The document investigates the effects of quantum diffraction on the linear and nonlinear behavior of electron plasma waves in a two-component electron-ion dense quantum plasma. Using a one-dimensional quantum hydrodynamic model, it is shown that quantum effects significantly affect the linear dispersion and group velocity of electron plasma waves. With immobile ions, quantum diffraction makes the electron plasma wave modulationally unstable in two distinct wavenumber regions, and the stability region shrinks with increasing quantum effects. Ion motion is also found to change the stability/instability domains at low wavenumbers.

1. Amplitude modulation of electron plasma waves in a quantum plasma
Basudev Ghosh,1,a͒
Swarniv Chandra,1
and S. N. Paul2
1
Department of Physics, Jadavpur University, Kolkata 700032, India
2
Swami Vivekananda Institute of Science and Technology, Dakshin Gobindapur, P.S.-Sonarpur,
Kolkata 700145, India
͑Received 8 October 2010; accepted 10 December 2010; published online 11 January 2011͒
Using the one dimensional quantum hydrodynamic model for a two-component electron-ion dense
quantum plasma the linear and nonlinear properties of electron plasma waves are studied including
ion motion. By using the standard method of multiple scales perturbation technique a nonlinear
Schrödinger equation containing quantum effects is derived. From this equation it is shown that with
immobile ions an electron plasma wave becomes modulationally unstable in two distinct regions of
the wavenumber. Numerical calculation shows that the stability domain of the wavevector shrinks
with the increase in quantum diffraction effect. It is also found that the growth rate of instability in
the high wavenumber region increases with the increase in quantum effect. Ion motion is found to
have significant effect in changing the stability/instability domains of the wavenumber in the low
k-region. © 2011 American Institute of Physics. ͓doi:10.1063/1.3533670͔
I. INTRODUCTION
Physics of quantum plasmas is now a very rapidly grow-
ing subject of plasma physics. Quantum effects in plasma
may become important in a variety of environments such as
in intense laser-solid density plasma interaction
experiments,1,2
in dense astrophysical and cosmological
environments,3,4
in ultrasmall electronic devices,5
and in
metal nanostructures.6
During the past few years there has
been a great deal of interest in the investigation of various
types of wave modes in dense quantum plasmas. Haas et al.7
have investigated the role of quantum diffraction in the
propagation of ion-acoustic waves. Formation of dark soli-
tons and vortices in quantum electron plasma has been re-
ported by Shukla and Eliasson.8
Modulational instability of
ion-acoustic waves has been investigated in electron-ion
quantum plasma by Misra and Bhowmik9
and in electron-
positron-ion quantum plasma by Bains et al.10
The linear
dispersion character of compressional magneto-acoustic
waves in a quantum magnetoplasma has been studied by
Shukla11
by taking into account the quantum Bohm potential
and the magnetization of electrons due to electron −1
2 spin
effect. Linear and nonlinear properties of dust ion-acoustic
waves in ultracold quantum dusty plasma have been studied
by Khan and Mushtaq12
and Misra and Roy Chowdhury.13
Quantum dust acoustic double layers in an unmagnetized
quantum dusty plasma have been studied by Moslem et al.14
The amplitude modulation of electron plasma oscillation in
unmagnetized dense electron-hole plasma has been investi-
gated by Misra and Shukla.15
Various interesting properties
of electrostatic and electromagnetic waves in semiconductors
have also been studied by a number of authors.16–19
A critical study of these referred works shows that the
quantum effects can alter significantly the linear and nonlin-
ear properties of plasma waves. The purpose of the present
paper is to investigate the effects of quantum diffraction on
the linear and nonlinear behavior of electron plasma waves
in a two-component electron-ion dense quantum plasma.
