The Physics of Spallation Processes

Chapter 4Theory/Models4.1 Transport equationPhenomena in radiation physics and particle transport of leptons, baryons, mesons andenergetic photons can be described by the Boltzmann integro-differential equation settledin 1872. The equation will be briefly described here, because Monte-Carlo and deterministicapproaches employ solutions of the equation for neutron and gamma-transport. Itis a continuity equation in phase space consisting of three space coordinates, the kineticenergy and the direction of motion. Solutions of the Boltzmann equation which hereare just briefly presented in order in introduce some nomenclature are evaluated in moredetail in the literature for reactor physics [Eme69] and for fusion technology [Dol82]. Thenon-relativistic Boltzmann equation can be written as( )1 ∂ ˙Φiv i ∂t=++I{ }} {− ⃗ Ω∇ ˙Φ iII{ }} {∑[∫ ∫jIII{ }}]{S[ ∂ ˙Φi∂E − ˙Φ i2Eσ ij(⃗x, E B → E, ⃗ Ω B → ⃗ Ω ) ˙Φj(⃗x, E B , ⃗ Ω B , t ) dE B d ⃗ Ω B ]− σ i (⃗x, E) ˙Φ iIV{ }} {ln 2− ˙Φi + ∑ ln 2b ijvE 1/2,i jV{ }} {+ Y i (⃗x, E, Ω,vE ⃗ t) (4.1)1/2,iwhere ˙Φ i (⃗x, E, ⃗ Ω, t) is the angular dependent unknown flux, i.e. the number of particlesof type i in the volume element dxdydz at ⃗x at time t, in the energy element dE at Ewith a direction of motion within dΩ at ⃗ Ω, multiplied by the speed v i of the particle. Itgives the number of particles per cm 2 , per MeV, per steradian and per second at a givenlocation at a given time.(I) the first term in Eq. 4.1 reflects the translation/reduction of the phase space−div [ ⃗ Ω ˙Φi (⃗x, E, ⃗ Ω, t) ] = − ⃗ Ω∇ ˙Φ i .(II) considers the particle nucleus interaction (energy, angle and particle type are changed).σ ij(⃗x, E B → E, ⃗ Ω B → ⃗ Ω ) is the macroscopic cross section for the production of i-35