Hi, 

I think you could find these notes attached useful to answer to your questions!

Suppose that you have two different scattering lengths in your system, the partial differential cross section takes care of the inelastic processes and of the 'dynamics' of the problem, but the spectrum will be composed by coherent and incoherent parts. This physically means that, coherent scattering measures the Fourier transform of the pair correlation function G(r,t) in the Structure neutron factor , taking into account the possible interference mechanisms, whereas the incoherent scattering measures the Fourier transform of the self correlation function, which does not consider interference effects. The former cross section, gives you info on Bragg peaks, spin waves, phonons,..., whereas the latter gives you info on atomic diffusion, vibrational density of states, and so on..

Elastic and inelastic, define processes in which energy is preserved / not preserved after the scattering process.

Best,

A.

https://www.ncnr.nist.gov/summerschool/ss12/pdf/SS2012_Gehring_InelasticScattering.pdf

Hi Angelo,

Thank you for your explanations. 

Does it mean that coherent scattering corresponds to equal-time G(r,t) and that incoherent scattering corresponds to equal-position G(r,t)?

Following your explanation, what is vibrational density of states? Is it equal to phonon density of states? From the experimental aspects, how to design coherent and incoherent scattering spectrometers? Based on my understanding, using TOF chopper spectrometer, we can get phonon dispersion relations with single crystal or phonon DOS with polycrystalline. Does this mean the former is coherent scattering and the later is incoherent scattering?

And what is atomic diffusion? What kind of information can we get from diffuse scattering? Does diffuse scattering have "elastic" and "inelastic" differences?

Hi Ren,

In a typical neutron scattering experiment, scattered neutrons are measured as a function of change in their direction (related to momentum transfer, Q) and energy (Energy transfer, deltaE). While Q dependence tells you the structure of the scattering material, the deltaE dependence tells you about the dynamics in the material. Further, the scattered neutron waves from different scattering centers may or may not interfere. Based on this information, you have the following.

1. Elastic scattering - Does not discriminate scattered neutrons for the energy transfer, I. e. integrate the detected signal over all energy transfer. This means you lose 'dynamics' information and are left with structural information alone.

2. Inelastic scattering - Measures scattered neutrons as a function of both energy and momentum transfer. This means you have both 'dynamical' as well as 'structural' information. Further, if you study the variation of scattered intensity at a given energy transfer as a function of Q, you get dispersion relation which can give you information on the geometry of the dynamical process that occurs at that energy transfer.

3. Coherent scattering - When you have interference between scattered neutron waves from different scattering centers (different atoms), you get information on the relative positions of these atoms. Thus coherent scattering gives you information on the relative atomic positions - structure of the material. In terms of G(r,t), the coherent scattering gives you the probability of finding an atom (say A) at position r, at time t, given that the same atom (A), or any other atom was present at origin (r=0) at t=0.

4. Incoherent scattering - When you have no interference between the scattered neutron waves from different centers, you are basically following a single atom. Thus, you are losing the information on the relative positions of atoms or structure. What you are studying is the movement of the same atom in time. In terms of G(r,t), the incoherent scattering gives you the probability of finding atom A at position r, at time t, given that the same atom A (and not any other atom, contrast this with coherent scattering case above) was present at origin (r=0). Thus, while incoherent inelastic scattering can trace single atom dynamics, the coherent counterpart would give you information on the collective motion of atoms.

I hope you find this useful.

With best regards.

Hi Siddharth,

Thank you very much for this comprehensive explanation. Now I have get the points of coherent scattering.

Could you provide some specific applications of incoherent elastic scattering and incoherent inelastic scattering? And how to design incoherent scattering spectrometers, or the incoherent measurements can be done on coherent scattering spectrometers? If the answer is yes, how to distinguish the coherent/incoherent signals?

In addition, is incoherent scattering equal to diffusing scattering?

Hi again Ren,

For elastic scattering, you'd be seldom interested in the incoherent part, mostly you are interested in structural information that can be obtained by the coherent part. In fact incoherent scattering in case of elastic scattering for studying structure is a nuisance you would like to avoid. In inelastic scattering of course, the incoherent signal is interesting as that would give you information on single-particle dynamics. In particular in quasielastic neutron scattering (QENS), that is scattering where an elastic line is broadened due to stochastic motion, incoherent contribution can help you understand self-diffusive motions.

