# Optical and acoustic modes

• aaaa202
In summary, the crystal needs to have more than atom per primitive basis in order for optical and acoustical modes to happen.

#### aaaa202

Basically phonons in crystals can either be acoustic or optical according to my book. But then it stresses that a necessary condition for this to hold is that the crystal has more than atom per primitive basis. The optical modes are then when neighbouring atoms oscillate out of phase (i.e. no motion of the center of mass) and the acoustical mode is the exact opposite. But I don't understand why it is so crucial that the crystal has two or more atoms per primitive basis. Can't optical and acoustical modes happen between neighbouring atoms regardless of this?

If there is only one atom per unit cell, how many degrees of freedom are available?

The unit cell is the repeating unit of a crystal; you should be able to define everything that occurs in the crystal in terms of the unit cell - this is especially convenient if you move to the reciprocal space.

I don't understand what you are hinting at :( I know the unit cell is the smallest repetitive unit. But what does have to do with the degrees of freedom? Maybe you can point to a drawing which shows what you are hinting at.
I am thinking if there is two atoms per unit cell they can oscillate relative to each other, which is a degree of freedom which is not there if there is only one atom per unit cell? But then why can't an atom just move relative to another atom in the neighbouring unit cell?

It can. You still can have an acoustic mode, but not the optic one(s).

why can't u have an optic mode, i.e. neighbouring atoms oscillating in opposite phase wrt each other?

aaaa202 said:
I know the unit cell is the smallest repetitive unit. But what does have to do with the degrees of freedom?

A perfect crystal contains a vast number of unit cells - think Avogadro's number for its gram molecular weight - and when you apply an impulse to the crystal, the rigidity of the system makes it felt everywhere.

If one unit cell responds with a particular motion, why should its immediate neighbors respond in some other way? Being identical means that they should have the same response ... they are physically the same.

The the degrees of freedom of the unit cell tell you what responses are possible.

This should be clear from a careful reading of an introductory text like Kittel.

aaaa202 said:
why can't u have an optic mode, i.e. neighbouring atoms oscillating in opposite phase wrt each other?

optical modes require atoms to be vibrating out of phase with each other along with different masses/charges to result in optical modes in a crystal.

Assume a crystal with one atom per primitive cell. It has only acoustic modes. The wavelength of the phonon determines the phase difference between atoms in the neighbouring cells. If the wavelength is twice the lattice parameter, the neighbours move out of phase. See animated picture of http://en.wikipedia.org/wiki/Phonon.

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okay I think I get it. But why is it then that the energy of the optical phonons is much greater than the energy of the acoustical ones?

You can consider the optical modes as internal vibrations of the molecules forming the basic unit in the primitive cell. The force constants between the atoms of a molecule are usually considerably stronger than the intermolecular forces. At the Brillouin zone boundary, however, the gap between the acoustic and optical mode frequencies/energies can be quite narrow sometimes.

ehild