Generalized displacement parameters

Let us now return to the basic relations (5.27) through (5.32), which play the role of generalized displacement parameters. To gain some insight into the significant feature of these quantities, we consider the zero-temperature limit We assume that only the vibrationless level of the electronic exdted state is occupied with an appreciable probability in thermal equiUhrium, and therefore u>, = 1 (ji = 1,2). Furthermore, we make the special assumption by setting = raj,. Under these circumstances, Equation 5.27 reduces to the simpler form... 

Difference maps also allow adjustments to be made to displacement parameters. If the displacement parameter of an atom in the model is too large, it will be spread over more space than necessary, and its peak height will be lower than it should be. As a result, there will be a positive peak at the atomic center in the difference map if the displacement factor is too small, a negative valley will appear in that position [Figure 9.8(b)]. The final difference Fourier map is generally not completely flat because it contains indications of both errors in the data ( Ft, 1) and inadequacies in the model ( Fc ), including, of course, the relative phase angle, a hkl)c-... 

The data obtained from an X-ray crystal structure determination are the unit cell dimensions, the space group, the atomic coordinates, the atomic displacement parameters (to be described) and the atomic occupancy factors (which are generally unity). 

This is the form used for calculating structure factors. One of the terms in Equation 13.4, B or (7, is generally used as the displacement parameter by crystallographers, and some form of this parameter is refined in the least-squares procedure. 

Yet another level of complexity of vibrational motion is taken into account by using the so-called anharmonic approximation of atomic displacement parameters. One of the commonly used approaches is the cumulant expansion formalism suggested by Johnson, in which the structure factor is given by the following general expression ... 

Just as in the case of the conventional anisotropic approximation, the maximum number of displacement parameters is only realized for atoms located in the general site position (site symmetry 1). In special positions some or all of the displacement parameters will be constrained by symmetry. For example, 7333, Yi 13, Y223 and Y123 for an atom located in the mirror plane perpendicular to Z-axis are constrained to 0. Furthermore, if an atom is located in the center of inversion, all parameters of the odd order anharmonic tensors (3, 5, etc.) are reduced to 0. 

Considering the most general expression of the structure amplitude, Eq. 2.107, its value is defined by population and displacement parameters of all... 

At this point, we may begin to refine atomic displacement parameters of the individual atoms. This is done by substituting individual B s (which were kept at 0) by the refined overall atomic displacement parameter. The overall B after this substitution should be set to 0. The distribution of atoms in the model of the crystal structure of LaNi4,85Sno.i5 is such that two of the sites, 2(c) and 3(g), see Table 7.2, are occupied by different types of atoms simultaneously. Generally, atomic displacement parameters of different atoms occupying the same crystallographic site should be constrained at identical values. ... 

Displacement parameters of atoms are also expected to be different as the temperature of the powder diffraction experiment varies. Furthermore, it is also feasible that atomic positions may change due to generally anisotropic thermal expansion of crystal lattices. These considerations are in addition to the most obvious cause (different lattice parameters) preventing combined refinement using powder diffraction data collected at different temperatures. In general, material may also be polymorphic but this is not the case here, as was established in Chapter 6, sections 6.10 and 6.11. 

Figure 2. Generalized coherent states. The superposition coefficients feW for a) versus displacement parameter amplitude a for (a) n = 0, (b) n = 1, and (c) n = 2 in the Hilbert spaces of different dimensionality s = 1 (dotted), s = 2 (dot-dashed), s = 3 (dashed), and s = oo (solid curves).

Figure 3. Generalized coherent states (black bars) versus truncated coherent states (white bars) photon-number distribution Ps(n) as a function of n in FD Hilbert spaces with s = 5,..., 50 for the same displacement parameters a = a = 4.

Figure 4. Generalized coherent states Wigner function Ws(n,Qm) in FD Hilbert space with s = 18 for a)p8j with different values of displacement parameter a, chosen as fractions of the quasiperiod T i T % 8.8. As in Fig. 1, higher values of Wigner function are depicted darker.

Figure 7. Generalized phase coherent states. Wigner function for (5,0)p8j with different phase displacement parameters p chosen to be fractions of the quasiperiod T — 7 — 8.8.

Figure 8. Generalized displaced number states. Wigner function for a, ) = a, l)(is) (a-c) and ot, 2)p8j (d-f) with different displacement parameters a given by fractions of the quasiperiod...

Generally, two parameters, such as deformation and stress, are necessary for the evaluation of the effective compliance, and it is almost impossible to determine the stress state if the materials contains many open cracks within. However, the total deformation of the cracked structure is easily measured. The crack opening displacements can be... 

Once least-squares methods came into general use it became standard practice to refine not only atomic positional parameters but also the anisotropic thermal parameters or displacement parameters (ADPs), as they are now called [22]. These quantities are calculated routinely for thousands of crystal structures each year, but they do not always get the attention they merit. It is true that much of the ADP information is of poor quality, but it is also true that ADPs from reasonably careful routine analyses based on modem point-by-point or area diffractometer measurements can yield physically significant information about atomic motions in solids. We may tend to think of crystal structures as static, but in reality the molecules undergo translational and rotational vibrations about their equilibrium positions and orientations, as well as internal motions. Cruickshank taught us in 1956 how analysis of ADPs can yield information about the molecular rigid-body motion [23], and many improvements and modifications have been introduced since then. In particular, various computer programs are available to estimate the amplitudes of simple postulated types of internal molecular motion e.g., torsion-... 

For every atom in the model that is located on a general position in the unit cell, there are three atomic coordinates and one or six atomic displacement parameters (one for isotropic, six for anisotropic models) to be refined. In addition there is one overall scale factor per structure (osf, or the first free variable in SHELXL see Section 2.7) and possibly several additional scale factors, like tbe batch scale factors in the refinement of twirmed structures, the Flack-x parameter for non-centrosymmetric structures, one parameter for extinction, etc. In addition to the overall scale factor, SHELXL allows for up to 98 additional free variables to be refined independently. These variables can be tied to site occupancy factors (see Chapter 5) and a variety of other parameters such as interatomic distances. 

In general, too small thermal parameters mean that the current model does not contain enough electrons at the place of an atom, while too large displacement parameters suggest the presence of a lighter atom than the one in the model. 

Generally, twinned crystals tend to have a poor effective data to parameter ratio, so they often require restraints in order to obtain a satisfactory refinement (Watkin, 1994). The following restraints can be useful distance restraints for chemically equivalent 1,2- and 1,3-distances, planarity restraints for groups such as phenyl rings, rigid bond ADP restraints (Hirshfeld, 1976 Rollett, 1970 Traeblood and Dunitz, 1983) and similar ADP restraints (Sheldrick, 1997b). Even when restraints are employed, the distribution of the displacement parameters (ORTEPplot) and residual features in a difference electron density map can be less satisfactory than for a normal structure determination. 

---