Bremsstrahlung - Wikipedia

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 ... Bremsstrahlung FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Typeofelectromagneticradiation Bremsstrahlungproducedbyahigh-energyelectrondeflectedintheelectricfieldofanatomicnucleus. Bremsstrahlung/ˈbrɛmʃtrɑːləŋ/[1](Germanpronunciation:[ˈbʁɛms.ʃtʁaːlʊŋ](listen)),frombremsen"tobrake"andStrahlung"radiation";i.e.,"brakingradiation"or"decelerationradiation",iselectromagneticradiationproducedbythedecelerationofachargedparticlewhendeflectedbyanotherchargedparticle,typicallyanelectronbyanatomicnucleus.Themovingparticleloseskineticenergy,whichisconvertedintoradiation(i.e.,photons),thussatisfyingthelawofconservationofenergy.Thetermisalsousedtorefertotheprocessofproducingtheradiation.Bremsstrahlunghasacontinuousspectrum,whichbecomesmoreintenseandwhosepeakintensityshiftstowardhigherfrequenciesasthechangeoftheenergyofthedeceleratedparticlesincreases. Broadlyspeaking,bremsstrahlungorbrakingradiationisanyradiationproducedduetothedeceleration(negativeacceleration)ofachargedparticle,whichincludessynchrotronradiation(i.e.photonemissionbyarelativisticparticle),cyclotronradiation(i.e.photonemissionbyanon-relativisticparticle),andtheemissionofelectronsandpositronsduringbetadecay.However,thetermisfrequentlyusedinthemorenarrowsenseofradiationfromelectrons(fromwhateversource)slowinginmatter. Bremsstrahlungemittedfromplasmaissometimesreferredtoasfree–freeradiation.Thisreferstothefactthattheradiationinthiscaseiscreatedbyelectronsthatarefree(i.e.,notinanatomicormolecularboundstate)before,andremainfreeafter,theemissionofaphoton.Inthesameparlance,bound–boundradiationreferstodiscretespectrallines(anelectron"jumps"betweentwoboundstates),whilefree–boundone—totheradiativecombinationprocess,inwhichafreeelectronrecombineswithanion. Contents 1Classicaldescription 1.1Totalradiatedpower 1.2Angulardistribution 2Simplifiedquantumdescription 3Thermalbremsstrahlung:emissionandabsorption 4Inplasma 4.1Relativisticcorrections 4.2Bremsstrahlungcooling 5Polarizationalbremsstrahlung 6Sources 6.1X-raytube 6.2Betadecay 6.2.1Innerandouterbremsstrahlung 6.2.2Radiationsafety 6.3Inastrophysics 6.4Inelectricdischarges 7Quantummechanicaldescription 8Electron–electronbremsstrahlung 9Seealso 10References 11Furtherreading 12Externallinks Classicaldescription[edit] Mainarticle:Larmorformula Fieldlinesandmodulusoftheelectricfieldgeneratedbya(negative)chargefirstmovingataconstantspeedandthenstoppingquicklytoshowthegeneratedBremsstrahlungradiation. Ifquantumeffectsarenegligible,anacceleratingchargedparticleradiatespowerasdescribedbytheLarmorformulaanditsrelativisticgeneralization. Totalradiatedpower[edit] Thetotalradiatedpoweris[2] P = q 2 γ 4 6 π ε 0 c ( β ˙ 2 + ( β ⋅ β ˙ ) 2 1 − β 2 ) , {\displaystyleP={\frac{q^{2}\gamma^{4}}{6\pi\varepsilon_{0}c}}\left({\dot{\beta}}^{2}+{\frac{\left({\boldsymbol{\beta}}\cdot{\dot{\boldsymbol{\beta}}}\right)^{2}}{1-\beta^{2}}}\right),} where β = v c {\textstyle{\boldsymbol{\beta}}={\frac{\mathbf{v}}{c}}} (thevelocityoftheparticledividedbythespeedoflight), γ = 1 1 − β 2 {\textstyle\gamma={\frac{1}{\sqrt{1-\beta^{2}}}}} istheLorentzfactor, β ˙ {\displaystyle{\dot{\boldsymbol{\beta}}}} signifiesatimederivativeof β {\displaystyle{\boldsymbol{\beta}}} ,andqisthechargeoftheparticle. Inthecasewherevelocityisparalleltoacceleration(i.e.,linearmotion),theexpressionreducesto[3] P a ∥ v = q 2 a 2 γ 6 6 π ε 0 c 3 , {\displaystyleP_{a\parallelv}={\frac{q^{2}a^{2}\gamma^{6}}{6\pi\varepsilon_{0}c^{3}}},} where a ≡ v ˙ = β ˙ c {\displaystylea\equiv{\dot{v}}={\dot{\beta}}c} istheacceleration.Forthecaseofaccelerationperpendiculartothevelocity( β ⋅ β ˙ = 0 {\displaystyle{\boldsymbol{\beta}}\cdot{\dot{\boldsymbol{\beta}}}=0} ),forexampleinsynchrotrons,thetotalpoweris P a ⊥ v = q 2 a 2 γ 4 6 π ε 0 c 3 . {\displaystyleP_{a\perpv}={\frac{q^{2}a^{2}\gamma^{4}}{6\pi\varepsilon_{0}c^{3}}}.} Powerradiatedinthetwolimitingcasesisproportionalto γ 4 {\displaystyle\gamma^{4}} ( a ⊥ v ) {\displaystyle\left(a\perpv\right)} or γ 6 {\displaystyle\gamma^{6}} ( a ∥ v ) {\displaystyle\left(a\parallelv\right)} .Since E = γ m c 2 {\displaystyleE=\gammamc^{2}} ,weseethatforparticleswiththesameenergy E {\displaystyleE} thetotalradiatedpowergoesas m − 4 {\displaystylem^{-4}} or m − 6 {\displaystylem^{-6}} ,whichaccountsforwhyelectronsloseenergytobremsstrahlungradiationmuchmorerapidlythanheavierchargedparticles(e.g.,muons,protons,alphaparticles).ThisisthereasonaTeVenergyelectron-positroncollider(suchastheproposedInternationalLinearCollider)cannotuseacirculartunnel(requiringconstantacceleration),whileaproton-protoncollider(suchastheLargeHadronCollider)canutilizeacirculartunnel.Theelectronsloseenergyduetobremsstrahlungatarate ( m p / m e ) 4 ≈ 10 13 {\displaystyle(m_{p}/m_{e})^{4}\approx10^{13}} timeshigherthanprotonsdo. Angulardistribution[edit] Themostgeneralformulaforradiatedpowerasafunctionofangleis:[4] d P d Ω = q 2 16 π 2 ε 0 c | n ^ × ( ( n ^ − β ) × β ˙ ) | 2 ( 1 − n ^ ⋅ β ) 5 {\displaystyle{\frac{dP}{d\Omega}}={\frac{q^{2}}{16\pi^{2}\varepsilon_{0}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}}}} where n ^ {\displaystyle{\hat{\mathbf{n}}}} isaunitvectorpointingfromtheparticletowardstheobserver,and d Ω {\displaystyled\Omega} isaninfinitesimalbitofsolidangle. Inthecasewherevelocityisparalleltoacceleration(forexample,linearmotion),thissimplifiesto[4] d P a ∥ v d Ω = q 2 a 2 16 π 2 ε 0 c 3 sin 2 ⁡ θ ( 1 − β cos ⁡ θ ) 5 {\displaystyle{\frac{dP_{a\parallelv}}{d\Omega}}={\frac{q^{2}a^{2}}{16\pi^{2}\varepsilon_{0}c^{3}}}{\frac{\sin^{2}\theta}{(1-\beta\cos\theta)^{5}}}} where θ {\displaystyle\theta} istheanglebetween a {\displaystyle\mathbf{a}} andthedirectionofobservation. Simplifiedquantumdescription[edit] Thissectiongivesaquantum-mechanicalanalogofthepriorsection,butwithsomesimplifications.Wegiveanon-relativistictreatmentofthespecialcaseofanelectronofmass m e {\displaystylem_{e}} ,charge − e {\displaystyle-e} ,andinitialspeed v {\displaystylev} deceleratingintheCoulombfieldofagasofheavyionsofcharge Z e {\displaystyleZe} andnumberdensity n i {\displaystylen_{i}} .Theemittedradiationisaphotonoffrequency ν = c / λ {\displaystyle\nu=c/\lambda} andenergy h ν {\displaystyleh\nu} .Wewishtofindtheemissivity j ( v , ν ) {\displaystylej(v,\nu)} whichisthepoweremittedper(solidangleinphotonvelocityspace*photonfrequency),summedoverbothtransversephotonpolarizations.Wefollowthecommonastrophysicalpracticeofwritingthisresultintermsofanapproximateclassicalresulttimesthefree-freeemissionGauntfactorgffwhichincorporatesquantumandothercorrections: j ( v , ν ) = 8 π 3 3 ( e 2 4 π ϵ 0 ) 3 Z 2 n i c 3 m e 2 v g f f ( v , ν ) {\displaystylej(v,\nu)={8\pi\over3{\sqrt{3}}}\left({e^{2}\over4\pi\epsilon_{0}}\right)^{3}{Z^{2}n_{i}\overc^{3}m_{e}^{2}v}g_{\rm{ff}}(v,\nu)} Ageneral,quantum-mechanicalformulafor g f f {\displaystyleg_{\rm{ff}}} existsbutisverycomplicated,andusuallyisfoundbynumericalcalculations.Wepresentsomeapproximateresultswiththefollowingadditionalassumptions: Vacuuminteraction:weneglectanyeffectsofthebackgroundmedium,suchasplasmascreeningeffects.Thisisreasonableforphotonfrequencymuchgreaterthantheplasmafrequency ν p e ∝ n e 1 / 2 {\displaystyle\nu_{\rm{pe}}\propton_{\rm{e}}^{1/2}} with n e {\displaystylen_{e}} theplasmaelectrondensity.Notethatlightwavesareevanescentfor ν < ν p e {\displaystyle\nu Z 2 E h {\displaystyleT_{e}>Z^{2}E_{\rm{h}}} ,and ℏ ω p / T e = 0.1 {\displaystyle\hbar\omega_{\rm{p}}/T_{e}=0.1} . Thissectiondiscussesbremsstrahlungemissionandtheinverseabsorptionprocess(calledinversebremsstrahlung)inamacroscopicmedium.Westartwiththeequationofradiativetransfer,whichappliestogeneralprocessesandnotjustbremsstrahlung: 1 c ∂ t I ν + n ^ ⋅ ∇ I ν = j ν − k ν I ν {\displaystyle{\frac{1}{c}}\partial_{t}I_{\nu}+{\hat{\mathbf{n}}}\cdot\nablaI_{\nu}=j_{\nu}-k_{\nu}I_{\nu}} I ν ( t , x ) {\displaystyleI_{\nu}(t,\mathbf{x})} istheradiationspectralintensity,orpowerper(area*solidangleinphotonvelocityspace*photonfrequency)summedoverbothpolarizations. j ν {\displaystylej_{\nu}} istheemissivity,analogousto j ( v , ν ) {\displaystylej(v,\nu)} definedabove,and k ν {\displaystylek_{\nu}} istheabsorptivity. j ν {\displaystylej_{\nu}} and k ν {\displaystylek_{\nu}} arepropertiesofthematter,nottheradiation,andaccountforalltheparticlesinthemedium-notjustapairofoneelectronandoneionasinthepriorsection.If I ν {\displaystyleI_{\nu}} isuniforminspaceandtime,thentheleft-handsideofthetransferequationiszero,andwefind I ν = j ν k ν {\displaystyleI_{\nu}={j_{\nu}\overk_{\nu}}} Ifthematterandradiationarealsointhermalequilibriumatsometemperature,then I ν {\displaystyleI_{\nu}} mustbetheblackbodyspectrum: B ν ( ν , T e ) = 2 h ν 3 c 2 1 e h ν / k B T e − 1 {\displaystyleB_{\nu}(\nu,T_{e})={\frac{2h\nu^{3}}{c^{2}}}{\frac{1}{e^{h\nu/k_{\rm{B}}T_{e}}-1}}} Since j ν {\displaystylej_{\nu}} and k ν {\displaystylek_{\nu}} areindependentof I ν {\displaystyleI_{\nu}} ,thismeansthat j ν / k ν {\displaystylej_{\nu}/k_{\nu}} mustbetheblackbodyspectrumwheneverthematterisinequilibriumatsometemperature–regardlessofthestateoftheradiation.Thisallowsustoimmediatelyknowboth j ν {\displaystylej_{\nu}} and k ν {\displaystylek_{\nu}} onceoneisknown–formatterinequilibrium. Inplasma[edit] NOTE:thissectioncurrentlygivesformulasthatapplyintheRayleigh-Jeanslimit ℏ ω ≪ k B T e {\displaystyle\hbar\omega\llk_{\rm{B}}T_{e}} ,anddoesnotuseaquantized(Planck)treatmentofradiation.Thusausualfactorlike exp ⁡ ( − ℏ ω / k B T e ) {\displaystyle\exp(-\hbar\omega/k_{\rm{B}}T_{e})} doesnotappear.Theappearanceof ℏ ω / k B T e {\displaystyle\hbar\omega/k_{\rm{B}}T_{e}} in y {\displaystyley} belowisduetothequantum-mechanicaltreatmentofcollisions. Inaplasma,thefreeelectronscontinuallycollidewiththeions,producingbremsstrahlung.AcompleteanalysisrequiresaccountingforbothbinaryCoulombcollisionsaswellascollective(dielectric)behavior.AdetailedtreatmentisgivenbyBekefi,[5]whileasimplifiedoneisgivenbyIchimaru.