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GSAS Tutorial - Rietveld Refinement

Instructions in Using GSAS Rietveld Software for Quantitative X-ray Diffraction Analysis of Portland Clinker and Cement

Leslie Struble (University of Illinois) and Paul Stutzman (National Institute of Standards and Technology)

Version 1 (March 2015)

Rietveld Refinement

In Rietveld refinement, as described by Young[1], the powder pattern is calculated according to the following equation:

yci = SΣKLK|FK|2φ(2θi - 2θK)PKA + ybi


yci is calculated intensity at point i

K represents index (hkl) of Bragg reflections contributing to intensity at point i

S is phase scale factor

LK contains Lorentz, polarization, and multiplicity factors

FK is structure factor for Bragg reflection

φ is profile function (diffractometer effect)

PK is preferred orientation

A is absorption

ybi is background at point i.

Then the calculated pattern is refined to optimize the fit to the measured pattern according to the following equation:

Q = ∑iwi(yoi - 2yci)2


Q is residual difference

yoi is observed intensity at point i

yci is calculated intensity at point i

wi is a weighting factor, 1/yoi[1]

Several criteria for the refinement are recognized. In these instructions we are using chi squared (Χ2). This and other criteria are discussed by Toby[9]. Χ2 has a value never less than unity and is defined in the following equation:

χ2 = Q⁄N


Q is residual difference, as above, using the standard deviation weighting factor

N is the number of data points (or the degrees of freedom, a related parameter).

Created May 9, 2014, Updated June 2, 2021