Models of the diffraction profile

Gałdecka, E.

doi:10.1107/97809553602060000597

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

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. C. ch. 5.3, pp. 517-519

Section 5.3.3.3.1. Models of the diffraction profile

E. Gałdecka^a

^a Institute of Low Temperature and Structure Research, Polish Academy of Sciences, PO Box 937, 50-950 Wrocław 2, Poland

5.3.3.3.1. Models of the diffraction profile

| top | pdf |

Every measurement is based on a certain model of its object. By `model' we understand here² all the systematized a priori knowledge concerning the given measurement, necessary for planning and performing the experiment and for estimating parameters being determined. The use of an incorrect model results in a bias, i.e. an additional systematic error that may appear aside from physical and geometric aberrations. Therefore, the choice of a well founded model is essential in accurate measurements.

In the case of lattice-parameter determination, the object of direct measurements is a diffraction profile, already mentioned in Subsection 5.3.1.1, and the quantity that is directly determined from the experiment is the Bragg angle $[\theta]$ .

The a priori information about the diffraction profile should define: (i) the way in which the Bragg angle $[\theta]$ is related to the measured profile $[h(\omega)]$ , i.e. a measure of location; (ii) the mean values of the measured intensities within the profile; and (iii) their variances.

(i) In traditional photographic methods, the Bragg angle is determined from the measurement of distance on the film, where points or lines of the most intense blackening are taken into account. The blackening, which corresponds to the recorded intensity, may be estimated qualitatively (`by eye') or quantitatively, by means of a special device. In the second case, the intensity is determined as a function of the coordinates on the photograph, which, in turn, are related to the angular positions of diffracted beams. The distribution so obtained, i.e. the line profile or the diffraction profile, allows more precise measurements of the distances and the determination of $[\theta]$ angles, if a definition of the point $[(\theta_0, h_0)]$ of the profile $[h(\theta)]$ , corresponding to the Bragg angle, i.e. a measure of location, is accepted. The analogous situation appears when the diffraction profile is recorded by means of the counter diffractometer. Then the intensities are measured by a counter, while the angular positions of the detector ( $[2\theta]$ scan) or the sample ( $[\theta]$ scan), or both ( $[\omega]$ – $[2\theta]$ scan), are controlled by stepping motor. The device is normally combined with a computer, which facilitates the data processing.

There are various measures of location of the diffraction profile (Wilson, 1965; Thomsen & Yap, 1968). The most popular are:

(1) the centroid or the centre of gravity, defined as $[\theta_c=\textstyle\int\limits^{\Omega_2}_{\Omega_1} \theta h(\theta)\,{\rm d}\theta\Big/\int\limits^{\Omega_2}_{\Omega_1}\,h(\theta)\,{\rm d}\theta, \eqno (5.3.3.6)]$ where $[\Omega_1]$ and $[\Omega_2]$ are the selected truncation limits;
(2) the median, the value $[\theta_m]$ that equally divides some specified portion of the line profile, i.e. $[\textstyle\int\limits^{\theta_m}_{\Omega_1} h(\theta)\,{\rm d}\theta=\int\limits^{\Omega_2}_{\theta_m} h(\theta)\,{\rm d}\theta; \eqno (5.3.3.7)]$
(3) the geometrical peak – the abscissa value $[\theta_p]$ for which the maximum occurs, i.e. $[[{\rm d} h(\theta)/{\rm d}\theta]_{\theta=\theta_p}=0; \eqno (5.3.3.8)]$
(4) the extrapolated peak or the midchord peak, introduced by Bearden (1933) – the point θ_ep of intersection of two curves, one of them approximating the midpoints of chords drawn through the profile parallel to the abscissa axis (or to the background) and the other approximating the data points (Fig. 5.3.3.2 );

Figure 5.3.3.2| top | pdf |
The extrapolated-peak procedure (after Bearden, 1933).
(5) the single midpoint of a chord θ_mc drawn horizontally at the defined height, $[\alpha H]$ , where H is the peak height and α is the truncation level, $[0\lt\alpha\lt1]$ .

