Rietveld

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 analysis of siUcon carbide involves identification, chemical analysis, and physical testing. For identification, x-ray diffraction, optical microscopy, and electron microscopy are used (136). Refinement of x-ray data by Rietveld analysis allows more precise deterrnination of polytype levels (137). 

Rietveld analysis indicates that S4N4 undergoes a transition to a new... 

For a long time the structural classification of the mineral todorokite was uncertain, until Turner and Buseck [4] could demonstrate by HRTEM investigations that the crystal structure of that mineral consists of triple chains of edge-sharing octahedra, which form [3 x 3] tunnels by further corner-sharing. These tunnels are partially filled by Mg2+, Ca2+, Na+, K+, and water (according to the chemical analysis of natural todorokites). In 1988 Post and Bish could perform a Rietveld structure determination from XRD data taken for a sample of natural todorokite [25], This diffraction study confirmed the results of Turner and Buseck. The cations... 

Michel, V., Ildefonse, P. and Morin, G. 1995 Chemical and structural changes in Cervus elephus tooth enamels during fossilization (Lazaret Cave) a combined IR and XRD Rietveld analysis. Applied Geochemistry 10 145-159. 

Fig. 4.8 Powder X-ray diffraction pattern of pure single-phase Prb-IVand Rietveld refinement.

Fig. 1. Quantification of framework Ti sites and unit cell expansion versus Ti contents using XPS 2p transition lines ( O ). XANES profiles at Ti K-edge (A) and Rietveld refinement of the XRD powder patterns ( ). Solid data points for TS-1 prepared as in ref [9].

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. 

In this section, we will describe the principal features of the main methods and the information that the Rietveld analysis generally delivers, especially related to the investigations on poor crystallites and multiphase compounds. This introduction is not exhaustive there are hundreds of articles that explore practically all of the aspects of this kind of analysis and now, 40 years after the first article [43], new improvements and results are still being published. 

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. 

Even if the main intent of the Rietveld analysis is the structure refinement in material science, sometimes the information relative to the structure is not the heart of the matter. 

The study of the peak shape gives important information relative to the microstructure of the sample even when it is included in the Rietveld code. Actually, in order to perform the Warren-Averbach analysis or other mi-crostructural studies, it is not necessary to use the Rietveld analysis, but it is often sufficient to operate with less complex, non-structural peak fitting procedures. 

There is, in any case, an important advantage in using the Rietveld analysis, which is the possibility of quanti-f5ing the phase to which we are ascribing the microstructure properties. 

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 ... 

This formula works only if all the phases in the sample are crystalline and are refined in the Rietveld procedure so that the relative scale, cell volume and cell composition are known. 

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). 

Figure 57.13. The Rietveld XRD refinement for CA53 after being leached 45 min at rt for the determination of the Ni fee (A), Ni/Al bcc (B) and the Ni2Al3 (C) contents. The residual error analysis of the curve fit is displayed in hne D.

Figure 57.14. The wt.% of Ni fee, Ni/Al bcc and Ni2Al3 as Al was leached from the CA53 as calculated by the Rietveld refinement method.

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