  Certain multi-wavelength observations of neutron stars, such as intermittent
radio emissions from rotation-powered pulsars beyond the pair-cascade death
line, the pulse profile of the magnetar SGR 1900+14 after its 1998 August 27
giant flare, and X-ray spectral features of PSR J0821-4300 and SGR 0418+5729,
suggest that the magnetic fields of non-accreting neutron stars are not purely
dipolar and may contain higher-order multipoles. Here, we calculate the
ellipticity of a non-barotropic neutron star with (i) a quadrupole
poloidal-toroidal field, and (ii) a purely poloidal field containing arbitrary
multipoles, deriving the relation between the ellipticity and the multipole
amplitudes. We present, as a worked example, a purely poloidal field comprising
dipole, quadrupole, and octupole components. We show the correlation between
field energy and ellipticity for each multipole, that the l=4 multipole has the
lowest energy, and that l=5 has the lowest ellipticity. We show how a mixed
multipolar field creates an observationally testable mismatch between the
principal axes of inertia (to be inferred from gravitational wave data) and the
magnetic inclination angle. Strong quadrupole and octupole components (with
amplitudes ~ 10^2 times higher than the dipole) in SGR 0418+5729 still yield
ellipticity ~ 10^{-8}, consistent with current gravitational wave upper limits.
The existence of higher multipoles in fast-rotating objects (e.g., newborn
magnetars) has interesting implications for the braking law and hence phase
tracking during coherent gravitational wave searches.
