Ion-acoustic envelope modes in a degenerate relativistic electron-ion
plasma
McKerr, M., Haas, F., & Kourakis, I. (2016). Ion-acoustic envelope modes in a degenerate relativistic electron-
ion plasma.
Physics of Plasmas
,
23
(5), [152120]. https://doi.org/10.1063/1.4952774
Published in:
Physics of Plasmas
Document Version:
Publisher's PDF, also known as Version of record
Queen's University Belfast - Research Portal:
Link to publication record in Queen's University Belfast Research Portal
Publisher rights
© 2016 AIP Publishing.
General rights
Copyright for the publications made accessible via the Queen's University Belfast Research Portal is retained by the author(s) and / or other
copyright owners and it is a condition of accessing these publications that users recognise and abide by the legal requirements associated
with these rights.
Take down policy
The Research Portal is Queen's institutional repository that provides access to Queen's research output. Every effort has been made to
ensure that content in the Research Portal does not infringe any person's rights, or applicable UK laws. If you discover content in the
Research Portal that you believe breaches copyright or violates any law, please contact openaccess@qub.ac.uk.
Download date:10. Aug. 2022
Ion-acoustic envelope modes in a degenerate relativistic electron-ion plasma
M. McKerr, F. Haas, and I. Kourakis
Citation: Physics of Plasmas 23, 052120 (2016); doi: 10.1063/1.4952774
View online: http://dx.doi.org/10.1063/1.4952774
View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/23/5?ver=pdfcov
Published by the AIP Publishing
Articles you may be interested in
Dust-ion-acoustic envelopes and modulational instability with relativistic degenerate electrons
Phys. Plasmas 22, 123705 (2015); 10.1063/1.4938041
Modulation of ion-acoustic waves in a nonextensive plasma with two-temperature electrons
Phys. Plasmas 22, 092124 (2015); 10.1063/1.4931074
Amplitude modulation of quantum-ion-acoustic wavepackets in electron-positron-ion plasmas: Modulational
instability, envelope modes, extreme wavesa)
Phys. Plasmas 22, 022305 (2015); 10.1063/1.4907247
Oblique modulation of ion-acoustic waves and envelope solitons in electron-positron-ion plasma
Phys. Plasmas 16, 062305 (2009); 10.1063/1.3142473
Nonlinear Zakharov–Kuznetsov equation for obliquely propagating two-dimensional ion-acoustic solitary waves
in a relativistic, rotating magnetized electron-positron-ion plasma
Phys. Plasmas 12, 072306 (2005); 10.1063/1.1946729
Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 143.117.193.82 On: Mon, 31 Oct
2016 16:29:24
Ion-acoustic envelope modes in a degenerate relativistic electron-ion plasma
M. McKerr,
1
F. Haas,
2
and I. Kourakis
1
1
Centre for Plasma Physics, School of Mathematics and Physics, Queen’s University Belfast, BT7 1NN Belfast,
Northern Ireland, United Kingdom
2
Instituto de F
ısica, Universidade Federal do Rio Grande do Sul, Av. Bento Gonc¸alves 9500, Porto Alegre,
RS, Brazil
(Received 3 March 2016; accepted 16 May 2016; published online 27 May 2016)
A self-consistent relativistic two-fluid model is proposed for one-dimensional electron-ion plasma
dynamics. A multiple scales pertur bation technique is employed, leading to an evolution equation for
the wave envelop e, in the form of a nonlinear Schr
€
odinger type equation (NLSE). The inclusion of rel-
ativistic effects is shown to introduce density-dependent factors, not present in the non-relativistic
case—in the conditions for modulational instability. The role of relativistic effects on the linear disper-
sion laws and on envelope soliton solutions of the NLSE is discussed. Published by AIP Publishing.
[http://dx.doi.org/10.1063/1.4952774]
I. INTRODUCTION
The elucidation of the dynamics of ultra-high density
plasmas in one-dimensional (1D) geometry
1
is recognized as
a challenging area of research among researchers in the last
decade. Dense plasmas occur in “extreme” astrophysical envi-
ronments, such as white dwarfs or neutron stars
2–4
and in the
core of giant planets (e.g., Jovian planets).
