Crystal Vibrations: Impurity in a monoatomic chain

[Perses]

Homework Statement

Hello everyone, I'm a bit stuck on this particular problem and i was hoping somebody could give me a couple tips or hints!

"consider a linear monoatomic chain where all atoms have a mass M except one which has a small mass m. The force constant is C."
First I'm asked to write the equation of motion for the relevant atoms. Then, I'm asked to solve it assuming a displacement function for atom n in the form of

Un = Aexp(-ax)exp(i((omega)t - Kx))

Let the chain be:

vs-2 -- vs-1 --- u --- vs --- vs+1

where u is the impurity

The Attempt at a Solution

I'm currently lost at the beginning of the problem, finding the equation of motion. What i first thought was to treat this almost like it were the case of a diatomic chain.

Mdvs/dt = C(vs+1 + u - 2vs)

mdu/dt = C(vs-1 + vs - 2u)

however, upon continuing in the problem I'm not sure that this is correct. Could anyone offer any advice as to how i should start this problem?

 

[Gokul43201]

Your notation is a little distracting. Simply call the displacement of the n'th atom (from its equilibrium position) $u_n$. Let the impurity be at n=k.

Secondly. you've made a mistake in constructing your equation of motion, which is a second order linear DE (not first order).

See that you should get something more like :

$$m_n \ddot{u_n} = c(u_{n+1} -2u_n+u_{n-1})$$

From here, I'm not sure what's the best approach. One possibility is to Fourier transform to k-space and hope the equations become decoupled. Odds are they will. Also, you will likely find that the solution in k-space is the FT of the localized wavepacket provided to you.