# 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.

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- Nature 01/1939; 144:130-131. · 38.60 Impact Factor
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##### Article: How to model quantum plasmas

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**ABSTRACT:**Traditional plasma physics has mainly focused on regimes characterized by high temperatures and low densities, for which quantum-mechanical effects have virtually no impact. However, recent technological advances (particularly on miniaturized semiconductor devices and nanoscale objects) have made it possible to envisage practical applications of plasma physics where the quantum nature of the particles plays a crucial role. Here, I shall review different approaches to the modeling of quantum effects in electrostatic collisionless plasmas. The full kinetic model is provided by the Wigner equation, which is the quantum analog of the Vlasov equation. The Wigner formalism is particularly attractive, as it recasts quantum mechanics in the familiar classical phase space, although this comes at the cost of dealing with negative distribution functions. Equivalently, the Wigner model can be expressed in terms of $N$ one-particle Schr{\"o}dinger equations, coupled by Poisson's equation: this is the Hartree formalism, which is related to the `multi-stream' approach of classical plasma physics. In order to reduce the complexity of the above approaches, it is possible to develop a quantum fluid model by taking velocity-space moments of the Wigner equation. Finally, certain regimes at large excitation energies can be described by semiclassical kinetic models (Vlasov-Poisson), provided that the initial ground-state equilibrium is treated quantum-mechanically. The above models are validated and compared both in the linear and nonlinear regimes.06/2005; - [Show abstract] [Hide abstract]

**ABSTRACT:**Recently, magnetic fields of 0.7({+-}0.1) gigaGauss (GG) have been observed in the laboratory in laser plasma interactions. From scaling arguments, it appears that a few gigaGauss magnetic fields may be within reach of existing petawatt lasers. In this paper, the equations of state (EOS) are calculated in the presence of these very large magnetic fields. The appropriate domain for electron degeneracy and for Landau quantization is calculated for the density-temperature domain relevant to laser plasma interactions. The conditions for a strong Landau quantization, for a magnetic field in the domain of 1-10 GG, are obtained. The role of this paper is to formulate the EOS in terms of those that can potentially be realized in laboratory plasmas. By doing so, it is intended to alert the experimental laser-plasma physics community to the potential of realizing Landau quantization in the laboratory for the first time since the theory was first formulated.Physics of Plasmas 05/2005; · 2.38 Impact Factor

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

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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

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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

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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)

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Page 6

where, the normalized relativity parameter R0 and the relativistic diffraction param-

eter, H=

?mi/2me(?ωpi)/(mec2), are related through the simple relation H=

e??ρcR3

Φe=minc

0/miπ/(2m3/2

e c2). Then in normalized form we have

8

(2γ)5/2

? ?

1

ne(R,γ)

?∂

∂η

?η

0

H−1/2(q)

√1 + 2γqdq

?dη

dR

?

dR, η =R2

2γ.

(11)

This gives a remarkably simple form as Φe=√1 + R2= εFefor any quantization level. This

remarkable feature indicates that in the classical-limit (n = n∞

e = (R/R0)3) the effective

potential is completely independent of the magnetic field strength and only depends on the

fermion number-density. However, in the quantum-limit (n = n0

e= γR/R3

0) the effective

potential depends strictly on the fractional field-parameter, γ, i.e.

Φe=

?1 + R2

1 +

0ρ2/3l = ∞

R6

0ρ2

γ

?

l = 0

(12)

where, we have redefined R0= (n0/nc)1/3= (ρ0/ρc)1/3with ρc= minc≃ 1.98 × 106gr/cm3.

On the other hand, for the interacting Coulomb pressure we have [36]

PC(R,γ) = −18π2m4

ec5

h3

?α5Z2/3

10L4

?

, L = α

?

3π

8ne(R,γ)

?1/3

.(13)

where, the parameters, α = e2/?c ≃ 1/137 and Z are the fine-structure constant and the

atomic number, respectively. Hence, we can write for both l = ∞ and l = 0 regimes

?

Note that this effective potential introduces a negative force in the momentum equation

Φint=

1

n(R,γ)

∂PC(R,γ)

∂R

dR = −βR0ρ1/3, β =8α(3Z)2/3

5π1/3

.(14)

unlike the electron degeneracy pressure.

III.SOLITARY EXCITATIONS IN QUANTIZED DEGENERACY REGIMES

In order to obtain a stationary soliton-like solution we follow the conventional pseudopo-

tential approach. In the moving frame with speed M we take the coordinate transformation

ξ = x − Mt which brings us to the co-moving soliton frame. Introducing the new coordi-

nates into the normalized hydrodynamic equations, integrating by use of the appropriate

boundary conditions, lim

ξ→±∞ρ = 1 and lim

ξ→±∞uc= 0, we solve the continuity relation to give

u = M (1/ρ − 1). Eventually, by making use of this relation in the momentum equation and

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defining the new variable, ρ = A2, we arrive at the following single differential equation for

the classical-limit, we have

H2

A

∂2A

∂ξ2=M2

2

(1 − A−2)2− M2(1 − A−2) +

?

