Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-19T04:17:26.164Z Has data issue: false hasContentIssue false
  • English
  • Français

Data Collection Strategies for Constant Wavelength Rietveld Analysis*

Published online by Cambridge University Press:  10 January 2013

R. J. Hill  and
I. C. Madsen
R. J. Hill
Affiliation:
Division of Mineral Chemistry, CSIRO, PO Box 124, Port Melbourne, Victoria 3207, Australia
I. C. Madsen
Affiliation:
Division of Mineral Chemistry, CSIRO, PO Box 124, Port Melbourne, Victoria 3207, Australia
Get access Rights & Permissions [Opens in a new window]

Abstract

In the Rietveld method for the analysis of powder diffraction data the entire pattern is calculated using a model for the positions of the peaks (the unit cell parameters), their intensities (dependent on the atomic positional and thermal parameters, preferred orientation, etc.) and their widths and shapes, together with a description of the background. The calculated pattern is then compared with the observed step profile, point by point, and the model parameters are adjusted by least-squares methods.

In order to ensure the best possible outcome of the refinement, a number of critical decisions must be made prior to the collection of the step intensities. For constant wavelength diffractometers, these decisions relate to the wavelength, beam collimation, range of diffraction angles, angular distance between steps, and counting time (X-rays) or monitor setting (neutrons) at each step. The effect of the first three of these factors is well known, but the selection of appropriate values for the counting time T (in effect, the intensity) and for the step interval (which, for a given scan range, determines the number of steps, N) is not straightforward. In fact, N and T are usually chosen more by tradition or by pressure of instrument time than by a consideration of their possible effect on the results of the analysis.

Since each measured step intensity is used in the Rietveld method, the number of ‘observations’, N, in a scan can be made arbitrarily large (independent of the number of Bragg reflections) by decreasing the step interval. It is, however, the intensities of the Bragg peaks that are the fundamental quantities in any structure analysis, not the step intensities themselves. Thus, although the precision of the peak intensity measurement is improved by increasing N or T, this only occurs up to the point where counting variance becomes negligible in relation to other sources of error; further increases provide no additional structural information.

Systematic studies of the effect of variations in T indicate that the optimum value of the maximum step intensity is only a few thousand counts. If significantly larger numbers of counts are accumulated, the accuracy of the structural parameters is not improved, time is wasted, and the usual weighting scheme based on counting variance becomes inappropriate (i.e., the parameter esd's reflect their precision rather than their accuracy). In the case of step width, the optimum value is between one-fifth and one-half the minimum full-width at half-maximum (FWHM) of well-resolved peaks, the exact value depending on T and the complexity of the diffraction pattern. Smaller steps provide little or no improvement in parameter accuracy (especially when step intensities are large) and, at the same time, introduce serial correlation between adjacent residuals in the profile, again leading to wasted time and corruption of the esd's.

In practice, it is the combination of N and T chosen for the experiment that is of greatest importance in determining the efficiency of the data collection strategy. If the pattern has many overlapping peaks, N should be large, corresponding to a step interval of about FWHM/5, to provide adequate peak resolution, and T should be correspondingly small to minimize serial correlation. For a fixed total data collection time, a given level of Rietveld precision can be achieved more efficiently by the use of large N and small T than by combinations of small N and large T. The implications of these restrictions for ‘real-time’ data collection are noteworthy.

Type
Research Article
Information
Powder Diffraction , Volume 2 , Issue 3 , September 1987 , pp. 146 - 162
Copyright
Copyright © Cambridge University Press 1987

