Time‐dependentclose‐couplingmethodsforelectron‐atom/

moleculesca6ering

JamesColgan1,MitchPindzola2

1Theore9calDivision,LosAlamosNa9onalLaboratory,LosAlamos,NM2DepartmentofPhysics,AuburnUniversity,Auburn,AL

Outlineoftalk•  Theore9calapproachestoelectron‐impactioniza9on–briefdescrip9ons–  Time‐dependentclose‐coupling(TDCC)method

•  Strengthsandweaknesses•  Considera9onsofconvergenceissues

–  Distorted‐wave(DW)approaches•  EIIcrosssec9ons:overviewofcalcula9onsperformedtodate–  Selectedexamplesofioniza9onofneutralatoms,ionsandafewmolecules

•  EIEcrosssec9ons–lessdoneinthisareausingTDCCbuttherearesomeexamplesofLiandBeq+ions

•  Inclusionofioniza9ondatainpublicdatabases•  Conclusions&outstandingissues

– Whats9llneedstobedone–  Apathforward?

TDCC approach to electron-impact ionization •  The TDCC method centers around solution of the time-dependent Schrödinger

equation for two interacting electrons

•  The electron-electron interaction between the outgoing electrons from an ionization process is treated exactly; a close-coupling, or non-perturbative approach

•  In electron-impact ionization all possible LS final states are possible (1,3S, 1,3P, 1,3D, …), and one needs to monitor carefully the convergence with respect to the number of states –  For moderate projectile energies, the number of partial waves which is required for

convergence is usually quite manageable. •  Probabilities for ionization may be extracted at some final time via standard

projection methods •  For multi-electron targets, the interaction of the remaining electrons with the

two outgoing electrons is treated through additional direct and local exchange potential terms

i!PLS

(r1, r2, t)

!t= T

l1l2PLS(r1, r2, t)+ V

l1l2l '1 l '2PLS(r1, r2, t)

l '1 l '2

"

Why is a time-dependent approach useful? •  The TDCC approach avoids the troublesome boundary condition

associated with the three-body Coulomb problem by propagating the equations that describe the electron motion until the interactions are small, and so that converged probabilities for a given scattering process may be extracted.

Plots of the radial component of the total wavefunction show how the two electrons interact

t=0 t=25

•  Suchissuesinclude–  Radialmeshchosenbyuser–  Numberofpar9alwavesretainedinexpansion– Whethertop‐up(withDWpar9alcrosssec9ons)wasusedandatwhatpar9alwavethesewereapplied

–  Choiceofcorepoten9alforoneortwo‐electroncalcula9onsoutsideofafrozencore

•  Anexperiencedusershouldbeabletominimizetheuncertaintyarisingfromsuchconsidera9ons,buts9lltediousand9me‐consumingtes9ngistheonlywaytochecktheseissues

•  Theseare“numerical”issues;othermore“physics‐based”issuesalsoexist

UncertaintyissuesinaTDCCcalcula=on

UncertaintyissuesinaTDCCcalcula=on

•  Theinterac9onofoutgoing(ac9ve)electronswiththeremainingcoreelectronscanleadtouncertaintyinthetotalcrosssec9onpredic9onfromTDCCcalcula9ons

•  TheTDCCapproachtoioniza9onisaconfigura9on‐averagedapproachandsotermdependenceoftheinterac9onoftheoutgoingelectronswiththecoreisnotproperlyincluded

•  ExamplesofthiswillbepresentedforNe

•  Thedistorted‐waveapproachtoelectron‐impactioniza9on(orexcita9on)isawell‐establishedmethod–  Theincident,sca[ered,andejectedelectronicwavefunc9onsareall‘distorted’bythepoten9al

ofthetargetatom(includingnuclearandelectronpoten9alterms)–  Theradialdistortedwavesareevaluatedusingaone‐electronHamiltonianofaformsimilarto

