Rigid bodies are a very powerful way to simplify crystallographic models by fixing the geometry of a group of atoms. There is an easy way to understand how the simplification works: The location for an individual atom usually has three degrees of freedom – meaning three refineable parameters, the x, y and z coordinates for the atom. (For now, let’s ignore atoms on special positions). Note that fewer parameters are needed to describe the placement of a group of atoms held at a fixed geometry with respect to each other. For that group we need three degrees of freedom for the location of the group and, at most, three degrees of freedom for the orientation of the group. That drops down to two degrees of freedom for a linear group of atoms. Thus, in the case of a two-atom group, we replace 6 degrees of freedom with 5 and in the general case of a non-linear group of \(n\) atoms, we replace \(3n\) degrees of freedom with 6.
Rigid bodies are employed in two ways in GSAS-II. They are used for ab initio structure determination, in the Monte-Carlo/Simulated Annealing structure solution module where they serve to reduce the number of variables that need to be generated to find an approximate structure that then can be used in minimization. The second way, and the subject of this chapter, is that they are used as a type of constraint, to reduce the complexity of a model when performing refinements. (Constraints have been introduced in Chapter 24.) Note that there are many chemical moieties that have very minimal changes across large numbers of known crystal structures. A crystallographic model where the atoms in this moiety are constrained to have the known geometry is almost always going to be a more accurate description for the actual average structure for the material than one where the model is not very sensitive to exact atomic positions and the fitting allows the moiety to have some distortions. The Cambridge Crystallographic Database (CSD) has some excellent tools for exploring the conformation for a group of atoms across the very large collection of crystal structures in that database.
GSAS-II offers two ways to describe the geometry for a group of atoms, which are called a Vector Rigid Body and a Residue Rigid Body. A third treatment for rigid bodies, Spinning Bodies, is for moieties that are partly disordered, but do maintain some order with respect to the underlying lattice. The vector and residue types of rigid bodies will be described in subsequent sections, but in atom positions are described in a Cartesian coordinate system where the axes are orthogonal and the coordinate units are in Å.