Solid State: Mean Square Lattice Strain

Itserpol

1. The problem statement, all variables and given/known data

Okay, this is from Kittel's Introduction to Solid State Physics (8th ed.) and it's driving me crazy.

The problem is: In the Debye approximation, consider [latex]\langle (\tfrac{\partial R}{\partial x})^2 \rangle =\tfrac{1}{2}\Sigma K^2u^2_0[/latex] as the mean square strain, and show that it is equal to [latex]\tfrac{\hbar \omega^2_DL}{4MNv^3}[/latex] for a line of N atoms each of mass M, counting longitudinal modes only.

2. Relevant equations

If there is anything relevant to this problem, I'm missing it.

3. The attempt at a solution

The solution manual says: [latex]\tfrac{1}{2}\Sigma K^2u^2_0=\tfrac{\hbar}{2MNv}\Sigma K=\tfrac{\hbar}{2MNv}(\tfrac{K_D^2}{2})=\tfrac{\hbar\omega_D^2}{4MNv^3}[/latex]

Now, the last step is obvious since [latex]K=\tfrac{\omega}{v}[/latex] is just the Debye approximation, but all of the preceding steps are like a wizard waving his hands. They make absolutely no sense, and I can't find anything in the book that could possibly lead me to this.

I don't mean to be so whiny, but this horrible text book combined with my professor's nearly indecipherable Chinese accent have made this one of the most frustrating courses I've ever been in. Please, if anyone out there can help me, I really need it.