Parametric resonant states of charged fermions in the magnetar’s crust

We have presently derived the positive-energy solutions to the Dirac equation minimally coupled to a depth-dependent spatially harmonic tangential magnetostatic field to the magnetar crust, similar to the one proposed by Wareing and Hollerbach. It turns out that, for ultra-relativistic fermions and time-intervals much less the characteristic time (comparable to the average Ohmic timescale in the crust), the corresponding linearly independent modes get their depth-dependent amplitudes expressed in terms of Mathieu’s functions and therefore, non-trivial resonances arise, leading to instabilities in the system, for computable ranges of the model parameters. In order to detail these features, we have also discussed the current density components, pointing out the regions for which the particle density has a double bounded modulation. Finally as the magnetic field induction is increasing, the instability range gets larger triggering the exponential growth of the amplitudes, once the imaginary part of the Mathieu Characteristic Exponent becomes more and more dominant.