Along with having vibrational energy, do both crystalline and amorphous solids also have translational energy?

I ask because I've always understood solids to have just vibrational motion/energy. But I've heard it said that, along with having vibrational energy, solids also have translational energy (because temperature is defined as a measure of the average molecular translational energy in a system). If this is true, how can solids have translational energy if they don't have translational motion?

## Best Answer

The same way any substance can have transnational energy.

It arises because the energy of motion is proportional to the square of the velocity so even if the average is zero, the average of the square is not.

If the mean velocity of the particles is $\bar v$ then the instantaneous transnational energy would be:

$$\frac12 m\sum_i (v_i-\bar v)^2$$

$$\frac12 m\sum_i (v_i^2 -2v_i\,\bar v + \bar v^2)$$

$$\frac12 m \sum_i v_i^2- m \sum_i v_i\,\bar v + \frac12 \, n \, m \, \bar v^2$$

$$\frac12 m \sum_i v_i^2- m \, n \, \bar v^2 + \frac12 \, n \, m \, \bar v^2$$

$$\frac12 m \sum_i v_i^2- \frac12 m \, n \, \bar v^2$$

So this is the kinetic energy of all the particles minus the kinetic energy of the bulk movement. Even when the bulk movement is zero there's still the vibrational velocity that contributes to the energy.