Atomic displacement parameters

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. 

Structure Determination from a Powder Pattern. In many cases it is possible to determine atomic positions and atomic displacement parameters from a powder pattern. The method is called the Rietveld method. Single-crystal stmcture deterrnination gives better results, but in many situations where it is impossible to obtain a suitable single crystal, the Rietveld method can produce adequate atomic and molecular stmctures from a powder pattern. 

In practice, the choice of parameters to be refined in the structural models requires a delicate balance between the risk of overfitting and the imposition of unnecessary bias from a rigidly constrained model. When the amount of experimental data is limited, and the model too flexible, high correlations between parameters arise during the least-squares fit, as is often the case with monopole populations and atomic displacement parameters [6], or with exponents for the various radial deformation functions [7]. 

Macromolecular crystallographic refinement is an example of a restrained optimization problem. Standard refinement programs adjust the atomic positions and, typically, also their atomic displacement parameters of a given model with the... 

High resolution (between 1.4 and 2.0 A) Automated model building with ARP/wARP should work with most phase sets. RESOLVE, which uses a template-based rather than atom-based approach, should also perform well but may be computationally more consuming. Refinement can best be carried out with REEMAC or PHENIX using isotropic ADPs since the amount of data is no longer sufficient for an anisotropic description of atomic displacement parameters. The use of TLS (Winn et ah, 2003) is highly recommended. A use of NCS restraints should be critically evaluated and in most cases the refinement can proceed without them. Double conformations of side chains should be visible and modelled. Ordered solvent can be modelled automatically. 

S is the scattering vector, Mj is the atomic displacement parameter in this simplified notation assumed to be isotropic, 6 is the scattering angle, and 1 the wavelength of the incident radiation. The atomic displacement depends on the temperature, and hence so does the Debye-Waller factor. If an atom is modeled by a classical oscillator, then the atomic displacement would change linearly with temperature ... 

The typical behavior of an atomic displacement parameter is represented by the curve plotted in Fig. 2. This trend tells us that below the turn point (0e/2) atomic vibrations are not only smaller but also quite constant. 

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. 

Uncorrelated motion is most likely to be found when the bonds are weak, for example, when the cation is a large alkali metal. In this case the mean square amplitude, A, is given by the sum of the components of the atomic displacement parameters, U, of the two atoms along the bond direction, that is. 

Figure 19 Crystal structure of ( )-55a. Atomic displacement parameters obtained at 233 K are drawn at a 30 % probability level [63], Reproduced by permission of the Royal...

ADP atomic displacement parameters HFSC heavy-fermion superconductor... 

NdOs4Sbi2 may undergo a displacive-type phase transition at -86 °C in which the Nd atoms freeze at off center positions (Evers et al., 1995). This transition was proposed on the basis of scanning calorimetry measurements and the huge room temperature value for the Nd atomic displacement parameter (Beq = 4 A2). 

Fig. 10. X -ray crystal structure of expanded [6]radialene 30b H-atoms have been omitted for clarity. Atomic displacement parameters at 203 K are drawn at the 20 % probability level. The perethynylated C6o-core is highlighted in bold it adopts a chair-like conformation [29d].

Table 1. Fractional coordinates, occupancies, g, and isotropic atomic displacement parameters, U, of Na CoOVvHjO (x 0.35, y 1.3).

In BLH-NhaCo()2, the guest species in the galleries are highly disordered. MEM is quite effective for detailed structure analysis of such an intercalation compound. In MEM-based whole-pattern fitting (MPF), crystal structures are expressed not by structure parameters such as fractional coordinates and atomic displacement parameters but by electron densities in the unit cell. Therefore, MPF allows us to represent the disordered atomic configuration in a more appropriate way than conventional Rietveld analysis adopting a split-atom model. The... 

Brian C. Sales, David G. Mandrus, and Bryan C. Chakoumakos, Use of Atomic Displacement Parameters in Thermoelectric Materials Research... 

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. 

Yang, Q.-C., Richardson, M. F. and Dunitz, J. D. (1989). Conformational polymorphism of dimethyl 3,6-dichloro-2,5- dihydroxyterephthalate. I. Structures and atomic displacement parameters between 100 and 350 K for three crystal forms. Acta Crystallogr. B, 15, 312-23. [215]... 

There is, however, an alternative (but still indirect) way to view these molecules. It involves studies of crystalline solids and the use of the phenomenon of diffraction. The radiation used is either X rays, with a wavelength on the order of 10 cm, or neutrons of similar wavelengths. The result of analyses by these diffraction techniques, described in this volume, is a complete three-dimensional elucidation of the arrangement of atoms in the crystal under study. The information is obtained as atomic positional coordinates and atomic displacement parameters. The coordinates indicate the position of each atom in a repeat unit within the crystal, while the displacement parameters indicate the extent of atomic motion or disorder in the molecule. From atomic coordinates, it is possible to calculate, with high precision, interatomic distances and angles of the atomic components of the crystal and to learn about the shape (conformation) of molecules in the crystalline state. 

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. 

Dunitz, J. D., Schomaker, V., and Trueblood, K. N. Interpretation of atomic displacement parameters from diffraction studies of crystals. J. Phys. Chem. 92, 856-867 (1988). 

Peaks occur in a difference map in positions in the unit cell where the model did not include enough electron density valleys appear in places where the model contained too much electron density. This information may be used to obtain more precise atomic positions, atomic displacement parameters, or atomic numbers. For example, in the last category, the identities of atoms (carbon or nitrogen) in a tricyclic molecule were established by setting all atoms to one type (carbon in this case) in the structure factor calculation. A difference map was calculated with the calculated phases and examined for excess electron density at atomic positions (Table 9.2). It was found to be possible to distinguish between nitrogen (seven electrons) and carbon (six electrons), even though these atoms are adjacent in the Periodic Table. 

Values of the e.s.d.s of parameters can be obtained, as shown in Figure 10.13, in the least-squares refinement from values of the diagonals of the inverse matrix. Similarly, any correlations between parameters, such as is often found to occur between occupancy and atomic displacement parameters, can be identified and taken into account in the description of the resulting molecular structure. The e.s.d.s for the refined parameters can then be used to calculate e.s.d.s of derived parameters, such as distances, angles, and torsion angles. ... 

The reader should appreciate that because atoms are always vibrating, the atomic positions found in an X-ray diffraction experiment represent the average positions of the atoms during vibration. The atomic parameters include atomic displacement parameters (described in Chapter 13) which give some measure of the amplitude of this... 

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). 

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). 

Now that fairly precise measures of electron density can be made, atomic displacement parameters can be refined so that the best possible fit to the experimental electron-density profiles of each atom is obtained. This is done by the introduction of additional atomic parameters, one parameter if the displacements are isotropic, six if they are anisotropic. When this least-squares refinement of displacement parameters is completed, the crystallographer is then left with the problem of explaining the atomic displacement parameters so obtained in terms of vibration, static disorder, dynamic disorder, or a combination of these. 

Different molecular components of a crystal structure may have ver -different atomic displacement parameters such that the relative influences of these different components on the diffraction pattern will also differ. 

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