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. The amount the wavelength increases by is called the Compton shift. Although nuclear compton scattering exists, what is meant by Compton scattering usually is the interaction involving only the electrons of an atom. Compton effect was observed by Arthur Holly Compton in 1923, for which he earned the 1927 Nobel Prize in Physics.

The effect is important because it demonstrates that light cannot be explained purely as a wave phenomenon. Thomson scattering, the classical theory of charged particles scattered by an electromagnetic wave, cannot explain any shift in wavelength. Light must behave as if it consists of particles in order to explain the Compton scattering. Compton's experiment convinced physicists that light can behave as a stream of particles whose energy is proportional to the frequency.

The interaction between high energy photons and electrons 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.

Compton scattering occurs in all materials and predominantly with photons of medium energy, i.e. about 0.5 to 3.5 MeV. It is also observed that high-energy photons; photons of visible light or higher frequency, for example, have sufficient energy to even eject the bound electrons from the atom (Photoelectric effect).

The Compton shift formula

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For differential cross section of Compton scattering, see Klein-Nishina formula.

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.

• Light as a particle, as noted previously in the photoelectric effect.
• Relativistic dynamics Special Theory of Relativity
• Trigonometry - Law of cosines

The final result gives us the Compton scattering equation:

$\backslash lambda\_2\; =\; \backslash frac\{h\}\{m\_e\; c\}(1-\backslash cos\{\backslash theta\})\; +\; \backslash lambda\_1$

where

$\backslash lambda\_1$ is the wavelength of the photon before scattering,

$\backslash lambda\_2$ is the wavelength of the photon after scattering,

m_e is the mass of the electron,

h/(m_ec) is known as the Compton wavelength,

θ is the angle by which the photon's heading changes,

h is Planck's constant, and

c is the speed of light.

Collectively, the Compton wavelength is 2.43×10^-12 meter.

Derivation

We use that:

$E\_\{\backslash gamma\}\; +\; E\_\{e\}\; =\; E\_\{\backslash gamma\text{'}\}\; +\; E\_\{e\text{'}\}\backslash ,$

(Conservation of energy, where $E\_\{\backslash gamma\}$ is the energy of a photon before the collision and $E\_e$ is the energy of an electron before collision — its rest mass). The variables with a prime are used for those after the collision.
And:

$\backslash vec\; p\_\{\backslash gamma\}\; +\; \backslash vec\{p\_\{e\}\}\; =\; \backslash vec\{p\_\{\backslash gamma\text{'}\}\}\; +\; \backslash vec\{p\_\{e\text{'}\}\}\backslash ,$

(Conservation of momentum, with the $p\_e=0$ because we assume that the electron is at rest.)
We then use $E\; =\; hf\; =\; pc$:

$\backslash vec\{p\_\{e\text{'}\}\}\; =\; \backslash vec\{p\_\{\backslash gamma\}\}\; -\; \backslash vec\{p\_\{\backslash gamma\text{'}\}\}\backslash ,$

$\{\backslash vec\{p\_\{e\text{'}\}\}\}^2\; =\; \{(\backslash vec\{p\_\{\backslash gamma\}\}\; -\; \backslash vec\{p\_\{\backslash gamma\text{'}\}\})\}^2$

$\{\backslash vec\{p\_\{e\text{'}\}\}\}^2\; =\; \backslash vec\{p\_\{\backslash gamma\}\}^2\; -\; 2\backslash cdot\backslash vec\{p\_\{\backslash gamma\}\}\backslash cdot\backslash vec\{p\_\{\backslash gamma\text{'}\}\}\; +\; \backslash vec\{p\_\{\backslash gamma\text{'}\}\}^2$

$\backslash vec\{p\_\{e\text{'}\}\}\; \backslash cdot\; \backslash vec\{p\_\{e\text{'}\}\}\; =\; \backslash vec\{p\_\{\backslash gamma\}\}\; \backslash cdot\; \backslash vec\{p\_\{\backslash gamma\}\}-\; 2\backslash cdot\backslash vec\{p\_\{\backslash gamma\}\}\backslash cdot\backslash vec\{p\_\{\backslash gamma\text{'}\}\}\; +\; \backslash vec\{p\_\{\backslash gamma\text{'}\}\}\; \backslash cdot\; \backslash vec\{p\_\{\backslash gamma\text{'}\}\}$

$\{p\_\{e\text{'}\}\}^2\; \backslash cdot\; \backslash cos(0)\; =\; p\_\{\backslash gamma\}^2\; \backslash cdot\; \backslash cos(0)\; -\; 2\; \backslash cdot\; p\_\{\backslash gamma\}\; \backslash cdot\; p\_\{\backslash gamma\text{'}\}\; \backslash cdot\; \backslash cos(\backslash theta)\; +\; p\_\{\backslash gamma\text{'}\}^2\backslash cdot\; \backslash cos(0)$

The $\backslash cos(\backslash theta)$ term appears because the momenta are spatial vectors, all of which lie in a single 2D plane, thus their inner product is the product of their norms multiplied by the cosine of the angle between them.
substituting $p\_\{\backslash gamma\}$ with $\backslash frac\{hf\}\{c\}$ and $p\_\{\backslash gamma\text{'}\}$ with $\backslash frac\{hf\text{'}\}\{c\}$, we derive