5–8
Such plasmas
may also occur in the next generation of laser-based matter
compression schemes.
8,9
The topic has gained momentum
recently, thanks to its relevance to high-power laser-assisted
energy production (fusion) research and in particular, to the
target normal sheath acceleration (TNSA) mechanism
10
dur-
ing the irradiation of solid targets with a high-intensity laser
beam.
11
Other applications of (1D) degenerate plasmas
include the dense quantum diode,
12
the electron-hole plasma
in quantum wires,
13
the 1D fermionic Luttinger liquid,
14
and
1D semiconductor quantum wells,
15
to mention a few.
In such extreme plasma environments, magnetic fields
can be extremely strong, effectively varying over many
orders of magnitude, from a few kilogauss to gigagauss (or
even petagauss) in white dwarfs (neutron stars, respectively),
hence effectively confining particle motion to one dimension
(1D). On the other hand, temperatures can be quite high,
comparable to fusion plasma (10
8
K).
9
In such conditions,
quantum degeneracy and relativity effects are ubiquitous,
since the de-Broglie wavelength may approach, or even
exceed, the inter-particle (fermion) distance. At extremely
high densities, the electron Fermi energy E
Fe0
can exceed by
far than thermal energy, hence the electron thermal pressure
may be neglig ible, compared to the Fermi degeneracy pres-
sure; the latter arises due to the combined effect of Pauli’s
exclusion principle and Heisenberg’s uncertainty principle.
From a nonlinear dynamical point of view, ultrahigh-
density plasmas pose a real challenge; their rich and varied
dynamics may sustain a wide range of excitations, from
breather-mode oscillations in 1D semiconductors
15
and
Lagrangian structures in dense 1D plasmas
16
to 1D nonlinear
envelope modes in dense electron-positron-ion plasmas,
17
quasi-1D solitons,
18
wakefields in quantum wires,
19
among
others. It may be added that the study of the dynamics of 1D
plasmas is certainly not restricted to dense systems only. In
Ref. 20, kinetic theoretical arguments have been employed
to found the possibility of reconnection between Langmuir
and Alfv
en modes in a strongly magnetized, non-degenerate,
relativistic pair plasma.
For ultra-high plasma densities, relativistic effects need
to be included in plasma modeling, since the relativistic
parameter p
F
=mc
21
(p
F
is the Fermi momentum, m is the elec-
tron mass, and c is the light speed) acquires large values, thus
modifying the equation of state and hence the dynamical
plasma profile. Many authors have considered the problem of
relativistically dense plasma before, from different angles.
Problems such as the formation of electrostatic shocks within
an electron-ion plasma,
22
the existence of arbitrary-amplitude
solitary structures,
23
and small-amplitude envelope modes
24
within an electron-positron-ion plasma have been studied in
the past. Stationary profile electrostatic pulses and Langmuir-
type excitations have been investigated in Refs. 25 and 26,
respectively. However, the majority of works tacitly apply
Chandrasekhar’s (three-dimensional, 3D) equation of state,
27
a reasonable assumption since the environments under consid-
eration in the above (e.g., white dwarf stars) certainly occupy
three dimensions. What we aim for, in the study at hand, is an
understanding of envelope modes in a dense, 1D plasma, such
as is used as a model for the study of target normal sheath
acceleration (TNSA).
10
We shall here focus on a relativistic one-dimensional
model for dense plasmas. Our aim is to propose a self-
consistent fully relativistic theoretical framework for low
(ionic) frequency electrostatic modulated envelope structures
propagating in unmagnetized electron-ion plasma. The model
comprises an inertialess electron fluid, which is described by a
quantum-mechanical degenerate distribution function, and a
classical inertial ion fluid. A fully relativistic fluid model is
adopted for both components. The use of Fermi-Dirac statis-
tics in the description of the electron fluid forces us to counte-
nance the exclusion principle. In the case of high densities, a
significant overlap of the electrons’ position-wavefunctions
leads to a pressure, which, according to Pauli’s exclusion
1070-664X/2016/23(5)/052120/8/$30.00 Published by AIP Publishing.23, 052120-1
PHYSICS OF PLASMAS 23, 052120 (2016)
Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 143.117.193.82 On: Mon, 31 Oct
2016 16:29:24
principle, exists to resist degeneracy as would occur if two
electrons were to share the same state. Such a pressure results
in a considerable (density-dependent) momentum near the
Fermi surface (the surface in momentum-space below which
all states are occupied) of the electron gas, the magnitude
of which demands a relativistic treatment. To this end, an
equation of state is employed that is similar to that of
Chandrasekhar
27
but is essentially that of the one-dimensional
“water-bag” distribution.
