5.3.4 Peak Fitting

The observed peaks were asymmetric with a pronounced tail towards
lower energies. After correction for detection efficiency functions
of the following type [14,27,28] were fitted
to every peak in energy space:

(5.12) |

(5.13) |

the total area under equals to:

(5.14) |

This is used to normalize the function to an area of 1:

(5.15) |

According to [28] the parameters and are
now substituted by

introducing two new constants and .
This choice will be justified below. The function
may now be written as

where denotes the decay constant of the exponential function in and the width of the gaussian function .

Shifting the function
along the x-axis
by
such that the mean value is equal to 0
yields:

(5.23) | |||

(5.24) | |||

(5.25) |

and again substituting and :

(5.26) |

Shifting the function so that the mean value is equal to (note: not ) and scaled to an enclosed area of yields:

with the following meaning of the parameters:

The peak width of is calculated using Equations 5.17 and 5.18:

(5.28) |

Instead of using the mean value of
it is also possible to use the mean value of the gaussian part
of
using Equations 5.16
and 5.18:

(5.29) |

(5.30) |

Using the erf function instead of erfc and some transformations one obtains:

The advantage of this type of function is that values of interest
such as mean value, peak width etc. may all be directly calculated
from the fit parameters. As a further feature this type of function
allows the separation of the symmetrical from the asymmetrical part
of the fitted peak. This makes it possible to deconvolve the approximately
symmetrical contribution of the energy distribution of the primary
beam using the assumption that the energy distribution of the primary
beam is gaussian. Additionally if molecules are used as primary particles
the broadening due to dissociation can be estimated. With the definitions
used in Equation 5.22 the symmetrical part can
be rewritten as convolution of two gaussian functions:

(5.32) |

denotes the energy, the approximately gaussian distribution of the primary energy, the combined gaussian distributed energy loss in front of the surface and the broadening due to dissociation of molecules (see below for a more accurate calculation).

The width of the symmetrical part is then given by:

(5.33) |

This decomposition works well for atoms as primary particles where is approximately half the value of . For primary molecules dominates. In this case the gaussian approximation might no longer be valid. A simple model for the broadening due to dissociation is presented in the next chapter but it is clear that these simple calculations have their limitations.

March 2001 - Martin Wieser, Physikalisches Institut, University of Berne, Switzerland