[6]InthissectionwefollowBekefi'sdielectrictreatment,withcollisionsincludedapproximatelyviathecutoffwavenumber, k m a x {\displaystylek_{\rm{max}}} . Considerauniformplasma,withthermalelectronsdistributedaccordingtotheMaxwell–Boltzmanndistributionwiththetemperature T e {\displaystyleT_{e}} .FollowingBekefi,thepowerspectraldensity(powerperangularfrequencyintervalpervolume,integratedoverthewhole 4 π {\displaystyle4\pi} srofsolidangle,andinbothpolarizations)ofthebremsstrahlungradiated,iscalculatedtobe d P B r d ω = 8 2 3 π [ e 2 4 π ε 0 ] 3 1 ( m e c 2 ) 3 / 2 [ 1 − ω p 2 ω 2 ] 1 / 2 Z i 2 n i n e ( k B T e ) 1 / 2 E 1 ( y ) , {\displaystyle{dP_{\mathrm{Br}}\overd\omega}={8{\sqrt{2}}\over3{\sqrt{\pi}}}\left[{e^{2}\over4\pi\varepsilon_{0}}\right]^{3}{1\over(m_{e}c^{2})^{3/2}}\left[1-{\omega_{\rm{p}}^{2}\over\omega^{2}}\right]^{1/2}{Z_{i}^{2}n_{i}n_{e}\over(k_{\rm{B}}T_{e})^{1/2}}E_{1}(y),} where ω p ≡ ( n e e 2 / ε 0 m e ) 1 / 2 {\displaystyle\omega_{p}\equiv(n_{e}e^{2}/\varepsilon_{0}m_{e})^{1/2}} istheelectronplasmafrequency, ω {\displaystyle\omega} isthephotonfrequency, n e , n i {\displaystylen_{e},n_{i}} isthenumberdensityofelectronsandions,andothersymbolsarephysicalconstants.Thesecondbracketedfactoristheindexofrefractionofalightwaveinaplasma,andshowsthatemissionisgreatlysuppressedfor ω < ω p {\displaystyle\omega ω p {\displaystyle\omega>\omega_{\rm{p}}} .Thisformulashouldbesummedoverionspeciesinamulti-speciesplasma. Thespecialfunction E 1 {\displaystyleE_{1}} isdefinedintheexponentialintegralarticle,andtheunitlessquantity y {\displaystyley} is y = 1 2 ω 2 m e k m a x 2 k B T e {\displaystyley={1\over2}{\omega^{2}m_{e}\overk_{\rm{max}}^{2}k_{\rm{B}}T_{e}}} k m a x {\displaystylek_{\rm{max}}} isamaximumorcutoffwavenumber,arisingduetobinarycollisions,andcanvarywithionspecies.Roughly, k m a x = 1 / λ B {\displaystylek_{\rm{max}}=1/\lambda_{\rm{B}}} when k B T e > Z i 2 E h {\displaystylek_{\rm{B}}T_{\rm{e}}>Z_{i}^{2}E_{\rm{h}}} (typicalinplasmasthatarenottoocold),where E h ≈ 27.2 {\displaystyleE_{\rm{h}}\approx27.2} eVistheHartreeenergy,and λ B = ℏ / ( m e k B T e ) 1 / 2 {\displaystyle\lambda_{\rm{B}}=\hbar/(m_{\rm{e}}k_{\rm{B}}T_{\rm{e}})^{1/2}} [clarificationneeded]istheelectronthermaldeBrogliewavelength.Otherwise, k m a x ∝ 1 / l C {\displaystylek_{\rm{max}}\propto1/l_{\rm{C}}} where l C {\displaystylel_{\rm{C}}} istheclassicalCoulombdistanceofclosestapproach. Fortheusualcase k m = 1 / λ B {\displaystylek_{m}=1/\lambda_{B}} ,wefind y = 1 2 [ ℏ ω k B T e ] 2 . {\displaystyley={1\over2}\left[{\frac{\hbar\omega}{k_{\rm{B}}T_{e}}}\right]^{2}.} Theformulafor d P B r / d ω {\displaystyledP_{\mathrm{Br}}/d\omega} isapproximate,inthatitneglectsenhancedemissionoccurringfor ω {\displaystyle\omega} slightlyabove ω p {\displaystyle\omega_{\rm{p}}} . Inthelimit y ≪ 1 {\displaystyley\ll1} ,wecanapproximate E 1 {\displaystyleE_{1}} as E 1 ( y ) ≈ − ln ⁡ [ y e γ ] + O ( y ) {\displaystyleE_{1}(y)\approx-\ln[ye^{\gamma}]+O(y)} where γ ≈ 0.577 {\displaystyle\gamma\approx0.577} istheEuler–Mascheroniconstant.Theleading,logarithmictermisfrequentlyused,andresemblestheCoulomblogarithmthatoccursinothercollisionalplasmacalculations.For y > e − γ {\displaystyley>e^{-\gamma}} thelogtermisnegative,andtheapproximationisclearlyinadequate.Bekefigivescorrectedexpressionsforthelogarithmictermthatmatchdetailedbinary-collisioncalculations. Thetotalemissionpowerdensity,integratedoverallfrequencies,is P B r = ∫ ω p ∞ d ω d P B r d ω = 16 3 [ e 2 4 π ε 0 ] 3 1 m e 2 c 3 Z i 2 n i n e k m a x G ( y p ) G ( y p ) = 1 2 π ∫ y p ∞ d y y − 1 2 [ 1 − y p y ] 1 2 E 1 ( y ) y p = y ( ω = ω p ) {\displaystyle{\begin{aligned}P_{\mathrm{Br}}&=\int_{\omega_{\rm{p}}}^{\infty}d\omega{dP_{\mathrm{Br}}\overd\omega}={16\over3}\left[{e^{2}\over4\pi\varepsilon_{0}}\right]^{3}{1\overm_{e}^{2}c^{3}}Z_{i}^{2}n_{i}n_{e}k_{\rm{max}}G(y_{\rm{p}})\\G(y_{p})&={1\over2{\sqrt{\pi}}}\int_{y_{\rm{p}}}^{\infty}dy\,y^{-{\frac{1}{2}}}\left[1-{y_{\rm{p}}\overy}\right]^{\frac{1}{2}}E_{1}(y)\\y_{\rm{p}}&=y(\omega=\omega_{\rm{p}})\end{aligned}}} G ( y p = 0 ) = 