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Characteristics of Quantum Magnetosonic-Wave Dispersion

Using the quantum magnetohydrodynamics (QMHD) model, linear dispersion of magnetosonic waves are studied in a quasineutral quantum electron-ion plasma in two distinct regimes of nonrelativistic and relativistic degeneracies considering also the plasma composition effect. It is shown that the paramagnetic spin effects of the degenerated electrons plays a key role in dynamics of magnetosonic waves. The linear wave-speed is found to have minimum value at some degeneracy parameter in such plasmas. This is due to delicate interplay between relativistic degeneracy and the Pauli spin-magnetization. It is also revealed that the plasma composition has significant effect on the linear dispersion in the relativistic degeneracy limit contrary to that of nonrelativistic case and Zeeman energy has significant effect in nonrelativistic degeneracy regime unlike that of relativistic one in the linear perturbation-limit. Current findings can have important applications in both inertial plasma confinement and astrophysical degenerate plasmas.


I. INTRODUCTION
A quantum plasma is usually referred to as a degenerate dense and cold ionized matter.
Despite the high temperatures observed for astrophysical compact objects, it has been shown [1] that these objects can be regarded as fully degenerate cold quantum plasmas. This is because the temperature of plasma species in these objects stay below a well defined Fermi temperature of order 10 4 K. Vast majority of the recent studies on quantum plasmas [2][3][4][5][6][7][8][9][10][11] have been inspired by the pioneering works of Bohm, Pines and Levine et.al [12][13][14] and the earlier works of Chandrasekhar. Distinction between an ordinary and quantum plasma (also known as the zero-temperature Fermi-plasma) is that, in a quantum plasma contrary to the ordinary one, the inter-fermion distances are much lower than the characteristic de Broglie thermal wavelength, h/(2πmk B T ) 1/2 so that the particle waves overlap causing a new type collective phenomenon through the electron degeneracy and tunneling effects. These properties which are encountered only in quantum plasmas, for instance in ordinary metals and compact astrophysical objects, lead to distinct features of wave dynamics when the electron number-density or the ambient magnetic field strength or even the system dimensionality and the degrees of freedom of fermions is varied [15][16][17]. The electron degeneracy is a direct consequence of the Pauli exclusion principle and effectively rules the thermodynamical properties of a dense matter [18].
Most of current investigation on quantum plasmas are motivated due to the applicability of the quantum magnetohydrodynamics (QMHD) to semiconductors [19]. Quantum optical and electronic transport phenomena are also known as main ingredients in the study of metallic and semiconductor nano-structured materials such as, nano-particles, quantumwells, quantum-wires and quantum-dots [20]. Therefore, the subject of dense cold ionized matter under strong magnetic field can promise wide applications in both laboratory as well as naturally occurring plasmas such as astrophysical compact objects [21,22]. One of the differences of the QMHD model from those of classical counterparts is the inclusion Bohm tunneling effect. It has been shown that the increase in the electron number density enhances this feature significantly altering the dynamics of linear and nonlinear wave propagations [23]. More recently, Marlkund and Brodin have extended QMHD model and incorporated the electron spin-1/2 effects [24] by introducing a generalized term for the quantum force a feature which is expected to be dominant in superdense plasmas such as Fermi-Dirac. It has been estimated that relativistic electrons in a white dwarf can generate magnetic fields of the order of 10 7 G and higher [25], and that in a neutron star the self-field may be of order 10 13 − 10 14 G [26]. On the other hand, it has been shown that, in a perfectly conductive quantum plasma, the spin magnetization term introduced in quantum magnetohydrodynamics (QMHD) equations acts as a negative pressure-like entity significantly affecting the dynamics of spin-induced nonlinear magnetosonic excitations [27].
