IntegratePeaksMD v2¶

See Also¶

CentroidPeaksMD, IntegratePeaksHybrid, IntegratePeaksMDHKL, IntegratePeaksUsingClusters, IntegratePeaksCWSD

Similar algorithms¶

See IntegrateEllipsoids v3 for a ways of integrating peaks from data collected in EventWorkspace. PeakIntensityVsRadius v1 is meant to help determine an appropriate value for PeakRadius.

Inputs¶

The algorithms takes two input workspaces:

• A MDEventWorkspace containing the events in multi-dimensional space. This would be the output of ConvertToDiffractionMDWorkspace v3.
• As well as a PeaksWorkspace containing single-crystal peak locations. This could be the output of FindPeaksMD v1
• The OutputWorkspace will contain a copy of the input PeaksWorkspace, with the integrated intensity and error found being filled in.

Calculations¶

Integration is performed by summing the weights of each MDEvent within the provided radii. Errors are also summed in quadrature.

IntegratePeaksMD_graph1.png

• All the Radii are specified in \(\AA^{-1}\)
• A sphere or ellipsoid of radius PeakRadius is integrated around the center of each peak.
  • This gives the summed intensity \(I_{peak}\) and the summed squared error \(\sigma I_{peak}^2\).
  • The volume of integration is \(V_{peak}\).
  • An ellipsoidal shape is integrated when the Ellipsoid option is enabled (the following also applies to BackgroundInnerRadius and BackgroundOuterRadius):
    • When only one value for PeakRadius is given, then the ellipsoidal shape is calculated automatically with axes proportional to the sqrt of the eigenvalues of the covariance matrix. If FixMajorAxisLength = true then with the ellipsoid will be scaled such that the radius along the major axis is equal to the radius of the spherical region over which the covariance was estimated. In addition the covariance can be estimated iteratively (by setting MaxIterations > 1) - points outside of 3 standard deviations will be exluded and the covariance will be estimated again. If the ellipsoid is large enough to fully contain the spherical region then the spherical region will be used instead.
    • Three values for PeakRadius can be given, which correspond to the semi-axis lengths (a,b,c) of the ellipsoid.
• If BackgroundOuterRadius is specified, then a shell, with radius r where BackgroundInnerRadius < r < BackgroundOuterRadius, is integrated.
  • This gives the summed intensity \(I_{shell}\) and the summed squared error \(\sigma I_{shell}^2\).
  • The volume of integration is \(V_{shell}\).
  • BackgroundInnerRadius allows you to give some space between the peak and the background area.
  • If the option UseOnePercentBackgroundCorrection is enabled, which it is by default, then the top one percent of the background events are removed so that there are no intensity spikes near the edges.
• AdaptiveQMultiplier can be used for the radius to vary as a function of the modulus of Q. If the AdaptiveQBackground option is set to True, the background radius also changes so each peak has a different integration radius. Q includes the 2*pi factor.
  • PeakRadius + AdaptiveQMultiplier * |Q|
  • BackgroundOuterRadius + AdaptiveQMultiplier * |Q|
  • BackgroundInnerRadius + AdaptiveQMultiplier * |Q|

Background Subtraction¶

The background signal within PeakRadius is calculated by scaling the background signal density in the shell to the volume of the peak:

\(I_{bg} = I_{shell} \frac{V_{peak}}{V_{shell}}\)

with the error squared on that value:

\(\sigma I_{bg}^2 = (\frac{V_{peak}}{V_{shell}})^2 \sigma I_{shell}^2\)

This is applied to the integrated peak intensity \(I_{peak}\) to give the corrected intensity \(I_{corr}\):

\(I_{corr} = I_{peak} - I_{bg}\)

with the errors summed in quadrature:

\(\sigma I_{corr}^2 = \sigma I_{peak}^2 + \sigma I_{bg}^2\)

If BackgroundInnerRadius is Omitted¶

If BackgroundInnerRadius is left blank, then BackgroundInnerRadius = PeakRadius, and the integration is as follows:

IntegratePeaksMD_graph2.png

IntegrateIfOnEdge option¶

Edges for each bank or pack of tubes of the instrument are defined by masking the edges in the PeaksWorkspace instrument. e.g. For TOPAZ pixels 0 and 255 in both directions for the Rectangular Detector. Q in the lab frame for every peak is calculated, call it C For every point on the edge, the trajectory in reciprocal space is a straight line, going through:

\(\vec{O}=(0,0,0)\)

Calculate a point at a fixed momentum, say k=1. Q in the lab frame:

\(\vec{E}=(-k \cdot \sin(\theta) \cdot \cos(\phi),-k \cdot \sin(\theta) \cdot \sin(\phi),k-k \cdot \cos(\phi))\)

Normalize E to 1:

\(\vec{E}=\vec{E} \cdot (1./\left|\vec{E}\right|)\)

The distance from C to OE is given by:

\(dv=\vec{C}-\vec{E} \cdot (\vec{C} \cdot \vec{E})\)

If:

\(\left|dv\right|<PeakRadius\)

for the integration, one of the detector trajectories on the edge is too close to the peak This method is also applied to all masked pixels. If there are masked pixels trajectories inside an integration volume, the peak must be rejected.

CorrectIfOnEdge option¶

This is an extension of what was calculated for the IntegrateIfOnEdge option. It will only be calculated if this option is true and the minimum dv is less than PeakRadius or BackgroundOuterRadius.

For the background if

\(\left|dv\right|_{\text{min}}<BackgroundOuterRadius\)

\(h = BackgroundOuterRadius - \left|dv\right|_{\text{min}}\)

From the minimum of dv the volume of the cap of the sphere is found:

\(V_{cap} = \pi h^2 / 3 (3 \cdot BackgroundOuterRadius - h)\)

The volume of the total sphere is calculated and for the background the volume of the inner radius must be subtracted:

\(V_{shell} = 4/3 \pi (BackgroundOuterRadius^3 - BackgroundInnerRadius^3)\)

The integrated intensity is multiplied by the ratio of the volume of the sphere divided by the volume where data was collected

\(I_{bkgMultiplier} = V_{shell} / (V_{shell} - V_{cap})\)

For the peak assume that the shape is Gaussian. If

\(\left|dv\right|_{\text{min}}<PeakRadius\)

\(\sigma = PeakRadius / 3\)

\(h = PeakRadius \cdot \exp(-\left|dv\right|_{min}^2 / (2 \sigma^2)\)

From the minimum of dv the volume of the cap of the sphere is found:

\(V_{cap} = \pi h^2 / 3 (3 \cdot PeakRadius - h)\)

and the volume of the sphere is calculated

\(V_{sphere} = 4/3 \pi PeakRadius^3\)

The integrated intensity is multiplied by the ratio of the volume of the sphere divided by the volume where data was collected

\(I_{peakMultiplier} = V_{sphere} / (V_{sphere} - V_{cap})\)