Introduction

Introduction

Structure refinement is an essential part of practical crystallography and must be distinguished from structure solution. Refinement as the name suggests implies taking an approximate model of the structure and refining it so that diffraction data calculated from the model structure has a closer resemblance to the observed (i.e. measured) data. It cannot be emphasised strongly enough that it is the model structure that is refined and not the data: it is not uncommon to hear people (some senior academics included) who talk about refinement of their powder diffraction data; worse still they even say it in writing!!!

The process of refinement does not solve structures. The majority of crystallographers treat the process of refinement as a "black box" process as summarized in the flow diagram below where measured data and model are used for input and refined crystal structure (when things work smoothly) results as output.

Model Structure
+
Diffraction Data

→

"Black Box"

→

Refined Crystal Structure

Although several programs exist for the refinement of crystal structures from diffraction data, the basic concepts behind all of them generally remain the same. Thus the simple and highly-developed small-molecule single-crystal refinement programs have much in common with the more complex protein single-crystal and Rietveld powder refinement packages. The common theme throughout these programs is that they all use a least-squares procedure to refine the initial structure model in order to improve the agreement between the observed diffraction data and that calculated from the model.

An understanding of how the various programs work is valuable on those occasions when the least-squares refinement program goes astray. This is particularly true today, since the improvements in crystallographic software now enable the non-specialist crystallographer to both solve and refine many crystal structures in a semi-automatic fashion.

The Minimized Function

The diffraction experiment provides a set of observable values, Y(obs), and the crystallographic model provides a corresponding set of calculated values, Y(calc). The quantity Y varies according to the nature of the refinement program. In single-crystal refinement programs, the most common observables Y are the modulus of the structure factor of the hkl^th reflection, |F(hkl)|, or better still, the square of the structure factor, F²(hkl). In refinement programs that are used with powder diffraction data, the quantity Y can be the intensity of the individual profile points in the 2θ scan, y_i, or the total intensity of groups of reflections, Σ I(hkl).

For reasons that go well beyond the scope of the present discussion, the quantity that is minimized is:

Σ i	{ Y_i(obs) - Y_i(calc)}²

which gives rise to the expression "least-squares" minimisation. The next page discusses how crystallographic structures were refined by least-squares procedures from powder diffraction data prior to the pioneering work of Rietveld as published in the famous paper of 1969.

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Author(s): Jeremy Karl Cockcroft