I was refining my XRD data in FullProf. Highest intensity was 400000. The fitting was good but still the Chi^2 value is very high (0.174E+04) I divided the intensity by some constant factor (say 50) and Chi^ reduced very significantly though the fitted profile is not as good as earlier (visually). What is the reason ? Is there any way to reduce Chi^2 without dividing the intensity? Thanks in advance !!!
Dear Biprojit Sana ,
when dividing by a large factor you carelessly modify the noise distribution of the counting statistics of your XRD pattern.
An example:
for a counted number of N you have a noise expectation of about sqrt(N).
When numerically scaling down the N to 1/50 ( N to N/50) the noise goes down to 1/50*sqrt(N)= 0,02*sqrt(N).
But when measuring only 1/50 of the time you will get also N/50 counts as expected value but the noise now is
sqrt(N/50) = Sqrt(1/50)*sqrt(N) ~ 0,14*sqrt(N).
So when numerically dividing your XRD pattern by 50 you lead the fitting software to believe a 7 times (=sqrt(50)) smaller noise, which has direct consequences onto the Chi².
Dear Biprojit Sana ,
when dividing by a large factor you carelessly modify the noise distribution of the counting statistics of your XRD pattern.
An example:
for a counted number of N you have a noise expectation of about sqrt(N).
When numerically scaling down the N to 1/50 ( N to N/50) the noise goes down to 1/50*sqrt(N)= 0,02*sqrt(N).
But when measuring only 1/50 of the time you will get also N/50 counts as expected value but the noise now is
sqrt(N/50) = Sqrt(1/50)*sqrt(N) ~ 0,14*sqrt(N).
So when numerically dividing your XRD pattern by 50 you lead the fitting software to believe a 7 times (=sqrt(50)) smaller noise, which has direct consequences onto the Chi².
Here actually the reduction of background-noise (absolute values, not relative, that remains constant for division by a constant factor) is helping to reduce χ2.
Apart from χ2, there are a number of other parameters too, that determine the goodness of fit including Rexp, Rwp etc that need some attention. x2 gives an overall measure of goodness of fit, but the residual factors have to be considered for a publishable fitting data.
Thank you Gerhard Martens , Volker Kahlenberg and Souvik Bhattacharjee .
The equation for chi square is normally X2=Rwp2/Rexp2. When yoor pattern has a "high" intensity the counting statistics to do that the Rexp decreases and the X2 has a high value despite of the good agreement factors. Dividing the pattern the Rexp increases and the reduced chi square decreases with the same good agreement factors. Ideally to measure the powder pattern is necesary "to control" the intensisty, changing the exposition time of the sample, and avoid this type of problem.
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