Tham khảo tài liệu 'diffusion solids fundamentals diffusion controlled solid state episode 1 part 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Vacancy Mechanism of Self-diffusion 113 from giving a comprehensive review of all methods. Instead we rather strive for a physical understanding of the underlying ideas we consider explicitely low vacancy concentrations and cubic coordination lattices. Then the averages in Eq. refer to one complete encounter. Since for a given value of n there are n j pairs of jump vectors separated by j jumps and since all vacancy-tracer pairs immediately after their exchange are physically equivalent we introduce the abbreviation cos Sj cos Si j and get f 1 lim 2ầí j cosSj . n n j 1 Here cos Sj is the average of the cosines of the angles between all pairs of vectors separated by j jumps in the same encounter. With increasing j the averages cos Sj converge rapidly versus zero. Executing the limit n TO Eq. can be written as f 1 2 f cos Sj j i 1 2 cos Si cos S2 . . To get further insight we consider - for simplicity reasons - the x-dis-placements of a series of vacancy-tracer exchanges. For a suitable choice of the x-axis only two x-components of the jump vector need to be considered2 which are equal in length and opposite in sign. Since then cos Sj 1 we get from Eq. f 1 2 p p- where pt p- denote the probabilities that tracer jump j occurs in the same opposite direction as the first jump. If we consider two consecutive tracer jumps say jumps 1 and 2 the probabilities fulfill the following equations p p-p- p- p p- p-pt. Introducing the abbreviations tj pt p- and t1 t we get t2 pt pt p p- pt p t2 . - - y - - y t t From this we obtain by induction the recursion formula tj tj . 2 Jumps with vanishing x-components can be omitted. 114 7 Correlation in Solid-State Diffusion The three-dimensional analogue of Eq. was derived by Compaan AND Haven 20 and can be written as cos6j cosff j where 6 is the angle between two consecutive tracer jumps. With this recursion expression we get from Eq. f 1 2 cos6 cosff cosỚ 2 . . The .
