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A method for obtaining the error of a peak position due to counting statistics

Published online by Cambridge University Press:  10 January 2013

Bing H. Hwang  and
S. F. Tu
Bing H. Hwang
Affiliation:
Institute of Materials Science and Engineering, National Sun Yat-Sen University, Kaohsiung, Taiwan, Republic of China
S. F. Tu
Affiliation:
Institute of Materials Science and Engineering, National Sun Yat-Sen University, Kaohsiung, Taiwan, Republic of China
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Abstract

A method is described to determine the standard deviation of a peak position due to counting statistics. In this method, the statistical noise for a peak is first simulated based on the model of Poisson distribution and then imposed on the peak. The peak position is then determined by a profile fitting routine. By repeated application of these processes on a given peak, one obtains a number of slightly different peak positions and hence their corresponding standard deviation. This method can be applied to a peak described by any analytical function or a set of any number of digitized points. Investigation of the precision of peak positions was carried out by applying this method on various analytical peaks located at low, medium, and high 2θ angles. The results showed that the standard deviation of a peak position decreases with increasing P/B and P/σ, where P is the net peak count above the background B and σ=(P + B)½ is the estimated standard deviation for noise. It increases with increasing peak width and Δ2θ, the step size used to digitize the peak profiles. It decreases as the 2θ range of simulation and fitting increases up to 2θ=3 FWHMs (full width at half maximum) and remains at a constant value thereafter. The standard deviation of the position of a real peak obtained by ten repeated measurements is 0.0034°; the corresponding result obtained by the current method is 0.0035°. This peak is best fitted by a Pearson-VII function and the standard deviation obtained by this method is 0.013° when only the head portion with intensity ≥85% maximum net intensity is simulated and fitted. The corresponding result obtained by the traditional method of parabola fitting is 0.0055°. Therefore, the precision of a peak position also depends on the peak shape.

Keywords

counting statisticsPoisson distributionprofile fittingsimulationstandard deviation
Type
Research Article
Information
Powder Diffraction , Volume 10 , Issue 3 , September 1995 , pp. 148 - 151
Copyright
Copyright © Cambridge University Press 1995

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References

Devine, T. J., and Cohen, J. T. (1986). “Profile Fitting in Residual Stress Determination,” Adv. X-ray Anal. 29, 89–101.Google Scholar
Huang, T. C., and Parrish, W. (1984). “A Combined Derivative Method for Peak Search Analysis,” Adv. X-ray Anal. 27, 45–52.Google Scholar
Huang, T. C. (1988). “Precision Peak Determination in X-ray Powder Diffraction,” Aust. J. Phys. 41, 201–212.CrossRefGoogle Scholar
James, M. R., and Cohen, J. B. (1977). Adv. X-ray Anal. 20, 291–307.Google Scholar
Kelly, C. F., and Short, M. A. (1971). Residual Stress Measurement by X-ray Diffraction-SAE J784a, edited by Hilley, M. E., Larson, J. A., Jatczak, C. F., and Ricklefs, R. E. (Society of Automotive Engineers, Warrendale, PA), 2nd ed., pp. 51 and 52.Google Scholar
Siemens Application Laboratory (1989). Software Package of Diffrac-AT, Fitting Option (Karlsruhe, German).Google Scholar
Wilson, A. J. C. (1963). Mathematical Theory of X-ray Powder Diffractometry, Philips Technical Library (Eindhoven, The Netherlands).Google Scholar
Wilson, A. J. C. (1967). “Statistical Variance of Line Profile Parameters, Measures of Intensity, Location and Dispersion,” Acta Crystallogr. 23, 888–898.CrossRefGoogle Scholar
Young, R. A., and Wiles, D. B. (1982). “Profile Shape Functions in Rietveld Refinements,” J. Appl. Crystallogr. 15, 430–438.CrossRefGoogle Scholar