28
Unlike the original Chandrasekhar
equation of state,
27
which was developed for one-dimensional
(1D) propagation (in fact, in the radial direction) within a
spherical-symmetric geometry, our equations of state is suita-
ble for modeling strictly 1D propagation dynamics.
25
The electrons will be treated as “cold,” so as to avail of
the zero-temperature Fermi-Dirac distribution. Such an
approximation is justified under certain conditions that
depend on the density, entering the algebraic description via
the relativistic electron Fermi energy E
Fe;rel
, viz.,
k
B
T
e
E
Fe;rel
¼ m
e
c
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ p
2
Fe
=m
2
e
c
2
q
m
e
c
2
: (1)
Here, k
B
is Boltzmann’s constant, T
e
is the electron (thermal)
temperature, c is the speed of light in vacuo, and p
Fe
and m
e
are, respectively, the local Fermi momentum and the
rest mass of the electron. The local Fermi momentum is
expressed in terms of the local density, n
e
, and Planck’s con-
stant, h,asp
Fe
¼ hn
e
=4.
The layout of this article goes as follows. In Section
II, a self-consistent, relativistic fluid model is introduced.
The evolution equation for the plasma state variables is
then scaled, and a dimensionless system is presented in
Section III. A multiple scale perturbation technique is
employedinSectionIV andthenanalyzedinthelowest
(linear) and h igher (nonlinear) order(s) in Sections V and
VI, respectively. The modulational wavepacket profile is
outlined in Section VII. Localized envelope structures are
introduced in Section VIII. A parametric analysis is pre-
sented in Section IX, and the results are summarized in
Section X.
II. FLUID MODEL
We are interested in investigating ion dynamics in a
degenerate relativistic plasma. We shall adopt the quantum
hydrodynamic description,
29
by introducing a fluid model
that is described in the following. The ion fluid is described
by its particle (number) density, n
i
, and velocity, v
i
.Itisa
“cold,” fully ionized fluid of singly charged, positive ions,
whose dynamics is dominated by electric forces deriving
from an electrosta tic potential, /ðx; tÞ. A magnetic field has
not been considered, for the sake of simplicity.
The electron fluid constitutes an inertialess background
to the ion dynamics. It is characterized by a number density
n
e
and a fluid velocity, v
e
, directed along the xaxis. The
electrons are considered to be relativistically degenerate, and
therefore the appropriate equation of state to govern their
motion is provided by the expression for relativistic degener-
acy pressure in one dimension
28,30
P
e
¼
2m
2
e
c
3
h
n
e
n
2
e
þ 1
1=2
sinh
1
n
e
; (2)
where n
e
¼ p
Fe
=ðm
e
cÞ is a dimensionless parameter measuring
the effect of relativistic electroneffects.Thelatterequationof
state is a consequence of the Pauli exclusion principle and is valid
for arbitrary strength of relativistic effects. Note that an expansion
of the pressure (2) for low density—n
0
1—yields the non-
relativistic 1D Fermi pressure, P
e
¼ 2E
Fe0
n
0
ðn
e
=n
0
Þ
3
=3.
Similarly, an ultrarelativistic (n
e
1) approximation is found to
be P
e
¼ cp
Fe0
n
0
ðn
e
=n
0
Þ
2
=2, where p
Fe0
¼ hn
0
=4.
The model comprises five equations, namely, the fluid-
dynamical equations expressing continuity (number density
conservation) and momentum conservation for the ion and
electron fluid(s), with the system closed by Poisson’s equa-
tion for the electrostatic potential /, which essentially cou-
ples the dynamical variables to one another.
