Non-stationary hopping kinetics

The analytical formalism just discussed has two shortcomings first, the usage of quite particular hop length distribution and, secondly, the restriction to the steady-state properties. The Torrey model becomes inadequate for point defects in crystals, where single hop lengths A between the nearest lattice sites takes place, (p r) = 5(r — A) in equation (4.3.4). This results in the 

In order to analyze the temporal process of steady-state formation in a course of tunnelling reactions in crystals, Kotomin [85] solved numerically equation (4.3.28) and the relevant equation (4.1.19) for the reaction rate K t). It is clearly seen in Fig. 4.11 that the steady-states formed after the transient 

This trend continues as A decreases further (Fig. 4.13). At last, for small A O.OSro the reaction profile practically coincides with the limiting case of continuous diffusion, equation (4.3.11). It is seen in Fig. 4.14 how fastly the transient time required to reach the steady-state increases with decreasing A. For A ro this happens very rapidly, whereas for small A (continuous diffusion) it goes very slowly. 

Qualitatively speaking, as the hop length tends to zero, the transient time can be estimated in the spirit of equation (4.2.21) as 

As it follows from equation (4.3.29), in another extreme case, A oo, ftr tends to r, as it is indicated by a broken line in the insert of Fig. 4.14. It is also demonstrated in [85] that the continuous diffusion approximation, equation (4.1.63), gives quite reasonable reaction rates up to A Stq. This comes from the fact that K t) is a convolution of the correlation function and the exponentially decaying reaction probability cr r), that is, the essential deviation of the reaction profile from the diffusion limit does not affect the reaction rate considerably. 

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