Question

An X-ray source of unknown wavelength is directed at a carbon sample. An electron is scattered with a speed of$\mathbf{4}\mathbf{.}\mathbf{5}\mathbf{\times }{\mathbf{10}}^{\mathbf{7}}\mathit{m}\mathbf{/}\mathit{s}$ at an angle of.Determine the wavelength of the X-ray source.

Short answer

The Wavelength of the X-ray source directed at the carbon sample is 0.213 nm.

Step-by-step solution

Concept Introduction

The expression for the Compton scattering can be expressed as,

$\mathit{\lambda }\mathbf{\text{'}}\mathbf{-}\mathit{\lambda }\mathbf{=}\frac{\mathbf{h}}{{\mathbf{m}}_{\mathbf{e}}\mathbf{c}}\left(1-cos\theta \right)$

Analysis of given data and arriving at the scattered wavelength.

According to the given data, the scattered electron has a speed of,

$v=4.5\times {10}^{7}m/s$

We know that $KE=\frac{1}{2}m{v}^2$

And also for the scattered electron of $\lambda \text{'},KE=\frac{hc}{\lambda \text{'}}$

Thus,

role="math" localid="1659080762269" $$\begin{gathered} \frac{1}{2}m{v}^2=\frac{hc}{\lambda \text{'}} \\ \lambda \text{'}=\frac{2hc}{m{v}^2} \\ \lambda \text{'}=\frac{2\times \left(6.625\times {10}^{-34}J·s\right)\left(3\times {10}^{8}m/s\right)}{\left(9.1\times {10}^{-31}kg\right)\times {\left(4.5\times {10}^{7}m/s\right)}^2} \\ \lambda \text{'}=21.5\times {10}^{-}m \end{gathered}$$

Using the Compton effect equation.

We have,

$$\begin{gathered} \lambda =\lambda \text{'}-\frac{h}{mc}\left(1-\cos \theta \right) \\ \lambda =21.5\times {10}^{-11}m-\frac{625\times {10}^{-34}J·s}{\left(9.1\times {10}^{-31}kg\right)\times 3\times {10}^{8}m/s}\left(1-\cos \theta \right) \\ \lambda =21.5\times {10}^{-11}m-\left(0.242\times {10}^{-11}m\times \left(05\right)\right) \\ \lambda =21.379\times {10}^{-11} \\ \lambda =0.213nm \end{gathered}$$