# Difference between Phonons and heat?

1, if these two were the different then how we differentiate with one another?. 2, if these two were the same, then what is really vibrating?, atom in the lattice or electron in the atom or the bond between two atoms or an entire medium(sound)? The main problem is to find frequency of the phonon in a solid. many people used dispersion relation. but the omega symbol is mainly used for angular frequency. and the unit of dispersion relation is one over centimetre. which means it is might be wave number k. there are so many things confusing me to find the frequency of phonons in metal.

sorry for my english it's my second language.

I will try to clear up your confusion step by step.

First of all: Your main question.

Phonons describe a collective excitation of atoms/molecules in solids. Think of vibrations in a material. They can be treated like particles which is why phonons are often called quasiparticles.

Heat is a form of energy which is an entirely different quantity. You are comparing apples and oranges when you ask whether phonons and heat are the same thing. Usually, phonons strongly influence the heat capacity in a solid but they are not identical.

Now, let's get to the sub-questions.

Heat is energy stored in the (random) motion of particles. In a solid, the important particles are free electrons and the ions they leave behind. You could consider the motion of each ion separately, but it is usually more convenient to consider a collective motion of the ions at the same frequency. That's a phonon. That's also why there's an $\omega$ in the dispersion relation. That $\omega$ is an angular frequency --- the frequency the ions are oscillating at.
Phonons can be thought of as waves, and the dispersion relation tells you the relation between the wavevector (which can have units of $\left[cm\right]^{-1}$) of the wave and its frequency. In general, these relations can be quite complicated (especially compared to many waves in undergraduate physics, such as light in a vacuum, which has a linear dispersion relation $\omega = c k$). A good starting point is to understand the dispersion relation for a one-dimensional lattice. (For example, see the wikipedia page for phonons.) It shows an example of a non-linear dispersion relation $\omega \propto \sqrt{1 - \cos{k a}}$, where $a$ is the spacing between the atoms in the lattice.