A photoelectric material having work-function ϕ 0 is illuminated with a light of wavelength λ λ < h c ϕ 0…

According to the equation of photoelectric effect,

$K{E}_{max}=\frac{hc}{\lambda }- {\varphi }_0$

Kinetic energy in terms of momentum is given as $KE=\frac{p^2}{2m}$

$\frac{p^2}{2m}=\frac{hc}{\lambda }-{\varphi }_0$

$\frac{{\left(\frac{h}{{\lambda }_d}\right)}^2}{2m}=\frac{hc}{\lambda }-{\varphi }_0$

Assuming small changes, differentiating both sides,

$\frac{h^2}{2m}\left(-\frac{2d{\lambda }_d}{{\lambda }_d^3}\right)= -\frac{hc}{{\lambda }^2} d\lambda$

$\frac{d{\lambda }_d}{d\lambda } \propto \frac{{\lambda }_d^3}{{\lambda }^2}$