question_answer

Consider the process of pair production.

It is possible for a photon to materialize into an electron and a positron. In this way, electromagnetic energy is converted into matter. Both energy and linear momentum are conserved when an electron-positron pair is created near an atomic nucleus. In absence of nucleus, pair production cannot occur in empty space because

A) it is impossible to conserve momentum without presence of nucleus

B) it is impossible to conserve energy without presence of nucleus

C) Both a and b are correct

D) pair production is possible even in empty space

Correct Answer: C

Solution :

Following is the vector diagram of the momenta involved if a photon were to materialise into an electron-positron pair in empty space.

For momentum to conserve,

\[\frac{hf}{c}=2p\cos \theta \]

\[\Rightarrow \]   \[hf=2pc\cdot \cos \theta \]

(here, f= frequency of incident photon)

As, \[p=mv\]

\[hf=2m{{c}^{2}}\left( \frac{v}{c} \right)\cos \theta \]

As \[\frac{v}{c}<1\]land \[\cos \theta \le 1\]

We can say that,
\[hf<2m{{c}^{2}}\]		... (i)
But energy conservation required
\[hf=2m{{c}^{2}}\]	... (ii)
Hence, it is impossible to satisfy both
Eqs. (i) and (ii) simultaneously.
But if there is some other object is present to carry away part of initial momentum, then both condition (i) and (ii) can be satisfied.