How does the Compton effect change the wavelength of an X-ray or gamma-ray photon?

The Compton effect describes the increase in the wavelength of an X-ray or gamma-ray photon when it undergoes an elastic collision with a free or loosely bound electron.12

During this interaction, the incident photon transfers a portion of its energy and momentum to the electron, causing the electron to recoil. Because a photon's energy is inversely proportional to its wavelength, the loss of energy results in the scattered photon having a longer wavelength than the incident photon.134

The magnitude of this wavelength shift ($\Delta \lambda$) depends on the angle ($\theta$) at which the photon is scattered, according to the following relation:

$$\Delta \lambda ={\lambda }^{\prime }-\lambda =\frac{h}{mc}(1-\cos \theta )$$

In this equation:
* ${\lambda }^{\prime }$ is the wavelength of the scattered photon.13
* $\lambda$ is the wavelength of the incident photon.13
* $h$ is Planck’s constant.13
* $m$ is the rest mass of the electron.13
* $c$ is the speed of light.13
* $\frac{h}{mc}$ is the Compton wavelength, a constant approximately equal to $2.43\times {10}^{-12}$ metres.14