Isotropic displacement factor

Table 2 Atomic coordinates of a TDAE-C60 at room temperature and isotropic displacement factors.The atoms assigned with a prime belong to the TDAE molecule whilethe others belong to the C60 molecule...

FIGURE 6.22. Isotropic displacement factors for oxygen, atomic number 8. Note that the larger the value of B, the greater the falloff in intensity as a function of sinO/X. but that the values of / when sin = 0 is always 8.0. 

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 ... 

For vanadium, the ratios are smaller, and the dynamic density maps do not show a distinct maximum in the cube direction. The difference is attributed to anharmonicity of the thermal motion. Thermal displacement amplitudes are larger in V than in Cr, as indicated by the values of the isotropic temperature factors, which are 0.007 58 and 0.00407 A2 respectively. As in silicon, the anharmonic displacements are larger in the directions away from the nearest neighbors, and therefore tend to cancel the asphericity of the electron density due to bonding effects. 

Atomic parameters A set of numbers that specifies the position of an atom in the unit cell (atomic coordinates), the extent of its displacement about an equilibrium position (vibration), and an occupancy factor (generally 1.0). Three parameters define position, one parameter can be used to define isotropic displacements, or six to define anisotropic displacements and one parameter defines the site occupancy. 

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. 

Temperature factor An exponential expression by which the scattering of an atom is reduced as a consequence of vibration (or a simulated vibration resulting from static disorder). For isotropic motion the exponential factor is exp(—5iso sin 0/A ), where Biso is the isotropic temperature factor. It equals 87r (ti ), where (ti ) is the mean-square displacement of the atom from its equilibrium position. For anisotropic motion the exponential expression usually contains six parameters, the anisotropic vibration or displacement parameters, which describe ellipsoidal rather than isotropic (spherically symmetrical) motion or average static displacements. 

When sinO/A, increases, both the atomic scattering functions and temperature factors decrease exponentially (see sections 2.11.4 and 2.11.3, respectively). Thus, the unreasonably high isotropic displacement parameters of selected atoms indicate that the scattering ability of the respective sites is reduced. In the title compound, the only reasonable explanation is that the suspected sites are partially occupied. 

We require only three parameters not available through the model, the root mean square thermal displacement amplitude and estimates of and cop, in order to recreate the entire fiber diffractogram. For our universal isotropic thermal displacement factor we employed the B value used successfully by Noitholt (1) for the experimental analysis of scattering from PPTA ... 

Element symbol, atom identifier for this atom, oxidation state, number of positions, Wyckoff notation, atomic coordinates xyz, isotropic or anisotropic displacement factors, site occupation, all values with s.u. (standard uncertainty) Conditions of measurement (by defined acronyms) ... 

The terms involving the subscript j represents the contribution of atom j to the computed structure factor, where nj is the occupancy, fj is the atomic scattering factor, and Ris the coordinate of atom i. In Eq. (13-4) the thermal effects are treated as anisotropic harmonic vibrational motion and U =< U U. > is the mean-square atomic displacement tensor when the thermal motion is treated as isotropic, Eq. (13-4) reduces to ... 

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 ... 

Linear elastic fracture mechanics (LEFM) approach can be used to characterize the fracture behavior of random fiber composites. The methods of LEFM should be used with utmost care for obtaining meaningful fracture parameters. The analysis of load displacement records as recommended in method ASTM E 399-71 may be subject to some errors caused by the massive debonding that occurs prior to catastrophic failure of these composites. By using the R-curve concept, the fracture behavior of these materials can be more accurately characterized. The K-equa-tions developed for isotropic materials can be used to calculate stress intensity factor for these materials. 

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