Bremsstrahlung (in German pronounced as /ˈbʁɛms.ʃtʁaːlʊŋ/), from German: bremsen "to brake" and German: Strahlung "radiation"; i.e., "braking radiation" or "deceleration radiation", is electromagnetic radiation produced by the deceleration of a charged particle when deflected by another charged particle, typically an electron by an atomic nucleus. The moving particle loses kinetic energy, which is converted into radiation (i.e., photons), thus satisfying the law of conservation of energy. The term is also used to refer to the process of producing the radiation. Bremsstrahlung has a continuous spectrum, which becomes more intense and whose peak intensity shifts toward higher frequencies as the change of the energy of the decelerated particles increases.
Broadly speaking, bremsstrahlung or braking radiation is any radiation produced due to the deceleration (negative acceleration) of a charged particle, which includes synchrotron radiation (i.e. photon emission by a relativistic particle), cyclotron radiation (i.e. photon emission by a nonrelativistic particle), and the emission of electrons and positrons during beta decay. However, the term is frequently used in the more narrow sense of radiation from electrons (from whatever source) slowing in matter.
Bremsstrahlung emitted from plasma is sometimes referred to as free–free radiation. This refers to the fact that the radiation in this case is created by electrons that are free (i.e., not in an atomic or molecular bound state) before, and remain free after, the emission of a photon. In the same parlance, bound–bound radiation refers to discrete spectral lines (an electron "jumps" between two bound states), while free–bound one—to the radiative combination process, in which a free electron recombines with an ion.
See main article: Larmor formula.
If quantum effects are negligible, an accelerating charged particle radiates power as described by the Larmor formula and its relativistic generalization.
The total radiated power is^{[1]}
P=
q^{2}\gamma^{4}  
6\pi\varepsilon_{0}c 
\left(
\beta 
^{2}+
\left(\vec{\beta  
⋅ 
\vec{\beta 
where
\vec\beta=
\vecv  
c 
\gamma=
1\sqrt{1\beta  
^{2}} 
\vec\beta 
\vec\beta
P_{a}=
q^{2}a^{2}\gamma^{6}  
6\pi\varepsilon_{0}c^{3} 
,
where
a\equiv
v 
=
\beta 
c
\vec{\beta} ⋅
\vec{\beta 
P_{a}=
q^{2}a^{2}\gamma^{4}  
6\pi\varepsilon_{0}c^{3} 
.
Power radiated in the two limiting cases is proportional to
\gamma^{4}
\left(a\perpv\right)
\gamma^{6}
\left(a\parallelv\right)
E=\gammamc^{2}
E
m^{4}
m^{6}
(m_{p/m}
4  
e) 
≈ 10^{13}
The most general formula for radiated power as a function of angle is:^{[3]}
dP  
d\Omega 
=
q^{2}  
16\pi^{2}\varepsilon_{0}c 
\left\hat{n  
x 
\left(\left(\hat{n}\vec{\beta}\right) x
\vec{\beta 
\hat{n}
d\Omega
In the case where velocity is parallel to acceleration (for example, linear motion), this simplifies to^{[3]}
dP_{a}  
d\Omega 
=
q^{2a}^{2}  
16\pi^{2}\varepsilon_{0}c^{3} 
\sin^{2}\theta  
(1\beta\cos\theta)^{5} 
\theta
\vec{a}
This section gives a quantummechanical analog of the prior section, but with some simplifications. We give a nonrelativistic treatment of the special case of an electron of mass
m_{e}
e
v
Ze
n_{i}
\nu=c/λ
h\nu
j(v,\nu)
j(v,\nu)={8\pi\over3\sqrt3}\left({e^{2\over}
3  
4\pi\epsilon  
0}\right) 
2n  
{Z  
i 
\over
2v}g  
c  
\rmff 
(v,\nu)
A general, quantummechanical formula for
g_{\rm}
\nu_{\rm}\propto
1/2  
n  
\rme 
n_{e}
\nu<\nu_{\rm}
h\nu\ll
2/2  
m  
ev 
With these assumptions, two unitless parameters characterize the process:
η_{Z}\equivZe^{2/\hbar}v
η_{\nu}\equiv
2  
h\nu/2m  
ev 
η_{Z\ll}1
g_{\rm}={\sqrt3\over\pi}ln{1\overη_{\nu}}
In the opposite limit
η_{Z\gg}1
g_{\rm}={\sqrt3\over\pi}\left[ln\left({1\overη_{Zη}_{\nu}\right)}\gamma\right]
where
\gamma ≈ 0.577
1/η_{Zη}_{\nu=m}
3/\pi  
ev 
Ze^{2\nu}
h
A semiclassical, heuristic way to understand the Gaunt factor is to write it as
g_{\rm} ≈ ln(b_{\rm}/b_{\rm})
b_{max}
b_{\rm}
b_{\rm}=v/\nu
b_{\rm}
≈ h/m_{ev}
≈
2/4\pi\epsilon  
e  
0m 
2  
ev 
The above results generally apply as long as the argument of the logarithm is large, and break down when it is less than unity. Namely, the Gaunt factor becomes negative in this case, which is unphysical. A rough approximation to the full calculations, with the appropriate Born and classical limits, is
g_{\rm} ≈ max\left[1,{\sqrt3\over\pi}ln\left[{1\over
\gammaη  
η  
Z)}\right] 
\right]
This section discusses bremsstrahlung emission and the inverse absorption process (called inverse bremsstrahlung) in a macroscopic medium. We start with the equation of radiative transfer, which applies to general processes and not just bremsstrahlung:
{1\overc}\partial_{tI}_{\nu+\hat}n ⋅ \nablaI_{\nu}=j_{\nuk}_{\nu}I_{\nu}
I_{\nu(t,\vec}x)
j_{\nu}
j(v,\nu)
k_{\nu}
j_{\nu}
k_{\nu}
I_{\nu}
I_{\nu={j}_{\nu}\overk_{\nu}}
If the matter and radiation are also in thermal equilibrium at some temperature, then
I_{\nu}
B_{\nu(\nu,}T_{e)}=
2h\nu^{3}  
c^{2} 
1  

