The MSC Process in Geant4 Geant4 does not use the detailed algorithm where all the collisions/interaction experienced by the particle are simulated. G4 uses condensed simulation algorithm. The model of MSC used by G4 ( G4UrbanMscModel ) simulates the scattering of the particle after a given step and also computes the path length correction and the lateral displacement. There are few general steps of MSC simulation at G4.

Multiple Scattering (1) zd zd/2 Silvia Pozzi Muon

1/λk = 2πnα∫ [1-Pk(cosχ)] (dσ(χ)/dΩ) d(cosχ) (1) 1) The step length of the particle is determined by the physics process or the geometry of detector. For particle (muon) with E,p the stepping algorithm checks all the step length of possible physical process and takes the minimal one (t)p – ”physics” step length. 2)Has to be compared with the geometrical step length determined by detector geometry (z)d – geometrical step length (values are imposed when cross the volume boundaries). To do this it needs the transformation (t)(z). The properties of MSC completely determined by the transport mean free path: 1/λk = 2πnα∫ [1-Pk(cosχ)] (dσ(χ)/dΩ) d(cosχ) (1) dσ(χ)/dΩ – the differencial cross section of the scattering, nα –number of atoms per volume, Pk(cosχ) – (k) Legandre polinome. Need only λ1 ,λ2, which are tabulated in G4 for 15 materials but only for e- and e+ at low energies, but can be lineary extrapolated in masses and β= v/c of other particles. < z > = λ1 [1-exp(- t/λ1)] - path length correction (t z transformation) (2) but when step small and E loss can be neglected. If not, λ 1 = λ1(t), decreases along the step: λ 1(t) =λ10 (1- αt), α = (λ10 -λ11)/tλ10 (3) λ10 – transport mean.free.p. at the beginning of the step, α – const. But for high energies: (3a)

t(z) = <t> = - λ1log(1-z/λ1) , for small steps or (4) 3) Only after ”physics” step length (t) transformation in (z) we can compare (z)p with (z)g and take min as the actual step length. 4) After the determination of actual step and particle relocation we need to compute the energy loss and scattering parameters ( θ(t) and lateral displacement ), which are functions of true step length (t). So, we need to do the inverse transformation (z) (t): t(z) = <t> = - λ1log(1-z/λ1) , for small steps or (4) t(z) = 1/α[1 - (1 – αwz)]^1/w , for big steps, with E loss taken in account (5), where w = 1 + 1/(αλ10) . 5) The scattering angle θ (t) is computed according to model function: g(u) = pg1(u) + (1-p)g2(u), (6) where u = cos(θ), 0 ≤ p ≤ 1, gi - functions of cos (θ), normalized at -1≤ u ≤ 1, g1(u) = C1 exp [-a(1-u)], g2 (u) = C2 1/(b – 1) ^2 , (7) Ci –normalization constants, a, b>0 – model parameters.

θ0 =(13.6 Mev/ βcp)·Zch√ t/X0 ·[1 + 0.038 ln (t /X0) ] (PDG) (10) 5.1) determination of parameters a,b,p, which are not independent. a = 0.5/(1-cos(θ0))  g1(u), (8) θ0 =(13.6 Mev/ βcp)·Zch√ t/X0 ·[1 + 0.038 ln (t /X0) ] (PDG) (10) t/X0 –is the true path length in the radiation length unit . b = 1/<cos (θ)>  g2(u) (11) <cos (θ)> = exp [- t/λ1] l.e. , h.e. (12) 6) After the simulation of scattering angle θ, G4 computes the lateral displacement and its direction: (13) where k =λ1 /λ2 , τ= t/ λ1 , vx and vy are x and y components of components of the directional unit vector. So, G4 simulates the θ angular distribution from model functions, while for the the lateral displacement and lateral correlations only mean values are used.

7) Step limitation algorithm. In G4 particle doesn’t penetrate in a new volume without stopping at the boundary. It restricts the step size when the particle entering a new volume to a value: fr · max{ r, λ1} (14) r – the range of the particle, fr [0,1] is a parameter (constant for a low energies). For high energies for particles with λ1 >λlim G4 uses the effective value feff = fr · [1 –sc + sc· (λ1 /λlim)] (15) (sc=0.25, λlim = 1mm). In order to prevent a particle from crossing a volume in just one step G4 makes additional limitation: after entering a volume the step size cannot be bigger than dgeom/fg , (16) dgeom is the distance to the next boundary, fg is a constant. At the start of the track the restriction is 2dgeom/fg (17)

8) Boundary crossing algorithm. The new algorithm was implemented (2006) to improve the simulation around interfaces. It doesn’t allow a “big” last step in a volume and “big” first step in the next volume. The s.l. around the boundary crossing s.l. < λelastic (mean free path of elastic scattering in the given material). After this small step the particle scattering according to a single scattering law (no MSC very close to the boundary or at the boundary). The key parameter of algorithm is ‘skin’. When ‘skin’ ≤ 0, algorithm doesn’t work. When ‘skin’ >0, it works in layers of thickness (skin ·λelastic ) before boundary and of thikness (skin-1) ·λelastic after the boundary. In this areas the particle performs step of length ≤ λelastic. For these small steps there are no path length correlations and lateral displasement computation. In other words the program works in this thin layer in “microscopic” mode. The elastic mean free path is estimated as: Λelastic = λ1·rat(Ekin ) rar(Ekin) = (0.001(MeV)^2)/T(T + 10 MeV). (14) The single scattering by angle θis determined by distribution: (15)

u = cos(θ), -screening parameter (16) where Z – atomic number, τ- kinetik energy measured in particle mass unit, α – constant.