@ c
i
n
i
ðÞ
@t
þ
@
@x
c
i
n
i
v
i
ðÞ
¼ 0;
@ c
e
n
e
ðÞ
@t
þ
@
@x
c
e
n
e
v
e
ðÞ
¼ 0;
@ c
i
v
i
ðÞ
@t
þ v
i
@ c
i
v
i
ðÞ
@x
þ
e
m
i
@/
@x
¼ 0;
e
@/
@x
c
e
n
e
@P
e
@x
þ
v
e
c
2
@P
e
@t
¼ 0;
@
2
/
@x
2
þ
e
0
c
i
n
i
c
e
n
e
ðÞ
¼ 0:
(3)
Note that electron inertia has been neglected, according to the
underlying assumptions of our model, as discussed above.
Adopting the electrostatic approximation, we have suppressed
(neglected) magnetic field generation; hence, the remaining
Maxwell relations were omitted. As expected in a relativistic
model, the factor c
e;i
¼ 1=
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 v
2
e;i
=c
2
q
appears in the fluid-
dynamical equations, as a result of Lorentz transformations
and relations between quantities, such as the electron and ion
number density (functions), between different inertial frames.
It is understood that the validity of our model equations
(3) above assumes that assumption (1) holds, i.e., for suffi-
ciently high density.
III. DIMENSIONLESS MODEL
It is appropriate to derive a dimensionless model, by scal-
ing by appropriate quantities. A natural speed scale in our phys-
ical problem is the characteristic quantity c
s
¼ð2E
Fe0
=m
i
Þ
1=2
,
where E
Fe0
¼ h
2
n
2
0
=ð32m
e
Þ is the non-relativistic electron
Fermi energy: This is the equivalent of the ion “sound speed”
in classical plasma dynamics. Accordingly, well adopt the fol-
lowing scaling:
x !
x
pi
x
c
s
; t ! x
pi
t;
n
e;i
!
n
e;i
n
0
; v
e;i
!
v
e;i
c
s
; / !
e/
m
i
c
2
s
:
(4)
Note that n
e0
¼ n
i0
¼ n
0
from the quasi-neutrality condition
(obtained upon considering Poisson’s relation at equilibrium).
052120-2 McKerr, Haas, and Kourakis Phys. Plasmas 23, 052120 (2016)
Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 143.117.193.82 On: Mon, 31 Oct
2016 16:29:24
Finally, a natural pressure scale P
0
¼ e/
0
n
0
is considered.
The evolution equations take the form:
@c
i
n
i
@t
þ
@
@x
c
i
n
i
v
i
ðÞ
¼ 0;
@c
e
n
e
@t
þ
@
@x
c
e
n
e
v
e
ðÞ
¼ 0;
@c
i
v
i
@t
þ v
i
@c
i
v
i
@x
þ
@/
@x
¼ 0;
@/
@x
c
e
n
e
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ n
2
0
n
2
e
q
@n
e
@x
þ av
e
@n
e
@t
¼ 0;
@
2
/
@x
2
þ c
i
n
i
c
e
n
e
¼ 0;
(5)
where
c
e;i
¼ 1
c
2
s
c
2
v
2
e;i
1=2
: (6)
With this scaling, there is only one free parameter left:
n
0
¼ hn
0
=4m
e
c. The electron Fermi energy can be expressed
as E
Fe0
¼ m
e
c
2
n
2
0
=2 and so can a as a result
a ¼
c
2
s
c
2
¼
2E
Fe0
m
i
c
2
¼
m
e
n
2
0
m
i
: (7)
We may, where appropriate, still retain the notation for a
below, recalling (rather than substituting with) the exact
expression above, for the sake of analytical tractability.
Concluding this section, we note that the essential
physics of our model is elegantly “hidden” in the parameter
n
0
, which incorporates the relativistic effect, here manifested
in terms of the (high) plasma density.
IV. MULTISCALE PERTURBATION SCHEME
A multiple-scales technique will be employed in the fol-
lowing.
