78 Chapter 2. Interaction of Radiation with Matter
The annihilation process in itself does not have any threshold energy. That is,
the process can also happen even if the electron and positron are at rest. However to
produce particles other than photons, the electron and the positron must be allowed
to collide with each other at very high energies. This is actually done, for example at
Large Electron Positron collider (or LEP) at CERN in Switzerland where particles
are accelerated to center of mass energies reaching at 45− 100 GeV before collision.
The result of the collision is generation of large number of particles, which are then
tracked down and identified to get clues about the fundamental particles and their
interactions.
At low electron-positron energies, at least two photons are produced in the an-
nihilation process. The reason why only one photon can not be emitted lies in the
law of conservation of momentum. According to this law, the total momentum of
the emitted photons must be equal to that of the total momentum of electron and
positron. Now, if the electron and the positron move in opposite directions with
equal kinetic energies or are at rest before annihilation, the net momentum before
collision will be zero. This implies that after the annihilation the total momentum
must also be zero. We can not have a zero net momentum with only one photon
and hence we conclude that at least two photons must be produced. Hence by using
a simple argument, we have set a lower limit on the number of photons produced.
Let us now suppose that an electron-positron annihilation process produces only
two equal energy photons. If the net momentum before collision was zero then
the photons must travel in opposite direction to each other since only then the
momentum conservation can be guaranteed. The energy these photons carry can be
deduced from the law of conservation of energy, which states that the total energy
of the system before and after collision must be equal. The reader may recall that
the total energy of a particle is the sum of its kinetic and rest mass energies, such
as
E=T+m 0 c^2 , (2.2.1)
whereTis the kinetic energy,m 0 is the rest mass, andcis the velocity of light. The
total energy before collision can then be written as
Etotal = Ee+Ee+
= Te+m 0 c^2 +Te++m 0 c^2
= Te+Te++2m 0 c^2 ,where we have made use of the fact that both the electron and the positron have
equal rest masses. The kinetic energies of the two particles will be zero if they were
at rest before the annihilation. Hence the total energy before the collision will be
Etotal=2m 0 c^2 =2× 511 keV.This shows that to conserve energy, each of the two photons produced must carry
an energy of 511keV. Hence we can conclude that if the conservation of momentum
and energy are to be satisfied then at the minimum there must be two photons each
carrying 511keV of energy and traveling in opposite directions. This concept is also
graphically depicted in Fig.2.2.1
An interesting point to note here is that the arguments we gave in the preceding
paragraphs do not exclude the possibility of one-photon annihilation process. The
assumption of zero net momentum before collision is not always true and therefore it