1 + R2

0A4/3−

?

1 + R2

0− βR0A2/3+ βR0,

(15)

and, for quantum-limit we may write

H2

A

∂2A

∂ξ2=M2

2(1 − A−2)2−M2(1−A−2)+

?

1 +R6

0A4

γ2

−

?

1 +R6

0

γ2−βR0A2/3+βR0. (16)

Now multiplying of the Eq. (15) and Eq. (16) by the quantity dA/dξ and integrating with

respect to, ξ with the mentioned boundary conditions, lead us to the well-known energy

integral of the forms given below in terms of plasma mass-density

(dξρ)2/2 + U(ρ) = 0.(17)

Hence, for the classical-limit, we have

U∞(ρ) =

+2R2

1

4H2R3

0

?

0(4ρ − 1) − 3ρ?1 + R2

4M2R3

0(ρ − 1)2+ R0ρ

?

3

??1 + R2

0−

?1 + R2

0ρ2/3ρ1/3?

??

0

??1 + R2

0ρ2/3+ βR0(1 − 4ρ + 3ρ4/3)

−3ρsinh−1R0+ 3ρsinh−1(R0ρ1/3)?,

and, for the quantum-limit it follows that

(18)

U0(ρ) =

1

2H2R3

0

?

R3

0

?

2M2(1 − ρ)2− 2γ−1ρ?γ2+ R6

0ρ2+ 4ρ2γ−1?γ2+ R6

0ρ2

+R0βρ(1 − 4ρ + 3ρ4/3) − 2ρ2γ−1?γ2+ R6

+2γρ?sinh−1(γ−1R3

Existence of the localized density profiles requires the following basic conditions to met all

0ρ2?

0) − sinh−1(ργ−1R3

0)??.

(19)

together

U(ρ)|ρ=1=dU(ρ)

dρ

????

ρ=1

= 0,

d2U(ρ)

dρ2

????

ρ=1

< 0.(20)

Evidently, the first two requirements are satisfied for the pseudopotential given by Eqs. (18)

and (19). The third condition, however, can be directly evaluated using the pseudopotentials

for both quantized degeneracy. The soliton, therefore, exists if the there is pseudopotential

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root other than at the unstable point ρ = 0 which will be examined later. In such conditions,

the spatial extension of the solitary wave in then given as

ξ − ξ0= ±

?ρm

1

dρ

?−2U(ρ). (21)

we now return to the evaluation of the second derivative of the Sagdeev potential in both

degeneracy limits, at unstable density ρ = 1, which leads to the following expression for the

classical-limit

d2U∞(ρ)

dρ2

????

ρ=1

=

2

3H2

?

3M2+ R0

?

β −

R0

?1 + R2

0

??

,(22)

and the following relation for quantum-limit

d2U0(ρ)

dρ2

????

ρ=1

=

2

3H2

?

3M2+ R0

?

β −

3R5

0

γ?γ2+ R6

0

??

.(23)

The soliton Mach-range is therefore defined through the following critical Mach-values in

each degeneracy limit. For classical-limit, we have

M∞

cr=

?

R2

0

3?1 + R2

0

−R0β

3

. (24)

Note that, the given relation reduces appropriately to the one in Ref. [38, 39] in the field-free

(β = 0) limit. On the other hand, for the quantum-limit, we obtain

M0

cr=

?

R6

0

γ?γ2+ R6

0

−R0β

3

. (25)

Now let us examine the possibility of a root other than ρ = 1 which is other essential

condition for existence of potential valley. For the classical-limit, it is found that

lim

ρ→0U∞(ρ) =M2

which indicates that a rarefactive solitary excitation always exists with Mach-values below

H2> 0, lim

ρ→∞U∞(ρ) = −∞ × sgn(1 − β),(26)

that of critical value defined above, and, for the quantum-limit, we have

lim

ρ→0U0(ρ) =M2

revealing also the fact that only rarefactive density profile is possible in this limit. In the

H2> 0, lim

ρ→∞U0(ρ) = −∞, (27)

next section we bring into the attention the possible differences in soliton characteristics

caused by distinct magnetic degeneracy limits and discuss the variations of solitary profiles

in terms of change in various fractional plasma parameters.

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IV.NUMERICAL INSPECTION

In this section we examine the characteristics of soliton dynamics in the two magnetic

degeneracy limits under the influence of change in many plasma parameters such as fractional

field-parameter, γ, the normalized relativity parameter, R0, and the atomic-number, Z.

Figure 2 shows the soliton stability volume in classical-limit, i.e. when a large number of

Landau orbital-levels are filled. It is observed that the increase in the fractional Fermi-

momentum R0widens the soliton Mach-range, while, the change in the atomic number does

not affect the soliton stability significantly, in this regime. Comparing this figure with Fig. 3

which is the corresponding volume in the case of quantum-limit (l = 0), reveals that, in this

case the atomic number has significant effect on the soliton stability. In fact it is observed

that, for this regime at low relativity parameter region, there are unstable atomic-number

values for a given value of R0. It is remarked that for l = 0 limit the less the atomic-number

the more stable density structures are.

