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Crystal and magnetic structures of magnetic topological insulators MnBi$_2$Te$_4$ and MnBi$_4$Te$_7$

Using single crystal neutron diffraction, we present a systematic investigation of the crystal structure and magnetism of van der Waals topological insulators MnBi$_2$Te$_4$ and MnBi$_4$Te$_7$, where rich topological quantum states have been recently predicted and observed. Structural refinements reveal that considerable Bi atoms occupied on the Mn sites in both materials, distinct from the previously reported antisite disorder. We show unambiguously that MnBi$_{2}$Te$_{4}$ orders antiferromagnetically below 24 K featured by a magnetic symmetry $R_I$-${3c}$ while MnBi$_{4}$Te$_{7}$ is antiferromagnetic below 13 K with a magnetic space group $P_c$-${3c1}$. They both present antiferromagnetically coupled ferromagnetic layers with spins along the $c$-axis. We put forward a stacking rule for the crystal structure of an infinitely adaptive series MnBi$_{2n}$Te$_{3n+1}$ (n$\geq$1) with the building unit of [Bi$_2$Te$_3$]. A comparison of magnetic properties between MnBi$_{2}$Te$_{4}$ and MnBi$_{4}$Te$_{7}$, together with the recent density-functional theory calculations, enables us to draw that a two-dimensional magnetism limit might be realized in the derivatives. Our work may promote the theoretical studies of topological magnetic states in the series of MnBi$_{2n}$Te$_{3n+1}$.