Field-Induced Degeneracy Regimes in Quantum Plasmas

Abstract

It is shown that in degenerate magnetized Fermi-Dirac plasma where the
electron-orbital are quantized distinct quantum hydrodynamic (QHD) limits exist
in which the nonlinear density waves behave differently. The Coulomb
interaction among degenerate electrons affect the electrostatic nonlinear wave
dynamics more significant in the ground-state Landau quantization or the
so-called quantum-limit ($l=0$) rather than in the classical-limit
($l=\infty$). It is also remarked that the effective electron quantum potential
unlike the number-density and degeneracy pressure is independent of the applied
magnetic field in the classical-limit plasma, while, it depends strongly on the
field strength in the quantum-limit. Current findings are equally important in
the study of wave dynamics in arbitrarily-high magnetized astrophysical and
laboratory dense plasmas.

Full text

Page 1

arXiv:1201.6571v1 [physics.plasm-ph] 31 Jan 2012
Field-Dependent Degeneracy Regimes in Quantum Plasmas
M. Akbari-Moghanjoughi
Azarbaijan University of Tarbiat Moallem, Faculty of Sciences,
Department of Physics, 51745-406, Tabriz, Iran
(Dated: February 1, 2012)
Abstract
It is shown that in degenerate magnetized Fermi-Dirac plasma where the electron-orbital are
quantized distinct quantum hydrodynamic (QHD) limits exist in which the nonlinear density waves
behave differently. The Coulomb interaction among degenerate electrons affect the electrostatic
nonlinear wave dynamics more significant in the ground-state Landau quantization or the so-
called quantum-limit (l = 0) rather than in the classical-limit (l = ∞). It is also remarked that
the effective electron quantum potential unlike the number-density and degeneracy pressure is
independent of the applied magnetic field in the classical-limit plasma, while, it depends strongly
on the field strength in the quantum-limit. Current findings are equally important in the study of
wave dynamics in arbitrarily-high magnetized astrophysical and laboratory dense plasmas.
PACS numbers: 52.30.Ex, 52.35.-g, 52.35.Fp, 52.35.Mw
1

Page 2

I.THEORETICAL REVIEW
Plasmas under extreme conditions such as high pressure and magnetic field has well-
known applications in astrophysics [1–4]. Since the nobel work of Chandrasekhar, it is well
understood that the thermodynamic properties of degenerate matter can be fundamentally
altered due to relativistic consideration caused by gigantic gravity force [5]. It has been
shown that the relativistic degeneracy can affect the whole thermodynamics [6] as well as
linear and nonlinear wave dynamics [7–9] of Fermi-Dirac plasmas. In fact the remarkable
property of relativistic degeneracy has been shown to be the lead cause of complete chain
of stellar evolution and existence of variety of compact stars in our Universe [10]. The
associated magnetic field with some highly dense stellar objects [11] its inevitable role in
star forming molecular clouds [12–14] has brought the attention to the problems such as
behavior of atomic structure and Fermi-electrons under an arbitrary strength magnetic field.
The earliest extensive study of this kind belongs to Canuto and Chiu [15–22]. It has been
shown [17, 23] that the electronic density of state, hence all thermodynamics quantities,
become quantized due to the familiar Landau orbital quantization.
Obviously, the significant change in equation of state of ordinary matter has yet to be
observed in laboratory due to high magnetic fields close to critical value, Bc= m2
ec3/e? ≃
4.41 × 1013G, required, however, the oscillatory nature [24] of quantization given rise to De
Haasvan Alphen effect in many experiments is an undeniable proof of the Landau theory. a
complete review of the properties of degenerate matter under arbitrary magnetic field may be
found in literature [1, 25]. One of the remarkable predictions based on the Landau extension
of Fermi-Dirac statistics is the existence a metastable ferromagnetic state called LOFER [26–
30] which might explain the unsolved origin of high magnetic field in dense astrophysical
plasmas [27, 30]. The studies based on LOFER condition estimates magnetic field of order
108G for white dwarf and of 1013G for neutron star electron-number density ranges [31].
Based on the magnetohydrodynamic formalism it has been recently shown that the LOFER
concept for the astrophysical density ranges can lead to quantum collapse where the Fermi-
Dirac degeneracy pressure is totaly canceled by the Landau spin-orbit magnetization pressure
[32, 33]. Future progress in intense laser technology may facilitate the observation of Landau
quantization in strong laser-matter interaction experiments [34, 35]. However it is thankful
of nature for providing us with the astrophysical laboratory to observe phenomena in which
2

