Question

Compton used X-rays of 0.071nm wavelength. Some of the carbon’s electrons are too tightly bound to be stripped away by these X-rays, which accordingly interact essentially with the atom as a whole. In effect m_ein equation (3-8) is replaced by carbon’s atomic mass. Show that this explains why some X-rays of the incident wavelength were scattered at all angles.

Short answer

The concept of Compton scattering explains that light is scattering at all angles instead of just the outer electrons.

Step-by-step solution

 Writing the Compton scattering equation, explaining its terms.

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

Where $\mathit{\lambda }\mathbf{\text{'}}$is the wavelength of the scattered photon, $\mathit{\lambda }$ is the wavelength of the incident photon, and θ is angle with which the photon scatters with its incident direction.

Finding the scattered wavelength.

From the Compton scattering equation,

$\lambda \text{'}=\lambda +\frac{h}{mc}\left(1-\cos \theta \right)$

Substituting λ = 0.071 nm, mass of carbon= 12 amu = 1.99×10^-26kg.

Hence,

$\begin{array}{rcl}\lambda \text{'}& =& \left(0.071\times {10}^{-9}m\right)+\left(\frac{6.625\times {10}^{-34}J·s}{\left(1.99\times {10}^{-26}kg\right)\times \left(3\times {10}^{8}m/s\right)}\right)\left(1-\cos \theta \right)\\ & =& 7.1\times {10}^{-11}m+1.11\times {10}^{-16}\left(1-\cos \theta \right)\\ & =& 7.1\times {10}^{-11}m\end{array}$

Analysis of scattering angle.

We can see from the above equation that 1.11×10^-16 is much smaller compared to 7.1×10^-11. Hence for all values of θ, the cosθ value does not change much.

Therefore the wavelength of the scattered photon does not change much compared to the incident X-ray wavelength if the mass of an electron is replaced by the mass of carbon.