Fowler-Nordheim Electron Cold Emission Formalism in Presence of Strong Magnetic Field

Formalisms for both non-relativistic as well as relativistic versions of field emission of electrons in presence of strong quantizing magnetic field, relevant for strongly magnetized neutron stars or magnetars are developed. In the non-relativistic scenario, where electrons obey Schr{$\ddot{\rm{o}}$}dinger equation, we have noticed that when Landau levels are populated for electrons in presence of strong quantizing magnetic field the transmission probability exactly vanishes unless the electrons are spin polarized in the opposite direction to the external magnetic field. On the other hand, the cold electron emission under the influence of strong electrostatic field at the poles is totally forbidden from the surface of those compact objects for which the surface magnetic field strength is $\gg 10^{15}$G (in the eventuality that they may exist). Whereas in the relativistic case, where the electrons obey Dirac equation, the presence of strong quantizing magnetic field completely forbids the emission of electrons from the surface of compact objects if $B >10^{13}$G.