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Bremsstrahlung

From Wikipedia, the free encyclopedia
Electromagnetic radiation due to deceleration of charged particles
Bremsstrahlung produced by a high-energy electron deflected in the electric field of an atomic nucleus

Inparticle physics,bremsstrahlung (/ˈbrɛmʃtrɑːləŋ/;[1]German:[ˈbʁɛms.ʃtʁaːlʊŋ]ⓘ; from German bremsen 'to brake' and Strahlung 'radiation') iselectromagnetic radiation produced by thedeceleration of acharged particle when deflected by another charged particle, typically anelectron by anatomic nucleus. The moving particle loseskinetic energy, which is converted into radiation (i.e.,photons), thus satisfying thelaw of conservation of energy. The term is also used to refer to the process of producing the radiation. Bremsstrahlung has acontinuous 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 orbraking radiation is any radiation produced due to the acceleration (positive or negative) of a charged particle. This includessynchrotron radiation (i.e., photon emission by arelativistic particle),cyclotron radiation (i.e. photon emission by a non-relativistic particle), and the emission of electrons andpositrons duringbeta decay. However, the term is frequently used in the more narrow sense of radiation produced when electrons (from whatever source) decelerate in matter.

Bremsstrahlung emitted fromplasma is sometimes referred to asfree–free radiation – that is, created by electrons that arefree (i.e., not in an atomic or molecularbound state) before, and remain free after, the emission of a photon. In the same parlance,bound–bound radiation refers to discretespectral lines (an electron "jumps" between two bound states), whilefree–bound radiation refers to theradiative combination process, in which a free electronrecombines with an ion.

This article uses SI units, along with the scaled single-particle chargeq¯≡q/(4πϵ0)1/2{\displaystyle {\bar {q}}\equiv q/(4\pi \epsilon _{0})^{1/2}}.

Classical description

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Main article:Larmor formula
Field lines and modulus of the electric field generated by a (negative) charge first moving at a constant speed and then stopping quickly to show the generated Bremsstrahlung radiation.

Ifquantum effects are negligible, an accelerating charged particle radiates power as described by theLarmor formula and its relativistic generalization.

Total radiated power

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The total radiated power is[2]P=2q¯2γ43c(β˙2+(β⋅β˙)21−β2),{\displaystyle P={\frac {2{\bar {q}}^{2}\gamma ^{4}}{3c}}\left({\dot {\beta }}^{2}+{\frac {\left({\boldsymbol {\beta }}\cdot {\dot {\boldsymbol {\beta }}}\right)^{2}}{1-\beta ^{2}}}\right),}whereβ=vc{\textstyle {\boldsymbol {\beta }}={\frac {\mathbf {v} }{c}}} (the velocity of the particle divided by the speed of light),γ=1/1−β2{\textstyle \gamma ={1}/{\sqrt {1-\beta ^{2}}}} is theLorentz factor,ε0{\displaystyle \varepsilon _{0}} is thevacuum permittivity,β˙{\displaystyle {\dot {\boldsymbol {\beta }}}} signifies a time derivative ofβ{\displaystyle {\boldsymbol {\beta }}}, andq is the charge of the particle.In the case where velocity is parallel to acceleration (i.e., linear motion), the expression reduces to[3]Pa∥v=2q¯2a2γ63c3,{\displaystyle P_{a\parallel v}={\frac {2{\bar {q}}^{2}a^{2}\gamma ^{6}}{3c^{3}}},}wherea≡v˙=β˙c{\displaystyle a\equiv {\dot {v}}={\dot {\beta }}c} is the acceleration. For the case of acceleration perpendicular to the velocity (β⋅β˙=0{\displaystyle {\boldsymbol {\beta }}\cdot {\dot {\boldsymbol {\beta }}}=0}), for example insynchrotrons, the total power isPa⊥v=2q¯2a2γ43c3.{\displaystyle P_{a\perp v}={\frac {2{\bar {q}}^{2}a^{2}\gamma ^{4}}{3c^{3}}}.}

Power radiated in the two limiting cases is proportional toγ4{\displaystyle \gamma ^{4}}(a⊥v){\displaystyle \left(a\perp v\right)} orγ6{\displaystyle \gamma ^{6}}(a∥v){\displaystyle \left(a\parallel v\right)}. SinceE=γmc2{\displaystyle E=\gamma mc^{2}}, we see that for particles with the same energyE{\displaystyle E} the total radiated power goes asm−4{\displaystyle m^{-4}} orm−6{\displaystyle m^{-6}}, which accounts for why electrons lose energy to bremsstrahlung radiation much more rapidly than heavier charged particles (e.g., muons, protons, alpha particles). This is the reason a TeV energy electron-positron collider (such as the proposedInternational Linear Collider) cannot use a circular tunnel (requiring constant acceleration), while a proton-proton collider (such as theLarge Hadron Collider) can utilize a circular tunnel. The electrons lose energy due to bremsstrahlung at a rate(mp/me)4≈1013{\displaystyle (m_{\text{p}}/m_{\text{e}})^{4}\approx 10^{13}} times higher than protons do.

Angular distribution

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The most general formula for radiated power as a function of angle is:[4]dPdΩ=q¯24πc|n^×((n^−β)×β˙)|2(1−n^⋅β)5{\displaystyle {\frac {dP}{d\Omega }}={\frac {{\bar {q}}^{2}}{4\pi c}}{\frac {\left|{\hat {\mathbf {n} }}\times \left(\left({\hat {\mathbf {n} }}-{\boldsymbol {\beta }}\right)\times {\dot {\boldsymbol {\beta }}}\right)\right|^{2}}{\left(1-{\hat {\mathbf {n} }}\cdot {\boldsymbol {\beta }}\right)^{5}}}}wheren^{\displaystyle {\hat {\mathbf {n} }}} is aunit vector pointing from the particle towards the observer, anddΩ{\displaystyle d\Omega } is an infinitesimalsolid angle.

