Constraints & Restraints

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Concepts

While the words "constraint" and "restraint" are virtually synonymous in common English usage, they have a very precise and well-defined meaning with regard to crystallographic refinements:

A constraint implies a mathematical equality as in x = y;

A restraint implies equality but within some experimental error, e.g. d_C-H ≈ 1.05 Å or d_C-H = 1.05 ± 0.02 Å

Early versions of the Rietveld program (and many single-crystal programs) gave the user the opportunity to employ constraints during the refinement, but rarely offered any way of applying restraints. Today the use of restraints is common, particularly for the refinement of organic and organometallic structures from both single-crystal and powder diffraction data.

Why do we need to employ constraints and restraints during the refinement of a crystal structure? The usual reason is that the use of constraints and/or restraints reduces the number of variable parameters. In addition, many constraints prevent correlations between parameters, e.g. the correlation in the coordinates of atoms related by symmetry.

Constraints have the effect of reducing the size of the least-squares matrix in the minimisation procedure. By constrast, the use of restraints leaves size of the matrix unchanged, but makes systematic modifications to the matrix elements in order to improve its stability with respect to inversion. In effect, restraints provide data additional to that provided by the diffraction experiment.

Constraints are used in most Rietveld refinements, often without any explicit use: lattice parameters should be automatically constrained according to the appropriate crystal system, hkl indices are constrained to be integer, and so on. By contrast, restraints are nearly always explicitly defined by the user as will be evident in the following pages.

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