–  ThesedistortedwavesarethenusedtocomputeSlaterintegralswhicharesummedappropriatelytocalculatethesca[eringprobability

–  Theincident/sca[ered/ejectedelectronradialorbitalseachcanbecomputedineitheraVNorVN‐1poten9al•  AcommonchoiceistousetheVNpoten9alfortheincidentandsca[eredorbitalsandaVN‐1

poten9alfortheejectedorbital•  Alterna9vesincludecompu9ngallorbitalsinaVN‐1poten9al,thisleadstobe[er

orthogonalityproper9esforthewavefunc9ons•  Athighenergiesorforhighlyionizedtargets,eitherchoiceproducessimilarioniza9oncross

sec9ons–  Atlowenergies,orforneutral/near‐neutralsystems,thetwochoicescanproducedifferent

results–  Distorted‐wavemethodsareconsidered‘perturba9ve’inthattheinterac9onbetweenthe

electronsisnottreatedtoallorders

Distorted‐waveapproachtoelectron‐impactioniza=on

Electron‐impactioniza0onofH:TotalCrosssec0ons

•  EarliestTDCCcalcula9onsweremadefor(e,2e)ofH

•  Close‐couplingcalcula9ons(CCCandthenTDCC)wereshowntoprovidesignificantlybe[eragreementwithmeasurementthandistorted‐wave(DW)calcula9ons

•  ForH,theDWcalcula9onsare~20‐30%largerthanthemeasurements;atrendthatpersistsformanyneutralsystems

•  Progressmadesincethiswork‐includingaccuratedescrip9onsoftheangulardistribu9onsoftheoutgoingelectrons(TDCS)–leadstotheacceptancethat(e,2e)ofH(andH‐likesystems?)isasolvedproblem

Pindzola&Robicheaux,PRA54,2142(1996)

H(1s)

•  Latercalcula9onsdrama9callyshowedhowdistorted‐wavemethodsbecomelessaccurateformoreexcitedstatesofatoms

•  StudywasmadeforHandH‐likesystems

•  Thisworkalsoclearlyshowshowthecrosssec9onfromexcitedatomicstatesismuchlargerthanfromthegroundstate

Griffinetal,JPB38,L199(2005)

Electron‐impactioniza0onofH:TotalCrosssec0ons H(ns)

•  Alsoinves9gatedweresomeH‐likeions(Li2+);

•  DW1(mixedpoten9als)doesverywellforgroundstate

•  Althoughthisisnotnecessarilytrueforexcitedstates,whereDWapproachesappearlessaccurate

Electron‐impactioniza0onofLi2+:TotalCrosssec0ons

Griffinetal,JPB38,L199(2005)

Li2+(ns)

•  TDCCcalcula9onsforelectron‐impactioniza9onofHefindsimilarconclusionstotheHioniza9onwork

•  AgainDWover‐es9matesmeasurementforneutralsystemsby~20‐30%

Pindzola&Robicheaux,PRA61052707(2000)

Electron‐impactsingleioniza0onofHe:TotalCrosssec0ons He(1s2)

•  Cau9onarytale:–  Experimentisnotguaranteedtobeperfect(!)

•  Forioniza9onfromexcited‐stateHe,measuredcrosssec9onshavebeenques9oned:–  Threeclose‐couplingcalcula9onsareingoodagreementwitheachotherbutnotwithmeasurement•  MeasurementbyDixon,Harrison&Smithmadein1976

•  Anewmeasurementherewouldbemosthelpful

Colganetal,PRA66,062707(2002)

Electron‐impactsingleioniza0onofHe(1s2s) He(1s2s)

•  Cau9onarytale:–  Experimentisnotguaranteedtobeperfect(!)

