Solid state physics: phonon density of state

[Pistike]

Homework Statement

There is a Ar cube crystal with Lenard jones potential: V$^{L.J.}$=-$\alpha$/r$^{6}$+$\beta$/r$^{12}$

We know the volume V, the parameters of the L.J. potential, and the mass of the Ar atom (m$_{Ar}$).
Determine the phonon density of state. (D(ω)=?)

Homework Equations

D(ω)=Ʃδ(ω-ω$_{i}$)

The Attempt at a Solution

I have no idea what to do with the given lenard jones potential, and its parameters. How should I calculate the dispersion relation?

 

[mark726]

Thank you for your post. I understand your confusion about how to calculate the phonon density of state for the given system. Let me help guide you through the process.

First, we need to understand that the Lenard Jones potential is a model used to describe the interaction between two atoms in a crystal lattice. It takes into account the repulsive and attractive forces between the atoms.

To calculate the phonon density of state, we need to know the dispersion relation, which is the relationship between the frequency (ω) and the wave vector (k) for phonons in the crystal. This can be obtained by solving the equations of motion for the atoms in the crystal using the given Lenard Jones potential.

Once we have the dispersion relation, we can use the equation D(ω)=Ʃδ(ω-ω_{i}) to calculate the phonon density of state. This equation basically sums up the contribution of all phonon modes with frequency ω to the total density of states. The δ(ω-ω_{i}) term represents the contribution of a single phonon mode with frequency ω_{i}.

I hope this helps you understand the steps involved in calculating the phonon density of state. If you have any further questions, please feel free to ask. Good luck!