  The use of highly efficient and solarblind GaN photocathodes as part of
multichannel plate UV detectors for applications in astronomy would strongly
benefit from the direct growth of GaN on typical window materials with high
transmission down to the deep UV range. GaN growth on MgF$_2$ substrates by
plasma-assisted molecular beam epitaxy has recently been demonstrated. Here, we
report an extensive scanning transmission electron microscopy study of the thin
film microstructure for growth at 525 $^{\circ}$C and 650 $^{\circ}$C on (100)
MgF$_2$. These results are systematically supported by X-ray diffraction
reciprocal space maps. For both growth temperatures predominant cubic (111),
(115) and (110) GaN is found with no preferred nucleation on the substrate and
typical grain sizes of 100-200 nm. All observed orientations can be understood
as the result of first and second order twins on different $\{111\}$ planes
related to the underlying substrate. The higher growth temperature shows a
strongly increased twin density along with a higher surface roughness.
Furthermore, grains with cubic (110) GaN growth show a reduced density and a
reduced size of about 20 nm. Furthermore, in-diffusion of Mg and F into the GaN
is observed, which is accompanied by the formation of cavities in the MgF$_2$
directly at the interface.
