Effect of statistical counting errors on wavelength criteria for x-ray spectra

Abstract

Various features of the spectral profil e of an x-ray on e ca n be meas ured with an uncertainty whi ch is onl y a s mall fraction of the observe d line width. With rece nt improve me nts in me asure me nt tech· niqu es, statisti cal errors due to th e ra ndom Au ctuations of th e intensities in counter recordin gs may beco me signifi cant. The prese nt study co nsiders th e effec t of s uch errors on se ver al features of the line profil e whi ch could be used for definition of its wave length. These may be broadly classified into three groups, viz , th e peak, the centroid , and the median. In th e prese nt an alysis the s tatistical eu ors associated with these features are co mpared theore ticall y, with the ass umption of negligible error in angul ar meas ureme nt. Certai n sys temati c errors are also brie Ay exa min ed. The effects of trun cation range, asymmetry, and bac kground intensit y are co ns idered, as well as possibl e optimi z.ation of th e data-taking procedu re.In general, fT , the standa rd dev iation of the wave length , is give n by fT/W = F/(lvn' /2, where IV is the full width at half-maximum intensity, Iv the peak intensit y, T the total countin g ti me, and F a dim ensionless fac tor of the order of unity. Thu s. F may be rega rd ed as a fac tor of me rit for comparing th e various cases, a low valu e of F bein g des irable . Wh en the form of th e on e pro fil e is known a priori , it is usually bes t to make use of thi s kn owl edge; e.g. , a Lore ntzi an ca n be thu s fitted with F "" 0.8 for any of the three wavelength features. Us in g optimize d truncation ranges a nd in cludin g the error in locating end points, one obtain s ap prox im ate ly thi s sa me F for the ce ntroid or medi an even without prior knowl edge of the profil e. In th e latt er case the val ue of F for the pea k usuall y ra nges from abo ut 1.6 to 2. 1. Howeve r, the peak is less subject to c ertain syste mati c errors a nd is prefera ble from th e view point of sim plic ity and hi storical precede nt. It is reco mmen ded that use of the pea k be co ntinued at present; further stud y of th e proble m from the viewpoi nt of atomi c e nergy level interpretation would be desirable.

The Measurement of Residual Stresses by X-Ray Diffraction Techniques

“…Treatments of counting statistics for the different numerical methods for determining line-profiles can be found in the literature (Wilson, 1967;Thomsen and Yap, 1968).…”

The Monolithic Double-Crystal Spectrometer: Theory and Design Principles

“…Thus, it should be possible, in principle, to determine the absolute energy scale to an equal accuracy using the MDCS. In practice, however, several sources of errors like the measurements statistics (Thomsen & Yap, 1968), the residual asymmetry due to miscuts of the crystal's surface with respect to the reflecting planes (Deutsch & Hart, 1982), the centroid shift of the crystal reflection curves due to nonzero absorption (Thomsen, 1974) etc. will limit the actual accuracy achieved in a particular measurement.…”

Description and peak-position determination of a single X-ray diffraction profile for high-accuracy lattice-parameter measurements by the Bond method. I. An analysis of descriptions available

“…This causes the results of polynomial approximation to be strongly dependent on the scanning range and the degree of the polynomial (cf. Wilson, 1965;Thomsen & Yap, 1968;Grosswig, J~ickel & Kittner, 1986). To obtain accurate results, the scanning range and/or the degree of the polynomial should be carefully selected, based on reasonable criteria resulting from a statistical model of recorded counts (discussed in § 2.6 below).…”

“…To discuss the shape of the diffraction profile h(to) separately from the actual parameters of the profile the maximum intensity H, the peak position tOp and the half-width toh -the profile can be described using the following standarization [used by Thomsen & Yap (1968) and generalized by the author of the present paper]:…”

“…Originally, ~0 was proportional to tan 0; here, according to assumption (2) in § 2.1, the dependence on 0 is not taken into account. For moderate asymmetry, a simpler linear factor (Thomsen & Yap, 1968),…”