A thermal neutron in a graphite moderator travels a mean free path between isotropic…

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Computational Physics · Monte Carlo and lattice simulation of physical systems

Computational Physics, Ch Drawing numbers: random walks and Monte Carlo

Problem

A thermal neutron in a graphite moderator travels a mean free path ℓ=2.5 cm\ell = 2.5\,\mathrm{cm} between isotropic elastic scatters, moving at a constant speed v=2200 m s−1v = 2200\,\mathrm{m\,s^{-1}}. Idealise its path as a walk on a cubic lattice: after each collision it hops a fixed distance alat=ℓa_{\rm lat} = \ell along one of the six directions ±x,±y,±z\pm x, \pm y, \pm z, chosen with equal probability 16\tfrac16 each, and the time between hops is τ=ℓ/v\tau = \ell/v. (a) The two-dimensional lattice walk gives ⟨r2⟩=4Dt\langle r^2\rangle = 4Dt with D=alat2/(4τ)D = a_{\rm lat}^2/(4\tau), because each step moves one of two axes with probability 12\tfrac12 each. Generalise the argument to three dimensions, where each step moves one of three axes with probability 13\tfrac13 each, to show that ⟨r2⟩=6Dt\langle r^2\rangle = 6Dt with D=alat2/(6τ)D = a_{\rm lat}^2/(6\tau). (b) Compute DD in m2 s−1\mathrm{m^2\,s^{-1}} for the graphite numbers given. (c) Find the elapsed time tt and the root-mean-square displacement ⟨r2⟩\sqrt{\langle r^2\rangle} from the starting point after N=1000N = 1000 collisions. (d) A simulation of 10410^{4} such neutrons run for 10001000 collisions each measures ⟨r2⟩/(6Dt)=1.02±0.03\langle r^2\rangle/(6Dt) = 1.02 \pm 0.03. State whether this is consistent with the exact value 11, quoting the discrepancy in standard errors.

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