A New Look at Energy — Question 10

Question 10

Pair-production threshold depends on the photon collision angle

Two photons of energies $E_1$ and $E_2$ meet at angle $\theta$ between their propagation directions. They may produce two particles each of invariant mass m. Use $|\mathbf p_i|=E_i/c$ and ignore additional selection rules.

1. Derive the initial invariant mass and the pair-production threshold inequality.

2. Compare head-on and parallel photons, and find the minimum $E_2$ for head-on collision with known $E_1$.

Question 10 - Solution

Pair-production threshold depends on the photon collision angle

(a) Method and calculation

$$M^2c^4=(E_1+E_2)^2-c^2|\mathbf p_1+\mathbf p_2|^2=2E_1E_2(1-\cos\theta).$$$$M\ge2m\quad\Rightarrow\quad\boxed{E_1E_2(1-\cos\theta)\ge2m^2c^4}.$$

(b) Interpretation and further calculation

$$\theta=\pi:\quad\boxed{E_{2,\min}=m^2c^4/E_1}.$$$$\theta=0:\quad M=0<2m\quad\Rightarrow\quad\boxed{\text{pair production impossible}}.$$

Physical check

Large total laboratory energy alone is insufficient; enough invariant center-of-momentum energy is required.