Magnetic Deformation of Atoms in the Crust of Magnetars

Some special features of magnetically deformed multi-electron atoms in the crustal region of strongly magnetized neutron stars (magnetars) are studied with the relativistic version of semi-classical Thomas-Fermi model. An analytical solution of Poisson’s equation with cylindrical symmetry is obtained. The variation of atomic dimension with magnetic field strength is also studied. An expression for atomic quadrupole moment, appears because of magnetic deformation is derived. The effect of atomic quadrupole-quadrupole interaction in the equation of state of crustal matter of magnetar is briefly discussed. E-mail:[email protected] E-mail:[email protected] Permanent address 1 From the observational evidence of some compact stellar objects with extremely high surface magnetic field [1], the study of the effect of ultra strong magnetic field on the geometrical structure (electronic orbits) of atoms in the crustal region have got a new dimension. The possibility of such ultra-high magnetic field in these stellar objects has given re-birth to this ”the old subject” of quantum mechanics. These strongly magnetized objects are known as magnetars [2]. The measured value of magnetic field strength at the magnetar surface can be as high as ≥ 10G. These objects are also supposed to be the sources of soft gamma repeaters and anomalous X-ray pulsars. Soft gamma repeaters are a class of compact stellar objects which periodically emit bursts of gamma rays, in addition to persistent X-rays. These two classes of objects appear to be very similar (aside from the presence of bursts in SGR’s). The measured period lie in the range 5 − 12 secs. The dipole fields at the surface inferred from the spin down rate are 10G, which are much larger than the surface field strength of known strongly magnetized conventional neutron stars (≃ 10G). It has long been suggested that strong magnetic energy may be the ultimate source of both the bursts and the persistent radiation [2]. From the observational data it is believed that such objects (magnetars) are strongly magnetized young neutron stars. If the strength of surface magnetic field is ≥ 10G, then it would be interesting to investigate the effect of such strong magnetic field on the properties of crustal matter. Some of the properties of such low density matter in presence of strong magnetic field have already been studied [3]. The other interesting effect of ultra-strong magnetic field on the crustal matter, which has not been explored in greater detail, which we are going to discuss in this article is the possibility of magnetic deformation of atoms at the lattice points in the crustal region. Although the physical problem related to the atoms placed in an external magnetic field is one of the oldest subject of quantum mechanics, this particular exotic phenomena – the magnetic deformation of atoms, heavier than hydrogen, have not been studied. The effect of strong magnetic field on the stability of hydrogen atoms and on the multi-proton bound states have been investigated in detail [4–7]. In a recent work, the magnetic deformation of hydrogen atom and the appearance of its quadrupole moment have been studied [8]. In the present 2 report, we shall consider the effect of strong magnetic field on the geometrical structure of multi-electron atoms (much heavier than hydrogen atom). We do believe that such physical situation can not be ruled out at the crustal region of magnetars. We shall also investigate the effect of strong magnetic field on the quadrupole moment of deformed atoms at the crustal lattice points. It is believed that the shape of an atom is almost spherical at B ≤ B0 and a cylindrical at B ≥ B0, where B0 = m 2 ee 3 = 2.35×10G, me and e are respectively the mass and electric charge of the electron (in our calculation we have assumed h̄ = c = 1). The critical parameter γ = B/B0 has significance in the non relativistic theory. In our case B ≥ 4.4× 10G which is the relativistic region and we consider the parameter ζ = B/Bc, where Bc = 4.4×10 G, the typical field strength at which the Landau levels of electrons are populated. Since fully relativistic calculation of a many body quantum mechanical system of finite size in presence of ultra-strong magnetic field is a very difficult task, even numerically, we are considering for the multi-electron atoms in presence of strong magnetic field B ≥ Bc, a simple semi-classical approachthe relativistic version of Thomas-Fermi model in presence of strong magnetic fields. We shall now consider the basic formalism of relativistic Thomas-Fermi model in presence of strong magnetic field. The electrostatic potential φ satisfies the equation [3], ∇ φ = 4πnee− 4πZeδ(r − rn) (1) Where ne is the electron density, Z is the atomic number and rn is nuclear radius. Since rn ∼ fm ≪ atomic radius ∼ A , the nuclear contribution on the right hand side can therefore be neglected for r > rn. Now assuming that only the zeroth Landau level for the electrons is populated, the chemical potential of electron is given by μ = (pF 2 +me ) − eφ(r) = constant (2) This gives, pF = [(μ+ eφ) 2 −me ] (3)