  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.
