netoplasma was theoretically investigated by Masood and Mushtaq (2008). They found that propagation character-istics of the EA SWs are significantly affected by the presence of quantum corrections and the ratio of hot to cold electron concentration. Recently, Hossen and Mamun (Hossen, 2011c) have theoretically investigated the nonlinear positron-acoustic (PA) waves propagating in the fully relativistic electron-positron-ion plasma and found that the effects of relativistic degeneracy of electrons and positrons, static heavy ions, plasma particles velocity, and enthalpy, etc. have significantly modified the ba-sic properties of the PA SWs. Using a fully relativistic set of two fluids plasma equations, Lee and Choi (Lee, 2007), Tribeche and Boukhalfa (2011), Saberian et al, (2011), and Akbari-Moghanjoughi (2011) have studied the characteristics of the nonlinear IA waves in different fully relativistic plasmas. Therefore, in our present work, we have examined the basic properties of the El acoustic waves propagating in a degenerate quantum plasma composed of relativistically magnetized quantum electrons and light ions in the presence of stationary heavy ions. We have considered the quantum mechanical (such as tunneling associated with the Bohm potential) effects for both electrons and light ions which abundantly occurs in different astrophysical situations (viz. white dwarfs, neutron stars, active galactic nuclei, etc. (Woolsey, 2004; Shapiro, 1983; Garcia-Berro, 2010; Mamun, 2010), and laboratory plasmas like intense laser-solid matter interaction experiment (5. 6). The manuscript is organized as in Sec. II, the basic equations governing our plasma model are presented; in Sec. Ill and IV, the K-dV and mK-dV equations along with their solutions are derived; in Sec. V, a brief discussion is given.
MODEL EQUATIONS
We consider a collision less plasma system with an ambient magnetic field directed along the z axis, i.e., (Bo =
z
ˆ
B
o). wherez
ˆ
is a unit vector in the z direction. The obliquely propagation of the nonlinear acoustic SWs through our considered plasma system is assumed such that the wave vector lies in the x-z plane. At equilibrium, the quasi-neutrality condition can be expressed as,
0
io eoho h
Z
where
ho,
io, and
eo are the equilibrium number densities of immobile heavy ions, light ions, and electrons, respectively, and Zhis the immobile heavy ions charge state. The unnormalized dynamic equations for the consideredmagnetized quantum El plasmas are as follows:
0
,
.
*
s s
s
N
U
T
N
(1)
*
.
s s
U
T
U
0*
1
B
U
c
m
q
U
se s s
,
1
2
2 * *
2 * *
ss e
s e
s
N
n
m
N
m
P
(2)
,
4
2 *
h h e
i
N
Z
N
N
e
(3)
is
*
s
P
and the full expression for it is (Mon, 1935)
s s s s
e
s
R
R
R
R
h
c
m
P
3 2 1/2 2 15 4 *
sinh
3
3
2
1
3
(4)
in which the relativity parameter
R
s
pF
s/
m
ec
andpF
s
3 / 1 3
)
8
/
3
(
h
n
s
is the Fermi relativistic momentum of s'th species.
Taking
N
s
N
s0
N
s1,
N
s1 is the perturbation of the s'th species withN
s1Nso and the Taylor expansion up to second order, Eq. (4) turns to
3 1 0 2
* 2 1
0 * 0
*
2
1
s Ns Ns s s s
Ns Ns s s s
s
N
N
P
N
N
P
P
P
'
9
4
3
3
2
1 2 0 3
0 2 0 1 0
0 s
s s
s s
s s
s
N
N
F
R
N
F
P
(5)
where
2 ' 0 0
1
s s
R
/
,
0
0
P
m
c
R
s
s eP
s0
(
3
h
3n
s0/
8
)
1/3, and
(
3
)
/
2
.
3 / 2 2 2
e s
s
n
m
F
In order to normalize the basic Eqs. (l)-(3), we use the following normalized parameters
e s s e e
pe pe
uF
u
U
F
e
uF
t
T
,
2
,
* ,
e pe s
e s ps
s s s
F
H
m
n
e
n
n
N
2
,
4
,
02
0
pe c
e e
s
s
c
c
m
eB
c
m
F
uF
2
,
0,
.