1 {\displaystyleG(y_{\rm{p}}=0)=1} anddecreaseswith y p {\displaystyley_{\rm{p}}} ;itisalwayspositive.For k m a x = 1 / λ B {\displaystylek_{\rm{max}}=1/\lambda_{\rm{B}}} ,wefind P B r = 16 3 ( e 2 4 π ε 0 ) 3 ( m e c 2 ) 3 2 ℏ Z i 2 n i n e ( k B T e ) 1 2 G ( y p ) {\displaystyleP_{\mathrm{Br}}={16\over3}{\left({\frac{e^{2}}{4\pi\varepsilon_{0}}}\right)^{3}\over(m_{e}c^{2})^{\frac{3}{2}}\hbar}Z_{i}^{2}n_{i}n_{e}(k_{\rm{B}}T_{e})^{\frac{1}{2}}G(y_{\rm{p}})} Notetheappearanceof ℏ {\displaystyle\hbar} duetothequantumnatureof λ B {\displaystyle\lambda_{\rm{B}}} .Inpracticalunits,acommonlyusedversionofthisformulafor G = 1 {\displaystyleG=1} is[7] P B r [ W / m 3 ] = Z i 2 n i n e [ 7.69 × 10 18 m − 3 ] 2 T e [ eV ] 1 2 . {\displaystyleP_{\mathrm{Br}}[{\textrm{W}}/{\textrm{m}}^{3}]={Z_{i}^{2}n_{i}n_{e}\over\left[7.69\times10^{18}{\textrm{m}}^{-3}\right]^{2}}T_{e}[{\textrm{eV}}]^{\frac{1}{2}}.} Thisformulais1.59timestheonegivenabove,withthedifferenceduetodetailsofbinarycollisions.SuchambiguityisoftenexpressedbyintroducingGauntfactor g B {\displaystyleg_{\rm{B}}} ,e.g.in[8]onefinds ε f f = 1.4 × 10 − 27 T 1 2 n e n i Z 2 g B , {\displaystyle\varepsilon_{\mathrm{ff}}=1.4\times10^{-27}T^{\frac{1}{2}}n_{e}n_{i}Z^{2}g_{\rm{B}},\,} whereeverythingisexpressedintheCGSunits. Relativisticcorrections[edit] Relativisticcorrectionstotheemissionofa30-keVphotonbyanelectronimpactingonaproton. Forveryhightemperaturestherearerelativisticcorrectionstothisformula,thatis,additionaltermsoftheorderof k B T e / m e c 2 . {\displaystylek_{\rm{B}}T_{e}/m_{e}c^{2}\,.} [9] Bremsstrahlungcooling[edit] Iftheplasmaisopticallythin,thebremsstrahlungradiationleavestheplasma,carryingpartoftheinternalplasmaenergy.Thiseffectisknownasthebremsstrahlungcooling.Itisatypeofradiativecooling.Theenergycarriedawaybybremsstrahlungiscalledbremsstrahlunglossesandrepresentsatypeofradiativelosses.Onegenerallyusesthetermbremsstrahlunglossesinthecontextwhentheplasmacoolingisundesired,ase.g.infusionplasmas. Polarizationalbremsstrahlung[edit] Polarizationalbremsstrahlung(sometimesreferredtoas"atomicbremsstrahlung")istheradiationemittedbythetarget'satomicelectronsasthetargetatomispolarizedbytheCoulombfieldoftheincidentchargedparticle.[10][11]Polarizationalbremsstrahlungcontributionstothetotalbremsstrahlungspectrumhavebeenobservedinexperimentsinvolvingrelativelymassiveincidentparticles,[12]resonanceprocesses,[13]andfreeatoms.[14]However,thereisstillsomedebateastowhetherornottherearesignificantpolarizationalbremsstrahlungcontributionsinexperimentsinvolvingfastelectronsincidentonsolidtargets.[15][16] Itisworthnotingthattheterm"polarizational"isnotmeanttoimplythattheemittedbremsstrahlungispolarized.Also,theangulardistributionofpolarizationalbremsstrahlungistheoreticallyquitedifferentthanordinarybremsstrahlung.[17] Sources[edit] X-raytube[edit] SpectrumoftheX-raysemittedbyanX-raytubewitharhodiumtarget,operatedat60kV.Thecontinuouscurveisduetobremsstrahlung,andthespikesarecharacteristicKlinesforrhodium.Thecurvegoestozeroat21pminagreementwiththeDuane–Huntlaw,asdescribedinthetext. Mainarticle:X-raytube InanX-raytube,electronsareacceleratedinavacuumbyanelectricfieldtowardsapieceofmetalcalledthe"target".X-raysareemittedastheelectronsslowdown(decelerate)inthemetal.TheoutputspectrumconsistsofacontinuousspectrumofX-rays,withadditionalsharppeaksatcertainenergies.Thecontinuousspectrumisduetobremsstrahlung,whilethesharppeaksarecharacteristicX-raysassociatedwiththeatomsinthetarget.Forthisreason,bremsstrahlunginthiscontextisalsocalledcontinuousX-rays.[18] TheshapeofthiscontinuumspectrumisapproximatelydescribedbyKramers'law. TheformulaforKramers'lawisusuallygivenasthedistributionofintensity(photoncount) I {\displaystyleI} againstthewavelength λ {\displaystyle\lambda} oftheemittedradiation:[19] I ( λ ) d λ = K ( λ λ min − 1 ) 1 λ 2 d λ {\displaystyleI(\lambda)\,d\lambda=K\left({\frac{\lambda}{\lambda_{\min}}}-1\right){\frac{1}{\lambda^{2}}}\,d\lambda} TheconstantKisproportionaltotheatomicnumberofthetargetelement,and λ min {\displaystyle\lambda_{\min}} istheminimumwavelengthgivenbytheDuane–Huntlaw. Thespectrumhasasharpcutoffat λ min {\displaystyle\lambda_{\min}} ,whichisduetothelimitedenergyoftheincomingelectrons.Forexample,ifanelectroninthetubeisacceleratedthrough60kV,thenitwillacquireakineticenergyof60keV,andwhenitstrikesthetargetitcancreateX-rayswithenergyofatmost60keV,byconservationofenergy.