We report polarized- and unpolarized- neutron inelastic scattering measurements of the magnetic excitation spectrum in the spin-charge ordered phase of La3/2Sr1/2NiO4. Up to energies of ~30 meV we observe broad magnetic modes characteristic of a near checkerboard ordering. A linear spin-wave model for an ideal checkerboard ordering with a single antiferromagnetic exchange interaction J = 5.8 +/- 0.5 meV between next-nearest-neighbour spins on Ni2+ sites, together with a small XY-like single-ion anisotropy, provides a reasonable description of the measured dispersion. Above 30 meV the excitations are not fully consistent with the linear spin-wave model, with modes near the two-dimensional reciprocal space wavevector (0.5,0.5) having an anomalously large intensity. Furthermore, two additional dispersive modes not predicted by spin wave theory were observed, both of which are probably magnetic. One disperses away from (0.5,0.5) in the energy range between 50-56 meV, and the other appears around (h,k) type positions (h,k = integer) in the energy range 31-39 meV. We propose a model in which these anomalous features are explained by the existence of discommensurations in the checkerboard ordering. At low energies there is additional diffuse scattering centred on the magnetic ordering wavevector. We associate this diffuse scattering with dynamic antiferromagnetic correlations between spins attached to the doped holes.
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