CERN Program Library Long Writeup W5013 - CERNLIB ...

where
λ 0
= ¯h/p Compton wavelength of the electron
2π
r TF = 0.885a 0 Z − 1 3 Thomas-Fermi radius of the atom
a 0 = ¯h2
me 2 Bohr’s radius
α = 1/137.035 ... the fine-structure constant
so that we have
χ 2 α = m2 e 4 [
1.13
p 2¯h 2 (0.885) Z 2 2 3 1+3.34
( ) ] 2 αZZinc
For a compound/mixture the following rule holds:
β
.
ln(χ 2 α) =
=
N∑
N∑
n i Z i (Z i +1)ln(χ 2 αi)
i=1
i=1
=
N∑
N∑
n i Z i (Z i +1)
i=1
i=1
N∑
i=1
= ln
+
p i
A i
Z i (Z i +1)ln
(
m 2 e 4 )
1.13
p 2¯h 2 −
(0.885) 2
N∑
i=1
p i
A i
Z i (Z i +1)ln(χ 2 αi)
p i
A i
Z i (Z i +1)
{
m 2 e 4 1.13
p 2¯h 2 (0.885) 2 Z 2 3
i
[1+3.34
N∑
i=1
N∑
[
p i
Z i (Z i +1) 1+3.34
A i
N∑
i=1
p i
A i
Z i (Z i +1)
p i
Z i (Z i +1)lnZ − 2 3
i
A i
i=1
N∑ p i
Z i (Z i +1)
A
i=1 i
( ) ]
αZi Z 2 inc
p i
A i
Z i (Z i +1)
β
( ) ]}
αZi Z 2 inc
β
To understand the transformations in the above formulae, the following should be noted. Let n i be the
number of atoms of type i in a compound/mixture, that is the number of moles of element i in a mole of
material, and p i the proportion by weight of that element. We have the following relation:
p i =
n i A i
∑ Nj=1
n j A j
that is n i =
so that we can simplify expressions in the following way:
⎛
N∑
⎝
j=1
n j A j
⎞
⎠ p i
A i
= W p i
A i
∑ Ni=1
n i ...
W
N∑
= (p i /A i ) ...
i=1
PHYS325 – 4 245