Since
j_{\nu}
k_{\nu}
I_{\nu}
j_{\nu/k}_{\nu}
j_{\nu}
k_{\nu}
NOTE: this section currently gives formulas that apply in the RayleighJeans limit
\hbar\omega\llk_{\rm}T_{e}
\exp(\hbar\omega/k_{\rm}T_{e)}
\hbar\omega/k_{\rm}T_{e}
y
In a plasma, the free electrons continually collide with the ions, producing bremsstrahlung. A complete analysis requires accounting for both binary Coulomb collisions as well as collective (dielectric) behavior. A detailed treatment is given by Bekefi,^{[4]} while a simplified one is given by Ichimaru.^{[5]} In this section we follow Bekefi's dielectric treatment, with collisions included approximately via the cutoff wavenumber,
k_{\rm}
Consider a uniform plasma, with thermal electrons distributed according to the Maxwell–Boltzmann distribution with the temperature
T_{e}
4\pi
{dP_{Br}\overd\omega}={8\sqrt2\over3\sqrt\pi}\left[{e^{2}\over4\pi\varepsilon_{0}}\right]^{3}{1\over
2)  
(m  
ec 
^{3/2}
where
\omega_{p}\equiv
2/\varepsilon  
(n  
0m 
1/2  
e) 
\omega
n_{e,}n_{i}
\omega<\omega_{\rm}
\omega>\omega_{\rm}
The special function
E_{1}
y
y={1\over2}{\omega^{2}m_{e}\over
2  
k  
\rmmax 
k_{\rm}T_{e}}
k_{\rm}
k_{\rm}=1/λ_{\rm}
k_{\rm}T_{\rm}>
2  
Z  
i 
E_{\rm}
E_{\rm} ≈ 27.2
λ_{\rm}=\hbar/(m_{\rm}k_{\rm}T_{\rm})^{1/2}
k_{\rm}\propto1/l_{\rm}
l_{\rm}
For the usual case
k_{m}=1/λ_{B}
y={1\over2}\left[
\hbar\omega  
k_{\rm}T_{e} 
\right]^{2.}
The formula for
dP_{Br/d\omega}
\omega
\omega_{\rm}
In the limit
y\ll1
E_{1}
E_{1(y)} ≈ ln[ye^{\gamma]}+O(y)
\gamma ≈ 0.577
y>e^{\gamma}
The total emission power density, integrated over all frequencies, is
\begin{align} P_{Br}&=
infty  
\int  
\omega_{\rm} 
d\omega{dP_{Br\over}d\omega}={16\over3}\left[{e^{2}\over4\pi\varepsilon_{0}}\right]^{3}{1\over
2c  
m  
e 
^{3}}
2  
Z  
i 
n_{i}n_{e}k_{\rm}G(y_{\rm})\\ G(y_{p)}&={1\over2\sqrt{\pi}}
infty  
\int  
y_{\rm} 
dy
 