(ThisupperlimitcorrespondstotheelectroncomingtoastopbyemittingjustoneX-rayphoton.Usuallytheelectronemitsmanyphotons,andeachhasanenergylessthan60keV.)Aphotonwithenergyofatmost60keVhaswavelengthofatleast21pm,sothecontinuousX-rayspectrumhasexactlythatcutoff,asseeninthegraph.Moregenerallytheformulaforthelow-wavelengthcutoff,theDuane-Huntlaw,is:[20] λ min = h c e V ≈ 1239.8 V  pm/kV {\displaystyle\lambda_{\min}={\frac{hc}{eV}}\approx{\frac{1239.8}{V}}{\text{pm/kV}}} wherehisPlanck'sconstant,cisthespeedoflight,Visthevoltagethattheelectronsareacceleratedthrough,eistheelementarycharge,andpmispicometres. Betadecay[edit] Mainarticle:Betadecay Betaparticle-emittingsubstancessometimesexhibitaweakradiationwithcontinuousspectrumthatisduetobremsstrahlung(seethe"outerbremsstrahlung"below).Inthiscontext,bremsstrahlungisatypeof"secondaryradiation",inthatitisproducedasaresultofstopping(orslowing)theprimaryradiation(betaparticles).ItisverysimilartoX-raysproducedbybombardingmetaltargetswithelectronsinX-raygenerators(asabove)exceptthatitisproducedbyhigh-speedelectronsfrombetaradiation. Innerandouterbremsstrahlung[edit] The"inner"bremsstrahlung(alsoknownas"internalbremsstrahlung")arisesfromthecreationoftheelectronanditslossofenergy(duetothestrongelectricfieldintheregionofthenucleusundergoingdecay)asitleavesthenucleus.Suchradiationisafeatureofbetadecayinnuclei,butitisoccasionally(lesscommonly)seeninthebetadecayoffreeneutronstoprotons,whereitiscreatedasthebetaelectronleavestheproton. Inelectronandpositronemissionbybetadecaythephoton'senergycomesfromtheelectron-nucleonpair,withthespectrumofthebremsstrahlungdecreasingcontinuouslywithincreasingenergyofthebetaparticle.Inelectroncapture,theenergycomesattheexpenseoftheneutrino,andthespectrumisgreatestataboutonethirdofthenormalneutrinoenergy,decreasingtozeroelectromagneticenergyatnormalneutrinoenergy.Notethatinthecaseofelectroncapture,bremsstrahlungisemittedeventhoughnochargedparticleisemitted.Instead,thebremsstrahlungradiationmaybethoughtofasbeingcreatedasthecapturedelectronisacceleratedtowardbeingabsorbed.Suchradiationmaybeatfrequenciesthatarethesameassoftgammaradiation,butitexhibitsnoneofthesharpspectrallinesofgammadecay,andthusisnottechnicallygammaradiation. Theinternalprocessistobecontrastedwiththe"outer"bremsstrahlungduetotheimpingementonthenucleusofelectronscomingfromtheoutside(i.e.,emittedbyanothernucleus),asdiscussedabove.[21] Radiationsafety[edit] Insomecases,e.g.32P,thebremsstrahlungproducedbyshieldingthebetaradiationwiththenormallyuseddensematerials(e.g.lead)isitselfdangerous;insuchcases,shieldingmustbeaccomplishedwithlowdensitymaterials,e.g.Plexiglas(Lucite),plastic,wood,orwater;[22]astheatomicnumberislowerforthesematerials,theintensityofbremsstrahlungissignificantlyreduced,butalargerthicknessofshieldingisrequiredtostoptheelectrons(betaradiation). Inastrophysics[edit] Thedominantluminouscomponentinaclusterofgalaxiesisthe107to108kelvinintraclustermedium.Theemissionfromtheintraclustermediumischaracterizedbythermalbremsstrahlung.ThisradiationisintheenergyrangeofX-raysandcanbeeasilyobservedwithspace-basedtelescopessuchasChandraX-rayObservatory,XMM-Newton,ROSAT,ASCA,EXOSAT,Suzaku,RHESSIandfuturemissionslikeIXO[1]andAstro-H[2]. BremsstrahlungisalsothedominantemissionmechanismforHIIregionsatradiowavelengths. Inelectricdischarges[edit] Inelectricdischarges,forexampleaslaboratorydischargesbetweentwoelectrodesoraslightningdischargesbetweencloudandgroundorwithinclouds,electronsproduceBremsstrahlungphotonswhilescatteringoffairmolecules.Thesephotonsbecomemanifestinterrestrialgamma-rayflashesandarethesourceforbeamsofelectrons,positrons,neutronsandprotons.[23]TheappearanceofBremsstrahlungphotonsalsoinfluencesthepropagationandmorphologyofdischargesinnitrogen-oxygenmixtureswithlowpercentagesofoxygen.[24] Quantummechanicaldescription[edit] ThecompletequantummechanicaldescriptionwasfirstperformedbyBetheandHeitler.[25]Theyassumedplanewavesforelectronswhichscatteratthenucleusofanatom,andderivedacrosssectionwhichrelatesthecompletegeometryofthatprocesstothefrequencyoftheemittedphoton.Thequadruplydifferentialcrosssectionwhichshowsaquantummechanicalsymmetrytopairproduction,is: d 4 σ = Z 2 α fine 3 ℏ 2 ( 2 π ) 2 | p f | | p i | d ω ω d Ω i d Ω f d Φ | q | 4 × [ p f 2 sin 2 ⁡ Θ f ( E f − c | p f | cos ⁡ Θ f ) 2 ( 4 E i 2 − c 2 q 2 ) + p i 2 sin 2 ⁡ Θ i ( E i − c | p i | cos ⁡ Θ i ) 2 ( 4 E f 2 − c 2 q 2 ) + 2 ℏ 2 ω 2 p i 2 sin 2 ⁡ Θ i + p f 2 sin 2 ⁡ Θ f ( E f − c | p f | cos ⁡ Θ f ) ( E i − c | p i | cos ⁡ Θ i ) − 2 | p i | | p f | sin ⁡ Θ i sin ⁡ Θ f cos ⁡ Φ ( E f − c | p f | cos ⁡ Θ f ) ( E i − c | p i | c 1 cos ⁡ Θ i ) ( 2 E i 2 + 2 E f 2 − c 2 q 2 ) ] . {\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|c1\cos\Theta_{i}\right)}}\left(2E_{i}^{2}+2E_{f}^{2}-c^{2}\mathbf{q}^{2}\right)\right].