Figure 98: Periodic distortion of a one dimensional lattice with lattice constant 2 a.bands), if there is an odd number of atoms per unit cell, the material will be a metal.
Fig. 99 shows dispersion relations of a material which has one electron per unit cell (band is half
filled). The left plot shows a periodic function (one dimensional lattice of atoms), the right plot a
periodic distortion. The dispersion relation of the normal periodic function looks like a parabola, the
dispersion relation on the right side shows a particular gap atk = 2 πa. The gap arises because the
energy on the left side ofk = 2 πa decreases, while the energy on the right side ofk = 2 πa increases
(fig. 100 ).
So all in all for small displacements the systemdecreasesthe energy because there is an increase of
the elastic energy (∆^2 ) and a decrease of the electronic energy with∆. The linear term always wins
for small displacements.
Figure 99: Material with one elctron per unit cell. left: peridoc function, right: periodic distortion of
a one dimensional lattice