  Two-dimensional van der Waals MnBi$_{2n}$Te$_{3n+1}$ (n = 1, 2, 3, 4)
compounds have been recently found to be intrinsic magnetic topological
insulators rendering quantum anomalous Hall effect and diverse topological
states. Here, we summarize and compare the crystal and magnetic structures of
this family, and discuss the effects of chemical composition on their
magnetism. We found that a considerable fraction of Bi occupies at the Mn sites
in MnBi$_{2n}$Te$_{3n+1}$ (n = 1, 2, 3, 4) while Mn is no detectable at the
non-magnetic atomic sites within the resolution of neutron diffraction
experiments. The occupancy of Mn monotonically decreases with the increase of
n. The polarized neutron diffraction on the representative MnBi$_{4}$Te$_{7}$
reveals that its magnetization density is exclusively accumulated at the Mn
site, in good agreement with the results from the unpolarized neutron
diffraction. The defects of Bi at the Mn site naturally explain the
continuously reduced saturated magnetic moments from n = 1 to n = 4. The
experimentally estimated critical exponents of all the compounds generally
suggest a three-dimensional character of magnetism. Our work provides
material-specified structural parameters that may be useful for band structure
calculations to understand the observed topological surface states and for
designing quantum magnetic materials through chemical doping.

Strongly Correlated Electrons;Materials Science