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I have to calculate the following:
a) The speed of the acoustic phonons
b) Which phonons(frequency and wavevector) can be observed for scattering angles between 0° and 180° and how much of the Brillouin zone they cover.

The problem statement is as follows: Light from an Argon laser($\lambda=514.5nm$) is scattered by a NaCl crystal(refraction index=1.544). The scattered light is analyzed perpendicular to the beam direction(scattering angle $\theta=90°$). Two pais of lines are observed, the frequency displacement of this lines w.r.t the lasers frequency is $\Delta\nu=19.26GHz$ and $\Delta\nu=10.25GHz$

For part a) I did the following:
because the wavenumber of the incident and scattered radiation is much smaller than the 1BZ the additive reciprocal lattice vector must be zero.
Let k be the phonon wavevector and q the laser wavevector.

From a "scattering diagram" we get the following:

$k=\frac{2.n.\omega.sin(\theta/2)}{c}$

Where $n$ is the refraction index, $\omega$ the angular frequency and $c$ the speed of light.

Using conservation of energy and the Debye approximation$\omega(k)=c_s.k$:

$\Delta \omega=c_s.n.\omega/c.sin(\theta/2)$

$\omega=2.\pi.\nu$ Solve for $c_s$ and plug in the values of $\Delta\omega$

For part b) I really dont know where to start, at first I thought about using the Bragg's condition but I don't see how it can make things easier. Im stuck. I would appreciate some help.

Cheers!

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