Effect of Statistical Counting Errors on Wavelength Criteria for X-Ray Spectra

Various features of the spectral profile of an x-ray line can be measured with an uncertainty which is only a small fraction of the observed line width. With recent improvements in measurement techniques, statistical errors due to the random fluctuations of the intensities in counter recordings may become significant. The present study considers the effect of such errors on several features of the line profile which could be used for definition of its wavelength. These may be broadly classified into three groups, viz, the peak, the centroid, and the median. In the present analysis the statistical errors associated with these features are compared theoretically, with the assumption of negligible error in angular measurement. Certain systematic errors are also briefly examined. The effects of truncation range, asymmetry, and background intensity are considered, as well as possible optimization of the data-taking procedure. In general, σ, the standard deviation of the wavelength, is given by σ/W = F/(I T)^1/2, where W is the full width at half-maximum intensity, I the peak intensity, T the total counting time, and F a dimensionless factor of the order of unity. Thus F may be regarded as a factor of merit for comparing the various cases, a low value of F being desirable. When the form of the line profile is known a priori, it is usually best to make use of this knowledge; e.g., a Lorentzian can be thus fitted with F ≈ 0.8 for any of the three wavelength features. Using optimized truncation ranges and including the error in locating end points, one obtains approximately this same F for the centroid or median even without prior knowledge of the profile. In the latter case the value of F for the peak usually ranges from about 1.6 to 2.1. However, the peak is less subject to certain systematic errors and is preferable from the viewpoint of simplicity and historical precedent. It is recommended that use of the peak be continued at present; further study of the problem from the viewpoint of atomic energy level interpretation would be desirable.