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ahtee, M., Unonius, L., Nurmela, M. & Suortti, P. (1984). J. Appl. Crystallogr. 17, 352–357.CrossRefGoogle Scholar
Albinati, A. & Willis, B. T. M. (1982). J. Appl. Crystallogr. 15, 361–374.Google Scholar
Bacon, G. E. & Lisher, E. J. (1980). Acta Crystallogr. B36, 1908–1916.Google Scholar
Baerlocher, C. (1984). Proc. 6th Int. Zeolite Conf. Reno, USA, July, 1983.Google Scholar
Baharie, E. & Pawley, G. S. (1983). J. Appl. Cryst. 16, 404–406.CrossRefGoogle Scholar
Caglioti, G., Paoletti, A. & Ricci, F. P. (1958). Nucl. Instrum, 3, 223–228.CrossRefGoogle Scholar
Cheetham, A. K. & Taylor, J. C. (1977). J. Solid State Chem. 21, 253–275.Google Scholar
Christensen, A. N., Lehmann, M. S. & Nielsen, M. (1985). Aust. J. Phys. 38, 497–505.CrossRefGoogle Scholar
Cooper, M. J. (1982). Acta Crystallogr. A38, 264–269.Google Scholar
Cooper, M. J. (1983). Z. Kristallogr. 164, 64157–158.CrossRefGoogle Scholar
Cooper, M. J. & Glasspool, A. V. (1976). J. Appl. Crystallogr. 9, 63–67.Google Scholar
Cooper, M. J., Rouse, K. D. & Sakata, M. (1981). Z. Kristallogr. 157, 101–117.Google Scholar
Cox, D. E., Hastings, J. B., Cardoso, L. P. & Finger, L. W. (1986). Mat. Sri. Forum (in press).Google Scholar
Cox, D. E., Hastings, J. B., Thomlinson, W. & Prewitt, C. T. (1983). Nucl. Instrum. Methods, 208, 573–578.CrossRefGoogle Scholar
David, W. I. F. & Matthewman, J. C. (1984). Rutherford-Appleton Laboratory Rep. RAL 84–064.Google Scholar
Dollase, W. A. (1986). J. Appl. Crystallogr. 19, 267–272.Google Scholar
Durbin, J. & Watson, G. S. (1950). Biometrika, 37, 409–428.Google Scholar
Durbin, J. & Watson, G. S. (1951). Biometrika, 38, 159–178.Google Scholar
Durbin, J. & Watson, G. S. (1971). Biometrika, 58, 1–19.Google Scholar
Flack, H. D. (1985). Crystallographyic Computing 3: Data Collection, Structure Determination, Proteins, and Databases, eds. Sheldrick, G. M., Kruger, C., and Goddard, R., 18–27. Oxford: Clarendon Press.Google Scholar
Flack, H. D., Vincent, M. G. & Vincent, J. A. (1980). Acta Crystallogr. Sect. A, 36, 495–496.CrossRefGoogle Scholar
Greaves, C. (1985). J. Appl. Crystallogr. 18, 48–50.CrossRefGoogle Scholar
Hall, M. M. Jr., Veeraraghavan, V. G., Rubin, H. & Winchell, P. G. (1977). J. Appl. Crystallogr.. 10, 66–68.Google Scholar
Hastings, J. B., Thomlinson, W. & Cox, D. E. (1984). J. Appl. Crystallogr.. 17, 85–95.CrossRefGoogle Scholar
Hewat, A. W. (1975). Nucl. Instrum. Methods, 127, 361–370.Google Scholar
Hewat, A. W. (1980). NBS Spec. Publ. (U.S.) 567, 111–141. National Bureau of Standards, Gaithersburg, MD.Google Scholar
Hewat, A. W. (1986). Chemica Scripta 26 A, 119–130.Google Scholar
Hewat, A. W. & Sabine, T. M. (1981). Aust. J. Phys. 34, 707–712.CrossRefGoogle Scholar
Hill, R. J. (1982). Mater. Res. Bull. 17, 769–784.CrossRefGoogle Scholar
Hill, R. J. (1984). Am. Mineral. 69, 937–942.Google Scholar
Hill, R. J. & Flack, H. D. (1986). Am. Crystallogr. Assoc. Prog. Abstr., McMaster Univ., Hamilton, Ontario, V2, 76.Google Scholar
Hill, R. J. & Howard, C. J. (1985).J. Appl. Crystallogr. 18, 173–180.Google Scholar
Hill, R. J. & Howard, C. J. (1986). A Computer Program for Rietveld Analysis of Fixed Wavelength X-ray and Neutron Powder Diffraction Patterns. Rept. No. M 112, Australian Atomic Energy Commission, Lucas Heights Research Laboratories, PMB, Sutherland, New South Wales, Australia.Google Scholar
Hill, R. J. & Madsen, I. C. (1984). J. Appl. Crystallogr. 17, 297–306.Google Scholar