\end{aligned}}} There Z {\displaystyleZ} istheatomicnumber, α fine ≈ 1 / 137 {\displaystyle\alpha_{\text{fine}}\approx1/137} thefinestructureconstant, ℏ {\displaystyle\hbar} thereducedPlanck'sconstantand c {\displaystylec} thespeedoflight.Thekineticenergy E kin , i / f {\displaystyleE_{{\text{kin}},i/f}} oftheelectronintheinitialandfinalstateisconnectedtoitstotalenergy E i , f {\displaystyleE_{i,f}} oritsmomenta p i , f {\displaystyle\mathbf{p}_{i,f}} via E i , f = E kin , i / f + m e c 2 = m e 2 c 4 + p i , f 2 c 2 , {\displaystyleE_{i,f}=E_{{\text{kin}},i/f}+m_{e}c^{2}={\sqrt{m_{e}^{2}c^{4}+\mathbf{p}_{i,f}^{2}c^{2}}},} where m e {\displaystylem_{e}} isthemassofanelectron.Conservationofenergygives E f = E i − ℏ ω , {\displaystyleE_{f}=E_{i}-\hbar\omega,} where ℏ ω {\displaystyle\hbar\omega} isthephotonenergy.Thedirectionsoftheemittedphotonandthescatteredelectronaregivenby Θ i = ∢ ( p i , k ) , Θ f = ∢ ( p f , k ) , Φ = Anglebetweentheplanes  ( p i , k )  and  ( p f , k ) , {\displaystyle{\begin{aligned}\Theta_{i}&=\sphericalangle(\mathbf{p}_{i},\mathbf{k}),\\\Theta_{f}&=\sphericalangle(\mathbf{p}_{f},\mathbf{k}),\\\Phi&={\text{Anglebetweentheplanes}}(\mathbf{p}_{i},\mathbf{k}){\text{and}}(\mathbf{p}_{f},\mathbf{k}),\end{aligned}}} where k {\displaystyle\mathbf{k}} isthemomentumofthephoton. Thedifferentialsaregivenas d Ω i = sin ⁡ Θ i   d Θ i , d Ω f = sin ⁡ Θ f   d Θ f . {\displaystyle{\begin{aligned}d\Omega_{i}&=\sin\Theta_{i}\d\Theta_{i},\\d\Omega_{f}&=\sin\Theta_{f}\d\Theta_{f}.\end{aligned}}} Theabsolutevalueofthevirtualphotonbetweenthenucleusandelectronis − q 2 = − | p i | 2 − | p f | 2 − ( ℏ c ω ) 2 + 2 | p i | ℏ c ω cos ⁡ Θ i − 2 | p f | ℏ c ω cos ⁡ Θ f + 2 | p i | | p f | ( cos ⁡ Θ f cos ⁡ Θ i + sin ⁡ Θ f sin ⁡ Θ i cos ⁡ 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}}} TherangeofvalidityisgivenbytheBornapproximation v ≫ Z c 137 {\displaystylev\gg{\frac{Zc}{137}}} wherethisrelationhastobefulfilledforthevelocity v {\displaystylev} oftheelectronintheinitialandfinalstate. Forpracticalapplications(e.g.inMonteCarlocodes)itcanbeinterestingtofocusontherelationbetweenthefrequency ω {\displaystyle\omega} oftheemittedphotonandtheanglebetweenthisphotonandtheincidentelectron.KöhnandEbertintegratedthequadruplydifferentialcrosssectionbyBetheandHeitlerover Φ {\displaystyle\Phi} and Θ f {\displaystyle\Theta_{f}} andobtained:[26] d 2 σ ( E i , ω , Θ i ) d ω d Ω i = ∑ j = 1 6 I j {\displaystyle{\frac{d^{2}\sigma(E_{i},\omega,\Theta_{i})}{d\omega\,d\Omega_{i}}}=\sum\limits_{j=1}^{6}I_{j}} with I 1 = 2 π A Δ 2 2 + 4 p i 2 p f 2 sin 2 ⁡ Θ i ln ⁡ ( Δ 2 2 + 4 p i 2 p f 2 sin 2 ⁡ Θ i − Δ 2 2 + 4 p i 2 p f 2 sin 2 ⁡ Θ i ( Δ 1 + Δ 2 ) + Δ 1 Δ 2 − Δ 2 2 − 4 p i 2 p f 2 sin 2 ⁡ Θ i − Δ 2 2 + 4 p i 2 p f 2 sin 2 ⁡ Θ i ( Δ 1 − Δ 2 ) + Δ 1 Δ 2 ) × [ 1 + c Δ 2 p f ( E i − c p i cos ⁡ Θ i ) − p i 2 c 2 sin 2 ⁡ Θ i ( E i − c p i cos ⁡ Θ i ) 2 − 2 ℏ 2 ω 2 p f Δ 2 c ( E i − c p i cos ⁡ Θ i ) ( Δ 2 2 + 4 p i 2 p f 2 sin 2 ⁡ Θ i ) ] , I 2 = − 2 π A c p f ( E i − c p i cos ⁡ Θ i ) ln ⁡ ( E f + p f c E f − p f c ) , I 3 = 2 π A ( Δ 2 E f + Δ 1 p f c ) 4 + 4 m 2 c 4 p i 2 p f 2 sin 2 ⁡ Θ i × ln ⁡ [ ( [ E f + p f c ] [ 4 p i 2 p f 2 sin 2 ⁡ Θ i ( E f − p f c ) + ( Δ 1 + Δ 2 ) ( [ Δ 2 E f + Δ 1 p f c ] − [ Δ 2 E f + Δ 1 p f c ] 2 + 4 m 2 c 4 p i 2 p f 2 sin 2 ⁡ Θ i ) ] ) [ ( E f − p f c ) ( 4 p i 2 p f 2 sin 2 ⁡ Θ i [ − E f − p f c ] + ( Δ 1 − Δ 2 ) ( [ Δ 2 E f + Δ 1 p f c ] − ( Δ 2 E f + Δ 1 p f c ) 2 + 4 m 2 c 4 p i 2 p f 2 sin 2 ⁡ Θ i ] ) ] − 1 × [ − ( Δ 2 2 + 4 p i 2 p f 2 sin 2 ⁡ Θ i ) ( E f 3 + E f p f 2 c 2 ) + p f c ( 2 [ Δ 1 2 − 4 p i 2 p f 2 sin 2 ⁡ Θ i ] E f p f c + Δ 1 Δ 2 [ 3 E f 2 + p f 2 c 2 ] ) ( Δ 2 E f + Δ 1 p f c ) 2 + 4 m 2 c 4 p i 2 p f 2 sin 2 ⁡ Θ i − c ( Δ 2 E f + Δ 1 p f c ) p f ( E i − c p i cos ⁡ Θ i ) − 4 E i 2 p f 2 ( 2 [ Δ 2 E f + Δ 1 p f c ] 2 − 4 m 2 c 4 p i 2 p f 2 sin 2 ⁡ Θ i ) ( Δ 1 E f + Δ 2 p f c ) ( [ Δ 2 E f + Δ 1 p f c ] 2 + 4 m 2 c 4 p i 2 p f 2 sin 2 ⁡ Θ i ) 2 + 8 p i 2 p f 2 m 2 c 4 sin 2 ⁡ Θ i ( E i 2 + E f 2 ) − 2 ℏ 2 ω 2 p i 2 sin 2 ⁡ Θ i p f c ( Δ 2 E f + Δ 1 p f c ) + 2 ℏ 2 ω 2 p f m 2 c 3 ( Δ 2 E f + Δ 1 p f c ) ( E i − c p i cos ⁡ Θ i ) ( [ Δ 2 E f + Δ 1 p f c ] 2 + 4 m 2 c 4 p i 2 p f 2 sin 2 ⁡ Θ i ) ] , I 4 = − 4 π A p f c ( Δ 2 E f + Δ 1 p f c ) ( Δ 2 E f + Δ 1 p f c ) 2 + 4 m 2 c 4 p i 2 p f 2 sin 2 ⁡ Θ i − 16 π E i 2 p f 2 A ( Δ 2 E f + Δ 1 p f c ) 2 ( [ Δ 2 E f + Δ 1 p f c ] 2 + 4 m 2 c 4 p i 2 p f 2 sin 2 ⁡ Θ i ) 2 , I 5 = 4 π A ( − Δ 2 2 + Δ 1 2 − 4 p i 2 p f 2 sin 2 ⁡ Θ i ) ( [ Δ 2 E f + Δ 1 p f c ] 2 + 4 m 2 c 4 p i 2 p f 2 sin 2 ⁡ Θ i ) × [ ℏ 2 ω 2 p f 2 E i − c p i cos ⁡ Θ i × E f ( 2 Δ 2 2 [ Δ 2 2 − Δ 1 2 ] + 8 p i 2 p f 2 sin 2 ⁡ Θ i [ Δ 2 2 + Δ 1 2 ] ) + p f c ( 2 Δ 1 Δ 2 [ Δ 2 2 − Δ 1 2 ] + 16 Δ 1 Δ 2 p i 2 p f 2 sin 2 ⁡ Θ i ) Δ 2 2 + 4 p i 2 p f 2 sin 2 ⁡ Θ i + 2 ℏ 2 ω 2 p i 2 sin 