PENELOPE 2003 - OECD Nuclear Energy Agency

216 Appendix A. Collision kinematics
photons by the projectile is not possible 1 . When the energy E of the projectile increases,
Q − decreases and tends to the photon line when β tends to unity. Hence, emission
of photons by ultrarelativistic projectiles in low-density media is barely prevented by
energy and momentum conservation. Generally speaking, as the interaction involves the
exchange of a virtual photon, the DCS increases as the photon becomes more real, that
is as we approach the photon line. For a dilute gas, this causes a gradual increase of the
cross section with the projectile energy when β → 1.
The scattering angle θ is related to the energy loss through
The recoil angle θ r between p and q is given by
which can also be written in the form
cos 2 θ r =
cos θ = (cp)2 + (cp ′ ) 2 − Q(Q + 2m e c 2 )
. (A.40)
2(cp)(cp)
cos θ r = (cp)2 − (cp ′ ) 2 + (cq) 2
, (A.41)
2(cp)(cq)
W 2 /β 2 (
1 + Q(Q + 2m ec 2 ) − W 2 ) 2
. (A.42)
Q(Q + 2m e c 2 ) 2W (E + mc 2 )
For heavy (m ≫ m e ) high-energy projectiles and collisions such that Q ≪ E and
W ≪ E,
cos 2 W 2 /β 2
θ r ≃
Q(Q + 2m e c 2 ) ≃ Q −(Q − + 2m e c 2 )
Q(Q + 2m e c 2 ) .
(A.43)
1 In a condensed medium, ultrarelativistic projectiles can emit real photons (Cerenkov radiation)
under certain, quite restricting circumstances (see e.g. Jackson, 1975).