Question

X rays with initial wavelength 0.0665 nm undergo Compton scattering. What is the longest wavelength found in the scattered x rays? At which scattering angle is this wavelength observed?

Short answer

The longest wavelength found in the scattered x rays is . The wavelength is observed at $180°$.

Step-by-step solution

Equation used for Compton effect and formula for the longest wavelength of the scattered photon

$\mathit{\lambda }\mathbf{\text{'}}\mathbf{-}\mathit{\lambda }\mathbf{=}\frac{\mathbf{h}}{\mathbf{m}\mathbf{c}}\mathbf{(}\mathbf{1}\mathbf{-}\mathit{c}\mathit{o}\mathit{s}\mathit{\varphi }\mathbf{)}$ (1)

$\mathit{\lambda }\mathbf{\text{'}}\mathbf{-}\mathit{\lambda }\mathbf{+}\frac{\mathbf{2}\mathbf{h}}{\mathbf{mc}}$ (2)

Calculate the longest wavelength

From equation (1), the longest wavelength found in the scattered x rays occurs when $\lambda \text{'}-\lambda$is maximum. And this maximum occurs when $1-\cos \varphi$is maximum and it occurs when $c\mathrm{os}\varphi =-1$

So, plug the values in equation (1),

$$\begin{gathered} \mathit{\lambda }\mathbf{\text{'}}\mathbf{=}\left(0.0665*{10}^{-9}m\right)\mathbf{+}\frac{\mathbf{2}\left(6.626*{10}^{-34}Js\right)}{\left(9.11*{10}^{-31}kg\right)\left(3.00*{10}^{8}m/s\right)\mathbf{}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{071}\mathit{n}\mathit{m} \end{gathered}$$

Calculate the scattering angle

This occurs when $\mathit{c}\mathit{o}\mathit{s}\mathit{\varphi }\mathbf{=}\mathbf{0}$.

Thus,

$\mathit{\varphi }\mathbf{=}{\mathbf{cos}}^{\mathbf{-}\mathbf{1}}\mathbf{=}\mathbf{180}\mathbf{°}$

Therefore, the longest wavelength found in the scattered x rays is 0.071nm. The wavelength is observed at $\mathbf{180}\mathbf{°}$.