A beam of x-rays with wavelength λ = 0.300 nm is directed toward a sample in which the x-rays scatter off of electrons that are effectively free. The wavelength of the outgoing electrons is measured as a function of scattering angle, where a scattering angle of 0 means the direction of the x-rays was unchanged when passing through the sample. When looking at all possible scattering angles, what are the longest and shortest wavelengths that the scattered x-rays can have?

The problem relates to Compton Effect in which electrons are scattered due to external radiation . The electron is scattered out and photons relating to radiation also undergo scattering at angle θ .

The formula relating to Compton Effect is as follows

[tex]\lambda_f-\lambda_i=\frac{h}{m_0c} (1-cos\theta)[/tex]

Here [tex]\lambda_i[/tex] = 3 0 x 10⁻¹¹

For longest [tex]\lambda_f[/tex] θ =180°

[tex]\lambda_f[/tex] = [tex]\lambda_i + \frac{2\times h}{m_0c}[/tex]

= .3 x 10⁻⁹ + [tex]\frac{2\times6.6\times 11^{-34}}{9\times10^{-31}\times3\times10^8}[/tex]

= .348 nm

For shortest wavelength θ = 0

Putting this value in the given formula

[tex]\lambda_f=\lambda_i[/tex]

[tex]\lambda_f[/tex] = .3 nm