Quantum effects are shown to have significant effects on the
linear dispersion character and group velocity of electron
plasma waves. In an unbounded classical plasma electron
plasma waves are modulationally stable for all
wavelengths.20
Here it is shown that with immobile ions
quantum diffraction effects make the electron plasma wave
modulationally unstable in two distinct regions of the wave-
number. The range of the wavenumber corresponding to the
stability of the wave is found to decrease with increase in
quantum diffraction effect. Thus quantum effects can alter
the stability and instability domains of the wavenumber sig-
nificantly. The quantum effect can also significantly change
the instability growth rate. In the high wavenumber region it
is shown that the growth rate of modulational instability in-
creases with the increase in quantum effects. Ion motion
though found to be unimportant in high k-region has signifi-
cant effect in changing the stability/instability domains of the
wavenumber.
II. BASIC EQUATIONS
We consider a one dimensional quantum hydrodynamic
͑QHD͒ model to describe the dynamics of electron plasma
waves in a two-component homogeneous plasma consisting
of electrons and ions immersed in an essentially infinite mag-
netic field directed along the x-axis. This magnetic field con-
tains the electrons and ions to move along the x-direction.
We assume that the plasma particles behave as a one dimen-
sional Fermi gas at zero temperature and therefore the pres-
sure law21
is
pj =
mjVFj
2
3nj0
2 nj
3
, ͑1͒
where j=e for electron and j=i for ions, mj is the mass,a͒
Electronic mail: bsdvghosh@gmail.com.
PHYSICS OF PLASMAS 18, 012106 ͑2011͒
1070-664X/2011/18͑1͒/012106/5/$30.00 © 2011 American Institute of Physics18, 012106-1

2. VFj =ͱ2kBTFj /mj is the Fermi thermal speed, TFj is the Fermi
temperature, and kB is the Boltzmann constant. nj is the num-
ber density with the equilibrium value nj0.
For slow waves the set of QHD equations governing the
dynamics of electron plasma waves in a two-component
plasma is given by
‫ץ‬nj
‫ץ‬t
+
‫ץ‬͑njuj͒
‫ץ‬x
= 0, ͑2͒
‫ץ‬uj
‫ץ‬t
+ uj
‫ץ‬uj
‫ץ‬x
= −
qj
mj
‫ץ‬␾
‫ץ‬x
−
1
mjnj
‫ץ‬pj
‫ץ‬x
+
ប2
2mj
2
‫ץ‬
‫ץ‬x
ͩ 1
ͱnj
‫ץ‬2ͱnj
‫ץ‬x2 ͪ, ͑3͒
‫ץ‬2
␾
‫ץ‬x2 = 4␲e͑ne − ni͒, ͑4͒
where uj, qj, and pj are, respectively, the fluid velocity,
charge, and pressure of the jth species, qe=−e, qi=e,ប is the
Planck’s constant divided by 2␲, and ␾ is the electrostatic
wave potential. We now introduce the following normaliza-
tion:
x → x␻pe/VFe, t → t␻pe, ␾ → e␾/2kBTFe,
nj → nj/n0 and uj → uj/VFe,
where ␻pe=ͱ4␲n0e2
/me the electron plasma oscillation fre-
quency and VFe is the Fermi thermal speed of electrons.
Using the above normalization Eqs. ͑2͒–͑4͒ can be writ-
ten as
‫ץ‬nj
‫ץ‬t
+
‫ץ‬͑njuj͒
‫ץ‬x
= 0, ͑5͒
ͩ‫ץ‬
‫ץ‬t
+ ue
‫ץ‬
‫ץ‬x
ͪue =
‫ץ‬␾
‫ץ‬x
− ne
‫ץ‬ne
‫ץ‬x
+
H2
2
‫ץ‬
‫ץ‬x
ͩ 1
ͱne
‫ץ‬2ͱne
‫ץ‬x2 ͪ, ͑6͒
ͩ‫ץ‬
‫ץ‬t
+ ui
‫ץ‬
‫ץ‬x
ͪui = − ␮
‫ץ‬␾
‫ץ‬x
− ␴ni
‫ץ‬ni
‫ץ‬x
+
␮2
H2
2
‫ץ‬
‫ץ‬x
ͩ 1
ͱni
‫ץ‬2ͱni
‫ץ‬x2 ͪ, ͑7͒
‫ץ‬2
␾
‫ץ‬x2 = ͑ne − ni͒, ͑8͒
where H=ប␻pe/2kBTFe is a nondimensional quantum param-
eter proportional to the quantum diffraction, ␮=me/mi, and
␴=TFi/TFe. The parameter H is proportional to the ratio be-
tween the plasma energy ប␻pe ͑energy of an elementary ex-
citation associated with an electron plasma wave͒ and the
Fermi energy kBTFe.