Page 3

quantum, relativity and magnetism effects are all involved.
In current research we examine the effect of such extreme conditions on nonlinear wave
dynamics incorporating the relativistic Coulomb interactions which is most relevant to highly
degenerate plasmas such as the crystalized core of white dwarfs or neutron stars. It has been
shown that the magnetic field and Coulomb interactions may lead to minor deviations from
the Chandrasekhar mass-radius relation [36]. There are some other effects such as electron
distribution non-uniformity, exchange and correlation effects which are of less importance
compared to the Coulomb interaction [37] and are neglected in the present work. The pre-
sentation of the article is as follows. The basic normalized plasma equations are introduced
in Sec. II. A general nonlinear arbitrary-amplitude solitary solution is given in Sec. III.
The numerical presentation and discussion is given in Sec. IV and conclusions are drawn in
section V.
II. QUANTIZED FERMI-DIRAC PLASMA MODEL
In this section we present simple quasineutral quantum hydrodynamics model for a com-
pletely degenerate quantized Fermi-Dirac plasma of with interacting electrons. For simplic-
ity, we consider only perturbations parallel to an arbitrary-strength uniform magnetic field
applied in the z direction to probe the possible magnetic quantization effect on nonlinear
density structures propagating in z direction parallel to the existing field. Based on the
standard quantum hydrodynamics (QHD) model, the continuity and momentum equations
for electrons and ions ignoring the Bohm force on ions, can be combined to give a complete
set of equations in the plasma center of mass frame reference, as
∂ρ
∂t+ ∇ · (ρu) = 0, (1)
where, ρ = (mini+ mene) ≃ miniand, u = (nemeue+ nimiui)/ρ are the center of mass
density and velocity of quantum plasma, and
∂2√ne
∂x2
ρdu
dt= −∂Ptot
∂x
+
ρ?2
2memi
∂
∂x
?
1
√ne
?
,(2)
where, Rtot= Pe?+ Pi+ Pintwith Pe?, Pi, Pintbeing the parallel component of the field-
dependent degeneracy pressure of electrons, classical ion pressure and interaction pressure
such as Coulomb, respectively and ? is the scaled Plank constant. It is known that in the
3

Page 4

Fermi-Dirac plasma among the wide variety of interactions the Coulomb interaction is the
most dominant [36] which will be considered in our problem. We further use the following
scaling to reduce to normalized model
¯t
ωpi, ρ → ¯ ρρ0, u → ¯ ucs,
where the normalizing factors, ρ0, ωpi= e/mi√4πρ0and cs= c?me/midenote the equi-
librium plasma mass-density, characteristic ion plasma-frequency and ion quantum sound-
x →
cs
ωpi¯ x, t → (3)
speed, respectively. The bar notations denote the dimensionless quantities, hence, ignored in
forthcoming algebra for simplicity. The normalization of the continuity equation is straight.
On the other hand, the quasineutrality condition (ne ≃ ni = n) along with the scalings
introduced in Eq. (3) gives rise to the following equation for the momentum
∂2√ρ
∂x2
du
dt= −1
ρ
∂Ptot
∂x
+ H2∂
∂x
?1
√ρ
?
,(4)
where H =
?mi/2me(?ωpi)/(mec2) is the relativistic quantum diffraction parameter. The
thermodynamic quantities of a Fermi-Dirac gas in an arbitrary strength magnetic-field is
given by Canuto and Chiu [17, 36]. The quantized fermion number-density (not normalized)
reads as
ne(εFe,γ) = ncγ
lm
?
l=0
(2 − δl,0)
?
ε2
Fe− 2lγ − 1, (5)
where, nc= m3
ec3/2π2?3, γ = B0/Bcwith Bc= m2
√1 + R2= EFe/mec2is the normalized Fermi-energy and
ec3/e? ≃ 4.41×1013G being the fractional
critical-field parameter, εFe =
R = pFe/mec is the normalized Fermi-momentum the so-called relativity parameter. Also,
lmis the maximum filled Landau-level defined as lm= (ε2
Fe−1)/2γ ≥ l. The parameter δl,0
is one for the quantum-limit (l = 0) and zero for other l-values. On the other hand, the (not
normalized) quantized pressure can be written as
Pe?(εFe,γ) =1
2ncγmec2
?
lm ?
l=0(2 − δl,0)
?
εFe
?ε2
Fe− 2lγ − 1
−(1 + 2lγ)ln
εFe+√
ε2
Fe−2lγ−1
√2lγ+1
??
.
(6)
The small-field expansion for the critical-field parameter, γ, can be written as [36]
ne(R,γ) =2
3nc
?
R3+γ2
??R√1 + R2(2R2− 3) + 3sinh−1R?
+ 2sinh−1R −
4R+ O(γ4)
?
,
Pe?(R,γ) =ncmec2
+γ2?√1+R2
12
R
?
1 +
1
R(R+√1+R2)
??
4
+ O(γ4)
?
,
(7)

