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The interparticle interaction and crossover in critical lines on field-temperature plane in Pr$_{0.5}$Sr$_{0.5}$MnO$_{3}$ nanoparticles

The magnetic properties and the effects of interparticle interaction on it have been studied in nanoparticles of half doped Pr$_{0.5}$Sr$_{0.5}$MnO$_{3}$. Three samples consisting of nanoparticles of different average particle sizes are synthesized to render the variation in interparticle interaction. Though all the samples crystallize in the same structure to that of their bulk compound, the low temperature ferromagnetic-antiferromagnetic transition, which is present in bulk compound, is not evident in the nanoparticles. Linear as well as nonlinear ac susceptibility coupled with dc magnetic measurements have shown the superparamagnetic behavior of these nanoparticles where the blocking temperature increases with the increasing particle size. Presence of interparticle interaction is confirmed from the temperature variation of coercive field and the analysis of frequency dependent ac susceptibility. We have identified the nature of this interaction to be of dipolar type, and show that its strength decreases with the increasing particle size. The effect of this dipolar interaction on magnetic properties is intriguing as the compounds exhibit crossover from de Almeida-Thouless to Gabay-Toulouse like critical lines on field-temperature plane above their respective interaction field. In agreement with theoretical prediction, we infer that this crossover is induced by the unidirectional anisotropy arising from interparticle interaction, and this is confirmed from the presence of exchange bias phenomenon.