Spin paramagnetic deformation of a neutron star

Quantum mechanical corrections to the hydromagnetic force balance equation, derived from the microscopic Schr\"{o}dinger-Pauli theory of quantum plasmas, modify the equilibrium structure and hence the mass quadrupole moment of a neutron star. It is shown here that the dominant effect --- spin paramagnetism --- is most significant in a magnetar, where one typically has $\mu_{B}|\boldsymbol{B}|\gtrsim k_B T_e$, where $\mu_{B}$ is the Bohr magneton, $\boldsymbol{B}$ is the magnetic field, and $T_e$ is the electron temperature. The spin paramagnetic deformation of a nonbarotropic magnetar with a linked poloidal-toroidal magnetic field is calculated to be up to ${{\sim 10}}$ times greater than the deformation caused solely by the Lorentz force. It depends on the degree of Pauli blocking by conduction electrons and the propensity to form magnetic domains, processes which are incompletely modelled at magnetar field strengths. The star becomes more oblate, as the toroidal field component strengthens. The result implies that existing classical predictions underestimate the maximum strength of the gravitational wave signal from rapidly spinning magnetars at birth. Turning the argument around, future gravitational-wave upper limits of increasing sensitivity will place ever-stricter constraints on the physics of Pauli blocking and magnetic domain formation under magnetar conditions.