Atoms vibrating, displacement

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... 

In a review of the subject, Ubbelohde [3] points out that there is only a relatively small amount of data available concerning the properties of solids and also of the (product) liquids in the immediate vicinity of the melting point. In an early theory of melting, Lindemann [4] considered that when the amplitude of the vibrational displacements of the atoms of a particular solid increased with temperature to the point of attainment of a particular fraction (possibly 10%) of the lattice spacing, their mutual influences resulted in a loss of stability. The Lennard-Jones—Devonshire [5] theory considers the energy requirement for interchange of lattice constituents between occupation of site and interstitial positions. Subsequent developments of both these models, and, indeed, the numerous contributions in the field, are discussed in Ubbelohde s book [3]. 

We demonstrate that the spectral function of valence harmonic vibrations of a diatomic group that effects rotational reorientations is broadened by w. The vector of atom C displacements relative to the atom B (see Fig. A2.1) may be represented as x(t)e(t), where x(t) is the change in the length of the valence bond oriented at the time t along the unit vector e(/). Characteristic periods of valence vibrations are much shorter than periods of changes in unit vector orientations. As a consequence, the GF of the displacements defined by Eq. (4.2.1) can be expressed approximately as ... 

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. 

From Table IV, it may be seen that two frequencies are calculated at 887 and 901 cm-1 that could be attributed to an axial C-l-H vibration. In fact, many atoms were displaced for each vibration. The most prominent vibrations for these two calculated frequencies are shown in Fig. 3. It may be seen from Fig. 3b that the vibration at 901 cm-1 contains coupling of CH2 and C-l-H motions, whereas the calculated frequency of 887 cm 1 exhibits, in Fig. 3c, coupling of CH2 and C-5-H groups, with no observed contributions from C-l-H or C-O-H bending. These observations, combined with... 

The double helix has unbalanced charge on many atoms. The displacements associated with vibrational modes generate oscillating dipole moments. These moments are constructed from our eigenvector displacements and models of atomic net charge taken from the literature (2, 3). These dipole moment matrix elements have been calculated (4). 

Deformation density, as originally conceptualized represents the difference between crystallographically observed electron density and calculated densities of the spherical atoms, which consitute the so-called promolecule. The effects of vibrational displacement, represented by ellipsoids in Figure 5.19, and ignored when defining a promolecule by spherical atoms, are most likely... 

Wave Propagation in a One-dimensional Crystal Lattice.—Let us consider N atoms, each of mass m, equally spaced along a line, with distance d between neighbors. Let the x axis be along the line of atoms. We may conveniently take the positions of the atoms to be at x = d, 2d, 3d,. . . Nd, with y = 0, z = 0 for all atoms. These are the equilibrium positions of the atoms. To study vibrations, we must assume that each atom is displaced from its position of equilibrium. Consider the jth atom, which normally has coordinates x = jdy y = z 0, and assume that it is displaced to the position x = jd + /, y = Vi, z = f so that Vi, f / are the three components of the displacement of the atom. If the neighboring atoms, the (j — l)st and the j -f- l)st, are undisplaced, we assume that the force acting on the jth atom has the components... 

If it is assumed that the interaction is an exponential repulsion [equation (1)], where x is the distance between the impinging atom and the molecule, and that the amplitude of the vibrational displacement is much less than /, then one may show that [13]... 

When an atom is displaced from its equilibrium position in a molecule, it is subject to a restoring force which increases with the displacement. A spring follows the same law (Hooke s law) a chemical bond is therefore formally similar to a spring that has weights (atoms) attached to its two ends. A mechanical system of this kind possesses a natural vibrational frequency which depends on the masses of the weights and the stiffness of the spring. 

In the standard treatment of rigid molecules, we define a rigid reference configuration of the atomic nuclei of the molecule with respect to which we measure the vibrational displacements of the atomic nuclei It is of course necessary to introduce a set of constraints on these displacements so that external large-amplitude motions (such as translation and overall rotation) are not accounted for as vibrational motions. 

In the treatment of non-rigid molecules, we can avoid the above-mentioned difficulties if we define a non-rigid reference configuration of the atomic nuclei which essentially follows the large-amplitude motions. Vibrational displacements measured with respect to the non-rigid reference configuration remain therefore small the large-amplitude problem is removed from the vibrational part of the Hamiltonian. 

Polar tensors have been developed by Biarge et al. (1961) to aid in the calculation of vibrational intensities (see also Sec. 5.2.9.1.2). They are derived for each atom by displacing the atom in small increments along the cartesian coordinates and noting the change in dipole moment. 

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. 

Let us now consider in detail the possible modes of vibration of the carbon dioxide molecule. In the deformation vibration, the carbon atom is displaced away from the axis of the molecule in one direction and the oxygen atoms are displaced in the opposite direction (see Figure 33),... 

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... 

---