Figs. 3 and 4 depict the variations in pseudopotentials for the cases of l = ∞ and l = 0

while one of the plasma parameters is changed and the others are fixed. The comparison

of the figures reveal the fundamental differences on plasma parameters on potential profiles

in the two limits. Both potential width and depth are affected relatively strongly in the

quantum-limit degeneracy. The corresponding rarefactive solitons shown in Figs. 5 and 6

with identical parameter values to those in Figs. 3 and 4 for two regimes confirm the above

statement, clearly. While the effect of the plasma parameters on soliton profiles are similar

in the two cases, the nature of the effects are fundamentally different. For l = ∞ the increase

in atomic-number (Fig. 5(a)), relativity parameter (Fig. 5(b)) and fractional field-strength

(Fig. 5(c)) all widen the soliton width without altering the amplitude significantly. Also,

increase in the Mach-number decreases/increases the soliton amplitude/width slightly for

this case (e.g. see Fig. 5(d)).

On the other hand, for the case of l = 0, the increase in the atomic-number decreases

the soliton amplitude (Fig. 6(a)), while the increase in the normalized Fermi-momentum

increases the soliton amplitude (Fig. 6(b)). Also, it is observe from Fig. 6(c) that the in-

crease in fractional field strength decrease the soliton amplitude significantly. Moreover, the

effect of increase in soliton Mach value (Fig. 6(d)) strongly decreases the soliton amplitude

in this case. All these characteristics differ fundamentally in the two magnetic quantization

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limits. These findings are quite comparable to the distinctive features of nonlinear wave

dynamics in nonrelativistic and relativistic degeneracy limits reported previously for the

field-free Fermi-Dirac plasmas [38].

V. CONCLUSIONS

Based on quantum hydrodynamics formulation and using pseudopotential approach it was

shown that for a Coulomb interacting Fermi-Dirac plasma under arbitrary strength uniform

magnetic field two distinct degeneracy limits exist in which the parallel electrostatic solitary

wave dynamics are fundamentally different. It was further revealed that plasma parameter

such as atomic-number, fractional Fermi-momentum, fractional field-strength and soliton

Mach affect the wave characteristics significantly only in the quantum-limit degeneracy

compared to that of classical-limit. These findings can help better understand the physical

processes in astrophysical compact objects as well as the strong laser-matter interactions.

10

Page 11

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Page 13

FIGURE CAPTIONS

Figure-1

The monotonic and oscillatory components in fermion number-density and the exact den-

sity (thick curve) is shown in left plot, indicating a linear region of dependence on fractional

electron relativistic Fermi-momentum, R, (ne∝ γR) which corresponds to quantum-limit

Landau-quantization, l = 0. The plot to the right shows the fading of the oscillatory part and

asymptotic Chandrasekhar-limit approximation (ne∝ R3) when the Fermi-Dirac plasma is

in classical Landau quantization-limit (l = ∞).

Figure-2

Figure 2 shows a volume in M-Z-R0space for classical-limit, (l = ∞), in which a localized

magnetosonic density localized excitation can exist.

Figure-3

Figure 3 shows a volume in M-Z-R0space for quantum-limit, (l = 0), in which a localized

magnetosonic density localized excitation can exist.

Figure-4

The variations of Sagdeev pseudopotentials of localized density-structure for the classical-

limit magnetic Fermi-Dirac plasma with respect to change in each of several independent

plasma fractional parameters, namely, normalized soliton-speed, M, the atomic-number, Z,

fractional field-strength, γ, and the normalized relativity parameter, R0, while the other

three parameters are fixed. The dash-size of the profiles increase according to increase in

the varied parameter.

Figure-5

The variations of Sagdeev pseudopotentials of localized density-structure for the

quantum-limit Fermi-Dirac plasma with respect to change in each of several independent

plasma fractional parameters, namely, normalized soliton-speed, M, the atomic-number, Z,

fractional field-strength, γ, and the normalized relativity parameter, R0, while the other

13

Page 14

three parameters are fixed. The dash-size of the profiles increase according to increase in

the varied parameter.

Figure-6

The variations of rarefactive localized density-structure shape for the the classical-limit

magnetic Fermi-Dirac plasma with respect to change in each of several independent plasma

fractional parameters, namely, normalized soliton-speed, M, the atomic-number, Z, frac-

tional field-strength, γ, and the normalized relativity parameter, R0, while the other three

parameters are fixed. Identical values as Fig. 4 is used in this figure. The thickness of the

profiles increase according to increase in the varied parameter.

Figure-7

The variations of rarefactive localized density-structure shape for the quantum-limit mag-

netic Fermi-Dirac plasma with respect to change in each of four independent plasma frac-

tional parameters, namely, normalized soliton-speed, M, the atomic-number, Z, fractional

field-strength, γ, and the normalized relativity parameter, R0, while the other three param-

eters are fixed. Identical values as Fig. 5 is used in this figure. The thickness of the profiles

increase according to increase in the varied parameter.

14

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