The Rietveld method

It is worth noting that parameters identical to those listed in our examples can be expected only when the same computer codes are used to perform full profile refinement due to small but detectable differences in the implementation of the Rietveld method by various software developers. Furthermore, even when the same version of an identical computer program is employed to treat the same set of experimental data, small deviations may occur due to subjective decisions, such as when to terminate the refinement. In the latter case, however, the differences should be within a few least squares standard deviations. 

The development of the Rietveld method and especially subsequent work that showed its applicability to processing conventional x-ray powder diffraction data, began a remarkable era, which continues today, where 

Rietveld refinement programs are included in software products sold by Broker (http //vmw.bruker-axs.com/production/indexnn.htm), Philips (http //wvm-us.analytical. philips.com/products/xrd/), Rigaku/USA (http //vmw.RigakuMSC.com/xrd/index.shtinl), STOE Cie, Gmbh (http //vmw.stoe.com/products/index.htm). Reflex or Reflex+ modules in Materials Studio or Cerius2 suites from Accelrys Inc. (vmw.accelrys.com). Jade and jPOWD from MDI, Materials Data Inc. (vmw.materialsdata.com) and others. 

Malmros and J.O. Thomas, Least-squares structure refinement based on profile analysis of powder film intensity data measured on an automatic microdensitometer, J. Appl. Cryst. 

7 (1977) R.A. Young, P.E. Mackie, and R.B. Von Dreele, Application of the patternfitting structure-refinement method of x-ray powder diffractometer patterns, J. Appl. Cryst. 10, 262 (1977) C.P Khattak and D.E. Cox, Profile analysis of x-ray powder diffractometer data structural refinement of Lao.vsSro jsCrOs. J. Appl. Cryst. 10, 405 (1977). 

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Structure Determination from a Powder Pattern. In many cases it is possible to determine atomic positions and atomic displacement parameters from a powder pattern. The method is called the Rietveld method. Single-crystal stmcture deterrnination gives better results, but in many situations where it is impossible to obtain a suitable single crystal, the Rietveld method can produce adequate atomic and molecular stmctures from a powder pattern. 

The Rietveld analysis is mainly used for refining the structures of crystalline phases and to perform quantitative analysis of multiphase samples. The quantitative analysis is possible since the Rietveld method can easily deal with diffraction patterns with strongly overlapped peaks, while preferred orientation can be quantitatively treated. 

The Rietveld Fit of the Global Diffraction Pattern. The philosophy of the Rietveld method is to obtain the information relative to the crystalline phases by fitting the whole diffraction powder pattern with constraints imposed by crystallographic symmetry and cell composition. Differently from the non-structural least squared fitting methods, the Rietveld analysis uses the structural information and constraints to evaluate the diffraction pattern of the different phases constituting the diffraction experimental data. 

Quantitative Phase Analysis by Rietveld Refinement. The quantitative analysis can be performed through the Rietveld method because the number of the elemental cells of each phase is NccKV, where K and are the refined scale factor and the cell volumes, respectively. So, the weight fraction of the /th phase is given by ... 

R. A. Young, The Rietveld Method, Oxford University Press, Oxford, 1993. 

Applications The general applications of XRD comprise routine phase identification, quantitative analysis, compositional studies of crystalline solid compounds, texture and residual stress analysis, high-and low-temperature studies, low-angle analysis, films, etc. Single-crystal X-ray diffraction has been used for detailed structural analysis of many pure polymer additives (antioxidants, flame retardants, plasticisers, fillers, pigments and dyes, etc.) and for conformational analysis. A variety of analytical techniques are used to identify and classify different crystal polymorphs, notably XRD, microscopy, DSC, FTIR and NIRS. A comprehensive review of the analytical techniques employed for the analysis of polymorphs has been compiled [324]. The Rietveld method has been used to model a mineral-filled PPS compound [325]. 

The catalysts were characterized by inductively coupled plasma emission spectroscopy (ICP-ES Perkin Elmer Optima 3300RL) to determine cobalt content, x-ray diffraction (XRD Bruker A-500) with crystallite size determination using the Rietveld method, and temperature-programmed reduction (Zeton Altamira AMI-200) using 30 ml/min 10% H2/Ar and a ramp rate of 10°C/min. Surface area... 

Wilson, S.A., Raudsepp, M., Dipple G.M. 2006. Verifying and quantifying carbon fixation in minerals from serpentine-rich mine tailings using the Rietveld method with X-ray powder diffraction data. American Mineralogist, 91, 1331-1341. 

Although simple intensity correction techniques can be used to develop very adequate XRPD methods of quantitative analysis, the introduction of more sophisticated data acquisition and handling techniques can greatly improve the quality of the developed method. For instance, improvement of the powder pattern quality through the use of the Rietveld method has been used to evaluate mixtures of two anhydrous polymorphs of carbamazepine and the dihydrate solvatomorph [43]. The method of whole pattern analysis developed by Rietveld [44] has found widespread use in crystal structure refinement and in the quantitative analysis of complex mixtures. Using this approach, the detection of analyte species was possible even when their concentration was less than 1% in the sample matrix. It was reported that good quantitation of analytes could be obtained in complex mixtures even without the requirement of calibration curves. 

