Atomic oscillators

Equation (Bl.8.6) assumes that all unit cells really are identical and that the atoms are fixed hi their equilibrium positions. In real crystals at finite temperatures, however, atoms oscillate about their mean positions and also may be displaced from their average positions because of, for example, chemical inlioniogeneity. The effect of this is, to a first approximation, to modify the atomic scattering factor by a convolution of p(r) with a trivariate Gaussian density function, resulting in the multiplication ofy ([Pg.1366]

This implies that the two atoms oscillate about a mutual distance d = r(4>) that depends on the angle (j) and is given by an ellipse. Let... 

In order for these atoms to actually climb over the barrier from A to 6, they must of course be moving in the right direction. The number of times each zinc atom oscillates towards B is v/6 per second (there are six possible directions in which the zinc atoms can move in three dimensions, only one of which is from A to B). Thus the number of atoms that actually jump from A to B per second is... 

An elementary solid, such as silver, is regarded as composed of atoms oscillating about fixed centres. The total energy content is therefore partly kinetic and partly potential. Since the solid has a finite compressibility, the atoms may be supposed to be maintained at small distances apart by forces they exert upon one another, and these may be resolved into two sets, one of which opposes a closer approximation of the atoms, and the other tends to draw the latter together. Both are functions of the distance between the atoms, and for a given distance are equal, since the form of the body is altered by external forces alone. 

The cytochromes are iron-containing hemoproteins in which the iron atom oscillates between Fe + and Fe + during oxidation and reduction. Except for cytochrome oxidase (previously described), they are classified as dehydrogenases. In the respiratory chain, they are involved as carriers of electrons from flavoproteins on the one hand to cytochrome oxidase on the other (Figure 12-4). Several identifiable cytochromes occur in the respiratory chain, ie, cytochromes b, Cp c, a, and (cytochrome oxidase). Cytochromes are also found in other locations, eg, the endoplasmic reticulum (cytochromes P450 and h, and in plant cells, bacteria, and yeasts. 

Bloch (1933a,b) first pointed out that in the Thomas-Fermi-Dirac statistical model the spectral distribution of atomic oscillator strength has the same shape for all atoms if the transition energy is scaled by Z. Therefore, in this model, I< Z Bloch estimated the constant of proportionality approximately as 10-15 eV. Another calculation using the Thomas-Fermi-Dirac model gives I tZ = a + bZ-2/3 with a = 9.2 and b = 4.5 as best adjusted values (Turner, 1964). This expression agrees rather well with experiments. Figure 2.3 shows the variation of IIZ vs. Z. 

Bending vibrations out of plane (symbol f), in which one atom oscillates through... 

The changes in reorientation of surface atoms were explained using the dynamic model of the crystal space lattice. It was assumed that during anodic polarization, when the oxidation of adsorbed water is taking place, atoms oscillate mainly in a direction perpendicular to the electrode surface. This process leads to periodic separation of atoms in the first surface layer. Thus, the location of atoms in different orientations is possible. It was stated that various techniques of electrode pretreatment used for... 

The most simple, but general, model to describe the interaction of optical radiation with solids is a classical model, due to Lorentz, in which it is assumed that the valence electrons are bound to specific atoms in the solid by harmonic forces. These harmonic forces are the Coulomb forces that tend to restore the valence electrons into specific orbits around the atomic nuclei. Therefore, the solid is considered as a collection of atomic oscillators, each one with its characteristic natural frequency. We presume that if we excite one of these atomic oscillators with its natural frequency (the resonance frequency), a resonant process will be produced. From the quantum viewpoint, these frequencies correspond to those needed to produce valence band to conduction band transitions. In the first approach we consider only a unique resonant frequency, >o in other words, the solid consists of a collection of equivalent atomic oscillators. In this approach, coq would correspond to the gap frequency. 

This model of atomic oscillators, in which we assume bound valence electrons, is also perfectly valid for metals, except that in this case we must set coq = 0. 

Let us now analyze the interaction of a light wave with our collection of oscillators at frequency two- In this case, the general motion of a valence electron bound to a nucleus is a damped oscillator, which is forced by the oscillating electric field of the light wave. This atomic oscillator is called a Lorentz oscillator. The motion of such a valence electron is then described by the following differential equation ... 

Vibrations in a real crystal are described by the lattice dynamical theory, discussed in section 2.1, rather than by the atomic oscillator model. Each harmonic phonon mode with branch index k and wavevector q then has, analogous to Eq. 

In the Slater-type mode, oxygen atoms oscillate with the largest relative amplitude. As a result, we find that the criterion of the FE lattice instability in totally oxygen isotope-exchanged STO and KTO is ... 

The calculations of g(r) and C(t) are performed for a variety of temperatures ranging from the very low temperatures where the atoms oscillate around the ground state minimum to temperatures where the average energy is above the dissociation limit and the cluster fragments. In the course of these calculations the students explore both the distinctions between solid-like and liquid-like behavior. Typical radial distribution functions and velocity autocorrelation functions are plotted in Figure 6 for a van der Waals cluster at two different temperatures. Evaluation of the structure in the radial distribution functions allows for discussion of the transition from solid-like to liquid-like behavior. The velocity autocorrelation function leads to insight into diffusion processes and into atomic motion in different systems as a function of temperature. 