y 
\left[1{y_{\rm}\over
 
y}\right] 
E_{1(y)}\\ y_{\rm}&=y(\omega=\omega_{\rm}) \end{align}
G(y_{\rm}=0)=1
y_{\rm}
k_{\rm}=1/λ_{\rm}
P_{Br}={16\over3}{\left(
e^{2}  
4\pi\varepsilon_{0} 
\right)^{3}\over(m_{e}c^{2)}
 
\hbar}
2  
Z  
i 
n_{i}n_{e}(k_{\rm}
 
T  
e) 
G(y_{\rm})
Note the appearance of
\hbar
λ_{\rm}
G=1
P_{Br}[rm{W}/rm{m}^{3]}=
2  
{Z  
i 
n_{i}n_{e}\over\left[7.69 x 10^{18}rm{m}^{3}\right]^{2}}
 
T  
e[rm{eV}] 
.
g_{\rm}
\varepsilon_{ff}=1.4 x 10^{27}
 
T 
n_{e}n_{i}Z^{2}g_{\rm},
where everything is expressed in the CGS units.
For very high temperatures there are relativistic corrections to this formula, that is, additional terms of the order of
k_{\rm}T_{e/m}_{e}c^{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. It is a type of radiative cooling. The energy carried away by bremsstrahlung is called bremsstrahlung losses and represents a type of radiative losses. One generally uses the term bremsstrahlung losses in the context when the plasma cooling is undesired, as e.g. in fusion plasmas.
Polarizational bremsstrahlung (sometimes referred to as "atomic bremsstrahlung") is the radiation emitted by the target's atomic electrons as the target atom is polarized by the Coulomb field of the incident charged particle.^{[9]} ^{[10]} Polarizational bremsstrahlung contributions to the total bremsstrahlung spectrum have been observed in experiments involving relatively massive incident particles,^{[11]} resonance processes,^{[12]} and free atoms.^{[13]} However, there is still some debate as to whether or not there are significant polarizational bremsstrahlung contributions in experiments involving fast electrons incident on solid targets.^{[14]} ^{[15]}
It is worth noting that the term "polarizational" is not meant to imply that the emitted bremsstrahlung is polarized. Also, the angular distribution of polarizational bremsstrahlung is theoretically quite different than ordinary bremsstrahlung.^{[16]}
See main article: Xray tube. In an Xray tube, electrons are accelerated in a vacuum by an electric field towards a piece of metal called the "target". Xrays are emitted as the electrons slow down (decelerate) in the metal. The output spectrum consists of a continuous spectrum of Xrays, with additional sharp peaks at certain energies. The continuous spectrum is due to bremsstrahlung, while the sharp peaks are characteristic Xrays associated with the atoms in the target. For this reason, bremsstrahlung in this context is also called continuous Xrays.^{[17]}
The shape of this continuum spectrum is approximately described by Kramers' law.
The formula for Kramers' law is usually given as the distribution of intensity (photon count)
I
λ
I(λ)dλ=K\left(
λ  
λ_{min} 
1\right)
1  
λ^{2} 
dλ
The constant K is proportional to the atomic number of the target element, and
λ_{min}
The spectrum has a sharp cutoff at
λ_{min}
λ_{min}=
hc  
eV 
≈
1239.8  
V 
pm/kV
See main article: Beta decay. Beta particleemitting substances sometimes exhibit a weak radiation with continuous spectrum that is due to bremsstrahlung (see the "outer bremsstrahlung" below). In this context, bremsstrahlung is a type of "secondary radiation", in that it is produced as a result of stopping (or slowing) the primary radiation (beta particles). It is very similar to Xrays produced by bombarding metal targets with electrons in Xray generators (as above) except that it is produced by highspeed electrons from beta radiation.
