Photoinduced K-shell hollow atoms
J. Hoszowska
∗, J.-Cl. Dousse
DepartmentofPhysics,UniversityofFribourg,CH-1700Fribourg,Switzerland
ThemechanismsleadingtotheproductionofhollowKshellatomsviasinglephotonimpactwereinves- tigatedforavarietyoflightelementswith12≤Z≤23.Thedouble1svacancystateswereproducedby irradiatingthesampleswithintensemonoenergeticsynchrotronradiationbeams.Thedouble-to-single K-shellphotoionizationprobabilitiesPKKandtheabsolutedoubleK-shellphotoionizationcrosssections 2+weredeterminedbymeasuringwithahigh-resolutionbentvonHamoscrystalspectrometertheK˛h hypersatelliteX-rayemissionofthesamples.Themeasurementswereperformedoverawiderangeof incomingphotonenergiesfromthresholduptoenergiesbeyondthebroadmaximumofthedouble-to- singlephotoionizationcrosssectionratios.ThePKKand2+weredeterminedfromtherelativeyieldsof theresolvedK˛hhypersatellitelines.ForMg,AlandSi,thetwo-electronone-photon(TEOP)K˛˛htransi- tionswhichrepresentanalternativebutmuchweakerdecaychannelfordouble1svacancystatescould bealsoobserved,usingahighlyefficientflatcrystalwavelengthdispersivespectrometer.Thisobser- vationofsinglephoton-inducedTEOPtransitionshasshownthattheI(K˛h)/I(K˛˛h)branchingratios areverypoorlyreproducedbymostofexistingtheoreticalmodels.Besidestherelativeyieldsofthe hypersatelliteandTEOPtransitions,theenergiesandnaturallinewidthsoftheK˛handK˛˛hX-raylines werealsodetermined.Theenergiesarefoundtobeingoodagreementwithdifferenttheoreticalpredic- tions,whereasthelinewidthsaresignificantlyunderestimatedbythecalculations,exceptifnon-lifetime broadeningeffectssuchastheouter-shellionizationandtheopenvalenceconfigurationaretakeninto consideration.
1. Introduction
Inthelastyearsimportanteffortswereundertakenbydiffer- entexperimentalandtheoreticalgroupstobetterunderstandthe mechanismsinvolvedintheproductionofK-shellhollowatoms viasinglephotonimpactandthesubsequentradiativedecayofthe photoinduceddouble1svacancies.HollowK-shellatomsareatoms thatarecharacterizedbyanemptyinnermostshellandoccupied outershells.SinceinphotoabsorptionorinelasticX-rayscattering processes,theincomingphotoninteractswithasingleelectron,the ejectionofthetwo1selectronsisdrivenbymany-electroninterac- tions.Thelatterplayingakeyroleintheunderstandingofatomic structure(see[1,2]andreferencestherein),single-photondouble ionizationprocesseshavereceivedarenewedinterestinthelast decade.ThesameholdsforthedecayofK-shellhollowatomsin whichelectroncorrelationeffectsdoalsoplayacrucialrole.
Ascomparedtoatomiccollisionsinvolvingheavyions(HI),pho- toionization andinelasticX-rayscatteringcan beconsideredas soft collisionsina perturbationsense.Asaconsequence,rather pure K-shell hollow atoms are obtained, with no or only few
∗Correspondingauthor.Tel.:+41263009210.
E-mailaddress:joanna.hoszowska@unifr.ch(J.Hoszowska).
additionalvacanciesintheoutershells,whichmakesthecompar- isonwiththeoreticalcalculationseasierandmorereliable.Onthe otherhand,theprobabilityforcreatingdouble1svacancystatesby singlephotonimpactisquitelow,rangingfromafewpercentfor lightelementsdowntoabout10−5forheavyones.Inthisrespect, experimentalstudiesconcerninghollowK-shellatomshavegreatly benefited fromthehighintensityandenergy tunabilityof syn- chrotronradiationsources.Furthermore,therecentadventofvery intensehardX-rayfree-electronlaser(XFEL)beamspermittingto investigatethedynamicsofatomicelectronsinthefemtosecond timescalehavegivenanewboosttothedomain[3].
In single photon absorption, the creation of the second 1s vacancyisgenerallyassumedtobeduetotwocompetitivemech- anisms,namelytheshake-off(SO)andknock-out(KO)processes thatarebothrelatedtotheejectionoftheKphotoelectron.Inthe SOprocess[4,5],thesecond1selectronisexcitedintothecontin- uumduetothesuddenchangeoftheatomicpotentialresulting fromthefastremovaloftheprimaryelectron.TheSOprobability isproportionaltothesquaredoverlapintegraloftheinitialand finalstatewavefunctionsoftheshakenelectron[6],providedthe changeoftheatomicpotentialismuchfasterthantheatomicrelax- ationtime.IntheKOprocess,theoutgoingphotoelectronknocks outthesecond1selectroninan(e,2e)-likeelectronimpacthalf- collision.Althoughthedependenceofthetwomechanismsonthe
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which should be cited to refer to this work.
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Fig.1.SchematicoftheTEOP(left)andOEOP(right)transitions.
incomingphotonenergyisverydifferent,inmostcasesitishard todistinguishthecontributionsofthetwoprocessesandattempts toseparatethemhavegivenrisetointensiveresearch[7–13].
HollowK-shellatomscanbeidentifiedeitherbyrecordingthe KhypersatellitetransitionsinAugerelectronspectra(KK-KLL,KK- KLM,etc.)orinX-rayfluorescencespectra(KK-KL;KK-KM,etc.).
ThemostprobableKhypersatellitetransitionsarethoseforwhich oneofthetwoKvacanciesisfilledbyaL-electron,namelyKK-KLX (X=L,M,etc.)hypersatellitesinAugerelectronspectraandKK-KL (usuallynotedK˛h)hypersatellitesinX-rayfluorescencespectra.
Augerspectroscopyismorelikelyusedforlightatomsbecausethe fluorescenceyieldsofthelatterarelow.However,thistechniqueis limitedtogaseoustargets(see,e.g.,[14])orverythinsolidsamples, duetothestrongabsorptionoflow-energyelectronsinmatter.In additiontheanalysisofthemeasuredhypersatelliteAugerspec- trais noteasy becausemultipleAugerfinal statesare possible.
