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 Compton scattering Feynman diagrams s-channel u-channel Light-matter interaction Low energy phenomena Photoelectric effect Mid-energy phenomena Compton scattering High energy phenomena Pair production

In physics, Compton scattering or the Compton effect is the decrease in energy (increase in wavelength) of an X-ray or gamma ray photon, when it interacts with matter. Because of the change in photon energy, it is an inelastic scattering process. Inverse Compton scattering also exists, where the photon gains energy (decreasing in wavelength) upon interaction with matter. The amount the wavelength changes by is called the Compton shift. Although nuclear compton scattering exists[1], Compton scattering usually refers to the interaction involving only the electrons of an atom. The Compton effect was observed by Arthur Holly Compton in 1923 and further verified by his graduate student Y. H. Woo in the years following. Arthur Compton earned the 1927 Nobel Prize in Physics for the discovery.

The effect is important because it demonstrates that light cannot be explained purely as a wave phenomenon. Thomson scattering, the classical theory of an electromagnetic wave scattered by charged particles, cannot explain low intensity shift in wavelength (Classically, light of sufficient intensity for the electric field to accelerate a charged particle to a relativistic speed will cause radiation-pressure recoil and an associated Doppler shift of the scattered light, but the effect would become arbitrarily small at sufficiently low light intensities regardless of wavelength.). Light must behave as if it consists of particles in order to explain the low-intensity Compton scattering. Compton's experiment convinced physicists that light can behave as a stream of particle-like objects (quanta) whose energy is proportional to the frequency.

The interaction between electrons and high energy photons comparable to the rest energy of the electron (511 keV) results in the electron being given part of the energy (making it recoil), and a photon containing the remaining energy being emitted in a different direction from the original, so that the overall momentum of the system is conserved. If the photon still has enough energy left, the process may be repeated. In this scenario, the electron is treated as free or loosely bound. Experimental verification of momentum conservation in individual Compton scattering processes by Bothe and Geiger as well as by Compton and Simon has been important in disproving the BKS theory.

If the photon is of lower energy, but still has sufficient energy (in general a few eV to a few keV, corresponding to visible light through soft X-rays), it can eject an electron from its host atom entirely (a process known as the photoelectric effect), instead of undergoing Compton scattering. Higher energy photons (1.022 MeV and above) may be able to bombard the nucleus and cause an electron and a positron to be formed, a process called pair production.

## The Compton shift formulaEdit

Compton used a combination of three fundamental formulas representing the various aspects of classical and modern physics, combining them to describe the quantum behavior of light.

The final result gives us the Compton scattering equation:

$\lambda' - \lambda = \frac{h}{m_e c}(1-\cos{\theta})$

where

$\lambda$ is the wavelength of the photon before scattering,
$\lambda'$ is the wavelength of the photon after scattering,
$m_e$ is the mass of the electron,
$\theta$ is the angle by which the photon's heading changes (between 0° and 180°),
$h$ is Planck's constant, and
$c$ is the speed of light.
$\frac{h}{m_e c} = 2.43 \times 10^{-12}\,\text{m}$ is known as the Compton wavelength. $\lambda' - \lambda$ can be between 0 (for $\theta$ = 0°) and two times the Compton wavelength (for $\theta$ = 180°).

### DerivationEdit

Let $\gamma$ denote the photon before the collision, $\gamma'$ the photon after the collision, $e$ the electron before the collision, and $e'$ the electron after the collision.

From the conservation of energy,

$E_\gamma + E_e = E_{\gamma'} + E_{e'}. \qquad\qquad (1) \,$

From the conservation of momentum,

$\vec{p}_\gamma = \vec{p}_{\gamma'} + \vec{p}_{e'}, \qquad\qquad (2) \,$

assuming the initial momentum of the electron is zero.

The energy of a photon of frequency $f$ is given by $E = hf$, where $h$ is the Planck constant. The energy of an electron is given in special relativity by $E^2 = (pc)^2 + (m_ec^2)^2$. Therefore, from equation (1),

$hf + mc^2 = hf' + \sqrt{(p_{e'}c)^2 + (mc^2)^2}.\,$

Then

$p_{e'}^2c^2 = (hf + mc^2-hf')^2-m^2c^4. \qquad\qquad (3) \,$

By rearranging equation 2,

$\vec{p}_{e'} = \vec{p}_\gamma - \vec{p}_{\gamma'}.$

Then squaring it,

$p_{e'}^2 = (\vec{p}_\gamma - \vec{p}_{\gamma'}) \cdot (\vec{p}_\gamma - \vec{p}_{\gamma'})$
$p_{e'}^2 = p_\gamma^2 + p_{\gamma'}^2 - 2|p_{\gamma}||p_{\gamma'}|\cos\theta$
$p_{e'}^2c^2 = p_\gamma^2c^2 + p_{\gamma'}^2c^2 - 2c^2|p_{\gamma}||p_{\gamma'}|\cos\theta.$

The relation between the frequency and the momentum of a photon is $pc = hf$, so

$p_{e'}^2c^2 = (h f)^2 + (h f')^2 - 2(hf)(h f')\cos{\theta} \qquad\qquad (4)$

Now equating 3 and 4,

$\left(h f\right)^2 + \left(h f'\right)^2 - 2h^2 ff'\cos{\theta} = (hf + mc^2-hf')^2 -m^2c^4 \,$
$2 h^2 f f' \left( 1 - \cos \theta \right) = 2 h f m c^2 - 2 h f' m c^2. \,$

Then dividing both sides by $2 h f f' m c^2$,

$\frac{h}{mc^2}\left(1-\cos \theta \right) = \frac{1}{f^\prime} - \frac{1}{f}. \,$

Since $f\lambda = f'\lambda' = c$,

$\lambda'-\lambda = \frac{h}{mc}(1-\cos{\theta}). \,$

## ApplicationsEdit

### Compton scattering Edit

Compton scattering is of prime importance to radiobiology, as it happens to be the most probable interaction of high energy X rays with atoms in living beings and is applied in radiation therapy.[2]

In material physics, Compton scattering can be used to probe the wave function of the electrons in matter in the momentum representation.

Compton scattering is an important effect in gamma spectroscopy which gives rise to the Compton edge, as it is possible for the gamma rays to scatter out of the detectors used. Compton suppression is used to detect stray scatter gamma rays to counteract this effect.

### Inverse Compton scattering Edit

Inverse Compton scattering is important in astrophysics. In X-ray astronomy, the accretion disk surrounding a black hole is believed to produce a thermal spectrum. The lower energy photons produced from this spectrum are scattered to higher energies by relativistic electrons in the surrounding corona. This is believed to cause the power law component in the X-ray spectra (0.2-10 keV) of accreting black holes.

The effect is also observed when photons from the cosmic microwave background move through the hot gas surrounding a galaxy cluster. The CMB photons are scattered to higher energies by the electrons in this gas, resulting in the Sunyaev-Zel'dovich effect. Observations of the Sunyaev-Zel'dovich effect provide a nearly redshift-independent means of detecting galaxy clusters.

## Notes Edit

1. P Christillin (1986). "Nuclear Compton scattering". J. Phys. G: Nucl. Phys. 12: 837–851. doi:10.1088/0305-4616/12/9/008.
2. Camphausen KA, Lawrence RC. "Principles of Radiation Therapy" in Pazdur R, Wagman LD, Camphausen KA, Hoskins WJ (Eds) Cancer Management: A Multidisciplinary Approach. 11 ed. 2008.