\(\renewcommand\AA{\unicode{x212B}}\)

StretchedExpFT

Properties (fitting parameters)

Name	Default	Description
Height	0.1	Intensity at the origin
Tau	100.0	Relaxation time
Beta	1.0	Stretching exponent
Centre	0.0	Centre of the peak

Description

Provides the Fourier Transform of the Symmetrized Stretched Exponential Function

\[S(Q,E) = Height \int_{-\infty}^{\infty} dt/h \cdot e^{-i2\pi (E-Centre)t/h} \cdot e^{-|\frac{t}{Tau}|^{Beta}} )\]

with \(h\) Planck’s constant. If the energy units of energy are micro-eV, then tau is expressed in pico-seconds. If E-units are micro-eV then tau is expressed in nano-seconds.

Properties:

• Normalization \(\int_{-\infty}^{\infty} dE \cdot S(Q,E) = Height\)

• Maximum \(S(Q,E\equiv 0)=Height \cdot Tau \cdot Beta^{-1} \cdot \Gamma(Beta^{-1})\)

Usage

Note

To run these usage examples please first download the usage data, and add these to your path. In Mantid this is done using Manage User Directories.

Example - Fit to a QENS signal:

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

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

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 in MantidWorkbench, construct the model. - Tie parameter \(Height\) of StretchedExpFT to zero, then carry out the Fit. This will result in optimized elastic line and background. - Untie parameter \(Height\) of StretchedExpFT 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.

# 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=StretchedExpFT,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 StrechtedExpFT
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]")

Output:

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

Categories: FitFunctions | QuasiElastic

Source

Python: StretchedExpFT.py