-                      r98b30b4
+                      r4c42410
 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
-..  _Polydispersity_Distributions:
-Polydispersity Distributions
-----------------------------
-Calculates the form factor for a polydisperse and/or angular population of
-particles with uniform scattering length density. The resultant form factor
-is normalized by the average particle volume such that
-P(q) = scale*\<F*F\>/Vol + bkg
-where F is the scattering amplitude and the\<\>denote an average over the size
-distribution.  Users should use PD (polydispersity: this definition is
-different from the typical definition in polymer science) for a size
-distribution and Sigma for an angular distribution (see below).
-Note that this computation is very time intensive thus applying polydispersion/
-angular distrubtion for more than one paramters or increasing Npts values
-might need extensive patience to complete the computation. Also note that
-even though it is time consuming, it is safer to have larger values of Npts
-and Nsigmas.
-The following five distribution functions are provided
-*  *Rectangular_Distribution_*
-*  *Array_Distribution_*
-*  *Gaussian_Distribution_*
-*  *Lognormal_Distribution_*
-*  *Schulz_Distribution_*
-.. _Rectangular_Distribution:
-Rectangular Distribution
-------------------------
-.. image:: pd_image001.png
-The xmean is the mean of the distribution, w is the half-width, and Norm is a
-normalization factor which is determined during the numerical calculation.
-Note that the Sigma and the half width *w*  are different.
-The standard deviation is
-.. image:: pd_image002.png
-The PD (polydispersity) is
-.. image:: pd_image003.png
-.. image:: pd_image004.jpg
-.. _Array_Distribution:
-Array Distribution
-------------------
-This distribution is to be given by users as a txt file where the array
-should be defined by two columns in the order of x and f(x) values. The f(x)
-will be normalized by SasView during the computation.
-Example of an array in the file
-        0.1
-        0.3
-        0.4
-        0.5
-        0.6
-        0.7
-        0.9
-We use only these array values in the computation, therefore the mean value
-given in the control panel, for example Ã¢â¬Ëradius = 60Ã¢â¬â¢, will be ignored.
-.. _Gaussian_Distribution:
-Gaussian Distribution
----------------------
-.. image:: pd_image005.png
-The xmean is the mean of the distribution and Norm is a normalization factor
-which is determined during the numerical calculation.
-The PD (polydispersity) is
-.. image:: pd_image003.png
-.. image:: pd_image006.jpg
-.. _Lognormal_Distribution:
-Lognormal Distribution
-----------------------
-.. image:: pd_image007.png
-The /mu/=ln(xmed), xmed is the median value of the distribution, and Norm is a
-normalization factor which will be determined during the numerical calculation.
-The median value is the value given in the size parameter in the control panel,
-for example, Ã¢â¬Åradius = 60Ã¢â¬ï¿œ.
-The PD (polydispersity) is given by /sigma/
-.. image:: pd_image008.png
-For the angular distribution
-.. image:: pd_image009.png
-The mean value is given by xmean=exp(/mu/+p2/2). The peak value is given by
-xpeak=exp(/mu/-p2).
-.. image:: pd_image010.jpg
-This distribution function spreads more and the peak shifts to the left as the
-p increases, requiring higher values of Nsigmas and Npts.
-.. _Schulz_Distribution:
-Schulz Distribution
--------------------
-.. image:: pd_image011.png
-The xmean is the mean of the distribution and Norm is a normalization factor
-which is determined during the numerical calculation.
-The z = 1/p2Ã¢â¬â 1.
-The PD (polydispersity) is
-.. image:: pd_image012.png
-Note that the higher PD (polydispersity) might need higher values of Npts and
-Nsigmas. For example, at PD = 0.7 and radisus = 60 A, Npts >= 160, and
-Nsigmas >= 15 at least.
-.. image:: pd_image013.jpg
-.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
-.. _Smearing_Computation:
-Smearing Computation
---------------------
-Slit_Smearing_
-Pinhole_Smearing_
-D_Smearing_
-.. _Slit_Smearing:
-Slit Smearing
--------------
-The sit smeared scattering intensity for SAS is defined by
-.. image:: sm_image002.gif
-where Norm =
-.. 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 q point, respectively. Here, we assumes that the weighting
-function is described by a rectangular function, i.e.,
-.. image:: sm_image006.gif
-Equation 2
-and
-.. image:: sm_image007.gif
-Equation 3
-so that |inlineimage008| |inlineimage009| for |inlineimage010| and u.
-The |inlineimage011| and |inlineimage012| stand for
-the slit height (FWHM/2) and the slit width (FWHM/2) in the q space. Now the
-integral of Equation 1 is simplified to
-.. image:: sm_image013.gif
-Equation 4
-Numerical Implementation of Equation 4
---------------------------------------
-Case 1
-------
-For |inlineimage012| = 0 and |inlineimage011| = constant.
-.. image:: sm_image016.gif
-For discrete q values, at the q values from the data points and at the q
-values extended up to qN= qi + |inlineimage011| the smeared
-intensity can be calculated approximately
-.. image:: sm_image017.gif
-Equation 5
-|inlineimage018| = 0 for *Is* in *j* < *i* or *j* > N-1*.
-Case 2
-------
-For |inlineimage012| = constant and |inlineimage011| = 0.
-Similarly to Case 1, we get
-|inlineimage019| for qp= qi- |inlineimage012| and qN= qi+ |inlineimage012|. |inlineimage018| = 0
-for *Is* in *j* < *p* or *j* > *N-1*.
-Case 3
-------
-For |inlineimage011| = constant and
-|inlineimage011| = constant.
-In this case, the best way is to perform the integration, Equation 1,
