Electrodynamics and Radiave Processes I Lecture 7 ...

Lecture7–BremsstrahlungRadia4onI

Date:29thAugust2019

Bhaswa4Bha?acharyya

ElectrodynamicsandRadia4veProcessesI

August-September2019

IUCAA-NCRAGraduateSchool

[email protected]

Lecture-7Ques4onsraisedintheclass

1.  Ques4onsrelatedtobeaming?2.Unitsusedintherela4vis4cderiva4ons?3.  Lighthasweight?

BremsstrahlunginAstrophysics

BremsstrahlungisaGermanworddirectlydescribingtheprocess:"Strahlung"means"radia4on",and"Bremse"meansbreak.Alsoknownasbreakingradia4on.Alsoknownasfree-freeemission.


h?ps://apatruno.files.wordpress.com/2016/09/lecture52.pdf

BremsstrahlungInEverydaylife

X-raysusedinmedicine(radiology)arecreatedinBremsstrahlung.


Wheneverthereishotionisedgasinuniverseitemitsbremsstrahlung

ü  Electronsinaplasmaareacceleratedbyencounterswithmassiveions.

ü  Thisisthedominantcon4nuumemissionmechanisminthermalplasmas.

ü  Animportantcoolantforplasmasathightemperature.


Anincomingfreeelectroncangetclosetothenucleusofanatom(orotherchargedpar4cle),thestrongelectricfieldofthenucleuswilla?racttheelectron,thuschangingdirec4onandspeedoftheelectron–accelera4ngit.

Severalsubsequentinterac4onsbetweenoneandthesameelectronanddifferentnucleiarepossible.

Free-freebecausetheelectronisfreebeforeandfreeaber.


Wheneverthereishotionisedgasintheuniverseitemitsbremsstrahlung

Examples

Radio:HIIregionscompactregionsofhotionisedhydrogenatT~104KRadioemissionfromionisedwindsandjets

X-ray:BinaryX-raysourcesatT~107KDiffuseX-rayemissionfromintergalac4cregioninclusterofgalaxiesT~108K


Bremsstrahlungradia4onisemi?edwhenachargedpar4cleisdeflected(decelerated)byanothercharge.Bremsstrahlungemissioncanbedetectedfromop4callythinhotionisedplasma.




ClassicaltreatmentQuantumeffectRela4vis4ceffect

Radia4onduetotheaccelera4onofachargeinthecoulombfieldofanotherchargeIscalledbremsstrahlung.

Bremsstrahlung“Free-FreeEmission”“BreakingRadia4on”

θ

PathofelectronIon

Ø  Bremsstrahlungduetocollisionoflikepar4cles(e.g.electro-electronorproton-proton)iszerointhedipoleapproxima4on

Ø  Inelectron-ionbremsstrahlungtheelectronsaretheprimaryradiators,sincerela4veaccelera4onsareinverselypropor4onaltomasses.

Bremsstrahlung“Free-FreeEmission”“BreakingRadia4on”

Fullunderstandingoftheprocesswillrequireaquantumtreatment.

Classicaltreatmentisjus4fiedinsomeregimes(discussedlaterinthelecture)

ApproachWefirststatetheclassicaltreatmentandthenquantumresultsascorrec4ons.

Wefirsttreatnonrela4vis4cbremsstrahlungandthenconsiderrela4vis4ccorrec4ons.

HistoryofBremsstrahlung“Free-FreeEmission”“BreakingRadia4on”

1930:CurlAndersonfoundthationisa4onloss–rateisunderes4matedforrela4vis4celectrons(thoughwasnotedbyTeslain1880)Addi4onalenergylossmechanismwasassociatedwithradia4onofelectromagne4cwavesbecauseofaccelera4onoftheelectronintheelectrosta4cfieldofnucleusRadia4oncorrespondstotransi4onbetweenunboundstatesoftheelectroninthefieldofthenucleus1939calcula4ononrela4vis4candnonrela4vis4cspectrumbyBatheandHeilterImprovedtreatmentappropriatetoastrophysicalsitua4onsBlumenthaletal1970

BremsstrahlungLayout

(1)Emissionfromsinglespeedelectronpickrestframeofioncalculatedipoleradia4oncorrectforquantumeffects(Gauntfactor)(2)Emissionfromcollec4onofelectronThermalbremsstrahlungFree-FreeAbsorp4onNon-thermalbremsstrahlung(3)Rela4vis4cbremsstrahlung(VirtualQuanta)

BremsstrahlungEmissionfromasinglespeedelectron

Consideranelectronofcharge–emovingpastanionofchargeZewithimpactparameterb

b R

e-

Ze

Assume:electronmovesrapidlyanditspathisstraightline

Dipolemomentd=-eR 2ndderiva4veofdipolemoment

Fouriertransform


Collision4me:4meintervaloverwhichelectronandionarecloseenoughtointeract

Case-1ωτ>>1theexponen4aloftheintegraloscillatesrapidlyandintegralissmallCase-2ωτ<<1exponen4alisessen4allyunity

Δvchangeofvelocityduringcollision


RecallSpectrumofdipoleradia4on

SoSpectrumofBremsstrahlungradia4on


Consideringlinearpath,changeinvelocityisnormaltothepath.Integratecomponentofaccelera4onnormaltothepath.

v


Thusforsmallanglesca?eringspectraofemissionfromasinglecollisionis

BremsstrahlungEmissionfrommul4plesinglespeedelectron

Bunchofelectrons,allwiththesamespeed,v,whichinteractwithabunchofions.


