(a) The photoelectric effect is the absorption of a photon by an electron: It IS responsible for ejecting electrons from an illuminated metal surface and for the ionisation of atoms. However photoelectric absorption by isolated electrons is never observed. Why not?
(b) How are energy and momentum conserved in the ionisation of an atom, which can take place in isolation?
(c) The production of an electron-positron pair from a photon can occur when the photon energy exceeds the combined rest energy of the pair. The process is observed when gamma rays interact with matter. Pair production by isolated gamma rays is never observed. Explain why this is.
Solution : (a) Consider the putative absorption of the photon by the lone electron. The process is shown in the laboratory frame in figure
(a) Consider the putative absorption of the photon by the lone electron. The process is shown in the laboratory frame in figure
and transformed to the CM frame in figure
In the CM frame it is clear that momentum is conserved but relativistic energy is not since, comparing the energy before and after, hv'+ \( \gamma \)(mc2 > mc2 as \( \gamma \)> 1.
(b) When an isolated atom absorbs a photon its rest mass changes. This is not possible for the electron since, being elementary, it has no excited states.
(c) In the laboratory frame the putative process is shown in the figure. In the final state the total momentum is zero in the centre of mass frame. However, for the initial state, there is no frame in which the momentum is zero because a single photon cannot have a rest frame. It is therefore obvious that momentum cannot be conserved so the process cannot occur.