In a paper by Albinati and Willis (1982) the application of the Rietveld method in neutron and X-ray powder diffraction was discussed considering the different experimental techniques of obtaining the diffraction patterns. For a detailed description of the method and its applications see a reference publication (Young 1995). See also Jenkins and Snyder (1996). A frequently used calculation program for the... 

Most of the unknown structures is determined from single crystal diffraction and refined from powder diffraction. Refinement is done with the Rietveld method, which is a least square fitting of the computed pattern to the measured one, while structure parameters are treated as the primary fitting parameters. This is in contrast to the procedure in pattern decomposition, which is outlined above (where not the structure parameters, but the peak intensities were the primary fitting parameters). Beside the... 

Young RA, Ed. (1993), The Rietveld Method, lUCr Monography 5, Oxford Science Publication. 

When all the phases present were identified, we can quantify their volume fraction in the analyzed volume similarly to the way the Rietveld-method is used for phase analysis in XRD. A whole profile fitting is used in ProcessDifraction, modeling background and peak-shapes, and fitting the shape parameters, thermal parameters and volume fractions. Since the kinematic approximation is used for calculating the electron diffraction intensities, the grain size of both phases should be below 10 nm (as a rule of... 

The advantage of the Rietveld method for these applications is that it uses the enhre diffraction pattern instead of just a few diffraction peaks. Details of the methods used are given in several good books and references [27-30]. 

Izumi, F. (1993) Rietveld analysis program RIE-TAN and PREMOS and special applications. In Young, R.A. (ed.) The Rietveld Method, Oxford, Oxford University Press, 236-253 Jackson, B.P. Miller, W.P. (2000) Effectiveness of phosphate and hydroxide for desorption of arsenic and selenium species from iron oxides. Soil Sci. Soc. Am. J. 64 1616-1622 Jain, A. Raven, K.P. Loeppert, R.EI. (1999) Ar-senite and arsenate adsorption on ferrihy-drite Surface charge reductions and net OEI-release stoichiometry. Environ. Sci. Techn. 

Young RA (ed) (1993) The Rietveld method. International Union of Crystallography, Oxford David WIF (1999) J Appl Crystallogr 32 654 Visser JW (1969) J Appl Crystallogr 2 89... 

Syntheses of near-single phases of the lead-substituted thallium monolayer phases with up to 6 Cu-O layers i.e., Pb-doped 1212, 1223, 1234, 1245, and 1256, have been recently reported (21). Reactant mixtures of various proportions of Tl2Os, PbO, CaO, Ba02, and CuO were pelletized, wrapped in gold foil, and sintered at 860-900°C under flowing oxygen for 10-30 h. The Tc value reached a maximum of 121 K for the 1234 compound and declined with further increase in the number of Cu-O layers. X-ray powder diffraction data for the different phases were refined using the Rietveld method and a consistent increase in the c-axis accompanied the increase in number of Cu-O layers. 

A method known as Rietveld analysis has been developed for solving crystal structures from powder diffraction data. The Rietveld method involves an interpretation of not only the line position but also of the line intensities, and because there is so much overlap of the reflections in the powder patterns, the method developed by Rietveld involves analysing the overall line profiles. Rietveld formulated a method of assigning each peak a gaussian shape and then allowing the gaussians to overlap so that an overall... 

During the last five years, a powerful new method of getting crystal structural information from powder diffraction patterns has become widely used. Known variously as the Rietveld method, profile refinement1, or, more descriptively, whole-pattern-fitting structure refinement, the method was first introduced by Rietveld (X, 2) for use with neutron powder diffraction patterns. It has now been successfully used with neutron data to determine crystal structural details of more than 200 different materials in polycrystalline powder form. Later modified to work with x-ray powder patterns (3, X) the method has now been used for the refinement of more than 30 crystal structures, in 15 space groups, from x-ray powder data. Neutron applications have been reviewed by Cheetham and Taylor (5) and those for x-ray by Young (6). 

The x-//J-Si3N4 ratio is measured by X-ray diffraction [228-231]. In earlier investigations only some peaks where chosen [229-231]. New techniques based on the Rietveld method use the whole angle range of the XRD diffraction pattern [219,228]. This results in more accurate data even in textured samples [228], The amount of amorphous Si3N4 also can be analysed by this method using an internal standard [224]. 

Peplinski B, Schultze D, Wenzel J (2001) Interlaboratory comparison (round robin) on the application of the Rietveld method to quantitative phase analysis by X-ray and neutron diffraction. In Delhez R, Mittemeijer EJ (eds) Proc 7th European powder diffraction conference (EPDIC-7), Barcelona, Spain, 20th 23rd May 2000, Trans Tech Publications Ltd, Switzerland, p 124... 

In 1995, an elaborated method was developed for accurate structure analysis using X-ray powder diffraction data, that is, the MEM/Rietveld method [1,9]. The method enables us to construct the fine structural model up to charge density level, and is a self-consistent analysis with MEM charge density reconstruction of powder diffraction data. It also includes the Rietveld powder pattern fitting based on the model derived from the MEM charge density. To start the methods, it is necessary to have a primitive (or preliminary) structural model. The Rietveld method using this primitive structural model is called the pre-Rietveld analysis. It is well known that the MEM can provide useful information purely from observed structure factor data beyond a presumed crystal structure model used in the pre-Rietveld analysis. The flow chart of the method is shown in Fig. 2. 

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