•  Forioniza9onfromexcited‐stateHe,measuredcrosssec9onshavebeenques9oned:–  Threeclose‐couplingcalcula9onsareingoodagreementwitheachotherbutnotwithmeasurement•  MeasurementbyDixon,Harrison&Smithmadein1976

•  PWBcalcula9onsinthe70sagreedwellwiththemeasurement;seeminglyanunfortunatecoincidence

Colganetal,PRA66,062707(2002)

Electron‐impactsingleioniza0onofHe(1s2s) He(1s2s)

•  Electron‐impactioniza9onofLigroundstate;againtheoriesagree(TDCC/CCC)butarelowerthanmeasurement– Measurementmadein1960s–  Anothermeasurementwouldbeusefulalso

•  ForexcitedstateRMPS,TDCCandCCCcalcula9onsareallinexcellentagreementaswell–butnomeasurements

Colganetal,PRA63,062709(2001)

Electron‐impactsingleioniza0onofLi(1s22s)Li(1s22s)

•  Asystema9cstudywasperformedusingTDCCofelectron‐impactioniza9onofallBeions,includingexcitedstates

•  ForneutralBe,3methodsareinreasonableagreement– Butnomeasurements

Colganetal,PRA68,032712(2003)

Electron‐impactsingleioniza0onofBe(1s22s2) Be(1s22s2)

•  ForBe+measurementsdoexist

•  Threetheore9calapproachesagreewellwitheachotherbutareagainlowerthanmeasurement

•  Perhapsitis9meforacampaigntore‐measuretheioniza9onfrommanyoftheselightatomicsystems?

Electron‐impactsingleioniza0onofBe+(1s22s)

Colganetal,PRA68,032712(2003)

Be+(1s22s)

•  TDCCandRMPScalcula9onsinverygoodagreement

•  DWapproachesappearreasonableforthistwice‐ionizedion– ButDWlessaccurateforexcitedstateioniza9on

Electron‐impactsingleioniza0onofBe2+(1s2)

Colganetal,PRA68,032712(2003)

Be2+(1s2)

•  Electron‐impactioniza9onofB(1s22s22p)

•  TDCCcalcula9onsareconsiderablylowerthanDWcalcula9ons–althoughnomeasurementswithwhichtocompare–  DWcalcula9onswithmixedpoten9alsexhibitashaperesonanceatlowenergies(unphysical)

–  BinaryencountermethodduetoY‐KKiminreasonableagreementwithTDCCresults

Electron‐impactsingleioniza0onofboronanditsions

Berengutetal,PRA76,042704(2007)

B(1s22s22p)

•  OldermeasurementsmadebyJILAgroupofGordonDunn

•  Measurementsdidnotknowfrac9onofmetastablecomponent(1s22s2p3P)intheionbeam

•  NewmeasurementsatORNL(Fogle&Bannister)wereabletodetermineametastablefrac9onof~9%

•  TDCCandRMPScalcula9onsingoodagreementwiththenewmeasurements

•  Furthercalcula9ons(notshown)appeartoconfirmthe9%metastablecomponent

Berengutetal,PRA78,012704(2008)

B+(1s22s2)

Electron‐impactsingleioniza0onofB+

•  Singleioniza9onofneutralC•  TDCCcalcula9ons(squares)agreewellwithmeasurement

•  DWaround15%largerthanTDCCcalcula9onsatcrosssec9onpeak

•  AmorerecentTDCCcalcula9onbyAbdel‐Nabyetal,PRA87,022708(2013)alsoexaminedexcited‐stateioniza9on

Electron‐impactsingleioniza0onofcarbon C(1s22s22p2)

Pindzolaetal,PRA62,042705(2000)

•  Singleioniza9onofexcitedC+(1s22s23l)

•  TDCCcalcula9onspredictadirectioniza9oncrosssec9on~20%lowerthanDWcalcula9on

•  RMPScalcula9onssignificantlyhigherduetostrongexcita9on‐autoioniza9oncontribu9onfromtermsarisingfromthe1s22s2p3sconfigura9on

Electron‐impactsingleioniza0onofexcitedcarbonions C+(1s22s23s)