III. DERIVATION OF THE NONLINEAR SCHRÖDINGER
EQUATION „NLSE…
We assume that in the linear approximation only elec-
trons participate, while the motion of ions enters through
nonlinear interaction. That is, ions respond to the slowly
varying part of the field quantities. This type of consideration
about ion motion was made previously by several authors
͑e.g., Ref. 22͒. Keeping this in mind we make the following
Fourier expansion of the field quantities:
΄
ne
ue
␾
΅=
΄
1
0
0
΅+ ␧2
΄
ne0
ue0
␾0
΅+ ͚s=1
ϱ
␧s
Ά΄
nes
ues
␾s
΅· exp͑is␺͒
+
΄
nes
‫ء‬
ues
‫ء‬
␾s
‫ء‬ ΅· exp͑− is␺͒
· ͑9a͒
ͫni
ui
ͬ= ͫ1
0
ͬ+ ␧2
ͫni0
ui0
ͬ ͑9b͒
in which ␺=kx−␻t ͑␻, k being the normalized wave wave-
number, respectively͒, the field quantities neo, ueo, ␾0, nes,
ues, ␾s, nio, and uio are assumed to vary slowly with x and t,
i.e., they are supposed to be functions of
␰ = ␧͑x − cgt͒ and ␶ = ␧2
t, ͑10͒
where ␧ is a smallness parameter and cg is the normalized
group velocity.
Now substituting the expansion ͑9͒ in Eqs. ͑5͒–͑8͒ and
then equating from both sides the coefficients of exp͑i␺͒,
exp͑i2␺͒ and terms independent of ␺ we obtain three sets of
equations which we call I, II, and III. To solve these three
sets of equations we make the following perturbation expan-
sion for the field quantities neo, ueo, ␾0 nes, ues, ␾s, nio, and
uio which we denote by W:
W = W͑1͒
+ ␧W͑2͒
+ ␧2
W͑3͒
+ ... . ͑11͒
Solving the lowest order equations obtained from the set of
equations I after substituting the expansion ͑11͒ we get
ne1
͑1͒
= − k2
␾1
͑1͒
, ue1
͑1͒
= − ␻k␾1
͑1͒
͑12͒
and the linear dispersion relation
␻2
= 1 + k2
ͩ1 +
k2
H2
4
ͪ. ͑13͒
In the dimensional form the dispersion relation is
␻2
= ␻pe
2
+ k2
VFe
2
ͩ1 +
k2
VFe
2
H2
4␻pe
2 ͪ. ͑14͒
It describes the quantum counterpart of the classical electron
plasma wave dispersion relation with a correction from
quantum diffraction effect. The normalized wave dispersion
relation ͑13͒ is plotted in Fig. 1 for different values of quan-
tum diffraction H. Figure 1 shows that the wave frequency ␻
increases with increase in quantum diffraction H for a given
wavenumber k ͑The change in the values of ␻ is significant
even in the semiclassical regime HϽ1͒.
012106-2 Ghosh, Chandra, and Paul Phys. Plasmas 18, 012106 ͑2011͒

3. The group velocity cg=d␻/dk is obtained from the dis-
persion relation ͑13͒ as
cg =
k +
H2
k3
2
ͱ1 + k2
ͩ1 +
H2
k2
4
ͪ
. ͑15͒
Figure 2 shows the plot of group velocity ͑cg͒ against the
wavenumber k for different values of the quantum diffraction
H. In the small wavenumber region the group velocity cg is
practically independent of quantum diffraction effects. Be-
yond certain critical value of k, cg increases significantly
with the increase in k for higher values of the quantum dif-
fraction parameter.