The "inner" bremsstrahlung (also known as "internal bremsstrahlung") arises from the creation of the electron and its loss of energy (due to the strong electric field in the region of the nucleus undergoing decay) as it leaves the nucleus. Such radiation is a feature of beta decay in nuclei, but it is occasionally (less commonly) seen in the beta decay of free neutrons to protons, where it is created as the beta electron leaves the proton.
In electron and positron emission by beta decay the photon's energy comes from the electronnucleon 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, decreasing to zero electromagnetic energy at normal neutrino energy. Note that in the case of electron capture, bremsstrahlung is emitted even though no charged particle is emitted. Instead, the bremsstrahlung radiation may be thought of as being created as the captured electron is accelerated toward being absorbed. Such radiation may be at frequencies that are the same as soft gamma radiation, but it exhibits none of the sharp spectral lines of gamma decay, and thus is not technically gamma radiation.
The internal process is to be contrasted with the "outer" bremsstrahlung due to the impingement on the nucleus of electrons coming from the outside (i.e., emitted by another nucleus), as discussed above.^{[20]}
In some cases, e.g., the bremsstrahlung produced by shielding the beta 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;^{[21]} as the atomic number is lower for these materials, the intensity of bremsstrahlung is significantly reduced, but a larger thickness of shielding is required to stop the electrons (beta radiation).
The dominant luminous component in a cluster of galaxies is the 10^{7} to 10^{8} kelvin intracluster medium. The emission from the intracluster medium is characterized by thermal bremsstrahlung. This radiation is in the energy range of Xrays and can be easily observed with spacebased telescopes such as Chandra Xray Observatory, XMMNewton, ROSAT, ASCA, EXOSAT, Suzaku, RHESSI and future missions like IXO https://web.archive.org/web/20080303062108/http://constellation.gsfc.nasa.gov/ and AstroH https://web.archive.org/web/20071112015825/http://www.astro.isas.ac.jp/future/NeXT/.
Bremsstrahlung is also the dominant emission mechanism for H II regions at radio wavelengths.
In electric discharges, for example as laboratory discharges between two electrodes or as lightning discharges between cloud and ground or within clouds, electrons produce Bremsstrahlung photons while scattering off air molecules. These photons become manifest in terrestrial gammaray flashes and are the source for beams of electrons, positrons, neutrons and protons.^{[22]} The appearance of Bremsstrahlung photons also influences the propagation and morphology of discharges in nitrogenoxygen mixtures with low percentages of oxygen.^{[23]}
The complete quantum mechanical description was first performed by Bethe and Heitler.^{[24]} They assumed plane waves for electrons which scatter at the nucleus of an atom, and derived a cross section which relates the complete geometry of that process to the frequency of the emitted photon. The quadruply differential cross section which shows a quantum mechanical symmetry to pair production, is:
\begin{align} d^{4\sigma}={}&
 