Ballanceetal,PRA84,062713(2011)

•  Jointtheory/experimentalstudyofsingleioniza9onofC2+(1s22s2)

•  Experimentwasabletoes9matefrac9onofionbeaminametastablestate(around60%)

•  RMPS/TDCC/CCCcalcula9ons,allmadeforbothgroundandmetastablestates,agreewellwithmeasurementusingthe40/60%ionbeammixture

Electron‐impactsingleioniza0onofC2+ C2+(1s22s2)

Lochetal,PRA71,012716(2005)

•  O1+toO4+

•  Müllergroup(Giessen,Germany)performedaseriesofioniza9onmeasurementsfromlow‐chargedOions

•  ForO+,TDCCcalcula9onsareingoodagreementwiththemeasurements

Lochetal,PRA67,042714(2003)

Electron‐impactsingleioniza0onofoxygenions O+(1s22s22p3)

•  TDCCcalcula9onsaresystema9callylargerthanthemeasurements,asareDWcalcula9ons

•  Howeverterm‐dependentDWcalcula9ons(dashedline)lowerthecomputedcrosssec9ontoclosetotheexperimentalresult–  Thisterm‐dependentcalcula9onusedthe

2p5kd1PHFpoten9alsintheDWcalcula9on

–  insteadofthe2p6configura9on‐averagedpoten9al

•  Demonstratesthattheterm‐dependenceoftheoutgoingelectronswiththe2p5coreneedstobetreatedaccurately

•  ItwouldbedesirabletoincludesuchHartree‐Fockinterac9onsinaTDCCapproach

Electron‐impactsingleioniza0onofneon

Pindzolaetal,PRA62,042705(2000)

Ne(1s22s22p6)

Electron‐impactsingleioniza0onofneon Ne(1s22s22p53s)

Ballanceetal,JPB37,4779(2004)

•  Excited‐state2p53s3PtermofNe

•  RMPSdoeswellcomparedtomeasurement;whenaveragedoverterms,RMPSagreesverywellwithTDCC

•  Excited‐state2p53s3PtermofNe

•  RMPSdoeswellcomparedtomeasurement;whenaveragedoverterms,RMPSagreesverywellwithTDCC

Electron‐impactsingleioniza0onofneon Ne(1s22s22p53s)

Ballanceetal,JPB37,4779(2004)

•  GroundstateofSi•  DWsignificantlyover‐es9matesmeasuredioniza9oncrosssec9on

•  BEBmethodduetoY‐KKiminreasonableagreement

•  NoTDCCmeasurements(yet)

Electron‐impactsingleioniza0onofsilicon

Si(1s22s22p63s23p2)

Colganetal,PRA77,062704(2008)

•  GroundandexcitedstatesofSi7+

•  Highlyionizedion–wecomputecontribu9onsfrom2s&2psubshells

•  DWcalcula9onisinverygoodagreementwithmeasurement

Electron‐impactsingleioniza0onofsiliconions

Si7+(1s22s22p3)

Colganetal,PRA77,062704(2008)

•  VeryrecentTDCCcalcula9onsweremadeforioniza9onofNaandMg

•  Calcula9onsmadeusinganewerTDCCversionwithabe[errepresenta9onofnon‐ionizedcoreelectrons

•  Tocompareagainstmeasuredangulardistribu9ons(TDCS)madebyManchestergroup–  Goodagreementfoundinbinary&recoilpeaksintheTDCS

•  Totalcrosssec9onagreedwellwithpreviousmeasurements

Electron‐impactioniza0onofNa&Mg

Armstrongetal,PRA88,042713(2013)

Na(1s22s22p63s)

•  3‐electronTDCCcalcula9onsweremadefordoubleioniza9onoftheH‐ion

•  Notveryrelevantformodeling,butagoodexampletodemonstratehowtheorycandiscriminatebetweenconflic9ngmeasurements

