Question

A radioactive nucleus of mass $M$ emits a photon of frequency $\upsilon$ and the nucleus recoils. The recoil energy will be

• A. $Mc^2-h\upsilon$
• B. $h^2\upsilon^2 /2 Mc^2$
• C. zero
• D. h$\upsilon$

Answer 1

Answer: (B)

$h^2\upsilon^2 /2 Mc^2$

Answer 2

$F _{ ex }=\frac{ d P }{ dt }=0$

$\Rightarrow dP =0$

$\Rightarrow P =$ constant

$\vec{ P }_{ i }=\vec{ P }_{ f }$

$0=\vec{ P }_{ Nu }+\vec{ P }_{ Ph }$

$\left|\vec{ P }_{ Nu }\right|=\left|\vec{ P }_{ Ph }\right|=\frac{ h }{\lambda}=\frac{ hv }{ c }$

Recoil $K.E$. of nucleus $K . E _{ Nu }=\frac{ P _{ Nu }^{2}}{2 M _{ Nu }}$

$K.E _{\cdot Nu }=\frac{( hv / c )^{2}}{2 M }=\frac{ h ^{2} v^{2}}{2 Mc ^{2}}$