Homogeneous to multi-scale inhomogeneous complex magnetism in Ba3LnRu2O9 (Ln = Gd, Ho), Revealed by muSR and Machine-Learned Muon-Site Calculations

We report a comprehensive muon spin rotation and relaxation (muSR) study of the hexagonal 6H-perovskites Ba3LnRu2O9 (Ln = Gd, Ho) to investigate the role of rare-earth anisotropy in the magnetic ground state of 4d-4f coupled systems. Ba3HoRu2O9 exhibits two successive magnetic orderings at TN1 = 46 K and TN2 = 10 K, leading to a complex magnetic ground state, whereas Ba3GdRu2O9, with Gd3+ (L = 0) S-state ions, shows primarily single magnetic ordering at TN = 14.5 K. Zero-field (ZF) and longitudinal-field (LF) muSR measurements reveal markedly different magnetic behavior despite their structural similarity. Ba3HoRu2O9 displays a multi-scale inhomogeneous magnetic state characterized by two-component exponential relaxation, broad local-field distributions, incomplete LF decoupling, and persistent spin dynamics. The muSR results identify two characteristic magnetic energy scales near 46 and 10 K, associated with correlated Ru-Ho interactions and quasistatic magnetic ordering, respectively. Redfield analysis further demonstrates strong local magnetic fields and distributed spin fluctuation timescales, confirming magnetic inhomogeneity. Machine-learning-assisted muon-site and dipolar-field calculations show that the broad field distribution naturally arises from muons sensing distinct coexisting magnetic domains at different structural sites. In contrast, Ba3GdRu2O9 exhibits a comparatively homogeneous antiferromagnetic ground state below 14.5 K, with Gaussian relaxation consistent with a narrow static field distribution. These results establish that the rare-earth ion in Ba3LnRu2O9 is a powerful tuning parameter, driving the ground state from homogeneous to complex, multi-scale inhomogeneous behavior through competing 4d-4f magnetic interactions.