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## Consequences

### Diffraction by a Lattice

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]).

### Bloch Theory

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.

### Bethe-Sommerfeld conjecture

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.

### Physical Results

Metals, Semi-conductors and Insulators
Since we still have to deal with a gas of independent electrons the same rules as in the Sommerfeld theory hold for the formation of the multi-electron state. But now in the process of filling up electron levels below a certain energy for the ground state two completely distinct situations can arise:
• All the bands are either completely occupied (valence bands) or completely empty (conduction bands).
• The highest occupied band (also called conduction band) is not completely filled up.
It's possible to give a semiclassical description of electronic transport extending the Drude transport equation and replacing 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.

Valencies
The description above also explains why only a certain fraction of the valence electrons of the free atom participate in electronic transport. Furthermore, since a filled band carries no current, a nearly filled band carries a current directed just opposite to the one carried by a nearly empty band. This enlightens the mistery of positive signs in the Hall transport.

Scaling of the Sommerfeld results
At room temperatures and below because of the similarity of the Fermi distribution with the zero-temperature case its possible to apply certain approximations in the calculations of e. g. the specific heat which just lead to a replacement of a factor by the density of states at the Fermi energy. This factor cancels out in the coefficient of the Wiedemann-Franz law.

Magnetoresistance etc.
The semiclassical transport theory for the independent electron gas also explains the deviations of measured magnetoresistance from the Sommerfeld result in many cases.

Next: Failures Up: Independent Electron Model Previous: Theory

Michael Gruber
Michael.Gruber@Uni-Augsburg.DE