$\chi ^{2}$ Minimisation

$\chi ^{2}$ is a measure of the deviation of the theoretical spectrum and the experimental spectrum, giving a best fit when it is at a minimum. The analysis programs vary the parameters of the theoretical spectrum to find a minimum in $\chi ^{2}$ according to the following formula:

$$\chi^{2} = \frac{1}{N-n} \sum^{N}_{i=1} \left[ \frac{E_{i}-T_{i}}{\sqrt{E_{i}}} \right]^{2}$$ (4.1)

where $N$ is the number of channels (576 in this case) and $n$ is the number of parameters that are free to vary. $E_{i}$ and $T_{i}$ are the number of counts in the $i$th channel of the experimental and theoretical spectra respectively.