$$

BackToBackExponential

Description

A back-to-back exponential peakshape function is defined as:

$$I\frac{AB}{2(A+B)}[\exp \left(\frac{A[A{S}^2+2(x-X0)]}{2}\right)\text{erfc}\left(\frac{A{S}^2+(x-X0)}{S\sqrt{2}}\right)+\exp \left(\frac{B[B{S}^2-2(x-X0)]}{2}\right)\text{erfc}\left(\frac{[B{S}^2-(x-X0)]}{S\sqrt{2}}\right)].$$

This peakshape function represent the convolution of back-to-back exponentials and a Gaussian function and is designed to be used for the data analysis of time-of-flight (TOF) neutron powder diffraction data, see Ref. 1.

The parameters $A$ and $B$ represent the absolute value of the exponential rise and decay constants (modelling the neutron pulse coming from the moderator) and $S$ represent the standard deviation of the Gaussian. The parameter $X0$ is the location of the peak; more specifically it represent the point where the exponentially modelled neutron pulse goes from being exponentially rising to exponentially decaying. In general $X0$ does not conicide with the maximum of the peak (it is typically at slightly lower TOF than the maximum, by an amount that depends on $A$, $B$ and $S$). $I$ is the integrated intensity.

For information about how to convert Fullprof back-to-back exponential parameters into those used for this function see CreateBackToBackParameters.

References

1. R.B. Von Dreele, J.D. Jorgensen & C.G. Windsor, J. Appl. Cryst., 15, 581-589, 1982

The figure below illustrate this peakshape function fitted to a TOF peak:

Properties (fitting parameters)

Name	Default	Description
I	0.0	integrated intensity of the peak
A	1.0	exponential constant of rising part of neutron pulse
B	0.05	exponential constant of decaying part of neutron pulse
X0	0.0	peak position
S	1.0	standard deviation of gaussian part of peakshape function

Note

the initial default guesses for in particular A and B are only based on fitting a couple of peaks in a dataset collected on the ISIS’s HRPD instrument.

Categories: FitFunctions | Peak

Source

C++ header: BackToBackExponential.h

C++ source: BackToBackExponential.cpp