MonteCarloAbsorption v1¶

See Also¶

MayersSampleCorrection, CarpenterSampleCorrection, PearlMCAbsorption, VesuvioCalculateMS

Input Workspace Requirements¶

The algorithm will compute the correction factors on a bin-by-bin basis for each spectrum within the input workspace. The following assumptions on the input workspace will are made:

• X units are in wavelength
• the instrument is fully defined
• properties of the sample and optionally its environment have been set with SetSample

By default the beam is assumed to be the a slit with width and height matching the width and height of the sample. This can be overridden using SetBeam.

Method¶

By default, the material for the sample & containers will define the values of the cross section used to compute the absorption factor and will include contributions from both the total scattering cross section & absorption cross section. This follows the Hamilton-Darwin [1], [2] approach as described by T. M. Sabine in the International Tables of Crystallography Vol. C [3].

The algorithm proceeds as follows. For each spectrum:

1. find the associated detector position

2. find the associated efixed value (if applicable) & convert to wavelength (\(\lambda_{fixed}\))

3. for each event in NEvents

   • loop over the bins.

     • If ResimulateTracksForDifferentWavelengths = True then generate tracks using the following procedure for each wavelength step, where the size of each wavelength step is defined by NumberOfWavelengthPoints. If ResimulateTracksForDifferentWavelengths = false generate one set of tracks and define a step size of 1 ie all bins are visited. At the moment there are no wavelength dependent effects in the simulation that affect the simulation of the track geometry so the default value for ResimulateTracksForDifferentWavelengths is false. For each step (\(\lambda_{step}\))

     • generate a random point on the beam face defined by the input height & width. If the point is outside of the area defined by the face of the sample then it is pulled to the boundary of this area

     • generate a random point within the sample or container objects as the scatter point and create a Track from the selected position on the beam face to the scatter point

     • test for intersections of the track & sample/container objects, giving the number of subsections and corresponding distances within the object for each section, call them \(l_{1i}\)

     • form a second Track with the scatter position as the starting point and the direction defined by detPos - scatterPos

     • test for intersections of the track & sample/container objects, giving the number of subsections and corresponding distances within the object for each section, call them \(l_{2i}\)

     • define \(\lambda_1\) as the wavelength before scattering & \(\lambda_2\) as wavelength after scattering:

       • Direct: \(\lambda_1 = \lambda_1\), \(\lambda_2 = \lambda_{step}\)
       • Indirect: \(\lambda_1 = \lambda_{step}\), \(\lambda_2 = \lambda_{fixed}\)
       • Elastic: \(\lambda_1 = \lambda_2 = \lambda_{step}\)

     • compute the self-attenuation factor for all intersections as \(\prod\limits_{i} \exp(-(\rho_{1i}\sigma_{1i}(\lambda_{1i})l_{1i} + \rho_{2i}\sigma_{2i}(\lambda_{2i})l_{2i}))\) where \(\rho\) is the mass density of the material & \(\sigma\) the absorption cross-section at a given wavelength

     • accumulate this wavelength-specific factor across all NEvents

4. average the accumulated attentuation factors over NEvents and assign this as the correction factor for this \(\lambda_{step}\). Calculate the error as the standard deviation of the accumulated attenuation factors divided by \(\sqrt{NEvents}\)

5. finally, if ResimulateTracksForDifferentWavelengths = True, interpolate through the unsimulated wavelength points using the selected method

The algorithm generates some statistics on the number of scatter points generated in the sample and each environment component if the logging level is set to debug.

Interpolation¶

The default linear interpolation method will produce an absorption curve that is not smooth. CSpline interpolation will produce a smoother result by using a 3rd-order polynomial to approximate the original points.

The errors that the Monte Carlo simulation calculates for different wavelength points in a single spectrum may or may not be independent. If the same set of tracks have been used for different wavelengths (ResimulateTracksForDifferentWavelengths=False) then the errors will be correlated. A worst case positive correlation has been assumed giving an error on the interpolated point that is approximately the same as the surrounding simulated points.

If ResimulateTracksForDifferentWavelengths=True then the errors on the simulated points will be independent and the errors can be combined using standard formulae for combining errors on independent variables.

The error propagation through the spline interpolation is complex because each cubic polynomial is usually expressed as a function of the known y values and a derivative of those y values at the same points (some texts use the first derivative others the second). The error in y and the derivative of y are correlated at a particular x value and the derivatives at different x values are also correlated. So some extra covariances are required in addition to the error (variance) of each y value. The method followed [4] involves inverting a symmetric tridiagonal matrix and an analytic calculation for the inversion has been implemented to minimize the run time [6]

Sparse instrument¶

The simulation may take long to complete on instruments with a large number of detectors. To speed up the simulation, the instrument can be approximated by a sparse grid of detectors. The behavior can be enabled by setting the SparseInstrument property to true.

The sparse instrument consists of a grid of detectors covering the full instrument entirely. The figure below shows an example of a such an instrument approximating the IN5 spectrometer at ILL.

Absorption corrections for IN5 spectrometer interpolated from the sparse instrument shown on the right. The sparse instrument has 6 detector rows and 22 columns, a total of 132 detectors. IN5, on the other hand, has approximately 100000 detectors.

When the sparse instrument option is enabled, a sparse instrument corresponding to the instrument attached to the input workspace is created. The simulation is then run using the created instrument. Finally, the simulated absorption corrections are interpolated to the output workspace.

The interpolation is a two step process: first a spatial interpolation is done from the detector grid of the sparse instrument to the actual detector positions of the full instrument. Then, if ResimulateTracksForDifferentWavelengths = True the correction factors are interpolated over the missing wavelengths.