Forthesereasons,inthepresentstudywhichconcernssolidele- ments,thehollowK-shellatomswereinvestigatedbymeansof high-resolutionX-rayspectroscopy.
AsshowninFig.1,theradiativedecayofdouble1svacancystates mayproceedthroughone-electronone-photon(OEOP)andtwo- electronone-photon(TEOP)transitions.TheOEOPprocess,which correspondstotheK˛h(1s−2→1s−12p−1)hypersatellitetransition representstoalargeextentthepredominantradiativedecaychan- nelofhollowK-shellatoms.IntheTEOPK˛˛h (1s−2→2s−12p−1) transition,thetwoK-shellcoreholesarefilledsimultaneouslyviaa correlatedtwo-electronjumpandasinglephotonhavinganenergy whichisapproximatelytwicethatoftheparentK˛diagramlineis emitted.Despitetheirextremelyweakintensitiesascomparedto thoseofhypersatellites,TEOPtransitionsareofinterestbecause theycorrespondtocorrelatedmultielectronprocesses.
ThetheoreticalpredictionofTEOPtransitionsdatesbackto1925 [15]butthefirstexperimentalevidenceforthiscorrelatedtwo- electron decaychannel wasreported onlyabout 50 years later [16].Actually,thisfirstobservationwasdonesomewhatacciden- tallywhileanalyzingaHI-inducedKX-rayspectrum,inwhicha veryweakX-raylinehavingthesameenergy astheoneof the TEOPtransitionwasfound.TheK˛htoK˛˛hbranchingratioisnot expectedtodependontheexcitationmode.However,multiple electronionizationinHIcollisionschangetheelectronicconfig- urationsandaffecttheintensitiesand energiesofthemeasured transitions.ThusdatafromHIcollisionexperimentsshowawide spreadofvalues [17–21],makingcomparisonwiththeoryoften inconclusive. In this respect,photon impact dataprovide more reliable results and a more stringent test for atomic structure
calculations.Incounterpart,single-photondoubleK-shellioniza- tioncrosssectionsare102–103smallerthaninHIcollisions.Thus photoionization experiments are more challenging and, to the best of our knowledge, all attempts to measure photoinduced TEOPtransitionshavebeenunsuccessful(seee.g.,[22])untilvery recently[23].NotethattheTEOPanalogousKK-LLX(X=L,M,etc.) three-electron Auger transitions corresponding to the simulta- neousfillingofthetwoKvacanciesbythecorrelatedjumpoftwo L-electronsandthetransferoftheentiretransitionenergyonthe AugerelectronXhaveneverbeenobservedsofar.However,simi- larthree-electronAugertransitionsofthetypeLL-MMMcouldbe detectedinArasaresultoflow-energycollisionswithseveralheavy ions[24].
In this paper we present an overview of the experimental and theoreticalefforts donein thelast decadein theendeavor ofbringingnewinsightstoelectron–electroninteractionsandin understandingthedoublephotoionizationinmany-bodysystems.
Inparticular,thephotonenergyevolutionofthedouble-to-single photoionizationcrosssectionratiosforseverallightelementswith 12≤Z≤23arereportedforawide photonenergyrange.TheZ- dependenttrendsand scalingpropertiesoftheseratiosand the double photoionization (DPI) cross sections are examined.The energiesandlinewidthsoftheK˛hhypersatelliteX-raytransitions andtheK˛h1/K˛h2intensityratiosarereportedandcomparedtothe- oreticalcalculationsaswellastootheravailableexperimentaldata.
ForMg,AlandSi,thecorrelatedtwo-electronone-photontransi- tionsinsingle-photonK-shelldoubleionizationcouldbeobserved recentlyforthefirsttime[23].TheTEOPenergiesandK˛htoK˛˛h branchingratiosobtainedinthisexperimentarealsopresentedand discussed.
2. Experimentalmethod
InX-rayspectroscopytheKhypersatellitelinescanbeeither observed directly with high resolution wavelength-dispersive spectrometersorindirectlybymeasuringincoincidencetheemit- tedK˛hhypersatelliteandsubsequentK˛LsatelliteX-rays,using twoenergy-dispersivedetectors.Thecoincidencetechnique[12]is preferabletohighresolutionX-rayspectroscopyincaseofheavy elementsforwhichtheK˛hhypersatelliteslieonthehigh-energy tailsofthe∼105strongerparentK˛diagramlines.Thedisadvan- tageofthecoincidencemethodisthat,duetothepoorresolutionof theenergy-dispersivedetectors,onlytheratioofdoubletosingleK- shellionizationcrosssectionscanbeextractedaccuratelyandnot theenergyofthehypersatellitetransitions,northeirnaturalwidth.
Many wavelengthdispersivespectrometers havebeendesigned andconstructedinthelastyears.ForsoftandtenderX-rays,Bragg- type crystal spectrometers areusually employed.Setups based oncylindricallyor sphericallybentcrystalsorarrays ofcrystals arranged in theJohann [25–27], Johansson [28] or von Hamos geometry [29,30]and setups using flat crystalscombined with half-lensepolycapillaryX-rayoptics[31]weredeveloped.Mostof thesecrystalspectrometerswereoptimizedforspecifictechniques such as resonant inelastic X-ray scattering (RIXS), highenergy resolution fluorescence detectedX-ray absorptionspectroscopy (HERFD-XAS)andX-rayRamanspectroscopy(XRS)appliedtosolid, liquid andgaseoussamples.Toa smallerextent,moreversatile and transportable instrumentsdesigned for thehigh-resolution measurementoftheX-rayfluorescenceresultingfromtheatomic core-levels’excitationbyimpactwithphotons,lightchargedpar- ticlesandheavyionswerealsodeveloped[29,28].
