Frenkel law

This Wilson-Frenkel law is in fact valid even more generally the essential assumption for the derivation was that the surface structure of the growing interface does not change from the equilibrium one. Every rough surface should then be able to grow on average according to this Wilson-Frenkel law ... 

For a small driving force this growth law is indeed slower than the Wilson-Frenkel law (33) with Fwf but incomparably much larger than that of the nucleation process on faceted surfaces, (24), with V exp(—c/A/.i), where c is a positive constant. Therefore, the... 

The electric field dependence is, in all systems, very well described by a Poole-Frenkel law ... 

In this expression, ptoo, Ao, fipp and T are constants. The name of Eq. (8.85) derives from its similarity to the Poole-Frenkel law [50]. 

This is the Wilson-Frenkel rate. With that rate an individual kink moves along a step by adsorbing more atoms from the vapour phase than desorbing. The growth rate of the step is then simply obtained as a multiple of Zd vF and the kink density. For small A/i the exponential function can be hnearized so that the step on a crystal surface follows a linear growth law... 

A linear law is valid above the roughening temperature and the Wilson-Frenkel rate is an upper hmit to the growth rate which is achieved in the case of very fast surface diffusion. This is illustrated in Fig. 2. 

The equilibrium concentration of defects is obtained by applying the law of mass action to Eq. (7) or (8). This leads in the case of Frenkel disorder to... 

The equilibrium thus established is a Frenkel defect. In both the Schottky and Frenkel equilibria, the stoichiometry of the crystal is unaltered (figure 4.2). Assuming that the thermodynamic activity of the various species obeys Raoult s law, thus corresponding to their molar concentrations (denoted hereafter by square brackets), the constant of the Schottky process is reduced to... 

These characteristic features are manifested also in the generation-recombination processes of the Frenkel defects in real crystals. However, in contrast to the box model, in a crystal statistical screening of the recombing particles occurs in coordinate space that leads to a complex spatial distribution of vacancies v and interstitial atoms i. This distribution depends on the law whereby the probability of recombination varies with the distance r between complementary particles. Usually one approximates this law by the step-distribution cr(r) = 1 (r ro), er(r) = 0 (r > ro), ro is a recombination radius. 

Similarly, in thermal equilibrium, some ionic crystals at a temperature above absolute zero enclose a certain number of Frenkel pair defects, that is, anion and cation interstitials in the structure. Since the concentration of Frenkel pair defects at equilibrium at an absolute temperature, T, obeys the mass action law, then [16]... 

Similar to a Frenkel disorder, or to the dissociation equilibrium in water, a mass action law according to... 

Numerous measurements over a large range of electric fields and temperatures have established that, in many materials, the carrier mobility can be described by a universal law bearing the Poole-Frenkel like form of the electric field dependence... 

Santos-Lemus and Hirsch (1986) measured hole mobilities of NIPC doped PC. Over a range of concentrations, fields, and temperatures, the transport was nondispersive. The field and temperature dependencies followed logn / El/2 and -(T0IT)2 relationships. For concentrations of less than 40%, a power-law concentration dependence was reported. The concentration dependence was described by a wavefunction decay constant of 1.6 A. To explain a mobility that shows features expected for trap-free transport with a field dependence predicted from the Poole-Frenkel effect, the authors proposed a model based on field-enhanced polaron tunneling. The model is based on an earlier argument of Mott (1971). 

In addition to the examples already mentioned we shall discuss the brominatlon of silver and Ag-Cd alloys. In all cases a parabolic rate law was observed above 200°C. Contrary to the previous example, the AgBr surface layer is an ionic conductor characterized by a Frenkel-type disorder, i.e., n g. n Ag we must write t t g B Hr 0> and consequently t becomes the rate determining factor in Bq. (27). According to Frenkel ... 

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