-numerically for both slit height and width. However, the numerical integration
-is not correct enough unless given a large number of iteration, say at least
-by 10000 for each element of the matrix, W, which will take minutes and
-minutes to finish the calculation for a set of typical SAS data. An
-alternative way which is correct for slit width << slit hight, is used in
-SasView. This method is a mixed method that combines method 1 with the
-numerical integration for the slit width.
-.. image:: sm_image020.gif
-Equation 7
-for qp= qi- |inlineimage012| and
-qN= qi+ |inlineimage012|. |inlineimage018| = 0 for
-*Is* in *j* < *p* or *j* > *N-1*.
-.. _Pinhole_Smearing:
-Pinhole Smearing
-----------------
-The pinhole smearing computation is done similar to the case above except
-that the weight function used is the Gaussian function, so that the Equation 6
-for this case becomes
-.. image:: sm_image021.gif
-Equation 8
-For all the cases above, the weighting matrix *W* is calculated when the
-smearing is called at the first time, and it includes the ~ 60 q values
-(finely binned evenly) below (\>0) and above the q range of data in order
-to cover all data points of the smearing computation for a given model and
-for a given slit size. The *Norm*  factor is found numerically with the
-weighting matrix, and considered on *Is* computation.
-.. _2D_Smearing:
-D Smearing
------------
-The 2D smearing computation is done similar to the 1D pinhole smearing above
-except that the weight function used was the 2D elliptical Gaussian function
-.. image:: sm_image022.gif
-Equation 9
-In Equation 9, x0 = qcos/theta/ and y0 = qsin/theta/, and the primed axes
-are in the coordinate rotated by an angle /theta/ around the z-axis (below)
-so that xÃ¢â¬â¢0= x0cos/theta/+y0sin/theta/ and yÃ¢â¬â¢0= -x0sin/theta/+y0cos/theta/.
-Note that the rotation angle is zero for x-y symmetric elliptical Gaussian
-distribution. The A is a normalization factor.
-.. image:: sm_image023.gif
-Now we consider a numerical integration where each bins in /theta/ and R are
-*evenly* (this is to simplify the equation below) distributed by /delta//theta/
-and /delta/R, respectively, and it is assumed that I(xÃ¢â¬â¢, yÃ¢â¬â¢) is constant
-within the bins which in turn becomes
-.. image:: sm_image024.gif
-Equation 10
-Since we have found the weighting factor on each bin points, it is convenient
-to transform xÃ¢â¬â¢-yÃ¢â¬â¢ back to x-y coordinate (rotating it by -/theta/ around z
-axis). Then, for the polar symmetric smear
-.. image:: sm_image025.gif
-Equation 11
-where
-.. image:: sm_image026.gif
-while for the x-y symmetric smear
-.. image:: sm_image027.gif
-Equation 12
-where
-.. image:: sm_image028.gif
-Here, 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.
-In the control panel, the higher accuracy indicates more and finer binnng
-points so that it costs more in time.
-.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
-.. _Polarisation_Magnetic_Scattering:
-Polarisation/Magnetic Scattering
---------------------------------
-Magnetic scattering is implemented in five (2D) models
-*  *SphereModel*
-*  *CoreShellModel*
-*  *CoreMultiShellModel*
-*  *CylinderModel*
-*  *ParallelepipedModel*
-In general, the scattering length density (SLD) in each regions where the
-SLD (=/beta/) is uniform, is a combination of the nuclear and magnetic SLDs and
-depends on the spin states of the neutrons as follows. For magnetic scattering,
-only the magnetization component, *M*perp, perpendicular to the scattering
-vector *Q* contributes to the the magnetic scattering length.
-.. image:: mag_vector.bmp
-The magnetic scattering length density is then
-.. image:: dm_eq.gif
-where /gamma/ = -1.913 the gyromagnetic ratio, /mu/B is the Bohr magneton, r0
-is the classical radius of electron, and */sigma/* is the Pauli spin. For
-polarised neutron, the magnetic scattering is depending on the spin states.
-Let's consider that the incident neutrons are polarized parallel (+)/
-anti-parallel (-) to the x' axis (See both Figures above). The possible
-out-coming states then are + and - states for both incident states
-Non-spin flips: (+ +) and (- -)
-Spin flips:     (+ -) and (- +)
-.. image:: M_angles_pic.bmp
-Now, let's assume that the angles of the *Q*  vector and the spin-axis (x')
-against x-axis are /phi/ and /theta/up, respectively (See Figure above). Then,
-depending upon the polarisation (spin) state of neutrons, the scattering length
-densities, including the nuclear scattering length density (/beta/N) are given
-as, for non-spin-flips
-.. image:: sld1.gif
-for spin-flips
-.. image:: sld2.gif
-where
-.. image:: mxp.gif
-.. image:: myp.gif
-.. image:: mzp.gif
-.. image:: mqx.gif
-.. image:: mqy.gif
-Here, the M0x, M0y and M0z are the x, y and z components of the magnetization
-vector given in the xyz lab frame. The angles of the magnetization, /theta/M
-and /phi/M as defined in the Figure (above)
-.. image:: m0x_eq.gif
-.. image:: m0y_eq.gif
-.. image:: m0z_eq.gif
-The user input parameters are M0_sld = DMM0, Up_theta = /theta/up,
-M_theta = /theta/M, and M_phi = /phi/M. The 'Up_frac_i' and 'Up_frac_f' are
-the ratio
-(spin up)/(spin up + spin down)
-neutrons before the sample and at the analyzer, respectively.
-*Note:* The values of the 'Up_frac_i' and 'Up_frac_f' must be in the range
-between 0 and 1.
-.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
 .. _Key_Combinations:

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SasView

Changeset 4c42410 in sasview

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src/sas/perspectives/fitting/media/fitting_help.rst

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