The experiments discussed in the present paper were per- formed at the European Synchrotron Radiation Facility (ESRF), using intense, monochromatic and energy-tunable synchrotron radiationbeamstoproducethedouble1svacancystatesinavariety
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oflightelements(Mg,Al,Si,Cl,K,Ca,ScandV).Toprobetheevo- lutionofthedouble-to-singlephotoionizationcross-sectionratios over wide photonbeam energyranges upto∼3 times theDPI thresholdenergies,themeasurementsoftheKhypersatelliteand diagramX-raytransitionswerecarriedoutattwoundulatorbeam- lines(ID21andID26),andatabendingmagnetbeamline(BM5).The measurementswereperformedbymeansofhigh-resolutionX-ray spectroscopy,usingforallmeasurementsexcepttheTEOPones theFribourgvonHamosBragg-typecurvedcrystalspectrometer [29,32].High-energyresolutionwasmandatorybecausetheMg,Al andSiKhypersatellitesarepartlyorcompletelyoverlappingwith theL-satellitesofthediagramKˇ(1s−1 →3p−1)lines,andinthe caseofCl,K,andCatheK˛hypersatellitesneedtoberesolvedfrom thecloselyingKMMradiativeAugertransitions[33,34].
TheprincipalelementsofthevonHamosspectrometerofFri- bourgare aneffective X-raysource,a crystalbent cylindrically toanominalradiusof25.4cm,andaposition-sensitivedetector, locatedonthecrystalaxisofcurvature.ThevonHamosgeome- trypermitsatonepositioningoftheelements,datacollectionover anenergybandwidthlimitedprimarilybythedetectorlength.The effectiveX-rayfluorescencesourceviewedbythecrystalisusually definedby arectangularslitwithanadjustablewidth.Alterna- tively,theeffectivesourcesizemaybedefinedbyafocusedbeam spotonthesample,andtheslitisleftwideopen.Thisso-called slit-lessoperationmoderesultsinahigheroveralldetectioneffi- ciency. Thesample,crystaland detectorareallcontainedinan evacuatedstainlesssteelchamber.Inthepresentexperimentsthe vonHamosspectrometerwasequippedwithfourdifferentcrystals, namelyaTlAP(001)(2d=25.772 ˚A),anADP(101)(2d=10.642 ˚A), aLiF(200)(2d=4.028 ˚A),andaGe(200)(2d=4.000 ˚A)crystal.The diffractedX-rayswererecordedwithathermoelectricallycooled (−45 to−50◦C) back-illuminated charge coupleddevice (CCD) cameraconsistingof1340columnsand400rowswithapixelsize of20m×20m.Bysettingappropriateenergywindows,theCCD detectorallowsdiscriminationagainsthigher-ordercrystalreflec- tionsandalsoarejectionofbackgroundevents.
AtthebeamlineID21wheretheMgandAlmeasurementswere performedthevonHamosspectrometerwasinstalleddownstream oftheScanningTransmissionX-rayMicroscope(STXM)chamber.
Monochromaticphotonbeamsrangingfrom2.7to8.0keVforMg andfrom3.1to7.0keVforAlwith∼10eVbandwidthwereobtained usingthedoubleNi/B4Cmultilayermonochromator.Forupperhar- monicsrejection,aNi-coatedmirrorwasemployed.Thebeamsize was definedby means of a 1mm in diameterpinhole. For the measurementsofCl,K,CaandV,thespectrometerwasinstalled at theBM5 beamline, in the firstexperimental hutch.The pri- maryX-raybeamwasmonochromatizedbymeansofa[Ru/B4C]70 double-multilayermonochromatorwithanenergyresolutionE/E of∼2×10−2andanharmonicsrejectionrateof1.8×10−4inthe 6–30keVphotonenergyrange.Thebeamsizeonthesamplewas definedbymeansofa2mmhighand5mmwiderectangularslit placedinfrontofthespectrometerbeamport.TheSiandScX-ray spectraaswellasdataathigherbeamenergiesforMg,Al,Caand VwerecollectedatthebeamlineID26.Forphotonenergiesupto 16keVtheSi(111)double-crystalmonochromatorwasemployed, whereasforhigherenergiestheSi(311)monochromatorwasused.
Dependingonthephotonenergy,doubleSi,Cr-andPd-coated,and doublePt-coatedmirrorssuppressedtheupperharmonics,andfor incidentbeamenergiesinthe3–5keVrangethemonochromator crystalswereadditionallydetuned.Theupperharmonicsrejection efficiencywas∼10−4–10−5.Themirrorsservedalsotofocusthe beamhorizontallyonthesampleto∼250m,permittingthereby tooperatethespectrometerintheslit-lessgeometry.
The incident photon flux was∼1–3×1012ph/s at the three beamlines.TheexposuretimeoftheCCDwaschosendependingon thecountrate.ForthediagramX-raytransitions,acquisitiontimes
of1sperimagewerechosen,andtoavoidmultiple-hiteventson onepixeltheincomingphotonfluxeswereattenuatedwithappro- priateabsorbers.ThehypersatelliteX-rayspectrawerecollectedin shortsuccessivescansoffewhundredCCDimageswithacquisition timesof2–10sperimage.FortheX-rayhypersatellitespectraof Cl,KandCa,twoorthreeoverlappingCCDregionsweremeasured toincludetheKMMradiativeAugertransitionsand Kˇdiagram lines.Fornormalizationpurposes,thenumberofincomingpho- tonswasdeterminedwithaphoto-diodeatthebeginningandthe endofeachX-rayemissionspectrummeasurement.Thisallowed tomonitorthephotonfluxandtocorrecttheX-rayspectraoff-line foranybeamintensityfluctuations.
TheTEOPmeasurementswereperformedatthebeamlineID21, usingthewavelengthdispersivespectrometer(WDS)[31]which wasinstalledrecentlyattheScanningTransmissionX-rayMicro- scope(STXM)chamber.TheWDSconsistsmainlyofapolycapillary opticsforthecollectionofthesampleX-rayfluorescence,aflatcrys- talandaflowgasX-raydetector.Duetotheultralowintensities oftheTEOPtransitionsandthepresenceofstrongerdiagramX-ray linesfromtraceimpuritiesinthemeasuredspectra,high-efficiency andgoodenergyresolutionwereindeedprerequisiteforthischal- lengingexperiment.FortheMgandAlmeasurements,theWDS spectrometerwasequippedwithaSi(111)crystal(2d=6.271 ˚A), whereas for the Si ones a Ge(220) crystal (2d=4.000 ˚A) was employed.TheenergycalibrationoftheWDSwasdeterminedfrom measurementsofseveraldiagramtransitions(Rh,Ru,Cl,Pd,Ag,Sn, K,andSc),usingfortheenergiesofthereferencetransitionstheval- uesreportedin[35].Thesemeasurementsalsoservedtodetermine thefullwidthathalfmaximum(FWHM)oftheGaussianinstrumen- talresponsefunction.TheFWHMwasfoundtovary,dependingon theenergy,between7and10eV.