In 1935 Chandrasekhar [28], in a pioneering work, proved that the gigantic inward gravitational pressure can lead to the new state of relativistic degeneracy for electrons and may cause the ultimate collapse of the star in extreme degeneracy cases [29]. The sudden collapse of the white dwarfs was shown to be due to softening of the degeneracy pressure of electrons relativistic motion. This effect has been shown to alter the whole thermodynamical properties of the star [30]. Current investigation is an attempt to trace possible differences in linear magnetosonic wave dynamics caused due to change in the plasma degeneracy state such as nonrelativistic and relativistic ones and explore the composition effects on linear dispersion.
The presentation of the article is as follows. The QMHD plasma model including the spin contribution is introduced in Sec. II. The linear dispersion of spin-induced magnetoacoustic waves are derived in Sec. III for different plasma degeneracy regimes and the numerical results are presented in Sec. IV.

II. QMHD MODEL INCLUDING SPIN EFFECTS
Let us consider a collisionless dense degenerate quasineutral electron-ion perfectly conducting plasma with electrons and ions immersed in a spatially varying magnetic field in a fixed direction. We further assume that the magnitude of the external field is very high and the dynamics of wave propagation is strongly affected by spin contribution from fermions.
In the center of mass frame of plasma the continuity relation reads as where, ρ = m e n e +m i n i ≃ m i n i and V = (n e m e V e +n i m i V i )/ρ are the plasma center of mass density and velocity. The momentum equation, taking into account the quantum diffraction [3] and spin pressure [24] effects, in the center of mass frame takes the form [27,31] ∂V ∂t where, the quantum force F Q = F B +F S is a collective contribution of quantum Bohm-force, F B , and the so-called spin-force, F S , namely, The quantities J, B and P are the magnetization current, magnetic field and scalar total center of mass plasma pressure, respectively. The magnetization current, J, is related to the spin-magnetization (Pauli-magnetization) per unit volume M P = (3µ 2 B n e /2k B T F e ) B (T F e being the electron Fermi-temperature) through; m e c 2 ) has been evaluated in Ref. [32], which is as follows where, γ and P F e are the s-wave interaction parameter and the Fermi relativistic momentum, respectively. It can be shown that in the nonrelativistic limit with γ = 0 the susceptibility given in Eq. (4) reduces to the well-known Pauli paramagnetic susceptibility. In order to complete the set of QMHD fluid equations, we use the generalized Faraday law without the Hall term in the form where, η is the plasma resistivity. Moreover, with ρ = ρ(x, t) and V = V (x, t), and assuming the propagation of the magnetoacoustic wave to be in x direction perpendicular to the magnetic field B(x, t) along the z-axis, we may write simplified one-dimensional version of the complete QMHD equations as [27] ∂ρ On the other hand, in a relativistically degenerate zero-temperature Fermi-Dirac plasma one can ignore the ion-pressure compared to that of the dominant electron degeneracy pressure [33] which can be expressed in the following general form [1] in which R = (P F e )/(m e c) = (ρ/ρ c ) 1/3 (ρ c ≃ 1.97 × 10 6 gr/cm 3 ) is the relativity parameter with P F e being the electron Fermi relativistic momentum. The relativistic degeneracy due to the Pauli exclusion principle increases with decrease in inter-fermion distances in degenerated plasmas. It is noted that, although the relativistic effects arise due to increase in fermion number-density in a dense degenerate plasma, however, unlike for the low-pressure relativistic plasmas the degeneracy pressure in the fermion momentum fluid equation usually dominates the relativistic (electron/ion) dynamic effects in super-dense plasma state and the relativistic electron/ion mass effects may be ignored. It has been shown [28] that for a zerotemperature dense degenerate Fermi-gas such as a white-dwarf with a mass-density, ρ, the degeneracy pressure turns from P e ∝ ρ 5/3 (with polytropic index 3) dependence for normal degeneracy in the limit R ≪ 1 to P e ∝ ρ 4/3 (with polytropic index 2/3) dependence for relativistic degeneracy case in the limit R ≫ 1. On the other hand, the relativistic degeneracy starts at mass density of about 4.19 × 10 6 (gr/cm 3 ) of the order in the core of a 0.3M ⊙ white dwarf, which corresponds to a Fermi-momentum P F e ∼ 1.29m e c (equivalent to the relativistic degeneracy parameter value of R ∼ 1.29) or the threshold velocity of u F e ∼ 0.63c (the Fermi relativistic factor γ F e ∼ 1.287). This may justify our nonrelativistic treatment of electron mass effects in above arguments.