,
,
2
00 0
*
e h
s Fe
s s
n
n
Fi
Fe
n
P
P
(6)
The normalized equations for the ion quantum fluid can be written as
0
,
.
i i
i
n
u
t
n
(7)
i i i
c i
i i
n
P
z
u
u
u
t
u
)
ˆ
(
.
,
1
)
1
(
2
2 3
/ 2 2
ii h
n
n
Z
H
(8)
and the normalized equations for the electron quantum fluid can be written as
0
,
.
e e
e
n
u
t
n
(9)
e e e
c e
e e
n
P
z
u
u
u
t
u
(
ˆ
)
.
,
1
2
2 2
pe
n
n
H
(10)
and the normalized form of the Poisson equation is
.
)
1
(
2
h h i
e
n
Z
Z
n
(11)
K-dV EQUATION
),
(
2 / 1
t
V
l
l
l
z
y
z
p
(12)
,
2 / 3
t
(13)
where
V
p[
/
(
is angular frequency and k is the wave vector)) is the wave phase speed,
(
0
1
)
is a measure of the solitary wave amplitude i.e., a measure of the weakness of the dispersion or of the nonlinear effect,l
x,
l
y,
andl
z are the directioncosines of the wave vector k along the x, y, and z axes, respectively, so that
1
.
2 2
xx
l
l
We then expand the perturbed
quantities
n
i,
n
e,
u
sx,y,u
sz'and
about their equilibrium values in power series of
as...,
1
(1)
2 (2)
3 4(3)
n
n
n
n
i i i(14)
...,
1
()
2 (2)
3 (3)
ei e ee
n
n
n
n
(15)
...,
) 3 (
. 2 / 5 ) 2 (
. 2 ) 1 (
. 2 / 3
,y
sxy
sxy
sxy
sx
u
u
u
u
(16)
...,
) 3 ( 3 ) 2 ( 2 ) 1 (
sz sz szsz
u
u
u
u
(17)
...
) 3 ( 3 ) 2 ( 2 ) 1 (
(18)
Now, substituting Eqs. (12)-(18) in Eqs. (7)-(ll), and equating the coefficients for the lowest order of
, we obtain the first order continuity equations, the z -component of the momentum equations, and Poisson's equation, which after simplification, we can write as,
3
3
(1) 2 2 02 0 )
1 (
z p i
z i i
l
V
l
n
(19)
,
3
3
(1)2 2 0
0 )
1 (
z p i
p z i iz
l
V
V
l
u
(20)
,
3
3
(1) 2 20 2
2 0 )
1 (
e p z
z e e
V
l
l
n
(21)
,
3
3
(1)2 0 2
2 0 )
1 (
p e z
p z e ez
V
l
V
l
u
(22)
, ) 2
( 3
) (
0 0
2 0 2 0
h i
e
h z i z e p
Z Z l l l
v
(23)
We define lz = cos
,. where
is the angle between the directions of the wave propagation vector k and the external magneticfield B0- The Eq. (23) represents the dispersion relation for the acoustic type electrostatic waves in the degenerate quantum plasma under consideration. We can write the first order x— and y—components of the momentum equations as
,
)
3
(
3
(1)2 2 0
2 0 )
1 (
z p i e
p i x iy
l
V
V
l
u
,
)
3
(
3
(1)2 2 0 2 0 ) 1 (
z p i c p i y ixl
V
V
l
u
(25),
)
3
(
3
(1)2 0 2 2 0 ) 1 (
p e z c p e yV
l
V
l
u
ey (26),
)
3
(
3
(1)2 0 2 2 0 ) 1 (
p e z c p e yV
l
V
l
u
ez (27)Eqs. (24)-(27), represent the x— and y—components of the ion-electron electric field drift, respectively. Again, substituting Eqs. (12)-(18) in Eq. (8) and Eq. (10) and using Eqs. (19)-(27), we obtain the next higher order x-and y—components of the momentum equations as