•  TDCCcalcula9onsinmuchbe[eragreementwithmeasurementsofDefrancegroup(1992)thanwitholdermeasurementsfromPeart&Dolder(1971)

•  Otherthree‐electronTDCCcalcula9onshavebeenmadefordoubleioniza9onofHe&Be

Electron‐impactdoubleioniza0onofatoms:H‐

Pindzolaetal,JPB39,L127(2006)

•  FirstmolecularTDCCcalcula9onofelectron‐impactioniza9onexaminedH2

+,thenH2•  TDCCcalcula9oninexcellentagreementwithmeasurement

•  Angulardistribu9onsalsoavailableforioniza9onofH2byelectron‐impact–  Thesecomparequitewellwithcoincidencemeasurements

Electron‐impactioniza0onofmolecules:H2

Pindzolaetal,PRA73,052706(2006)

•  TDCCcalcula9onwellsuitedtoprovidingexcita9oncrosssec9onsforone‐electronsystems

•  ComprehensivestudyofLiexcita9oncrosssec9onsdemonstratedgoodagreementbetweenthreeclose‐couplingmethods:RMPS/TDCC/CCC

•  ConvergenceinRMPScalcula9onswasalsoinves9gated

Electron‐impactexcita0onofLi

Griffinetal,PRA64,032718(2001)

•  RMPScalcula9onsprovidedcomprehensivesetofdataforallBeions

•  TDCCwasusedtobenchmarkthecrosssec9onsforBe+

•  Agreementagainverygoodbetweenallthreemethods

Ballanceetal,PRA68,062705(2003)

Electron‐impactexcita0onofBe+

ConclusionsofqualityofTDCCioniza=ondataformodeling

•  TheAuburngrouphasdemonstratedthatuseofimproved(close‐coupling)crosssec9onsforHandHedoesmakeadifferencetoplasmatransportstudies–  Theyalsoinves9gatedtheapproxima9onofusingsemi‐classicalapproachesorDWapproachesinsteadofclose‐couplingdata

–  Thisstudyallowedanassessmentofhowwellsuchapproachesmightworkforcaseswherenoclose‐couplingdataareavailable

–  Thesemi‐classicalapproachwaspreferredforneutralsbuttheDWapproachwaspreferredforions

Lochetal,PPCF51,105006(2009)

Ioniza=ondatawithindatabases•  Allioniza9ondata(fromTDCCandothersources)forlightelementsfromHthroughBhavebeenincludedintherecentversionsoftheADASdatabase–  GeneralizedRatecoefficientsaswellasrawcrosssec9onsarestored

–  Excita9onratesalsoincluded–  Publishedinaseriesofdetaileddatapapers:

•  Lochetal,PPCF51,105006(2009);“Theroleofexcitedstateioniza9ondataonHandHegeneralizedcollisionalradia9vecoefficients”

•  Lochetal,ADNDT92,813(2006);Generalisedcollisional‐radia9vemodelforlightelements.A:datafortheLiisonuclearsequence

•  Lochetal,ADNDT94,257(2008);Generalizedcollisionalradia9vemodelforlightelements:B:datafortheBeisonuclearsequence

•  Lochetal,JPhysConfSeries(2014);Generalizedcollisionalradia9vemodelforlightelements:C:datafortheBisonuclearsequence

•  Papersforthcomingontheratesforthecarbonisonuclearsequence

ConclusionsofuseofTDCCioniza=ondataformodeling

•  Wecandis9nguishbetweenthreeareasofioniza9ondataforatoms:

– Neutralatoms– Excited‐statesofneutrals&near‐neutrals– Mul9ply‐chargedIons

ConclusionsofuseofTDCCioniza=ondataformodeling

•  Neutralatoms–  Forone‐electronandtwo‐electronsystemsTDCCperformswellandhasbeenshowntogiveaccurateresults

–  Compareswelltootherclose‐couplingapproaches– DWapproaches,invariousforms,typicallyover‐es9matecrosssec9onby20‐50%,dependingontarget

