  We present a comprehensive study of the crystal structure of the thin-film,
ferromagnetic topological insulator (Bi, Sb)$_{2-x}$V$_x$Te$_3$. The
dissipationless quantum anomalous Hall edge states it manifests are of
particular interest for spintronics, as a natural spin filter or pure spin
source, and as qubits for topological quantum computing. For ranges typically
used in experiments, we investigate the effect of doping, substrate choice and
film thickness on the (Bi, Sb)$_2$Te$_3$ unit cell using high-resolution X-ray
diffractometry. Scanning transmission electron microscopy and energy-dispersive
X-ray spectroscopy measurements provide local structural and interfacial
information. We find that the unit cell is unaffected in-plane by vanadium
doping changes, and remains unchanged over a thickness range of 4--10 quintuple
layers (1 QL $\approx$ 1 nm). The in-plane lattice parameter ($a$) also remains
the same in films grown on different substrate materials. However, out-of-plane
the $c$-axis is reduced in films grown on less closely lattice-matched
substrates, and increases with the doping level.
