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Exercise 6
The aim of this exercise is to implement a function to fit the output data
from Exercise 4. For simplicity a solution file, 11001_deltaE.nxs,
is provided with the training data.
The peak can be fairly well approximated using a Lorentz function:
\[\LARGE \frac{A}{π}(\frac{\frac{Γ}{2}}{(x-c)^2 + (\frac{Γ}{2})^2})\]
where A is the amplitude, Γ is the full width at half maximum and
c is the peak centre. We will first define this as a simple 1D function
and then improve it to use the peak function capabilities.
Simple 1D
- Define a new 1D function called
Lorentz.
- It should have 3 parameters corresponding to the parameters described above.
- Write the
function1D method that evaluates the required values from the
input x data using the definition as above. (Hint: you can use the python
math module for
pi).
Test this implementation:
- Load the data file.
- Plot the spectrum.
- Use the fit browser (using the fit function tool button (looks like a peak
with a vertical red line on top)).
- Right click on plot and select “Add other function…”.
- You’ll want to see how the fit progresses so set the log level in the
Messages Box to information by right clicking in the
window and selecting Log Level->Information. This will display additional
information as the fit proceeds.
- You may need to adjust the parameter initial values in the Fit Function window
Analytical Derivative
Extend the above 1D function and add an analytical derivative by adding a
functionDeriv1D method. The derivatives w.r.t to each of the parameters
are as follows:
\[ \begin{align}\begin{aligned}\LARGE A \longrightarrow \frac{2}{π}\frac{Γ}{Γ^2 + 4(x - c)^2}\\\LARGE c \longrightarrow \frac{A}{π}\frac{Γ(x - c)}{[\{\frac{Γ}{2}\}^2 + (x - c)^2]^2}\\\LARGE Γ \longrightarrow - \frac{2A}{π}\frac{Γ^2 - 4(x - c)^2}{[Γ^2 + 4(x - c)^2]^2}\end{aligned}\end{align} \]
Re-run the fit using the above steps.
Peak Function
- Make a copy of the Lorentz function and rename it to LorentzPeak.
- Make this class an
IPeakFunction instead of IFunction1D and change
the methods from function1D to functionLocal and functionDeriv1D
to functionDerivLocal.
- Add the required methods
for Mantid to interact with this as a peak function.
- Retest using the steps above with the exception that when you right click on
the plot choose the “Add peak…” menu rather than “Add other function…”.
- You should now have interactivity in the GUI where you can set the initial
values using the tools and get a quicker fit.
Once finished check your answer with the provided Exercise 6 Solutions
