Isotropic displacement amplitude

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. 

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 ... 

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 ... 

Here the atom is assumed to vibrate with isotropic amplitude in all directions. As atoms in crystals seldom have isotropic environment, their displacements can be better approximated by an ellipsoid, that is, a symmetrical tensor U with six independent components llij ... 

The elastic radius delimits a zone inside which the displacements are irreversible. For a given medium, the size of this zone increases with explosion yield and decreases with depth. Observed from long distances compared with this radius, that is, beyond a few kilometres, an underground nuclear explosion can be represented, in seismic terms, by a single point emitting an isotropic seismic wave whose amplitude and frequency content are governed by the explosion yield, the depth of the zero point and the mechanical characteristics of the material surrounding this point. 

Isotropic temperature parameter An atomic displacement parameter (q.v.) that represents an equal amplitude of vibration of an atom in all directions through a crystal. 

The modes of vibration in a cylindrical coordinate system can be obtained through the method of separation of variables (Kausel 1974 Tassoulas 1981). Using the modes of vibration, equivalent nodal loads and nodal displacements can be evaluated at the cylindrical boundary r= ro of the region r>ro and the dynamic stiffness of the transmitting boundary in the water-saturated transversely isotropic layered strata can be obtained. The amplitudes of the equivalent nodal loads are given as follows ... 

Recently, LEED f(V) analysis has also been used to determine the thermal displacements of the ions of the NaQ(OOl) surface at various temperatures between 25 and 230 K in the approximation of isotropic thermal motions parallel and perpendicular to the surface [32] (Figure 16.4). According to this study, the thermal motions of the Na and the Cl ions in the first layer have nearly the same root MSAs, while in the second layer, the hghter cation has a larger amphtude than the anion, the amplitude of the latter being close to the corresponding bulk value. 

Let the electric field vary harmonically, i.e. in the complex notation let it be given by E (t) = E exp(icot), where E is the amplitude and co the angular frequency of variation. For an isotropic system, the time dependence of the electric displacement is also harmonic D (t) = DSexp(icot- eo), although and the phase difference are frequency dependent. Equation (4.3) now becomes... 

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