An Interlude — Question 4

Question 4

Annihilation at rest requires momentum-balanced photons

An electron and positron annihilate at rest into exactly two photons. Use $m_ec^2=.511$ MeV and $hc=1240$ eV nm.

1. Find photon energies, relative directions and wavelengths.

2. Explain quantitatively why annihilation into a single photon in otherwise empty space is impossible.

Question 4 - Solution

Annihilation at rest requires momentum-balanced photons

(a) Method and calculation

$$\mathbf p_1+\mathbf p_2=0,\quad E_1+E_2=2m_ec^2\Rightarrow\boxed{E_1=E_2=.511\,\mathrm{MeV}}.$$$$\boxed{\theta_{12}=180^\circ,\quad\lambda=hc/E=1240/511000=.00242661\,\mathrm{nm}}.$$

(b) Interpretation and further calculation

$$P_i^2c^2=(2m_ec^2)^2>0,\quad P_\gamma^2c^2=E_\gamma^2-p_\gamma^2c^2=0.$$$$\boxed{P_i^\mu\ne P_\gamma^\mu}\quad\text{for a lone final photon}.$$

Physical check

An extra body can absorb recoil and alter the allowed final-state kinematics. The two-photon result assumes negligible initial kinetic energy.