  In the Al-Co-Cu alloy system, both the decagonal quasicrystal with the space
group of $P\overline{10}m2$ and its approximant Al$_{13}$Co$_4$ phase with
monoclinic $Cm$ symmetry are present around 20 at.% Co-10 at.% Cu. In this
study, we examined the crystallographic features of prepared Al-(30-x) at.%
Co-x at.% Cu samples mainly by transmission electron microscopy in order to
make clear the crystallographic relation between the decagonal quasicrystal and
the monoclinic Al$_{13}$Co$_4$ structure. The results revealed a coexistence
state consisting of decagonal quasicrystal and approximant Al$_{13}$Co$_4$
regions in Al-20 at.% Co-10 at.% Cu alloy samples. With the help of the
coexistence state, the orientation relationship was established between the
monoclinic Al$_{13}$Co$_4$ structure and the decagonal quasicrystal. In the
determined relationship, the crystallographic axis in the quasicrystal was
found to be parallel to the normal direction of the (010)$_{\rm m}$ plane in
the Al$_{13}$Co$_4$ structure, where the subscript m denotes the monoclinic
system. Based on data obtained experimentally, the state stability of the
decagonal quasicrystal was also examined in terms of the Hume-Rothery (HR)
mechanism on the basis of the nearly-free-electron approximation. It was found
that a model based on the HR mechanism could explain the crystallographic
features such as electron diffraction patterns and atomic arrangements found in
the decagonal quasicrystal. In other words, the HR mechanism is most likely
appropriate for the stability of the decagonal quasicrystal in the Al-Co-Cu
alloy system.