Totalspectrumforamediumwithiondensitynielectrondensityneandfixedelectronspeedv

Fluxofelectrons(electronsperunitareaperunit4me)incidentononeionisnevTheelementofareais2πbdbaboutasingleion.bminisminimumvalueofimpactparameter


Forb<<v/ω

Wherebmaxissomevalueofbbeyondwhichb<<v/ωisnotapplicableandcontribu4ontointegralisnegligible


Valueofbmincanbees4matedintwowaysFirstwecantakethevalueatwhichstraightlineapproxima4onisnolongervalid.

Secondvalueforbminisquantuminnatureandcomesfromuncertaintyprinciple


b1min>>b2minaclassicaldescrip4onofsca?eringisvalid

b2min>>b1minquantumtreatmentrequired

Wechoosewhicheverofthesevaluesofbministhelargerforthephysicalcondi4onsoftheproblem


Foranyregimetheexactresultsarestatedintermsofacorrec4onfactorgff

Gauntfactor(correc4onfactor)

Gauntfactorisafunc4onofenergyoftheelectronandoffrequencyofemission.

ThermalBremsstrahlungEmission

Approximateanaly4calformulaforgauntfactorgff(ν,T)forthermalbremsstrahlung(Rybicki&Lightman)

Foranyregimetheexactresultsarestatedintermsofacorrec4onfactorgff


Useofformulasderivedforsinglevelocityofchargedpar4cleintheirapplica4ontothermalbremsstrahlung

Weaveragethederivedsinglespeedexpressionoverarangeofthermaldistribu4onofspeeds


TheprobabilitydPthatapar4clehasvelocityintheranged3V

Lowerlimitofelectronvelocityfromthecondi4on


Thusthetotalemissionperunit4meperunitvolumeperunitfrequencyforarangeofveloci4esforiondensityniandelectrondensityne

Limitsofintegra4on:0<v<αButatfrequencyνtheincidentvelocitymustbeatleastsuchthathν<<(1/2)mv2otherwiseaphotonofenergyhνcannotbecreated

Thiscutoffinthelowerlimitoftheintegra4onoverelectronveloci4esiscalledaPhotondiscretenesseffect


Thusthetotalemissionperunit4meperunitvolumeperunitfrequencyforarangeofveloci4esforiondensityniandelectrondensityne

Recapthetotalemissionperunit4meandperunitvolumeandperunitfrequencyforsinglevelocityelectronswithelectrondensityneconsideringiondensityni


Totalemissionperunit4meperunitvolumeperunitfrequencyforarangeofveloci4esforiondensityniandelectrondensityne

InC.G.S.unitswehaveFree-freeemissioncoefficienti.e.totalemissionperunitvolumeperunitfrequency

Unitergs-1cm-3Hz-1VelocityaveragedGauntfactor

Numericalvaluesofgauntfactor

Numericalvaluesofgauntfactorgff(ν,T).Frequencyvariableu=4.8x1011ν/T


Integrateoverfrequency

Numerically,totalemissionperunitvolumeperunit4meinC.G.S.unit

Unitergs-1cm-3

FrequencyaverageofthevelocityaveragedGauntfactorvaluerangesfrom1.1to1.5

ThermalBremsstrahlungEmissivity

Ratherflatspectruminthelog-logplotUptoacutoffatabouthν=KTTrueforop4callythinsources,notconsideringabsorp4onofphotonsbyfree-freeabsorp4on

ü  is~constantwithhνatlowfrequenciesffνε

ü  fallsoffexponen4allyathν~kTffνε

ThermalBremsstrahlungspectra

Spectraforthermalbremsstrahlungattwodifferenttemperatures(thoughsamedensity)

ThermalBremsstrahlungspectra


Thespectrumwillbeflat,exceptwhenexp(-h*nu/kT)becomesdominant.Thishappenswhenthethermalenergyofelectronsisinsufficienttogeneratehighenergyphotons.

Considerachargedpar4cleataspecificimpactparameter(b)andvelocity(v).Whenachargedpar4cleacceleratesitemitsradia4on.Accelera4onisafunc4onofb,vandZ.Accelera4onasafunc4onof4meintensityspectrumviaFourierTransform.Integrate(exactdetailstricky–givesrisetotheGauntFactorgff,whichisafunc4onofν,T,Z).Includetermforcollisionrate(dependsonnumberdensi4esneandni).Integrateoverv.AssumeplasmainthermalequilibriumàMaxwelliandistribu4onofv.

ThermalBremsstrahlungRecap

Lecture-7Ques4onsfornextclass

1.  Whyfree-freeemissionfromlikepar4cles(e-e,p-p)iszero?2.Whyelectronradiateinelectron-ionbremsstrahlung?

EndofLecture7

NextLecture:3rdSeptemberTopicofnextLecture:BremsstrahlungRadia4on-II(Chapter5ofRybicki&Lightman)

Electrodynamics and Radiave Processes I Lecture 7 ...

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