Self-diffusion in crystals

In real crystals it is necessary to take some account of the three-dimensional nature of the diffusion process. An easy way of doing this is to add a geometrical factor, g, into the equation for D so that it becomes  

In the one-dimensional case, the factor 1/2 is a geometrical term to account for the fact that an atom jump can be in one of two directions  

In a cubic structure diffusion can occiu along six equivalent directions, and a value of g of 1/6 is appropriate  

In the foregoing discussion, every possible atom jump is allowed. This may not be true in real crystals. For example, in the case of vacancy diffusion, no movement is possible if the vacancy population is zero. Equation (7.8) for the number of successful jumps ignores this, and should contain a term pj, that expresses the probability that the jump is possible from a structural point of view  

For example, in the A1 (face-centred cubic) structure of magnesium, each metal atom is surrounded by 12 nearest neighbours. If on average throughout 

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The mechanism by which the self-diffusion in the relaxed state occurs is not firmly established at present. However, there are reasons to believe that for certain atoms in glassy systems, self-diffusion occurs by a direct collective mechanism and is not aided by point defects in thermal equilibrium as in the vacancy mechanism for self-diffusion in crystals (Section 8.2.1).2 These reasons include ... 

Cohen and Turnbull s critical free-volume fluctuations picture of selfdiffusion in dense liquids is similar to the vacancy model of self-diffusion in crystals. However, in crystals individual vacancies exist and retain their identity over long periods of time, whereas in liquids the corresponding voids are ephemeral. The free volume is distributed statistically so that at any given instance there is a certain concentration of molecule-sized voids in the liquid. However, each such void is short-lived, being created and dying in continual free-volume fluctuations. The Frenkel hole theory of liquids ignores this ephemeral, statistical character of the free volume. 

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