Coherent scattering of neutrons It is said that neutrons have small scattering angles in case of elastic coherent scattering. I am having some queries regarding the same.


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*Why do we call it coherent? I mean, we say that for coherent scattering, there needs to be an absorption and re-radiation by atomic nuclei (like in case of photons scattered by nuclei). Does it mean that the neutron gets absorbed by the atomic nucleus and then again it is re-radiated?

*Why do we call it small angle scattering?
I have attached this slide from (http://meetings.chess.cornell.edu/ACABioSAS/TrackA/SAXSandSANS-Biospecifics_Trewhella_ACA_2015fin.pdf). What I don't understand is that since nucleus is much bigger than the neutron, classically, the neutron can even bounce back after the collision and thus the scattering angle would be quite large. So how do we explain the small scattering angle?
 A: When we are talking of scattering neutrons we are in the quantum mechanical regime. The scattering crossection ideally is completely defined if we know the wave function of the problem "neutron +Mass ---> neutron +Mass" both for elastic and inelastic scattering.
Inelastic scattering means that the neutron transfers part of its  energy by hitting a nucleus , whereas elastic means that it does not lose energy but only changes direction. This is explained in the link you gave.

What I don't understand is that since nucleus is much bigger than the neutron, classically, the neutron can even bounce back after the collision and thus the scattering angle would be quite large.

For a small percentage of elastic scattering, depending on the energy and the type of mass on which neutrons scatter, there is a probability of scattering backwards, see here for example. For the setup described in the link you give,  the small angle elastic scatters are chosen because  the phases are known  (coherent) , so there can be a superposition of the neutron wave functions . The cross section is the complex conjugate squared of the wave function and thus interference can appear.

So how do we explain the small scattering angle?

It is the choice for the study at hand, small scattering angles give useful interference patterns, and also are more probable in the quantum mechanical probability estimate. The inelastic scatters  of  neutrons on nuclei (strong interaction, not electromagnetic)  contribute noise. The elastic ones keep the phases and can transfer useful information.
A: In addition to the answer of anna v, I would like to clarify some points of your question. Suppose the interaction describing the neutron scattering on the matter (say, atomic nuclei). Independently on the type of the interactions (I'll tell about them below), they can be characterized by the quantity $q^{2} = q_{\mu}q^{\mu}$, where $q_{\mu}$ is the 4-momentum transferred from the neutron to the matter. As is discussed here, $q^{2}$ is intimately related to the distance $r$ of the interaction; namely $q^{2}\simeq r^{-2}$. 
In general, one needs to take into account all possible values of $q^{2}$. As long as $r \gg r_{n}$, where $r_{n}$ is the neutron radius, the neutron can be interpreted as point-like particle, and the scattering is typically elastic; equivalently, $r \gg r_{n}$ means that $q^{2} \ll r_{n}^{-2}$. In the center of mass frame (CM), 
$$
\tag 1 q^{2} = -4|\mathbf p_{\text{CM}}|^{2}\sin^{2}\left(\frac{\theta_{\text{CM}}}{2}\right),
$$ 
where $|\mathbf p_{\text{CM}}|$ is the neutron 3-momentum in the CM frame, while $\theta_{\text{CM}}$ is the CM scattering angle. From $(1)$ you see that the smallness of $q^{2}$ means, for fixed $|\mathbf p_{\text{CM}}|$, the smallness of $\theta_{\text{CM}}$; the more $|\mathbf p_{\text{CM}}|$ is, the less $\theta_{\text{CM}}$ must be in order $q^{2}$ to be small. But smallness of $\theta_{\text{CM}}$ simply means that the scattered neutron moves in almost the same direction as the incoming one. This is what can be called the coherent scattering.
Let's now briefly talk about the types of interactions. The neutron interacts with the matter through emission of interaction carriers, which depending on the value of $q^{2}$ can be photons, $W,Z$-bosons, gluons, pions and other; the neutron doesn't need to be absorbed and re-radiated in order to interact. The carriers all can be classified as strong, weak and electromagnetic interaction carriers. In dependence of the type of the interaction between the neutron and the matter, the dominant contribution into the total scattering cross-section are given by different values of $q^{2}$. The strong and weak interactions typically are characterized by large $q^{2}$ typically larger than $r_{n}^{-2}$, while electromagnetic interaction dominates for small $q^{2}$; that's why we call it long-ranged interaction. The latter qualitatively explains existence of small scattering angles interaction.