Hill, R. J. & Madsen, I. C. (1986). J. Appl. Crystallogr. 19, 10–18.CrossRefGoogle Scholar
Howard, C. J. (1982). J. Appl. Crystallogr. 15, 615–620.Google Scholar
Howard, C. J., Ball, C. J., Davis, R. L. & Elcombe, M. M. (1983). Aust. J. Phys. 36, 507–518.CrossRefGoogle Scholar
Immirzi, A. (1980). Acta Crystallogr. B36, 2378–2385.CrossRefGoogle Scholar
International Tables for X-ray Crystallography Vol. II. (1967). 330. Birmingham: Kynoch Press. (Present distributor D. Reidel, Dordrecht).Google Scholar
Keijser, de T. H., Mirtemeijer, E. J. & Rozendaal, H. C. F. (1983). J. Appl. Crystallogr. 16, 309–316.CrossRefGoogle Scholar
Khattak, C. P. & Cox, D. E. (1977). J. Appl. Crystallogr. 10, 405–411.CrossRefGoogle Scholar
Klug, H. P. & Alexander, L. E. (1974). X-ray Diffraction Procedures for Polycrystalline and Amorphous Materials. New York: Wiley.Google Scholar
Lewis, J., Schwarzenbach, D. & Flack, H. D. (1982). Acta Crystallogr. A38, 733–739.Google Scholar
Miyake, M., Minato, I., Morikawa, H. & Iwai, S. (1978). Am. Mineral. 63, 506–510.Google Scholar
Parrish, W., Hart, M. & Huang, T. C. (1986). J. Appl. Crystallogr. 19, 92–100.Google Scholar
Pawley, G. S. (1980). J. Appl. Crystallogr. 13, 630–633.CrossRefGoogle Scholar
Pawley, G. S. (1981). J. Appl. Crystallogr. 14, 357–361.Google Scholar
Pesonen, A. (1979). J. Appl. Crystallogr. 12, 460–463.Google Scholar
Prince, E. (1980). NBS. Tech. Note (U.S.) 1142, 20–23. National Bureau of Standards, Gaithersburg, MD.Google Scholar
Prince, E. (1981). J. Appl. Crystallogr. 14, 157–159.CrossRefGoogle Scholar
Retief, J. J., Engel, D. W. & Boonstra, E. G. (1985). J. Appl. Crystallogr. 18, 150–155.CrossRefGoogle Scholar
Rietveld, H. M. (1967). Acta Crystallogr. 22, 151–152.CrossRefGoogle Scholar
Rietveld, H. M. (1969). J. Appl. Crystallogr. 2, 65–71.Google Scholar
Rollett, J. S. (1982). Computational Crystallography, ed. Sayre, D., Oxford: Clarendon Press.Google Scholar
Sabine, T. M. (1980). Aust. J. Phys. 33, 565–572.Google Scholar
Sakata, M. & Cooper, M. J. (1979). J. Appl. Crystallogr. 12, 554–563.Google Scholar
Santoro, A. (1983). Solid State Ionics, 9–10, 31–40.Google Scholar
Scott, H. G. (1983). J. Appl. Crystalbgr. 16, 159–163.Google Scholar
Sonneveld, E. J. & Visser, J. W. (1975). J. Appl. Crystallogr. 8, 1–7.Google Scholar
Suortti, P., Ahtee, M. & Unonius, L. (1979). J. Appl. Crystallogr. 12, 365–369.Google Scholar
Taylor, J. C. (1985). Aust. J. Phys. 38, 519–538.CrossRefGoogle Scholar
Theil, H. & Nagar, A. L. (1961). J. Am. Stat. Assoc. 56, 793–806.CrossRefGoogle Scholar
Thompson, P. & Wood, I. G. (1983). J. Appl. Crystallogr. 16, 458–472.CrossRefGoogle Scholar
Toraya, H. (1985). J. Appl. Crystallogr. 18, 351–358.Google Scholar
Von Dreele, R. B., Jorgensen, J. D. & Windsor, C. G. (1982). J. Appl. Crystallogr. 15, 581–589.Google Scholar
Wertheim, G. K., Butler, M. A., West, K. W. & Buchanan, D. N. E. (1974). Rev. Sri. Instrum. 45, 1369–1371.Google Scholar
Will, G., Parrish, W. & Huang, T. C. (1983). J. Appl. Crystallogr. 16, 611–622.Google Scholar
Wilson, A. J. C. (1973). J. Appl Crystallogr. 6, 230–237.CrossRefGoogle Scholar
Young, R. A. & Wiles, D. B. (1981). Adv. X-ray Anal., ed. Smith, D. K., Barrett, C. S., Leyden, D. E., and Predecki, P. K., 24, 1–23. New York: Plenum.Google Scholar
Young, R. A. & Wiles, D. B. (1982). J. Appl. Crystallogr. 15, 430–438.CrossRefGoogle Scholar
Young, R. A., Prince, E. & Sparks, R. A. (1982). J. Appl. Crystallogr. 15, 357–359.Google Scholar