It turns out that in most calculations concerned with momenta a reciprocal lattice 
plays a fundamental rôle. It is defined by
A rough analysis of the conditions for constructive interference shows that an incoming wave with wave vector 
can be diffracted into an outgoing wave with wave vector 
(with same modulus, i. e. with same energy) if and only if there's a reciprocal lattice vector 
such that
This allows to determine the position of the peaks in X-ray diffraction patterns in excellent agreement with experimental data.
In addition it shows that the vectors in the reciprocal lattice have something to do with momentum which can be carried by the crystal itself though still working in the framework of a static lattice model.
Beyond that the Laue formulation of the diffraction condition shows up again in Skriganov's proof of the Bethe-Sommerfeld conjecture (see [3]).
Define the Wigner-Seitz cell or Dirichlet fundamental domain of the direct lattice by
Let 
be the Dirichlet fundamental domain of the reciprocal lattice (first Brillouin zone).
Bloch theory states that there's a direct integral decomposition
with respect to Lebesgue measure and a corresponding decomposition of the periodic Schrödinger operator
such that the decomposed operators have compact resolvent. The corresponding ordered multiplicity-counted eigenvalues 
(band functions) are continous, piece-wise differentiable functions on 
which extend continously, 
-periodic to all of 
. Furthermore we have
with the bands 
. The components of 
are called gaps.
A mathematically and physically very important quantity is the integrated densitiy of states which is defined by means of the level counting functions
through
and reduces in the case of the finite crystal just to a multiple of the ordinary level counting function for the whole Schrödinger operator.
The formalism of band theory settles a natural question: Are there infinitely many gaps (which obviously cannot accumulate and therefore would be located at arbitrary high energies) or is there a bound above which there are no gaps (i. e. no forbidden regions) which one would expect classically? The conjectured answer is contributed to Bethe and Sommerfeld ([2]): In dimensions two and above there are only finitely many gaps. This conjecture has now been proven along the lines of the ideas due to Bethe and Sommerfeld in a series of papers by M. M. Skriganov; for references see [3].
In dimension one there are infinitely many gaps for generic periodic potentials. The exceptional potentials are special functions characterized by the number and location of gaps.
by 
. One result is that electronic transport can be carried only by partially filled bands. So in the second situation described above we can expect the solid to be a conducting metal. But in the first case we clearly have an insulating material at zero temperature. At finite temperature because of the fermi function being smeared out (see fig. 1) a spectral region of half width 
centered at 
controls the transport properties: If the gap between the two bands just below and above the Fermi energy is considerably larger then 
, then no thermally activated transport can occur, and the material remains an insulator. If on the other hand the gap width is comparable to or less than 
then the conductivity is non-neglibile and increasing with increasing temperature, which is the case of an (intrinsic) semi-conductor.
by the density of states 
at the Fermi energy. This factor cancels out in the coefficient of the Wiedemann-Franz law.
