The application of the Rietveld refinement technique to synchrotron X-ray data collected from a capillary sample of AI20 3 in Debye-Scherrer geometry is described. The data were obtained at the Cornell High Energy Synchrotron Source (CHESS) with an Si(111) double-crystal monochromator and a Ge(111) crystal analyzer. Fits to a number of well resolved individual peaks demonstrate that the peak shapes are very well described by the pseudo-Voigt function, which is a simple approximation to the convolution of Gaussian and Lorentzian functions. The variation of the Gaussian and Lorentzian half widths, F~ and Ft+, with Bragg angle can be approximated quite closely by the functions V tan 0 and X/cos 0 which represent the contributions from instrumental resolution and particle-size broadening respectively. Rietveld refinement based on this model yields generally satisfactory results. The refined values of V and X are consistent with the expected vertical divergence (~_ 0-1 mrad) and the nominal particle size (~_ 0-3 I.tm). In particular, the use of a capillary specimen virtually eliminates preferred orientation effects, which are highly significant in flat-plate samples of this material.

A set of general guidelines for structure re®nement using the Rietveld (whole-pro®le) method has been formulated by the International Union of Crystallography Commission on Powder Diffraction. The practical rather than the theoretical aspects of each step in a typical Rietveld re®nement are discussed with a view to guiding newcomers in the ®eld. The focus is on X-ray powder diffraction data collected on a laboratory instrument, but features speci®c to data from neutron (both constant-wavelength and time-of-¯ight) and synchrotron radiation sources are also addressed. The topics covered include (i) data collection, (ii) background contribution, (iii) peak-shape function, (iv) re®nement of pro®le parameters, (v) Fourier analysis with powder diffraction data, (vi) re®nement of structural parameters, (vii) use of geometric restraints, (viii) calculation of e.s.d.'s, (ix) interpretation of R values and (x) some common problems and possible solutions.

A previously unreported ferroelectric phase has been discovered in a highly homogeneous sample of PbZr0.52Ti0.48O3 by high-resolution synchrotron x-ray powder diffraction measurements. At ambient temperature the sample has tetragonal symmetry (at=4.037 Å, ct=4.138 Å), and transforms below ∼250 K into a phase which, unexpectedly, has monoclinic symmetry (am=5.717 Å, bm=5.703 Å, cm=4.143 Å, β=90.53°, at 20 K). The intensity data strongly indicate that the polar axis lies in the monoclinic ac plane close to the pseudocubic [111] direction, which would be an example of the species m3m(12)A2Fm predicted on symmetry grounds by Shuvalov.

Analysis of a crystal structure using the Rietveld profile technique requires a suitable description of the shape of the peaks. In general, modern refinement codes include accurate formulations for most effects; however, the functions used for peak asymmetry are semi‐empirical and take very little account of diffraction optics. The deficiencies in these methods are most obvious for high‐resolution instruments. This study describes the implementation of powder diffraction peak profile formulations devised by van Laar & Yelon [J. Appl. Cryst. (1984), 17, 47–54]. This formalism, which describes the asymmetry due to axial divergence in terms of finite sample and detector sizes, does not require any free parameters and contains intrinsic corrections for the angular dependence of the peak shape. The method results in an accurate description of the observed profiles for a variety of geometries, including conventional X‐ray diffractometers, synchrotron instruments with or without crystal analyzers and neutron diffractometers.

The unusual magnetic properties of La 0.5 Ca 0.5 MnO 3 were found to be associated with structural and magnetic ordering phenomena, resulting from the close interplay between charge, orbital, and magnetic ordering. Analysis of synchrotron x-ray and neutron powder diffraction data indicates that the anomalous and hysteretic behavior of the lattice parameters occurring between T C ϳ225 K and T N ϳ155 K is due to the development of a Jahn-Teller ͑J-T͒ distortion of the MnO 6 octahedra, the d z 2 orbitals being oriented perpendicular to the orthorhombic b axis. We observed an unusual broadening of the x-ray Bragg reflections throughout this temperature region, suggesting that this process occurs in stages. Below T N , the development of well-defined satellite peaks in the x-ray patterns, associated with a transverse modulation with qϭ[1/2Ϫ,0,0], indicates that quasicommensurate ͑ϳ0͒ orbital ordering occurs within the a-c plane as well. The basic structural features of the charge-ordered low-temperature phase were determined from these satellite peaks. The lowtemperature magnetic structure is characterized by systematic broadening of the magnetic peaks associated with the ''Mn ϩ3 '' magnetic sublattice. This phenomenon can be explained by the presence of magnetic domain boundaries, which break the coherence of the spin ordering on the Mn ϩ3 sites while preserving the coherence of the spin ordering on the Mn ϩ4 sublattice as well as the identity of the two sublattices. The striking resemblance between these structures and the structural ''charge ordering'' and ''discommensuration'' domain boundaries, which were recently observed by electron diffraction and real-space imaging, strongly suggests that these two types of structures are the same and implies that, in this system, commensurate long-range charge ordering coexists with quasicommensurate orbital ordering.

Synchrotron x-ray powder diffraction measurements have been performed on unpoled ceramic samples of (1Ϫx)Pb(Mg 1/3 Nb 2/3)O 3 ϪxPbTiO 3 ͑PMN-xPT͒ with 30%рxр39% as a function of temperature around the morphotropic phase boundary, which is the line separating the rhombohedral and tetragonal phases in the phase diagram. The experiments have revealed very interesting features previously unknown in this or related systems. The sharp and well-defined diffraction profiles observed at high and intermediate temperatures in the cubic and tetragonal phases, respectively, are in contrast to the broad features encountered at low temperatures. These peculiar characteristics, which are associated with the monoclinic phase of M C-type previously reported by Kiat et al. ͓Phys. Rev. B 65, 064106 ͑2000͔͒ and Singh and Pandey ͓J. Phys. Condens Matter 13, L931 ͑2001͔͒, can only be interpreted as multiple coexisting structures with M C as the major component. An analysis of the diffraction profiles has allowed us to properly characterize the PMN-xPT phase diagram and to determine the stability region of the monoclinic phase, which extends from xϭ31% to xϭ37% at 20 K. The complex lansdcape of observed phases points to an energy balance between the different PMN-xPT phases which is intrinsically much more delicate than that of related systems such as PbZr 1Ϫx Ti x O 3 or (1Ϫx)Pb(Zn 1/3 Nb 1/3)O 3 ϪxPbTiO 3. These observations are in good accord with an optical study of xϭ33% by Xu et al. ͓Phys. Rev. B 64, 020102 ͑2001͔͒, who observed monoclinic domains with several different polar directions coexisting with rhombohedral domains, in the same single crystal.