  The structural properties of LaCu$_{6-x}$Au$_x$ have been studied using
neutron diffraction, x-ray diffraction, and heat capacity measurements. The
continuous orthorhombic-monoclinic structural phase transition in LaCu$_{6}$ is
suppressed linearly with Au substitution until a complete suppression of the
structural phase transition occurs at the critical composition, $x_{c}$ = 0.3.
Heat capacity measurements at low temperatures indicate residual structural
instability at $x_c$ that extends well into the orthorhombic phase. The
instability is ferroelastic in nature, with density functional theory (DFT)
calculations showing negligible coupling to electronic states near the Fermi
level. The data and calculations presented here are consistent with the zero
temperature termination of a continuous structural phase transition suggesting
that the LaCu$_{6-x}$Au$_x$ series hosts an elastic quantum critical point.
