2 The Interaction of Radiation with Matter
31
This procedure is carried out k times until the desired thickness 2 k x is reached.
Because of the tail of f (1) (E) towards large energy transfers, the numerical
convolution is performed on a logarithmic grid. More details of the implementation
can be found in Refs. [75, 76].
2.5.3 Laplace Transforms
In the Laplace domain, Eq. (2.35) becomes
F (s, x) = L{f ( x)} = e
−−n
∞
n=0
n
n!
L{f
(1) ()}
n
= exp
⎡
⎣ −Nx
∞
0
dE
1 − e
−sE
dσ
dE
⎤
⎦ .
Following Landau [20], we split the integral in the exponent in two parts,
Nx
∞
0
dE
1 − e
−sE
dσ
dE
= Nx
E 1
0
dE
1 − e
−sE
dσ
dE
+ Nx
∞
E 1
dE
1 − e
−sE
dσ
dE
,
where E 1 is chosen to be large compared to the ionisation threshold while at the
same time satisfying sE 1 1. For energy transfers exceeding E 1 , the differential
cross section is assumed to be given by the asymptotic expression for close
collisions (2.11); for E < E 1 , it is not specified.
Using exp (−sE) ∼ 1 − sE, we obtain for the first term
I 1 = Nx
E 1
0
dE
dσ
dE
1 − e
−sE
∼ Nxs
E 1
0
dE
dσ
dE
E.
We can therefore evaluate I 1 by subtracting the contribution due to energy transfers
between E 1 and E max according to Eq. (2.11) from the total average energy loss
xdE/dx = =
I 1 ∼ s − sξ
ln
E max
E 1
− β
2
,
where we have introduced the variable
ξ = x
2πz 2 (α ¯
hc)
2 NZ
mc 2 β 2
.