PENELOPE 2003 - OECD Nuclear Energy Agency

C.1. Tracking particles in vacuum. 227
expressed in the form (adapted from Bielajew, 1988)
r(t) = r 0 + tv 0⊥ + 1 a
[
cosh (act) − 1 + v 0‖
c sinh (act) ]
Ê,
(C.21)
where v 0‖ and v 0⊥ are the components of v 0 parallel and perpendicular to the direction
of the field,
v 0‖ = (v 0·Ê)Ê, v 0⊥ = v 0 − (v 0·Ê)Ê (C.22)
and
The velocity of the particle is
a ≡ Z 0eE
m e c 2 γ 0
= Z 0eE
E 0
. (C.23)
v(t) = v 0⊥ + [ c sinh (act) + v 0‖ cosh (act) ] Ê
= v 0 + { c sinh (act) + v 0‖ [cosh (act) − 1] } Ê. (C.24)
Since the scalar potential for the constant field is ϕ(r) = −E ·r, the kinetic energy of
the particle varies with time and is given by
E(t) = E 0 − Z 0 eE·[r 0 − r(t)] .
(C.25)
1 0
a b c
5
z (cm)
0
E
- 5
θ
v 0
f g h
- 1 0
0 1 0 2 0 3 0
x
(cm)
Figure C.1: Trajectories of electrons and positrons in a uniform electric field of 511 kV/cm.
Continuous curves represent exact trajectories obtained from eq. (C.21). The dashed lines are
obtained by using the first-order numerical tracking method described by eqs. (C.14)-(C.20)
with δ E = δ E = δ v = 0.02. The displayed trajectories correspond to the following cases. a:
positrons, E 0 = 0.1 MeV, θ = 135 deg. b: positrons, E 0 = 1 MeV, θ = 135 deg. c: positrons,
E 0 = 10 MeV, θ = 135 deg. f: electrons, E 0 = 0.2 MeV, θ = 90 deg. g: electrons, E 0 = 2
MeV, θ = 90 deg. h: electrons, E 0 = 20 MeV, θ = 90 deg.