Compton scattering

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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 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 followed. 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 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).

Contents

[edit] The Compton shift formula

A photon of wavelength comes in from the left, collides with a target at rest, and a new photon of wavelength emerges at an angle .
A photon of wavelength \lambda \, comes in from the left, collides with a target at rest, and a new photon of wavelength \lambda ' \, emerges at an angle \theta \,.
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.

The final result gives us the Compton scattering equation:

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

where

λ is the wavelength of the photon before scattering,
λ' is the wavelength of the photon after scattering,
me is the mass of the electron,
θ is the angle by which the photon's heading changes,
h is Planck's constant, and
c is the speed of light.
h/(mec)=2.43×10-12 meters, is known as the Compton wavelength.

[edit] Derivation

Begin with energy and momentum conservation:

E_\gamma + E_e = E_{\gamma^\prime} + E_{e^\prime} \quad \quad (1) \,
\vec p_\gamma = \vec{p}_{\gamma^\prime} + \vec{p}_{e^\prime} \quad \quad \quad \quad \quad (2) \,
where
E_\gamma \, and p_\gamma \, are the energy and momentum of the photon and
E_e \, and p_e \, are the energy and momentum of the electron.

[edit] Solving (1)

Now we fill in for the energy part:

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

We solve this for pe':

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

[edit] Solving (2)

Rearrange equation (2)

\vec{p}_{e'} = \vec{p}_\gamma - \vec{p}_{\gamma'} \,

and square it to see

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

[edit] Putting it together

Then we have two equations for p_{e'}^2 (eq 3 & 4), which we equate:

\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} \,

Now, one simplifies. First by multiplying both sides by c2:

h^2 f^2 + h^2 f'^2 - 2h^2 ff' \cos \theta = (hf + mc^2 - hf')^2 - m^2c^4 . \,

Next, multiply out the right-hand side:

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 .\,

A few terms cancel from both sides, so we have

-2h^2ff'\cos{\theta} = -2h^2ff'+2h(f-f')mc^2 .\,

Then divide both sides by ' − 2h' to see

hff'\cos{\theta} = hff'-(f-f')mc^2 \,
(f-f')mc^2 = hff'(1-\cos{\theta}) .\,

Now divide both sides by mc2 and then by ff^\prime:

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

Now the left-hand side can be rewritten as simply

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

This is equivalent to the Compton scattering equation, but it is usually written using λ's rather than f's. To make that switch use

f=\frac{c}{\lambda} \,

so that finally,

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

[edit] Applications

[edit] 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 Scatter 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.

[edit] 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 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.

[edit] See also


electron | positron | photon
self-energy | vacuum polarization | vertex function
Gupta-Bleuler formalism | ξ gauge | Ward-Takahashi identity
Compton scattering | Bhabha scattering | Møller scattering
anomalous magnetic dipole moment
bremsstrahlung | positronium


[edit] External links

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