If we want to retain it, we will have to translate it back into the new 1 st Brillouin Zone. At first glance, this seems to mean we have lost the information contained in the region. the size of the Brillouin Zone is halved). Any normal mode can be represented by a wavevector in the first Brillouin Zone.įor a diatomic chain, the repeat distance is doubled in real space, so it is halved in k-space (i.e. The nth Brillouin Zone is the set of points reached from the origin in reciprocal space by crossing n-1 Bragg planes and no fewer.Įach Brillouin Zone has the same volume. The first Brillouin Zone is the set of points closer to the origin in reciprocal space than any other reciprocal lattice point. For that reason, it also helps to remember this alternate definition. The standard definition of the BZ is that it is “the Wigner-Seitz unit cell in reciprocal space”, but there’s no point just quoting this if you don’t understand (or can’t readily explain) what it means. Rule of thumb- number of modulations = number of further neighbour interactions considered.īrillouin Zones: all the physics of a system is contained within a Brillouin Zone. Going back to nearest neighbours, try a solution of the form If we wanted to include more distant neighbour interactions (up to the Nth), our equation at this point would have the form In fact, having done all that, we could have just started from this point by remembering Hooke’s Law F=Cx (where x is displacement from equilibrium).Īside: just to check that we aren’t fiddling our force constants to make everything neat, we can use Newton III to show that C n=C -n Positions r n, equilibrium atomic separation r n-r n-1=aĪlso note that Φ’’(a) can be written as the force constant, C. One-dimensional monatomic chain: we’ll start by only including nearest neighbour forces. Harmonic approximations- terms of order higher than u 2 in the interatomic potential are neglected.
Assume the amplitude of the vibrations is small.Adiabatic approximation- assume electrons are attached rigidly to the nucleus.