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Table of Contents
Calculates a multiple scattering correction using a Monte Carlo method
MayersSampleCorrection, CarpenterSampleCorrection, VesuvioCalculateMS
This algorithm is also known as: Muscat
Name | Direction | Type | Default | Description |
---|---|---|---|---|
InputWorkspace | Input | MatrixWorkspace | Mandatory | The name of the input workspace. The input workspace must have X units of wavelength. This is used to supply the detector positions and the x axis range to calculate corrections for |
SofqWorkspace | Input | MatrixWorkspace | Mandatory | The name of the workspace containing S’(q). The input workspace must contain a single spectrum and have X units of momentum transfer. |
OutputWorkspace | Output | WorkspaceGroup | Mandatory | Name for the WorkspaceGroup that will be created. Each workspace in the group contains a calculated weight for a particular number of scattering events. The number of scattering events varies from 1 up to the number supplied in the NumberOfScatterings parameter. The group will also include an additional workspace for a calculation with a single scattering event where the absorption post scattering has been set to zero |
ScatteringCrossSection | Input | MatrixWorkspace | A workspace containing the scattering cross section as a function of k, \sigma_s(k) | |
NumberOfWavelengthPoints | Input | number | Optional | The number of wavelength points for which a simulation is attempted if ResimulateTracksForDifferentWavelengths=true |
EMode | Input | string | Elastic | The energy mode (default: Elastic). Allowed values: [‘Elastic’, ‘Direct’, ‘Indirect’] |
NeutronPathsSingle | Input | number | 1000 | The number of “neutron” paths to generate for single scattering |
NeutronPathsMultiple | Input | number | 1000 | The number of “neutron” paths to generate for multiple scattering |
SeedValue | Input | number | 123456789 | Seed the random number generator with this value |
NumberScatterings | Input | number | 2 | Number of scatterings |
Interpolation | Input | string | Linear | Method of interpolation used to compute unsimulated values. Allowed values: [‘Linear’, ‘CSpline’] |
SparseInstrument | Input | boolean | False | Enable simulation on special instrument with a sparse grid of detectors interpolating the results to the real instrument. |
NumberOfDetectorRows | Input | number | 5 | Number of detector rows in the detector grid of the sparse instrument. |
NumberOfDetectorColumns | Input | number | 10 | Number of detector columns in the detector grid of the sparse instrument. |
ImportanceSampling | Input | boolean | False | Enable importance sampling on the Q value chosen on multiple scatters based on Q.S(Q) |
MaxScatterPtAttempts | Input | number | 5000 | Maximum number of tries made to generate a scattering point within the sample. Objects with holes in them, e.g. a thin annulus can cause problems if this number is too low. If a scattering point cannot be generated by increasing this value then there is most likely a problem with the sample geometry. |
This algorithm calculates a Multiple Scattering correction using a Monte Carlo integration method. The method uses a structure function for the sample to determine the probability of a particular q value for each scattering event and it doesn’t therefore rely on an assumption that the scattering is isotropic.
The structure function that the algorithm takes as input is a linear combination of the coherent and incoherent structure factors:
S'(Q) = \frac{1}{\sigma_b}(\sigma_{coh} S(Q) + \sigma_{inc} S_s(Q))
If the sample is a perfectly coherent scatterer then S'(Q) = S(Q)
The algorithm is based on code which was originally written in Fortran as part of the Discus program [1]. The code was subsequently resurrected and improved by Spencer Howells under the Muscat name and was included in the QENS MODES package [2] These original programs calculated multiple scattering corrections for inelastic instruments but an elastic diffraction version of the code was also created and results from that program are included in this paper by Mancinelli [3].
This algorithm so far only considers the elastic case but the intention is to extend this to include inelastic at a later date following the approach in the Discus\MODES programs.
