In
particle physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
, the Klein–Nishina formula gives the
differential cross section (i.e. the "likelihood" and angular distribution) of
photons
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are Massless particle, massless ...
scattered
Scattered may refer to:
Music
* ''Scattered'' (album), a 2010 album by The Handsome Family
* "Scattered" (The Kinks song), 1993
* "Scattered", a song by Ace Young
* "Scattered", a song by Lauren Jauregui
* "Scattered", a song by Green Day from ' ...
from a single free
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have n ...
, calculated in the lowest order of
quantum electrodynamics
In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
. It was first derived in 1928 by
Oskar Klein
Oskar Benjamin Klein (; 15 September 1894 – 5 February 1977) was a Swedish theoretical physicist.
Biography
Klein was born in Danderyd outside Stockholm, son of the chief rabbi of Stockholm, Gottlieb Klein from Humenné in Kingdom of Hung ...
and
Yoshio Nishina
was a Japanese physicist who was called "the founding father of modern physics research in Japan". He led the efforts of Japan to develop an atomic bomb during World War II.
Early life and career
Nishina was born in Satoshō, Okayama. He rece ...
, constituting one of the first successful applications of the
Dirac equation
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac par ...
. The formula describes both the
Thomson scattering
Thomson scattering is the elastic scattering of electromagnetic radiation by a free charged particle, as described by classical electromagnetism. It is the low-energy limit of Compton scattering: the particle's kinetic energy and photon frequen ...
of low energy photons (e.g.
visible light
Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 t ...
) and the
Compton scattering
Compton scattering, discovered by Arthur Holly Compton, is the scattering of a high frequency photon after an interaction with a charged particle, usually an electron. If it results in a decrease in energy (increase in wavelength) of the photon ...
of high energy photons (e.g.
x-ray
An X-ray, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation. Most X-rays have a wavelength ranging from 10 picometers to 10 nanometers, corresponding to frequencies in the range 30&nb ...
s and
gamma-ray
A gamma ray, also known as gamma radiation (symbol γ or \gamma), is a penetrating form of electromagnetic radiation arising from the radioactive decay of atomic nuclei. It consists of the shortest wavelength electromagnetic waves, typically sh ...
s), showing that the total cross section and expected deflection angle decrease with increasing photon energy.
Formula
For an incident unpolarized photon of energy
, the
differential cross section is:
[
]
:
where
*
is the
classical electron radius
The classical electron radius is a combination of fundamental physical quantities that define a length scale for problems involving an electron interacting with electromagnetic radiation. It links the classical electrostatic self-interaction energ ...
(~2.82
fm,
is about 7.94 × 10
−30 m
2 or 79.4
mb)
*
is the ratio of the wavelengths of the incident and scattered photons
*
is the scattering angle (0 for an undeflected photon).
The angular dependent photon wavelength (or energy, or frequency) ratio is
:
as required by the conservation of
relativistic energy and momentum (see
Compton scattering
Compton scattering, discovered by Arthur Holly Compton, is the scattering of a high frequency photon after an interaction with a charged particle, usually an electron. If it results in a decrease in energy (increase in wavelength) of the photon ...
). The dimensionless quantity
expresses the energy of the incident photon in terms of the electron rest energy (~511
keV Kev can refer to:
Given name
* Kev Adams, French comedian, actor, screenwriter and film producer born Kevin Smadja in 1991
* Kevin Kev Carmody (born 1946), Indigenous Australian singer-songwriter
* Kev Coghlan (born 1988), Scottish Grand Prix moto ...
), and may also be expressed as
, where
is the
Compton wavelength
The Compton wavelength is a quantum mechanical property of a particle. The Compton wavelength of a particle is equal to the wavelength of a photon whose energy is the same as the rest energy of that particle (see mass–energy equivalence). It was ...
of the electron (~2.42 pm). Notice that the scatter ratio
increases
monotonically with the deflection angle, from
(forward scattering, no energy transfer) to
(180 degree backscatter, maximum energy transfer).
In some cases it is convenient to express the classical electron radius in terms of the Compton wavelength:
, where
is the
fine structure constant
In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by (the Greek letter ''alpha''), is a fundamental physical constant which quantifies the strength of the electromagnetic interaction between ele ...
(~1/137) and
is the ''reduced'' Compton wavelength of the electron (~0.386 pm), so that the constant in the cross section may be given as:
:
Polarized photons
If the incoming photon is polarized, the scattered photon is no longer isotropic with respect to the azimuthal angle. For a linearly polarized photon scattered with a free electron at rest, the differential cross section is instead given by:
:
where
is the azimuthal scattering angle. Note that the unpolarized differential cross section can be obtained by averaging over
.
Limits
Low energy
For low energy photons the wavelength shift becomes negligible (
) and the Klein–Nishina formula reduces to the classical
Thomson expression:
:
which is symmetrical in the scattering angle, i.e. the photon is just as likely to scatter backwards as forwards. With increasing energy this symmetry is broken and the photon becomes more likely to scatter in the forward direction.
High energy
For high energy photons it is useful to distinguish between small and large angle scattering. For large angles, where
, the scatter ratio
is large and
:
showing that the (large angle) differential cross section is inversely proportional to the photon energy.
The differential cross section has a constant peak in the forward direction:
:
independent of
. From the large angle analysis it follows that this peak can only extend to about
. The forward peak is thus confined to a small solid angle of approximately
, and we may conclude that the total small angle cross section decreases with
.
Total cross section
The differential cross section may be integrated to find the total cross section.
In the low energy limit there is no energy dependence and we recover the
Thomson cross section (~66.5 fm
2):
:
History
The Klein–Nishina formula was derived in 1928 by
Oskar Klein
Oskar Benjamin Klein (; 15 September 1894 – 5 February 1977) was a Swedish theoretical physicist.
Biography
Klein was born in Danderyd outside Stockholm, son of the chief rabbi of Stockholm, Gottlieb Klein from Humenné in Kingdom of Hung ...
and
Yoshio Nishina
was a Japanese physicist who was called "the founding father of modern physics research in Japan". He led the efforts of Japan to develop an atomic bomb during World War II.
Early life and career
Nishina was born in Satoshō, Okayama. He rece ...
, and was one of the first results obtained from the study of
quantum electrodynamics
In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
. Consideration of relativistic and quantum mechanical effects allowed development of an accurate equation for the scattering of radiation from a target electron. Before this derivation, the electron cross section had been classically derived by the British physicist and discoverer of the
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have n ...
,
J.J. Thomson
Sir Joseph John Thomson (18 December 1856 – 30 August 1940) was a British physicist and Nobel Laureate in Physics, credited with the discovery of the electron, the first subatomic particle to be discovered.
In 1897, Thomson showed that ...
. However, scattering experiments showed significant deviations from the results predicted by the Thomson cross section. Further scattering experiments agreed perfectly with the predictions of the Klein–Nishina formula.
See also
*
Synchrotron radiation
*
Yoshio Nishina
was a Japanese physicist who was called "the founding father of modern physics research in Japan". He led the efforts of Japan to develop an atomic bomb during World War II.
Early life and career
Nishina was born in Satoshō, Okayama. He rece ...
*
Oskar Klein
Oskar Benjamin Klein (; 15 September 1894 – 5 February 1977) was a Swedish theoretical physicist.
Biography
Klein was born in Danderyd outside Stockholm, son of the chief rabbi of Stockholm, Gottlieb Klein from Humenné in Kingdom of Hung ...
References
Further reading
*
*
*
{{DEFAULTSORT:Klein-Nishina formula
Quantum electrodynamics
Scattering