The wavelength of $K_{\alpha }$ X-rays for lead isotopes Pb²⁰⁸, Pb²⁰⁶, ,Pb²⁰⁴ are ${\lambda }_1, {\lambda }_2$ and  ${\lambda }_3$, respectively. Then

Wavelength of $K_{\alpha }$ lines for given isotopes of lead can be given by a general expression,

$\frac{1}{\lambda }=R(Z-1)\left(\frac{1}{1^2}-\frac{1}{2^2}\right)$

where, R is Rydberg's constant and Z is atomic number of the isotopes.

$\frac{1}{{\lambda }_1}=R(82-1{)}^2\left(\frac{1}{1^2}-\frac{1}{2^2}\right)$

$\frac{1}{{\lambda }_1}=\frac{3R}{4}\times (81{)}^2$

Similarly,  $\frac{1}{{\lambda }_2}=\frac{1}{{\lambda }_3}=\frac{3}{4}R\times (81{)}^2$

$\Rightarrow {\left(\frac{1}{{\lambda }_2}\right)}^2=\frac{1}{{\lambda }_1}\times \frac{1}{{\lambda }_3}$

$\Rightarrow {\lambda }_2=\sqrt{{\lambda }_1{\lambda }_3}$