Exponential healing

A simple parameterization of these spectral shapes is provided by the exponential healing [73] or NMR layer model [74]. The size of the metal particles (or, more precisely, their size distribution) can be measured by electron microscopy. Using the cubo-octahedral model, we can then calculate what fraction of atoms is in the surface layer, in the subsurface layer, and so on. The local density of states in all sites of the surface layer is not quite the same, but nevertheless clearly different from those in the subsurface layer, and so on. Therefore, the spectrum is decomposed as a superposition of (for convenience) Gaussians, each representing the collection of sites in a layer. The integrals must be proportional to the fraction of atoms in the layer. The maximum of the Gaussian corresponding to the nth layer... 

According to the exponential healing theorem the Knight shift of the n-th layer (/f ) heals back to the bulk value K ), exponentially as a function of the layer number (n). Therefore, the Knight shift of the n-th layer can be written as... 

The thermal healing has been studied most extensively for one-dimensional gratings. Above roughening, the gratings acquire, for small amplitude to wavelength ratios, a sinusoidal form, as predicted by the classical continuum theory of Mullins and confirmed by experimenf-s and Monte Carlo simulations. - The decay of the amplitude is, asymptotically, exponential in time. This is true for both evaporation dynamics and (experimentally more relevant) surface diffusion. 

Figure 7. Arrhenius pre-exponential factors as functions of temperature for a model hydrogen-atom abstraction with various barrier heights (a), tunneling correction only, Vo == Vir (a ), complete isotope effect, Vo = Vh (a"), complete isotope effect, Vo = V + 1. The curves are labeled with V in heal/ mole. (NT indicates that no tunneling correction is included.) Numbers at the top of each graph are InCn for =10, as a function of T p . Dashed lines areatAo = % andy/2 (22).

Crack-healing can be expressed as a function of oxygen partial pressure, P02, and healing temperature, T. For this purpose, two assumptions were employed. One is that the crack-healing rate, vh, which can be assumed to be the exponential function of P02, and the exponential index, corresponding to the reaction order of oxygen, are independent of temperature. The other is that the rate constant, which is a proportional constant of crack-healing rate versus n-th powered Poi, obeys Arrhenius law. 

The inverse of the minimum healing time, 1/rnMin, represents the crack-healing rate for complete strength recovery for the given temperature (vn). The erack-healing rate can be expressed as the following exponential function of the Po2-... 

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