In the case where velocity is parallel to acceleration (for example, linear motion), this simplifies to[4]dPa∥vdΩ=q¯2a24πc3sin2⁡θ(1−βcos⁡θ)5{\displaystyle {\frac {dP_{a\parallel v}}{d\Omega }}={\frac {{\bar {q}}^{2}a^{2}}{4\pi c^{3}}}{\frac {\sin ^{2}\theta }{(1-\beta \cos \theta )^{5}}}}whereθ{\displaystyle \theta } is the angle betweenβ{\displaystyle {\boldsymbol {\beta }}} and the direction of observationn^{\displaystyle {\hat {\mathbf {n} }}}.

Simplified quantum-mechanical description

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The full quantum-mechanical treatment of bremsstrahlung is very involved. The "vacuum case" of the interaction of one electron, one ion, and one photon, using the pure Coulomb potential, has an exact solution that was probably first published byArnold Sommerfeld in 1931.[5] This analytical solution involves complicated mathematics, and several numerical calculations have been published, such as by Karzas and Latter.[6] Other approximate formulas have been presented, such as in recent work by Weinberg[7] and Pradler and Semmelrock.[8]

This section gives a quantum-mechanical analog of the prior section, but with some simplifications to illustrate the important physics. We give a non-relativistic treatment of the special case of an electron of massme{\displaystyle m_{\text{e}}}, charge−e{\displaystyle -e}, and initial speedv{\displaystyle v} decelerating in the Coulomb field of a gas of heavy ions of chargeZe{\displaystyle Ze} and number densityni{\displaystyle n_{i}}. The emitted radiation is a photon of frequencyν=c/λ{\displaystyle \nu =c/\lambda } and energyhν{\displaystyle h\nu }. We wish to find the emissivityj(v,ν){\displaystyle j(v,\nu )} which is the power emitted per (solid angle in photon velocity space * photon frequency), summed over both transverse photon polarizations. We express it as an approximate classical result times the free−free emissionGaunt factorgff accounting for quantum and other corrections:j(v,ν)=8π33Z2e¯6nic3me2vgff(v,ν){\displaystyle j(v,\nu )={8\pi \over 3{\sqrt {3}}}{Z^{2}{\bar {e}}^{6}n_{i} \over c^{3}m_{\text{e}}^{2}v}g_{\rm {ff}}(v,\nu )}j(ν,v)=0{\displaystyle j(\nu ,v)=0} ifhν>mv2/2{\displaystyle h\nu >mv^{2}/2}, that is, the electron does not have enough kinetic energy to emit the photon. A general, quantum-mechanical formula forgff{\displaystyle g_{\rm {ff}}} exists but is very complicated, and usually is found by numerical calculations. We present some approximate results with the following additional assumptions:

With these assumptions, two unitless parameters characterize the process:ηZ≡Ze¯2/ℏv{\displaystyle \eta _{Z}\equiv Z{\bar {e}}^{2}/\hbar v}, which measures the strength of the electron-ion Coulomb interaction, andην≡hν/2mev2{\displaystyle \eta _{\nu }\equiv h\nu /2m_{\text{e}}v^{2}}, which measures the photon "softness" and we assume is always small (the choice of the factor 2 is for later convenience). In the limitηZ≪1{\displaystyle \eta _{Z}\ll 1}, the quantum-mechanicalBorn approximation gives:gff,Born=3πln⁡1ην{\displaystyle g_{\text{ff,Born}}={{\sqrt {3}} \over \pi }\ln {1 \over \eta _{\nu }}}

In the opposite limitηZ≫1{\displaystyle \eta _{Z}\gg 1}, the full quantum-mechanical result reduces to the purely classical resultgff,class=3π[ln⁡(1ηZην)−γ]{\displaystyle g_{\text{ff,class}}={{\sqrt {3}} \over \pi }\left[\ln \left({1 \over \eta _{Z}\eta _{\nu }}\right)-\gamma \right]}whereγ≈0.577{\displaystyle \gamma \approx 0.577} is theEuler–Mascheroni constant. Note that1/ηZην=mev3/πZe¯2ν{\displaystyle 1/\eta _{Z}\eta _{\nu }=m_{\text{e}}v^{3}/\pi Z{\bar {e}}^{2}\nu } which is a purely classical expression without the Planck constanth{\displaystyle h}.

A semi-classical, heuristic way to understand the Gaunt factor is to write it asgff≈ln⁡(bmax/bmin){\displaystyle g_{\text{ff}}\approx \ln(b_{\text{max}}/b_{\text{min}})} wherebmax{\displaystyle b_{\max }} andbmin{\displaystyle b_{\min }} are maximum and minimum "impact parameters" for the electron-ion collision, in the presence of the photon electric field. With our assumptions,bmax=v/ν{\displaystyle b_{\rm {max}}=v/\nu }: for larger impact parameters, the sinusoidal oscillation of the photon field provides "phase mixing" that strongly reduces the interaction.bmin{\displaystyle b_{\rm {min}}} is the larger of the quantum-mechanical de Broglie wavelength≈h/mev{\displaystyle \approx h/m_{\text{e}}v} and the classical distance of closest approach≈e¯2/mev2{\displaystyle \approx {\bar {e}}^{2}/m_{\text{e}}v^{2}} where the electron-ion Coulomb potential energy is comparable to the electron's initial kinetic energy.

