  We perturb the SC, BCC, and FCC crystal structures with a spatial Gaussian
noise whose adimensional strength is controlled by the parameter a, and analyze
the topological and metrical properties of the resulting Voronoi Tessellations
(VT). The topological properties of the VT of the SC and FCC crystals are
unstable with respect to the introduction of noise, because the corresponding
polyhedra are geometrically degenerate, whereas the tessellation of the BCC
crystal is topologically stable even against noise of small but finite
intensity. For weak noise, the mean area of the perturbed BCC and FCC crystals
VT increases quadratically with a. In the case of perturbed SCC crystals, there
is an optimal amount of noise that minimizes the mean area of the cells.
Already for a moderate noise (a>0.5), the properties of the three perturbed VT
are indistinguishable, and for intense noise (a>2), results converge to the
Poisson-VT limit. Notably, 2-parameter gamma distributions are an excellent
model for the empirical of of all considered properties. The VT of the
perturbed BCC and FCC structures are local maxima for the isoperimetric
quotient, which measures the degre of sphericity of the cells, among space
filling VT. In the BCC case, this suggests a weaker form of the recentluy
disproved Kelvin conjecture. Due to the fluctuations of the shape of the cells,
anomalous scalings with exponents >3/2 is observed between the area and the
volumes of the cells, and, except for the FCC case, also for a->0. In the
Poisson-VT limit, the exponent is about 1.67. As the number of faces is
positively correlated with the sphericity of the cells, the anomalous scaling
is heavily reduced when we perform powerlaw fits separately on cells with a
specific number of faces.