–  TDCCismoredifficultforatomsinwhichsignificanttermdependenceisexhibitedintheini9alstate•  i.e.whereconfigura9on‐averageapproachislessapplicable

ConclusionsofuseofTDCCioniza=ondataformodeling

•  Excited‐statesofneutrals&near‐neutrals–  TDCC(alsootherclose‐couplingcalcula9ons)areowenmoredifficultforexcitedstates•  Duetoslowerconvergenceproper9eswithrespecttoradialmesh,par9alwaveexpansions,etc

–  Calcula9onsperformedtocompletenessshouldbeaccurateforone‐electronandtwo‐electronsystems

– DWcalcula9onshavebeenshowntobecomelessaccurateforexcitedstatesofneutralsandfew‐chargedions

ConclusionsofuseofTDCCioniza=ondataformodeling

•  Mul9ply‐chargedIons– DWapproachesrapidlybecomemoreaccurateasthechargeontheionincreases

– Lessneedforclose‐couplingapproacheswhenthenuclearchargedominates

– Broadlyspeaking,thisistrueforionsthataremorethantwiceionized•  Butsuchassump9onsmustbechecked–especiallyifexcitedstateioniza9onisimportant!

BroaderConclusionsforioniza=ondata(1)

•  Usuallyweassumethatexperimentistheul9matearbitertodis9nguishbetweencalcula9onsandtoassessaccuracyofdata

•  Howeverinsomecasesthereiscompellingevidencethatsomemeasurementsneedtobelookedatagain–  Forcasesforwhichtwoorthreeclose‐couplingapproachesagreewellwith

eachother,butnotwithmeasurement–  OccursforneutralHe*,Li,Be*,and….

•  Weencouragemoremeasurements,especiallyforneutralsystems•  Thesearedifficultmeasurementsthatmustbeperformedcarefully

–  determina9onofmetastablecomponentofionbeamisachallenge–  alsofundingconstraintsareasignificanthurdle

•  Howeveritisdifficulttobetrulyconfidentinadatasetunlessseveralcalcula9onsandseveralmeasurementshavebeenperformed–  Isthiswishunrealis9ctoday?!

BroaderConclusionsforioniza=ondata(2)

•  Tungsten–  Wehavemostlyignoredheavierelementsinthistalksofar–  However,Wisofprimeimportancetofusionmodeling,andtransi9onmetals

areowenkeytounderstandingastrophysicalplasmas–  DWcalcula9onshavebeenperformedforallionsintheWisonuclear

sequence•  S.D.Loch,etal,Phys.Rev.A72,052716(2005)

–  Similarcalcula9onsalsomadefortheArisonuclearsequence–  TheDWcalcula9onsarelikelytobeaccurateformoderatelyandhighly‐

chargedions,butmuchlessaccurateforneutral&near‐neutral•  Someevidencethationiza9onofW4+orW5+ionsisreasonablywelldescribed

–  However,noclose‐couplingmethodhasbeenshowntobefeasibleforneutral/near‐neutralW

–  Thesignificantcomplexityoftheatomicstructureofsuchionstagesisthemainreason

–  TDCCmethodsareunlikelytobeappliedtosuchionstagesinthenearfuture

–  RMPScalcula9onsmightbepossiblebutares9llextremelydaun9ng!

Pathforward

•  Abenchmarkstudywouldbeausefulwayforwardfortheatomiccollisioncommunity– TDCCmethodscancontributetosuchastudyviaaccurateioniza9oncrosssec9ons,givenasuitablesystem.

– Unfortunately,ionsthatmightbeofmostinteresttofusion,suchaslow‐chargedTungstenions,areoutofreachfromTDCC(andotherclose‐coupling?)calcula9onsatpresent

– AlowerZneutralatomorlow‐chargedionmightbemostviablecandidateforsuchabenchmarkstudy