The above approximations generally apply as long as the argument of the logarithm is large, and break down when it is less than unity. Namely, these forms for the Gaunt factor become negative, which is unphysical. A rough approximation to the full calculations, with the appropriate Born and classical limits, isgff≈max[1,3πln⁡[1ηνmax(1,eγηZ)]]{\displaystyle g_{\text{ff}}\approx \max \left[1,{{\sqrt {3}} \over \pi }\ln \left[{1 \over \eta _{\nu }\max(1,e^{\gamma }\eta _{Z})}\right]\right]}

Thermal bremsstrahlung in a medium: emission and absorption

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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:1c∂tIν+n^⋅∇Iν=jν−kνIν{\displaystyle {\frac {1}{c}}\partial _{t}I_{\nu }+{\hat {\mathbf {n} }}\cdot \nabla I_{\nu }=j_{\nu }-k_{\nu }I_{\nu }}

Iν(t,x){\displaystyle I_{\nu }(t,\mathbf {x} )} is the radiation spectral intensity, or power per (area ×solid angle in photon velocity space× photon frequency) summed over both polarizations.jν{\displaystyle j_{\nu }} is the emissivity, analogous toj(v,ν){\displaystyle j(v,\nu )}defined above, andkν{\displaystyle k_{\nu }} is the absorptivity.jν{\displaystyle j_{\nu }} andkν{\displaystyle k_{\nu }} are properties of the matter, not the radiation, and account for all the particles in the medium – not just a pair of one electron and one ion as in the prior section. IfIν{\displaystyle I_{\nu }} is uniform in space and time, then the left-hand side of the transfer equation is zero, and we findIν=jνkν{\displaystyle I_{\nu }={j_{\nu } \over k_{\nu }}}

If the matter and radiation are also in thermal equilibrium at some temperature, thenIν{\displaystyle I_{\nu }} must be theblackbody spectrum:Bν(ν,Te)=2hν3c21ehν/kBTe−1{\displaystyle B_{\nu }(\nu ,T_{\text{e}})={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{h\nu /k_{\text{B}}T_{\text{e}}}-1}}}Sincejν{\displaystyle j_{\nu }} andkν{\displaystyle k_{\nu }} are independent ofIν{\displaystyle I_{\nu }}, this means thatjν/kν{\displaystyle j_{\nu }/k_{\nu }} must be the blackbody spectrum whenever the matter is in equilibrium at some temperature – regardless of the state of the radiation. This allows us to immediately know bothjν{\displaystyle j_{\nu }} andkν{\displaystyle k_{\nu }} once one is known – for matter in equilibrium.

In plasma: approximate classical results

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NOTE: this section currently gives formulas that apply in the Rayleigh–Jeans limitℏω≪kBTe{\displaystyle \hbar \omega \ll k_{\text{B}}T_{\text{e}}}, and does not use a quantized (Planck) treatment of radiation. Thus a usual factor likeexp⁡(−ℏω/kBTe){\displaystyle \exp(-\hbar \omega /k_{\rm {B}}T_{\text{e}})} does not appear. The appearance ofℏω/kBTe{\displaystyle \hbar \omega /k_{\text{B}}T_{\text{e}}} iny{\displaystyle y} below is due to the quantum-mechanical treatment of collisions.

Bekefi's classical result for the bremsstrahlung emission power spectrum from a Maxwellian electron distribution. It rapidly decreases for largeω{\displaystyle \omega }, and is also suppressed nearω=ωp{\displaystyle \omega =\omega _{\rm {p}}}. This plot is for the quantum caseTe>Z2Eh{\displaystyle T_{\text{e}}>Z^{2}E_{\text{h}}}, andℏωp/Te=0.1{\displaystyle \hbar \omega _{\text{p}}/T_{\text{e}}=0.1}. The blue curve is the full formula withE1(y){\displaystyle E_{1}(y)}, the red curve is the approximate logarithmic form fory≪1{\displaystyle y\ll 1}.

In aplasma, 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,[9] while a simplified one is given by Ichimaru.[10] In this section we follow Bekefi's dielectric treatment, with collisions included approximately via the cutoff wavenumber,kmax{\displaystyle k_{\text{max}}}.

Consider a uniform plasma, with thermal electrons distributed according to theMaxwell–Boltzmann distribution with the temperatureTe{\displaystyle T_{\text{e}}}. Following Bekefi, the power spectral density (power per angular frequency interval per volume, integrated over the whole4π{\displaystyle 4\pi }sr of solid angle and summed over the polarizations) of the bremsstrahlung radiated, is calculated to bedPBrdω=823πe¯6(mec2)3/2[1−ωp2ω2]1/2Zi2nine(kBTe)1/2E1(y),{\displaystyle {dP_{\mathrm {Br} } \over d\omega }={\frac {8{\sqrt {2}}}{3{\sqrt {\pi }}}}{{\bar {e}}^{6} \over (m_{\text{e}}c^{2})^{3/2}}\left[1-{\omega _{\rm {p}}^{2} \over \omega ^{2}}\right]^{1/2}{Z_{i}^{2}n_{i}n_{\text{e}} \over (k_{\rm {B}}T_{\text{e}})^{1/2}}E_{1}(y),}whereωp≡(nee2/ε0me)1/2{\displaystyle \omega _{p}\equiv (n_{\text{e}}e^{2}/\varepsilon _{0}m_{\text{e}})^{1/2}} is the electron plasma frequency,ω{\displaystyle \omega } is the photon frequency,ne,ni{\displaystyle n_{\text{e}},n_{i}} is the number density of electrons and ions, and other symbols arephysical constants. The second bracketed factor is the index of refraction of a light wave in a plasma, and shows that emission is greatly suppressed forω<ωp{\displaystyle \omega <\omega _{\rm {p}}} (this is the cutoff condition for a light wave in a plasma; in this case the light wave isevanescent). This formula thus only applies forω>ωp{\displaystyle \omega >\omega _{\rm {p}}}. This formula should be summed over ion species in a multi-species plasma.