31
We anticipate a solution that comprises a fast car-
rier wave and a slowly evolving envelope amplitude
u uðX
1
; X
2
; :::; T
1
; T
2
Þe
iðkX
0
xT
0
Þ
; (8)
where T
r
¼
r
t and X
r
¼
r
x; >0 is a small, free parameter
(it is independent of X
r
and T
r
).
The state functions are expanded around their equilib-
rium values as
n
i
1 þ n
i1
þ
2
n
i2
þ
3
n
i3
n
e
1 þ n
e1
þ
2
n
e2
þ
3
n
e3
v
i
v
1
þ
2
v
2
þ
3
v
3
v
e
v
e1
þ
2
v
e2
þ
3
v
e3
/ /
1
þ
2
/
2
þ
3
/
3
:
(9)
Furthermore, each of the functions is decomposed into
Fourier components; for instance, for the velocity contribu-
tion in order
n
u
n
¼
X
n
l¼n
u
ðlÞ
n
e
ilðkX
0
xT
0
Þ
: (10)
This relation holds 8 n ¼ 1 ; 2; 3; :::, hence
•
l ¼1; 0; 1 for n ¼1,
•
l ¼2; 1; 0; 1; 2 for n ¼2,
and so on. Since these functions are real-valued, it must be
imposed that
u
ðrÞ
n
¼
u
ðrÞ
n
:
Upon substituting into the model equations (3) above
and then isolating successive contributions (orders in ), this
perturbation/expansion scheme yields a system of polyno-
mials in whose coefficients are required to vanish inde-
pendently, since is free (arbitrary-valued). For any given
value of n (¼ 1; 2; :::), these coefficients can be decomposed
into their separate harmonics, expressed by the second index
l (taking values from n to n). Each decomposition suggests
a relation to be imposed between its constituent variables,
which provides the solution for the given harmonic (ampli-
tude). These expressions for the harmonics are then fed into
the next order in n, and so on. The tedious, but straightfor-
ward algebraic procedure, is presented in detail in Ref. 31.
As an example, consider Poisson’s equation at the sec-
ond or der of
@
2
/
2
@X
2
0
þ 2
@
2
/
1
@X
0
@X
1
þ n
i2
n
e2
þ
a
2
v
2
i1
v
2
e1
¼ 0:
This can be split into equations for the “zeroth,” first, and
second harmonics, respectively
n
ð0Þ
i2
n
ð0Þ
e2
þ aðv
ð1Þ
i1
v
ð1Þ
i1
v
ð1Þ
e1
v
ð1Þ
e1
Þ¼0
k
2
/
1
ðÞ
2
þ 2ik
@/
1
ðÞ
1
@X
1
þ n
1
ðÞ
i2
n
1
ðÞ
e2
¼ 0
4k
2
/
2
ðÞ
2
þ n
2
ðÞ
i2
n
2
ðÞ
e2
þ
a
2
v
1
ðÞ
i1
2
v
1
ðÞ
e1
2
¼ 0:
Analogous equations are obtained at all expansion and har-
monic order(s), thus providing explicit solutions for the
harmonic amplitudes. The tedious details of the algebraic
procedure are omitted here: in the following, we shall pro-
vide the main steps. The relevant expressions for the har-
monic amplitudes are reported in the Appendix.
V. LINEAR RESPONSE AND DISPERSION RELATION
At the first order, there is only a first harmonic to inves-
tigate. The equations are presented below
xn
i1
þ kv
i1
¼ 0;
xn
e1
þ kv
e1
¼ 0;
xv
i1
þ k/
1
¼ 0;
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ n
2
0
q
/
1
þ n
e1
¼ 0;
k
2
/
1
þ n
i1
n
e1
¼ 0:
(11)
The electrons’ equation of motion is used to eliminate n
e1
from Poisson’s relation. The electrons’ equation of continu-
ity is used to find v
e1
but contains no other state variables.
052120-3 McKerr, Haas, and Kourakis Phys. Plasmas 23, 052120 (2016)
Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 143.117.193.82 On: Mon, 31 Oct
2016 16:29:24
