In my book, when it discusses Compton Scattering it provides the following momentum equations, where $\gamma$ is Lorentz factor, $m$ is the mass, $u$ is the velocity, $h$ is Plank constant, and $\lambda$ is the wavelength:

$$\text{x-component: }\frac{h}{\lambda} = \frac{h}{\lambda'}\cos\theta + \gamma m u\cos\phi$$ $$\text{y-component: }0 = \frac{h}{\lambda'}\sin\theta - \gamma mu\sin\phi.$$

However, it does not elaborate on why it divides the equation into an x-compoenent and a y-compoenent. What exactly is those two equations telling us in regards to the geometry?

  • $\begingroup$ The fact that we have defined the $x$ direction to be the direction of propagation of the incoming photon (with the $e^{-}$ at rest). If you see, the initial y-momentum = $0$ $\endgroup$ – insomniac Mar 20 '20 at 9:16

Look at this drawing

enter image description here
(image from KJ 3055: X Ray Spectrometry and Radiochemical Methods)

For our convenience we have put the $xyz$ coordinate frame in such a way, that the $x$-axis is in the direction of the incident photon, and the scattered electron and photon are moving in the $xy$-plane.

From the drawing above we can read off the various momentum vectors (each with $x$, $y$ and $z$ component):

  • The incident photon has momentum with an $x$ component only: $$\vec{p}_{\gamma,\text{incident}} =\frac{h}{\lambda}\begin{pmatrix}1\\0\\0\end{pmatrix} \tag{1a}$$
  • The initial electron is at rest and therefore has zero momentum: $$\vec{p}_{e,\text{initial}} =\begin{pmatrix}0\\0\\0\end{pmatrix} \tag{1b}$$
  • The scattered photon has momentum with an $x$ and $y$ component: $$\vec{p}_{\gamma,\text{scattered}} =\frac{h}{\lambda'}\begin{pmatrix}\cos\theta\\\sin\theta\\0\end{pmatrix} \tag{1c}$$ (Here we used the geometric definition of $sin$ and $cos$.)
  • The scattered electron has momentum with an $x$ and $y$ component: $\vec{p}_{e,\text{scattered}} =\gamma m u\begin{pmatrix}\cos\phi\\-\sin\phi \\0\end{pmatrix} \tag{1d}$ (Here again we used the geometric definition of $sin$ and $cos$.)

During Compton scattering the total momentum is conserved. So we have the vector equation: $$\vec{p}_{\gamma,\text{incident}} + \vec{p}_{e,\text{initial}} = \vec{p}_{\gamma,\text{scattered}} + \vec{p}_{e,\text{scattered}} \tag{2}$$

Plugging in the four momenta from (1a-d) we get $$\frac{h}{\lambda}\begin{pmatrix}1\\0\\0\end{pmatrix} + \begin{pmatrix}0\\0\\0\end{pmatrix} = \frac{h}{\lambda'}\begin{pmatrix}\cos\theta\\\sin\theta\\0\end{pmatrix} + \gamma m u\begin{pmatrix}\cos\phi\\-\sin\phi\\0\end{pmatrix} \tag{3}$$

Writing vector equation (3) in components $(x,y,z)$ we get: $$\begin{align} \frac{h}{\lambda} &= \frac{h}{\lambda'}\cos\theta + \gamma mu\cos\phi \\ 0 &= \frac{h}{\lambda'}\sin\theta - \gamma mu\sin\phi \\ 0 &= 0 \end{align} \tag{4}$$


Momentum is a vector, so 'conservation of momentum' implies 3 equations.

These can be taken along any set of axes you choose. You can call them $x,y,z$ or $e_1,e_2,e_3$, whatever.

However it makes life simple if you choose the $x$ axis to be the initial direction of the photon. And then the $y$ axis to be the orthogonal axis that gives the scattering plane. So, by construction, neither the electron nor the scattered photon has any $z$ component of momentum. The third equation, for $z$, is just $0=0$ which is automatically satisfied and you only have to bother with two.


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