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International Journal of Physics
Volume 1, Issue 1, June 2015, Pages 11-19
Journal home page: http://iosrd.org/journals
Electron-acoustic solitary structures in finite temperature quantum
plasmas containing electrons at two different temperatures
Swarniv Chandra1, 3*
, Sibarjun Das2
, Basudev Ghosh3
1
Department of Physics, Techno India University, Kolkata, India
2
Department of FTBE, Jadavpur University, Kolkata, India
3
Department of Physics, Jadavpur University, Kolkata, India
*Corresponding Author E-mail Address: swarniv147@gmail.com
Article history: Received 14 March 2015, Accepted 6 May 2015, Available online 25 June 2015
Abstract: Using the quantum hydrodynamic (QHD) model for quantum plasma at finite temperature the linear and nonlinear properties and
solitary wave structures of electron-acoustic waves is investigated by deriving a Kortewrg de Vries equation equation. It was found that the
electron degeneracy parameter significantly affects the linear and nonlinear properties of electron plasma waves in quantum plasma.
Keywords: Quantum hydrodynamic model, Electron acoustic Waves, Finite Temperature Model, Solitary wave structures, Quantum
Plasma
1. Introduction
In the recent years the study of ultradense matter has been carried out quite extensively and intensively. Such matter is found
in metal nanostructures, neutron stars, white dwarfs and other astronomical bodies as well as in laser plasma interaction
experiments. In such situations the average inter-Fermionic distance is comparable or even less than the thermal de Broglie
wavelength and as a result the quantum degeneracy becomes important. In such extreme conditions of density the thermal
pressure of electrons may be negligible as compared to the Fermi degeneracy pressure which arises due to the Pauli exclusion
principle. However in this case the quantum effect can’t be neglected and proper mathematical modeling becomes necessary.
Such quantum effects are generally studied with the help of two well known formulations, the Wigner-Poisson and the
Schrodinger Poisson formulation. The former one studies the quantum kinetic behavior of plasmas while the latter describes
the hydrodynamic behavior of plasma waves. The quantum hydrodynamic (QHD) model is derived by taking the velocity
space moments of the Wigner equations. The QHD model modifies the classical fluid model for plasmas with the inclusion of
a quantum correction term generally known as the Bohm potential. The model also incorporates the quantum statistical effect
through the equation of state. The model has been widely used to study quantum behavior of plasma. A survey of the
available literature [1-12] shows that most of the works done in quantum plasma in order to study the nonlinear behavior of
different plasma waves uses ultra low temperature approximation. But in most practical cases the temperature is not zero but
finite. In this paper we have used the model developed by Eliasson and Shukla [13] to study the linear and nonlinear behavior
of electron acoustic waves in a quantum plasma at finite temperature consisting of two groups of electrons, one group
comparatively hot and another colder or slightly inertial and stationary ions forming a neutralizing background.
2. The finite temperature model and formulations
2.1 The finite temperature quantum plasma model
Based on the 3D equilibrium Fermi-Dirac distribution for electrons at an arbitrary temperature a set of fluid equations which
are valid both in the large and low temperature limits were derived. When a plane longitudinal electron plasma wave
propagates in collisionless quantum plasma, it leads to adiabatic compression thereby causing temperature anisotropy in the
electron distribution. In quantum plasma the classical compressibility of the electron phase fluid is violated due to quantum
mechanical tunneling. However to a first approximation it can be assumed that the electron phase fluid is incompressible.

International Journal of Physics (http://iosrd.org/journals)
Further the chemical potential (µ) remains constant during the nonequilibrium dynamics of the plasma. Under such
assumptions the nonequilibrium particle density is given by:
  
  
3 2 3 21 2
0 3 22 2 2 3 2 20
1 2 1 2 3
exp
2 exp 1 2 2
m E dE m
n Li
E

    

    
 
     
     
     
 
(2.1-1)
whereLiγ(y) is the polylogarithm function.
In the ultracold limit i.e T→0, we have β→∞ and µ→EF. The temperature anisotropy is given by:
     ex e0 ex 0 e
x,t T / T x,t n / n x,t      . (2.1-2)
The Fermi energy is given by:
   
2 32 2
0
3 / 2F
E n m  (2.1-3)
Now using the zeroth and first moments of the Wigner equation with the Fermi-Dirac distribution function and assuming
that the Bohm potential is independent of the thermal fluctuations in a finite temperature plasma one can derive the continuity
and momentum equation in the following form:
( )
0e e ex
n n u
t x
 
 
 
(2.1-4)
 
3 22 2
,, , 0,0
, 2 2
, ,
1
2
e jex j e j jex Te
ex j
e e j e e j
nu n nu n Ve
u G
t x m x n x m x xn
    
   
     
 
 
  

