  As opposed to the conventional, approximate theory of electrical conduction
in solids, which is based on energy band, quasi-particle states in infinite
lattices, a rigorous theory exists that can be used to explain transport
phenomena, in finite lattices, at reduced temperature, through the effects of a
broken gauge symmetry: The loss of translational invariance with respect to
Galilean transformations that maintain particle-particle separation.
Implications of this result in areas related to the transport of hydrogen (H)
and its isotopes in nano-crystalline structures of palladium (Pd) and of
neutral, coherent atomic waves in finite, optical lattices are presented. These
include: 1. The prediction of a novel variant of a known, phenomenon,
Zener/Electronic Breakdown in insulators, in which ions in nano-scale
palladium-deuteride (PdD) crystals (as opposed to electrons in insulating
crystals) that initially, effectively, are confined to particular regions of
space, begin to move, spread-out, and conduct charge after they are subjected
to an applied, external (constant) electric field for a sufficiently long
period of time; and 2. A rigorous treatment of scattering at low temperatures
that can be used to identify critical time- and length- scales for problems
related to the transport of neutral atomic matter waves in finite, optical
lattices, in the presence of gravitational fields.
  Keywords: Generalized Multiple Scattering, Atomic Bose Einstein Condensate,
Optical Lattice, Periodic Order, Bloch's Theorem, Broken Gauge Symmetry
