Inelastic

From DANSE

Table of contents
1 Reference Material for Inelastic Scattering
2 Experimental inelastic neutron scattering
3 Theory of inelastic neutron scattering
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Reference Material for Inelastic Scattering

  • [Textbook (http://www.cacr.caltech.edu/projects/danse/doc/Inelastic_Book_sm.pdf)] Experimental Inelastic Neutron Scattering. Acrobat pdf book that explains theory, experiment, and software for inelastic scattering with chopper spectrometers (approx 300 pages).
  • [Structure of reduction software (http://wiki.cacr.caltech.edu/danse/index.php/ARCS_alpha_package)] for the alpha release of the software for data reduction for chopper spectrometers.
  • [Software components (http://drchops.caltech.edu/)] for reduction and analysis tasks—build your own application, publish your analysis component.
  • Descriptions of instruments and related software: simulations, beamline components, etc.
  • Science done with inelastic scattering
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Experimental inelastic neutron scattering

right
Figure: Inelastic neutron scattering data, S(Q,E), from the Pharos spectrometer at Los Alamos. The sample was a polycrystalline Ni-Fe alloy. Note the phonon dispersions emanating from the Bragg diffractions and extending to positive and negative energy transfers. The diffraction pattern from the fcc structure is indexed at the elastic line (the bright horizontal line at E=0).

Inelastic neutron scattering is used to study dynamical processes in materials, molecules, and condensed matter. "Dynamical processes" include the motions of atoms in a sample, such as their vibrations or phonons. The motions of electron spins such as magnons are also studied by inelastic neutron scattering.

Inelastic neutron scattering measures the transfer of energy, E, and momentum, Q, from neutrons to a sample. The spectrum of these transfers is

S(\vec{Q}, E)

where \vec{Q} is a vector in the general case. Typical measurements are in the energy range from micro-eV (electron-Volt) to eV, and on momentum scales from tenths to tens of inverse Angstroms. Of course new techniques and instrument designs are constantly pushing these boundaries!

Various methods for measuring neutron energy and momentum have been devised over the years, and a whole zoo of inelastic instruments have been built at neutron sources world-wide to exploit these techniques. Each class of instruments has its own computational methods for reducing data, although common software components can describe the composition and structure of inelastic instruments both for data reduction and instrument simulation.

The primary computational tasks for inelastic neutron scattering are:

  • reducing data from the raw intensity measured in lab coordinates to the S(\vec{Q}, E) function expressed in parameters appropriate to the underlying dynamical process. This includes
    • correcting for instrumental artifacts (spurious scattering, background, ...)
    • correcting for physical artifacts (multiple scattering, absorption, ...)
  • comparing the measured S(\vec{Q}, E) with dynamics models
  • simulating the performance of spectrometers and the scattering by a sample


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Theory of inelastic neutron scattering

The theory of inelastic neutron scattering is based primarily on the van Hove correlation function.

The Van Hove correlation function is the maximum information that inelastic neutron scattering can give about the motions and positions of atoms or spins in a sample. The relationship between the experimental data (involving momentum and energy) and the Van Hove correlation function (involving positions and time) is presented here.

After data reduction and correction for characteristics of the instrument, a successful inelastic scattering experiment can provide the intensity as a function of momentum and energy transfer from the neutron to the sample:

S(\vec{Q},E) \ .

This intensity is related to a quantity that describes dynamics in the sample, a quantity that is a function of space and time. It may involve, for example, the positions of atoms \vec{r} and the periods of the atom vibrations. The actual positions and motions of atoms are described by the form factor:

f(\vec{r}, t ) \ ,

which gives the positions of the scatterers as they move. This function f(\vec{r}, t ) contains all information about the motions of the nuclei or spins that scatter the neutrons.


Unfortunately, inelasic neutron scattering spectra cannot provide the form factor itself. The fundamental problem is that experiments measure intensities at the detector, not neutron wavefunctions. Instead, an anlysis of the measured S(\vec{Q},E) can provide the overlap of f(\vec{r}, t ) \ with itself in space and time.

More precisely, an analysis of the experimental S(\vec{Q},E) can provide the Van Hove correlation function G(\vec{r}, t):

G(\vec{r}, t)\equiv \iint_{-\infty}^{+\infty} f^{*}(\vec{r}^{\prime}, t^{\prime} ) f(\vec{r}+\vec{r}^{\prime}, t+ t^{\prime}) \, d^{3}\vec{r}^{\prime } d t^{\prime } \; .

The Van Hove function G(\vec{r}, t) and the experimental intensity S(\vec{Q},E) are related through a double Fourier transformation in space and time:

S(\vec{Q}, E)=\iint_{-\infty}^{+\infty} G(\vec{r}, t) \, e^{-i(\vec{Q}\cdot{\vec{r}} - \omega t)} \, d^{3}\vec{r} \, d t \; .

The Van Hove correlation function is all the physical information that can be obtained about atoms positions and motions from an experimental measurement of S(\vec{Q},E).

These relationships are explained more thoroughly in the book Experimental Inelastic Neutron Scattering. It is available for free, and can be [downloaded here (http://www.cacr.caltech.edu/projects/danse/doc/Inelastic_Book_sm.pdf)].

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