(2.1-5)
where ne,j and vex ,j are respectively the ‘particle density’ and ‘fluid velocity’ of jth
species of electron, (j=c for cold, =h for
hot);  is the electrostatic wave potential and /Te B Te e
v k T m is the thermal speed. G is the ratio of two polylogarithm
functions given by:
 
 
5 2
3 2
exp[ ]
exp[ ]
Li
G
Li





(2.1-6)
The system is closed by the Poisson’s equation,
 
2
2
4 ec eh i ie n n Z n
x



  

(2.1-7)
As the degeneracy parameter G determines the transition from ultracold to thermal cases it is important to know its value.
Table-2.1 shows the values of G for certain practical plasmas.
Table: 2.1 Degeneracy Parameter for Different Kinds of Plasma
Type of Plasma Tokamak Inertial Confinement Fusion Metal and Metal clusters Jupiter White Dwarf
Density (m-3
) 1020
1032
1028
1032
1035
Temperature (K) 1018
108
104
104
108
G 1 1 1.4 1.4 4

2.2 The normalized set of dynamic equations
We now introduce the following normalization:
0 0
, , , , and
2
j jc i
c j i j
sh B Fh j i sh
n ux ne
x t t n n u
c k T n n c
 
       in which 2
04 /ec ec en e m  is the
cold electron plasma frequency, 2 /sh B Feh ec k T m is the electron-acoustic speed. Equations (2.1-4,5,7) can be recast in
the following normalized form as:
, , ,( )
0e h e h ex hn n u
t x
 
 
 
(2.2-2)
, , ,( )
0e c e c ex cn n u
t x
 
 
 
(2.2-3)
22
,,
, , , 2
,
1
3
2
e he hh
ex h ex h e h
e h
nnG H
u u n
t x x x x xn


     
      
         
(2.2-4)
22
,,
, , , 2
,
1
3
2
e ce cc
ex c ex c e c
e c
nnG H
u u n
t x x x x xn


     
      
         
(2.2-5)
2
1
2
eh
ec i
n
n n
x

 
  
   
  
(2.2-6)
where  
2
, ,Fe h Te hV V  . H is the non-dimensional quantum diffraction parameter defined as / 2ec B FehH k T  , where
TFeh is the Fermi temperature for hot electrons; 0 0/ec ehn n  and 1 0 0/i i ehZ n n  , in which nec0 , neh0 and ni0 are the
equilibrium number densities of cold electrons, hot electrons and ions respectively.
3. Linear and nonlinear analysis
3. 1 Linear dispersion relation
In order to investigate the nonlinear behavior of electrostatic wave modes we make the following perturbation expansions for
the field quantities neh, nec, ueh, ueh and  about their equilibrium values:
(1) (2)
(1) (2)
2(1) (2)
(1) (2)
(1) (2)
1
0
......1
0
0
eh eh eh
eh eh eh
ec ec ec
ec ec ec
n n n
u u u
n n n
u u u
 
  
      
      
      
         
      
      
             
(3.1-1)
Linearizing the set of normalized equations (2.2-2)-(2.2-6) and assuming that all the field variables vary as
( )i kx t
e 
we get
the following dispersion relation of electrostatic wave modes in quantum plasma including quantum diffraction and statistical
effects:
 
2 4 2 4
2 2 2 2
1 1/
4 4
eh ec
H k H k
F k F k  
   
         
   
(3.1-2)
In the dimensional form the dispersion relation is

2 4 4 2 4 42
2 2 2 2 2 2
2 2
1 1/ 1
4 4
Fe Fepe
h Fe c Fe
pe pe
H k V H k V
F V k FV k


 
 
 
   
(3.1-3)
where  3eh hF G  and  3ec cF G  (3.1-4)
Equation (3.1-3) is a quadratic equation in ω2
and has the following solutions:
2 2
1 4 2B B C    
 
(3.1-5a)
2 2
2 4 2B B C    
 
(3.1-5b)
Where,
 
2 4
2
2 4 2 4 2 4
2 2 2
1
2
1
4 4 4
eh ec
ec
eh ec eh
H k
B k F F
FH k H k H k
C k F k F k F



 
   
      
  
         
             
       
(3.1-6)
Equations (3.1-5) indicate that two stable linear modes for EAWs are possible when one considers inertial and restoring force
due to finite temperature effects of both groups of electrons.
EAWs are high frequency electrostatic electron oscillations where the restoring force comes from the hot electron pressure
and the cold electrons provide the inertia. If we neglect the inertia of hot electrons and assume that the pressure is solely due
to the hot electrons then the dispersion relation (3.1-2) reduces to:
 
 
2 2 2 2 4
2
2 2 2
3 4
41 3 4
h
h
k G H k H k
k G H k
 

 
   
