*J. Appl. Cryst.*
(1994). **27**, 440-441

**Pp. x + 298. Oxford:
International Union of Crystallography/Oxford University Press, 1993.
Price £45.00. ISBN 0-19-855577-6**

In 1969, H. M. Rietveld published his seminal paper on structure refinement based on the complete powder diffraction profile rather than a limited amount of low-angle integrated intensity data. This paper laid the foundation for a dramatic renaissance of powder diffraction, widely regarded at the time as a valuable tool for phase identification and qualitative analysis but of little use for quantitative structure determination. The coming-of-age of Rietveld refinement, as it is now known, was celebrated in June 1989 at an International Workshop on the Rietveld Method, hosted by the Netherlands Energy Research Foundation at Rietveld's home institution in Petten and organized by the Commission on Powder Diffraction of the International Union of Crystallography.

*The
Rietveld method* is an outcome of the Petten meeting
but is certainly not a Proceedings in the accepted sense. The
contributing authors, all of them leaders in the field, were requested
to write articles suitable for a book aimed primarily at those with some
experience of the technique but also providing some introductory
material for beginners. The first drafts of these chapters underwent
considerable revision under the guidance and encouragement of the editor
and the end result is not only an authoritative and coherent text but
also an excellent reference work covering the literature up to the start
of 1991 or thereabouts.

The first chapter (R. A. Young) provides an excellent introduction to the Rietveld method. It contains an account of the basic mathematical procedures used in the fitting to the diffraction pattern, the types of parameters to be refined, a list of the peak-shape functions commonly used, corrections for preferred orientation (one of the more troublesome systematic errors afflicting X-ray powder data), some of the popular computer programs currently available, criteria of fit and precision and accuracy, and a very helpful section on refinement strategy.

Following a brief but interesting historical account in
Chapter 2, by H. M.
Rietveld, of the development and acceptance of the method that now bears
his name, some of the mathematical aspects of Rietveld refinement are
summarized in Chapter 3 (E. Prince), including the method of least
squares and its application to the Rietveld model, weights, constrained
models, refinement procedures and estimates of uncertainty. In Chapter
4, T. M. Sabine considers processes that
modify the flow of radiation in a polycrystalline material and derives
simple expressions for absorption, multiple scattering and extinction
that can be readily incorporated into the Rietveld model in the form of
two refinable parameters: the effective specimen size and the size of
the mosaic blocks. Chapter 5 (R.
J. Hill) is the longest in the book and gives a
comprehensive and pragmatic account of data-collection strategies for
Rietveld refinement. Among the many important topics covered are the
relative merits of laboratory X-ray, synchrotron X-ray and neutron
diffractometers for different types of experiment, the choice of
wavelength and resolution, the basic requirements of pattern analysis,
how the choices of step increment and counting statistics affect the
precision and accuracy of the refinement, and the application of
Rietveld refinement to the quantitative analysis of multiphase
materials. Background modelling in Rietveld analysis is addressed in
Chapter 6, by J. W. Richardson Jr, who describes how a Fourier filtering
technique can be used to correct time-of-flight neutron data with broad
oscillations in the background caused by the presence of a
noncrystalline component in the sample. Some procedures for analytical
fitting of laboratory X-ray peak profiles in the application of Rietveld
analysis are discussed in Chapter 7 by R. L. Snyder. The convolution of
sample, spectral and instrumental contributions to the observed profile
can be satisfactorily modelled by a split Pearson-VII function. Chapter
8 (R. Delhelz *et al*.) contains a
detailed account of the effect of crystal imperfections on the shape and
breadth of the peak profiles in a powder diffraction pattern and of the
ways in which information about size and strain can be extracted by
Rietveld refinement or pattern decomposition. In Chapter 9, P. Suortti
shows how the instrumental profile function can be calculated from the
known scattering geometry by ray-tracing or phase-space analysis
techniques and also how the background contribution can be divided into
an incoherent part that can be calculated explicitly and a coherent part
that can be represented by a radial correlation function incorporating
the salient features of the thermal and disorder diffuse scattering. In
Chapter 10, C. Baerlocher points out how soft constraints, or
restraints, can be used to improve the quality of the refinement of
complex structures such as zeolites. These restraints, implemented in
the form of approximate geometrical relationships in the least-squares
minimization process, are now incorporated into several widely used
programs. Chapter 11 (W. I. F. David and J. D. Jorgensen) reviews
Rietveld refinement with time-of-flight neutron powder data. The power
of this technique with a high-resolution diffractometer, such as the
HRPD at the ISIS pulsed neutron source, is strikingly illustrated by
examples of anisotropic line broadening in LaNbO_{4} and a
high-precision refinement of the structure of benzene. There are also
advantages when special sample environments such as high-pressure cells
and furnaces are required. In Chapter 12, R. B. Von Dreele (a co-author
with A. C. Larson of the very versatile and widely used
*GSAS* program) describes the extension
of Rietveld refinement to a combination of X-ray and neutron powder
diffraction data and illustrates how multiple data sets of this type may
be the only way to determine the distribution of different atomic
species among a number of crystallographic sites. F. Izumi, the author
of another very versatile Rietveld refinement program
(*RIETAN*),
widely used in Japan, describes some of its features in Chapter 13,
including the refinement of incommensurate structures from powder
diffraction data and the use of multiple data sets of different types,
as exemplified by a refinement of the modulated structure of the
high-*T _{c}* superconductor
Bi

In a book like this, it is inevitable that there is some unevenness in the length and degree of detail in the chapters and that some important topics receive inadequate attention. I would like to see, for example, a more exhaustive discussion of specimen preparation and diffraction geometry, accuracy and significance of the results, the tendency towards empiricism in modelling the peak shapes and the use of symmetry-adapted spherical harmonics to correct for preferred orientation and anisotropic line broadening. However, these are not serious criticisms and I recommend this book as a necessity for any diffraction library or for the personal collection of anyone with a serious interest in the application of the Rietveld technique.

**David E. Cox**

*
Physics Department
Brookhaven National Laboratory
Upton
NY
11973
USA
*

**Copyright © 1997 International Union of Crystallography**