How to find a neutron dispersion curve

[Ayotte]

A neutron is at rest inside a crystalline solid. Can a phonon come along and give up all of its energy to the neutron? Address the question with a graphical solution, i.e. overlay the dispersion curves for neutrons and phonons. For simplicity assume that the solid has a basis of a single atom and have only one acoustic branc

In the textbook, it says that a neutron's dispersion curve can only be found through experiment. Because neutron's don't really interact with much, I'm stumped about how to find such a curve in order to overlay the two.

 

[Nate810]

However, a simplified graphical solution could be to draw a horizontal line representing the energy of the neutron at rest, and then draw a curved line representing the phonon dispersion curve. The idea would be to show that the phonon's energy is distributed over different wave vectors (represented by the curvature of the line), and that its energy is lower than that of the neutron at rest. This would illustrate that it is not possible for the phonon to give up all of its energy to the neutron.