Inordertoobtainthehighestpossiblefluxonthesamples,theSR fromtwoundulatorswasused.Theintensebeamwasmonochro- matizedusingthedoubleNi/B4Cmultilayermonochromatorand focusedonthesampleswithaKirkpatrick–BaezopticsforAlandSi, andpolycapillaryopticsforMg.Upperharmonicswererejectedby meansofNi-coatedmirrorssettoanangleof7.5mrad.Themicro- focusedincidentphotonfluxwas∼2–3.5×1012photons/s.Photon beamenergiesof3.364keVforMgand4.620keVforAlandSiwere employedtoproducethesamplefluorescence.Theseenergieswere chosentomatchtheK-shelldoublephotoionizationcross-section maximareportedin[2].Self-supportedmetallicfoilsofMg,Al,and ac-Siwereemployed.TheAlandSisamplepuritywas99.999%, andthatofMg99.9%.TheX-rayspectrawerecollectedinsucces- sivescansof∼0.5–1heach,withtotalacquisitiontimesof∼42h,
∼51h,and∼17h,forMg,AlandSi,respectively.Fornormalization purposesthephotonfluxwasrecordedwithaphoto-diodeatthe beginningandtheendofeachscan.
Since for allthree elements it wasnot possible to measure withthesamecrystalthehypersatelliteandTEOPtransitions,the branchingratiosI(K˛h)/I(K˛˛h)werederivedfromthemeasured intensityratiosbetweentheTEOPandclose-lyingreferenceKX- raydiagramtransitions.Thisapproachbenefitedfromwellknown valuesofthesingle[37]andthedouble[2]K-shellphotoionization cross sectionsand presented theadditionaladvantage tomini- mizethecorrectionsrelatedtotheenergydependentpolycapillary transmission.
3. Resultsanddiscussion
3.1. DoubleK-shellphotoionization
3.1.1. Double-to-singlephotoionizationcrosssectionratios
Thedouble-to-single photoionizationcrosssectionratiosPKK were obtained from the relative intensities of the resolved
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Fig.2.Double-to-singleK-shellphotoionizationratiosPKKforAl,Ca,andVversusthephotonbeamenergy.ForCaandVthepresentexperimentaldataarecomparedto thoseofOuraetal.[38]andHuotarietal.[13].ThesolidblacklinescorrespondtothebestfitstoourdatawiththeSO–KOempiricalmodel.
hypersatelliteK˛h(1s−2 →1s−12p−1)tothediagramK˛(1s−1→ 2p−1)X-raytransitions:
PKK= IK˛h
IK˛
ωK
ωKK, (1)
whereωKandωKK arethefluorescenceyieldsforthesingle-and double-hole states [36],respectively. Theevolution of thedou- bleK-shellphotoionizationratiosPKK withthephotonenergyfor selectedelementsispresentedinFig.2.Itshouldbepointedout, thatincontrasttoL-shellX-raysatellitelines,theM-satellitescan- notberesolved,theirenergyshiftbeingsmallerthanthenatural linewidthsoftheparentdiagramorhypersatellitelines.Theinten- sitiesoftheseM-satellitesarethereforeincludedinthemeasured IK˛andIK˛hyields.AstheM-shellshakeprobabilityisexpectedto behigherforatomswithadouble1svacancyintheinitialstatethan forthosewithasingle1svacancy,thePKKratioscalculatedwithEq.
(1)mightbesomewhatoverestimated.However,thereisnoexper- imentalevidencethattheK-shellshakeortheknock-outprocess takesplacepriortotheM-shellshake,andinfirstapproximationit isreasonabletoconsidertheproductionofthesecond1svacancy andtheM-shellvacancyasquasi-simultaneous.Wearetherefore inclinedtobelievethatthesystematicerrorrelatedtothediffer- enceintheM-shellshakeprobabilitiesresultingfromthesingleand doubleK-shellphotoionizationissmallinourcase.Thisassump- tionseemstobeconfirmedbythefactthat,despiteaverygood instrumentalresolution,noasymmetrywasobservedinthefitted hypersatellitetransitions,indicatingthatthecontaminationofthe hypersatellitetransitionsbyunresolvedM-satellitewasweakfor themeasuredelements.
Becausethephotoninteractswithonlyoneelectronandvan- ishes,insingle-photonK-shelldoublephotoionizationtheremoval ofthetwoinnermostelectronsproducingaK-shellhollowatomis duetoelectroncorrelations.TwomechanismsdominatetheK-shell doublephotoionization, namely,theshake-off(SO)process[39]
andtheinelasticelectron–electronscattering(knock-out).These electron–electroninteractionsareofquantumandclassicalnature.
ThedoubleK-shellphotoionizationviashakeisaconsequenceof thechangeoftheself-consistentfieldandelectron–electroncor- relations[39].Intheknock-out(KO)theoutgoingphotoelectron knocksoutthesecond1selectroninan(e,2e)-likeelectronimpact half-collision[40].Further,ground-stateelectroncorrelationsare importantfortheshake-off,whilstthefinal-stateelectroninter- actionsgovernthedielectronicprocess.Althoughqualitativelythe twomechanismshaveverydifferentphotonenergydependences anddifferentelectroninteractiontimes,theseparationofKOand SOandquantificationoftheinterferencesisnotstraightforward (seee.g.,[7,10,41,42,11,43]).IncomparisontoHeandHe-likeions, muchlesstheoreticalefforthasbeendevotedtotheDPIofneu- tralatoms.Thesingle-photonK-shellDPIwasaddressedwithin thelowestorderperturbationtheory[11],andafirstsystematic studyusinganabinitiononperturbativeclose-couplingapproach
wasperformedbyKheifetsetal.[42]showingdifficultiesofsucha calculationwhichisparticularlydemandingtotheaccuracyofthe groundstatewave-function.Incontrasttoabinitioapproachesin whichtheSOandKOaretreatedcoherently,alternativetheoretical modelsbasedonanincoherentpictureofthe(e,2e)-likeprocessand shake-offwereproposedbySamson[44]andelaboratedbyPattard andBurgdörfer[45].Atheoreticalmodelforanincoherentsepara- tionofSOandKOwasalsodevelopedbySchneideretal.[8,9].The latterisbasedonaquasi-classicalformulationoftheKOandthe purelyquantummechanicalnatureofSO,i.e.,SOisviewedasa quantumcorrectiontothequasi-classicallycalculateddoublepho- toionization.ForHe,anexcellentagreementwiththeexperimental datawasobtained,suggestingthatinterferencesplayonlyaminor role.
