Do neutrons change their wavelengths when diffracted? Both neutrons and X-rays are used in diffraction. In XRD, the diffracted beam is of the same wavelength of the incident beam. The reason is that X-rays are EM waves which cause electrons to vibrate. The vibrating electrons then emit X-rays of the same frequency. Neutrons on the other hand collide with atomic nuclei like billiard balls. Because the masses of neutrons and atomic nuclei (especially hydrogen) are quite similar, there should be significant kinetic energy transfer. Does that mean the wavelength will increase after diffraction?
 A: X- rays can be treated with classical electromagnetic equations, and  diffraction presupposes elastic scattering, so no (or very small) loss in energy.
Neutrons are quantum mechanical entities and the  wavelength is not evident on individual scatters, it will appear in the accumulation of scatters due to the probability distribution to the quantum mechanical solution of the problem "neutron+material". Again in order to get diffraction the scattering should be elastic, otherwise the pattern will not appear.

Neutron diffraction is a form of elastic scattering where the neutrons exiting the experiment have more or less the same energy as the incident neutrons. The technique is similar to X-ray diffraction but the different type of radiation gives complementary information. A sample to be examined is placed in a beam of thermal or cold neutrons and the intensity pattern around the sample gives information of the structure of the material.

A: In diffraction, the neutron is scattering coherently from many nuclei in the lattice.  This coherent scattering can't create nuclear excited states, because multiple nuclei can't be excited in a coherent way. The neutron can scatter coherently but inelastically from the lattice, which is generally best modeled as interactions with phonons or magnons or other collective condensed-matter excitations.
Note that "the lattice" can also include diffraction from liquids.  Your favorite neutron scattering textbook will have an enthusiastic section about the relationship between diffraction patterns and structure functions.
You only get diffraction if the wavelength of the neutrons is comparable to or larger than the spacing between nuclei.  It's a little incongruous to think of neutron diffraction as "billiard ball" scattering:  diffraction is solidly a wave effect.  However, you can also have incoherent scattering from individual nuclei. In the Born approximation, the incoherently scattered neutron is in an s-wave final state, with spherical symmetry in the center-of-mass frame.  Incoherent scattering explains why hydrogen functions as a neutron temperature moderator, and why polyethylene makes reasonable neutron shielding.
Beware that neutrons in a diffraction device may have come from a cold moderator, because longer- wavelength neutrons give larger diffraction patterns.  A cold neutron interacting inelastically with a room-temperature sample will tend to gain, not lose, energy. Moderation is a thermal process.
A commenter mentions inelastic scattering from nuclear excited states. I wrote above that scattering from nuclear excited states can't contribute coherently to diffraction.  But nuclei at room temperature are overwhelmingly in their ground states, and thermal neutrons don't have the energy to access MeV-scale nuclear excitations.  Inelastic nuclear scattering involves nuclear reactions,  like $\rm (n,\gamma)$ capture, $\rm (n,2n)$ fragmentation in lithium-7, or $\rm (n,p)$ transmutation on nitrogen-14.
