In the case where the rates P(x, t), Q(x, t) are suffi­ciently smooth, it is reasonable to approximate them by continuous functions P(x, t), Q(x, t) and to replace the right-hand sides of eqns [18] and [19] by contin­uous functions of two variables, J(x, t) and f (x, t). The Fokker-Plank equation can be obtained from the ME by expanding the right-hand side of eqn [18] in Tailor series, omitting derivatives higher than the second order

@fS(x>t) л-e/ .лі

@t = G(x) — @4 V (x’t )f (x’t)]

@2

+ gXX2^D(x, t)f (x’t)] [33]

where

V(x, t) = P(x, t) — Q(x, t)

D(x)=2[^(x’t)+q (x >t) [34]

The first term in eqn [33] describes the hydrodynamic­like flow of clusters, whereas the second term accounts for their diffusion in the size-space. Note that for clusters of large enough sizes, when the cluster evolution is mainly driven by the hydrody­namic term, the functions P(x, t), Q(x, t) are smooth; hence the ME and F-P equations are equally accurate. For sufficiently small cluster sizes, when the diffusion term plays a leading role, eqn [33]) provides only poor description.67,68,83 As the cluster nucle — ation normally takes place at the beginning of irradi­ation, that is, when the clusters are small, the results obtained using F-P equation are expected to be less accurate compared to that of ME.

1.13.4.4.1 Mean-size approximation

In eqn [24], the term with V(x, t) is responsible for an increase of the mean cluster size, while the term with D(x) is responsible for cluster nucle — ation and broadening of the SDF. For large mean cluster size, most of the clusters are stable and the diffusion term is negligible. This is the case when the nucleation stage is over, and the cluster density does not change significantly with time. A reasonably accurate description of the cluster evolution is then given in the mean-size approxima­tion, when fc(x, t) = Nc8(x — (x(t))) where 8(£) is the Kronecker delta and Nc is the cluster density. The rate of change of the mean size in this case can be calculated by omitting the last term in the right-hand side of eqn [24], multiplying both sides by x, integrating over x from 0 to infinity, and taking into account that f (x = 1, t) = 0 and f (x = 0, t) = 0

d(x) = V((x), t) [35]