\textbf{Geometric Magnetic Frustration in Li}$_{\mathrm{\mathbf{3}}}$\textbf{Mg}$_{\mathrm{\mathbf{2}}}$\textbf{OsO}$_{\mathrm{\mathbf{6}}}$\textbf{ Studied with Muon Spin Relaxation}

Geometric frustration manifests when the spatial arrangement of ions inhibits magnetic order. Typically associated with antiferromagnetically (AF)-correlated moments on triangular or tetrahedral lattices, frustration occurs in a variety of structures and systems, resulting in rich phase diagrams and exotic ground states. As a window to exotic physics revealed by the cancellation of normally dominant interactions, the research community has taken great interest in frustrated systems. One family of recent interest are the rock-salt ordered oxides A$_{\mathrm{5}}$BO$_{\mathrm{6}}$, in which the B sites are occupied by magnetic ions comprising a network of interlocked tetrahedra, and nonmagnetic ions on the A sites control the B oxidation state through charge neutrality. Here we will discuss studies of Li$_{\mathrm{3}}$Mg$_{\mathrm{2}}$OsO$_{\mathrm{6}}$ using muon spin relaxation ($\mu $SR), a highly sensitive local probe of magnetism. Previous studies of this family included Li$_{\mathrm{5}}$OsO$_{\mathrm{6}}$, which exhibits AF order below 50K with minimal evidence for frustration, and Li$_{\mathrm{4}}$MgReO$_{\mathrm{6}}$, which exhibits glassy magnetism. Li$_{\mathrm{3}}$Mg$_{\mathrm{2}}$RuO$_{\mathrm{6}}$, meanwhile, exhibits long-range AF, with the ordering temperature suppressed by frustration. But its isoelectronic twin, Li$_{\mathrm{3}}$Mg$_{\mathrm{2}}$OsO$_{\mathrm{6}}$ (5$d^{\mathrm{3}}$ vs. 4$d^{\mathrm{3}})$ exhibits very different behavior, revealed by $\mu $SR to be a glassy ground state below 12K. Understanding why such similar systems exhibit diverse ground-state behavior is key to understanding the nature of geometric magnetic frustration.