  The mechanical, thermal and vibrational properties of defective crystals are
important in many different contexts, from metallurgy and solid-state physics
to, more recently, soft matter and colloidal physics. Here we study two
different models of disordered fcc crystal lattices, with randomly-removed
bonds and with vacancies, respectively, within the framework of non-affine
lattice dynamics. We find that both systems feature the same scaling of the
shear modulus with the newly defined inversion-symmetry breaking (ISB)
parameter, which shows that local inversion-symmetry breaking around defects is
the universal root source of the non-affine softening of the shear modulus.
This finding allows us to derive analytical relations for the non-affine
(zero-frequency) shear modulus as a function of vacancy concentration in
excellent agreement with numerical simulations. Nevertheless, due to the
different microstructural disorder, the spatial fluctuations of the local ISB
parameter are different in the vacancy and bond-depleted case. The vacancy fcc
exhibits comparatively a more heterogenous microstructural disorder (due to the
broader distribution of coordination number $Z$), which is reflected in a
different scaling relation between boson peak frequency in the DOS and the
average $\bar{Z}$. These differences are less important at low vacancy
concentrations, where the numerical DOS of the vacancy fcc can be well
described theoretically by coherent-potential approximation, presented here for
the bond-depleted fcc lattice in 3d.