The functionE1{\displaystyle E_{1}} is theexponential integral, and the unitless quantityy{\displaystyle y} isy=12ω2mekmax2kBTe{\displaystyle y={\frac {1}{2}}{\omega ^{2}m_{\text{e}} \over k_{\text{max}}^{2}k_{\text{B}}T_{\text{e}}}}

kmax{\displaystyle k_{\text{max}}} is a maximum or cutoff wavenumber, arising due to binary collisions, and can vary with ion species. Roughly,kmax=1/λB{\displaystyle k_{\text{max}}=1/\lambda _{\text{B}}} whenkBTe>Zi2Eh{\displaystyle k_{\text{B}}T_{\text{e}}>Z_{i}^{2}E_{\text{h}}} (typical in plasmas that are not too cold), whereEh≈27.2{\displaystyle E_{\text{h}}\approx 27.2} eV is theHartree energy, andλB=ℏ/(mekBTe)1/2{\displaystyle \lambda _{\text{B}}=\hbar /(m_{\text{e}}k_{\text{B}}T_{\text{e}})^{1/2}}[clarification needed] is the electronthermal de Broglie wavelength. Otherwise,kmax∝1/lC{\displaystyle k_{\text{max}}\propto 1/l_{\text{C}}} wherelC{\displaystyle l_{\text{C}}} is the classical Coulomb distance of closest approach.

For the usual casekm=1/λB{\displaystyle k_{m}=1/\lambda _{B}}, we findy=12[ℏωkBTe]2.{\displaystyle y={\frac {1}{2}}\left[{\frac {\hbar \omega }{k_{\text{B}}T_{\text{e}}}}\right]^{2}.}

The formula fordPBr/dω{\displaystyle dP_{\mathrm {Br} }/d\omega } is approximate, in that it neglects enhanced emission occurring forω{\displaystyle \omega } slightly aboveωp{\displaystyle \omega _{\text{p}}}.

In the limity≪1{\displaystyle y\ll 1}, we can approximateE1{\displaystyle E_{1}} asE1(y)≈−ln⁡[yeγ]+O(y){\displaystyle E_{1}(y)\approx -\ln[ye^{\gamma }]+O(y)} whereγ≈0.577{\displaystyle \gamma \approx 0.577} is theEuler–Mascheroni constant. The leading, logarithmic term is frequently used, and resembles the Coulomb logarithm that occurs in other collisional plasma calculations. Fory>e−γ{\displaystyle y>e^{-\gamma }} the log term is negative, and the approximation is clearly inadequate. Bekefi gives corrected expressions for the logarithmic term that match detailed binary-collision calculations.

The total emission power density, integrated over all frequencies, isPBr=∫ωp∞dωdPBrdω=163e¯6me2c3Zi2ninekmaxG(yp)G(yp)=12π∫yp∞dyy−1/2[1−ypy]1/2E1(y)yp=y(ω=ωp){\displaystyle {\begin{aligned}P_{\mathrm {Br} }&=\int _{\omega _{\text{p}}}^{\infty }d\omega {\frac {dP_{\mathrm {Br} }}{d\omega }}={\frac {16}{3}}{\frac {{\bar {e}}^{6}}{m_{\text{e}}^{2}c^{3}}}Z_{i}^{2}n_{i}n_{\text{e}}k_{\text{max}}G(y_{\text{p}})\\[1ex]G(y_{p})&={\frac {1}{2{\sqrt {\pi }}}}\int _{y_{\text{p}}}^{\infty }dy\,y^{-{1}/{2}}\left[1-{y_{\text{p}} \over y}\right]^{1/2}E_{1}(y)\\[1ex]y_{\text{p}}&=y({\omega \!=\!\omega _{\text{p}}})\end{aligned}}}

G(yp=0)=1{\displaystyle G(y_{\text{p}}=0)=1} and decreases withyp{\displaystyle y_{\text{p}}}; it is always positive. Forkmax=1/λB{\displaystyle k_{\text{max}}=1/\lambda _{\text{B}}}, we find

PBr=163e¯6(mec2)32ℏZi2nine(kBTe)12G(yp){\displaystyle P_{\mathrm {Br} }={16 \over 3}{{\bar {e}}^{6} \over (m_{\text{e}}c^{2})^{\frac {3}{2}}\hbar }Z_{i}^{2}n_{i}n_{\text{e}}(k_{\rm {B}}T_{\text{e}})^{\frac {1}{2}}G(y_{\rm {p}})}

Note the appearance ofℏ{\displaystyle \hbar } due to the quantum nature ofλB{\displaystyle \lambda _{\rm {B}}}. In practical units, a commonly used version of this formula forG=1{\displaystyle G=1} is[11]PBr[W/m3]=Zi2nine[7.69×1018m−3]2Te[eV]12.{\displaystyle P_{\mathrm {Br} }[\mathrm {W/m^{3}} ]={Z_{i}^{2}n_{i}n_{\text{e}} \over \left[7.69\times 10^{18}\mathrm {m^{-3}} \right]^{2}}T_{\text{e}}[\mathrm {eV} ]^{\frac {1}{2}}.}

This formula is 1.59 times the one given above, with the difference due to details of binary collisions. Such ambiguity is often expressed by introducingGaunt factorgB{\displaystyle g_{\rm {B}}}, e.g. in[12] one findsεff=1.4×10−27T12neniZ2gB,{\displaystyle \varepsilon _{\text{ff}}=1.4\times 10^{-27}T^{\frac {1}{2}}n_{\text{e}}n_{i}Z^{2}g_{\text{B}},\,}where everything is expressed in theCGS units.

Relativistic corrections

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Relativistic corrections to the emission of a 30 keV photon by an electron impacting on a proton.

For very high temperatures there are relativistic corrections to this formula, that is, additional terms of the order ofkBTe/mec2{\displaystyle k_{\text{B}}T_{\text{e}}/m_{\text{e}}c^{2}}.[13]

Bremsstrahlung cooling

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If the plasma isoptically thin, the bremsstrahlung radiation leaves the plasma, carrying part of the internal plasma energy. This effect is known as thebremsstrahlung cooling. It is a type ofradiative cooling. The energy carried away by bremsstrahlung is calledbremsstrahlung losses and represents a type ofradiative losses. One generally uses the termbremsstrahlung losses in the context when the plasma cooling is undesired, as e.g. infusion plasmas.

