Displacement, atomic vibrational

Force constant calculations are normally done in Cartesian coordinates. Suppose we have N atoms whose position vectors are Ri, R2,. .., Ra - Each of the atoms vibrates about its equilibrium position Ri g, Ri.e, , R v,e-The first step in our treatment is to define mass-weighted displacement coordinates... 

Assuming isotropic and harmonic vibration, the thermal parameter B becomes the quantity shown in equation 3.6, where u2 is the mean square displacement of the atomic vibration ... 

Close to this limit the displacements of the two types of atom have opposite sign and the two types of atom vibrate out of phase, as illustrated in the lower part of Figure 8.10. Thus close to q = 0, the two atoms in the unit cell vibrate around their centre of mass which remains stationary. Each set of atoms vibrates in phase and the two sets with opposite phases. There is no propagation and no overall displacement of the unit cell, but a periodic deformation. These modes have frequencies corresponding to the optical region in the electromagnetic spectrum and since the atomic motions associated with these modes are similar to those formed as response to an electromagnetic field, they are termed optical modes. The optical branch has frequency maximum at q = 0. As q increases slowly decreases and... 

That the vibrational displacements of the valence shell electrons may be smaller than those of the core electrons can be qualitatively understood by considering the vibrations of two identical, strongly bonded atoms. When the atoms vibrate in phase, they behave as a rigid body, so all shells will vibrate equally. But when they vibrate out of phase, the density near the center of the bond will be stationary, assuming the average static overlap density to be independent of the vibrations. This apparently invariant component of the valence density would contribute to a lowering of the outer-shell temperature... 

The obtained Ao gi = 5.7 x 10 is even larger than the value of Acr (Cu) X (= 4.7 X 10 A ), and of the hypothetical Co—Cu crystal with intermediate elastic properties than bulk cobalt and copper (4.1 x lO" A ). The derived effect of the effect of the lower coordination of the surface atoms on the mean-square relative displacement (perpendicular vs. parallel motions) is 1.4 times larger amplitude of the perpendicular vs. parallel motions, in agreement with lattice dynamics calculations. This SEXAFS study has produced a measure of the surface effect on the atomic vibrations. This has been possible due to the absence of surface or adsorbate reconstruction (i.e. no changes in bond orientations with respect to the bulk) and of intermixing. 

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. 

There are two totally symmetric ( ,) normal modes and one b2 normal mode. (The convention is to use lowercase letters for the symmetry species of the normal modes.) The symmetry species of the normal modes have been found without solving the vibrational secular equation. Moreover, since there is only one b2 normal mode, the form of this vibration must be determined from symmetry considerations together with the requirement that the vibration have no translational or rotational energy associated with it. Thus (Fig 6.1), any bent XYX molecule has a b2 normal mode with the X atoms vibrating along the X—Y bonds and the Y atom vibrating in the plane of the molecule and perpendicular to the symmetry axis. On the other hand, there are two ax symmetry coordinates and the two ax normal vibrations are linear combinations of the ax symmetry coordinates, where the coefficients are dependent on the nuclear masses and the force constants. Thus the angles between the displacement vectors of the X atoms and the X—Y bonds for the ax modes of a bent XYX molecule vary from molecule to molecule. 

Such a potential energy function gives rise to the famihar parabolic curve (Figure 22) where the curvature of the function is related to the force constant. The success of this simple harmonic model in treating surface atom vibrations lies in the relatively small displacement of surface atoms during a period of vibration. For some crystal properties, such as thermal expansion at elevated temperature, anharmoitic contributions to the potential must be included for an accurate description. 

Figure 14 Atomic displacements and vibrational frequencies of the 4-R opening vibration computed in Refs. 97 (a), 98 (b), and 153 (c).

Atomic vibrations are displacements from equilibrium positions, with frequencies of such vibration typically of the order of 10 per second. The frequencies of X rays (velocity of light/wavelength of X rays) are much faster, of the order of (3 x 10 cm/sec)/(1.5 x 10 cm) = 2 x 10 per second. As a result, the atom may vibrate and be viewed by X rays... 

Atomic displacement parameters Atomic vibrations are displacements from... 

The term M p,is the eph coupling constant, and ba is the annihilation operator of the mode a, whose frequency and normal mode coordinate are represented by Q,a and Qp, respectively. The sites for electrons i( T) coupled with phonons are restricted to the C region or a subpart of C. The focused modes should be sufficiently localized on the molecule in term of their definition. Practically, these internal modes can be calculated by means of a frozen-phonon approximation, where displaced atoms are atoms in the c region (or its subpart) denoted as a vibrational box though a check for convergence to the size of the vibrational box is necessary [90]. 

We have illustrated two types of vibration, one in which the atoms vibrate together in an equal distance (shown as a longitudinal vibration) and the other where the atoms vibrate an unequal distance, relative to one another (here shown as a transverse vibration). Note however, that the atoms vibrate in phase, regardless of the direction and the distance. We can write equations using these terms by considering u as a coordinate and p as a displacement, in terms of the total force required. Ft, and the individual forces on each atom, Fi, as a form of Hookes Law ... 

It is easily seen that, for the acoustical modes of lattice phonon vibration, both types of atoms vibrate together (the displacement is in the same direction - which may account for the lower energy required). In contrast, in the higher energy optical modes, each type of atom vibrates together, but in opposite direction to the other, i.e.- the displacement is opposite... 

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