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Fermionic matter under the effects of high magnetic fields and its consequences in white dwarfs

We investigate a recently proposed effect of strong magnetic fields in Fermionicmatter that is important to the structure of magnetic white dwarfs. This work is highly relevant in view of the recent observations of magnetized white dwarfs (B ∼ 108-9 G), and possible candidates for white dwarfs pulsars as an alternative descriptions for SGRs and AXPs. Here, we consider the matter inside white dwarfs composed by ions surrounded by an electron degenerate Fermi gas subject to a strong magnetic field. We investigate the effect of the Landau levels due to the huge magnetic field on the equation of state (EoS). We see that the behaviour of the equation of state as a function of the mass and energy density is much stiffer when only one Landau level is occupied. We also investigate the regime of lower magnetic fields where many Landau levels are occupied.


Introduction
The study of white dwarf allows to improve our understanding of nuclear matter under extreme densities and high magnetic fields [1] [2]. The interior of these stars offers an unique point of encounter among astrophysics and atomic physics, since the macroscopic properties of compact stars, such as mass, radius, rotation and thermal evolution depend on the microscopic composition of the stellar matter. This composition can change under the presence of strong magnetic fields, affecting the equation of state, and as a consequence the structure of the star.
Recently, Coelho et al. discussed some basic equilibrium properties of magnetized white dwarfs, in particular the condition for dynamical instability of the star in the presence of an extremely large magnetic field [3]. This analysis was done in the context of the virial theorem extended to include a magnetic term. Following the work of Chandrasekhar & Fermi of 1953, when the star magnetic energy W B exceeds its gravitational potential energy |W G | (W B > |W G |), the system becomes dynamically unstable [4]. In light of this, the new mass limit for very magnetized and spherical white dwarf of 2.58M , recently calculated by [5], should be considered carefully, since these objects are unstable and unbound. Furthermore, it was showed that the new mass limit was obtained neglecting several macro and micro physical aspects such as gravitational, dynamical stability, breaking of spherical symmetry, general relativity, inverse β decay, and pycnonuclear fusion reactions. These effects are relevant for the self-consistent description of the structure and assessment of stability of these objects. When accounted for, they lead to the conclusion that the existence of such ultramagnetized white dwarfs in nature is very unlikely due to violation of minimal requirements of stability, and therefore the canonical Chandrasekhar mass limit of white dwarfs has to be still applied.

Fermion matter
Our starting point will be the microscopic energy-momentum tensor obtained from the system Lagrangian. The Lagrangian density of a fermionic system in the presence of a magnetic field is given by where we used the notation usual / a = γ µ a µ and D µ = 1 2 ( − → ∂ − ← − ∂ ) + Γ µ + ı|q|A µ with Γ µ being the spin connection which is zero in flat space, and F µν is the field strength tensor of the electromagnetic field We have still the vector potential chosen as that produces a constant magnetic field in the Z direction.
Solving the Dirac equation from the Lagrangian density we determined the dispersion relation: where p z is the longitudinal momentum, q electric charge, m e the electron mass, c the speed light, and ν the Landau level given by being s = ±1.

Zero Temperature
The integration in momentum for a electron gas with charge |q| immersed in magnetic field, restricted to discrete Landau levels of magnetic field is [6]: is the degeneracy in each Landau level. At zero temperature the distribution function is given by a theta function to one-particle where µ is the chemical potential.

Basic Equations for Landau Level Systems of Degenerate Electrons
In term of the chemical potential µ, the maximum p z is defined Rewriting the equation above,introducting of the dimensionless parameter e c 3 ∼ 4.414 × 10 13 G, γ = B/B c , and ν max is the maximum number of Landau level given by [7], If the lowest Landau level ν = 0 is occupied, ν max = 1. Similarly, for two level system, when gthe lowest ν = 0, and first, ν = 1, levels are occupied, ν max = 2, and so on [2] .

Number Density Equation
The number density is given by where λ =h mec . Integrating the equation above we have:

Mass Density
We can relate the matter density with the electron number density n e from where Z is the atomic number, A the mass number, and m n the neutron mass. The matter density can be rewritten in the following way, being K 1 = 2γm N µe (2π) 2 λ 3 , However, we see that K1 changes with the magnetic field due to the term γ.

Energy Equation and Longitudinal Pressure
The total energy density to zero temperature is given by is the electron energy density of the magnetized fermi gas and k 2 = γmec 2 (2π) 2 λ 3 . Then, by integration in the zero temperature limit As the previous case, we obtain the equation of state for the pressure where the function ξ is given by

Results
In this section we present the numerical values for the longitudinal pressure, mass density and the mass-radius relations.
In Figure 1. we see the longitudinal pressure as a function of the density, we observe in graphs (a), (b), and (c) that for magnetic fields up to B = 10 13 G the equation of state with magnetic field behaves like non-magnetic equation of state, with no significant effect caused by Landau levels.

Conclusions
In this work we solve the equations of state for an degenerate electron Fermi gas under the presence of strong magnetic fields. We investigate the effect of the Landau levels due to a strong magnetic field in the equation of state (EOS) for several values of the magnetic field. We conclude that for strong magnetic fields the separation among Landau levels is large, so electrons with lower energy (non-relativistic) can only occupy the ground state. As the magnetic field decreases, the separation between Landau levels decreases, as shown in Fig. 2. Hence it becomes energetically favorable for electrons to jump to a higher level, so the number of occupied Landau levels increases accordingly. Likewise, in the case of relativistic electrons, if the magnetic field is low, the separation of Landau levels is comparable to the rest energy of the electrons and hence the electrons can pass freely between the highest Landau levels making the EOS similar to nonmagnetic case as we can see in the Fig. 1. This case is also shown in Fig. 1, which for low values of the magnetic field the Landau levels behave almost continuously and it is no longer possible to see the peaks shown in Fig. 2, that represent precisely the critical density for each Landau level.
We also conclude that for magnetic fields below B c the mass radius relation of the star is similar to the nonmagnetic with M = 1.44M as the mass limit case. Furthermore, for fields of the order B ∼ 10 15 G the mass exceeds the well-known limit for a white dwarf, may reach masses M = 2.58M . However, as has been discussed in [3], these white dwarfs are unstable and not reliable.

Acknowledgments
EO acknowledges the support by CAPES -Brazilian Federal Agency for Support and Evaluation of Graduate Education within the Ministry of Education of Brazil and MM acknowledges the financial support of CNPq and FAPESP (São Paulo state agency, thematic project #2013/26258-4).