Compton scattering
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2008-07-15T19:51:28Z
Washburnmav
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Reverted 1 edit by [[Special:Contributions/157.178.2.1|157.178.2.1]] identified as [[WP:VAND|vandalism]] to last revision by [[User:Borgx|Borgx]]. ([[WP:TW|TW]])
{| cellpadding="5" cellspacing="0" align="right" style="border:1px solid black; margin-left:1em"
! style="background-color:#ffdead;" | [[Feynman diagrams]]
|-
| align="center" | '''s-channel'''<br>[[Image:ComptonScattering-s.svg|220px]]
|-
| align="center" | '''u-channel'''<br>[[Image:ComptonScattering-u.svg|220px]]
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{| align="right" border="1"
| 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. '''Inverse Compton scattering''' also exists, where the photon gains energy (decreasing in wavelength) upon interaction with matter. The amount the wavelength increases by is called the '''Compton shift'''. Although [[nuclear compton scattering]] exists, Compton scattering usually refers to the interaction involving only the [[electron|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 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 electrons and high [[energy]] photons 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. If the photon is of lower energy, but still has sufficient energy (in general a few [[Electronvolt|eV]], right around the energy of [[visible light]]), it can eject an electron from its host atom entirely (a process known as the [[Photoelectric effect]]), instead of undergoing Compton scattering.
==The Compton shift formula==
[[Image:Compton-scattering.svg|frame|right|A photon of wavelength <math>\lambda \,</math> comes in from the left, collides with a target at rest, and a new photon of wavelength <math>\lambda ' \,</math> emerges at an angle <math>\theta \,</math>.]]
{{see also|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 relativity|special theory of relativity]]
* Trigonometry: [[law of cosines]]
The final result gives us the '''Compton scattering equation''':
:<math>\lambda' - \lambda = \frac{h}{m_e c}(1-\cos{\theta})</math>
where
:<math>\lambda\,</math> is the wavelength of the photon '''before''' scattering,
:<math>\lambda'\,</math> is the wavelength of the photon '''after''' scattering,
:<math>m_e</math> is the mass of the [[electron]],
:<math>\theta\,</math> is the angle by which the photon's heading changes,
:<math>h</math> is [[Planck's constant]], and
:<math>c</math> is the [[speed of light]].
:<math>\frac{h}{m_e c} = 2.43 \times 10^{-12}\,m</math> is known as the [[Compton wavelength]].
===Derivation===
Begin with conservation of energy and conservation of momentum:
::<math>E_\gamma + E_e = E_{\gamma^\prime} + E_{e^\prime} \quad \quad (1) \,</math>
::<math>\vec p_\gamma = \vec{p}_{\gamma^\prime} + \vec{p}_{e^\prime} \quad \quad \quad \quad \quad (2) \,</math>
:where
::<math>E_\gamma \,</math> and <math>p_\gamma \,</math> are the energy and momentum of the photon and
::<math>E_e \,</math> and <math>p_e \,</math> are the energy and momentum of the electron.
