Acta Cryst. (1967). 22, 151-152.

Line profiles of neutron powder-diffraction peaks for structure refinement.*

H. M. Rietveld,
Reactor Centrum Nederland, Petten, The Netherlands

Received 28 July 1966


A structure refinement procedure is described, which obtains a least-squares fit between the calculated and observed intensities measured at equal angular intervals on a neutron powder-diffractometer. It is shown that the structure parameters are significantly more reliable than those determined by a method using the integrated intensities of overlapping peaks.

While the structure determination from a well resolved powder diffraction diagram often proves to be quite feasible, the inherent presence of overlapping reflections generally prevents the full use of the available information to refine the structural parameters. An obvious solution is to include as observed data in the least-squares refinement the integrated intensities of the composite diffraction peaks (Rietveld, 1966). A major drawback of such a procedure, however, is the fact that any detail in the profiles of these peaks is lost.

In the case of neutron diffraction, a more direct method to solve this problem was found when it appeared that the peak shapes of the single diffraction peaks obtained from the powder-diffractometer at Petten were Gaussian. The profile of a composite peak can then be regarded as the sum total of the constituent Gaussian peaks representing the individual Bragg reflections. The contributions of each of these constituent peaks to the enveloping peak at position tex2html_wrap_inline52 can be written as tex2html_wrap_inline54 .

The quantities b and tex2html_wrap_inline58 in this expression, being respectively the full width at half height (halfwidth) and the position of the peak, can be obtained from an inspection of the diffraction diagram, the unit-cell parameters and the wavelength. The remaining quantity a is then the only unknown. It is proportional to tex2html_wrap_inline62 , where j is the multiplicity of the structure factor F, the precise relation being as follows.

Equating the area of the Gauss curve to the integrated intensity gives


from which




gives tex2html_wrap_inline68 , where c' represents an overall scale factor.


Fig. 1. Neutron powder diffraction diagram of tex2html_wrap_inline72 (intensity vs tex2html_wrap_inline74 ). The solid line indicates the calculated profile and the dots the measured intensities. The rectangles represent the integrated single-peak intensities, their different heights the agreement between calculated and observed values.

Finally, putting


where tex2html_wrap_inline76 is a measure of the contribution of structure factor tex2html_wrap_inline78 at position tex2html_wrap_inline80 to the intensity measured at position tex2html_wrap_inline52 , and corrected for background, we can say that


The summation is taken over all reflections tex2html_wrap_inline78 which can contribute significantly to tex2html_wrap_inline86 ; the effective range of a Gaussian peak for this purpose is set at three times its halfwidth.

In addition, the well resolved peaks are replaced by a tex2html_wrap_inline88 peak, i.e. all tex2html_wrap_inline90 's in this range are replaced by zero's except for one value which is made equal to the area of the Gaussian peak and


This procedure eliminates the (now unnecessary) introduction of an uncertainty due to the supposition of the idealized Gaussian peak shape.

To test the method the structure of tex2html_wrap_inline72 (Loopstra & Boldrini, 1966) was refined. It was found that the standard deviation of the least-squares

parameters improved by an average factor 2.3, compared with those obtained by the method of overlapping reflections (Rietveld, 1966). Fig. 1 shows the final agreement between calculated and observed values. The dots represent the intensities, measured at each step by the proportional counter and corrected for background, and the full line the calculated profile. The different heights of the rectangles representing the integrated intensities indicate the agreement between those values.


Loopstra, B. O. & Boldrini, P. (1966). Acta Cryst.21, 158.
Rietveld, H. M. (1966). Acta Cryst. 20, 508.

* Work supported jointly by Reactor Centrum Nederland and Institutt for Atomenergi, Kjeller, Norway.