In the elastic neutron scattering the energy of the incomming neutron does not change after being scattered from the sample whereas in inelastic neutron scattering the energy of the scattered neutrons are different from those of the incoming neutrons. By elastic neutron scattering positions and intensities you can get structural information including both crystal and magnetic structures whereas the inelastic neutron scattering can give you information about the phonon and magnon dynamics. Experimentally specking in elastic neutron scattering we usually use monochromatic neutrons and do not analyze the energy of the scattered neutrons whereas in inelastic neutron scattering we also use monochromatic neutrons but analyze the energy of the scattered neutrons by using an analyzer crystal. This is traditional way of measuring elastic and inelastic neutron scattering in a steady reactor source. In spallation neutron source the method of study of neutron scattering is different.

In the so-called elastic neutron diffraction experiments (say an angle dispersive powder diffractometer) we actually measure all the neutrons that are scattered from the sample elastically or inelastically. Only elastic scattered neutrons gives strong Bragg peaks but below and in between Bragg peaks there are plenty of other neutrons scattered inelastically. Since we do not analyze the energy they do not give rise to peaks but mege in the background showing sometimes much broader small humps that we often call diffuse scattering. We can include these diffuse scattering (after subtracting the background that do not come from the sample) and treat the total scattering by the pair-distribution function (PDF) analysis. Then we can learn something more than we learn just from the intensities and positions of Bragg peaks. But for this method to work you must measure the total diffraction intensities up to a very high Q (=4pi sintheta/lambda). The Bragg peak intensities and positions can give you average structure only whereas the PDF analysis can give you information about the local structure if properly interpreted. In fact it gives you only a time average of the local structure because we do not know the energies of the inelastically scattered neutrons. So this is somehow an intermediate step only. To get a complete picture of the startic and dynamics of your sample you must have complete information of the scattering function S(Q,E). Then of course you have to analyze the energies of neutrons scattered from your sample. How can we do it? We need a neutron spectrometer where we can analyze the energies of the scattered neutrons. We can use a triple axis (TAS) or a time-of-flight (TOF) spectrometers for mapping the complete S(Q,E).

Between elastic and inelastic neutron scattering there exists another type of scattering called quasielastic scattering. Here the scattering is broader than the instrument resolution limited elastic scattering. This is a very popular way to study the dynamics of hydrogen atoms. The hydroen has a very large incoherent scattering cross-section and can be very profitably used to study the H dynamics in hydrogen containing materials. These processes are slow and therefore we need a very good instrumental energy resolution of the order of a few micro-eV. The back-scattering neutron spectrometers are ideally suited for such studies. But for studying dynamics in polymers we need even better resolution and in such cases spin echo spectrometers are used.

I talked shortly about diffuse scattering before in connection with PDF analysis. This type of scattering was studied by mostly crystallographers to investigate disorder in the structure. The neutron experimments were done without analyzing the enegies of the scattered neutrons. But unless we have the evidence by other experiments we cannot distinguish between the static and dynamic disorder by this type of study.. The trouble is that we do not know the real Q values of the inelastically scattered neutrons. The interpretation of diffuse scattering is therefore often goes wrong. The best strategy is to measure the scattering function S(Q,E) to get a complete information about the static and dynamics of the system under study.

To sum up I must say that there alredy exists vast literature about elestic, inelastic, quasielastic and diffuse neutron scattering. Each topic mrerits a seperate book of considerable length and in fact there exist several books covering each of these topic only. 

Thank you so much sir.it helped me a lot.

The last topic I would like to touch is the Debye-Waller factor or atomic displacement parameters. The Q dependence of the intensity of the Bragg reflections gives you the Debye-Waller factor. This factor essentilly takes away intensity from the Bragg peaks. The question is where does this intensity go. The answer is the diffuse scattering under and inbetween Bragg peaks. So from the missing intensities from the Bragg peak you can learn something about the atomic displacements due to thermal motions of the atoms. There is a great book for this by B.T.M. Willis and A.W. Pryor "Thermal vibrations in crystallography", Cambridge University Press (1975).

Going back to how the original question was posed, diffraction in the sense of Bragg reflection, can be considered an elastic phenomenon. However, when talking of neutrons, as Tapan rightly points out, one talks of scattering rather than diffraction in order to distinguish among the various ways in which neutrons are "redirected" when going through matter. Elastic scattering closely represents the Bragg diffraction of neutrons which is investigated by the traditional crystallographic methods (space distribution of the scattered neutrons), whereas inelastic and quasi-elastic scattering are investigated by spectroscopic methods (energy distribution of the scattered neutrons).

I disagree with the definition stated above regarding the "elastic scattering" process.

1. elastic scattering:The neutron and the nuclide collide and share a part of their kinetic energies.  So the neutron does change its energy, but the ‘total kinetic energy’ before and after the collision remains the same. If the nucleus is stationary before collision, it will gain energy from the neutron and start moving, and the neutron gets slowed down due to loss of kinetic energy.

2. Inelastic scattering, the total kinetic energy is not conserved since a portion of it goes to excite the nucleus and that what makes the difference between the two process i.e. the total kinetic energy not the neutron one.

I'm afraid there's no disagreement between the two descriptions of the phenomena given above as to elastic and inelastic neutron scattering.

In fact, the first description, not a definition (RR April 16, 2015), implies conservation of energy in elastic scattering and modification of the energy distribution in inelastic scattering. Just like the second description infers (RM December 7th, 2016) based on the observation of the sole energetic principles involved without considering the angular momentum of any of the processes.

Hence there is no contradiction.