Derivation Of Compton Shift Equation



Compton Shift

Using the above diagram, derive the Compton Shift Equation.

Conservation of momentum gives:

$$\begin{aligned}p \, &= p’ \, \text{cos} \, \theta + P_{e} \, \text{cos} \, \phi \\ P_{e} \, \text{cos} \, \phi \, &= p \, – p’ \, \text{cos} \, \theta \end{aligned} $$

AND

$$\begin{aligned} 0 \, &= p’ \, \text{sin} \, \theta \, – P_{e} \, \text{sin} \, \phi \\ P_{e} \, \text{sin} \, \phi \, &= p’ \, \text{sin} \, \theta \end{aligned}$$

Combining the above two equations gives equation 1:  (Use $\text{sin}^{2} \, \phi + \text{cos}^{2} \, \phi = 1$)

$$\begin{aligned} P^{2}_{e} \, &= \left( p \, – p’ \, \text{cos} \, \theta \right)^{2} + p’^{2} \, \text{sin}^{2} \, \theta \\ P^{2}_{e} \, &= p^{2} + p’^{2} – 2pp’ \, \text{cos} \, \theta \end{aligned}$$

Conservation of energy gives:

$$ E + E_{o} \, = E’ + E_{e} $$

Using $E^{2} = m^{2} c^{4} + p^{2} c^{2}$ and $E = pc$ for photon, the above equation becomes:

$$\begin{aligned} pc + mc^{2} \, &= p’c + \left(m^{2} c^{4} + P^{2}_{e} c^{2} \right)^{\frac{1}{2}} \\ \left(p \, – p’ + mc \right)^{2} \, &= m^{2}c^{2} + P^{2}_{e} \end{aligned}$$

Using equation 1 to get rid of $P^{2}_{e}$ in the above equation, we obtain

$$\require{cancel} \begin{aligned} \left(p \, – p’ + mc \right)^{2} \, &= m^{2}c^{2} + \left(p^{2} + p’^{2} – 2pp’ \, \text{cos} \, \theta \right) \\ \cancel{p^{2}} + \cancel{p’^{2}} + \cancel{m^{2}c^{2}} + 2 \left( – pp’ + pmc \, – p’mc \right) \, &= \cancel{m^{2}c^{2}} + \cancel{p^{2}} + \cancel{p’^{2}} – 2pp’ \, \text{cos} \, \theta \\ pmc – p’mc \, &= pp’ \left( 1 \, – \, \text{cos} \, \theta \right) \\ \frac{mc}{p’} – \frac{mc}{p} \, &= 1 \, – \, \text{cos} \, \theta \\ \frac{mc \lambda’}{h} – \frac{mc \lambda}{h} \, &= 1 – \, \text{cos} \, \theta \\ \lambda’ – \lambda \, &= \frac{h}{mc} \left( 1 \, – \, \text{cos} \, \theta \right) \end{aligned}$$

 

 


Alternative Method:

 

Derivation of compton shift eqn 1

Derivation of compton shift eqn 2

 


Back To Compton Shift



Mini Physics

Administrator of Mini Physics. If you spot any errors or want to suggest improvements, please contact us. If you like the content in this site, please recommend this site to your friends!



Leave a Comment

This site uses Akismet to reduce spam. Learn how your comment data is processed.