  The local structures of Zn$_{1-x}$Mg$_x$O alloys have been studied by Raman
spectroscopy and by synchrotron x-ray pair distribution function (PDF)
analysis. Within the solid solution range ($0\leq{x}\leq{0.15}$) of
Zn$_{1-x}$Mg$_x$O, the wurtzite framework is maintained with Mg homogeneously
distributed throughout the wurtzite lattice. The $E_2^\mathrm{high}$ Raman line
of Zn$_{1-x}$Mg$_x$O displays systematic changes in response to the evolution
of the crystal lattice upon the Mg-substitution. The red-shift and broadening
of the $E_2^\mathrm{high}$ mode are explained by the expansion of hexagonal
$ab$-dimensions, and compositional disorder of Zn/Mg, respectively. Synchrotron
x-ray PDF analyses of Zn$_{1-x}$Mg$_x$O reveal that the Mg atoms have a
slightly reduced wurtzite parameter $u$ and more regular tetrahedral bond
distances than the Zn atoms. For both Zn and Mg, the internal tetrahedral
geometries are independent of the alloy composition.
