PENELOPE 2003 - OECD Nuclear Energy Agency

A.2. Inelastic collisions of charged particles 213
x
E' = E- W
E
2
1
θ
1
2
θ r
z
W
Figure A.2: Kinematics of elastic collisions.
In the case of collisions of particles with equal mass, m = M, this expression simplifies
to
W = E(E + 2mc2 ) sin 2 θ
E sin 2 θ + 2mc 2 if M = m. (A.26)
In this case, θ can only take values less than 90 deg. For θ = 90 deg, we have W = E (i.e.
the full energy and momentum of the projectile are transferred to the target). Notice
that for binary collisions of electrons and positrons (m = m e ), the relation (A.26)
becomes identical to (A.17).
For elastic collisions of electrons by atoms and ions, the mass of the target is much
larger than that of the projectile and eq. (A.25) becomes
W =
[
(E + mc 2 ) sin 2 θ + Mc 2 (1 − cos θ) ] E(E + 2mc 2 )
(E + Mc 2 ) 2 − E(E + 2mc 2 ) cos 2 θ
if M ≫ m.
(A.27)
The non-relativistic limit (c → ∞) of this expression is
W = 2m M (1 − cos θ)E if M ≫ m and E ≪ mc2 . (A.28)
A.2 Inelastic collisions of charged particles
We consider here the kinematics of inelastic collisions of charged particles of mass m
and velocity v as seen from a frame of reference where the stopping medium is at
rest (laboratory frame). Let p and E be the momentum and the kinetic energy of
the projectile just before an inelastic collision, the corresponding quantities after the
collision are denoted by p ′ and E ′ = E − W , respectively. Evidently, for positrons the
maximum energy loss is W max = E. In the case of ionization by electron impact, owing
to the indistinguishability between the projectile and the ejected electron, the maximum
energy loss is W max ≃ E/2 (see section 3.2). The momentum transfer in the collision is
q ≡ p − p ′ . It is customary to introduce the recoil energy Q defined by
Q(Q + 2m e c 2 ) = (cq) 2 = c 2 ( p 2 + p ′2 − 2pp ′ cos θ ) ,
(A.29)