Atoms vibrational frequencies

If, as above, the potential-energy barrier height is E, statistical mechanical considerations indicate that the atom will have sufficient thermal energy to pass over the barrier a fraction exp(— E/k T) of the time. If/is a characteristic atomic vibrational frequency, the probability p that during unit time the atom will pass the potential-energy barrier is given by... 

In Section 3.1.1, self-diffusion was analyzed by studying the diffusion of radioactive tracer atoms, which were isotopes of the inert host atoms, thereby eliminating any chemical differences. Possible effects of a small difference between the masses of the two species were not considered. However, this difference has been found to have a small effect, which is known as the isotope effect. Differences in atomic masses result in differences of atomic vibrational frequencies, and as a result, the heavier isotope generally diffuses more slowly than the lighter. This effect can—if migration is approximated as a single-particle process—be predicted from the mass differences and Eq. 7.14. If mi and m2 are the atomic masses of two isotopes of the same component, Eqs. 7.13 and 7.52 predict the jump-rate ratio,... 

To be able to diffuse, an atom must surmount the energy barrier for migration presented by its neighbors. If vq is a characteristic atomic vibrational frequency, the probability for jump per second, v, is expressed as v = pq exp(—E /kT). The atom makes pq passes at the barrier with a probability exp(—Era/kT) on each try of surmounting it by thermal energy (pq is the so-called attempt frequency). 

In this equation, Aads f corresponds to the enthalpy difference between occupied and unoccupied adsorption sites and contains Meads-s be difference of the solvation enthalpies of Meads and S. Asub is the sublimation enthalpy, which is related to the interaction enthalpy per Me bond, Hle-Me, approximated as y/i in the case of first nearest neighbors (cf. eqs. 2.2 and 2.3). The terms and Vads represent the mean vibrational volumes of an atom in a kink site position or in an adatom position, respectively [3.269]. They are related to the mean atomic vibration frequencies in the 3D Me bulk lattice and in the Meads overlayer, respectively. 

Raman spectroscopy, although it has different selection rules, similarly portrays atomic vibration frequencies that can be sensitive to conformational changes. The infrared and Raman transitions take place in times that are of the order of 10" to lO i s and so essentially freeze the distribution on this timescale, i.e. they show the nature of the interchanging states and the energy difference between them, but not the actual interchange rates. 

The factor tq was found to be 10 s, which coincides with the reciprocal of the atomic vibrational frequency. The parameter Uq is taken as the activation energy, and it decreases linearly with tensile stress. The coefficient -y is a stmcmral factor that describes the orientation of the material. Zhurkov defined -y as a coefficient that relates the activation volume, 14. and (p, the localized overstress on a bond, to the average stress in the specimen. The source of stress on the system can be from either mechanical- or thermo-chemical effects. 

N is the number of potentially mobile ions Q is the oxide volume per mobile ion V is the atomic vibration frequency W is the energy barrier to ion movement into the oxide q is the ionic charge Ze a is half the ion jump distance E is the electric field Ylx k is Boltzmann s constant T is the temperature... 

In Eq. (10), V is the atomic vibration frequency, p is the dislocation density, a is the area swept out by a dislocation during an event, b is the Burgers vector, and Q is the activation energy. [A mistake in the reference has been corrected in Eq. (10).] Following the development of the Dorn creep equation, Eq. (10) can be generalized to... 

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