Selecting incoherent or coherent scattering is a matter of choosing the right material to study, rather than the spectrometer. Of course, what you want to study would dictate your choice of spectrometer, but coherent/incoherent contribution to the scattering signal is determined by the chemical composition of your sample. It so happens that hydrogen has an incoherent neutron scattering cross-section that is an order of magnitude higher than incoherent or coherent neutron scattering cross-section of any other element. This means that if you have hydrogen or any hydrogen-bearing compound in your sample, your scattering signal would be dominated by the incoherent signal from hydrogen and you'll be blinded to see anything else. Depending on what your objective is, it may be a good thing or bad. For instance, if you want to see how this hydrogen-bearing molecule in your sample is moving, it is  easy to do a neutron inelastic measurement and extract the information you want. On the other hand if you are interested in structure, this hydrogen-bearing part in your sample becomes a big nuisance as it does not give you any information on structure and blinds you to the coherent signal you are looking for. Fortunately, neutron scattering cross-sections are isotope dependent. This makes this technique very useful. Deuterium (a heavier isotope of hydrogen) has a coherent ns cross-section which is more than twice the incoherent ns cross-section. So here, you have a good signal (coherent) and less noise (incoherent) from a structural perspective. So if you want to study the structure in a material, deuterating it (replacing all hydrogens with deuterium) helps. This way you are not changing the chemical properties of your sample, but making it easier to study using neutron scattering.

To distinguish between coherent and incoherent scattering signal from a material, polarization analysis can be used. A polarized neutron beam is scattered from the sample and the detector monitors the amount of neutrons detected that flip their spins after scattering, and those that do not. From this, one can calculate the ratio of coherent to incoherent contributions to the scattering.

A deviation from an ideal crystalline structure that is translation invariant would give diffuse scattering. There can be two contributions to this. (1) Local disorder in the system, caused for example by dislocations, and (2) Thermal motion of atoms that makes them move away from their lattice positions. So incoherent scattering is not equal to diffuse scattering.

I hope this answers your questions.

Good luck!

Hi,

the elastic/inelastic option is just referred to conservation/non-conservation of kinetic energy in the frame of the c.o.m.

The coherent/incoherent option is a bit trickier.

In the most general case (which is the inelastic), the typical output of a neutron scattering experiment - with the exception of spin-echo techniques - is the dynamic structure factor S(Q,omega), which can be related to the double-differential scattering cross-section, and is formally represented as the double FT of the van Hove correlation function G(r,t).

G(r,t) returns the amount of correlation between a particle placed in different positions at different instants. In the case of systems for which the symmetric or antisymmetric nature of the wavefunction is of little importance (Boltzmann statistics), G goes naturally splitted into a one-particle ('self') component  and a pair ('distinct') contribution.

Now, while Gself gives life to the incoherent part of the scattering (that is, the part concerning the correlation between the position of any atom at t0 and

the one of the same atom at time t), it is the entire G function to account for the coherent part (that is, the collective part dominated by interference terms). That's why experimentally telling coherent scattering apart from incoherent is not rarely arduous.

In the special case of a crystalline material, any coherent scattering that is not Bragg scattering is termed diffuse. Bragg scattering is generated by the 'average' lattice, and is always elastic; diffuse scattering contains info about any deviation from the average structure, and can be either elastic or inelastic depending on its source.

P.S: Vibrational density of states and phonon density of states are synonyms.

Spectrometers are employed for inelastic measurements (but including also the elastic component), and they're able to probe both coherent and incoherent scattering, except for Vesuvio (suited for deep inelastic scattering running under the Impulse Approximation, a special case of the incoherent approximation). Conversely, diffractometers are used for elastic scattering alone.

Dear Siddharth and Alexandra,

Thanks very much for your comprehensive explanations. Now I think I got some understanding on the diffuse scattering. Diffuse scattering can be either static (elastic) or dynamic (inelastic). For crystalline samples, point defects/dislocations (or some other effects) can bring about perturbations on the perfect long-range order. These deviation will bring scattering broadening in the Q-space. On the other hand, if these perturbations are dynamic, they also impose a perturbation on collective excitation. In this case, you can observe diffuse scattering signal in the E-space. 

Generally speaking, diffuse scattering is a kind of coherent scattering.

In coherent scattering, incident photon undergoes change in direction without change its frequency and obey conservation theorem of energy and momentum which is called coherent elastic scattering. Ex. Rayleigh scattering and Thomson scattering.

In incoherent scattering, incident photon undergoes change in direction with change its frequency and obey conservation theorem of energy and momentum which is called incoherent elastic scattering. Ex. Compton scattering.

Elastic coherent - collective atomic positions - e.g. lattice structure

Elastic incoherent - relative atomic positions - e.g. amorphous / liquid samples

Inelastic coherent - correlated motions - e.g. phonons, magnons

Inelastic incoherent - single atom motion - e.g. diffusion, vibrational density of states