,
)
3
(
3
2 ) 1 ( 2 2 2 0 2 0 3 ) 2 (
z p i c y i pl
V
l
V
u
iy (28),
)
3
(
3
2 ) 1 ( 2 2 2 0 2 0 3 ) 2 (
z p i c x i pl
V
l
V
u
iy (29),
)
3
(
3
2 ) 1 ( 2 2 0 2 2 0 3 ) 2 (
p e z c y e pV
l
l
V
u
ey (30),
)
3
(
3
2 ) 1 ( 2 2 0 2 2 0 3 ) 2 (
p e z c x e pV
l
l
V
u
ex (31)Eqs. (28)-(31), denote the y— and x—components of the ion-electron polarization drift, respectively. Further, substituting Eqs. (12)-(18) in Eqs. (7)-(ll), we obtain the next higher order continuity equations, the z- component of the momentum equations, and Poisson's equation, which can be given as
(1) (2) (2) (2) (2)1
iz z iy y ix z i pi
V
n
l
u
u
l
u
n
() ()
0
,
l
zn
ilu
izl
(32)
(2)0 ) 2 ( ) 1 ( ) 1 ( ) 2 ( ) 1 (
3
1
i i z z iz iz z iz piz
u
l
n
u
l
u
V
u
) ( ) ( 0 ) ( ) ( 3 0 2 03
9
4
3
l ili i z l i l i i i
z
R
n
n
l
n
n
l
,
0
4
3 ) 1 ( 3 2
i zn
l
H
(33)
(1) (2) (2) (2) (2)ez z ey y ex x e p
e
V
n
l
u
l
u
l
u
n
(1) (1)
0
,
l
zn
eu
ez
(2)0 ) 2 ( )
1 ( ) 1 ( ) 2 ( )
1 (
3
e
e z z
ez ez x ez p
ez
l
n
l
u
u
l
u
V
u
) 1 ( 3
0 0 2
9
)
4
3
(
e e
e
z
R
n
l
(1) (1)3 0 ) 1 (
3
e e e z
e
l
n
n
n
,
0
)
1
(
4
3) 1 ( 3 3 / 2 2
e
h
z
n
Z
l
H
(35)
) 2 ( )
2 ( 2
) 1 ( 2
)
1
(
h ie
Z
n
n
(36)
Now, simplifying Eqs. (32)-(36) by using Eqs. (19)-(31), and combining each other, we finally obtain our desired equation in the form
,
0
3 ) 1 ( 3 )
1 ( ) 1 ( ) 1 ( 2
B
A
(37)
Eq. (37) is the well known K-dV equation describing the dynamics of the SWs propagating in the degenerate quantum plasma. The nonlinear coefficient, A and the dispersion coefficient, B are given as
,
)
1
(
6
6
2 30 3
2 0
4 3 2 1
eV
pa
b
pV
pab
Z
hL
L
L
L
A
(38)
,
)
1
(
72
72
2 2 2 20 2
2 2 2
0
5 4 3 2 1
eV
pl
za
c iV
pl
zb
cZ
hK
K
K
K
K
B
(39)
Where
),
3
(
i0V
p2l
z2a
(
3
),
2 0 2
p e z
V
l
b
),
27
30 2 321 e
V
pa
l
zL
L
2
a
3l
z4(
3
R
e20
3
e20
4
),
),
1
(
27
30 2 3 23
iV
pb
l
zZ
h
L
),
1
)(
4
3
3
(
202 0 4 3
4
b
l
zR
i
iZ
h
L
,
4
2 2 21
a
b
cK
36
(
1
),
2 2 4 2
0
2 e
V
pa
l
zK
,
9
20 4 2 2 23 e
l
zH
a
cK
2 4 2
0 4
36
V
b
K
i p(
1
2),
z
l
(
1
Z
h
),
),
9
20 4 2 2 25 i
l
zH
b
cK
(
1
Z
h
)
1/3,
(40)
Now, to investigate the properties of the El acoustic SWs, we are interested in the SWs solution of the K-dV equation. To do so,
we introduce another stretched coordinates.
u
0.
After the coordinate transformation, the steady state(
/
0
)
solution of the SWs can be written as
,
sec
1 2 )
1 (
mh
(41)
where the amplitude
m
3
U
0/
A
and the width,
1
4
B
/
U
0.