Polarizational bremsstrahlung

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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.[14][15] Polarizational bremsstrahlung contributions to the total bremsstrahlung spectrum have been observed in experiments involving relatively massive incident particles,[16] resonance processes,[17] and free atoms.[18] 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.[19][20][21]

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.[22]

Sources

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X-ray tube

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Spectrum of the X-rays emitted by anX-ray tube with arhodium target, operated at 60kV. The continuous curve is due to bremsstrahlung, and the spikes arecharacteristic K lines for rhodium. The curve goes to zero at 21pm in agreement with theDuane–Hunt law, as described in the text.
Main article:X-ray tube

In anX-ray tube, electrons are accelerated in a vacuum by anelectric field towards a piece of material called the "target". X-rays are emitted as the electrons hit the target.

Already in the early 20th century physicists found out that X-rays consist of two components, one independent of the target material and another with characteristics offluorescence.[23] Now we say that the output spectrum consists of a continuous spectrum of X-rays with additional sharp peaks at certain energies. The former is due to bremsstrahlung, while the latter arecharacteristic X-rays associated with the atoms in the target. For this reason, bremsstrahlung in this context is also calledcontinuous X-rays.[24] The German term itself was introduced in 1909 byArnold Sommerfeld in order to explain the nature of the first variety of X-rays.[23]

The shape of this continuum spectrum is approximately described byKramers' law.

The formula for Kramers' law is usually given as the distribution of intensity (photon count)I{\displaystyle I} against thewavelengthλ{\displaystyle \lambda } of the emitted radiation:[25]I(λ)dλ=K(λλmin−1)dλλ2{\displaystyle I(\lambda )\,d\lambda =K\left({\frac {\lambda }{\lambda _{\min }}}-1\right){\frac {d\lambda }{\lambda ^{2}}}}

The constantK is proportional to theatomic number of the target element, andλmin{\displaystyle \lambda _{\min }} is the minimum wavelength given by theDuane–Hunt law.

The spectrum has a sharp cutoff atλmin{\displaystyle \lambda _{\min }}, which is due to the limited energy of the incoming electrons. For example, if an electron in the tube is accelerated through 60kV, then it will acquire a kinetic energy of 60keV, and when it strikes the target it can create X-rays with energy of at most 60 keV, byconservation of energy. (This upper limit corresponds to the electron coming to a stop by emitting just one X-rayphoton. Usually the electron emits many photons, and each has an energy less than 60 keV.) A photon with energy of at most 60 keV has wavelength of at least21 pm, so the continuous X-ray spectrum has exactly that cutoff, as seen in the graph. More generally the formula for the low-wavelength cutoff, the Duane–Hunt law, is:[26]λmin=hceV≈1239.8Vpm⋅kV{\displaystyle \lambda _{\min }={\frac {hc}{eV}}\approx {\frac {1239.8}{V}}\,\mathrm {pm{\cdot }kV} }whereh is thePlanck constant,c is thespeed of light,V is thevoltage that the electrons are accelerated through,e is theelementary charge,pm ispicometre, and in the rightmost expression the voltage is in units of kilovolts (kV).

Beta decay

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Main article:Beta decay

Beta particle-emitting 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 X-rays produced by bombarding metal targets with electrons inX-ray generators (as above) except that it is produced by high-speed electrons from beta radiation.

Inner and outer bremsstrahlung

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The "inner" bremsstrahlung (also known as "internal bremsstrahlung") arises from the creation of the electron and its loss of energy (due to the strongelectric 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 andpositron emission by beta decay 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 theneutrino, 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 softgamma radiation, but it exhibits none of the sharp spectral lines ofgamma 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.[27]

Radiation safety

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In some cases, such as the decay of32
P, the bremsstrahlung produced byshielding 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, such asPlexiglas (Lucite),plastic,wood, orwater;[28] 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).

In astrophysics

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The dominant luminous component in a cluster of galaxies is the 107 to 108 kelvinintracluster medium. The emission from the intracluster medium is characterized by thermal bremsstrahlung. This radiation is in the energy range of X-rays and can be easily observed with space-based telescopes such asChandra X-ray Observatory,XMM-Newton,ROSAT,ASCA,EXOSAT,Suzaku,RHESSI and future missions likeIXO[1] and Astro-H[2].

Bremsstrahlung is also the dominant emission mechanism forH II regions at radio wavelengths.

In electric discharges

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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 interrestrial gamma-ray flashes and are the source for beams of electrons, positrons, neutrons and protons.[29] The appearance of Bremsstrahlung photons also influences the propagation and morphology of discharges in nitrogen–oxygen mixtures with low percentages of oxygen.[30]

Quantum mechanical description

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The complete quantum mechanical description was first performed by Bethe and Heitler.[31] 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 topair production, is