In the same spirit, to assess the effect of outer shell elec- tronsand therelative importanceofinitial-state andfinal-state electron–electron correlations to the K-shell DPI, an empirical SO–KOmodelbasedonanincoherentsummationofthedouble- to-singlecrosssectionratiosfortheshakeprocessandknock-out wasproposed[1,2].InourSO–KOmodelthedouble-to-singlepho- toionizationratioasafunctionofthephotonenergyisgivenby:
PKK(E)=PSO(E)+PKO(E), (2) with
PSO(E)=R∞exp
− (rE+)2 15.32(E−E2+)
(3a)
and
PKO(E)=PKOmax
cosh
ˇln
E−E2+EKOmax
−1/ˇ. (3b)
ThePSO(E)correspondstotheexpressionofThomas[46]forshake- off.R∞standsfortheshake-offasymptotichigh-energylimit,i.e., whenthephotoelectronisinfinitelyfast,E+isthebindingenergy oftheremainingK-shellelectron,rrepresentsthedistancein ˚A traveledbytheKphotoelectronduringthetimetheatomicpoten- tialchanges,andE2+denotestheDPIthresholdenergy.Allenergies areineV.Becausetheshake-offasymptoticnon-relativistichigh- energy limit can be calculated quite accurately for the helium isoelectronicsequenceanditisalmostthesameforneutralatoms andHe-likeions[42,47],forR∞thevaluesfromForreyetal.[48]
wereused.Fortheknock-outprobabilityPKO(E),theanalyticalform oftheuniversalshapefunctionforelectronimpactionizationofH- likeionsofAicheleetal.[49]wasadopted.Thechoiceoftheshape functionwasbasedonthesimilarityofelectron-impactionization ofaH-likeiontotheKOpartofthedoublephotoionizationofthe correspondingHe-likeion[44,10,9,45].ThePKOmax correspondsto themaximumvalueofKO,EmaxKO totheexcessenergywherethe maximumoccurs,andthepowerˇ=0.4.
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Table1
Double-to-singlephotoionizationcrosssectionratiosinthepeakregionofthephotonenergyevolutionPKKmaxandthefittingparametersoftheSO–KOmodel.TheDPIthreshold energyE2+andthePmaxKO werederivedfromthefits,whileE+valuesandthephotoabsorptionasymptoticlimitsR∞werekeptfixed.TheeffectivenuclearchargeZ*wasdeduced usingthehydrogenicformulaE+=Z*2Ry,whereRy=13.6eV.ListedarealsotheMCDFpredictionsfortheDPIthresholdenergyE2+MCDF.TheωKK/ωKwerededucedfromthe valuesquotedbyChen[36].Note,thatforClandKtheX-rayemissionspectrawerecollectedatasinglephotonenergyof13.3keVintheregionofthebroadmaximumof thedouble-to-singlephotoionizationcross-sectionratios.TheobtainedPmaxKK is8.61(1.77)×10−4forCl,and9.24(1.23)×10−4forK.
Element Z Z* PmaxKK ωKK/ωK E2+(eV) EMCDF2+ (eV) E+(eV) R∞ PKOmax
Mg 12 10.4 2.03(19)×10−3 1.27 2741(35) 2776.6 1464.8 6.08×10−4 1.74×10−3
Al 13 11.3 1.83(20)×10−3 1.24 3189(23) 3294.0 1736.8 5.20×10−4 1.55×10−3
Si 14 12.2 1.43(14)×10−3 1.21 3788(42) 3882.5 2032.5 4.47×10−4 1.20×10−3
Ca 20 17.8 1.02(10)×10−3 1.08 8039(40) 8357.0 4324.7 2.22×10−4 9.42×10−4
Sc 21 18.8 7.84(74)×10−4 1.075 9060(53) 9297.4 4791.8 2.00×10−4 6.60×10−4
V 23 20.7 6.87(63)×10−4 1.065 11,277(110) 11283.1 5798.7 1.68×10−4 6.10×10−4
Thebestfitstothedouble-to-singlephotoionizationratiosof Mg,SiandScasafunctionofthescaledexcessenergyaredepicted inFig.3andresultsoftheleast-squaresfitstotheexperimentaldata withEq.(2)arelistedinTable1.OninspectionofFig.3thepreva- lenceofKOnearthresholdandforintermediateexcessenergiescan beobserved.AthighexcessenergiesKObecomesnegligibleand thePKKratiosapproachtheSOphotoabsorptionasymptoticlimit.
TheseresultsareinaccordwiththeconclusionsofKanteretal.
[12]andHuotarietal.[13]fortheprevalenceofKOinthenear- thresholdregionandforintermediatephotonenergiesofthePKK photonenergyevolution.Theverygoodagreementbetweenthe experimentaldataforthephotonenergydependenceofPKK and theSO–KOmodelfitsupportsthisphysicalpicturefortheK-shell doublephotoionizationforlow-Zneutralatoms.
Fig.3. Double-to-singleK-shellphotoionizationratiosasafunctionofthescaled excessenergy.ForE2+valuesfromthefitwereused.Resultsofbestfitstoourdata withtheSO–KOempiricalmodelarerepresentedbyblacksolidthicklines,whereas theKOcontributionsaredepictedbythinbluelinesandtheSObyreddashedlines.
(Forinterpretationofthereferencestocolorinthisfigurelegend,thereaderis referredtothewebversionofthearticle.)
