# Bremsstrahlung

File:Bremsstrahlung.svg
Bremsstrahlung produced by a high-energy electron deflected in the electric field of an atomic nucleus

Bremsstrahlung (pronounced Template:Audio-IPA, from German [bremsen] error: {{lang}}: text has italic markup (help) "to brake" and [Strahlung] error: {{lang}}: text has italic markup (help) "radiation", i.e. "braking radiation" or "deceleration radiation"), is electromagnetic radiation produced by the deceleration of a charged particle, such as an electron, when deflected by another charged particle, such as an atomic nucleus. The term is also used to refer to the process of producing the radiation. Bremsstrahlung has a continuous spectrum. The phenomenon was discovered by Nikola Tesla during high frequency research he conducted between 1888 and 1897.

Bremsstrahlung may also be referred to as free-free radiation. This refers to the radiation that arises as a result of a charged particle that is free both before and after the deflection (acceleration) that causes the emission. Strictly speaking, bremsstrahlung refers to any radiation due to the acceleration of a charged particle, which includes synchrotron radiation; however, it is frequently used (even when not speaking German) in the more narrow sense of radiation from electrons stopping in matter.

The word Bremsstrahlung is retained from the original German to describe the radiation which is emitted when electrons are decelerated or "braked" when they are fired at a metal target. Accelerated charges give off electromagnetic radiation, and when the energy of the bombarding electrons is high enough, that radiation is in the x-ray region of the electromagnetic spectrum. It is characterized by a continuous distribution of radiation which becomes more intense and shifts toward higher frequencies when the energy of the bombarding electrons is increased.

File:Brem cross section.png
Bremsstrahlung cross section for the emission of a photon with energy 30 keV by an electron impacting on a proton.

## Outer Bremsstrahlung

"Outer bremsstrahlung" is the term applied in cases where the energy loss by radiation greatly exceeds that by ionization as a stopping mechanism in matter. This is seen clearly for electrons with energies above 50 keV.

## Inner Bremsstrahlung

"Inner bremsstrahlung" is the term applied to the less frequent case of radiation emission during beta decay, resulting in the emission of a photon of energy less than or equal to the maximum energy available in the nuclear transition. Inner bremsstrahlung is caused by the abrupt change in the electric field in the region of the nucleus of the atom undergoing decay, in a manner similar to that which causes outer bremsstrahlung. In electron and positron emission the photon's energy comes from the electron/nucleon pair, with the spectrum of the bremsstrahlung decreasing continuously with increasing energy of the beta particle. In electron capture the energy comes at the expense of the neutrino, and the spectrum is greatest at about one third of the normal neutrino energy, reaching zero at zero energy and at normal neutrino energy.

Beta particle emitting substances sometimes exhibit a weak radiation with continuous spectrum that is due to both outer and inner bremsstrahlung, or to one of them alone.

Bremsstrahlung is a type of "secondary radiation", in that it is produced as a result of stopping (or slowing) the primary radiation (beta particles). In some cases, e.g. 32P, the Bremsstrahlung produced by shielding this radiation with the normally used dense materials (e.g. lead) is itself dangerous; in such cases, shielding must be accomplished with low density materials, e.g. Plexiglas, Lucite, plastic, wood, or water [1]; because the rate of deceleration of the electron is slower, the radiation given off has a longer wavelength and is therefore less penetrating.

Suppose that a particle of charge ${\displaystyle q}$ experiences an acceleration ${\displaystyle {\vec {a}}}$ which, for the sake of simplicity, is collinear with its velocity ${\displaystyle {\vec {v}}}$. Then, the relativistic expression for the angular distribution of the bremsstrahlung (considering only the dominant dipole radiation contribution), is

${\displaystyle {\frac {dP(\theta )}{d\Omega }}={\frac {\mu _{0}q^{2}a^{2}}{16\pi ^{2}c}}{\frac {\sin ^{2}{\theta }}{(1-\beta \cos {\theta })^{5}}}}$,
where ${\displaystyle \beta =v/c}$ and ${\displaystyle \theta }$ is the angle between ${\displaystyle {\vec {a}}}$ and the point of observation.

