Merge pull request #22971 from mantidproject/22966_IndirectDataAnalysisDocs

Indirect - Data Analysis Interfaces - Documentation Update

docs/source/concepts/ConvFit.rst

+.. _ConvFitConcept:
+
+Conv Fit
+========
+
+The measured data :math:`I(Q, \omega)` is proportional to the convolution of the
+scattering law :math:`S(Q, \omega)` with the resolution function :math:`R(Q,
+\omega)` of the spectrometer via :math:`I(Q, \omega) = S(Q, \omega) ⊗  R(Q,
+\omega)`. The traditional method of analysis has been to fit the measured
+:math:`I(Q, \omega)` with an appropriate set of functions related to the form of
+:math:`S(Q, \omega)` predicted by theory.
+
+* In quasielastic scattering the simplest form is when both the :math:`S(Q,
+  \omega)` and the :math:`R(Q, \omega)` have the form of a Lorentzian - a
+  situation which is almost correct for reactor based backscattering
+  spectrometers such as IN10 & IN16 at ILL. The convolution of two Lorentzians
+  is itself a Lorentzian so that the spectrum of the measured and resolution
+  data can both just be fitted with Lorentzians. The broadening of the sample
+  spectrum is then just the  difference of the two widths.
+* The next easiest case is when both :math:`S(Q, \omega)` and :math:`R(Q,
+  \omega)` have a simple functional form and the convolution is also a function
+  containing the parameters of the :math:`S(Q, \omega)` and R(Q,  \omega) functions.
+  The convoluted function may then be fitted to the data to provide the
+  parameters. An example would be the case where the :math:`S(Q, \omega)` is a
+  Lorentzian and the :math:`R(Q, \omega)` is a Gaussian.
+* For diffraction, the shape of the peak in time is a convolution of a Gaussian
+  with a decaying exponential and this function can be used to fit the Bragg
+  peaks.
+* The final case is where :math:`R(Q, \omega)` does not have a simple function
+  form so that the measured data has to be convoluted numerically with the
+  :math:`S(Q, \omega)` function to provide an estimate of the sample scattering.
+  The result is least-squares fitted to the measured data to provide values for
+  the parameters in the :math:`S(Q, \omega)` function.
+
+This latter form of peak fitting is provided by SWIFT. It employs a
+least-squares algorithm which requires the derivatives of the fitting function
+with respect to its parameters in order to be faster and more efficient than
+those algorithms which calculate the derivatives numerically. To do this the
+assumption is made that the derivative of a convolution is equal to the
+convolution of the derivative-as the derivative and the convolution are
+performed over different variables (function parameters and energy transfer
+respectively) this should be correct. A flat background is subtracted from the
+resolution data before the convolution is performed.
+
+Four types of sample function are available for :math:`S(Q, \omega)`:
+
+Quasielastic
+  This is the most common case and applies to both translational (diffusion) and
+  rotational modes, both of which have the form of a Lorentzian. The fitted
+  function is a set of Lorentzians centred at the origin in energy transfer.
+
+Elastic
+  Comprising a central elastic peak together with a set of quasi-elastic
+  Lorentzians also centred at the origin. The elastic peak is taken to be the
+  un-broadened resolution function.
+
+Shift
+  A central Lorentzian with pairs of energy shifted Lorentzians. This was
+  originally used for crystal field splitting data but more recently has been
+  applied to quantum tunnelling peaks. The fitting function assumes that the
+  peaks are symmetric about the origin in energy transfer both in position and
+  width. The widths of the central and side peaks may be different.
+
+Polymer
+  A single quasi-elastic peak with 3 different forms of shape. The theory behind
+  this is described elsewhere [1,2]. Briefly, polymer theory predicts 3 forms
+  of the :math:`I(Q,t)` in the form of :math:`exp(-at2/b)` where :math:`b` can
+  be 2, 3 or 4. The Full Width Half-Maximum (FWHM) then has a Q-dependence
+  (power law) of the form :math:`Qb`. The :math:`I(Q,t)` has been numerically
+  Fourier transformed into :math:`I(Q, \omega)` and the :math:`I(Q, \omega)`
+  have been fitted with functions of the form of a modified Lorentzian. These
+  latter functions are used in the energy fitting procedures.
+
+References:
+
+1. J S Higgins, R E Ghosh, W S Howells & G Allen, `JCS Faraday II 73 40 (1977) <http://dx.doi.org/10.1039/F29777300040>`_
+2. J S Higgins, G Allen, R E Ghosh, W S Howells & B Farnoux, `Chem Phys Lett 49 197 (1977) <http://dx.doi.org/10.1016/0009-2614(77)80569-1>`_
+
+
+.. categories:: Concepts