Rietveld method

Another tool useful in discriminating between several possible structural models is Rietveld refinement [24,25]. This 

A complete understanding of the structure of the material under study or application is a sine qua non condition for the successful research or use of the material. In the case of powders, the best way to decipher the structure of new materials is the Rietveld method. This methodology was initially developed by Hugo M. Rietveld in 1969 [23] as a procedure for refining crystal structures using neutron powder diffraction data. To implement the method in practice, certain information about the estimated crystal structure of the phase or phases of interest in the diffraction profile under test is necessary. 

The aim of the method is to find the best agreement between measured v) and calculated y diffraction patterns. In the refinement procedure, the model is assumed optimum when the sum of the squares of the differences between the experimental and the calculated patterns, R, is a minimum according to a nonlinear optimization process [29], The sum of the squares of the differences between the experimental and the calculated patterns is given by 

The Physical Chemistry of Materials Energy and Environmental Applications 

F kl is the structure factor, which includes the Debye-Waller factor 

9(29 - 29.) is the peak profile function describing the particle size broadening and other sources of peak broadening 

---

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. 

XRD analysis perfected by the application of Rietveld method [11,12] provided an average metal nanocluster diameter equal to 3.3 nm (Table 3). 

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]. 

Tagmatarchis N, Prato M (2004) Organofullerene Materials 109 1-39 Takata M, Nishibori E, Sakata M, Shinohara M (2004) Charge Density Level Structures of Endohedral Metallofullerenes by MEM/Rietveld Method 109 59-84 Takenobu T, see Margadonna S (2004) 109 127-164 Talarico G, see BudzelaarPHM (2003) 105 141-165 Taniguchi H, see Itoh M (2007) 124 89-118... 

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. 

Recent developments and prospects of these methods have been discussed in a chapter by Schneider et al. (2001). It was underlined that these methods are widely applied for the characterization of crystalline materials (phase identification, quantitative analysis, determination of structure imperfections, crystal structure determination and analysis of 3D microstructural properties). Phase identification was traditionally based on a comparison of observed data with interplanar spacings and relative intensities (d and T) listed for crystalline materials. More recent search-match procedures, based on digitized patterns, and Powder Diffraction File (International Centre for Diffraction Data, USA.) containing powder data for hundreds of thousands substances may result in a fast efficient qualitative analysis. The determination of the amounts of different phases present in a multi-component sample (quantitative analysis) is based on the so-called Rietveld method. Procedures for pattern indexing, structure solution and refinement of structure model are based on the same method. 

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. 

The ID diffraction profiles are also ideally suited to profile refinement, as illustrated in Figs. 9 and 13 in Sect. 4, from which atomic coordinates can be obtained using standard Rietveld methods. 

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. 

---