In an accelerator experiment on high-energy collisions of electrons with positrons, a certain event is interpreted as annihilation of an electron-positron pair of total energy $10.2\; BeV$ into two $\gamma$ -rays of equal energy. What is the wavelength associated with each $\gamma$ -ray? $\left(1\; BeV =10^{9}\; eV \right)$

Total energy of two $\gamma$ -rays: $E =10.2 \,BeV$

$=10.2 \times 10^{9} eV$

$=10.2 \times 10^{9} \times 1.6 \times 10^{-10}\, J$

Hence, the energy of each $\gamma$ -ray

$E^{\prime}=\frac{E}{2}$

$=\frac{10.2 \times 1.6 \times 10^{-10}}{2}=8.16 \times 10^{-10}\, J$

Plank's constant, $h=6.626 \times 10^{-34}\, Js$

Speed of light $c=3 \times 10^{8} \,m / s$

Energy is related to wavelength as:

$E^{\prime}=\frac{h c}{\lambda}$

$\lambda=\frac{h c}{E^{\prime}}$

$=\frac{6.626 \times 10^{-34} \times 3 \times 10^{8}}{8.16 \times 10^{-10}}=2.436 \times 10^{-16} \,m$

Therefore, the wavelength associated with each $\gamma$ -ray is $2.436 \times 10^{-16}\; \,m$