International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 5.2, pp. 491-492

## Section 5.2.1.4. Bragg angle: operational definitions

W. Parrish,a A. J. C. Wilsonb and J. I. Langfordc

aIBM Almaden Research Center, San Jose, CA, USA,bSt John's College, Cambridge CB2 1TP, England, and cSchool of Physics & Astronomy, University of Birmingham, Birmingham B15 2TT, England

#### 5.2.1.4. Bragg angle: operational definitions

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The Bragg angles are determined from the observations by a series of operations that are often quite complex.

For film cameras of diameter 57.3 or 114.6 mm, a simple measurement with a millimetre scale gives θ in degrees (1 mm = 1 or 0.5°). This determination is crude, and ordinarily the lines on the film would be measured with a low-power travelling microscope or a densitometer. The effective camera diameter is found from measurements of fiducial marks imprinted on the film, or by use of the Straumanis film mounting. References to detailed descriptions are given in Section 2.3.4 .

For Bragg–Brentano (Parrish) and Seemann–Bohlin diffractometers, rate-meter measurements with strip-chart recordings have time-constant errors, and precision measurements require step-scanning (Subsection 2.3.3.5 ). The data may be analysed to give one or more of the following measures of position:

 (a) The centroid of the reflection (Subsection 2.3.3.3 ). (b) The peak of the reflection (Subsection 2.3.3.3 ). The extrapolated mid-point of chords is a kind of modified peak determination, but the best method of locating peaks so far in operation is that called `peak search' (Subsection2.3.3.7 ). (c) Profile fitting (Subsection 2.3.3.8 ). In principle, profile fitting could give the Bragg angle corresponding to any desired feature of the diffraction maximum (centroid peak, median, ), but it has been used in practice mainly for locating the Bragg angle corresponding to the peak.

As usual, it is necessary to distinguish between the precision (reproducibility) of a measurement and its accuracy (extent to which it is affected by systematic errors). In principle, it does not matter if the Bragg angle obtained by any of the above operations is affected by systematic errors, as these can be calculated and allowed for, as described in the following paragraphs. The most precise methods are the peak-search and individual profile-fitting computer procedures. They are routinely capable of a precision of about for reasonably sharp reflections, and are free from the subjective effects that may influence, for example, film measurements or the graphical extrapolation of the mid-points of chords. As well as a measure of the peak position, the peak-search procedure gives a measure of the peak intensity, and the profile-fitting procedure gives a measure of the peak intensity and (if desired) a measure of the integrated intensity.