Energy difference, optic and acoustic branch

[Cheetox]

Homework Statement

Suggest how the energy difference arises between the optic and acoustic branches in a 1d diatomic chain.

Homework Equations

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.

 

[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.