The second harmonic quantities in the lowest order can
be obtained from the solutions of the lowest order equations
obtained from the set of equations II after substituting the
perturbation expansion ͑11͒. Thus we get
␾2
͑1͒
= − b2␾1
͑1͒2
,
ne2
͑1͒
= 4k2
b2␾1
͑1͒2
, ͑16͒
ue2
͑1͒
= ␻k͑4b2 − k2
͒␾1
͑1͒2
,
where
b2 =
k2
ͩ3␻2
+ k2
−
H2
k4
4
ͪ
8ͩ␻2
− k2
− H2
k4
−
1
4
ͪ
=
k2
ͩ3 + 4k2
+
H2
k4
2
ͪ
6͑1 − H2
k4
͒
. ͑17͒
The zeroth harmonic quantities are obtained from the solu-
tions of the lowest order equations obtained from the set of
equations III after substituting the perturbation expansion
͑11͒
␾0
͑1͒
= b0␾1
͑1͒2
,
ne0
͑1͒
= ni0
͑1͒
= b1␾1
͑1͒2
,
͑18͒
ue0
͑1͒
= ͑b1cg − 2␻k3
͒␾1
͑1͒2
,
ui0
͑1͒
= b1cg␾1
͑1͒2
,
where
b0 =
k2
ͩ1 + 4k2
+
3H2
k2
2
ͪ
ͫ1 +
␮͑cg
2
− 1͒
͑cg
2
− ␴͒
ͬ
,
b1 =
␮b0
͑cg
2
− ␴͒
. ͑19͒
The first harmonic quantities in the second order are obtained
from the solutions ͑12͒ by replacing −i␻ by ͓−i␻
−␧cg͑‫ץ‬/‫␰ץ‬͒+␧2
͑‫ץ‬/‫ץ‬␶͔͒ and ik by ͓ik+␧͑‫ץ‬/‫␰ץ‬͔͒ and then
picking out order ␧ terms. Thus we obtain
␾1
͑2͒
= 0,
ne1
͑2͒
= 2ik
‫ץ‬␾1
͑1͒
‫␰ץ‬
, ͑20͒
ue1
͑2͒
= i͑␻ + kcg͒
‫ץ‬␾1
͑1͒
‫␰ץ‬
.
Now collecting coefficients of ␧3
from both sides of the sets
of equations I after substituting the perturbation expansion
͑11͒ we get a set of equations for the first harmonic quantities
in the third order. Using the above solutions and after proper
elimination we obtain the following desired NLSE describ-
ing the nonlinear evolution of the wave amplitude:
FIG. 1. ͑Color online͒ Linear wave dispersion relation for different values
of the quantum diffraction parameter H with ␴=0.001.
FIG. 2. ͑Color online͒ Variation of group velocity ͑cg͒ with wavenumber ͑k͒
for different values of the quantum diffraction parameter H with ␴=0.001.
012106-3 Amplitude modulation of electron plasma waves… Phys. Plasmas 18, 012106 ͑2011͒

4. i
‫ץ‬␣
‫ץ‬␶
+ P
‫ץ‬2
␣
‫␰ץ‬2 = Q␣2
␣‫ء‬
, ͑21͒
where ␣=␾1
͑1͒
, the group dispersion coefficient
P =
1
2
dcg
dk
=
1
2␻
ͩ1 − cg
2
+
3
2
H2
k2
ͪ ͑22͒
and the nonlinear coefficient
Q =
␻
2
͓b1 + 8b2k2
− 3k4
͔ +
k
2
͓2b1cg + 4b2k␻ − 3k3
␻͔
+
k2
2␻
ͫb1 + ͩ4b2 −
H2
2
ͪk2
−
5
2
b2H2
k4
ͬ. ͑23͒
Note that both the coefficients P and Q depend on the quan-
tum parameter H.
