An Interlude — Question 7

Question 7

Two photons need enough invariant energy to create a massive pair

A high-energy photon collides with a photon of energy epsilon=1.00 eV. The two-photon invariant is $W^2=2E\epsilon(1-\cos\theta)$ in energy-squared units. Electron pair production requires $W\ge2m_ec^2$, where $m_ec^2=.511$ MeV.

1. Find the head-on threshold for E.

2. Find the threshold at a 90-degree opening angle and explain the collinear limit.

Question 7 - Solution

Two photons need enough invariant energy to create a massive pair

(a) Method and calculation

$$\theta=\pi:\quad4E\epsilon\ge4m_e^2c^4\Rightarrow E_{\mathrm{th}}=m_e^2c^4/\epsilon.$$$$\boxed{E_{\mathrm{th,head-on}}=(.511\times10^6)^2/(1.00)=2.61121\times10^{11}\,\mathrm{eV}}.$$

(b) Interpretation and further calculation

$$\boxed{E_{\mathrm{th,90^\circ}}=2E_{\mathrm{th,head-on}}=5.22242\times10^{11}\,\mathrm{eV}}.$$$$\theta\to0:\ W^2\to0\Rightarrow\boxed{\text{parallel photons cannot create a massive pair}}.$$

Physical check

Total laboratory energy alone does not determine the threshold. The opening angle determines how much of it is available as center-of-mass energy.