Interpreting magic-number and evaporation effects in cluster size distributions

The Smoluchowski equations describe the coalescence of clusters to form larger clusters. If the kernels or rate constants in these equations are homogeneous, meaning that K_{λj, λk} =λ^2ωK_jk (where j and k are cluster sizes), it can be shown that the populations n_k approach Ak^ae -^bk for large k and large time, where the constants a and b depend on the homogeneity parameter ω. Deviations of observed populations from this formula may be ascribed to magic-number and/or evaporation effects on the kernels. By integrating the Smoluchowski equations numerically for various choices of the kernels, we derive population distributions and show the effects of magic-number clusters and evaporation on the population distribution. Various methods are used to extract the value of ω, in order to determine the best way to extract the underlying value of ω from experimental data. Experimental populations for sodium metal clusters are then analyzed according to the same procedure, to extract the homogeneity parameter and explain the patterns in the population distribution.