I have some high quality synchrotron powder XRD data, gathered in a Debye-Scherrer geometry. I've calculated relative peak areas by fitting Gaussian/Lorentzian peaks using Igor and calculated structure factors using
I = |F|2p (1 + cos2 2θ) / (sin2θ cos θ)
where I is the relative intensity, F is the structure factor, p is the multiplicity factor and θ is the Bragg angle. (This is Eq. (4-19) in Cullity, 2nd Ed.)
I now need to estimate an uncertainty in these structure factors. Can anyone provide an expression for this, or point me towards some relevant references?
Dr. Ivanova,
Thanks for your response. I agree that the temperature factor could be included, and perhaps I should modify my expression for the intensity. I don't believe your first equation is correct, however, since it is the intensity that is proportional to e-2M and NOT the structure factor.
Your second equation and the references you list seem to deal with the error in atomic coordinates derived from XRD results and with statistical methods for locating atoms in protein molecules. Since these are not the type of calculations I'm trying to perform, they are irrelevant to my question about the uncertainty in the structure factor, F.
Any time you obtain parameters from a fitting procedure, you can derive numerical uncertainties from the covariance matrix. Question is, does your software give you access to this? In matlab, the lsqcurvefit routine can output the Jacobian, J, ==> the unscaled Covariance Matrix ==> The SDev of every parameter using a scale parameter S = residual/(# of independent points). This gets you uncertainties in I, theta. Then standard error propagation methods get you uncertainty in F (looks ugly).
There is so much stuff in Igor exchange, this must be in some fitting package somewhere. But I do not use Igor so cannot help you find it.
Two problems:
1 You have to estimate the # independent data points < actual number of data points,
2 The derived uncertainty might be small if the fit is well defined, but changing method e.g. background subtraction or fitted curve shape, might move the fit to different values, and this uncertainty, often larger, is not taken into account. (Only way I know is to try the fit several times with different choices, see what happens)
Thanks for your contribution. In fact, Igor spits out the uncertainties in both the peak position (i.e. theta) and the peak area (i.e. I). I'd hoped someone could point me towards a discussion of the error propagation - particularly in the more complicated case where the absorption factor and the temperature factor are included - but I see I have some maths to do! :-)
The major component in diffraction intensity measurements will be counting statistics, which is simply the square root of each intensity measurement. This statistical error must be propagated when calculating the integrated intensity of each peak. and then the structure factor. This statistical error has to be taken from the raw data before background correction. The other major cause of error in intensity would depend on the detector used. If it is a point detector then you may need dead time corrections. If it is an area detector than you must take into account dark current and readout noise and linearity of response. These errors will dominate over any uncertainty in 2theta position at a properly calibrated Synchrotron diffraction beamline.
One last thing. I'm not sure which Synchrotron you used, but if it was not in top-up mode (constant ring current) then the incident flux will be decreasing with time (unless the monochromator had feedback in constant intensity mode, although that normally causes a slow shift in energy as the heat load on the first crystal changes). Even in top-up mode with constant ring current, there can be fluctuations in incident flux due to the limits in monochromator stability. This may be stating the obvious which you have already done, but it important to normalize the intensity measurements to the incident flux monitor.
Was this in transmission? Hopefully the x-ray energy and sample thickness allows you to ignore dynamical effects, otherwise... ouch good luck.
OK, I was not sure what your problem was. The error propagation w/o absorption and temperature is relatively straight forward (though complicated) as long as errors are reasonably small. If you have the covariance matrix then you know the cross-correlation of the variables and you can use the standard equation (attached as pdf). If not you have to assume I, theta are independent and set the covariance to zero. The required partial derivatives of 1/f(theta) will be tedious to get correct - that is just what Mathematica is for.
The above assumes that the derived peak areas for adjacent peaks are independent. If there is overlap, this may not be the case. Then the error in Fn also depends on the cross-correlation of I(n-1), I(n) etc. In principle including such effects and including more variables (absorption, temperature) just puts more terms into the equation - all terms with similar forms. But the number of terms soon explodes, and in practice you really hope that some uncertainties and many cross-correlations are small enough to neglect.
Most texts on errors discuss this, but few are easy reading. You might try the attached link and the references there as a first step.
http://seismo.berkeley.edu/~kirchner/eps_120/Toolkits/Toolkit_05.pdf
One further comment about fitting high quality X-ray data:
Dr Margulies correctly notes that the major statistical uncertainty is the shot noise, the square root of the raw count. Most fitting software assumes that each data point has the same uncertainty, unless some weighting function is entered, and without that the fit will overweight the high data values (they have low fractional but high absolute uncertainties).
IF the background that has been subtracted is small, so the data is a reasonable estimate of the raw count, then this problem is removed by fitting the square root of the data, not the data itself.
(x+-dx)½ =~ x½(1 +- dx/2x). When dx = x½ this is x½(1 +- x½/2x) = x½ +- ½
and all points now have the same uncertainty.
I have never seen this trick of mine published anywhere.
Details of the structure factors calculation are given by Carmelo Giacovazzo (Editor) in "Fundamentals of Crystallography, IUCr texts, 1992" on pages 87-88 and Appendix 2.1.
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