The advantages and disadvantages of these measures of location have been widely discussed (Wilson, 1965, 1967, 1968, 1969; Thomsen & Yap, 1968; Segmüller, 1970; Kirk & Caulfield, 1977; Grosswig, Jäckel & Kittner, 1986; Gałdecka, 1994), the errors, both systematic (biases) and statistical (variances), resulting from each of these definitions being taken into account. The dependence of these errors on the scanning range (truncation limits) is of great importance. Such features of the definitions as their simplicity or current usage were also considered.

The geometrical peak of the least-squares parabola, approximating the data points near the top of the profile, distinguishes itself with the best precision but rather large bias (because of the asymmetry of the profiles met in practice); the extrapolated peak – commonly used in the case of the Bond (1960) method (definition 4) – permits location of the peak with better accuracy and omitting the dispersion error (cf. §5.3.3.4.3.2). The centre of gravity, very useful in theoretical considerations (Wilson, 1963), is strongly dependent on the truncation limits and requires a rather large scanning range. The choice of the definition of the measure of location is the first step of lattice-parameter calculations and also of systematic and statistical error estimation.

In the papers that appeared in the mid-1950's, and which were mainly concerned with powder samples, the centre of gravity as a measure of location was more often used than the peak, probably owing to its property of additivity (the total systematic error in the Bragg angle is a sum of the partial errors related to various physical and apparatus factors) and the estimated errors were consequently referred to this point. The papers were reviewed by Wilson (1963, 1980), one of the authors, in the form of a homogeneous mathematical theory of X-ray powder diffractometry. Some of the formulae describing corrections for displacements of the centroid caused by physical and geometrical factors (collected in convenient tables) proved to be useful for single-crystal methods as well (Smakula & Kalnajs, 1955; Kheiker & Zevin, 1963). Wilson (1963) derived the general formula for calculations of the peak displacements due to various factors. As results from this, the displacements are not additive and, in the case when at least one of the partial distributions is asymmetric, the convolution of the curves [see equation (5.3.1.6)] may lead to an appreciable peak shift, if the distributions are not known. The problem has been treated by Berger (1984, 1986a), who used computer modelling.

In later single-crystal methods, in particular in the Bond (1960) method, the peak position of the profile was determined rather than the centroid and the respective corrections referred to the peak (§5.3.3.4.3.2). As a rule, the corrections that related to the peak position were treated as being independent. In practice, this simplifying assumption can be sufficient in measurements with moderate and even high accuracy. However, if the highest accuracy, say of 1 part in 10⁷, is required, the joint effect of all the aberrations should be considered (the so-called `cross terms' are used besides the main terms). Such considerations [Härtwig & Grosswig, 1989; cf. §5.3.3.4.3.2, point (7)] must be based on a well-founded physical model of the diffraction profile.

(ii) As already mentioned in Subsection 5.3.1.1, the diffraction profile can be described as a convolution of several factors (distributions), namely the wavelength distribution, crystal profile and certain aberration profiles. To the so-obtained net profile [equation (5.3.1.6)], a background should be added – constant in the case of an $[\omega]$ scan (as in one-crystal spectrometers, for example), and more complex (but usually approximated with a straight line within a narrow angular range) in other cases. Thus, to describe accurately the distribution of the mean values of measured intensities, all individual distributions must be given.

Such complete syntheses of the diffraction profile are rarely performed, and only for the highest-accuracy absolute measurements (Härtwig, Hölzer, Förster, Goetz, Wokulska & Wolf, 1994). Since one of the basic factors of the convolution model is the wavelength distribution that characterizes a given source of radiation, its accurate determination and proper scaling in metric units is of primary importance in high-accuracy lattice-parameter measurements. At present, only a few such measurements are reported, which relate to the Cu Kα emission line (Berger, 1986b; Härtwig, Hölzer, Wolf & Förster, 1993; Härtwig, Bąk-Misiuk, Berger, Brühl, Okada, Grosswig, Wokulska & Wolf, 1994) and to the Cu Kβ line (the latter paper). Owing to a relatively simple analytical model proposed by Berger (1986b) to describe the $[K\alpha_{1,2}]$ doublet, the measurement results are easy to handle.

