Question

In a gamma decay process, the internal energy of a nucleus of mass M decreases, a gamma photon of energy E and linear momentum E/c is emitted and the nucleus recoils. Find the decrease in internal energy.

Answer 1

Let the nucleus recoils with a velocity v.

Applying the law of conservation of linear momentum, we get:
Linear momentum of recoiled nucleus = Linear momentum of gamma photon
⇒ mv = $\frac{E}{c}$

$\therefore$ $v=\frac{E}{mc}$

Kinetic energy of the recoiled nucleus = $\frac{1}{2}M{v}^2$
$\Rightarrow \mathrm{K}.\mathrm{E}.=\frac{1}{2}m{\left(\frac{E}{mc}\right)}^2=\frac{1}{2}\frac{E^2}{m{c}^2}$

Decrease in the internal energy = photon energy + the kinetic energy of the recoiled nucleus
⇒ Decrease in the internal energy = $E+\frac{E^2}{2m{c}^2}$