A single integral expression for the magnetisation of a textured superparamagnetic system

A superparamagnetic system consists of many single domain particles with relaxation times smaller than the time of measurement (t=100 s), with the result that thermal agitation leads to zero magnetisation in zero field,M(H=0)=0 for t>100 s. For particles in a fluid, an external field will result in a magnetisation given by the Langevin function as inparamagnetism. However, particles fixed into position in the presence of a field, obtain an orientational texture. We still find M(H=0)=0, but the magnetisation is not given by the Langevin function. The model of Chantrell et al. can yield an expression for M capable of including texture. This requires the calculation of the partition function of the system which is a double integral in the coordinate system used. Here we reduce this double integral to a single integral.