The problem of spin-induced magnetosonic has been thoroughly investigated by Masood et al. [34] for the case of the relativistic degeneracy, however, for completeness in current work we try to extend the previous study to the case of quantum plasma with the Coulomb interaction effect. In a dense quantum plasma the coupling parameter Γ representing the potential to kinetic energy ratio may exceed unity leading to strong coupling due to other many-body effects such as Thomas-Fermi screening, Coulomb interactions, electronexchange and ion-correlation. Salpeter [35] has examined these effects in the relativistically degenerate plasmas and concluded that the first-order correction to the degeneracy pressure can be that of Coulomb interactions and other effects may be negligible. Following the Salpeter expression for classical Coulomb plus Thomas-Fermi corrections, we may write where, α = e 2 / c ≃ 1/137 and Z are the fine-structure constant and the atomic number, respectively. The first term corresponds to the Coulomb negative pressure and the second term (proportional to α 2 ) is a relatively smaller contribution due to Thomas-Fermi screening, which will be neglected in current scheme.
On the other hand, the plasma resistivity, η, of a completely degenerate Fermi-Dirac plasma is supposed to be negligible, since, the electron-ion collisions are lowered by Pauliblocking mechanism, i.e., η ∼ 0. By using Eq. (6) this lead to the simple relation, B/B 0 = ρ/ρ 0 =ρ (the so-called frozen-in field condition). The new quantities B 0 , ρ 0 andρ denote the equilibrium values of magnetic field intensity, plasma mass-density and the normalized mass density, respectively. Therefore, the complete set of dimensionless equations governing the dynamics of magnetosonic waves reduce to ∂ρ ∂t + ∂ρV ∂x = 0, where, β = α(2 5/3 /5)(3Z 2 /π) 1/3 and we have used (1/ρ)∂ x P e (ρ) = ∂ x √ 1 + R 2 and (1/ρ)∂ x P C (ρ) = ∂ x (βR) and employed the following scalings in order to obtain the dimensionless equations ignoring the normalization bar notation where, c, ω pi = e 2 n e0 /(ε 0 m i ) and V s = c m e /m i are the vacuum light speed, the characteristic ion plasma frequency and ion sound-speed (this speed is much higher despite the name comparable to the Fermi-speed of an electron in a solid), respectively, and the parameter ρ 0 denotes the equilibrium plasma mass-density. We have also introduced new fractional plasma entities used frequently in forthcoming discission, namely, the quantum diffraction parameter, H = m i /2m e ( ω pi )/(m e c 2 ), relativistic degeneracy parameter, R 0 = (ρ 0 /ρ c ) 1/3 [33], and the normalized Zeeman energy, ǫ = µ B B 0 /(m e c 2 ).

