  We consider multi-dimensional Schr\"odinger operators with a weak random
perturbation distributed in the cells of some periodic lattice. In every cell
the perturbation is described by the translate of a fixed abstract operator
depending on a random variable. The random variables, indexed by the lattice,
are assumed to be independent and identically distributed according to an
absolutely continuous probability density. A small global coupling constant
tunes the strength of the perturbation. We treat analogous random Hamiltonians
defined on multi-dimensional layers, as well. For such models we determine the
location of the almost sure spectrum and its dependence on the global coupling
constant. In this paper we concentrate on the case that the spectrum expands
when the perturbation is switched on. Furthermore, we derive a Wegner estimate
and an initial length scale estimate, which together with Combes--Thomas
estimate allows to invoke the multi-scale analysis proof of localization. We
specify an energy region, including the bottom of the almost sure spectrum,
which exhibits spectral and dynamical localization. Due to our treatment of
general, abstract perturbations our results apply at once to many interesting
examples both known and new.
