Neutron powder diffraction study of NaMn$_{2}$O$_{4}$ and Li$_{0.92}$Mn$_{2}$O$_{4}$: Insights on spin-charge-orbital ordering

High-pressure synthesized quasi-one-dimensional NaMn$_{2}$O$_{4}$ and Li$_{0.92}$Mn$_{2}$O$_{4}$ are both antiferromagnetic insulators. Here their atomic and magnetic structures are investigated using neutron powder diffraction. The present crystal structural analyses of NaMn$_{2}$O$_{4}$ reveal that a Mn$^{3+}$/Mn$^{4+}$ charge-ordering state exists even at low temperature (down to 1.5 K). It is evident that one of the Mn sites shows a strongly distorted Mn$^{3+}$ octahedron due to the Jahn-Teller effect. Above TN=35 K, a two-dimensional short-range correlation is observed, as indicated by asymmetric diffuse scattering. Below TN, two antiferromagnetic transitions are observed: (i) a commensurate long-range Mn$^{3+}$ spin ordering below TN1=35 K and (ii) an incommensurate Mn$^{4+}$ spin ordering below TN2=11 K. Surprisingly, the two antiferromagnetic orders are found to be independent of each other. The commensurate magnetic structure (kC=0.5,0.5,0.5) follows the magnetic anisotropy of the local easy axes of Mn$^{3+}$, while the incommensurate Mn$^{4+}$ one shows a spin-density-wave or a cycloidal order with kIC=(0,0,0.216). For Li$_{0.92}$Mn$_{2}$O$_{4}$, on the other hand, the absence of a long-range spin-ordered state is confirmed down to 1.5 K.