Compton scattering

Fig. 3: Energies of a photon at 500 keV and an electron after Compton scattering.

A photon γ with wavelength λ collides with an electron e in an atom, which is treated as being at rest. The collision causes the electron to recoil, and a new photon γ' with wavelength λ' emerges at angle θ from the photon's incoming path. Let e' denote the electron after the collision. Compton allowed for the possibility that the interaction would sometimes accelerate the electron to speeds sufficiently close to the velocity of light as to require the application of Einstein's special relativity theory to properly describe its energy and momentum.

At the conclusion of Compton's 1923 paper, he reported results of experiments confirming the predictions of his scattering formula, thus supporting the assumption that photons carry momentum as well as quantized energy. At the start of his derivation, he had postulated an expression for the momentum of a photon from equating Einstein's already established mass-energy relationship of ${\displaystyle E=mc^{2}}$ to the quantized photon energies of ${\displaystyle hf}$, which Einstein had separately postulated. If ${\displaystyle mc^{2}=hf}$, the equivalent photon mass must be ${\displaystyle hf/c^{2}}$. The photon's momentum is then simply this effective mass times the photon's frame-invariant velocity c. For a photon, its momentum ${\displaystyle p=hf/c}$, and thus hf can be substituted for pc for all photon momentum terms which arise in course of the derivation below. The derivation which appears in Compton's paper is more terse, but follows the same logic in the same sequence as the following derivation.

The conservation of energy ${\displaystyle E}$ merely equates the sum of energies before and after scattering.

${\displaystyle E_{\gamma }+E_{e}=E_{\gamma '}+E_{e'}.}$

Compton postulated that photons carry momentum;[5] thus from the conservation of momentum, the momenta of the particles should be similarly related by

${\displaystyle \mathbf {p} _{\gamma }=\mathbf {p} _{\gamma '}+\mathbf {p} _{e'},}$

in which (${\displaystyle {p_{e}}}$) is omitted on the assumption it is effectively zero.

The photon energies are related to the frequencies by

${\displaystyle E_{\gamma }=hf}$

${\displaystyle E_{\gamma '}=hf'}$

where h is Planck's constant.

Before the scattering event, the electron is treated as sufficiently close to being at rest that its total energy consists entirely of the mass-energy equivalence of its (rest) mass ${\displaystyle m_{e}}$,

${\displaystyle E_{e}=m_{e}c^{2}.}$

After scattering, the possibility that the electron might be accelerated to a significant fraction of the speed of light, requires that its total energy be represented using the relativistic energy–momentum relation

${\displaystyle E_{e'}={\sqrt {(p_{e'}c)^{2}+(m_{e}c^{2})^{2}}}~.}$

Substituting these quantities into the expression for the conservation of energy gives

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

This expression can be used to find the magnitude of the momentum of the scattered electron,

$${\displaystyle p_{e'}^{\,2}c^{2}=(hf-hf'+m_{e}c^{2})^{2}-m_{e}^{2}c^{4}.}$$

(1)

Note that this magnitude of the momentum gained by the electron (formerly zero) exceeds the energy/c lost by the photon,

${\displaystyle {\frac {1}{c}}{\sqrt {(hf-hf'+m_{e}c^{2})^{2}-m_{e}^{2}c^{4}}}>{\frac {hf-hf'}{c}}~.}$

Equation (1) relates the various energies associated with the collision. The electron's momentum change involves a relativistic change in the energy of the electron, so it is not simply related to the change in energy occurring in classical physics. The change of the magnitude of the momentum of the photon is not just related to the change of its energy; it also involves a change in direction.

Solving the conservation of momentum expression for the scattered electron's momentum gives

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

Making use of the scalar product yields the square of its magnitude,

${\displaystyle {\begin{aligned}p_{e'}^{\,2}&=\mathbf {p} _{e'}\cdot \mathbf {p} _{e'}=(\mathbf {p} _{\gamma }-\mathbf {p} _{\gamma '})\cdot (\mathbf {p} _{\gamma }-\mathbf {p} _{\gamma '})\\&=p_{\gamma }^{\,2}+p_{\gamma '}^{\,2}-2p_{\gamma }\,p_{\gamma '}\cos \theta .\end{aligned}}}$

In anticipation of ${\displaystyle p_{\gamma }c}$ being replaced with ${\displaystyle hf}$, multiply both sides by ${\displaystyle c^{2}}$,

${\displaystyle p_{e'}^{\,2}c^{2}=p_{\gamma }^{\,2}c^{2}+p_{\gamma '}^{\,2}c^{2}-2c^{2}p_{\gamma }\,p_{\gamma '}\cos \theta .}$

