## Compton Scattering

### Key characteristics of Compton Scattering

- Compton scattering dominates
**at intermediate energies.** - It is the scattering of photons
**by atomic electrons** - Photons undergo a wavelength shift called
**the Compton shift.** - The energy transferred to the recoil electron can
**vary from zero to a large fraction**of the incident gamma ray energy

### Definition of Compton Scattering

Compton scattering is the inelastic or nonclassical scattering of a photon (which may be an X-ray or gamma ray photon) by a charged particle, usually an electron. In Compton scattering, the incident gamma ray photon is deflected through an angle Θ with respect to its original direction. This deflection results in a decrease in energy (decrease in photon’s frequency) of the photon and is called **the Compton effect**. The photon transfers a portion of its energy to **the recoil electron**. The energy transferred to the recoil electron can vary from zero to a large fraction of the incident gamma ray energy, because all angles of scattering are possible. The Compton scattering was observed by **A. H.Compton in 1923** at Washington University in St. Louis. Compton earned **the Nobel Prize in Physics in 1927** for this new understanding about the particle-nature of photons.

### Compton Scattering Formula

The Compton formula was published in 1923 in the Physical Review. Compton explained that the X-ray shift is caused by particle-like momentum of photons. Compton scattering formula is the mathematical relationship between the shift in wavelength and the scattering angle of the X-rays. In the case of Compton scattering the photon of frequency *f* collides with an electron at rest. Upon collision, the photon bounces off electron, giving up some of its initial energy (given by Planck’s formula E=hf), While the electron gains momentum (mass x velocity), the **photon cannot lower its velocity**. As a result of momentum conservetion law, the photon must lower its momentum given by:So the decrease in photon’s momentum must be translated into **decrease in frequency** (increase in wavelength Δ**λ = λ’ – λ**). The shift of the wavelength increased with scattering angle according to **the Compton formula**:where

**λ**is the initial wavelength of photon

**λ’**is the wavelength after scattering,

**h**is the Planck constant = 6.626 x 10

^{-34}J.s

**m**

**is the electron rest mass (0.511 MeV)**

_{e}**c**is the speed of light

**Θ**is the scattering angle.The minimum change in wavelength (

*λ′*−

*λ*) for the photon occurs when Θ = 0° (cos(Θ)=1) and is at least zero. The maximum change in wavelength (

*λ′*−

*λ*) for the photon occurs when Θ = 180° (cos(Θ)=-1). In this case the photon transfers to the electron as much momentum as possible.The maximum change in wavelength can be derived from Compton formula:The quantity h/m

_{e}c is known as the

**Compton wavelength**of the electron and is equal to

**2.43×10**.

^{−12}m### Compton Scattering – Cross-Sections

The probability of Compton scattering per one interaction with an atom increases linearly with atomic number Z, because it depends on the number of electrons, which are available for scattering in the target atom.** The angular distribution** of photons scattered from a single free electron is described by **the Klein-Nishina formula**:where ε = E

_{0}/m

_{e}c

^{2}and r

_{0}is the “classical radius of the electron” equal to about 2.8 x 10

^{-13}cm. The formula gives the probability of scattering a photon into the solid angle element dΩ = 2π sin Θ dΘ when the incident energy is E

_{0}.

### Compton Edge

In spectrophotometry, **the Compton edge** is a feature of the spectrograph that results from the Compton scattering in the scintillator or detector. This feature is due to photons that undergo Compton scattering with a scattering angle of 180° and then escape the detector. When a gamma ray scatters off the detector and escapes, only a fraction of its initial energy can be deposited in the sensitive layer of the detector. It depends on the scattering angle of the photon, how much energy will be deposited in the detector. This leads to a spectrum of energies. The Compton edge energy corresponds to full backscattered photon**.**

### Inverse Compton Scattering

**Inverse Compton scattering** is the scattering of low energy photons to high energies by relativistic electrons. Relativistic electrons can boost energy of low energy photons by a potentially enormous amount (even gamma rays can be produced). This phenomenon is very important in astrophysics.

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