Minimization of the Variance of Parameters Derived From X-Ray Powder Diffractometer Line Profiles

Wilson, A. J. C.; Thomsen, John S.; Yap, F. Y.

doi:10.1063/1.1754358

Cited by 7 publications

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“…This result is a special case of that given by Wilson, Thomsen and Yap [12]. From eqs (3.4) and (3.8) we may form the optimization ratio…”

Section: Optimization Of Statistical Errormentioning

confidence: 80%

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

Thomsen¹,

Yap²

1968

J. RES. NATL. BUR. STAN. SECT. A.

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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.

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“…This result is a special case of that given by Wilson, Thomsen and Yap [12]. From eqs (3.4) and (3.8) we may form the optimization ratio…”

Section: Optimization Of Statistical Errormentioning

confidence: 80%

“…Subsequently Wilson, Thomsen, and Yap [12] generalized this optimization procedure. The overall problem has been comprehensively treated in a thesis by Yap [13] , which forms the basis for the present paper.…”

Section: Figures In Bracket S Indi Cate the Literature References At mentioning

confidence: 99%

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

Thomsen¹,

Yap²

1968

J. RES. NATL. BUR. STAN. SECT. A.

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“…It has also been shown how the optimization in regard to statistics may be applied to the individual steps in a step-scanned profile (Wilson, Thomsen & Yap, 1965;Wilson, 1967).…”

Section: Optimization Of Effortmentioning

confidence: 99%

Present problems and future opportunities in precise intensity measurements with single-crystal X-ray diffractometers

Young¹

1969

Acta Cryst Sect A

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“…Procedures for optimizing counting times for the achievement of different purposes have been proposed by many authors, but it seems that they have usually been proposals never realized routinely. The following are some typical papers, in order of date: Mack & Spielberg (1958), Zevin, Umanskij, Khejker & Pan6enko (1961), Wilson, Thomsen & Yap (1965), Killean (1967), Wilson (1967), Shoemaker (1968), Thomsen & Yap (1968a,b), Shoemaker & Hamilton (1972), Mackenzie & Williams (1973), Killean (1973), Grant (1973) and Szabo (1978).…”

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confidence: 99%

Relationship between `observed' and `true' intensity; effect of various counting modes

Wilson¹

1980

Acta Cryst Sect A

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be neglected and the familiar formula for shortrange-order diffuse scattering (see e.g. Guinier, 1963, p. 269) is obtained. For values of a much greater than this many such diffuse curves corresponding to higher values of P must be included in the summation. Each of these will represent successively broader more diffuse peaks as r 2p approaches zero. The factor (k 2 t72)P/P! eventually goes to zero as P increases for any tr but for values of a ~_ 1 many terms must be included.References BARTLETT, M. S. (1967). J. R. Stat. Soc. A, 130, 467-477. BESAG, J. (1974). J. R. Stat. Soc. B, 36, 192-225. BR~i, MER, R. (1975 Expressions for the probability P(Ro) that a reflexion of 'true' intensity R will have an observed value R o (possibly negative) are obtained for four counting modes: fixed-time counting, equal and unequal times for total and background; fixed-count timing, equal and unequal counts for total and background. The distributions have a positive excess and are in general skew, though the skewness may be zero for particular choices of unequal times (counts). Deviations from the normal distribution with the same mean and variance may be considerable for IRol ~_ 0 and for IRol large, and may possibly be significant in some applications even for R o ~_ R. This apparent conflict with the central limit theorem is reconciled. 0567-7394/80/060929-08501.00In both single-crystal and powder diffractometry the integrated intensity of a reflexion is obtained as the difference between a counting rate averaged over a region of reciprocal space intended to include the reflected intensity, and a counting rate averaged over a neighbouring volume of reciprocal space intended to include only background. If the intentions are not effectively realized there will be a systematic error in the measured intensity, but the present concern is not with such systematic errors but with statistical fluctuations in the intensity as observed. Although an intensity can never be really negative, it is not uncommon for the measured background counting rate to be higher than the measured reflexion-plus-background rate, giving an

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Minimization of the Variance of Parameters Derived From X-Ray Powder Diffractometer Line Profiles

Cited by 7 publications

References 0 publications

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

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

Present problems and future opportunities in precise intensity measurements with single-crystal X-ray diffractometers

Relationship between `observed' and `true' intensity; effect of various counting modes

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