Integrating over all angles then gives the total power emitted as

${\displaystyle P={\frac {\mu _{0}q^{2}a^{2}\gamma ^{6}}{6\pi c}}}$,
where ${\displaystyle \gamma (v)}$ is the Lorentz factor. This basic treatment shows a very strong dependence on the Lorentz factor, gamma, meaning that the amount of bremsstrahlung emitted by the particle increases greatly with its speed, if the speed is at least semi-relativistic to begin with. This illustrates that, for a given fixed particle energy E, the amount of bremsstrahlung emitted by a particle has a strong dependence on the particle's mass, since ${\displaystyle \gamma =E/(mc^{2})}$. In this case, ${\displaystyle P\propto m^{-6}}$ for a fixed energy, so if an electron and muon have the same energy, the electron will emit ${\displaystyle (m_{\mu }/m_{e})^{6}}$ = 2076 = 7.87×1013 times more radiation than the muon. This is why muons have such high penetrating power — they lose very little energy via bremsstrahlung.[1]

The general expression for the total radiated power is[2]

${\displaystyle P={\frac {q^{2}\gamma ^{4}}{6\pi \epsilon _{0}c}}\left({\dot {\beta }}^{2}+{\frac {({\vec {\beta }}\cdot {\dot {\beta }})^{2}}{1-\beta ^{2}}}\right)}$

where ${\displaystyle {\dot {\beta }}}$ signifies a time derivative.

## From plasma (thermal Bremsstrahlung)

File:Bremsstrahlung power.png
The bremsstrahlung power spectrum rapidly decreases from being infinite at ${\displaystyle \omega =0}$ to zero as ${\displaystyle \omega \rightarrow \infty }$. This plot is for the quantum case ${\displaystyle T_{e}>Z^{2}27.2}$ eV and the constant K=3.17.

In a plasma the free electrons are constantly producing Bremsstrahlung in collisions with the ions. In a uniform plasma, with thermal electrons (distributed according to the Maxwell–Boltzmann distribution with the temperature ${\displaystyle T_{e}}$), the power spectral density (power per angular frequency interval per volume, integrated over the whole solid angle) of the Bremsstrahlung radiated, is calculated to be [3]

${\displaystyle {dP_{\mathrm {Br} } \over d\omega }={4{\sqrt {2}} \over 3{\sqrt {\pi }}}\left[n_{e}r_{e}^{3}\right]^{2}\left[{\frac {m_{e}c^{2}}{k_{B}T_{e}}}\right]^{1/2}\left[{m_{e}c^{2} \over r_{e}^{3}}\right]Z_{\mathrm {eff} }E_{1}(w_{m}),}$

where ${\displaystyle n_{e}}$ is the number density of electrons, ${\displaystyle r_{e}}$ is the classical radius of electron, ${\displaystyle m_{e}}$ is its mass, ${\displaystyle k_{B}}$ is the Boltzmann constant, and ${\displaystyle c}$ is the speed of light. Note that all but the third bracketed factor on the right-hand side are dimensionless. The "effective" ion charge state ${\displaystyle Z_{\mathrm {eff} }}$ is given by an average over the charge states of the ions:

${\displaystyle Z_{\mathrm {eff} }=\sum _{Z}Z^{2}{n_{Z} \over n_{e}}}$ ,

where ${\displaystyle n_{Z}}$ is the number density of ions with charge ${\displaystyle Z}$.

The special function ${\displaystyle E_{1}}$ is defined in the exponential integral article, and

${\displaystyle w_{m}={\omega ^{2}m_{e} \over 2k_{m}^{2}k_{B}T_{e}}}$

(${\displaystyle k_{m}}$ is a maximum or cutoff wavenumber). ${\displaystyle k_{m}=K/\lambda _{B}}$ when ${\displaystyle k_{B}T_{e}>Z^{2}}$ 27.2 eV (for a single ion species; 27.2 eV is twice the ionization energy of hydrogen) where K is a pure number and ${\displaystyle \lambda _{B}=\hbar /(m_{e}k_{B}T_{e})^{1/2}}$ is a thermal electron de Broglie wavelength. Otherwise, ${\displaystyle k_{m}\propto 1/l_{c}}$ where ${\displaystyle l_{c}}$ is the classical Coulomb distance of closest approach.

For the case ${\displaystyle k_{m}=K/\lambda _{B}}$, we find

${\displaystyle w_{m}={1 \over 2K^{2}}\left[{\frac {\hbar \omega }{k_{B}T_{e}}}\right]^{2}}$ .