By examining the size, shape, and orientation of the thermal ellipsoid associated with the bridging hydrogen atom, we concluded that the hydrogen atom in the bent Mo-H-Mo system is described preferably as an effectively symmetric atom oscillating around a single equilibrium point rather than being randomly distributed between two equilibrium positions in the crystal lattice. The thermal... 

Rg. 15.40 Postulated structures of (CjH Bc (a) Slipped sandwich in which one ring is pentahapto and the other is munohaplc (b) In solution the Cp rings appear to rock as the beryllhim atom oscillates. (From Fratten, J- Cooper. M. K.i Aroney. M. f., Filipcruk. S. W. J- Chi m. Si>c. Dalton franS. 1985. 1761-1765. Used with permission.)... 

To estimate the activation energy of diffusion at lower temperatures let us assume the pre-exponential factor Du to be independent of T and, by the order of magnitude, to be equal to D = A2vk % 10 4cm2s 1 (A % 10 8 cm is the characteristic value of a diffusion jump, vk 1012s 1 is the characteristic frequency of atomic oscillations in a solid). Then from the experimental values of D one can find the activation energy of diffusion Ea = 9 kcalmol 1... 

In this way the uniformity of the surface, although violated instantaneously, is retained on the average, provided the time element is large compared with the time of relaxation of the thermal motion on the surface. We should immediately note that this time of relaxation is by no means necessarily of the order of 10-13 sec (the period of atomic oscillations in the lattice) since not only is simple displacement of atoms about their equilibrium positions possible, but also much more complicated and slower processes—for example, exchange of foreign dissolved atoms between the surface and the bulk of the crystal. From our point of view such a surface can be called uniform if each atom of the surface has the same probability of being replaced... 

Atomic Theory of Diffusion. Diffusion Mechanisms. The atomic theory of diffusion describes how an atom moves from one part of a crystal to another. The lattice sites in a crystal are assumed to be fixed locations of the atoms making up the crystal. The atoms oscillate around these lattice sites, which are their equilibrium positions. These oscillations lead to finite... 

Time base Crystal or atomic Oscillator Escapement, motor, tuning fork Pyrotechnic burning rate Oscillator Rate of chemical reaction... 

The photodissociation of trifluoromethyl iodide, CF3I —> CF3 + I/I, which was briefly discussed in Section 6.4, seems to illustrate case (a) of Figure 9.4 while the photo dissociation of methyl iodide, CH3I —> CH3 + I/I, appears more to represent case (b). In both examples, the 1/2 umbrella mode, in which the C atom oscillates relative to the Irrespectively F3-plane, is predominantly excited. Following Shapiro and Bersohn (1980) the dissociation of CH3I and CF3I may be approximately treated in a two-dimensional, pseudo-linear model in which the vibrational coordinate r describes the displacement of the C atom from the H3-/F3-plane and the dissociation coordinate R is the distance from iodine to the center-of-mass of CH3/CF3 (see Figure 9.6).t... 

The more recent theories of chemical conversions [59-61] take into account the fact that the process of overcoming the activation barrier involves a cooperative change of more than one degree of freedom for the starting reagents subsystem. For the surface processes this is expected to lead to a need for considering the dynamics of the solid atom motion and, at least, the model should include information on Debye frequencies for its atoms (see, e.g., Ref. [62]). An additional inconvenience of the models for the elementary surface processes is associated with the fact that the frequencies of the surface atom oscillations differ from those inside the solid. Consideration of the multiphonon contributions to the probabilities that the elementary process can take place results in a significant modification of its rate constant up to the complete disappearance of the activation form of the temperature dependence [63,64]. 

Two types of vibrations occur in molecules stretches, where the distance between bonded atoms oscillates, and bends, where the bond angles oscillate. These oscillations are often coupled among several atoms. As examples, one stretching vibration and one bending vibration of the CH2 group are shown here ... 

Inspection of the MD simulation results shows that the O-H bond is broken, and the H atom reattaches to the up Si atom of the same dimer (dimer A2) to form a new H-Si bond. After 7ps, all atoms oscillate around their stable equilibrium positions. Another method of performing the MD simulations consists in setting an initial temperature T = 300 K without any heating and letting the system evolve at this temperature. Following this avenue, we arrive at the same conclusions as reported above. 

As regards the vibrational motion, the two atoms oscillate against each other. The molecule, therefore, possesses both potential and kinetic energy. This means that the energy of vibration involves two degrees of freedom. The vibrational motion in a molecule is, thus, associated with energy = 2 1/2 kT = kT per molecule or RT per mole. 

The theory of atomic oscillations recently advanced by Schroedinger is of extraordinary importance since it throws a new light on the problems of atomic structure and, at the same time, offers a convenient practical method for calculating the Heisenberg-Born intensity matrices. It seemed desirable to apply it to as many special cases as possible. A complete theory of the Stark effect in hydrogen was, therefore, developed. ... 

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