 Solution

ID: past-exam-of-the-particle-and-nuclear-physics-course-of-the-university-of-cambridge/2012/paper-1/1/b/solution

Neglect atomic binding energy and take the target electron at rest. Use natural units temporarily. The final state contains three electrons and one Positron, so its threshold invariant mass is

4 m_{e}

. For incident total energy

E

s = (p_{1} + p_{2})^{2} = 2 m_{e}^{2} + 2 m_{e} E = 16 m_{e}^{2}, E_{th} = 7 m_{e} c^{2} = 3.577 MeV ≃ 3.58 MeV .

The corresponding incident kinetic energy is

6 m_{e} c^{2} = 3.066 MeV

. At threshold the products have no relative momentum in the centre-of-momentum frame; their common laboratory motion requires more energy than merely the pair's rest energy.

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