PENELOPE 2003 - OECD Nuclear Energy Agency

230 Appendix C. Electron/positron transport in electromagnetic fields
4
b
2
x (cm)
0
- 2
a
v 0
- 4
θ
B
0 1 0 20 3 0 40
z
(cm)
Figure C.3: Trajectories of electrons and positrons in a uniform magnetic field of 0.2 tesla.
Continuous curves are exact trajectories calculated from eq. (C.30). The short-dashed lines
are obtained by using the numerical tracking method described in the text with δ v = 0.02.
Long-dashed curves are the results from the tracking algorithm with δ v = 0.005. a: electrons,
E 0 = 0.5 MeV, θ = 45 deg. b: positrons, E 0 = 3 MeV, θ = 45 deg.
makes the calculation slow. On the other hand, the action of the uniform magnetic field
is described by simple analytical expressions [eqs. (C.30) and (C.31)], that are amenable
for direct use in the simulation code. These arguments suggest the following obvious
modification of the tracking algorithm.
As before, we assume that the fields are essentially constant along each trajectory
step and write
r(s) = r 0 + sˆv 0 + (∆r) E + (∆r) B ,
(C.32)
where (∆r) E and (∆r) B are the displacements caused by the electric and magnetic
fields, respectively. For (∆r) E we use the first-order approximation [see eq. (C.14)],
(∆r) E = s 2 1 2
Z 0 e [E 0 − β0 2(E 0·ˆv 0 )ˆv 0 ]
. (C.33)
m e c 2 γ 0 β0
2
The displacement caused by the magnetic field is evaluated using the result (C.30), i.e.
(∆r) B = − s v 0
v 0⊥ + 1 ω [1 − cos(sω/v 0)] ( ˆω×v 0⊥ ) + 1 ω sin(sω/v 0)v 0⊥
(C.34)
with
ω ≡ − Z 0ecB 0
E 0
, and v 0⊥ = v 0 − (v 0· ˆω) ˆω. (C.35)
Similarly, the particle velocity along the step is expressed as
v(s) = v 0 + (∆v) E + (∆v) B
(C.36)