$p\_\{e\text{'}\}^2\; =\; \backslash frac\{h^2\; f^2\}\{c^2\}\; +\; \backslash frac\{h^2\; f\text{'}^2\}\{c^2\}\; -\; \backslash frac\{2h^2\; ff\text{'}\backslash cos\{\backslash theta\}\}\{c^2\}$

Now we fill in for the energy part:

$E\_\{\backslash gamma\}\; +\; E\_\{e\}\; =\; E\_\{\backslash gamma\text{'}\}\; +\; E\_\{e\text{'}\}\backslash ,$

$hf\; +\; mc^2\; =\; hf\text{'}\; +\; \backslash sqrt\{(p\_\{e\text{'}\}c)^2\; +\; (mc^2)^2\}\backslash ,$

We solve this for p_e':

$(hf\; +\; mc^2-hf\text{'})^2\; =\; (p\_\{e\text{'}\}c)^2\; +\; (mc^2)^2\backslash ,$

$\backslash frac\{(hf\; +\; mc^2-hf\text{'})^2\; -m^2c^4\}\{c^2\}=\; p\_\{e\text{'}\}^2\backslash ,$

Then we have two equations for $p\_\{e\text{'}\}^2$, which we equate:

$\backslash frac\{(hf\; +\; mc^2-hf\text{'})^2\; -m^2c^4\}\{c^2\}\; =\; \backslash frac\{h^2\; f^2\}\{c^2\}\; +\; \backslash frac\{h^2\; f\text{'}^2\}\{c^2\}\; -\; \backslash frac\{2h^2\; ff\text{'}\backslash cos\{\backslash theta\}\}\{c^2\}$

Now it's just a question of rewriting:

$h^2f^2+h^2f\text{'}^2-2h^2ff\text{'}+2h(f-f\text{'})mc^2\; =\; h^2f^2+h^2f\text{'}^2-2h^2ff\text{'}\backslash cos\{\backslash theta\}\backslash ,$

$-2h^2ff\text{'}+2h(f-f\text{'})mc^2\; =\; -2h^2ff\text{'}\backslash cos\{\backslash theta\}\backslash ,$

$hff\text{'}-(f-f\text{'})mc^2\; =\; hff\text{'}\backslash cos\{\backslash theta\}\backslash ,$

$hff\text{'}(1-\backslash cos\{\backslash theta\})\; =\; (f-f\text{'})mc^2\backslash ,$

$h\backslash frac\{c\}\{\backslash lambda\text{'}\}\backslash frac\{c\}\{\backslash lambda\}(1-\backslash cos\{\backslash theta\})\; =\backslash left(\backslash frac\{c\}\{\backslash lambda\}-\backslash frac\{c\}\{\backslash lambda\text{'}\}\backslash right)mc^2$

$h\backslash frac\{c\}\{\backslash lambda\text{'}\}\backslash frac\{c\}\{\backslash lambda\}(1-\backslash cos\{\backslash theta\})\; =\; \backslash left(\backslash frac\{c\backslash lambda\text{'}\}\{\backslash lambda\backslash lambda\text{'}\}-\backslash frac\{c\backslash lambda\}\{\backslash lambda\text{'}\backslash lambda\}\backslash right)mc^2$

$h(1-\backslash cos\{\backslash theta\})\; =\; \backslash frac\{\backslash lambda\text{'}\}\{c\}\backslash frac\{\backslash lambda\}\{c\}\backslash left(\backslash frac\{c\backslash lambda\text{'}\}\{\backslash lambda\text{'}\backslash lambda\}-\backslash frac\{c\backslash lambda\}\{\backslash lambda\backslash lambda\text{'}\}\backslash right)mc^2$

$h(1-\backslash cos\{\backslash theta\})\; =\; \backslash left(\backslash frac\{\backslash lambda\text{'}\}\{c\}-\backslash frac\{\backslash lambda\}\{c\}\backslash right)mc^2$

$\backslash frac\{h\}\{mc\}(1-\backslash cos\{\backslash theta\})\; =\backslash lambda\text{'}-\backslash lambda$

Applications

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

Compton scattering has on occasion been proposed as an alternative explanation for the phenomenon of the redshift by opponents of the Big Bang theory, although this is not generally accepted because the influence of the Compton scattering would be noticeable in the spectral lines of distant objects and this is not observed.

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

Compton Scatter is an important effect in Gamma spectroscopy, 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.

See also

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• Thomson scattering
• Photoelectric effect
• Timeline of cosmic microwave background astronomy
• Peter Debye
• Sunyaev Zel'dovich effect
• Walther Bothe
• List of astronomical topics
• List of physics topics

External links

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• Compton Effect (PDF file) by Michael Brandl for Project PHYSNET.

Astronomy | Atomic physics | Foundational quantum physics | X-rays

Comptonův jev | Compton-Effekt | Efecto Compton | Diffusion Compton | Efecto Compton | Effetto Compton | Compton-effect | コンプトン効果 | Zjawisko Comptona | Efeito Compton | Эффект Комптона | Comptonov pojav | Комптонов ефекат | Comptonin ilmiö | Comptonspridning | Hiệu ứng Compton