# Homework Help: Energy difference, optic and acoustic branch

1. May 10, 2010

### Cheetox

1. The problem statement, all variables and given/known data
Suggest how the energy diﬀerence arises between the optic and acoustic branches in a 1d diatomic chain.

2. Relevant equations

3. The attempt at a solution
I know that on the optic branch atoms oscillate in anti-phase within the unit cell and in phase in the unit cell on the acoustic branch. So does the energy difference come from the difference in energy between the symmetric and anti-symmetric wavefunctions required to describe the two situations? or does the extra energy in the optic branch come from the energy contained within the oscillating dipole that will be set up, or are these two answers intrinsically linked? any help much appreciated.

2. May 10, 2010

### nickjer

You are close. In the acoustic mode (and long wavelength limit), they all oscillate in phase. So the invisible springs attached to them never stretch or compress. So if you thought of potential energy of the springs as $V(x) = \tfrac12 k x^2$, then there would be no potential energy.

For the optical mode (again long wavelength), the all oscillate out of phase with respect to their neighbors so potential energy is a maximum in the spings. So there is an energy difference between the two phonon modes.