Monte Carlo capture times for the standard trap hierarchy. Log-log plots of tCAPT as a function of the target concentration CT. (a) Triangular lattice. (b) Cubic lattice. Points: Monte Carlo results for single targets (ST, circles), multiple targets (MT, +), a single set of traps and target (STt, triangles), and multiple sets of traps and targets (MTt, ×). Lines: theoretical values, tCAPT = tM for single targets (ST) from Eqs. 6, 7 and tCAPT = tM〈tESC〉 for a single set of traps and targets (STt). In these examples the total concentration of traps and targets CTt = 31CT, so the total concentrations can be high; in the MTt series the maximum concentrations are CT = 0.02861, CTt = 0.8868 in panel a and CT = 0.02827, CTt = 0.8764 in panel b.