3.1.2. Double-photoionizationcrosssections
TheexperimentaldoubleK-shellphotoionizationcrosssections 2+asafunctionofthephotonenergyaredepictedinFig.4.The valuesweredeterminedemployingtherelation
2+(E)=PKK(E)+(E), (4) whereEisthephotonenergyand+standsforthesingleK-shell photoionization crosssectiondeducedfromtheXCOMdatabase [37].Forallelementsthecrosssectionsshowacommonshapechar- acterizedbyasharpriseabovethethresholdfordoubleionization tothemaximumandasubsequentrapiddecreasewiththepho- tonenergy.Pattard[50]establishedanuniversalshapefunctionfor multipleionizationbyphotonsthatprovidesanexcellentparam- eterizationofphotoionizationcrosssectionsfordoubleionization ofHe-likeions.Theanalyticalformulareads:
2+(E)=max2+x˛
˛+7/2˛x+7/2
(˛+7/2), (5)
where max2+ denotes thecrosssection maximum,˛=1.056,and x=E/Emax.EandEmaxcorrespondtoexcessenergies.Least- squaresfitstotheexperimentaldatawithEq.(5)demonstratethat theshapefunctionisapplicabletoK-shellhollowatomsaswell(see Fig.4).Thephotonenergiesandtheexcessenergiescorresponding totheDPIcrosssectionmaximascaleasEmax(Z*)=25.03(11)Z*2.08(3) and Emax(Z*)=5.72(0.07)Z*2.00(7),respectively. A power-lawfit tothemaximumvaluesof2+asafunctionofeffectivenuclear charge Z* yieldsa 0.15(5)/Z*3.68(11)fall-off. Further,asshownin Fig. 5, the double photoionization cross sections in the range 2≤Z≤47exhibitanuniversalscalingbehaviorinreducedcoor- dinates2+Z*3.68against(E−E2+)/Z*2andcoincidewiththe2+Z*4 fortheHeisoelectronicseries[1,2].
Fig.4. DoubleK-shellphotoionizationcross-sectionsversusthephotonenergy.
SolidlinesshowthebestfitswiththeuniversalshapefunctionofPattard[50].
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Fig.5. ScaledexperimentalDPIcrosssectionsforMg,Al,Si,Ca,ScandVcompared tothescaleddataforHe[40]andexperimentaldataforTi[38],Cu[13]andAg[12]
asafunctionofthescaledexcessenergy.ForneutralatomsthescalingexponentX is3.68.Thecurves(solidlines)werededucedfromtheresultsofthebestfitsofPKK
withtheSO–KOmodel.
3.2. RadiativedecayofdoubleK-shellholestates
3.2.1. One-electronone-photonandtwo-electronone-photon X-raytransitions
FollowingtheK-shellDPI,theatomicdoubly-excitedcorestate decaysin a cascade of non-radiative Auger and radiativetran- sitions.Theradiativede-excitation ofK-shell doubleholestates via the one-electron-one-photon process corresponding to the K˛h(1s−2→1s−12p−1)hypersatellitetransitionisthemaindecay channel.De-excitationthroughtransitionsfromothersubshellsis alsopossible,butlessprobable.ThealternativedecaychannelTEOP inwhichthetwoK-shellcore-holesarefilledsimultaneouslyviaa correlatedtwo-electronjumpofone2sandone2pelectronand onephotonisemittedK˛˛h(1s−2→2s−12p−1)isevenfewordersof magnitudeweaker.
These transitions permit not only to investigate the double ionizationprocess,but alsogiveinsight in fundamentalaspects ofatomicphysicsasBreitinteraction,quantumelectrodynamics (QED)andrelativityeffects.SincetheK˛h1hypersatelliteoriginates fromthespin-fliptransition(3P1→1S0)whichisdipoleforbiddenin thepureL–S-couplingscheme,theIK˛h
1/IK˛h
2 intensityratioprobes theintermediacyofthecouplingschemeacrosstheperiodictable andthevariouseffectsthatinfluencethemixing,forinstanceBreit
interactionand relativity.In fact,theTEOPtransitions areeven more sensitivetotheBreitinteractionthan thehypersatellites.
Ofinterestarealsotheradiativelinewidthswhicharerelatedto themeanlifetimesofthedoubly-excitedstatesbytheHeisenberg uncertaintyrelation=,whereisthewidthandisthelife- timeofanexcitedatomicstate.Further,astheTEOPtransitions arecorrelatedmulti-electronprocessestheycanbeonlydescribed bymany-electronmodels.Thus,boththeOEOPandTOEPradia- tivetransitionsprovideastringenttestforthemulti-configuration calculations.
Forillustration,theOEOPandTEOPX-rayemissionspectraof Mg,AlandSiareshowninFig.6.Duetothechangeintheelectronic screeningofthenuclearcharge,theenergiesoftheOEOPtransitions areshiftedwithrespecttotheirparentX-raylinesdecayingsingly- ionizedstates.AlthoughforMgandAltheK˛h1X-raytransitionwas muchtooweaktobeobserved,itwasmeasuredforthefirsttime forSi.TheobtainedEK˛h
1
and K˛h2 hypersatelliteandEK˛˛h 2
TEOP energiesaresummarizedinTable2andcomparedtothefewexist- ingdataandthemostrecenttheoreticalcalculationsfromMartins etal.[52],Costaetal.[51],Natarajan[53]andSahaetal.[56].The energiesoftheK˛h2andK˛˛h2transitionsforlowerZelementsare ingoodtoverygoodagreementwithintheexperimentaluncer- taintieswithdifferenttheoreticalpredictions.AthigherZ,theory underestimatestheexperimentalenergiesforboththeK˛h2 and K˛h1lines,andthedifferencesarefoundtobegreaterfortheK˛h1 hypersatellites.
TheIK˛h 1/IK˛h
2 intensityratiosarelistedinTable3.For Si,the experimentalratiowasfoundtobeinexcellentagreementwith theMCDFtheoreticalpredictionsofCostaetal.[51]thatinclude boththeBreitandQEDcontributions.Indeed,thesecalculations predictthatforelementsZ<18theintensityratioofthehypersatel- litelinesdoesnotfollowthesameregulartrendasthatobserved forZ≥18,butpeaksatZ=15.Thiseffectisinterpretedasbeingdue totheinteractionbetweentheholeinthe2plevelandthe3pelec- tronsthatopensseveraldecaychannelsandleadstoanincreaseof theIK˛h
1/IK˛h
2 intensityratio.ForCaandScourdatacomparewell withintheexperimentaluncertaintieswiththeory[51],butinthe caseofVthecalculationsoverestimateourresult.Thepresentratio forV,however,agreesverycloselywiththeexperimentalandthe relativisticMCDF-calculatedvaluesreportedbyDiamantetal.[59].
