## 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,d**e**); deflection and energy/species d**e** 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