A new powder diffractometer for synchrotron radiation with six detector arms has been constructed. Five detector arms are attached radially at intervals of 25 ° to the 20 axis and form a multiple-detector system. Five scintillation counters coupled with flat Ge(111) crystal analyzers on the respective arms can simultaneously record the whole powder pattern divided into five segments, each with an equal 20 span. The optics design is based on flat-specimen reflection geometry using a parallel beam. The intensity data are collected using a 20 step-scan technique in asymmetric diffraction at a fixed incident angle. A sixth multi-purpose detector arm can be used in the conventional single-arm scan mode. It can be equipped with various kinds of analyzers such as long horizontal parallel slits, a fiat or channel-cut crystal analyzer, a receiving slit and a solid-state detector. Test operations of the multiple-detector system, conducted at the Photon Factory in Tsukuba, recorded a full width at half maximum of 0.022 ° and a peak maximum intensity of more than 40000 counts s -I for the (111) reflection from Si powder. The whole powder pattern of Mg2SiO4 over a 20 range of 130 ° could be step-scanned at a step interval of 0.004 ° (20) in just 4 h. Results of whole-powder-pattern decomposition and Rietveld refinement of the Mg2SiO4 pattern are given.

An ef®cient and accurate method to evaluate the theoretical diffraction peak pro®les from spherical crystallites with lognormal size distribution (SLN pro®le) is presented. Precise results can be obtained typically by an eight-term numerical integral for any values of the parameters, by applying an appropriate substitution of the variable to the integral formula. The calculated SLN pro®les have been veri®ed by comparison with those calculated by inverse Fourier transform from the exact analytical solution of the Fourier-transformed SLN pro®le. It has been found that the shape of the SLN pro®le strongly depends on the variance of size distribution. When the logarithmic standard deviation 3 of the size distribution is close to 0.76, the SLN pro®le becomes close to a Lorentzian pro®le, and`super-Lorentzian' pro®les are predicted for larger values of 3, as has been concluded by Popa & Balzar [J. Appl. Cryst. (2002), 35, 338±346]. The intrinsic diffraction peak pro®les of an SiC powder sample obtained by deconvolution of the instrumental function have certainly showǹ super-Lorentzian' line pro®les, and they are well reproduced by the SLN pro®le for the value 3 = 0.93.

The uncertainty in measured diffraction intensities caused by particle statistics, which originates from the limited number of crystallites satisfying the diffraction condition, has been evaluated by a step‐scan measurement about the rotation angle of a specimen‐spinning attachment of a laboratory powder X‐ray diffractometer. The residual statistical variance of the spinner‐scan intensity data, after subtraction of periodic drift and variance caused by counting statistics, was assigned to the variance caused by particle statistics. Particle statistics for a standard Si powder (NIST SRM640c) and three size fractions (nominally 3–7, 8–12 and 18–22 µm in Stokes diameter) of quartz powder separated by a sedimentation method have been analysed by scanning electron microscopy (SEM) and the spinner‐scan method using a powder X‐ray diffractometer. It has been confirmed that the observed ratio of the squared diffraction‐peak intensity to the variance caused by particle statistics is proportional to the multiplicity of reflections predicted by the crystal structure. The spinner‐scan intensity data for the standard Si powder (NIST SRM640c), the effective particle diameter of which was estimated at 5.6 µm by SEM image analysis, was used as the standard for crystallite‐size evaluation of quartz powder based on analysis of spinner‐scan data. The effective crystallite diameters of the three quartz powder samples have been estimated at 6.5 (2), 11.7 (2) and 22.8 (2) µm by the analysis of the spinner‐scan data, while the effective particle diameters evaluated by SEM image analysis are 7.1, 12 and 25 µm, respectively. Other possible applications of the analysis of particle statistics based on the spinner‐scan method are also discussed.

Key indicatorsSingle-crystal X-ray study T = 293 K Mean (Ru-O) = 0.009 Å R factor = 0.047 wR factor = 0.035 Data-to-parameter ratio = 30.4For details of how these key indicators were automatically derived from the article, see

A formula of the instrumental function for a high-resolution synchrotron X-ray diffractometer, equipped with a¯at crystal analyser and a set of Soller slits for limiting the axial divergence of the diffracted beam, has been derived. The formula incorporates the effects of (i) the axial divergence of the diffracted beam limited by the Soller slits, (ii) the Bragg angle of the¯at crystal analyser, and (iii) the tilt angle de®ned as the deviation of the normal direction of the analyser face from the goniometer plane. The model pro®le function given by the convolution of a Lorentzian function with the instrumental function has been applied to ®t the experimental diffraction peak pro®les of standard Si powder (NIST SRM640b) measured with a high-resolution synchrotron X-ray diffractometer, MDS, on beamline BL4B2 at the Photon Factory in Tsukuba. The convolution has been calculated by applying an ef®cient algorithm for numerical integration. The pro®le function reproduces not only the experimental pro®les measured with a well aligned crystal analyser, but also signi®cantly distorted pro®les arising from misalignment of the analyser, with R p values within 1.4%, by varying only the instrumental parameter for the tilt angle. It is suggested that further convolution with a Gaussian distribution is practically not necessary for the model instrumental function to ®t the data collected with MDS. More rapid computation can be achieved by applying an analytical formula of the pro®le function, when the tilt angle of the crystal analyser is within about 0.2 .