ThefinitelifetimegivestoanX-raytransitionlineaLorentzian shapewithanaturalwidthequaltothesumofthetotalradiative andnon-radiativewidthsoftheinitialandfinalstatesinvolvedin
a b c
e f d
Fig.6.One-electronone-photon(upperpanels)andtwo-electronone-photon(lowerpanels)transitionsofMg,AlandSi.TheTEOPspectracorrespondtoresidualsofthe measureddatasets.Thesolidlinesarethebestleast-squaresfitstothedatausingVoigtfunctions.
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Table2
Energiesoftheone-electron-one-photonandtwo-electron-one-photontransitionscomparedtootherexperimentaldataandtheoreticalpredictions.SinceCostaetal.[51]
presentenergyshiftsrelativetothecorrespondingdiagramlines,thelistedK˛h1andK˛h2transitionenergieswerecalculatedusingtherecommendedvaluesofDeslattesetal.
[35]fortheK˛1andK˛2lines.Thenotation1367.86(9/6)means1367.86±0.09eVwithanincludedstatisticalerrorfromthefitof0.06eV.
Z K˛h2(eV) K˛h1(eV) K˛˛h2(eV)
Experiment Theory Experiment Theory Experiment Theory
[52] [51] [53] [52] [51] [53]
12 1367.86(9/6) 1368.53 1367.71 1367.7 2586.7(4) 2585.45[52]
1367.8(2)[54]
1367.7(6)[55]
13 1610.38(4/2) 1611.75 1610.89 3056.5(9) 3056.54[52]
3057.49[56]
3058.68[57]
3055.99[57]a
14 1874.87(6/4) 1874 1873.99 1874.5 1881.20(12/8) 1880 1879.96 1881.6 3568.3(4) 3566[52]
1873.6(1)[58] 3567.43[56]
3569.37[57]
17 2787.80(10/8) 2785
19 3499.20(9/6) 3498 3497.77
20 3887.50(9/5) 3884.8 3885.93 3884.5 3899.80(15/9) 3896.39 3897.54 3896.3
3883.5(6)[58]
21 4296.31(8/7) 4294.16 4295.24 4309.6(7/6) 4306.27 4307.43
23 5177.65(8/6) 5174 5176.24 5192.0(9/7) 5188 5190.86
5178.1(5)[38]
5176.6(1)[59] 5191.7(1)[59]
aK˛˛hratesincludethecouplingoftheinitialandfinalstatevacancieswiththe3pelectron.
Table3
Linewidthsoftheone-electronone-photonandtwo-electronone-photontransitions,andtheIK˛h 1
/IK˛h 2
intensityratios.TheK˛h 1
forSi,CaandScwerefixedinthespectra fits.
Z K˛h
2
(eV) K˛h
1
(eV) K˛˛h
2
(eV) IK˛h
1
/IK˛h 2
Exp. [51]
12 1.49(8) 2.5(6) 0.00074
13 1.88(7) 2.9(1.7) 0.0087
14 1.86(9) 1.86 3.8(9) 0.03(1) 0.0288
17 2.86(34)
19 3.46(22) 0.0189
20 3.72(18) 3.72 0.035(13) 0.0274
21 3.88(17) 3.75 0.045(9) 0.051
23 5.54(19) 5.6(1.0) 0.077(15) 0.0989
5.5(1)[59] 6.0(6)[59] 0.08(1)[59]
thetransition.ThelinewidthsoftheOEOPandTEOPtransitionscan bethusapproximatedbythesumofthetotalatomiclevelwidthsof theinitial1s−2andfinaldouble-holeconfigurationstates1s−12p−1 and2s−12p−1,respectively:
K˛h KK+(KL−1
2,3+L2,3) (6)
and
K˛˛h KK+(L1+L2,3), (7) whereKKisthedouble-Kholestatewidth[36,53],Listhesingle- Lholestatewidth,andKL−1
2,3 isthereducedKlevelwidthdueto thepresenceoftheL-shellspectatorvacancy[62,63,58].
PresentexperimentalK˛h andK˛˛h linewidthsaresumma- rizedinTable3,andinFig.7theK˛h
2valuesarecomparedtothose calculatedusingEq.(6)andtheoreticalpredictionsofPolasiketal.
[60].Thesingle-anddouble-vacancylevelwidthsandotherexper- imentaldataarealsoshown.ThewidthsKKandKwereobtained byinterpolatingthevaluescorrespondingtotheatomicnumbers selectedbyChen[36].Theaccuracyoftheinterpolationprocedure isatthelevelof∼1–4%.TheKL−1 widthswereestimatedfollow- ingthescalingprocedureofLarkins[62]andusingfortheradiative andAugerKlevelwidthstheinterpolatedvaluesfromChen[36].
FortheL,therecommendedvaluesofCampbellandPapp[61]
wereadopted.
FromFig.7itcanbeseenthatforelementsinthe12≤Z≤30 range the K˛h linewidths calculated with Eq. (6) systemati- callyunderestimatetheexperimentalvalues.Likewise,theTEOP
Fig.7.ExperimentalOEOPlinewidthsandexperimentaldataofDiamantetal.[59]
asafunctionofZ(lefty-axis).PlottedarealsothevaluescalculatedwithEq.(6) andtheoreticalpredictionsofPolasiketal.[60].Ontherighty-axisshownarethe theoreticaltotallevelwidthsKKofChen[36]andCostaetal.[51].TheKandL2
correspond,respectively,tothelevelwidthsKandL2recommendedbyCampbell andPapp[61].
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Table4
TheK˛htoK˛˛hbranchingratiosforMg,AlandSi.Alsolistedaredifferenttheoretical predictions.TheK˛hratesfrom[53]areinthelengthgauge.
