PENELOPE 2003 - OECD Nuclear Energy Agency

C.2. Exact tracking in homogeneous magnetic fields 229
e +
e
−
B
Figure C.2: Trajectories of electrons and positrons in a uniform magnetic field. The two
particles start from the base plane with equal initial velocities.
where ˆv 0⊥ ≡ v 0⊥ /v 0⊥ and s ⊥ = sv 0⊥ /v 0 . Equivalently,
r(s) = r 0 + sˆv 0 − s v 0
v 0⊥ + 1 ω [1 − cos(sω/v 0)] ( ˆω×v 0⊥ ) + 1 ω sin(sω/v 0)v 0⊥ .
(C.30)
After the path length s, the particle velocity is
v(s) = v 0
dr
ds = v 0 + [cos(sω/v 0 ) − 1] v 0⊥ + sin(sω/v 0 )( ˆω×v 0⊥ ).
(C.31)
In fig. C.3 we compare exact trajectories of electrons and positrons in a uniform
magnetic field obtained from the analytical formula (C.30) with results from the firstorder
tracking algorithm [eqs. (C.14)-(C.20)] with δ B = δ E = δ v = 0.02. The field
strength is 0.2 tesla. The depicted trajectories correspond to 0.5 MeV electrons (a)
and 3 MeV positrons (b) that initially move in a direction forming an angle of 45 deg
with the field. We see that the numerical algorithm is quite accurate for small path
lengths, but it deteriorates rapidly for increasing s. In principle, the accuracy of the
algorithm can be improved by reducing the value of δ v , i.e. the length of the step length.
In practice, however, this is not convenient because it implies a considerable increase of
numerical work, which can be easily avoided.
C.2 Exact tracking in homogeneous magnetic fields
In our first-order tracking algorithm [see eqs. (C.14) and (C.16)], the effects of the
electric and magnetic fields are uncoupled, i.e. they can be evaluated separately. For
uniform electric fields, the algorithm offers a satisfactory solution since it usually admits
relatively large step lengths. In the case of uniform magnetic fields (with E = 0), the
kinetic energy is a constant of the motion and the only effective constraint on the
step length is that the change in direction |∆v|/v 0 has to be small. Since the particle
trajectories on the plane perpendicular to the field B are circles and the first-order
algorithm generates each step as a parabolic segment, we need to move in sub-steps of
length much less than the radius R (i.e. δ v must be given a very small value) and this