Structure of Short Range Magnetic Order in the half-doped layered perovskite La$_{1.5}$Sr$_{0.5}$CoO$_{4}$

We have investigated the short range magnetic ordering in La$_{1.5}$Sr$_{0.5}$CoO$_{4}$ by elastic neutron scattering. The magnetic scattering is diffuse, with broad peaks characteristic of short range correlations. Such scattering is often described by replacing the delta-functions in the Bragg scattering cross- section by the corresponding multi-dimensional Lorentzians. However, this approach is not quite satisfactory. First, because it extends the small-q Orstein-Zernike approximation beyond the region of its validity and second, because the multi-dimensional Lorentzians cannot be properly normalized. On the other hand, in the case of a disorder introduced by the linear topological defects such as magnetic disclinations, the scattering is described by a simple expression in the form of the factorized ``lattice Lorentzians'' (I. A. Zaliznyak, S.-H. Lee, in Y. Zhu (Ed.), Modern Techniques for Characterizing Magnetic Materials, Kluwer Academic, New York (2005)).We compare our data with both expressions and show that it is inconsistent with a system of linear magnetic disclinations.