III. COULOMB INSTABILITY AND DEGENERACY REGIMES
Now let us consider the special cases of ǫ R 0 ≪ 1 and ǫ R 0 ≫ 1 in the spin-contributed term in the momentum equation (Eq. (9)). These limits will be shown to lead to distinct features in dispersion of linear spin-induced magnetosonic waves. Since, the normalized Zeeman energy parameter is solely defined by the external magnetic field strength, the two extreme degeneracy limits, namely, the nonrelativistic (R 0 ≪ 1) and relativistic (R 0 ≫ 1) degeneracy can be distinguished for a given magnetic field strength value, say, B 0 = 10 5 T (ǫ ≃ 10 −5 ) typical of spin-dominated compact astrophysical objects. On the other hand, the reader should have already realized the elegant relationship between the quantum diffraction parameter, H, which is due to the electron tunneling and the relativistic degeneracy parameter, R 0 , due to the increase in the electron number density, namely, H = e ρ c R 3 0 /π/(2m 3/2 e c 2 ). Fourier analyzing Eq. (9) results in the following dispersion relation for the linear magnetosonic waves in the coupled relativistically degenerate plasma case The differences in dispersion relation (Eq. (11)) between the two cases of ǫ R 0 ≪ 1 (relativistic degeneracy) and ǫ R 0 ≫ 1 (normal degeneracy) is readily apparent from the dispersion relation Eq. (11). It is observed that, for the case of relativistic degeneracy the last term in dispersion relation is negligibly small and the phase speed of the linear magnetosonic-wave is almost independent from Zeeman energy, ǫ, while for the nonrelativistic plasma regime this is not the case. On the other hand, for the case of nonrelativistic degeneracy the first term in the dispersion relation is insignificant and the wave becomes dispersionless. The approximate dispersion regimes in extreme degeneracy limits can be summarized as below It is apparent from Eq. (11) that the instability can set in when This condition can be relaxed for the nonrelativistic (R 0 ≪ 1) long-wavelength (k ≪ 1) limit to the following instability condition Thus, for every plasma composition (every value of Z) there exists a R 0 -region in R 0 -ǫ plane for a given Zeeman parameter ǫ < ǫ m , where the instability sets-in for the magnetosonic linear waves and for ǫ > ǫ m the linear waves are completely stable. This effect has been discussed in detail in the next section, where, it has been shown that the width of instability region only loosely depends on the wavenumber, k (e.g. see Fig. 4).

IV. NUMERICAL RESULTS AN ANALYSIS
It is clearly observed from Eq. (11) that, in the nonrelativistic degeneracy limit of our model the diffraction parameter, H, is negligibly small, hence, the magnetosonic waves are almost dispersionless, while, for the relativistic degeneracy regime the values of, H are relatively large and the linear waves become dispersive. The dispersion in this regime becomes more pronounced for higher values of wave number, k. In Fig. 1, the variation of paramagnetic susceptibility χ p , given by Eq. (4), versus the normalized plasma massdensity, ρ 0 /ρ c is shown. It is observed that the increase in the mass-density (equivalently increase in the relativistic degeneracy parameter of the plasma) causes the increase in the relativistic susceptibility value. Figure 2 shows the variation of the magnetosonic wave dispersion relation due to variations in the relativistic degeneracy parameter, R 0 (presenting the normalized plasma mass-density) and the normalized Zeeman energy, ǫ (presenting the magnetic field-strength) in the two distinct nonrelativistic and relativistic degeneracy limits in the absence of Coulomb interactions (β = 0). From Fig. 2(a) it is apparent that in the nonrelativistic regime the speed of linear magnetosonic waves decrease with increase in the relativistic degeneracy parameter for a given fixed value of the magnetic field parameter, ǫ.
On the other hand, it is observed from Fig. 2(b) that (in the absence of Coulomb interaction) increase in the strength of the magnetic field leads to the increase in the wave speed in this regime. However, as it is revealed from Fig. 2(c), in the relativistic degeneracy regime, the increase in the normalized plasma mass-density (i.e. increase in the relativistic degeneracy parameter, R 0 ) results in an increase in the linear magnetosonic speed which is opposite to the case of nonrelativistic degeneracy case shown in Fig. 2(a). Furthermore, increase in the magnetic field strength in relativistic degeneracy regime in ineffective in changing the linear wave-speed for a fixed normalized R 0 -parameter which is contrary to the case of nonrelativistic degeneracy (e.g. compare Figs. 2(b) and 2(d)).
Distinguished feature is observed in Fig. 2 Thomas-Fermi plasmas [36,37]. crust of a neutron star or a white-dwarf core is believed to be iron-rich while they are also relativistically degenerate and might be highly magnetized. Also, the outer regions of a white-dwarf star may contain degenerate carbon or helium. There has been many compact star reports with strong internal or external magnetic fields [38][39][40][41].