${\displaystyle dP_{\mathrm {Br} }/d\omega }$ is infinite at ${\displaystyle \omega =0}$, and decreases rapidly with ${\displaystyle \omega }$. The resulting power density, integrated over all frequencies, is finite and equals

${\displaystyle P_{\mathrm {Br} }={8 \over 3}\left[n_{e}r_{e}^{3}\right]^{2}\left[{k_{B}T_{e} \over m_{e}c^{2}}\right]^{1/2}\left[{m_{e}c^{3} \over r_{e}^{4}}\right]Z_{\mathrm {eff} }\alpha K}$ .

Note the appearance of the fine structure constant ${\displaystyle \alpha }$ due to the quantum nature of ${\displaystyle \lambda _{B}}$. In practical units, a commonly used version of this formula is [4]

${\displaystyle P_{\mathrm {Br} }[{\textrm {W/m}}^{3}]=\left[{n_{e} \over 7.69\times 10^{18}{\textrm {m}}^{-3}}\right]^{2}T_{e}[{\textrm {eV}}]^{1/2}Z_{\mathrm {eff} }}$ .

This formula agrees with the theoretical estimate if we set K=3.17; the value K=3 is suggested by Ichimaru.

For very high temperatures there are relativistic corrections to this formula, that is, additional terms of order kBTe/mec2.[2]

If the plasma is optically thin, the Bremsstrahlung radiation leaves the plasma, carrying part of the internal plasma energy. This effect is known as the Bremsstrahlung cooling.

## In astrophysics

The dominant luminous component in a cluster of galaxies is the 107 to 108 Kelvin intracluster medium. The emission from the intracluster medium is characterized by thermal Bremsstrahlung. Thermal Bremsstrahlung radiation occurs when the particles populating the emitting plasma are at a uniform temperature and are distributed according to the Maxwell–Boltzmann distribution

${\displaystyle f(v)=4\pi \left({\frac {m}{2\pi kT}}\right)^{3/2}v^{2}\exp \left[{\frac {-mv^{2}}{2kT}}\right]}$

where speed, v, is defined as

${\displaystyle v={\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}}.}$

The bulk emission from this gas is thermal Bremsstrahlung. The power emitted per cubic centimeter per second can be written in the compact form[5]

${\displaystyle \epsilon _{\mathrm {ff} }=1.4\times 10^{-27}T^{1/2}n_{e}n_{i}Z^{2}g_{B}}$

with cgs units [erg cm-3 s-1] and where 'ff' stands for free-free, 1.4x10-27 is the condensed form of the physical constants and geometrical constants associated with integrating over the power per unit area per unit frequency, ne and ni are the electron and ion densities, respectively, Z is the number of protons of the bending charge, gB is the frequency averaged Gaunt factor and is of order unity, and T is the global x-ray temperature determined from the spectral cut-off frequency

${\displaystyle \hbar \nu =kT}$

above which exponentially small amount of photons are created because the energy required for creation of such a photon is available only for electrons in the tail of the Maxwell distribution.

This process is also known as Bremsstrahlung cooling since the plasma is optically thin to photons at these energies and the energy radiated is emitted freely into the universe.

This radiation is in the energy range of X-rays and can be easily observed with space-based telescopes such as Chandra X-ray Observatory, XMM-Newton, ROSAT, ASCA, EXOSAT, Astro-E2, and future missions like Con-X[3] and NeXT[4].

## General treatment

For a much more complete discussion, see Haug.[6]

## References

1. Introduction to Electrodynamics, 3rd edition, David J. Griffiths, pages 463–464.
2. A Plasma Formulary for Physics, Technology, and Astrophysics, D. Diver, p. 46-48.
3. Basic Principles of Plasmas Physics: A Statistical Approach, S. Ichimaru, p. 228.
4. NRL Plasma Formulary, 2006 Revision, p. 58.
5. Radiative Processes in Astrophysics, G.B. Rybicki & A.P. Lightman, p. 162.
6. Eberhard Haug & Nakel W (2004). The elementary process of Bremsstrahlung. River Edge NJ: World Scientific. p. Scientific lecture notes in physics, vol. 73. ISBN 9812385789.