d4σ=Z2αfine3ℏ2(2π)2|pf||pi|dωωdΩidΩfdΦ|q|4×[pf2sin2⁡Θf(Ef−c|pf|cos⁡Θf)2(4Ei2−c2q2)+pi2sin2⁡Θi(Ei−c|pi|cos⁡Θi)2(4Ef2−c2q2)+2ℏ2ω2pi2sin2⁡Θi+pf2sin2⁡Θf(Ef−c|pf|cos⁡Θf)(Ei−c|pi|cos⁡Θi)−2|pi||pf|sin⁡Θisin⁡Θfcos⁡Φ(Ef−c|pf|cos⁡Θf)(Ei−c|pi|cos⁡Θi)(2Ei2+2Ef2−c2q2)],{\displaystyle {\begin{aligned}d^{4}\sigma ={}&{\frac {Z^{2}\alpha _{\text{fine}}^{3}\hbar ^{2}}{(2\pi )^{2}}}{\frac {\left|\mathbf {p} _{f}\right|}{\left|\mathbf {p} _{i}\right|}}{\frac {d\omega }{\omega }}{\frac {d\Omega _{i}\,d\Omega _{f}\,d\Phi }{\left|\mathbf {q} \right|^{4}}}\\&{}\times \left[{\frac {\mathbf {p} _{f}^{2}\sin ^{2}\Theta _{f}}{\left(E_{f}-c\left|\mathbf {p} _{f}\right|\cos \Theta _{f}\right)^{2}}}\left(4E_{i}^{2}-c^{2}\mathbf {q} ^{2}\right)+{\frac {\mathbf {p} _{i}^{2}\sin ^{2}\Theta _{i}}{\left(E_{i}-c\left|\mathbf {p} _{i}\right|\cos \Theta _{i}\right)^{2}}}\left(4E_{f}^{2}-c^{2}\mathbf {q} ^{2}\right)\right.\\&{}\qquad +2\hbar ^{2}\omega ^{2}{\frac {\mathbf {p} _{i}^{2}\sin ^{2}\Theta _{i}+\mathbf {p} _{f}^{2}\sin ^{2}\Theta _{f}}{(E_{f}-c\left|\mathbf {p} _{f}\right|\cos \Theta _{f})\left(E_{i}-c\left|\mathbf {p} _{i}\right|\cos \Theta _{i}\right)}}\\&{}\qquad -2\left.{\frac {\left|\mathbf {p} _{i}\right|\left|\mathbf {p} _{f}\right|\sin \Theta _{i}\sin \Theta _{f}\cos \Phi }{\left(E_{f}-c\left|\mathbf {p} _{f}\right|\cos \Theta _{f}\right)\left(E_{i}-c\left|\mathbf {p} _{i}\right|\cos \Theta _{i}\right)}}\left(2E_{i}^{2}+2E_{f}^{2}-c^{2}\mathbf {q} ^{2}\right)\right],\end{aligned}}}

whereZ{\displaystyle Z} is theatomic number,αfine≈1/137{\displaystyle \alpha _{\text{fine}}\approx 1/137} thefine-structure constant,ℏ{\displaystyle \hbar } thereduced Planck constant andc{\displaystyle c} thespeed of light. The kinetic energyEkin,i/f{\displaystyle E_{{\text{kin}},i/f}} of the electron in the initial and final state is connected to its total energyEi,f{\displaystyle E_{i,f}} or itsmomentapi,f{\displaystyle \mathbf {p} _{i,f}} viaEi,f=Ekin,i/f+mec2=me2c4+pi,f2c2,{\displaystyle E_{i,f}=E_{{\text{kin}},i/f}+m_{\text{e}}c^{2}={\sqrt {m_{\text{e}}^{2}c^{4}+\mathbf {p} _{i,f}^{2}c^{2}}},}whereme{\displaystyle m_{\text{e}}} is themass of an electron.Conservation of energy givesEf=Ei−ℏω,{\displaystyle E_{f}=E_{i}-\hbar \omega ,}whereℏω{\displaystyle \hbar \omega } is the photon energy. The directions of the emitted photon and the scattered electron are given byΘi=∢(pi,k),Θf=∢(pf,k),Φ=Angle between the planes (pi,k) and (pf,k),{\displaystyle {\begin{aligned}\Theta _{i}&=\sphericalangle (\mathbf {p} _{i},\mathbf {k} ),\\\Theta _{f}&=\sphericalangle (\mathbf {p} _{f},\mathbf {k} ),\\\Phi &={\text{Angle between the planes }}(\mathbf {p} _{i},\mathbf {k} ){\text{ and }}(\mathbf {p} _{f},\mathbf {k} ),\end{aligned}}}wherek{\displaystyle \mathbf {k} } is the momentum of the photon.

The differentials are given asdΩi=sin⁡Θi dΘi,dΩf=sin⁡Θf dΘf.{\displaystyle {\begin{aligned}d\Omega _{i}&=\sin \Theta _{i}\ d\Theta _{i},\\d\Omega _{f}&=\sin \Theta _{f}\ d\Theta _{f}.\end{aligned}}}

Theabsolute value of thevirtual photon between the nucleus and electron is

−q2=−|pi|2−|pf|2−(ℏcω)2+2|pi|ℏcωcos⁡Θi−2|pf|ℏcωcos⁡Θf+2|pi||pf|(cos⁡Θfcos⁡Θi+sin⁡Θfsin⁡Θicos⁡Φ).{\displaystyle {\begin{aligned}-\mathbf {q} ^{2}={}&-\left|\mathbf {p} _{i}\right|^{2}-\left|\mathbf {p} _{f}\right|^{2}-\left({\frac {\hbar }{c}}\omega \right)^{2}+2\left|\mathbf {p} _{i}\right|{\frac {\hbar }{c}}\omega \cos \Theta _{i}-2\left|\mathbf {p} _{f}\right|{\frac {\hbar }{c}}\omega \cos \Theta _{f}\\&{}+2\left|\mathbf {p} _{i}\right|\left|\mathbf {p} _{f}\right|\left(\cos \Theta _{f}\cos \Theta _{i}+\sin \Theta _{f}\sin \Theta _{i}\cos \Phi \right).\end{aligned}}}

The range of validity is given by the Born approximationv≫Zc137{\displaystyle v\gg {\frac {Zc}{137}}}where this relation has to be fulfilled for the velocityv{\displaystyle v} of the electron in the initial and final state.

