Distributional fixed-point equations for island nucleation in one dimension: a retrospective approach for capture zone scaling

Paul Mulheran, Kenneth Peter O'Neill, Michael Grinfeld, Wilson Lamb

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The distributions of inter-island gaps and captures zones for islands nucleated on a one-dimensional substrate during submonolayer deposition are considered using a novel retrospective view. This provides an alternative perspective on why scaling occurs in this continuously evolving system. Distributional fixed point equations for the gaps are derived both with and without a mean field approximation for nearest neighbour gap size correlation. Solutions to the equations show that correct consideration of fragmentation bias justifies the
mean field approach which can be extended to provide closed-from equations for the capture zones. Our results compare favourably to Monte Carlo data for both point and extended islands using a range of critical island size i = 0, 1, 2, 3. We also find satisfactory agreement with theoretical models based on more traditional fragmentation theory approaches
Original languageEnglish
Article number051606
Number of pages8
JournalPhysical Review E
Issue number5
Publication statusPublished - 15 Nov 2012


  • gap size distribution
  • capture zone distribution
  • distributional
  • fixed-point equations
  • island nucleation
  • one dimension
  • retrospective approach
  • capture-zone scaling

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