An Interlude — Question 6

Question 6

A recoil partner slightly raises the threshold for photon pair creation

A photon strikes a stationary body of mass M and produces an electron-positron pair while the body survives unchanged. Let $m_ec^2=.511$ MeV and $Mc^2=1000$ MeV. Use the threshold invariant mass $M+2m_e$.

1. Derive the exact photon threshold energy.

2. Evaluate it and compare with $2m_ec^2$.

Question 6 - Solution

A recoil partner slightly raises the threshold for photon pair creation

(a) Method and calculation

$$sc^4=M^2c^4+2Mc^2E_\gamma,\quad s_{\mathrm{th}}c^4=(M+2m_e)^2c^4.$$$$\boxed{E_{\gamma,\mathrm{th}}=2m_ec^2\left(1+\frac{m_e}{M}\right)}.$$

(b) Interpretation and further calculation

$$\boxed{E_{\gamma,\mathrm{th}}=1.022(1+.511/1000)=1.02252224\,\mathrm{MeV}}.$$$$\boxed{\Delta E_{\mathrm{recoil}}=.000522242\,\mathrm{MeV}=522.242\,\mathrm{eV}}.$$

Physical check

At threshold all final particles share a common velocity in the lab. Pair rest energy alone is insufficient because incident momentum must be retained.