It is seen that the quantum parameter H is present in
1,
whereas
m is totally independent of these parameter.mK-DV EQUATION
The K-dV equation (Eq. (37)) is the result of the second order calculation in the smallness parameter
, where the quadraticnature has been revealed by the nonlinear term
/
,
) 1 ( ) 1 (
A
.For plasmas with more than two species as like our system, however, there can arise cases where A vanishes at a particular value of a certain parameter
, and Eq. (37) fails to describe nonlinear evolution of perturbation. So, higher order calculation is neededat this critical value
c.
For this reason, to derive the mK-dV equation, we apply the following stretched coordinates(Washimi, 1966)
),
(
2 / 1t
V
l
l
l
x
y
z
p
(42),
2 / 3t
(43)We then expand the variables , and
, in power series of
as...,
1
1/2 (1)
(2)
3/2 (3)
i i ii
n
n
n
n
(44)
...,
1
1/2 (1)
(2)
3/2 (3)
e e ee
n
n
n
n
(45)...,
) 3 ( . 2 ) 2 ( . 2 / 3 ) 1 ( ..y
szy
sxy
sxy
ix
u
u
u
u
(46)...,
) 3 ( 2 / 3 ) 2 ( ) 1 ( 2 / 1
sz sz szsz
u
u
u
u
(47)...,
) 3 ( 2 / 3 ) 2 ( ) 1 ( 2 / 1
(48)By using Eq. (42)-(48) in Eqs. (7)-(ll), we found the values of
) 1 ( . ) 1 ( , ) 1 ( ) 1 ( ) 1 ( ) 1 ( , , , ,
, e iz ez ixy exy
i n u u u u
n
and Vpas like as that of the K-dV equation.
To the next higher order of
, we obtain a set of equations, which, after using the values of) 1 ( . ) 1 ( , ) 1 ( ) 1 ( ) 1 ( ) 1 ( , , , ,
, e iz ez ixy exy
i n u u u u
n
and Vpcan
be simplified as
(1)23 2 2 0 3 0 2 0 6 3 0 4 2 ) 2 (
)
3
(
2
)
4
3
3
(
3
81
z p i i i z i z p il
V
R
l
l
V
n
) 2 ( 2 2 0 0 2)
3
(
3
z p i i zl
V
l
(49)
(1)2 3 2 0 2 2 0 2 0 6 3 0 4 2 ) 2 ()
3
(
2
)
4
3
3
(
3
81
p e z e e z e z p eV
l
R
l
l
V
n
) 2 ( 2 0 2 0 2)
3
(
3
p e z i zV
l
l
(50)
) 1 ( ) 1 ( 2 2 2 0 0 2 0 2 0 4 ) 2 ( 0 ) 2 ()
3
(
)
4
3
3
(
3
i c i p zi i z x i i c x iy
l
V
R
l
l
n
l
u
,
)
3
(
3
2 ) 1 ( 2 2 2 2 0 2 0 3
z p i c i y pl
V
l
V
(51) sz y sx ei
n
u
u
) 1 ( ) 1 ( 2 2 2 0 0 2 0 2 0 4 ) 2 ( 0 ) 2 ()
3
(
)
4
3
3
(
1
3
i c i p zi i z y i i c y ix
l
V
R
l
n
l
u
,
)
3
(
3
2 ) 1 ( 2 2 2 0 2 0 3
z p i c i y pl
V
l
V
(52)
) 1 ( ) 1 ( 2 2 0 2 0 2 0 2 0 4 ) 2 ( 0 ) 2 ()
3
(
)
4
3
3
(
1
3
i c z i pi i z y i i c y ex
V
l
l
R
l
n
l
u
,
)
3
(
3
2 ) 1 ( 2 2 0 2 2 0 3
p e z c e y pV
l
l
V
(53)
