CERN Program Library Long Writeup W5013 - CERNLIB ...

If now we set
N∑
Z s = n i Z i (Z i +1) =W
Z E =
Z x =
i=1
N∑
i=1
N∑
i=1
N∑
=W
n i Z i (Z i +1)lnZ −2/3
i
[
n i Z i (Z i +1)ln
i=1
[
p i
Z i (Z i +1)ln 1+3.34
A i
N∑
i=1
p i
A i
Z i (Z i +1)
=WZ ′ s
=W ∑ p i
Z i (Z i +1)lnZ −2/3
i =WZ E
′ A i
( ) ]
αZi Z 2 inc
1+3.34
β
( ) ]
αZi Z 2 inc
=WZ x
′ β
we can write:
(
m
ln(χ 2 2 α ) = ln e 4 )
1.13
p 2¯h 2 + Z x − Z E
(0.885) 2 Z s
χ 2 α =
m 2 e 4 1.13
p 2¯h 2 (0.885) 2 e(Zx−Z E)/Z s
= m2 e 4 1.13
p 2¯h 2 (0.885) 2 e(Z′ x −Z′ E )/Z ′ s
and finally:
χ 2 c
Ω 0 =
e 2γ−1 χ 2 α
=
1 4πe 4 Z s ρN Av Zinc 2 t p 2¯h 2 (0.885) 2
1.167 Wc 2 p 2 β 2 1.13m 2 e 4 e −(Z′ x−Z E ′ )/Z s
′
= b c Zinc
2 t
β 2
b c = 4πN Av¯h 2
m 2 c 2 (0.885) 2
1.13 × 1.167 ρZ′ s e(Z′ E −Z′ x)/Z ′ s
≈ 6702.33ρZ ′ s e(Z′ E −Z′ x)/Z ′ s
The quantity b c is calculated during initialisation by the routine GMOLI setting β =1and Z inc =1. This
variable is called OMC or OMCMOL and it is stored in Q(JPROB+21). It has, via the atomic electron screening
angle χ 2 α, a small dependence on β and Z inc in the term Z x . The quantity b c ρZ s ′ is re-evaluated during
tracking by the routine GMOLIO, called by GMULTS, only when necessary.
The dependence of Z x on β and Z inc is via a term of the form: 1+3.34 ( αZZ inc β −1) 2 , so it is not necessary
to recalculate it as long as 1+3.34 ( αZZ inc β −1) 2 ≈ 1+3.34 (αZ) 2 that is:
[ (Zinc ) 2
3.34(Zα) 2 − 1]
≪ 1 ⇒ Z 2 Z2 inc − β2
β
β 2 ≪ 3.34α −2 ≈ 5500
Z 2 (Zinc 2 − β 2 ) ≤ 50β 2
For mixtures/compounds this condition should be checked for every component and this would be unacceptable
from the point of view of computer time. So we simply make the condition more restrictive multiplying
the first term by the number of elements in the mixture/compound.
246 PHYS325 – 5