**Peierls transition** Peierls' Theorem: a one-dimensional equally spaced chain with one electron per ion is unstable.

When a lattice distortion causes the period to double from a to 2a, new band gaps will be introduced at multiples of k=pi/(2a).

Around k=pi/(2a), where the new band gap had formed, the band energy of the lowest energy band is lowered compared to the undimerised system without the band gap. These electrons with a k around pi/(2a) have lowered their energy and this is the souce of the Peierls effect.

The lattice distortion become energetically favourable when the energy savings due to the new band gaps outweigh the elastic energy cost of rearranging the ions. Therefore the Peierls effect will be noticed when the electrons are arranged close to their ground state, i.e. when thermal excitation is low, i.e. at low temperatures.

__Source:__ http://en.wikipedia.org/wiki/Peierls_transition In the following papers Peierls and Frohlich showed that a one-dimensional electron system in a deformable lattice is unstable against a modulation of the lattice of wavenumber 2kF (kF is the Fermi wavenumber):

Peierls, R. E., 1955, Quantum Theory of Solids (London: Oxford University Press), p.108;

Frohlich, H., 1954, Proc. R. Soc. A, 223, 296.