  Under sufficient permanent random covalent bonding, a fluid of atoms or small
molecules is transformed into an amorphous solid network. Being amorphous,
local structural properties in such networks vary across the sample. A natural
order parameter, resulting from a statistical-mechanical approach, captures
information concerning this heterogeneity via a certain joint probability
distribution. This joint probability distribution describes the variations in
the positional and orientational localization of the particles, reflecting the
random environments experienced by them, as well as further information
characterizing the thermal motion of particles. A complete solution, valid in
the vicinity of the amorphous solidification transition, is constructed
essentially analytically for the amorphous solid order parameter, in the
context of the random network model and approach introduced by Goldbart and
Zippelius [Europhys. Lett. 27, 599 (1994)]. Knowledge of this order parameter
allows us to draw certain conclusions about the stucture and heterogeneity of
randomly covalently bonded atomic or molecular network solids in the vicinity
of the amorphous solidification transition. Inter alia, the positional aspects
of particle localization are established to have precisely the structure
obtained perviously in the context of vulcanized media, and results are found
for the analogue of the spin glass order parameter describing the orientational
freezing of the bonds between particles.
