See also: Rietveld

The method of Rietveld is a method of analysis in Diffractométrie of x-rays on powder. It was developed in 1969 by the crystallographer Dutch Hugo Rietveld.

This method consists in simulating a diffractogram starting from a crystallographic model of the sample, then to adjust the parameters of this model so that the simulated diffractogram is nearest possible measured diffractogram. According to the properties in which one is interested and the number of parameter to be refined, the software can be more or less complex.

Description of the method

Position and surface of the peaks

The base of the method is the Théorie of diffraction on a crystal. If one knows:

• the structure of the crystal,
• nature and the position of the Atom S within the mesh
• the factors of atomic diffusion (coefficient of Rayleigh scatter of the X-rays on the atoms);
• the coefficients absorption;

then one is able to determine the position of the peaks of diffraction, like their surface except for a factor (this factor depend on the power of the radiation source, of the output of the detector…).

It is also possibly necessary to take into account the preferential Orientation (texture).

Form peaks

To simulate the shape of each peak, one can recourrir with a mathematical function a priori , without particular significance, for example a Gaussian function, or better a Pseudo function of Voigt or a Fonction of Pearson VII, and refine his width H . One has thus as many widths to refine peaks. However, it is known that the width follows a law depending on the position 2θ of the peak, one can thus bind the widths of the peaks belonging to the same phase by this law and refine the parameter of the law. The typical law for diffractometry X is

$H\; (2\; \backslash \; theta)\; =\; H\_0\; +\; \backslash \; frac\; \{H\_1\}\; \{\backslash \; cos\; \backslash \; theta\}\; +\; H\_2\; \backslash \; cdot\; \backslash \; tan\; \backslash \; theta$

for neutron diffraction, one in general uses the Loi of Cagliotti (1958):

$H\; (2\; \backslash \; theta)\; =\; \backslash \; sqrt\; \{U\; \backslash \; cdot\; \backslash \; tan^2\; \backslash \; theta\; +\; V\; \backslash \; cdot\; \backslash \; tan\; \backslash \; theta\; +\; W\}$

there are thus only three parameters of width to refine by phase, whatever the number of peaks.

One can also determine the shape of the peaks starting from the law of the geometrical Optique applied to the configuration of the diffractometer.

See the detailed article Form of a peak of diffraction.

Algorithm

The algorithm of adjustment of parameters, called fitting in English, used is an algorithm aiming at minimizing the standard deviation between the simulated curve and the experimental curve; one speaks about algorithm of minimization of the error.

See also: Adjustment of profile

One uses the factor of balanced reliability in general (weighted reliability Factor) R_wp :

$R\_\; \{wp\}\; =\; \backslash \; sqrt\; \{$

\ frac {\ sum_i w_i \ cdot (I_i^ {exp} - I_i^ {cal}) ^2} {\sum_i w_i \cdot {I_i^{exp}}^2} } where w_i is the weight allotted to the point I , which is worth 1 I_i^exp .

If simulation were perfect, the factor of reliability would have a value depending on the Rapport signal on noise. If the radiation source is a Tube with x-rays, it is known that the noise follows a Loi of Poisson: its standard deviation is equal to the square Racine number of blows cumulated at each point. One can thus define a minimal factor of reliability R ₀. The function R_wp / R ₀ must normally tend towards 1.

Applications

Quantification without standard

With this method, one can simulate a mixture of several phases. The proportion of each phase being one of the parameters to be refined, the method of Rietveld thus makes it possible to make quantification.

This method is known as “without standard” because, contrary to the traditional quantitative methods, it is useless to calibrate the apparatus while passing from the samples of known composition.

This method is particularly interesting when the phases have close peaks with many superpositions. On the other hand, contrary to a traditional quantitative method (based on the surface or the height of some peaks per phase), the method of Rietveld imposes a measurement on a great angular beach (typically from 20 to 90 °) and with a signal positive ratio on noise, therefore a relatively long time of acquisition (several tens of minutes to several hours according to the signal report/ratio on noise).

Determination and refinement of structure

One can use the method of Rietveld to determine the structure of a crystal; it is an alternative method with the stereotyped of Laue on monocrystals.

The first stage consists in determining the symmetry of the crystal starting from the positiond are peaks: the Réseau of Faced then the Groupe of space. There exist dedicated programs which proceed in general by test-error: the program reviews the various groups of possible space and determines the group of space which corresponds best. One also determines the parameters da mesh. This stage is called indexing , each peak of the diffractogram being then associated with a crystallographic plan of Indices of Miller ( HKL ).

The method of Rietveld is then used to determine the position of each atom within the mesh. To help the program to converge, one can indicate constraints:

• to force the position of certain atoms, with a tolérence;
• to force atoms to remain grouped like kinds of molecules, one speaks about “rigid bodies” ( rigid body ).

In the method known as of the reheated simulated ( simulated annealing ), one places the atoms randomly then one lets the algorithm converge; one remakes this operation several times and one chooses the solution giving the weakest factor of reliability.

Other applications

One can make use of the method of Rietveld to determine any refined parameter.

See too

External bonds

• The Rietveld method

• '' has Profiles Refinement Method for Nuclear and Magnetic Structures '', article original of Hugo Rietveld, J. Appl. Cryst. (1969). 2,65-71.
• Mailing list of the SDPD (Structure Determination by Powder Diffractometry) (addresses Yahoo!)
• SDPD - course Internet, Armel the Lease, Université of Maine (Mans)
• Programmes used by the SDPD
• the Lease method page of the CCP14 (Collaborative Computational Project Number 14 - Individual Hook and Powder Diffraction)
• '' Ab.initio '' Crystal Structure Determination with Powder Diffraction Dated
• free software
• Brass (Bremen Rietveld Analysis and Structure Continuation) , a free software of the University of Bremen
• FullProf 2000 (other software)
• Xfit-Koalariet, free software of Coehlo and Cheary for Microsoft Windows 95

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