Z Experiment Theory
12 1838(258) 667[65]
928[56]
2417[57,53]
13 2115(403) 758[65]
686[68]
999[56]
2617[57,53]
2359[57,53]a
14 2610(370) 833[65]
1126[56]
3007[57,53]
aK˛˛hratesincludethecouplingoftheinitialandfinalstatevacancieswiththe 3pelectron.
linewidthsare foundtobe∼1.6times largerthan thosecorre- spondingto thesumof theinitialand final statewidthsgiven byEq.(7).ForNe,ontheotherhand,themeasuredwidthofthe Auger KK-KLL hypersatellite of1.0(1)eV[14] is consistent with thesumKK+K+2Lof1.08eV.Recentlyanewapproachbased onelaborateMulti-Configuration-Dirac-Fock(MCDF)calculations andtakingintoaccounttheinfluenceoftheeffectofopen-valence configurationand theouter-shell ionizationand excitation was proposedtoresolve thediscrepancies [60].Indeed, forselected elementsinthe20≤Z≤30 range,theoreticalpredictionsof the effective K˛h linewidths werefound tobe in good agreement withtheexperiment(seeFig.7).Thus, itcanbeconcludedthat anaccountofnon-lifetimebroadeningeffectssuchasthecomplex multipletstructureoftheX-rayspectraresultingfromthemulti- configurationstatesandmultiple-vacancies,thesolid-stateeffects, andalsotothemultiplet splittingdue theexchangeinteraction betweenthecore-holesandtheincompletevalence-shellsinthe theoreticalX-rayspectraisprerequisiteforcomparisonwithexper- iment.Onthetheoreticalside,calculationsforlowerZelements andK˛˛htransitionsarecertainlycalledfor.Ontheexperimental side,measurementsofTEOPtransitionsbymeansofhighenergy resolutionX-rayspectroscopytechniquesarealsoneeded.
3.2.2. One-electronone-photontotwo-electronone-photon branchingratios
Theobtainedmeanvaluesofthebranchingratiosaresumma- rizedinTable4andplottedinFig.8alongwithdatafromheavy-ion collisionexperiments[17–21],andtheZ-dependenttrendsofdif- ferenttheoreticalapproaches.Todeterminethebranchingratios (BR)thefollowingexpressionwasemployed:
BR= Ir IK˛˛h
KK
rK n nr
ωKK
ωKr Fexp, (8) whereIK˛˛handIrstandfortheintensitiesoftheK˛˛handtheclose- lyingreferenceKX-raydiagramtransitions,respectively.KKand rK arethedouble-andsingle-K-shellphotoionizationcrosssec- tions,ωKKandωrKarethefluorescenceyieldsforthedouble-and single-holestates,andnandnrdenotethenumberofatomsperunit volume.Fexp istheexperimentalcorrectionfactoraccountingfor therelativedifferencesinthephotonflux,crystalreflectivity,detec- torefficiency,transmissionofpolycapillaryoptics,self-absorption, andrelativetransitionprobabilitiesoftheK-shellemissionlines.
ForωrK,valuesfromRef.[64]wereadopted,andforself-consistency thoseofωKKwererescaledaccordinglyfromtheωKK/ωKratios[2].
Inourapproachwetookadvantageofthewellknownvaluesofthe single[37]andthedouble[2]K-shellphotoionizationcrosssec- tions.ForeachelementtworeferenceKX-raytransitionswereused todeducetheBR,i.e.,forMgtheClK˛ofNaClandKClsamples,for
Fig.8.K˛htoK˛˛hbranchingratiosforMg,AlandSi(opencircles)togetherwith theoreticalpredictionsasafunctionoftheatomicnumberZ.Theplotscorrespondto power-lawfitstothedatasets,whereasthesolidlinerepresentsaZ2-dependence.
ExperimentalresultsfromHIcollisionexperimentsarealsoshownforcomparison.
AltheKK˛andClK˛ofKCl,andforSitheScK˛andKKˇ.The BRvaluesforMgof1880(373)and1800(357),Alof2040(541)and 2208(605),andSiof2625(512)and2594(536),werefoundtobe consistentwithintheexperimentaluncertainties.
Sincethetwo-electronone-photonradiativedecaycorresponds toatransitionbetweencorrelatedmulti-electroninitialandfinal atomicstates,thetransitionratesareverysensitivetoanaccurate theoreticaltreatmentoftheelectroninter-andintra-shellinter- actions.Indeed,theavailablepredictionsforthebranchingratios showimportantdifferences(seeFig.8).Thepredictionsof ˚Aberg etal.[65],GavrilaandHansen[66],Baptista[67],Costaetal.[68], and Sahaet al. [56]underestimate ourexperimentalbranching ratios.TheBRvaluescomparebesttothemostrecentrelativistic configurationinteraction(RCI)calculationsofKadrekarandNatara- jan[57,53]andtothemany-bodyperturbationtheorypredictions [69–71].NoteworthyisthegoodagreementoftheRCIcalculations [57,53]withtheexperimentalbranchingratioforAlwhenthecou- plingbetweentheinner-shellvacanciesandtheouterincomplete subshellsisincluded.Thereportedresultsgiveanimportantpoint of comparison fordifferenttheoretical modelsthataddress the many-bodyproblemanddemonstratethepotentialoftheTEOP radiativedecayofK-shellhollowatomstounravelelectroncorre- lations.
4. Concludingremarksandoutlook
Understandingelectron–electroninteractions isnotonlyone ofthekeyissuesofatomicphysics,butisalsoimportantfor an accuratetheoreticaldescriptionofcomplexsystemsandprocesses in the fields of physics and chemistry. Yet, onthe theoretical side,anexacttreatmentofelectroninteractionsinmany-electron systemsstillrepresentsaformidablechallenge.Single-photondou- ble ionization process producing core–shell hollow atoms and molecules[72,73]isagrowingfield,andthepossibilitytoinves- tigate experimentallyultrafast electron dynamics withinatoms withXFELs opensnewexplorationroutes [74–77].Young etal.
[3]reportedonhollowNeatomscreatedthrougharapidphoto- ejection of inner-shellelectronsin an ultra-intenseXFEL beam andtheintensity-inducedX-raytransparency,whilethenon-linear atomicresponsetointenseX-raypulseswasreportedbyDoumy etal.[78].ThesestudieswereperformedbymeansofAugerand photoelectronspectroscopy.Newaspectsofhollowatomforma- tionanddecaysuchasX-rayemissionfromresonantlypumped double-holeK-shellstatesofAlwereobservedbyVinkoetal.[79].
