Long-range magnetic order in quasi-one-dimensional NaCrSi$_2$O$_6$ and NaCrGe$_2$O$_6$ metal oxides

Formation of a long-range magnetic order is observed at low temperatures in NaCrSi$_2$O$_6$ and NaCrGe$_2$O$_6$ quasi-one-dimensional metal oxide compounds with a pyroxene structure. The first of these compounds, NaCrSi$_2$O$_6$, is an antiferromagnet with the Neel temperature $T_N=3$ K, while the second, NaCrGe$_2$O$_6$, is a ferromagnet with the Curie temperature $T_C=6$ K. From the measurements of magnetization and specific heat of these compounds, the main parameters of their magnetic subsystems are determined. In NaCrSi$_2$O$_6$, a spin-flip transition is observed. A change in the type of magnetic order that accompanies the replacement of Si by Ge can be attributed to a change in the parameters of the competing direct antiferromagnetic Cr-Cr and indirect ferromagnetic Cr-O-Cr interactions in isolated chains of CrO$_6$ octahedra.