2 ⁡ Θ i ( 2 Δ 1 Δ 2 p f c + 2 Δ 2 2 E f + 8 p i 2 p f 2 sin 2 ⁡ Θ i E f ) E i − c p i cos ⁡ Θ i + 2 E i 2 p f 2 ( 2 [ Δ 2 2 − Δ 1 2 ] [ Δ 2 E f + Δ 1 p f c ] 2 + 8 p i 2 p f 2 sin 2 ⁡ Θ i [ ( Δ 1 2 + Δ 2 2 ) ( E f 2 + p f 2 c 2 ) + 4 Δ 1 Δ 2 E f p f c ] ) ( Δ 2 E f + Δ 1 p f c ) 2 + 4 m 2 c 4 p i 2 p f 2 sin 2 ⁡ Θ i + 8 p i 2 p f 2 sin 2 ⁡ Θ i ( E i 2 + E f 2 ) ( Δ 2 p f c + Δ 1 E f ) E i − c p i cos ⁡ Θ i ] , I 6 = 16 π E f 2 p i 2 sin 2 ⁡ Θ i A ( E i − c p i cos ⁡ Θ i ) 2 ( − Δ 2 2 + Δ 1 2 − 4 p i 2 p f 2 sin 2 ⁡ Θ i ) , {\displaystyle{\begin{aligned}I_{1}={}&{\frac{2\piA}{\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\piAc}{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\piA}{\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\piAp_{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\piE_{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\piA}{\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\piE_{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 = Z 2 α fine 3 ( 2 π ) 2 | p f | | p i | ℏ 2 ω Δ 1 = − p i 2 − p f 2 − ( ℏ c ω ) 2 + 2 ℏ c ω | p i | cos ⁡ Θ i , Δ 2 = − 2 ℏ c ω | p f | + 2 | p i | | p f | cos ⁡ Θ i . {\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,amuchsimplerexpressionforthesameintegralcanbefoundin[27](Eq.2BN)andin[28](Eq.4.1). Ananalysisofthedoublydifferentialcrosssectionaboveshowsthatelectronswhosekineticenergyislargerthantherestenergy(511keV)emitphotonsinforwarddirectionwhileelectronswithasmallenergyemitphotonsisotropically. Electron–electronbremsstrahlung[edit] Onemechanism,consideredimportantforsmallatomicnumbers Z {\displaystyleZ} ,isthescatteringofafreeelectronattheshellelectronsofanatomormolecule.[29]Sinceelectron–electronbremsstrahlungisafunctionof Z {\displaystyleZ} andtheusualelectron-nucleusbremsstrahlungisafunctionof Z 2 {\displaystyleZ^{2}} ,electron–electronbremsstrahlungisnegligibleformetals.Forair,however,itplaysanimportantroleintheproductionofterrestrialgamma-rayflashes.[30] Seealso[edit] Beamstrahlung Cyclotronradiation Free-electronlaser HistoryofX-rays Listofplasmaphysicsarticles Nuclearfusion:bremsstrahlunglosses Radiationlengthcharacterisingenergylossbybremsstrahlungbyhighenergyelectronsinmatter Synchrotronlightsource References[edit] ^"Bremsstrahlung".Merriam-WebsterDictionary. ^APlasmaFormularyforPhysics,Technology,andAstrophysics,D.Diver,pp.46–48. ^IntroductiontoElectrodynamics,D.J.Griffiths,pp.463–465 ^abJackson,ClassicalElectrodynamics,Sections14.2–3 ^RadiationProcessesinPlasmas,G.Bekefi,Wiley,1stedition(1966) ^BasicPrinciplesofPlasmasPhysics:AStatisticalApproach,S.Ichimaru,p.228. ^NRLPlasmaFormulary,2006Revision,p.58. ^RadiativeProcessesinAstrophysics,G.B.Rybicki&A.P.Lightman,p.162. ^Rider,T.H.(1995).Fundamentallimitationsonplasmafusionsystemsnotinthermodynamicequilibrium(PhDthesis).MIT.p. 25.hdl:1721.1/11412. ^PolarizationBremsstrahlungonAtoms,Plasmas,NanostructuresandSolids,byV.Astapenko ^NewDevelopmentsinPhotonandMaterialsResearch,Chapter3:"PolarizationalBremsstrahlung:AReview",byS.Williams ^Ishii,Keizo(2006)."ContinuousX-raysproducedinlight-ion–atomcollisions".RadiationPhysicsandChemistry.ElsevierBV.75(10):1135–1163.doi:10.1016/j.radphyschem.2006.04.008.ISSN 0969-806X. ^Wendin,G.;Nuroh,K.(1977-07-04)."BremsstrahlungResonancesandAppearance-PotentialSpectroscopynearthe3dThresholdsinMetallicBa,La,andCe".PhysicalReviewLetters.AmericanPhysicalSociety(APS).39(1):48–51.doi:10.1103/physrevlett.39.48.ISSN 0031-9007. ^Portillo,Sal;Quarles,C.A.(2003-10-23)."AbsoluteDoublyDifferentialCrossSectionsforElectronBremsstrahlungfromRareGasAtomsat28and50keV".PhysicalReviewLetters.AmericanPhysicalSociety(APS).91(17):173201.doi:10.1103/physrevlett.91.173201.ISSN 0031-9007.PMID 14611345. ^Astapenko,V.A.;Kubankin,A.S.;Nasonov,N.N.;Polyanskiĭ,V.V.;Pokhil,G.P.;Sergienko,V.I.;Khablo,V.A.(2006)."Measurementofthepolarizationbremsstrahlungofrelativisticelectronsinpolycrystallinetargets".JETPLetters.PleiadesPublishingLtd.84(6):281–284.doi:10.1134/s0021364006180019.ISSN 0021-3640.S2CID 122759704. ^Williams,Scott;Quarles,C.A.(2008-12-04)."Absolutebremsstrahlungyieldsat135°from53−keVelectronsongoldfilmtargets".PhysicalReviewA.AmericanPhysicalSociety(APS).78(6):062704.doi:10.1103/physreva.78.062704.ISSN 1050-2947. ^Gonzales,D.;Cavness,B.;Williams,S.