Klein-Nishina cross-section

Klein – Nishina cross section for the scattering angle at different energies
(input from the left, i.e. at 180 °)

The Klein-Nishina cross-section is the cross-section that indicates the angular distribution of photons that are scattered by stationary, punctiform, charged particles ( Compton scattering ). It was calculated for the electron by Oskar Klein and Yoshio Nishina in 1929 and was one of the first results of quantum electrodynamics . It agrees with the experimental results. In this article the calculation for the electron is reproduced; for other punctiform particles the elementary charge and the electron mass have to be changed by appropriate parameters. ${\displaystyle e}$ ${\displaystyle m}$

The following formulas are not written in the SI system , but in a natural system of units adapted for particle physics . In the natural system of units: ${\displaystyle \varepsilon _{0}=\hbar =c=1}$

definition

In photon-particle scattering, the conservation of energy and momentum determine how the energy of the scattered photon depends on the scattering angle and the original photon energy. However, it does not follow from the conservation laws how often this or that scatter angle occurs. This frequency is indicated by the differential cross section , with the solid angle element . ${\displaystyle E'}$ ${\displaystyle \theta }$ ${\displaystyle E}$ ${\displaystyle \mathrm {d} \sigma /\mathrm {d} \Omega }$ ${\displaystyle \mathrm {d} \Omega =\mathrm {d} \cos \theta \,\mathrm {d} \phi }$

The Klein-Nishina cross-section in the laboratory system is:

${\displaystyle {\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}_{\text{ Klein-Nishina}}={\frac {1}{2}}{\frac {\alpha ^{2}}{m^{2}}}\left({\frac {E'}{E}}\right)^{2}\left[{\frac {E'}{E}}+{\frac {E}{E'}}-\sin ^{2}\theta \right]}$

Here is the fine structure constant . The formula given applies to unpolarized photons. The quotient of the energy of the scattered photon and the incident photon results correctly from a semi-classical calculation (see Compton effect ) ${\displaystyle \alpha ={\frac {e^{2}}{4\pi }}\approx {\frac {1}{137}}}$

${\displaystyle {\frac {E'}{E}}={\frac {1}{1+{\frac {E}{m}}(1-\cos \theta )}}}$.

An integration over the differential cross section provides the total cross section with the abbreviation : ${\displaystyle x=E/m}$

${\displaystyle \sigma =\int {\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}\mathrm {d} \Omega ={\frac {\pi \alpha ^{2}}{m^{2}}}{\frac {1}{x^{3}}}\left({\frac {2x(2+x(1+x)(8+x))}{(1+2x)^{2}}}+((x-2)x-2)\log(1+2x)\right)}$

Borderline cases

Low-energy borderline case

For photon energies that are small compared to the rest energy of the electron, the following applies because of the masslessness of the photon and thus ${\displaystyle E\to 0}$

${\displaystyle \lim _{E\to 0}{\frac {E'}{E}}=1}$;

then the Klein-Nishina cross section goes against the Thomson cross section

${\displaystyle {\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}_{\text{ Thomson}}={\frac {1}{2}}{\frac {\alpha ^{2}}{m^{2}}}\left(1+\cos ^{2}\theta \right)}$,

which Joseph Thomson had calculated for the scattering of an electromagnetic wave at a point charge . For small energies, forward scattering of the photon is just as likely as backward scattering; forward scattering becomes more likely only at higher energies (see figure).

For low-energy photons, the total cross-section after a simple integration over the solid angle up to a factor of 8/3 is the area of ​​a circular disk , the radius of which is the classic electron radius : ${\displaystyle d\Omega }$ ${\displaystyle r_{\text{e}}=\alpha \hbar /(cm_{\text{e}})}$

${\displaystyle \sigma _{\text{Thomson}}={\frac {8\pi }{3}}r_{\text{e}}^{2}}$

High-energy borderline case

The total cross-section in the high-energy borderline case results from a development in the parameter to ${\displaystyle E\to \infty }$ ${\displaystyle x}$

${\displaystyle \sigma ={\frac {\pi \alpha ^{2}}{Em}}\left({\frac {1}{2}}+\ln {\frac {2E}{m}}\right)}$.

It therefore falls with the energy at high photon energies.

