.. 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