(2011-11-29)."Angulardistributionofthick-targetbremsstrahlungproducedbyelectronswithinitialenergiesrangingfrom10to20keVincidentonAg".PhysicalReviewA.84(5):052726.arXiv:1302.4920.doi:10.1103/physreva.84.052726.ISSN 1050-2947.S2CID 119233168. ^S.J.B.Reed(2005).ElectronMicroprobeAnalysisandScanningElectronMicroscopyinGeology.CambridgeUniversityPress.p. 12.ISBN 978-1-139-44638-9. ^Laguitton,Daniel;WilliamParrish(1977)."ExperimentalSpectralDistributionversusKramers'LawforQuantitativeX-rayFluorescencebytheFundamentalParametersMethod".X-RaySpectrometry.6(4):201.Bibcode:1977XRS.....6..201L.doi:10.1002/xrs.1300060409. ^ReneVanGrieken;AndrzejMarkowicz(2001).HandbookofX-RaySpectrometry.CRCPress.p. 3.ISBN 978-0-203-90870-9. ^Knipp,J.K.;G.E.Uhlenbeck(June1936)."Emissionofgammaradiationduringthebetadecayofnuclei".Physica.3(6):425–439.Bibcode:1936Phy.....3..425K.doi:10.1016/S0031-8914(36)80008-1.ISSN 0031-8914. ^"Environment,Health&Safety"(PDF).Archivedfromtheoriginal(PDF)on2017-07-01.Retrieved2018-03-14. ^Köhn,C.;Ebert,U.(2015)."Calculationofbeamsofpositrons,neutrons,andprotonsassociatedwithterrestrialgammarayflashes".JournalofGeophysicalResearch:Atmospheres.120(4):1620–1635.Bibcode:2015JGRD..120.1620K.doi:10.1002/2014JD022229. ^Köhn,C.;Chanrion,O.;Neubert,T.(2017)."TheinfluenceofbremsstrahlungonelectricdischargestreamersinN2,O2gasmixtures".PlasmaSourcesScienceandTechnology.26(1):015006.Bibcode:2017PSST...26a5006K.doi:10.1088/0963-0252/26/1/015006. ^Bethe,H.A.;Heitler,W.(1934)."Onthestoppingoffastparticlesandonthecreationofpositiveelectrons".ProceedingsoftheRoyalSocietyA.146(856):83–112.Bibcode:1934RSPSA.146...83B.doi:10.1098/rspa.1934.0140. ^Köhn,C.;Ebert,U.(2014)."Angulardistributionofbremsstrahlungphotonsandofpositronsforcalculationsofterrestrialgamma-rayflashesandpositronbeams".AtmosphericResearch.135–136:432–465.arXiv:1202.4879.Bibcode:2014AtmRe.135..432K.doi:10.1016/j.atmosres.2013.03.012.S2CID 10679475. ^Koch,H.W.;Motz,J.W.(1959)."BremsstrahlungCross-SectionFormulasandRelatedData".ReviewsofModernPhysics.31(4):920–955.Bibcode:1959RvMP...31..920K.doi:10.1103/RevModPhys.31.920. ^Gluckstern,R.L.;Hull,M.H.,Jr.(1953)."PolarizationDependenceoftheIntegratedBremsstrahlungCrossSection".PhysicalReview.90(6):1030–1035.Bibcode:1953PhRv...90.1030G.doi:10.1103/PhysRev.90.1030. ^Tessier,F.;Kawrakow,I.(2008)."Calculationoftheelectron-electronbremsstrahlungcrosssectioninthefieldofatomicelectrons".NuclearInstrumentsandMethodsinPhysicsResearchB.266(4):625–634.Bibcode:2008NIMPB.266..625T.doi:10.1016/j.nimb.2007.11.063. ^Köhn,C.;Ebert,U.(2014)."Theimportanceofelectron-electronbremsstrahlungforterrestrialgamma-rayflashes,electronbeamsandelectron-positronbeams".JournalofPhysicsD.47(25):252001.Bibcode:2014JPhD...47y2001K.doi:10.1088/0022-3727/47/25/252001. Furtherreading[edit] EberhardHaug;WernerNakel(2004).Theelementaryprocessofbremsstrahlung.ScientificLectureNotesinPhysics.73.RiverEdge,NJ:WorldScientific.ISBN 978-981-238-578-9. Externallinks[edit] LookupbremsstrahlunginWiktionary,thefreedictionary. IndexofEarlyBremsstrahlungArticles vteQuantumelectrodynamicsConcepts Anomalousmagneticdipolemoment Probabilityamplitude Propagator QEDvacuum Self-energy Vacuumpolarization ξgauge Formalism Feynmandiagram Feynmanslashnotation Gupta–Bleulerformalism Pathintegralformulation Vertexfunction Ward–Takahashiidentity Interactions Bhabhascattering Bremsstrahlung Comptonscattering Lambshift Møllerscattering Particles Dualphoton Electron Photon Positron Positronium Virtualparticles Seealso:Template:Quantummechanicstopics Portals:PhysicsAstronomyScience AuthoritycontrolGeneral IntegratedAuthorityFile(Germany) Nationallibraries France(data) UnitedStates Japan Other MicrosoftAcademic SUDOC(France) 1 Retrievedfrom"https://en.wikipedia.org/w/index.php?title=Bremsstrahlung&oldid=1062093063" Categories:AtomicphysicsPlasmaphysicsScatteringQuantumelectrodynamicsHiddencategories:CS1:longvolumevalueArticleswithshortdescriptionShortdescriptionisdifferentfromWikidataArticleswithhAudiomicroformatsArticlescontainingGerman-languagetextWikipediaarticlesneedingclarificationfromMay2016ArticleswithGNDidentifiersArticleswithBNFidentifiersArticleswithLCCNidentifiersArticleswithNDLidentifiersArticleswithMAidentifiersArticleswithSUDOCidentifiers Navigationmenu Personaltools NotloggedinTalkContributionsCreateaccountLogin Namespaces ArticleTalk Variants expanded collapsed Views ReadEditViewhistory More expanded collapsed Search Navigation MainpageContentsCurrenteventsRandomarticleAboutWikipediaContactusDonate Contribute HelpLearntoeditCommunityportalRecentchangesUploadfile Tools WhatlinkshereRelatedchangesUploadfileSpecialpagesPermanentlinkPageinformationCitethispageWikidataitem Print/export DownloadasPDFPrintableversion Inotherprojects WikimediaCommons Languages العربيةAsturianuБеларускаяCatalàČeštinaDanskDeutschΕλληνικάEspañolفارسیFrançaisGaeilge한국어हिन्दीHrvatskiBahasaIndonesiaItalianoעבריתMagyarNederlands日本語NorskbokmålNorsknynorskPolskiPortuguêsРусскийSlovenčinaСрпски/srpskiSuomiSvenskaTagalogไทยTürkçeУкраїнськаاردو中文 Editlinks