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 ).
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
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 (): 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 .
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.