PrimStretchedExpFT.rst

PrimStretchedExpFT

.. index:: PrimStretchedExpFT

.. attributes::

.. properties::

Description

Provides the Fourier Transform of the Symmetrized Stretched Exponential Function integrated over each energy bin.

S(Q,E) = \int_{E-\delta E}^{E+\delta E} dE' StretchedExpFT(E', Centre, Tau, Beta)

with StretchedExpFT is fit function :ref:`StretchedExpFT <func-StretchedExpFT>`. Quantity \delta E is evaluated at run time. If we request to evaluate this function over a set of energy values E_0,..,E_N, then 2\delta E = \frac{E_N-E_0}{N}

In the picture below we evaluated function StretcheExpFT at the center of bins of width 0.4 \mu eV (label <i>center_evaluated</i>). However, one should compare the integral of ​the model within each bin against experimental QENS data, because the data is a histogram. Nevertheless, we usually compare the value that the model takes at the center of each bin instead of the integral. This approximation is acceptable when the model does not change much within each bin. This is specially true at high energies when StretcheExpFT is very smooth. Sometimes this center-of-the-bin evaluation is wrong, like in the central bin [-0.2, 0.2]ueV. The integral of StretchedExpFT in this bin is the red dot, and the value of StretchedExpFT at the center of the bin is the black dot at the top. Very different! This situation is likely to arise for samples with a relaxation rate h/Tau \ll \delta E. Finally, the blue line is the evaluation of PrimStretchedExpFT at each bin center, which by design amounts to the integral of StretchedExpFT within each bin. This curve coincides with the red dot for the central bin, as expected. Also, it agrees with the center-of-bin evaluation of StretchedExpFT at high energies, when the center-of-the-bin approximation is acceptable.

Usage

Example - Fit to a QENS signal:

The QENS signal is modeled by the convolution of a resolution function with elastic and PrimStretchedExpFT components. Noise is modeled by a linear background:

S(Q,E) = R(Q,E) \otimes (\alpha \delta(E) + PrimStretchedExpFT(Q,E)) + (a+bE)

Obtaining an initial guess close to the optimal fit is critical. For this model, it is recommended to follow these steps: - In the Fit Function window of a plot MantidWorkbench, construct the model. - Tie parameter Height of PrimStretchedExpFT to zero, then carry out the Fit. This will result in optimized elastic line and background. - Untie parameter Height of PrimStretchedExpFT and tie parameter Beta to 1.0, then carry out the fit. This will result in optimized model using an exponential. - Release the tie on Beta and redo the fit.

.. testcode:: ExamplePrimStretchedExpFT

   # Load resolution function and scattered signal
   resolution = LoadNexus(Filename="resolution_14955.nxs")
   qens_data = LoadNexus(Filename="qens_data_14955.nxs")

   # This function_string is obtained by constructing the model
   # with the Fit Function window of a plot in MantidWorkbench, then
   # Setup--> Manage Setup --> Copy to Clipboard
   function_string  = "(composite=Convolution,FixResolution=true,NumDeriv=true;"
   function_string += "name=TabulatedFunction,Workspace=resolution,WorkspaceIndex=0,Scaling=1,Shift=0,XScaling=1;"
   function_string += "(name=DeltaFunction,Height=1,Centre=0;"
   function_string += "name=PrimStretchedExpFT,Height=1.0,Tau=100,Beta=0.98,Centre=0));"
   function_string += "name=LinearBackground,A0=0,A1=0"

   # Carry out the fit. Produces workspaces  fit_results_Parameters,
   #  fit_results_Workspace, and fit_results_NormalisedCovarianceMatrix.
   Fit(Function=function_string,
      InputWorkspace="qens_data",
      WorkspaceIndex=0,
      StartX=-0.15, EndX=0.15,
      CreateOutput=1,
      Output="fit_results")

   # Collect and print parameters for PrimStrechtedExpFT
   parameters_of_interest = ("Tau", "Beta")
   values_found = {}
   ws = mtd["fit_results_Parameters"]  # Workspace containing optimized parameters
   for row_index in range(ws.rowCount()):
      full_parameter_name = ws.row(row_index)["Name"]
      for parameter in parameters_of_interest:
         if parameter in full_parameter_name:
            values_found[parameter] = ws.row(row_index)["Value"]
            break
   if values_found["Beta"] > 0.63 and values_found["Beta"] < 0.71:
      print("Beta found within [0.63, 0.71]")
   if values_found["Tau"] > 54.0 and values_found["Tau"] < 60.0:
      print( "Tau found within [54.0, 60.0]")

.. testcleanup:: ExamplePrimStretchedExpFT

   DeleteWorkspace("resolution")
   DeleteWorkspace("qens_data")
   DeleteWorkspace("fit_results_Parameters")
   DeleteWorkspace("fit_results_Workspace")
   DeleteWorkspace("fit_results_NormalisedCovarianceMatrix")

Output:

.. testoutput:: ExamplePrimStretchedExpFT

   Beta found within [0.63, 0.71]
   Tau found within [54.0, 60.0]

.. categories::

.. sourcelink::
   :py: Framework/PythonInterface/plugins/functions/PrimStretchedExpFT.py