ESTIMATES OF THE UNCERTAINTY OF LATTICE PARAMETERS REFINED FROM NEUTRON POWDER DIFFRACTION DATA

Burkhard Peplinski^a, Daniel M. Toebbens^b, Winfried Kockelmann^c, Richard Ibberson^c

^a Federal Institute for Materials Research and Testing (BAM),
Richard-Willstätter-Str. 11, D-12489 Berlin, Federal Republic of Germany
^b Hahn-Meitner-Institute (HMI), BENSC, SF2
Glienicker Str. 100, D-14109 Berlin, Federal Republic of Germany
^c ISIS facility, Rutherford Appleton Laboratory (RAL), Chilton, Didcot, OX11 0QX, UK

Lattice parameter refinements from neutron diffraction (ND) data may be preferred to those from diffraction data collected with conventional X-rays or synchrotron radiation, for example if investigations on coarse-grained powders or bulk samples, with volumes of up to several cubic centimetres, are meant to be representative for the whole specimen. For these kind of analyses ND takes advantage of the high penetration of neutrons for most elements (isotopes), allowing them to simultaneously probe the whole volume of a thick specimen with nearly no attenuation. Furthermore, with ND the observed intensities of Bragg reflections are averaged over a larger number of crystallite orientations than the high-resolution modes of the conventional X-ray or synchrotron radiation techniques usually allow. This is a consequence of the large equatorial and axial divergence normally employed in constant wavelength ND beam optics and of the wide wavelength spectrum and the large acceptance angles of the detector banks used in time-of-flight (ToF) ND. From the metrological point of view the following three aspects of the uncertainty of measurement are especially relevant:

1. If the lattice parameter refinement is carried out by the Rietveld method or by any other whole pattern technique, then the only available measures of the uncertainty of the refined lattice parameters are the estimated standard deviations (e.s.d.s) calculated by the full pattern fitting program. However, e.s.d.s are measures of precision rather than of accuracy and these two terms must not be confused.

2. e.s.d.s calculated by many Rietveld programs are not reliable measures of the ‘probable errors’, because, in many cases, they are systematically too small, due to ‘serial correlation’. Therefore, uncorrected e.s.d.s are not only no measure of accuracy, but even no reliable measure of the precision of refined lattice parameters. The e.s.d.s are calculated under the assumption that the values in the difference curve are independent observations. However, adjacent individual points in the difference curve are not independent but correlated by the used profile function. This correlation depends on the size of the 2θ steps used for data collection and evaluation. A formula for estimating the corrections that should be applied to the e.s.d.s has been given by Berar and Lennan [1] who established that e.s.d.s calculated by Rietveld programs without consideration of serial correlations are often too small by a factor of approximately two. Therefore, at the 65% confidence level, the probable errors are calculated from the uncorrected e.s.d.s by multiplying the latter by a factor of about two. Consequently, the U_95% uncertainties of measurement at the 95% confidence level are calculated by multiplying the uncorrected e.s.d.-values by a factor of about four.

3. Bragg’s law involves a 100% correlation between the d-values (and therefore the lattice parameters) of a crystallographic phase and the wavelength of the diffracted radiation.
Thus for any assumed value of the wavelength another set of lattice parameters results, whereas the agreement indices and the e.s.d.s remain the same. Since for all neutron diffractometers the wavelength is not known per se, as would be the case with characteristic X-ray radiation, even under the idealising assumption of the complete absence of any systematic errors from the model and from the observed data the lattice parameters are completely indeterminate. Therefore, a determination of accurate lattice parameters by constant wavelength ND or ToF ND necessarily includes calibration procedures with reference materials that have lattice parameters certified for a given temperature by independent analytical techniques. The propagation of the error resulting from the calibration procedure should be considered in the estimation of the total uncertainty of the lattice parameters of the actual sample under investigation! Consequently, in the case that the calibration procedure as well as the analysis of the actual sample are carried out by the Rietveld method or any other whole pattern technique the U_95% uncertainties of the final analysis results can be as large as the eight-fold of the uncorrected e.s.d.s from a single refinement!

Further aspects of the uncertainty of measurement (e.g. line shifts due to absorption or the question about preferences for the methods of internal or external standard) have to be analysed separately for each of the following three types of ND instruments:

a. High-resolution powder ToF-ND instruments, see e.g. [2]

b. High-resolution powder diffractometers for monochromatic neutrons, equipped with
parallel collimators placed in front of the detectors and restricting the equatorial
divergence of the diffracted beam (prototype: D1A and D2B at the ILL, see e.g. [3])

c. Powder diffractometers for monochromatic neutrons in Debye-Scherrer-geometry with
one (or several) position-sensitive detector(s) providing resolution in the equatorial plane

The determination of accurate lattice parameters through the refinement of diffraction data collected with any type of these instruments requires highly accurate mathematical modelling of the observed diffraction line profiles. For a constant wavelength ND instrument of type b this can be achieved by the ‘fundamental parameter approach’ which combines the individual contributions to the reflection profile by convolution integrals. This leads to the following writing of Bragg’s law:

(1)

with λ_eff. = effective value of the wavelength; θ_obs. = observed halved diffraction angle; EPS_1,eff = effective value of the zero correction for the detector bank; Δ = axial divergence; R = diffractometer radius; h_d = effective height of the detector; h_s = effective height of the specimen; FWHM = contribution to the reflection profile caused by the real structure of the specimen; is the ‘axial divergence correction function’ which can be determined only approximately due to strong correlations between some of the parameters.

By differentiating equation (1) a formula for the uncertainty of refined lattice parameters can be derived that contains a number of individual contributions, not all of which are additive. An analogous formula can be derived for ToF-ND. By the analysis of the individual contributions and by taking into account the results of the proficiency testing carried out at several ND instruments (see e.g. [4-5]) it can be shown that the lattice parameters of cubic, tetragonal, hexagonal and orthorhombic materials can be refined in the metric system and at the 95% confidence level with a relative accuracy of Δa_i/a_i ≈ 2-3•10^-5 using a high-resolution multi-collimator/multi-detector powder diffractometer for monochromatic neutrons or a dedicated high-resolution ToF-ND instrument.

The present study is based on the analysis of data published by a number of laboratories as well as on experimental results of the authors. It facilitates the statistically sound inter-pretation of differences between lattice parameters refined on the same sample using different ND instruments.

1. Berar J.F., Lennan P. (1991) J. Appl. Cryst. 24, 1-5

2. David W.I.F. et al. (1986) Mater. Sci. Forum 9, 89-101

3. Hewart A.W. (1986) Mater. Sci. Forum 9, 69-79

4. Toebbens D.M. et al. (2001) Mater. Sci. Forum 378-381, 288-293

5. Kockelmann W. et al. (2000) Mater. Sci. Forum 321-324, 332-337