The algorithm calculates a set of dimensionless weights J_n describing the probability of detection at an angle \theta after n scattering events given a total incident flux I_0 and a transmitted flux of T:
T_n(\theta,\lambda) = J_n I_0(\lambda)
The quantity J_n is calculated by performing the following integration:
The variables l_i^{max} represent the maximum path length before the next scatter given a particular phi and theta value (Q). Each l_i is actually a function of all of the earlier values for the l_i, \phi and Q variables ie l_i = l_i(l_1, l_2, ..., l_{i-1}, \phi_1, \phi_2, ..., \phi_i, Q_1, Q_2, ..., Q_i)
The following substitutions are then performed in order to make it more convenient to evaluate as a Monte Carlo integral:
t_i = \frac{1-e^{-\mu_T l_i}}{1-e^{-\mu_T l_i^{max}}}
u_i = \frac{\phi_i}{2 \pi}
2 k^2 = \frac{\sigma_S \int_0^{2k} Q_i S(Q_i) dQ}{\sigma_s(k)}
Using the new variables the integral is:
This is evaluated as a Monte Carlo integration by selecting random values for the variables t_i and u_i between 0 and 1 and values for Q_i between 0 and 2k. A simulated path is traced through the sample to enable the l_i^{\ max} values to be calculated. The path is traced by calculating the l_i, \theta and \phi values as follows from the random variables. The code keeps a note of the start coordinates of the current path segment and updates it when moving to the next segment using these randomly selected lengths and directions:
l_i = -\frac{1}{\mu_T}ln(1-(1-e^{-\mu_T l_i^{\ max}})t_i)
\cos\theta_i = 1 - Q_i^2/k^2
\phi_i = 2 \pi u_i
The final Monte Carlo integration that is actually performed by the code is as follows where N is the number of scenarios:
The purpose of replacing 2 k^2 with \int Q S(Q) dQ can now be seen because it avoids the need to multiply by an integration range across dQ when converting the integral to a Monte Carlo integration. This is useful in the inelastic version of this algorithm where the integration of the structure factor is over two dimensions Q and \omega and the area of Q\omega space that has been integrated over is less obvious.
This is similar to the formulation described in the Mancinelli paper except there is no random variable to decide whether a particular scattering event is coherent or incoherent.
The algorithm includes an option to use importance sampling to improve the results for S(Q) profiles containing spikes. Without this option enabled, the contribution from rare, high values in the structure factor is only visible at a very high number of scenarios.
The importance sampling is achieved using a further change of variables as follows:
v_i = P(Q_i) = \frac{I(Q_i)}{I(2k)} where I(x) = \int_{0}^{x} Q S(Q) dQ
With this approach the Q value for each segment is chosen as follows based on a v_i value randomly selected between 0 and 1:
Q_i = P^{-1}(v_i)
\cos\theta_i is determined from Q_i as before. The change of variables gives the following integral for J_n:
Finally, the equivalent Monte Carlo integration that the algorithm performs with importance sampling enabled is:
The algorithm outputs a workspace group containing the following workspaces:
The output can be applied to a workspace containing a real sample measurement in one of two ways:
The multiple scattering correction should be applied before applying an absorption correction.
The Discus manual describes a further method of applying an attenuation correction and a multiple scattering correction in one step using a variation of the factor method. To achieve this the real sample measurement should be multipled by J_1^{*}/(\sum_{n=1}^{\infty} J_n). Note that this differs from the approach taken in other Mantid absorption correction algorithms such as MonteCarloAbsorption because of the properties of J_{1}^{*}. J_{1}^{*} corrects for attenuation due to absorption before and after the simulated scattering event (which is the same as MonteCarloAbsorption) but it only corrects for attenuation due to scattering after the simulated scattering event. For this reason it’s not clear this feature from Discus is useful but it has been left in for historical reasons.
The sample shape can be specified by running the algorithms SetSample or LoadSampleShape on the input workspace prior to running this algorithm.
The algorithm can take a long time to run on instruments with a lot of spectra andor a lot of bins in each spectrum. The run time can be reduced by enabling the following interpolation features:
Both of these interpolation features are described further in the documentation for the MonteCarloAbsorption algorithm
[1] | M W Johnson, 1974 AERE Report R7682, Discus: A computer program for the calculating of multiple scattering effects in inelastic neutron scattering experiments |
[2] | WS Howells, V Garcia Sakai, F Demmel, MTF Telling, F Fernandez-Alonso, Feb 2010, MODES manual RAL-TR-2010-006, doi: 10.5286/raltr.2010006 |
[3] | R Mancinelli 2012 J. Phys.: Conf. Ser. 340 012033, Multiple neutron scattering corrections. Some general equations to do fast evaluations doi: 10.1088/1742-6596/340/1/012033 |
Categories: AlgorithmIndex | CorrectionFunctions
C++ header: DiscusMultipleScatteringCorrection.h
C++ source: DiscusMultipleScatteringCorrection.cpp