6.4 Appendix D: Energy and Momentum Conservation in a Two-Body Collision …
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6.4 Appendix D: Energy and Momentum Conservation
in a Two-Body Collision that Produces a Gamma-Ray
In Sect. 1.4, we examined the Joliot-Curies’ proposed gamma-producing reaction
4
2 He +
9
4 Be →
13
6 C + γ
(6.14)
The alpha-particle carries ~5.3 MeV of kinetic energy into the reaction, and
collides with the initially stationary Be nucleus. The quantities of interest in this
reaction are the energy and momentum of the emergent gamma-ray. In this section
we develop formulae for these quantities, assuming that the collision is head on.
Figure 6.2 illustrates the situation. Let the rest masses of the three nuclei be m A ,
m B , and m C . Nucleus A is presumed to bring kinetic energy K A into the reaction.
Product C emerges from the reaction with kinetic energy K C and the gamma-ray
with energy E γ . We do not refer to E γ as a kinetic energy as that term is usually
reserved for the motional energy of a particle of non-zero rest mass.
Begin with energy conservation, accounting for the kinetic energies of the
reactants as well as their relativistic rest energies:
K A + (m A + m B ) c
2
= K C + m C c
2
+ E γ .
(6.15)
For momentum conservation. we take the Newtonian momentum p =
√
2m K
for particles with mass, and, from Einstein, p = E/c for the gamma-ray, giving
2m A K A = ±
2m C K C + E γ
c,
(6.16)
where the upper (lower) sign is to be taken if C is moving to the right (left) after
the reaction; we assume that the gamma-ray is always moving forward after the
reaction. The direction of C after the reaction is dictated by energy and momentum
conservation. Let the mc
2 rest energies of the particles with mass be designated by
E’s, e.g., E A = m A c
2 . If we replace the masses in (6.16) by these rest energies, a
factor of c can be canceled, leaving
2E A K A = ±
2E C K C + E γ .
(6.17)
m A
m B
m C
γ -ray
K A
K C
E γ
Fig. 6.2 Head-on collision of two massive particles leading to production of a massive particle and
a gamma-ray