Structural disorder and magnetic properties of NaNi$_{0.5}$Mn$_{0.5}$O$_{2}$ and LiNi$_{0.5}$Mn$_{0.5}$O$_{2}$.*

Magnetic properties of layered O(3) compounds LiNi$_{0.5}$Mn$_{0.5}$O$_{2}$ and NaNi$_{0.5}$Mn$_{0.5}$O$_{2}$ are studied using AC susceptibility and DC magnetization techniques in order to elucidate magnetic interactions within transition metal (TM) layers and between them in compounds with various TM distributions. In ideal layered NaNi$_{0.5}$Mn$_{0.5}$O$_{2}$, antiferromagnetic (AF) ordering transition at 60 K and a spin-flop transition at 5 K in the magnetic field of 2.2 T are found. Upon lost of Na, AF order changes with ferrimagnetic, which may be caused by Ni$^{2+}$ migration to the Na layer. LiNi$_{0.5}$Mn$_{0.5}$O$_{2}$ with flower or zigzag TM order show ferrimagnetic ordering at around 100 K, and significant magnetization hysteresis below this temperature, indicating presence of Ni$^{2+}$ in the Li layer. Magnetic interactions in all compounds are analyzed and models of spin order at low temperatures are proposed.