   
(3.1-7)
It describes the finite temperature quantum counterpart of the classical electron acoustic dispersion relation with corrections
from quantum diffraction and finite temperature effects. We discuss the behavior of the mode represented by the Eq. (3.1-5a),
the behavior of the other mode being almost similar. The normalized wave dispersion relation for the mode denoted by Eq.
(3.1-5b) is plotted in Fig. 1-3 for different values of quantum diffraction parameter H, electron degeneracy parameter Geh and
equilibrium cold-to-hot electron concentration ratio δ. In the long wavelength limit (i.e. k→0) from Eq. (3.1-7) we get:
 3 hk G   (3.1-8)
It represents the long wave dispersion character of EAWs in a finite temperature-quantum plasma composed of inertialess hot
electrons, inertial cold electrons and stationary ions. The long wave phase speed is:
 3 hV k G    (3.1-9)
The results were calculated numerically using MATLAB software and the graphs are plotted. We show the dependence of
linear dispersion characteristics and solitary wave structures on the quantum diffraction parameter H , electron degeneracy
parameter G and equilibrium cold-to-hot electron concentration ratio δ. Here G is for the hot electrons (Gh).

Fig 1 Linear dispersion characteristicsfor different value of H. Here Gh=1.5 and δ=0.01
Fig 2 Linear dispersion characteristicsfor different value of Gh. Here H=2 and δ=0.01
Figures 1 and 2 shows that the linear dispersion graph becomes more steep with increasing in the value of quantum
diffraction parameter H and electron degeneracy parameter Gh. This is also evident from equation (3.1-7). However there is
no such prominent effect due to equilibrium cold-to-hot electron concentration ratio δ as evident from Fig. 3.

Fig 3 Linear dispersion characteristics for different value of δ. Here H=2 and Gh =1.5
3. 2 Non-linear analysis and solitary wave solutions
In order to derive the desired KdV equation describing the nonlinear behavior of electrostatic wave in plasma we use the
standard reductive perturbation technique. We introduce the usual stretching of the space and time variables:
( )
1
2
x Vt   and
3
2
t  (3.2-1)
where V is the linear long wave phase speed normalized by electron Fermi speed VFe and ε is a smallness parameter
measuring the weakness of dispersion and nonlinear effects. Equations (2.2-2)-(2.2-6) are written in terms of the stretched
coordinates ξ and τ and then the perturbation expansions (6.3-1) is substituted. Solving the lowest order equations in ε with
the appropriate boundary conditions, the following solutions are obtained:
(1) (1) (1) (1)
(1) (1) (1) (1)
2 2
, , ,eh eh ec ecn u n u
V V V V
   
      (3.2-2)
Going to the next higher order terms in ε and following the usual method we obtain the desired Korteweg de Vries (KdV)
equation:
3
3
0A B
  

  
  
  
  
(3.2-3)
Where,
 
3 3
2 2 3 h
A
V G 
    (3.2-4)
  
 
2 24 2 3 (1 ) / 4(1 ) / 4
2 2 3
h
h
G HV H
B
V G
  
 
  
  (3.2-5)
To find the steady state solution of equation (6.4-21) we transform the independent variables ξ and τ into one variable η:

η= ξ - M τ (3.2-6)
where M is the normalized constant speed of the wave frame. Applying the boundary conditions that as η → ± ∞;
, ,
2
2
 

 
 
 
→0 the possible stationary solution of equation (3.2-6) is obtained as:
sec 2
m h

 

 
  
 
(3.2-7)
where the amplitude m and width ∆ of the solitary structure are given by:
m 3M A  (3.2-8)
4B M  (3.2-9)
The solitary wave structure is formed due to the balance between the dispersive effect and nonlinear effect. Relative strength
of these two effects determines the characteristic of such solitary wave structure. The coefficients A and B thus play a crucial
role in determining the solitary wave structure. The solitary wave structure is formed due to a delicate balance between
dispersive and nonlinear effects. From Eqs. (3.2-4) and (3.2-5) it is clear that both the nonlinear and dispersion coefficients
get modified due to the inclusion of electron degeneracy effect whereas the quantum effect enters only into the dispersion
coefficient. Both these coefficients depend on the equilibrium cold-to-hot electron concentration ratio (δ). For a given H and δ
there exists a critical value of the relativity parameter Geh at which the dispersion coefficient vanishes. This critical value of
Geh is given by:
   1 (6 )h c
G H    (3.2-10)
No solitary structure is possible for  h h critical
G G . Note that the critical value of Gh depends on both δ and H.
The dependence of the solitary structures on H and Gh are shown for both positive and negative values of M. However δ has
very limited effects on the formation and characteristics of solitary wave structures of electron-acoustic waves in this category
of plasma.
Fig 4 Solitary wave profiles for different value of H. Here Gh=1.5, M=1 and δ=0.01

