PENELOPE 2003 - OECD Nuclear Energy Agency

C.1. Tracking particles in vacuum. 225
which we cast in the form
dv
dt = A,
A ≡ Z [
0e E − β 2 (E ·ˆv)ˆv + βˆv×B ] . (C.7)
m e γ
Notice that, for arbitrary fields E and B, the “acceleration” A is a function of the
particle’s position r, energy E and direction of motion ˆv.
Implicit integration of eq. (C.7) gives the equations of motion
v(t) = v 0 +
∫ t
0
A(r(t ′ ), E(t ′ ), ˆv(t ′ )) dt ′ ,
(C.8)
r(t) = r 0 +
∫ t
0
v(t ′ ) dt ′ .
(C.9)
Evidently, these equations are too complex for straight application in a simulation code
and we must have recourse to approximate solution methods. We shall adopt the approach
proposed by Bielajew (1988), which is well suited to transport simulations. The
basic idea is to split the trajectory into a number of conveniently short steps such that
the acceleration A does not change much over the course of a step. Along each step, we
then have
v(t) = v 0 + t A(r 0 , E 0 , ˆv 0 )
(C.10)
r(t) = r 0 + t v 0 + t 2 1 2 A(r 0, E 0 , ˆv 0 ),
(C.11)
where the subscript “0” indicates values of the various quantities at the starting point
(t = 0). The traveled path length s and the flying time t are related by
which to first order becomes
t =
∫ s
0
ds ′
v ,
t = s/v 0 .
Then, to first order in the electromagnetic force,
(C.12)
(C.13)
v(s) = v 0 + s A(r 0, E 0 , ˆv 0 )
cβ 0
r(s) = r 0 + s ˆv 0 + s 2 1 2
A(r 0 , E 0 , ˆv 0 )
.
c 2 β0
2
That is,
r(s) = r 0 + s ˆv 0 + s 2 1 2
Z 0 e [E 0 − β0 2(E 0·ˆv 0 )ˆv 0 + β 0ˆv 0 ×B 0 ]
. (C.14)
m e c 2 γ 0 β0
2
The particle’s velocity can be calculated directly from eq. (C.10), which to first order
gives
v(s) = v 0 + ∆v
(C.15)