====Solving (Part 1)====
Now we fill in for the energy part:<br />
:<math>E_{\gamma} + E_{e} = E_{\gamma'} + E_{e'}\,</math>
:<math>hf + mc^2 = hf' + \sqrt{(p_{e'}c)^2 + (mc^2)^2}\,</math>
We solve this for p<sub>e'</sub>:
:<math>(hf + mc^2-hf')^2 = (p_{e'}c)^2 + (mc^2)^2\,</math>
:<math>\frac{(hf + mc^2-hf')^2-m^2c^4}{c^2}= p_{e'}^2 \quad \quad \quad \quad \quad (3) \,</math>
====Solving (Part 2)====
Rearrange equation (2)
::<math>\vec{p}_{e'} = \vec{p}_\gamma - \vec{p}_{\gamma'} \,</math>
and square it to see
::<math>p_{e'}^2 = (\vec{p}_\gamma - \vec{p}_{\gamma'}) \cdot (\vec{p}_\gamma - \vec{p}_{\gamma'})</math>
::<math>p_{e'}^2 = p_{\gamma}^2 + p_{\gamma'}^2 - 2\vec{p_{\gamma}} \cdot \vec{p_{\gamma'}}</math>
::<math>p_{e'}^2 = p_\gamma^2 + p_{\gamma'}^2 - 2|p_{\gamma}||p_{\gamma'}|\cos(\theta) \,</math>
::<math>p_{e'}^2 = \left(\frac{h f}{c}\right)^2 + \left(\frac{h f'}{c}\right)^2 - 2\left( \frac{hf}{c} \right) \left(\frac{h f'}{c} \right) \cos{\theta} \quad \quad \quad (4) </math>
====Putting it together====
Then we have two equations for <math>p_{e'}^2</math> (eq 3 & 4), which we equate:
:<math> \left(\frac{h f}{c}\right)^2 + \left(\frac{h f'}{c}\right)^2 - \frac{2h^2 ff'\cos{\theta}}{c^2} = \frac{(hf + mc^2-hf')^2 -m^2c^4}{c^2} \,</math>
Now, one simplifies. First by multiplying both sides by ''c''<sup>2</sup>:
:<math>h^2 f^2 + h^2 f'^2 - 2h^2 ff' \cos \theta = (hf + mc^2 - hf')^2 - m^2c^4 . \,</math>
Next, multiply out the right-hand side:
:<math>h^2f^2+h^2f'^2-2h^2ff'\cos{\theta} = h^2f^2+m^2c^4+h^2f'^2-2h^2ff'+2h(f-f')mc^2 -m^2c^4 .\,</math>
A few terms cancel from both sides, so we have
:<math> -2h^2ff'\cos{\theta} = -2h^2ff'+2h(f-f')mc^2 .\,</math>
Then divide both sides by '<math>-2h</math>' to see
:<math>hff'\cos{\theta} = hff'-(f-f')mc^2 \,</math>
:<math>(f-f')mc^2 = hff'(1-\cos{\theta}) .\,</math>
Now divide both sides by <math>mc^2</math> and then by <math>ff^\prime</math>:
:<math>\frac{f-f^\prime}{f f^\prime} = \frac{h}{mc^2}\left(1-\cos \theta \right) . \,</math>
Now the left-hand side can be rewritten as simply
::{|cellpadding="2" style="border:2px solid #ccccff"
|<math> \frac{1}{f^\prime} - \frac{1}{f} = \frac{h}{mc^2}\left(1-\cos \theta \right) \,</math>
|}
This is equivalent to the '''Compton scattering equation''', but it is usually written using <math>\lambda</math>s rather than <math>f</math>s. To make that switch use
::<math>f=\frac{c}{\lambda} \,</math>
so that finally,
::{|cellpadding="2" style="border:2px solid #ccccff"
|<math>\lambda'-\lambda = \frac{h}{mc}(1-\cos{\theta}) \, </math>
|}
==Applications==
=== Compton scattering ===
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]].
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 ===
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 radiation|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]].
==See also==
* [[Thomson scattering]]
* [[Klein-Nishina formula]]
* [[Photoelectric effect]]
* [[Pair production]]
* [[Timeline of cosmic microwave background astronomy]]
* [[Peter Debye]]
* [[Walther Bothe]]
* [[List of astronomical topics]]
* [[List of physics topics]]
* [[Washington University in St. Louis]] (Site of discovery)
{{QED}}
==External links==
* [http://prola.aps.org/abstract/PR/v21/i5/p483_1 ''A Quantum Theory of the Scattering of X-Rays by Light Elements''] - the original 1923 ''[[Physical Review]]'' paper by Arthur H. Compton (on the [[American Institute of Physics|AIP]] website).
Take a look at this, i think in half a page it teaches you more detail about what is going on ....!
http://hyperphysics.phy-astr.gsu.edu/Hbase/quantum/comptint.html
http://hyperphysics.phy-astr.gsu.edu/Hbase/quantum/compdat.html#c1
[[Category:Astrophysics]]
[[Category:Observational astronomy]]
[[Category:Atomic physics]]
[[Category:Foundational quantum physics]]
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[[Category:Quantum electrodynamics]]
[[Category:Scattering]]
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