PENELOPE 2003 - OECD Nuclear Energy Agency

228 Appendix C. Electron/positron transport in electromagnetic fields
Fig. C.1 displays trajectories of electrons and positrons with various initial energies
and directions of motion in a uniform electric field of 511 kV/cm directed along the
positive z-axis. Particles start from the origin (r 0 = 0), with initial velocity in the
xz-plane forming an angle θ with the field, i.e. v 0 = (sin θ, 0, cos θ), so that the whole
trajectories lie in the xz-plane. Continuous curves represent exact trajectories obtained
from the analytical formula (C.21). The dashed curves are the results from the firstorder
tracking algorithm described above [eqs. (C.14)-(C.20)] with δ E = δ E = δ v = 0.02.
We show three positron trajectories with initial energies of 0.1, 1 and 10 MeV, initially
moving in the direction θ = 135 deg. Three trajectories of electrons that initially move
perpendicularly to the field (θ = 90 deg) with energies of 0.2, 2 and 20 MeV are also
depicted. We see that the tracking algorithm gives quite accurate results. The error can
be further reduced, if required, by using shorter steps, i.e. smaller δ-values.
C.1.2
Uniform magnetic fields
We now consider the motion of an electron/positron, with initial position r 0 and velocity
v 0 , in a uniform magnetic field B. Since the magnetic force is perpendicular to the
velocity, the field does not alter the energy of the particle and the speed v(t) = v 0 is a
constant of the motion. It is convenient to introduce the precession frequency vector ω,
defined by (notice the sign)
ω ≡ − Z 0eB
m e γc = −Z 0ecB
E 0
, (C.26)
and split the velocity v into its components parallel and perpendicular to ω,
Then, the equation of motion (C.7) becomes
v ‖ = (v· ˆω) ˆω, v ⊥ = v − (v· ˆω) ˆω. (C.27)
dv ‖
dt = 0,
dv ⊥
dt
= ω×v ⊥ . (C.28)
The first of these eqs. says that the particle moves with constant velocity v 0‖ along
the direction of the magnetic field. From the second eq. we see that, in the plane
perpendicular to B, the particle describes a circle with angular frequency ω and speed
v 0⊥ (which is a constant of the motion). The radius of the circle is R = v 0⊥ /ω. That
is, the trajectory is an helix with central axis along the B direction, radius R and pitch
angle α = arctan(v 0‖ /v 0⊥ ). The helix is right-handed for electrons and left-handed for
positrons (see fig. C.2).
In terms of the path length s = tv 0 , the equation of motion takes the form
r(s) = r 0 + s v 0
v 0‖ + R [1 − cos(s ⊥ /R)] ( ˆω׈v 0⊥ ) + R sin(s ⊥ /R)ˆv 0⊥ ,
(C.29)