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.

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

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.

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,

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

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.