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# TOFSANSResolutionByPixel v1¶

## Summary¶

Calculate the Q resolution for TOF SANS data for each pixel.

TOFSANSResolution

## Properties¶

Name

Direction

Type

Default

Description

InputWorkspace

Input

MatrixWorkspace

Mandatory

Name the workspace to calculate the resolution for, for each pixel and wavelength

OutputWorkspace

Output

Workspace

Mandatory

Name of the newly created workspace which contains the Q resolution.

DeltaR

Input

number

0

Virtual ring width on the detector (mm).

Input

number

0

Input

number

0

SigmaModerator

Input

MatrixWorkspace

Mandatory

Moderator time spread (microseconds) as afunction of wavelength (Angstroms).

CollimationLength

Input

number

0

Collimation length (m)

AccountForGravity

Input

boolean

False

Whether to correct for the effects of gravity

ExtraLength

Input

number

0

## Description¶

Calculates the Q-resolution per pixel according to Mildner and Carpenter equation

$(\sigma_Q )^2 = \frac{4\pi^2}{12\lambda^2} [ 3(\frac{R_1}{L_1})^2 + 3(\frac{R_2}{L_3})^2 + (\frac{\Delta R}{L_2})^2 ] + Q^2(\frac{\sigma_{\lambda}}{\lambda})^2$

where $$L1$$ and $$L2$$ are the collimation length and sample-to-detector distance respectively and

$\frac{1}{L_3} = \frac{1}{L_1} + \frac{1}{L_2}$

and

$(\sigma_{\lambda})^2 = (\Delta \lambda )^2 / 12 + (\sigma_{moderator})^2$

where $$\sigma_{\lambda}$$ is the overall effective standard deviation in wavelength. $$\Delta \lambda$$ values are found from the wavelength binning of the InputWorkspace, $$\sigma_{moderator}$$ is the moderator time spread (the variation in time for the moderator to emit neutrons of a given wavelength). Note that $$\Delta \lambda$$ may be imposed by wavelength steps set elsewhere in Mantid which should be at least as large as the equivalent time bins used in the original histogram data collection. For event mode data $$\Delta \lambda$$ is in theory very small, but in practice a histogram in time has to be generated (perhaps using monitor time bins or specifically set event-time-bins), before a rebinning into user provided wavelength steps in InputWorkspace. Again the latter steps should be the largest.

Q values needed here are calculated in the same way as for Q1D, including correction for gravity for which detector coordinates are assumed centred at zero wavelength.

$$\sigma_Q$$ is returned as the y-values of the InputWorkspace, and the remaining variables in the main equation above are related to parameters of this algorithm as follows:

• $$R_1$$ equals SourceApertureRadius

• $$R_2$$ equals SampleApertureRadius

• $$\Delta R$$ equals DeltaR

• $$\sigma_{moderator}$$ equals SigmaModerator

• $$\L_1$$ equals CollimationLength

$$\lambda$$ in the equation is the midpoint of wavelength histogram bin values of InputWorkspace.

Collimation length $$L_1$$ in metres in the equation here is the distance between the first beam defining pinhole (Radius $$R_1$$) and the sample aperture (radius $$R_2$$). (Beware that $$L_1$$ is more often the moderator to sample distance.)

For rectangular collimation apertures, size H x W, Mildner & Carpenter say to use $$R = \sqrt{( H^2 +W^2)/6 }$$. Note that we are assuming isotropically averaged, scalar $$Q$$, and making some small angle approximations. Results on higher angle detectors may not be accurate. For data reduction sliced in different directions on the detector (e.g. GISANS) adjust the calling parameters to suit the collimation in that direction.

Note that $$\Delta$$ is the full width of a rectangular distribution in radius or wavelength, for which the standard deviation is $$\sigma=\Delta/\sqrt{12}$$. For a Gaussian distribution the FWHM (full width at half maximum) is $$\sqrt{8\ln{2}}\sigma=2.35482\sigma$$. For an exponential decay $$e^{-t/\tau}$$, the standard deviation (and the mean) is $$\tau$$. For non-rectangular distributions these equations allow the equivalent $$\Delta$$ to be entered as $$\Delta=\sqrt{12}\sigma$$.

This version of the algorithm neglects wavelength-dependent detector detection depth effects.

Categories: AlgorithmIndex | SANS