A shape dependent modified curie-weiss equation for finite-size magnetic nanomaterials

Finite-size effects fundamentally reshape the magnetic properties of nanomaterials,where conventional bulk-based models such as the Curie-Weiss law fail to capture surface- andshape-dependent phenomena. Here we develop a modified Curie-Weiss framework that explicitly incorporates finite size, surface-to-volume ratios, and shape anisotropy, enabling predictive descriptions of nanoscale magnetic responses. By deriving analytical expressions for spherical and cubic nanoparticles, we quantify the fraction of surface atoms (????) as a function of particle dimensions, establishing that ???? = 3????/???? for spheres and ???? = 6????/???? for cubes, with ???? representing surface shell thickness. These expressions reveal that cubic nanostructures exhibit larger surface fractions than spheres of comparable size, amplifying disordered spin states and suppressing effective Curie temperatures. The generalized susceptibility expression integrates competing contributions from ordered cores and disordered surfaces, capturing experimentally observed tworegime magnetic behaviour in Eu2MnHfO6 nanoparticles. Beyond spheres and cubes, this framework provides a foundation for extending Curie-Weiss theory to anisotropic nanoparticle morphologies, highlighting morphology as a tuneable parameter for controlling coercivity, relaxation dynamics, and blocking phenomena. Our results establish a universal shape-dependent modification of Curie-Weiss theory, bridging classical magnetism with nanoscale quantum and surface effects, and opening new pathways for rationally designing functional magnetic nanomaterials.

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