Representations of displacement parameters

The simplest assumption to make about atomic displacements is that they are the same in all directions, that is, isotropic. Then only a single term is needed to describe them. The exponential factor is exp(-Bisosin 9jX ) where B so is the isotropic displacement parameter. 

FIGURE 13.6, Scattering curves for the host and guest molecules in a / -cyclodextrin (host) - 2iV-acetylphenylalanine methyl ester (guest) complex. Data for the host (J cyclodextrin) are indicated by solid lines, and for the guest (the methyl ester) by broken lines. The guest is somewhat disordered in the channel provided by the more ordered structure of the crystallized host (Ref. 24). 

Since t/,so = (m ), it is the best form of the displacement parameter to list (see Table 13.1) because it represents a physical feature of the structure, the mean-square amplitude of displacement. 

Atoms in crystals seldom have isotropic environments, and a better approximation (but still an approximation) is to describe the atomic motion in terms of an ellipsoid, with larger amplitudes of vibration in some directions than in others. Six parameters, the anisotropic vibration or displacement parameters, are introduced for each atom. Three of these parameters per atom give the orientations of the principal axes of the ellipsoid with respect to the unit cell axes. One of these principal axes is the direction of maximum displacement and the other two are perpendicular to this and also to each other. The other three parameters per atom represent the amounts of displacement along these three ellipsoidal axes. Some equations used to express anisotropic displacement parameters, which may be reported as 71, Uij, or jdjj, axe listed in Table 13.1. Most crystal structure determinations of all but the largest molecules include anisotropic temperature parameters for all atoms, except hydrogen, in the least-squares refinement. Usually, for brevity, the equivalent isotropic displacement factor Ueq, is published. This is expressed as  

Visual representations of the extent to which an atom is displaced in various directions are obtained from thermal ellipsoid plots of displacement parameters these are commonly drawn by use of the program ORTEP 

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