For practical applications (e.g. inMonte Carlo codes) it can be interesting to focus on the relation between the frequencyω{\displaystyle \omega } of the emitted photon and the angle between this photon and the incident electron. Köhn andEbert integrated the quadruply differential cross section by Bethe and Heitler overΦ{\displaystyle \Phi } andΘf{\displaystyle \Theta _{f}} and obtained:[32]d2σ(Ei,ω,Θi)dωdΩi=∑j=16Ij{\displaystyle {\frac {d^{2}\sigma (E_{i},\omega ,\Theta _{i})}{d\omega \,d\Omega _{i}}}=\sum \limits _{j=1}^{6}I_{j}}with

I1=2πAΔ22+4pi2pf2sin2⁡Θiln⁡(Δ22+4pi2pf2sin2⁡Θi−Δ22+4pi2pf2sin2⁡Θi(Δ1+Δ2)+Δ1Δ2−Δ22−4pi2pf2sin2⁡Θi−Δ22+4pi2pf2sin2⁡Θi(Δ1−Δ2)+Δ1Δ2)×[1+cΔ2pf(Ei−cpicos⁡Θi)−pi2c2sin2⁡Θi(Ei−cpicos⁡Θi)2−2ℏ2ω2pfΔ2c(Ei−cpicos⁡Θi)(Δ22+4pi2pf2sin2⁡Θi)],I2=−2πAcpf(Ei−cpicos⁡Θi)ln⁡(Ef+pfcEf−pfc),I3=2πA(Δ2Ef+Δ1pfc)4+4m2c4pi2pf2sin2⁡Θi×ln⁡[([Ef+pfc][4pi2pf2sin2⁡Θi(Ef−pfc)+(Δ1+Δ2)([Δ2Ef+Δ1pfc]−[Δ2Ef+Δ1pfc]2+4m2c4pi2pf2sin2⁡Θi)])[(Ef−pfc)(4pi2pf2sin2⁡Θi[−Ef−pfc]+(Δ1−Δ2)([Δ2Ef+Δ1pfc]−(Δ2Ef+Δ1pfc)2+4m2c4pi2pf2sin2⁡Θi])]−1×[−(Δ22+4pi2pf2sin2⁡Θi)(Ef3+Efpf2c2)+pfc(2[Δ12−4pi2pf2sin2⁡Θi]Efpfc+Δ1Δ2[3Ef2+pf2c2])(Δ2Ef+Δ1pfc)2+4m2c4pi2pf2sin2⁡Θi−c(Δ2Ef+Δ1pfc)pf(Ei−cpicos⁡Θi)−4Ei2pf2(2[Δ2Ef+Δ1pfc]2−4m2c4pi2pf2sin2⁡Θi)(Δ1Ef+Δ2pfc)([Δ2Ef+Δ1pfc]2+4m2c4pi2pf2sin2⁡Θi)2+8pi2pf2m2c4sin2⁡Θi(Ei2+Ef2)−2ℏ2ω2pi2sin2⁡Θipfc(Δ2Ef+Δ1pfc)+2ℏ2ω2pfm2c3(Δ2Ef+Δ1pfc)(Ei−cpicos⁡Θi)([Δ2Ef+Δ1pfc]2+4m2c4pi2pf2sin2⁡Θi)],I4=−4πApfc(Δ2Ef+Δ1pfc)(Δ2Ef+Δ1pfc)2+4m2c4pi2pf2sin2⁡Θi−16πEi2pf2A(Δ2Ef+Δ1pfc)2([Δ2Ef+Δ1pfc]2+4m2c4pi2pf2sin2⁡Θi)2,I5=4πA(−Δ22+Δ12−4pi2pf2sin2⁡Θi)([Δ2Ef+Δ1pfc]2+4m2c4pi2pf2sin2⁡Θi)×[ℏ2ω2pf2Ei−cpicos⁡Θi×Ef(2Δ22[Δ22−Δ12]+8pi2pf2sin2⁡Θi[Δ22+Δ12])+pfc(2Δ1Δ2[Δ22−Δ12]+16Δ1Δ2pi2pf2sin2⁡Θi)Δ22+4pi2pf2sin2⁡Θi+2ℏ2ω2pi2sin2⁡Θi(2Δ1Δ2pfc+2Δ22Ef+8pi2pf2sin2⁡ΘiEf)Ei−cpicos⁡Θi+2Ei2pf2(2[Δ22−Δ12][Δ2Ef+Δ1pfc]2+8pi2pf2sin2⁡Θi[(Δ12+Δ22)(Ef2+pf2c2)+4Δ1Δ2Efpfc])(Δ2Ef+Δ1pfc)2+4m2c4pi2pf2sin2⁡Θi+8pi2pf2sin2⁡Θi(Ei2+Ef2)(Δ2pfc+Δ1Ef)Ei−cpicos⁡Θi],I6=16πEf2pi2sin2⁡ΘiA(Ei−cpicos⁡Θi)2(−Δ22+Δ12−4pi2pf2sin2⁡Θi),{\displaystyle {\begin{aligned}I_{1}={}&{\frac {2\pi A}{\sqrt {\Delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}}\ln \left({\frac {\Delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}-{\sqrt {\Delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}\left(\Delta _{1}+\Delta _{2}\right)+\Delta _{1}\Delta _{2}}{-\Delta _{2}^{2}-4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}-{\sqrt {\Delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}\left(\Delta _{1}-\Delta _{2}\right)+\Delta _{1}\Delta _{2}}}\right)\\&{}\times \left[1+{\frac {c\Delta _{2}}{p_{f}\left(E_{i}-cp_{i}\cos \Theta _{i}\right)}}-{\frac {p_{i}^{2}c^{2}\sin ^{2}\Theta _{i}}{\left(E_{i}-cp_{i}\cos \Theta _{i}\right)^{2}}}-{\frac {2\hbar ^{2}\omega ^{2}p_{f}\Delta _{2}}{c\left(E_{i}-cp_{i}\cos \Theta _{i}\right)\left(\Delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\right)}}\right],\\I_{2}={}&-{\frac {2\pi Ac}{p_{f}\left(E_{i}-cp_{i}\cos \Theta _{i}\right)}}\ln \left({\frac {E_{f}+p_{f}c}{E_{f}-p_{f}c}}\right),\\I_{3}={}&{\frac {2\pi A}{\sqrt {\left(\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right)^{4}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}}\times \ln \left[\left(\left[E_{f}+p_{f}c\right]\right.\right.\\&\left.\left[4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\left(E_{f}-p_{f}c\right)+\left(\Delta _{1}+\Delta _{2}\right)\left(\left[\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right]-{\sqrt {\left[\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right]^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}\right)\right]\right)\\&\left[\left(E_{f}-p_{f}c\right)\left(4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\left[-E_{f}-p_{f}c\right]\right.\right.\\&{}+\left.\left.\left(\Delta _{1}-\Delta _{2}\right)\left(\left[\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right]-{\sqrt {\left(\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}\right]\right)\right]^{-1}\\&{}\times \left[-{\frac {\left(\Delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\right)\left(E_{f}^{3}+E_{f}p_{f}^{2}c^{2}\right)+p_{f}c\left(2\left[\Delta _{1}^{2}-4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\right]E_{f}p_{f}c+\Delta _{1}\Delta _{2}\left[3E_{f}^{2}+p_{f}^{2}c^{2}\right]\right)}{\left(\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}\right.\\&{}-{\frac {c\left(\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right)}{p_{f}\left(E_{i}-cp_{i}\cos \Theta _{i}\right)}}-{\frac {4E_{i}^{2}p_{f}^{2}\left(2\left[\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right]^{2}-4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\right)\left(\Delta _{1}E_{f}+\Delta _{2}p_{f}c\right)}{\left(\left[\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right]^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\right)^{2}}}\\&{}+\left.{\frac {8p_{i}^{2}p_{f}^{2}m^{2}c^{4}\sin ^{2}\Theta _{i}\left(E_{i}^{2}+E_{f}^{2}\right)-2\hbar ^{2}\omega ^{2}p_{i}^{2}\sin ^{2}\Theta _{i}p_{f}c\left(\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right)+2\hbar ^{2}\omega ^{2}p_{f}m^{2}c^{3}\left(\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right)}{\left(E_{i}-cp_{i}\cos \Theta _{i}\right)\left(\left[\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right]^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\right)}}\right],\\I_{4}={}&{}-{\frac {4\pi Ap_{f}c\left(\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right)}{\left(\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}-{\frac {16\pi E_{i}^{2}p_{f}^{2}A\left(\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right)^{2}}{\left(\left[\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right]^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\right)^{2}}},\\I_{5}={}&{\frac {4\pi A}{\left(-\Delta _{2}^{2}+\Delta _{1}^{2}-4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\right)\left(\left[\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right]^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\right)}}\\&{}\times \left[{\frac {\hbar ^{2}\omega ^{2}p_{f}^{2}}{E_{i}-cp_{i}\cos \Theta _{i}}}\right.\\&{}\times {\frac {E_{f}\left(2\Delta _{2}^{2}\left[\Delta _{2}^{2}-\Delta _{1}^{2}\right]+8p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\left[\Delta _{2}^{2}+\Delta _{1}^{2}\right]\right)+p_{f}c\left(2\Delta _{1}\Delta _{2}\left[\Delta _{2}^{2}-\Delta _{1}^{2}\right]+16\Delta _{1}\Delta _{2}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\right)}{\Delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}\\&{}+{\frac {2\hbar ^{2}\omega ^{2}p_{i}^{2}\sin ^{2}\Theta _{i}\left(2\Delta _{1}\Delta _{2}p_{f}c+2\Delta _{2}^{2}E_{f}+8p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}E_{f}\right)}{E_{i}-cp_{i}\cos \Theta _{i}}}\\&{}+{\frac {2E_{i}^{2}p_{f}^{2}\left(2\left[\Delta _{2}^{2}-\Delta _{1}^{2}\right]\left[\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right]^{2}+8p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\left[\left(\Delta _{1}^{2}+\Delta _{2}^{2}\right)\left(E_{f}^{2}+p_{f}^{2}c^{2}\right)+4\Delta _{1}\Delta _{2}E_{f}p_{f}c\right]\right)}{\left(\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}\\&{}+\left.{\frac {8p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\left(E_{i}^{2}+E_{f}^{2}\right)\left(\Delta _{2}p_{f}c+\Delta _{1}E_{f}\right)}{E_{i}-cp_{i}\cos \Theta _{i}}}\right],\\I_{6}={}&{\frac {16\pi E_{f}^{2}p_{i}^{2}\sin ^{2}\Theta _{i}A}{\left(E_{i}-cp_{i}\cos \Theta _{i}\right)^{2}\left(-\Delta _{2}^{2}+\Delta _{1}^{2}-4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\right)}},\end{aligned}}}

