Thermal displacement parameters

The C60 molecules were found to be executing large amplitude reorientations at room temperature, so that large anisotropic thermal displacement factors of the C60 carbon atoms were found. The thermal displacement parameters for some of the C60 carbon atoms at room temperature are, in fact, so large that the C60 atomic coordinates may well represent only an average over one or more disordered structures involving fractional atomic occupancy. On the other hand, the TDAE N and C atomic coordinates are well-defined already at room temperature. 

In Eq. 2.74, t is sample thickness. Ignoring the absorption correction, especially when p, and/or t are small, which means a weakly absorbing or thin sample, results in the vmderestimated calculated intensity at high Bragg angles. As a result, unphysical (negative) values of thermal displacement parameters are usually obtained. 

The elementary cell or lattice is the lowest structural level of a crystal. The lattice is characterized by a space symmetry group, atom positions and thermal displacement parameters of the atoms as well as by the position occupancies. In principle, the lattice is the smallest building block for creating an ideal crystal of any size by simple translations, and it is the lattice that is responsible for the fundamental parameter. Therefore, it is extremely important to perform the structure refinement of a crystal obtained, especially if the crystal represents a solid solution compound or demonstrates unusual properties or has unknown oxygen content or is assumed to form a new structure modification. 

The strong dependence between site occupancy and thermal displacement parameters led us to improve the stmctural model by adding titanium vacancies and fixing B factors to acceptable values. Finally both these parameters were refined simultaneously. The best fit... 

Step 11. At this point a computer program refines the atomic parameters of the atoms that were assigned labels. The atomic parameters consist of the three position parameters x,j, and for each atom. Also one or six atomic displacement parameters that describe how the atom is "smeared" (due to thermal motion or disorder) are refined for each atom. The atomic parameters are varied so that the calculated reflection intensities are made to be as nearly equal as possible to the observed intensities. During this process, estimated phase angles are obtained for all of the reflections whose intensities were measured. A new three-dimensional electron density map is calculated using these calculated phase angles and the observed intensities. There is less false detail in this map than in the first map. 

Figure 6.6 and Tables 6.4-6.6 give ranges for local thermal discomfort parameters for the three categories listed in Table 6.3. The acceptable mean air velocity is a function of local air temperature and turbulence intensity. 7 he turbulence intensity may vary between 30% and 60% in spaces with mixed flow air distribution. In spaces with displacement ventilation or without mechanical ventilation, the turbulence intensity may be lower. 

As has become clear in previous sections, atomic thermal parameters refined from X-ray or neutron diffraction data contain information on the thermodynamics of a crystal, because they depend on the atom dynamics. However, as diffracted intensities (in kinematic approximation) provide magnitudes of structure factors, but not their phases, so atomic displacement parameters provide the mean amplitudes of atomic motion but not the phase of atomic displacement (i.e., the relative motion of atoms). This means that vibrational frequencies are not directly available from a model where Uij parameters are refined. However, Biirgi demonstrated [111] that such information is in fact available from sets of (7,yS refined on the same molecular crystals at different temperatures. 

A special kind of dynamic information can be supplied by careful analysis of anisotropic displacement parameters. Dunitz and others have used the thermal parameters from X-ray to provide valuable information on intra-and intermolecular mobility [36]. 

Treating the positional parameters and the elements of the thermal displacement tensor as variables, a best fit of the observed to the calculated structure factors in a least squares sense is determined. As this is a non-linear procedure, it is essential to over determine the problem. For a routine structure... 

At ambient temperature, it is usually impossible to discern the smearing caused by thermal vibrations (averaged over time) from that caused by static displacements of atoms (averaged over the whole crystal), and the vibrational parameters in fact account for both. Therefore it is now recommended to use the term atomic displacement parameters (ADP) instead. A diffraction study of the same stmcture at different temperatures, however, will show the difference the genuine thermal vibrations diminish on cooling, but the ADP due to static disorder (e.g. in ferrocene ) will remain large. 

The atomic environment within the crystal is usually far from isotropic, and the next simplest model of atomic motion (after the isotropic model just described) is one in which the atomic motion is represented by the axes of an ellipsoid this means that the displacements have to be described by six parameters (three to define the lengths of three mutually perpendicular axes describing the displacements in these directions, and three to define the orientation of these ellipsoidal axes relative to the crystal axes), rather than just one parameter, as in the isotropic case. Atomic displacement parameters, and their relationship to thermal vibrations and spatial disorder in crystals are covered in more detail in Chapter 13. 

Because the diffraction experiment involves the average of a very large number of unit cells (of the order of 10 in a crystal used for X-ray diffraction analysis), minor static displacements of atoms closely simulate the effects of vibrations on the scattering power of the average atom. In addition, if an atom moves from one disordered position to another, it will be frozen in time during the X-ray diffraction experiment. This means that atomic motion and spatial disorder are difficult to separate from each other by simple experimental measurements of intensity falloff as a function of sm6/X. For this reason, atomic displacement parameter is considered a more suitable term than the terms that have been used historically, such as temperature factor, thermal parameter, or vibration parameter for each of the correction factors included in the structure factor equation. A displacement parameter may be isotropic (with equal displacements in all directions) or anisotropic (with different values in different directions in the crystal). 

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... 

Disorder in a crystal structure is frequently revealed by the shapes of the thermal ellipsoids obtained from the least-squares refinement of the anisotropic displacement parameters. An example is provided by the crystal structure determination of potassium dihydrogen isocitrate. One carboxyl oxygen atom is very anisotropic as a result of two possible hydrogen bonding schemes in which it can take part (Figure 13.10). 

---