) 1 ( ) 1 ( 2 2 0 2 0 2 0 2 0 4 ) 2 ( 0 ) 2 ()
3
(
)
4
3
3
(
1
3
e c z e pe e z x e e c y ex
V
l
R
l
n
l
u
,
)
3
(
3
2 ) 1 ( 2 2 0 2 2 0 3
p e z c e x pV
l
l
V
(54)
) 1 ( ) 1 ( 2 2 2 0 0 2 0 2 0 5 3 0 3 ) 2 ()
3
(
)
4
3
3
(
9
z p i i i i z i z p izl
V
R
l
l
V
u
,
3
) 2 ( 0 ) 2 ( i i p z p zn
V
l
V
l
(55)
) 1 ( ) 1 ( 2 2 0 2 0 2 0 2 0 5 3 0 3 ) 2 ()
3
(
)
4
3
3
(
9
p e z e e e z e z p ezV
l
R
l
l
V
u
,
3
) 2 ( 0 ) 2 ( e e p z p zn
V
l
V
l
(56)
0
,
2
1
(1)2 1 ) 2 (
A
(57) where
3
') )(
1 (
3 0 2 2 3
2 1 3 2 0 2 2 1 1 z p i h p e
z V l
P P Z V l T T A (58) Where
'
81
2 4 301
V
pl
z eT
3
(
3
3
4
)'
2 0 2
0 6
2
l
zR
e
e
T
'
81
2 4 301
V
pl
z iP
3
(
3
3
4
)'
3 0 2 6
2
l
zR
i
i
P
(59)
To the next higher order of
, we obtain a set of equations which after simplification as follows
(3)2 2 0 0 2 ) 1 ( 2 2 2 0 0 3 0 2 ) 3 (
)
3
(
3
)
3
(
18
z p i i z z i i i z p il
V
l
l
V
l
V
n
2 2 20 3 / 2 2 0 2 4 2 2 2 0 2 2 2 0 4
)
3
(
)
1
(
4
9
)
3
(
1
9
z p i h i z z p i c z i pl
V
Z
H
l
l
V
l
V
3 2 2 0 3 0 2 0 6 3 0 4 2 3 ) 1 ( 3 ) 3 ( 4 3 3 ( 3 81 z p i i i z i z p l V R l l V
4 2 2 0 7 6 5 0 5 2 2 0 4 3 2 1 ) 2 ( ) 1 ()
3
(
)
3
(
2
]
[
z p i i z pi
V
l
F
F
F
l
V
F
F
F
F
'
)
3
(
2
) 1 ( 2 ) 1 ( 5 2 2 0 9 8
z pi
V
l
F
F
(60)
(3)2 0 2 0 2 ) 1 ( 2 2 0 2 2 0 2 ) 3 (
)
3
(
3
)
3
(
18
p e z e z p e z e z p eV
l
l
V
l
l
V
n
3 ) 1 ( 3 2 2 2 0 2 2 0 2 4 2 2 2 0 2 2 2 2 0 4)
3
(
4
9
)
3
(
)
1
(
9
p e z e p e z c z e pV
l
H
l
V
l
l
V
) 2 ( ) 1 ( 3 2 0 2 3 0 2 0 6 3 0 4 2)
3
(
4
3
3
(
3
81
p e z e e z e z pV
l
R
l
l
V
4 2 0 2 7 6 5 0 5 2 0 2 4 3 2 1)
3
(
)
3
(
2
z e p el
z eV
pG
G
G
V
l
G
G
G
G
'
)
3
(
2
) 1 ( 2 ) 1 ( 5 2 0 2 9 8
p e zV
l
G
G
(61) ) 3 ( ) 3 ( ) 1 ()
1
(
h ie
Z
n
n
(62) where,
486
2 8 501
V
pl
z iF
18
(
3
3
4
)
2 0 2 0 2 0 10
2
l
z iR
i
i
F
,
),
4
3
3
(
81
2 8 20 20 203
V
pl
z iR
i
i
F
'
)
4
3
3
(
3
20 22 0 10
4
l
zR
i
i
F
,
81
2 6 405
V
pl
z iF
9
(
3
3
4
),
2 0 2 0 0 8
6
V
pl
z iR
i
i
F
),
4
3
3
(
3
202 0 0 8
7
l
z iR
i
i
F
5 0 6 4 8
729
V
pl
z iF
),
4
3
3
(
27
28 20 20 209
V
pl
z iR
i
i
F
,
486
2 8 501
V
pl
z eG
18
(
3
3
4
)'
2 0 2 0 2 0 10 2
2
l
z eR
e
e
G
)'
4
3
3
(
81
28 30 20 203
V
pl
z eR
e
e
G
'
)
4
3
3
(
3
2 10 20 20 24
l
zR
e
e
G
,
81
2 6 405
V
pl
z eG
G
6
9
V
pl
z8
e20(
3
R
e20
3
e20
4
),
),
4
3
3
(
3
8 0 20 207
l
z eR
e
e
G
G
8
729
V
p4l
z6
e50,
).