Page 5

It is noted that the γ = 0 limit reduces to the Chandrasekhar equation of state.
The above equation of state may also be written in terms of Hurwitz zeta-functions [23].
For instance, for the particle density and the pressure of Landau quantized Fermi-Dirac gas,
we may write
ne(R,γ) = nc(2γ)3/2H−1/2
Pe?(R,γ) =ncmec2
Hz(q) = h(z,{q}) − h(z,q + 1) −1
h(z,q) =
?
where h(z,{q}) is the Hurwitz zeta-function of order z with the fractional part of q as
argument. It has been shown [23] that the thermodynamics parameters may be separated
?
0
R2
2γ
?
H−1/2(q)
√1+2γqdq,
,
2
(2γ)5/2?R2
2γ
2q−z,
∞
n=0(n + q)−z.
(8)
into monotonic and oscillatory parts. Figure 1 (left plot) shows the exact quantized particle-
density (thick curve) along with the monotonic and oscillatory components for γ = 0.5. Also,
Fig. 1 (right plot) reveals that up-to R = 1 (l = 0) the dependence is linear and the step-like
feature introduced in density structure due to the Landau quantization is smeared-out in
the classical-limit, l = ∞. The expansion of quantized density given in Eq. (8) for the
classical-limit, R/2γ ≫ 1, [23] leads to
ne(R,γ) =2
3nc
?
R3+ γ2/4R +3
2(2γ)3/2h(−1/2,{R2/2γ}) + O(γ4)
?
. (9)
The first two terms correspond to the monotonic (shown in Eq. (7)) and the third term
belongs to the oscillatory one. There are distinct critical-field regimes which are apparent in
Fig. 1, namely, the quantum-limit, l = 0 (i.e. R/2γ ≪ 1) and the classical-limit, l = ∞ (i.e.
R/2γ ≫ 1). These regimes for γ = 0.5 case are denoted as R < 1 for the quantum-limit
and R ≥ 3 for the Chandrasekhar classical-limit, in Fig. 1. The magnetic-field and density
range shown in both limits in Fig. 1 reveal the importance of both in the astrophysical and
laboratory applications. By defining a dimensionless normalizing parameter R0= (n0/nc)1/3
we have for the classical Chandrasekhar-limit (l = ∞), n∞
quantum-limit, we obtain, n0
0R3
0= R3[15, 34] and for the
eR3
0= γR (e.g. see Eq. (5) for l = 0).
We now return to the normalized hydrodynamic equation in terms of effective potential
instead of pressure. Ignoring the ion pressure compared to that of degenerate electrons, we
have
du
dt= −∂Φtot
∂x
+ H2∂
∂x
?
1
√ρ
∂2√ρ
∂x2
?
, Φtot= Φe+ Φint, (10)
5