Combined Analysis

Chapter 2

Structure Refinement by Diffraction Profile Adjustment (Rietveld Method)

2.1. Principle of the Rietveld method

This approach starts with roughly estimated integrated intensities I_k⁰, in order to calculate new intensities at the cycle n+1, I_kⁿ⁺¹, using the expression:

[2.1] 

where y_ib is the background associated with the i^th measured intensity y_io.

The method is numerically more stable and efficient than the Pawley approach, and was initially proposed by Rietveld [RIE 67]. Many other algorithms also use it [TAY 85, LEB 92, ROD 93]. It is actually the most intensively used technique because of its efficiency in powder diffraction when the crystal structure of the sample is known. It was originally developed for monochromatic neutron powder diffraction analysis, and has been extended to monochromatic x-ray experiments and modified to allow time of flight neutron and x-ray energy dispersive data analyses.

The Rietveld algorithm uses all the information of the experimental spectral range, including information outside the diffraction peaks. The variables used are:

– the instrumental characteristics (resolution curve of the diffractometer, displacement parameters concerning goniometer misadjustments, experimental geometry, detector characteristics, etc.);

– the structural parameters (unit-cell parameters, atomic positions, atomic occupations, thermal vibrations, etc.);