Superexchange disruption and anomalous magnetic anisotropy in Fe_3-x Lu _x O₄ nanoparticles

Magnetite Fe₃O₄ nanoparticles are widely regarded as a prototypical ferrimagnetic spinel system; however, their magnetic response at the nanoscale increasingly reflects a non-ideal regime where exchange interactions, structural disorder, and surface-driven effects coexist in a strongly coupled manner. In this work, we investigate Fe_3-x Lu _x O₄ (0 ≤ x ≤ 0.3) nanoparticles to elucidate how non-magnetic rare-earth substitution modifies this coupled magnetostructural landscape. Using a combination of X-ray and neutron powder diffraction, ⁵⁷Fe Mössbauer spectroscopy, and magnetization measurements, we show that Lu³⁺ incorporation does not act as a simple magnetic diluent, but instead induces a restructuring of cation distribution, lattice strain, and local exchange topology. Lu³⁺ ions preferentially occupy octahedral B sites, leading to progressive disruption of J _AB superexchange connectivity and the emergence of locally decoupled A-site Fe³⁺ clusters. This disruption is accompanied by increased structural coherence length and enhanced magnetic inhomogeneity, reflected in the coexistence of blocked, superparamagnetic, and defect-associated spin populations. The magnetic response arises from the competition between exchange dilution, surface spin disorder, and strain-mediated anisotropy. Magnetometry reveals simultaneous suppression of saturation magnetization and enhancement of coercivity with increasing Lu content, indicating that magnetization reversal becomes increasingly governed by anisotropy barriers and pinning effects rather than coherent ferrimagnetic alignment. The effective anisotropy constant K ₁(T) exhibits strong deviations from Callen-Callen scaling, with anomalously low exponents (q = 1.26-1.83), highlighting the breakdown of bulk-like spin-orbit scaling under finite-size and surface-dominated conditions. Although K ₁(T) follows a Bloch-type behavior, consistent with spin-wave excitations, it is strongly renormalized by nanoscale disorder. These findings demonstrate that Lu substitution drives Fe₃O₄ nanoparticles into a non-ideal magnetic regime where ferrimagnetic order, surface spin frustration, and structural heterogeneity are inseparably intertwined through cation redistribution, exchange-path fragmentation, and surface-mediated anisotropy fluctuations.