Profiles connected with individual apparatus factors (collimation, for example) can also be, in principle, described analytically, under some simplifying assumptions. Examples of such profiles are distributions related to the vertical divergence of the beam (Eastabrook, 1952) and to the horizontal (in-plane) divergence (Urbanowicz, 1981a). These are general enough, so can be calculated for given apparatus parameters. While performing high-accuracy measurements, however, the validity of all respective accompanying assumptions must be carefully considered (Urbanowicz, 1981b; Härtwig & Grosswig, 1989; Härtwig et al., 1993).

In wider practice, there is a tendency towards using simpler descriptions of the diffraction profile. Often, one of the factors, apart from the spectral distribution, is dominant, and the influence of the other ones can be neglected. Berger (1986b), for example, neglecting small effects of both the vertical divergence and the crystal profile, obtained an analytical model of the measured Cu Kα emission spectrum, with several adjusted parameters, and so managed to determine the pure Cu Kα emission-spectrum profile without the necessity of calculating the deconvolution of the measured spectrum in relation to the horizontal-divergence profile.

The choice of model of the shape of the diffraction profile depends, of course, on the purpose for which it is applied. The simplest possible descriptions are used in low- or medium-accuracy measurements, in which first the measured values of Bragg angles are determined by approximation of the measured profiles with simple analytical functions (polynomials or so-called shape functions), the parameters of which have no physical meaning, and then all necessary corrections are calculated and subtracted from the measured Bragg angles – under the assumption of their additivity, mentioned in (i) – to obtain their true values. Another application of the simple models is just the estimation of systematic and statistical errors of the Bragg-angle determination. The choice and use of such simple models will be shown in §5.3.3.3.2.
(iii) The knowledge of variances (and covariances) of recorded counts is needed to evaluate the goodness of fit while approximating the measured profile with a given model function (appropriate criteria have been formulated by Gałdecka, 1993a,b) and to estimate the precision of the Bragg-angle determination.

Most often, one assumes that the variances of measured intensities are defined by the Poisson statistic, i.e. $[\sigma^2(h)=h, \eqno (5.3.3.9)]$ where h is the intensity in number of counts.

Other factors affecting the statistics of recorded counts and the validity of the assumption [equation (5.3.3.9)] have been taken into consideration by Bačkovský (1965) [see also equations (5.3.3.17) and (5.3.3.18) and the comments on these], Wilson (1965), and Gałdecka (1985). The factors are mostly errors in the angle setting and reading and also fluctuations of the primary-beam intensity, of the counting time, and of the temperature of the sample. The use of automatic scanning can cause correlations between intensities measured at different points in the profile (Gałdecka, 1985).

References

Bačkovský, J. (1965). On the most accurate measurements of the wavelengths of X-ray spectral lines. Czech. J. Phys. B15, 752–759.Google Scholar

Bearden, J. A. (1933). The wavelengths of the silver, molybdenum, copper, iron and chromium Kα₁ lines. Phys. Rev. 43, 92–97.Google Scholar

Berger, H. (1984). A method for precision lattice-parameter measurement of single crystals. J. Appl. Cryst. 17, 451–455.Google Scholar

Berger, H. (1986a). Systematic errors in precision lattice parameter determination of single crystals caused by asymmetric line profiles. J. Appl. Cryst. 19, 34–38.Google Scholar

Berger, H. (1986b). Study of the Kα emission spectrum of copper. X-ray Spectrom. 15, 241–243.Google Scholar

Bond, W. L. (1960). Precision lattice constant determination. Acta Cryst. 13, 814–818.Google Scholar

Eastabrook, J. N. (1952). Effect of vertical divergence on the displacement and breadth of X-ray powder diffraction lines. Br. J. Appl. Phys. 3, 349–352.Google Scholar

Gałdecka, E. (1985). The variances and covariances of measured intensities in precise lattice-constant determination by the Bond method. Structure & statistics in crystallography, edited by A. J. C. Wilson, pp. 137–149. New York: Adenine Press.Google Scholar

Gałdecka, E. (1993a). 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. Acta Cryst. A49, 106–115.Google Scholar

Gałdecka, E. (1993b). Description and peak-position determination of a single X-ray diffraction profile for high-accuracy lattice-parameter measurements by the Bond method. II. Testing and choice of description. Acta Cryst. A49, 116–126.Google Scholar