Fig 5 Solitary wave profiles for different value of Gh. Here H=2, M=1 and δ=0.01
Figures 4 shows that with increasing value of H there is no prominent effect in the amplitude and width of solitary structures
(here these are cpompressive solitary structures); however the width show slight broadening with increaing H. But both the
amplitude and width are affected by a variation in the electron degeneracy parameter Gh. The amplitude decreases but the
width increases with increase in Gh. This is also evident from equations (3.2-4) & (3.2-5). This was however when we
consider positive values of M. Figures 6 & 7 show the solitary profiles for negative value of M. The rarefactive solitons
become more wide but its amplitude remains constant with variation of H whereas it becomes more broadened and with
diminished amplitude for increasing value of Gh. However it was found that slight variation in solitary structures was
observed for variation in the equilibrium cold-to-hot electron concentration ratio (δ). With increase in δ amplitude decreases
and width increases as evident from Eqs. 3.2-8 and 3.2-9.
Fig 6 Solitary wave profiles for different value of H. Here Gh=1.5, M=-1 and δ=0.01

Fig 7 Solitary wave profiles for different value of Gh. Here H=2, M=-1 and δ=0.01
5. Conclusion
The findings in this paper are important in explaining many astrophysical and laboratory produced dense plasmas showing
quantum behavior. We have investigated the effects of quantum diffraction parameter H (which is a measure of the Bohm
potential), electron degeneracy parameter for warm electrons Gh and the equilibrium cold-to-hot electron concentration ratio δ
on the linear and nonlinear properties of electron-acoustic waves in a dense quantum plasma at finite temperature.
Acknowledgements
The authors would like to thank CSIR, Govt of India; DST, Govt of India, Institute for Plasma Research, Gandhinagar, India,
Jadavpur University & Techno India University, Kolkata, India and The Abdus Salam International Centre for Theoretical
Physics, Trieste, Italy for providing resources that helped in carrying out the work.
References
[1] Shukla, P. K. and Eliasson, B. (2010) Nonlinear aspects of quantum plasma physics, Physics-Uspekhi 53(1), 51-76
[2] Haas, F. , Manfredi, G. , and Feix, M. , (2000) ‘Multistream Model for Quantum Plasmas Physical Review E 62, 2763
[3] Ghosh, B., Chandra, S., & Paul, S. N. (2011)Amplitude modulation of electron plasma waves in a quantum plasma; Physics of
Plasmas 18, 012106
[4] Chandra, S., & Ghosh, B., (2012), Modulatioal Instability of Electron-Acoustic Waves in Relativistically Degenerate Quantum
Plasma, Astrophysics and Space Sciences, 342, 417-424
[5] Chandra, S., & Ghosh, B., (2012), Nonlinear Solitary Structures of Electron Plasma Waves in a Finite Temperature Quantum Plasma,
World Academy of Science Engineering and Technology, 6(10), 1094
[6] Chandra, S., & Ghosh, B., (2012), Relativistic effects on the nonliner propagation of electron plasma waves in dense quantum plasma
with arbitrary temperature, International Journal of Engineering Research and Development, 3(1), 51-57
[7] Chandra, S., & Ghosh, B., (2012), Modulational Instability of Electron Plasma Waves in Finite Temperature Quantum Plasma, World
Academy of Science Engineering and Technology, 6(11), 580
[8] Ghosh, B., Chandra, S., & Paul, S.N., (2012) Relativistic effects on the modulational instability of electron plasma waves in quantum
plasma, Pramana-Journal of Physics 78, 779-790
[9] Chandra, S., Paul, S.N., & Ghosh, B., (2012), Linear and nonlinear propagation of electron plasma waves in quantum plasma, Indian
Journal of Pure and Applied Physics, 50,314
[10]Chandra, S., Ghosh, B. & Paul, S.N.,(2013), Propagation of Electron-Acoustic Solitary Waves in Weakly Relativistically Degenerate
Fermi Plasma; World Academy of Science Engineering and Technology, 7(3), 611
[11]Chandra, S., & Ghosh, B., (2013) Nonlinear propagation of Electrostatic Waves in relativistic Fermi Plasma at arbitrary temperature,
Indian Journal of Pure and Applied Physics 51(9), 627-633.
[12]Chandra, S., Maji, P.S. & Ghosh, B., (2014), Propagation of Nonlinear Surface Waves in Relativistically Degenerate Quantum Plasma
Half-Space, International Journal of Mathematical, Computational, Physical and Quantum Engineering, 8(5), 831.
[13]Eliasson, B. & Shukla, P.K., (2008)Nonlinear quantum fluid equations for a Fermi plasma, Physica Scripta, 78, 025503

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