PENELOPE 2003 - OECD Nuclear Energy Agency

Appendix A
Collision kinematics
To cover the complete energy range of interest in radiation transport studies we use
relativistic kinematics. Let ˜P denote the energy-momentum 4-vector of a particle, i.e.
˜P = (Wc −1 , p),
(A.1)
where W and p are the total energy (including the rest energy) and momentum respectively
and c is the velocity of light in vacuum. The product of 4-vectors, defined
by
( ˜P ˜P ′ ) = WW ′ c −2 − p·p ′ , (A.2)
is invariant under Lorentz transformations. The rest mass m of a particle determines
the invariant length of its energy-momentum,
( ˜P ˜P ) = W 2 c −2 − p 2 = (mc) 2 . (A.3)
The kinetic energy E of a massive particle (m ≠ 0) is defined as
E = W − mc 2 ,
(A.4)
where mc 2 is the rest energy. The magnitude of the momentum is given by
(cp) 2 = E(E + 2mc 2 ).
(A.5)
In terms of the velocity v of the particle, we have
where
E = (γ − 1)mc 2 and p = βγmcˆv, (A.6)
β ≡ v c = √
γ2 − 1
γ 2 =
is the velocity of the particle in units of c and
√ E(E + 2mc2 )
(A.7)
(E + mc 2 ) 2
γ ≡
√
1
1 − β = E + mc2
2 mc 2
(A.8)