  The lack of the theory of the crystalline electric field (CEF) in rare-earth
based quasicrystal (QC) and approximant crystal (AC) has prevented us from
understanding the electronic states. Recent success of the formulation of the
CEF theory on the basis of the point charge model has made it possible to
analyze the CEF microscopically. Here, by applying this formulation to the QC
Au-SM-Dy (SM=Si, Ge, Al, and Ga) and AC, we theoretically analyze the CEF. In
the Dy$^{3+}$ ion with $4f^9$ configuration, the CEF Hamiltonian is
diagonalized by the basis set for the total angular momentum $J=15/2$. The
ratio of the valences of the screened ligand ions $\alpha=Z_{\rm SM}/Z_{\rm
Au}$ plays an important role in characterizing the CEF ground state. For
$0\le\alpha<0.30$, the magnetic easy axis for the CEF ground state is shown to
be perpendicular to the mirror plane. On the other hand, for $\alpha>0.30$, the
magnetic easy axis is shown to be lying in the mirror plane and as $\alpha$
increases, the easy axis rotates to the clockwise direction in the mirror plane
at the Dy site and tends to approach the pseudo 5 fold axis. Possible relevance
of these results to experiments is discussed.
