CERN Program Library Long Writeup W5013 - CERNLIB ...

7. Update the number of interaction lengths for all the processes and go back to (2) till the particle
either leaves the detector or falls below its energy threshold or beyond its time cut or disappears in an
interaction.
2 Distance evaluation
2.1 The interaction length
Let σ(E,Z,A) be the total microscopic cross section for a given interaction. The mean free path, λ, for a
particle to interact is given by:
λ = 1 Σ
(1)
where Σ is the macroscopic cross-section in cm −1 . This quantity is given for an element by:
Σ= N Avρσ(E,Z,A)
A
and for a compound or a mixture by:
Σ= N Avρ ∑ i n iσ(E,Z i ,A i )
∑
i n iA i
= N Av ρ ∑ i
(2)
p i
A i
σ(E,Z i ,A i ) (3)
N Av Avogadro’s number (6.02486 × 10 23 )
Z atomic number
A atomic weight
ρ density
σ total cross-section for the reaction
n i proportion by number of the i th element in the material
p i = n i A i / ∑ j n j A j , proportion by weight of the i th element in the material
For electromagnetic processes which depend linearly on the atomic number Z we can write:
Σ(E) = N Av ρ ∑ i
Z eff = ∑ i
= N Av ρf(E) ∑ i
p i
A i
Z i
p i
σ(E,Z i )=N Av ρ ∑ p i
Z i f(E)
A i A
i i
p i
Z i = N Av ρf(E)Z eff
A i
the value above of Z eff is calculated by GPROBI. This mean free path is tabulated at initialisation time as a
function of the kinetic energy of the particle, or, for hadronic interactions, it is calculated at tracking time.
Cross sections are tabulated in the energy range defined as: ELOW(1) ≤ E ≤ ELOW(NEK1) in NEK1
bins. These values can be redefined by the data record RANGE. Default values are ELOW(1) =10keV ,
ELOW(NEK1) =10TeV and NEKBIN = NEK1− 1=90. NEKBIN cannot be bigger than 199. The array ELOW
is in the common /GCMULO/.
Numerically, if we measure the microscopic cross section in b where 1b =10 −24 cm −2 , we can express the
macroscopic cross section as:
Σ[cm −1 ] = 6.02486 × 1023 ρ[g cm −3 ]σ(E,Z,A)[b] × 10 −24
(4)
A
= 0.602486 ρ[g cm−3 ]
σ(E,Z,A)[b] (5)
A
which is the formula mostly used in GEANT.
PHYS010 – 2 199