#### Question

The wavelengths of K_{α} and L_{α} X-rays of a material are 21.3 pm and 141 pm respectively. Find the wavelength of K_{β} X-ray of the material.

(Use Planck constant h = 6.63 × 10^{-34} Js= 4.14 × 10^{-15} eVs, speed of light c = 3 × 10^{8} m/s.)

#### Solution

Given:-

Wavelength of K_{α} X-ray, `lambda_1 = 21.3 "pm"`

Wavelength of L_{α} X-ray, `lambda_2 = 141 "pm"`

Energy of K_{α} X-ray (`E_1`) is given by

`E_1 = 1242/(21.3 xx 10^-3)`

= `58.309 xx 10^3 "eV"`

Energy of L_{α} X-ray (`E_2`) is given by

`E_2 = 1242/(141 xx 10^-5)`

= `8.8085 xx 10^3 "eV"`

Energy of K_{β} X-ray (`E_3`) will be

`E_3 = E_1 + E_2`

`E_3 = (58.309 + 8.809) xx 10^3 "eV"`

`E_3 = 67.118 xx 10^3 "eV"`

Wavelength of K_{β} X-ray (`lambda`) is given by

`lambda = (hc)/(E_3) = 1242/(67.118 xx 10^3)`

= `18.5 xx 10^-3 "nm" = 18.5 "pm"`