01 Mar Particle Transport

Particle Transport 2016: Thomas M. Jordan (1978-1982 revised)

Attenuation between points x and x’, isotropic emission/reception

N(x,x’) = exp(-sk)/(4πs 2); s = |x’ – x|, u = (x’ – x)/s k = total cross section, explictly k = k(Q'(Q,s)) sk = mean free path integral, uses explicit cross sections

N(Q,Q’) = exp(-sk)/(4πs 2) p(Q’|Q,s); Q=x,u,e,t; dQ=dV dU dE dT x,u,e,t = position, direction, particle energy/species, time p(Q’|Q,s) Dirac-deltas/straggling, other coordinate changes

N = N(i,j) = average over δQ_i, integral over δQ’_j of N(Q,Q’) δQ_i, counted phase-space intervals covering all phase-space

Collision: changes direction, energy, species; isotropic=ck/(4π)

C(Q,Q’) = dk/dU p(Q’|Q,u); deflection u=u.u’, U = solid angle = d2 k/dUdE p(Q’|Q,u,de); deflection and energy/species de p(…) usually a delta function for other coordinate changes

C = C(i,j) = average over δQ_i, integral over δQ’_j of C(Q,Q’) for example, multigroup cross sections as used for: a) coupled neutron/gamma-ray transport b) coupled electron/photon transport for the energy/species/direction components of the counted phase-space vector elements δQ_i and δQ’_j

Neumann Series: using counted phase-space interval matrices

M = N + NCN + NCNCN + … = N + NCM = N + MCN = N + NBN

B = C + CNC + CNCNC + … = C + CNB = C + BNC = C + CMC

Green’s Functions: one group; N=1/k,C=ck,M=1/(k(1-c)),B=ck/(1-c)

M -1 = N-1 – C; B -1 = C-1 – N; right-hand sides are closed-form/known

Transport Problem Solution: S=S(i)=source integral over δQ_i, D=D(j)=average detector response over δQ’_j, R=integral response R = SMD = FD = SW; F=SM = flux/fluence; W=MD = weight/importance