Derivation

The fundamental process that leads to the Klein-Nishina cross-section is Compton scattering . Denotes the momentum of the incoming electron and that of the incoming (outgoing) photon (the momentum of the outgoing electron is determined by the law of conservation of energy and momentum and is not an independent quantity), the spin-averaged squared matrix element of the scattering matrix reads: ${\displaystyle \gamma e^{-}\rightarrow \gamma e^{-}}$ ${\displaystyle p}$ ${\displaystyle k(k')}$ ${\displaystyle p'}$

${\displaystyle {\overline {\left|{\mathcal {M}}\right|^{2}}}=2e^{4}\left[{\frac {pk'}{pk}}+{\frac {pk}{pk'}}+2m^{2}\left({\frac {1}{pk}}-{\frac {1}{pk'}}\right)+m^{4}\left({\frac {1}{pk}}-{\frac {1}{pk'}}\right)^{2}\right]}$

For the calculation of the differential cross section from the Lorentz invariant matrix element a reference system has to be chosen, in the case of the Klein-Nishina cross section the rest system of the electron. Furthermore, the coordinates can be chosen so that the incident photon propagates in the direction. Then with and as well as for the matrix element ${\displaystyle z}$ ${\displaystyle p=(m,0,0,0)}$ ${\displaystyle k=(E,0,0,E)}$ ${\displaystyle k'=(E,E\sin \theta ,0,E\cos \theta )}$

${\displaystyle {\overline {\left|{\mathcal {M}}\right|^{2}}}=2e^{4}\left[{\frac {E'}{E}}+{\frac {E}{E'}}+2m\left({\frac {1}{E}}-{\frac {1}{E'}}\right)+m^{2}\left({\frac {1}{E}}-{\frac {1}{E'}}\right)^{2}\right]}$

The quotient of the energies of the scattered and incident photons is obtained from the energy-momentum conservation law using

${\displaystyle m^{2}=p'^{2}=(p+k-k')^{2}=m^{2}+2m(E-E')-2EE'(1-\cos \theta )}$

as already postulated above

${\displaystyle {\frac {E'}{E}}={\frac {1}{1+{\frac {E}{m}}(1-\cos \theta )}}}$.

The differential cross-section now results from quantum field theory

${\displaystyle \int \mathrm {d} \sigma ={\frac {1}{2E_{A}\,2E_{B}\Delta v}}\int \mathrm {d} \Pi {\overline {|{\mathcal {M}}|^{2}}}}$

with the energies of the scatter partners, the speed difference and the phase space integral ${\displaystyle E_{A},E_{B}}$ ${\displaystyle \Delta v=|v_{A}-v_{B}|}$

${\displaystyle \int \mathrm {d} \Pi =\left(\prod _{f}\int {\frac {\mathrm {d} ^{3}p_{f}}{(2\pi )^{3}}}{\frac {1}{2E_{f}}}\right)(2\pi )^{4}\delta ^{(4)}(\sum p_{i}-\sum p_{f})}$,

where stand for the four -momentum of the incoming (outgoing) particles and the delta distribution ensures the conservation of energy and momentum. ${\displaystyle p_{i}(p_{f})}$

In the case of Compton scattering, the phase space integral finally results in

${\displaystyle \int \mathrm {d} \Pi ={\frac {1}{16\pi ^{2}}}\int \mathrm {d} \Omega {\frac {E'^{2}}{Em}}}$

as well as trivially because of the constancy of the speed of light . ${\displaystyle \Delta v=1}$

Put everything together and partially simplified with the help of the energy-momentum conservation law, this finally results in the Klein-Nishina cross-section

${\displaystyle \mathrm {d} \sigma ={\frac {1}{2}}{\frac {\alpha ^{2}}{m^{2}}}\left({\frac {E'}{E}}\right)^{2}\left[{\frac {E'}{E}}+{\frac {E}{E'}}-\sin ^{2}\theta \right]\mathrm {d} \Omega }$

literature

• Otto Nachtmann: Phenomena and concepts of elementary particle physics. Vieweg Braunschweig, 1986, ISBN 3-528-08926-1 .
• O. Klein and Y. Nishina: About the scattering of radiation by free electrons according to the new relativistic quantum mechanics according to Dirac. In: Journal of Physics. 52, 1929, pp. 853-868, doi : 10.1007 / BF01366453 .
• Michael D. Peskin and Daniel V. Schroeder: An Introduction to Quantum Field Theory , Perseus Books Publishing 1995, ISBN 0-201-50397-2