Question

4) Threshold for pair production in the field of a nucleus: Consider the inte action of a photon with a proton at rest, producing an electron-positron pair, i. p+7+p+e+ +e. a) Write down the expressions for the four-vector momenta of the proton (at rest Pp, the photon Pph, and the electron and positron, Pu: 5 b) Evaluate the four-vector scalar product (pp + Duh) in the frame in which the proton is at rest. (5] c) Fvaluate the four-vector scalar product (Pl+p++p_) in the center-of-momentum frame at threshold, i.e., where all particles are at rest. 15] d) Set the two expressions from b) and c) equal to find the threshold energy for a photon to undergo electron-positron pair production in the field of a proton at rest.

Answer 1

Answer:

(a) $p_p=(m_pc,\:\mathbf{0})$ , $p_{ph}=(p_{ph}c,\:\mathbf{p_{ph}})$ , $p_+=(E_+/c,\:\mathbf{p_+})$ and $p_-=(E_-/c,\:\mathbf{p_-})$ .

(b) The proton is at rest in the Lab frame. So, we will do calculations in the Lab frame in this part of the problem.

$(p_p+p_{ph})^2=(m_pc+p_{ph}c,\:\mathbf{p_{ph}})^2=(m_pc+p_{ph}c)^2-\mathbf{p_{ph}}^2=m_p^2c^2+p_{ph}^2c^2+2m_pp_{ph}c^2-\mathbf{p_{ph}}^2=m_p^2c^2+2m_pp_{ph}c^2$

[ Since, the rest mass of the photon is zero, so, $p_{ph}^2c^2-\mathbf{p_{ph}}^2=0$ ]

(c) In the Center-of-Momentum frame at threshold, i.e., where all the particles are at rest, we have:

$(p_{p'}+p_++p_-)^2=(\:(m_{p}+m_++m_-)c,\:\mathbf{0})^2=(m_{p}+m_++m_-)^2c^2$

(d) Equating the final results from (b) and (c), we have:

$m_p^2c^2+2m_pp_{ph}c^2=(m_{p}+m_++m_-)^2c^2$

So, $p_{ph}c=\frac{(m_{p}+m_++m_-)^2-m_p^2}{2m_p}\times c=E_{threshold}$ (Answer)

Cheers!