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.
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