Gałdecka, E. (1994). The extrapolated-peak method for the peak-position determination of an X-ray diffraction profile, and the accuracy of the Bragg-angle measurements. Sci. Bull. Tech. Univ. Łódź Branch Bielsko-Biała, 22(4), 1–24.Google Scholar

Grosswig, S., Jäckel, K.-H. & Kittner, R. (1986). Peak position determination of X-ray diffraction profiles in precision lattice parameter measurements according to the Bond-method with help of the polynomial approximation. Cryst. Res. Technol. 21, 133–139.Google Scholar

Härtwig, J., Bąk-Misiuk, J., Berger, H., Brühl, H.-G., Okada, Y., Grosswig, S., Wokulska, K. & Wolf, J. (1994). Comparison of lattice parameters obtained from an internal monocrystal standard. Phys. Status Solidi A, 142, 19–26.Google Scholar

Härtwig, J. & Grosswig, S. (1989). Measurement of X-ray diffraction angles of perfect monocrystals with high accuracy using a single crystal diffractometer. Phys. Status Solidi A, 115, 369–382.Google Scholar

Härtwig, J., Hölzer, G., Förster, E., Goetz, K., Wokulska, K. & Wolf, J. (1994). Remeasurement of characteristic X-ray emission lines and their application to line profile analysis and lattice parameter determination. Phys. Status Solidi A, 143, 23–34.Google Scholar

Härtwig, J., Hölzer, G., Wolf, J. & Förster, E. (1993). Remeasurement of the profile of the characteristic Cu Kα emission line with high precision and accuracy. J. Appl. Cryst. 26, 539–548.Google Scholar

Kheiker, D. M. & Zevin, L. S. (1963). Rentgenowskaya diffraktometriya (X-ray diffractometry), Chap. 4. Moscow: Fizmatgiz.Google Scholar

Kirk, D. & Caulfield, P. B. (1977). Location of diffractometer profiles in X-ray stress analysis. Adv. X-ray Anal. 20, 283–289.Google Scholar

Segmüller, A. (1970). Automated lattice parameter determination on single crystals. Adv. X-ray Anal. 13, 455–467.Google Scholar

Smakula, A. & Kalnajs, J. (1955). Precision determination of lattice constants with a Geiger-counter X-ray diffractometer. Phys. Rev. 99, 1737–1743.Google Scholar

Thomsen, J. S. & Yap, Y. (1968). Effect of statistical counting errors on wavelengths criteria for X-ray spectra. J. Res. Natl Bur. Stand. Sect. A, 72, 187–205.Google Scholar

Urbanowicz, E. (1981a). The influence of in-plane collimation on the precision and accuracy of lattice-constant determination by the Bond method. I. A mathematical model. Statistical errors. Acta Cryst. A37, 364–368.Google Scholar

Urbanowicz, E. (1981b). The influence of in-plane collimation on the precision and accuracy of lattice-constant determination by the Bond method. II. Verification of the mathematical model. Systematic errors. Acta Cryst. A37, 369–373.Google Scholar

Wilson, A. J. C. (1963). Mathematical theory of X-ray powder diffractometry. Philips Technical Library. Eindhoven: Centrex Publishing Company.Google Scholar

Wilson, A. J. C. (1965). The location of peaks. Br. J. Appl. Phys. 16, 665–674.Google Scholar

Wilson, A. J. C. (1967). Statistical variance of line-profile parameters. Measures of intensity, location and dispersion. Acta Cryst. 23, 888–898.Google Scholar

Wilson, A. J. C. (1968). Statistical variance of line-profile parameters. Measures of intensity, location and dispersion: Corrigenda. Acta Cryst. A24, 478.Google Scholar

Wilson, A. J. C. (1969). Statistical variance of line-profile parameters. Measures of intensity, location and dispersion: Addendum. Acta Cryst. A25, 584–585.Google Scholar

Wilson, A. J. C. (1980). Accuracy in methods of lattice-parameter measurement. Natl Bur. Stand. (US) Spec. Publ. No. 567. Proceedings of Symposium on Accuracy in Powder Diffraction, NBS, Gaithersburg, MD, USA, 11–15 June 1979.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 5.3, pp. 517-519