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Another Scattering Question

So I have this Bravais Lattice of sites R vibrating with some normal mode with a small displacement amplitude $u_o$, some wave vector k and some frequency $\omega$. We can clearly describe $u(R)=u_o \cos(k R - \omega t)$. If we scatter a beam of radiation from the lattice with a change of wave vector q, the amplitude at the detector is

$$\psi(t)=\Lambda e^{-i \Omega t} \sum_{\vec{R}} e^{i\vec{q}\vec{r}(\vec{R},t)}$$

Where $\Omega$ is the frequency of this incident wave, $\vec{r} (\vec{R},t)$ is the position of atom R and time t. So I need to find the wave vectors q that are coherently scattered as a result of vibrations. $u_o$ is assumed small, so we can say, for all q of interest, $q u_o << 1$.

So we are interested in Thompson Scattering here. Can some one give me a hint on how to set my parameters and start on this? What confuses me is how to operationalize finding all the wave vectors q and making sure that they are only the q that have been coherently scattered; I think this means I will have conditions on q I must set, then evaluate for my wave vectors. If they are coherently scattered, I think this means I want the set of wave vectors that contribute to the peaks constructive interference.


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Let us treat the "coherent" part first. Neutrons scatter off nuclei, and the strength of the scattering thus depends not only on atomic number, but also which isotope we are dealing with. If one of the elements in our sample has two or more common isotopes, these isotopes will be randomly distributed throughout the crystal, since the chemistry of different isotopes is identical. The net result of this is that we have a random variation of the scattering factors throughout the sample. These random variations give rise to a background signal which is homogenuously distributed in $q$, and this background is termed incoherent scattering. The part of the scattering that is due to the "constantness" of the scattering factors is termed coherent scattering. So when your problem asks for coherently scattered neutrons, it just means we should ignore this constant background.

Now, you seem to be describing the process where a neutron scatters inelastically off a phonon, annihilating the phonon. This type of scattering is frequently used to probe phonon dispersion relations. For this process we must require conservation of energy, $$ E_\text{neutron}^\text{in} + \hbar \omega_\text{phonon} = E_\text{neutron}^\text{out} $$ and conservation of momentum, $$ \mathbf{k}_\text{neutron}^\text{in} + \mathbf{k}_\text{phonon} = \mathbf{k}_\text{neutron}^\text{out} .$$ The neutron energy can easily be expressed in terms of the magnitude of the wave vector by using the de Broglie wavelength, $$ k_\text{neutron} = \frac{1}{\hbar}\sqrt{\frac{2E_\text{neutron}}{m_\text{neutron}}}.$$ With these equations you should have enough to solve for both $\mathbf{q} = \mathbf{k}_\text{neutron}^\text{out} - \mathbf{k}_\text{neutron}^\text{in}$ and $E_\text{neutron}^\text{out}$, provided you know the energy (or wavelength) of the incoming neutrons $E_\text{neutron}^\text{in}$.

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