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:

y_ci = SΣ_KL_K|F_K|²φ(2θ_i - 2θ_K)P_KA + y_bi

where

y_ci is calculated intensity at point i

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

S is phase scale factor

L_K contains Lorentz, polarization, and multiplicity factors

F_K is structure factor for Bragg reflection

φ is profile function (diffractometer effect)

P_K is preferred orientation

A is absorption

y_bi 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 = ∑_iw_i(y_oi - 2y_ci)²

where

Q is residual difference

y_oi is observed intensity at point i

y_ci is calculated intensity at point i

w_i is a weighting factor, 1/y_oi^[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

where

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).