and

A=Z2αfine3(2π)2|pf||pi|ℏ2ωΔ1=−pi2−pf2−(ℏcω)2+2ℏcω|pi|cos⁡Θi,Δ2=−2ℏcω|pf|+2|pi||pf|cos⁡Θi.{\displaystyle {\begin{aligned}A&={\frac {Z^{2}\alpha _{\text{fine}}^{3}}{(2\pi )^{2}}}{\frac {\left|\mathbf {p} _{f}\right|}{\left|\mathbf {p} _{i}\right|}}{\frac {\hbar ^{2}}{\omega }}\\\Delta _{1}&=-\mathbf {p} _{i}^{2}-\mathbf {p} _{f}^{2}-\left({\frac {\hbar }{c}}\omega \right)^{2}+2{\frac {\hbar }{c}}\omega \left|\mathbf {p} _{i}\right|\cos \Theta _{i},\\\Delta _{2}&=-2{\frac {\hbar }{c}}\omega \left|\mathbf {p} _{f}\right|+2\left|\mathbf {p} _{i}\right|\left|\mathbf {p} _{f}\right|\cos \Theta _{i}.\end{aligned}}}

However, a much simpler expression for the same integral can be found in[33] (Eq. 2BN) and in[34] (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.

Electron–electron bremsstrahlung

[edit]

One mechanism, considered important for small atomic numbersZ{\displaystyle Z}, is the scattering of a free electron at the shell electrons of an atom or molecule.[35] Since electron–electron bremsstrahlung is a function ofZ{\displaystyle Z} and the usual electron-nucleus bremsstrahlung is a function ofZ2{\displaystyle Z^{2}}, electron–electron bremsstrahlung is negligible for metals. For air, however, it plays an important role in the production ofterrestrial gamma-ray flashes.[36]

See also

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References

[edit]
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