Optical Glauber Model

(a) Woods–Saxon

ρ(r)=ρ0/[1+exp((r−R)/a)]

R is the nuclear radius and a is the surface diffuseness parameter.

(b) Hard sphere

ρ(r)=ρ0, r ≤ R
ρ(r)=0, r > R

For both profiles, ρ₀ is fixed by normalizing the density to the mass number A.

4π∫r2ρ(r)dr=A

T(s)=∫ρ(√(s2+z2)) dz

The thickness function projects the three-dimensional nuclear density onto the transverse plane by integrating along the beam direction z.

TA(s)d2s

gives the expected number of nucleons in a small transverse area element d²s around position s.

It satisfies the normalization condition

∫TA(s)d2s = A.

TAB(b)=∫TA(s)TB(|s−b|)d2s

This quantity measures the geometrical overlap of the two nuclei in the transverse plane at impact parameter b.

The overlap function controls the average number of binary nucleon-nucleon encounters.

∫TAB(b)d2b = AB

Large values correspond to central collisions, while small values indicate peripheral collisions.

Ncoll(b)=σNNinelTAB(b)

This gives the optical Glauber mean number of binary nucleon-nucleon collisions at fixed impact parameter b.

σ_NN^inel is the inelastic nucleon-nucleon cross section.

N_coll increases strongly for central collisions where the nuclear overlap is largest.

Npart(b) = ∫ [TA(s)PB(s−b) + TB(s−b)PA(s)] d2s

This gives the mean number of participant nucleons from both nuclei at impact parameter b.

PB(s−b)=1−exp[−σNNinelTB(s−b)]

is the probability that a nucleon from nucleus A at position s interacts at least once with nucleus B.

PA(s)=1−exp[−σNNinelTA(s)]

is the corresponding probability for a nucleon from nucleus B to interact with nucleus A.

c(b)=[σABinel]-1∫0b2πb′Pinel(b′)db′

This defines the centrality percentile corresponding to collisions with impact parameter smaller than b.

σABinel = ∫ 2πb Pinel(b) db

This total inelastic nucleus-nucleus cross section is used for normalization.

Small centrality percentiles correspond to central collisions, while large percentiles correspond to peripheral collisions.

Centrality class boundaries are obtained by inverting the cumulative relation c(b).