  The distribution of capture zones formed during the nucleation and growth of
point islands on a one-dimensional substrate during monomer deposition is
considered for general critical island size $i$. A fragmentation theory
approach yields the small and (for $i=0$) large size asymptotics for the
capture zone distribution (CZD) under the assumption of no neighbour-neighbour
gap size correlation. These CZD asymptotic forms are different to those of the
Generalised Wigner Surmise which has recently been proposed for island
nucleation and growth models, and we discuss the reasons for the discrepancies.