(2\pi)^{2} 
\leftp_{f\right}  
\leftp_{i\right} 
d\omega  
\omega 
d\Omega_{i}d\Omega_{f}d\Phi  
\leftq\right^{4} 
\\ &{} x \left[
 

2  
\left(4E  
i 
c^{2q}^{2\right)}+
 

2  
\left(4E  
f 
c^{2q}^{2\right) }\right.\\ &{}+2\hbar^{2\omega}^{2}
 
(E_{f}c\leftp_{f\right}\cos\Theta_{f)\left(E}_{i}c\leftp_{i\right}\cos\Theta_{i\right)} 
\\ &{}2\left.
\leftp_{i\right}\leftp_{f\right}\sin\Theta_{i}\sin\Theta_{f}\cos\Phi  
\left(E_{f}c\leftp_{f\right}\cos\Theta_{f\right)\left(E}_{i}c\leftp_{i\right}c1\cos\Theta_{i\right)} 
2  
\left(2E  
i 
+
2  
2E  
f 
c^{2q}^{2\right) }\right]. \end{align}
There
Z
\alpha_{fine ≈ }1/137
\hbar
c
E_{kin,i/f}
E_{i,f}
p_{i,f}
E_{i,}=E_{kin,}+m_{e}c^{2}=
2  
\sqrt{m  
e 
c^{4}+
2  
p  
i,f 
c^{2}, }
where
m_{e}
E_{f}=E_{i}\hbar\omega,
where
\hbar\omega
\begin{align} \Theta_{i}&=\sphericalangle(p_{i,}k),\\ \Theta_{f}&=\sphericalangle(p_{f,}k),\\ \Phi&=Anglebetweentheplanes(p_{i,}k)and(p_{f,}k), \end{align}
where
k
The differentials are given as
\begin{align} d\Omega_{i}&=\sin\Theta_{i d\Theta}_{i,\\ }d\Omega_{f}&=\sin\Theta_{f d\Theta}_{f. \end{align}}
The absolute value of the virtual photon between the nucleus and electron is
\begin{align} q^{2}={}&
2  
\leftp  
i\right 

2  
\leftp  
f\right 
\left(
\hbar  
c 
\omega\right)^{2 }+
2\leftp  

\omega\cos\Theta_{i }
2\leftp  

\omega\cos\Theta_{f}\\ &{}+2\leftp_{i\right}\leftp_{f\right }\left(\cos\Theta_{f\cos\Theta}_{i}+\sin\Theta_{f\sin\Theta}_{i\cos\Phi\right). \end{align}}
The range of validity is given by the Born approximation
v\gg
Zc  
137 
where this relation has to be fulfilled for the velocity
v
For practical applications (e.g. in Monte Carlo codes) it can be interesting to focus on the relation between the frequency
\omega
\Phi
\Theta_{f}
d^{2\sigma}(E_{i,}\omega,\Theta_{i)}  
d\omegad\Omega_{i} 
=
6  
\sum\limits  
j=1 
I_{j }
with
\begin{align} I_{1}={}&
2\piA  

and
\begin{align} A&=
 
(2\pi)^{2} 
\leftp_{f\right}  
\leftp_{i\right} 
\hbar^{2}  
\omega 
\\ \Delta_{1}&=
2  
p  
i 

2  
p  
f 
\left(
\hbar  
c 
\omega\right)^{2}+2
\hbar  
c 
\omega\leftp_{i\right\cos\Theta}_{i,}\\ \Delta_{2}&=2
\hbar  
c 
\omega\leftp_{f\right}+2\leftp_{i\right\leftp}_{f\right\cos\Theta}_{i. \end{align}}
However, a much simpler expression for the same integral can be found in ^{[26]} (Eq. 2BN) and in ^{[27]} (Eq. 4.1).
An analysis of the doubly differential cross section above shows that electrons whose kinetic energy is larger than the rest energy (511 keV) emit photons in forward direction while electrons with a small energy emit photons isotropically.
One mechanism, considered important for small atomic numbers
Z
Z
Z^{2}