After replacing the photon momentum terms with ${\displaystyle hf/c}$, we get a second expression for the magnitude of the momentum of the scattered electron,

$${\displaystyle p_{e'}^{\,2}c^{2}=(hf)^{2}+(hf')^{2}-2(hf)(hf')\cos {\theta }~.}$$

(2)

Equating the alternate expressions for this momentum gives

${\displaystyle (hf-hf'+m_{e}c^{2})^{2}-m_{e}^{\,2}c^{4}=\left(hf\right)^{2}+\left(hf'\right)^{2}-2h^{2}ff'\cos {\theta },}$

which, after evaluating the square and canceling and rearranging terms, further yields

${\displaystyle 2hfm_{e}c^{2}-2hf'm_{e}c^{2}=2h^{2}ff'\left(1-\cos \theta \right).}$

Dividing both sides by ${\displaystyle 2hff'm_{e}c}$ yields

${\displaystyle {\frac {c}{f'}}-{\frac {c}{f}}={\frac {h}{m_{e}c}}\left(1-\cos \theta \right).}$

Finally, since fλ = f ' λ' = c,

$${\displaystyle \lambda '-\lambda ={\frac {h}{m_{e}c}}(1-\cos {\theta })~.}$$

(3)

It can further be seen that the angle φ of the outgoing electron with the direction of the incoming photon is specified by

$${\displaystyle \cot \varphi =\left(1+{\frac {hf}{m_{e}c^{2}}}\right)\tan(\theta /2)~.}$$

(4)

Compton scattering is of prime importance to radiobiology, as it is the most probable interaction of gamma rays and high energy X-rays with atoms in living beings and is applied in radiation therapy.[6] [7]

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.

Magnetic Compton scattering is an extension of the previously mentioned technique which involves the magnetisation of a crystal sample hit with high energy, circularly polarised photons. By measuring the scattered photons' energy and reversing the magnetisation of the sample, two different Compton profiles are generated (one for spin up momenta and one for spin down momenta). Taking the difference between these two profiles gives the magnetic Compton profile (MCP), given by ${\displaystyle J_{\text{mag}}(\mathbf {p} _{z})}$ - a one-dimensional projection of the electron spin density.

$${\displaystyle J_{\text{mag}}(\mathbf {p} _{z})={\frac {1}{\mu }}\iint _{-\infty }^{\infty }(n_{\uparrow }(\mathbf {p} )-n_{\downarrow }(\mathbf {p} ))d\mathbf {p} _{x}d\mathbf {p} _{y}}$$

where ${\displaystyle \mu }$ is the number of spin-unpaired electrons in the system, ${\displaystyle n_{\uparrow }(\mathbf {p} )}$ and ${\displaystyle n_{\downarrow }(\mathbf {p} )}$ are the three-dimensional electron momentum distributions for the majority spin and minority spin electrons respectively.

Since this scattering process is incoherent (there is no phase relationship between the scattered photons), the MCP is representative of the bulk properties of the sample and is a probe of the ground state. This means that the MCP is ideal for comparison with theoretical techniques such as density functional theory. The area under the MCP is directly proportional to the spin moment of the system and so, when combined with total moment measurements methods (such as SQUID magnetometry), can be used to isolate both the spin and orbital contributions to the total moment of a system. The shape of the MCP also yields insight into the origin of the magnetism in the system.[8]

Inverse Compton scattering is important in astrophysics. In X-ray astronomy, the accretion disk surrounding a black hole is presumed 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 surmised to cause the power law component in the X-ray spectra (0.2–10 keV) of accreting black holes.[9]

The effect is also observed when photons from the cosmic microwave background (CMB) 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.

Some synchrotron radiation facilities scatter laser light off the stored electron beam. This Compton backscattering produces high energy photons in the MeV to GeV range[10][11] subsequently used for nuclear physics experiments.

Non-linear inverse Compton scattering (NICS) is the scattering of multiple low-energy photons, given by an intense electromagnetic field, in a high-energy photon (X-ray or gamma ray) during the interaction with a charged particle, such as an electron.[12] It is also called non-linear Compton scattering and multiphoton Compton scattering. It is the non-linear version of inverse Compton scattering in which the conditions for multiphoton absorption by the charged particle are reached due to a very intense electromagnetic field, for example the one produced by a laser.[13]

Non-linear inverse Compton scattering is an interesting phenomenon for all applications requiring high-energy photons since NICS is capable of producing photons with energy comparable to the charged particle rest energy and higher.[14] As a consequence NICS photons can be used to trigger other phenomena such as pair production, Compton scattering, nuclear reactions, and can be used to probe non-linear quantum effects and non-linear QED.[12]