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Smearing Functions

Sometimes it will be necessary to correct reduced experimental data for the physical effects of the instrumental geometry in use. This process is called desmearing. However, calculated/simulated data - which by definition will be perfect/exact - can be smeared to make it more representative of what might actually be measured experimentally.

SasView provides the following three smearing algorithms:

  • Slit Smearing
  • Pinhole Smearing
  • 2D Smearing

Slit Smearing

This type of smearing is normally only encountered with data from X-ray Kratky cameras or X-ray/neutron Bonse-Hart USAXS/USANS instruments.

The slit-smeared scattering intensity is defined by

sm_image002.gif

where Norm is given by

sm_image003.gif

[Equation 1]

The functions inlineimage004 and inlineimage005 refer to the slit width weighting function and the slit height weighting determined at the given q point, respectively. It is assumed that the weighting function is described by a rectangular function, such that

sm_image006.gif

[Equation 2]

and

sm_image007.gif

[Equation 3]

so that inlineimage008 inlineimage009 for inlineimage010 and u.

Here inlineimage011 and inlineimage012 stand for the slit height (FWHM/2) and the slit width (FWHM/2) in q space.

This simplifies the integral in Equation 1 to

sm_image013.gif

[Equation 4]

which may be solved numerically, depending on the nature of inlineimage011 and inlineimage012 .

Solution 1

For inlineimage012 = 0 and inlineimage011 = constant.

sm_image016.gif

For discrete q values, at the q values of the data points and at the q values extended up to qN= qi + inlineimage011 the smeared intensity can be approximately calculated as

sm_image017.gif

[Equation 5]

where inlineimage018 = 0 for Is when j < i or j > N-1.

Solution 2

For inlineimage012 = constant and inlineimage011 = 0.

Similar to Case 1

inlineimage019 for qp = qi - inlineimage012 and qN = qi + inlineimage012

[Equation 6]

where inlineimage018 = 0 for Is when j < p or j > N-1.

Solution 3

For inlineimage011 = constant and inlineimage011 = constant.

In this case, the best way is to perform the integration of Equation 1 numerically for both slit height and slit width. However, the numerical integration is imperfect unless a large number of iterations, say, at least 10000 by 10000 for each element of the matrix W, is performed. This is usually too slow for routine use.

An alternative approach is used in SasView which assumes slit width << slit height. This method combines Solution 1 with the numerical integration for the slit width. Then

sm_image020.gif

[Equation 7]

for qp = qi - inlineimage012 and qN = qi + inlineimage012

where inlineimage018 = 0 for Is when j < p or j > N-1.

Pinhole Smearing

This is the type of smearing normally encountered with data from synchrotron SAXS cameras and SANS instruments.

The pinhole smearing computation is performed in a similar fashion to the slit- smeared case above except that the weight function used is a Gaussian. Thus Equation 6 becomes

sm_image021.gif

[Equation 8]

2D Smearing

The 2D smearing computation is performed in a similar fashion to the 1D pinhole smearing above except that the weight function used is a 2D elliptical Gaussian. Thus

sm_image022.gif

[Equation 9]

In Equation 9, x0 = q cos(|theta|), y0 = q sin(|theta|), and the primed axes, are all in the coordinate rotated by an angle |theta| about the z-axis (see the figure below) so that x'0 = x0 cos(|theta|) + y0 sin(|theta|) and y'0 = -x0 sin(|theta|) + y0 cos(|theta|). Note that the rotation angle is zero for a x-y symmetric elliptical Gaussian distribution. The A is a normalization factor.

sm_image023.gif

Now we consider a numerical integration where each of the bins in |theta| and R are evenly (this is to simplify the equation below) distributed by |bigdelta||theta| and |bigdelta|R, respectively, and it is further assumed that I(x',y') is constant within the bins. Then

sm_image024.gif

[Equation 10]

Since the weighting factor on each of the bins is known, it is convenient to transform x'-y' back to x-y coordinates (by rotating it by -|theta| around the z axis).

Then, for a polar symmetric smear

sm_image025.gif

[Equation 11]

where

sm_image026.gif

while for a x-y symmetric smear

sm_image027.gif

[Equation 12]

where

sm_image028.gif

The current version of the SasView uses Equation 11 for 2D smearing, assuming that all the Gaussian weighting functions are aligned in the polar coordinate.

Weighting & Normalization

In all the cases above, the weighting matrix W is calculated on the first call to a smearing function, and includes ~60 q values (finely and evenly binned) below (>0) and above the q range of data in order to smear all data points for a given model and slit/pinhole size. The Norm factor is found numerically with the weighting matrix and applied on the computation of Is.

Note

This help document was last changed by Steve King, 01May2015

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