IV. MODULATIONAL INSTABILITY
We now analyze the possibility of amplitude modulation
of electron plasma waves described by the NLS Eq. ͑21͒
with quantum corrections. This type of equation has been
studied extensively in connection with the nonlinear propa-
gation of waves of various types and it has been found that
under certain conditions, conversion of an initially uniform
wavetrain into a spatially modulated wave proves to be en-
ergetically favorable. This effect is known as modulational
instability. It is well known that a uniform plasma wavetrain
is modulationally stable or unstable depending on whether
PQϾ0 or Ͻ0. In case of instability the growth rate depends
on the wavenumber of modulation. The growth rate attains a
maximum value, gm=͉Q͉·␣0
2
corresponding to the wavenum-
ber lm=͉Q/P͉1/2
·͉␣0͉ of the modulation. Thus we find that
the instability condition depends on the sign of the product
PQ. Numerical computation of P and Q by using the expres-
sions ͑22͒ and ͑23͒ for different values of k in terms of the
system parameters shows that in the classical limit H=0,
PQϾ0 for all values of k. This means that in a classical
plasma electron plasma waves are modulationally stable for
all wavelengths. This result is in agreement with that ob-
tained by earlier authors.20
In the quantum region numerical
computation of P and Q, with immobile ions show that for a
given value of quantum diffraction parameter ͑H͒ and the
Fermi temperature ratio ͑␴͒ a uniform electron plasma
wavetrain becomes modulationally unstable ͑PQϽ0͒ for
two distinct regions of the wavenumber ͑k͒: ͑i͒ kϽkc1 and
͑ii͒ kϾkc2, where kc1 and kc2 are two critical wavenumbers
separating stable and unstable regions in the wavenumber
domain. The wave is modulationally stable in the wavenum-
ber range kc1ϽkϽkc2. Numerically it is also found that the
value of kc2 decreases, the value of kc1 increases, and as a
result the stability region kc2−kc1 decreases with the increase
in quantum diffraction effects ͑Fig. 3͒. Thus the stability re-
gion of the wave in the wavenumber domain shrinks with the
increase in quantum diffraction effect. Also, it is found
through numerical calculations that in the high wavenumber
region ͑kϾkc2͒ the maximum growth rate of instability ͑gm͒
increases with the increase in quantum diffraction parameter
H ͑Fig. 4͒. Thus the effect of quantum diffraction or quantum
correlation of density fluctuations is to make the electron
plasma wave more unstable giving higher growth rate of
instability. Numerical calculations including ion motion
show that the ion motion has very little effect on the wave
motion in the high k-region but in the low k-region it has
significant effect in changing the stability and instability do-
mains of the wavenumber. This is due to the fact that the
ions, because of their heavier mass, become unable to follow
the rapid changes at higher frequencies but at low frequen-
cies ion motion plays an important role on the plasma dy-
namics. In the presence of ion motion and quantum diffrac-
tion effect the value of kc2 remains almost unchanged but the
domain of k below kc2 breaks up into two stable and unstable
regions for high values of the quantum diffraction parameter
H. In the modulationally unstable case an initial disturbance
to a uniform plane wave will grow. If at certain stage of
FIG. 3. ͑Color online͒ Variation of stability region with different values of
the quantum diffraction parameter H with ␴=0.001.
FIG. 4. ͑Color online͒ Variation of the growth rate of instability with wave-
number ͑k͒ for different values of the quantum diffraction parameter H with
␴=0.001.
012106-4 Ghosh, Chandra, and Paul Phys. Plasmas 18, 012106 ͑2011͒

5. evolution nonlinearity is balanced by the dispersive effect a
stable nonlinear wave structure, called envelop soliton, will
be formed. Otherwise, it may collapse until it is absorbed by
the plasma particles or converted into other waves.
