.. sm_help.rst
.. This is a port of the original SasView html help file to ReSTructured text
.. by S King, ISIS, during SasView CodeCamp-III in Feb 2015.
.. |inlineimage004| image:: sm_image004.gif
.. |inlineimage005| image:: sm_image005.gif
.. |inlineimage008| image:: sm_image008.gif
.. |inlineimage009| image:: sm_image009.gif
.. |inlineimage010| image:: sm_image010.gif
.. |inlineimage011| image:: sm_image011.gif
.. |inlineimage012| image:: sm_image012.gif
.. |inlineimage018| image:: sm_image018.gif
.. |inlineimage019| image:: sm_image019.gif
.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
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*
.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
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
.. image:: sm_image002.gif
where *Norm* is given by
.. image:: 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
.. image:: sm_image006.gif
**[Equation 2]**
and
.. image:: 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
.. image:: 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.**
.. image:: sm_image016.gif
For discrete *q* values, at the *q* values of the data points and at the *q*
values extended up to *q*\ :sub:`N`\ = *q*\ :sub:`i` + |inlineimage011| the smeared
intensity can be approximately calculated as
.. image:: sm_image017.gif
**[Equation 5]**
where |inlineimage018| = 0 for *I*\ :sub:`s` when *j* < *i* or *j* > *N-1*.
Solution 2
^^^^^^^^^^
**For** |inlineimage012| **= constant and** |inlineimage011| **= 0.**
Similar to Case 1
|inlineimage019| for *q*\ :sub:`p` = *q*\ :sub:`i` - |inlineimage012| and *q*\ :sub:`N` = *q*\ :sub:`i` + |inlineimage012|
**[Equation 6]**
where |inlineimage018| = 0 for *I*\ :sub:`s` 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
.. image:: sm_image020.gif
**[Equation 7]**
for *q*\ :sub:`p` = *q*\ :sub:`i` - |inlineimage012| and *q*\ :sub:`N` = *q*\ :sub:`i` + |inlineimage012|
where |inlineimage018| = 0 for *I*\ :sub:`s` when *j* < *p* or *j* > *N-1*.
.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
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
.. image:: sm_image021.gif
**[Equation 8]**
.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
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
.. image:: sm_image022.gif
**[Equation 9]**
In Equation 9, *x*\ :sub:`0` = *q* cos(|theta|), *y*\ :sub:`0` = *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'*\ :sub:`0` = *x*\ :sub:`0` cos(|theta|) +
*y*\ :sub:`0` sin(|theta|) and *y'*\ :sub:`0` = -*x*\ :sub:`0` sin(|theta|) +
*y*\ :sub:`0` cos(|theta|). Note that the rotation angle is zero for a x-y symmetric
elliptical Gaussian distribution. The *A* is a normalization factor.
.. image:: 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
.. image:: 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
.. image:: sm_image025.gif
**[Equation 11]**
where
.. image:: sm_image026.gif
while for a *x-y* symmetric smear
.. image:: sm_image027.gif
**[Equation 12]**
where
.. image:: 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.
.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
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 *I*\ :sub:`s`.
.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
.. note:: This help document was last changed by Steve King, 01May2015