Electron-photon scattering cross section
In
particle physics
, the
Klein?Nishina formula
gives the
differential cross section
(i.e. the "likelihood" and angular distribution) of
photons
scattered
from a single free
electron
, calculated in the lowest order of
quantum electrodynamics
. It was first derived in 1928 by
Oskar Klein
and
Yoshio Nishina
, constituting one of the first successful applications of the
Dirac equation
.
[1]
The formula describes both the
Thomson scattering
of low energy photons (e.g.
visible light
) and the
Compton scattering
of high energy photons (e.g.
x-rays
and
gamma-rays
), showing that the total cross section and expected deflection angle decrease with increasing photon energy.
Formula
[
edit
]
For an incident unpolarized photon of energy
, the
differential cross section
is:
[2]
where
- is the
classical electron radius
(~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
). The dimensionless quantity
expresses the energy of the incident photon in terms of the electron rest energy (~511
keV
), and may also be expressed as
, where
is the
Compton wavelength
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
(~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
[
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]
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
[
edit
]
Low energy
[
edit
]
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
[
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]
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
[
edit
]
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
[
edit
]
The Klein?Nishina formula was derived in 1928 by
Oskar Klein
and
Yoshio Nishina
, and was one of the first results obtained from the study of
quantum electrodynamics
. 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
,
J.J. Thomson
. 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.
[
citation needed
]
See also
[
edit
]
References
[
edit
]
- ^
Klein, O; Nishina, Y (1929). "Uber die Streuung von Strahlung durch freie Elektronen nach der neuen relativistischen Quantendynamik von Dirac".
Z. Phys
.
52
(11?12): 853 and 869.
Bibcode
:
1929ZPhy...52..853K
.
doi
:
10.1007/BF01366453
.
S2CID
123905506
.
- ^
Weinberg, Steven (1995).
The Quantum Theory of Fields
. Vol. I. pp. 362?9.
Further reading
[
edit
]
- Evans, R. D. (1955).
The Atomic Nucleus
. New York: McGraw-Hill. pp. 674?676.
OCLC
542611
.
- Melissinos, A. C. (1966).
Experiments in Modern Physics
. New York: Academic Press. pp. 252?265.
ISBN
0-12-489850-5
.
- Klein, O.; Nishina, Y. (1994). "On the Scattering of Radiation by Free Electrons According to Dirac's New Relativistic Quantum Dynamics". In Ekspong, Gosta (ed.).
The Oskar Klein Memorial Lectures, Vol. 2: Lectures by Hans A. Bethe and Alan H. Guth with Translated Reprints by Oskar Klein
. Singapore: World Scientific. pp. 113?139.
Bibcode
:
1994okml.book.....E
.
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