V. DISCUSSIONS AND CONCLUSIONS
Using the one dimensional QHD model we have inves-
tigated quantum effects on the linear and nonlinear properties
of electron plasma waves in a two-component electron-ion
dense quantum plasma including the effects of ion motion.
The QHD model as used in the paper includes two different
quantum effects: ͑i͒ quantum diffraction due to quantum cor-
relation of density fluctuations is taken into account by the
term proportional to ប2
, ͑ii͒ quantum statistical effect is in-
cluded in the model through the equation of state ͓Eq. ͑1͔͒,
which takes into account the fermionic character of the
plasma particles. In general the quantum effects in plasma
become important when thermal de Broglie wavelength be-
comes much larger than the average interparticle distance
and the temperature is lower than the so-called Fermi tem-
perature. The de Broglie wavelength roughly represents the
spatial extension of the particle’s wave function due to quan-
tum uncertainty. For cold and dense plasma the de Broglie
wavelength may become large and there is overlapping of
the wave functions and quantum interference. Because of
smaller mass the quantum behavior is reached much more
easily for the electrons than for ions. Since ions are much
heavier than electrons the quantum diffraction effect for ions
is almost negligible in most practical situations. For this we
have not considered the quantum diffraction effects of ions.
Regarding ion motion we have assumed that the ions respond
only to the low frequency nonlinear plasma dynamics. The
quantum effects are found to make the linear dispersion
properties of the electron plasma waves significantly differ-
ent from those of a classical plasma. In the classical limit
H→0, the dispersion relation reduces to the well-known
electron plasma dispersion relation. Both the phase velocity
and group velocity of the electron plasma wave increase sig-
nificantly with quantum diffraction parameter H in the high
k-region. But in the low k-region both are almost indepen-
dent of H. To describe the nonlinear evolution of electron
plasma waves a general type NLS equation has been derived
including quantum effects and ion motion. Both the disper-
sion ͑P͒ and nonlinear ͑Q͒ coefficients of the NLSE are
found to be modified by the quantum effects. As the system
parameters enter into the expression for P and Q, not in a
straightforward manner, we have analyzed them numerically.
Numerical computation of the coefficients with immobile
ions shows that the electron plasma waves become modula-
tionally unstable in two distinct regions of the wavenumber
separated by a stable region. The stability domain is found to
shrink with the increase in quantum diffraction effects. This
may be considered as the nonlinear effect introduced by the
quantum effects because Kakutani and Sugimoto20
showed
that the electron plasma wave is modulationally stable for all
wavelengths in an unbounded classical plasma. This type of
nonlinear effect introduced by quantum diffraction is similar
to that introduced by finite geometry on the propagation of
electron plasma waves.23
The growth rate of modulational
instability in the high wavenumber region is also calculated
with different values of quantum diffraction parameter H. It
is found that the quantum effect increases the instability
growth rate as compared to the classical case. The quantum
effect thus increases the tendency of a stable electron plasma
wave to become unstable and in case of instability it in-
creases the growth rate. A similar result was also obtained by
Bains et al.10
for ion-acoustic wave in magnetized quantum
electron-positron-ion plasmas. Thus our analytical and nu-
merical results show that the linear and nonlinear properties
of electron plasma waves are affected significantly by the
quantum effects. Ion motion is found to have a significant
effect on the stability/instability domains of the wavenumber
in the low k-region. A close scrutiny of the expression ͑18͒
for the low frequency field components reveals that some
sort of resonant interaction is possible when cg=ͱ␴ or in
dimensional form cg=csi where csi=͑2kBTFi/me͒1/2
is the
ion-acoustic speed. So for waves with cgӷcsi the effects of
ion motion may be neglected but the effect may become
particularly important when the wave propagates with a ve-
locity close to the quantum ion-acoustic speed. In fact when
cg=csi our evolution equation will lose its validity. The
present results may be found useful for understanding the
origin of amplitude modulation of electron plasma waves
and its stability/instability regimes in quantum plasmas.
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