4
3
3
(
27
28 20 20 209
V
pl
z eR
e
e
G
(63)
30
,
) 1 ( 3 ) 1 ( 2 ) 1 ( )
1 (
C
B
(64)where
)
1
(
18
18
2 20 2 2
2 0 2
2 2
e z p i z hp
a
l
V
b
l
Z
V
b
a
C
,2 2
1 2
2 5
9 8 4 2 0 5
1 5
9 8 4 2 0 5
1
a F F a R a
R Z b
G G b K b
K
e h
e
(65)
Where
,
4 3 2 1
1
G
G
G
G
K
K
2
G
5
G
6
G
7,
,
4 3 2 1
1
F
F
F
F
R
R
2
F
5
F
6
F
7.
(64)Now, taking the same stretching as like the solution of the K-dV SWs, the stationary SWs solution of Eq. (64) can be directly given as
, sec
2 )
1 (
h m
(67)
where the amplitude,
m
6
U
0/
C
and the width,
2
B
/
U
0 , where
2 depends H. On the other hand,
m is totallyindependent of these parameter.
DISCUSSION
The properties of the acoustic SWs in degenerate quantum plasmas are discussed in this section numerically. The effects of quantum diffraction are related not only in degenerate astrophysical plasmas but also may be meaningful in some dense laboratory plasma containing electrons and light ions. in presence of stationary heavy ions.
[image:10.595.114.509.494.751.2]We emphasize the effects of the propagation angle, quantum pressure, relativistic factors, immobile heavy ions charge state, and densities of the plasma components on the acoustic SWs in such degenerate quantum plasmas. For our purpose, we have derived the K-dV and
FIG. 1: Showing the variation of
V
p with
for U0 0.01,
= 40,
e0= 1.33,
i0
1
.
29
and Zh= 2.
FIG. 2: Showing the variation of
1(K-dV) with H for U0 = 0.01. e0
= 1.33,
i0= 1.29,
= 40,Z
h
1
,
= 1.1. and
= 0.1. The dotted line for
c
0
.
3
, the solid line for
c
0
.
4
, and the dotdashed line for
c
0
.
5
.FTG. 3: Showing the effects of
on the K-dV solitary profiles for U0 = 0.01. e0
= 1.33,
i0= 1.29,
= 0.4,
= 40,
c = 0.4,Z
h
1
, H = 0.3.5
.
2
e
R
andR
i
2
.
2
. The lower red dotted line for
= 1.1. the lower blue solid line for
= 1.13, the upper green dotdashed line for
= 1.16, and the upper black solid line for
= 1.19.FIG. 4: Showing the effects of Ri on the (K-dV) SWs profiles for U0 = 0.01. e0
= 1.33,
i0= 1.29,
= 0.4,
= 40,
c
0
.
4
,1
h
Z
FIG. 5: Showing the effects of
con the mK-dV SWs profiles for U0 = 0.01. e0
= 1.33,
i0= 1.29,
= 0.2,
= 60,Z
h
1
, H = 0.3, Re = 2.5, Ri = 2.2, and
= 1.6. The dotted line for3
.
0
c, the solid line for
c = 0.4, and the dotdashed line for
c
0
.
5
.FIG. 6: Showing the effects of
on the mK-dV SWs profiles for U0 = 0.01. e0
= 1.33,
i0= 1.29,
c = 0.4,
= 60,Z
h
1
, H = 0.3.R
e
2
.
5
andR
i
2
.
2
and
= 1.8. The dotted line for
= 0.1, the solid line for
= 0.2. and the dotdashed line for
= 0.3.FIG. 7: Showing the mK-dV SWs profiles for U0 = 0.01. c
= 0.3,
= 0.1,
= 60,Z
h
1
, H = 0.3. and
= 1.16. The solid line fornon-relativistic case
(
e0= 1,
i0= 1,R
e
0
andR
i
0
)
, and the dotted line for relativistic case (
e0 = 1.33,
i0 = 1.29,R
e
2.5,and
R
i
= 2.2).68249 International Journal of Current Research, Vol. 10, Issue, 04, pp.68237-68250, April, 2018
