Multiconfiguration Dirac-Hartree-Fock calculations of transi(2)

Inthesecondstage,thecon guration-interactioncalculations(i.e.,withnochangestotheradialwavefunctions)wereperformed,withmulticon gurationexpansionstailoredinsuchaway,astocapturethedominantelectroncorrelationcontributionstotheexpectationvalues.Thevalenceandcore-valencee ectsconstitutethedominantelectroncorrelationcontributionsintheoscillatorstrengthcalculations[37],thereforeallsingleanddoublesubstitutionswereallowedfromseveralcoreshellsandbothvalenceshells(i.e.,7s2,7s7p,or7s6d,dependingonthestate)toallvirtualshells,withthesamerestrictionasabove,i.e.thatatmostoneelectronmaybepromotedfromcoreshells.Thevirtualsetwassystematicallyincreasedfromoneto velayers,untiltheconvergenceoftransitionrateswasobtained.Inasimilarmanner,severalcoreshellsweresystematicallyopenedforelectronsubstitutions—fromtheoutermost6pto5s5p5d6s6pshells.Thee ectsofsubstitutionsfrom4s4p4d4fshellswereneglected.Weestimatedthemseparatelyforthreestatesanddiscoveredthattheychangethecalculatedvaluesoftransitionratesbynomorethanafractionofapercent.Thetransitionrateswerecalculatedwiththebiorthonormaltechnique[38,39],whichpermitstheapplicationofstandardRacahalgebra,whileretainingtheadvantageofwavefunctionsseparatelyoptimizedforeachstate.ExperimentalvaluesoftransitionenergiesfromMoore’stables[8]wereusedincalculationsoftransitionrates.

4Results

4.1Themetastable7s7p3P0state

Inprincipletherearethreepossibledecaychannelsofthe7s7p3P0state.Itcandecayto(theonlylowerlying)groundstatethrough(1)ablackbodyradiationinduceddecay,(2)a2-photonE1M1transition,orthrough(3)ahyper neinducedtransition.The rstisbeyondthescopeofthepresentpaper,sinceitdependsontheambienttemperature.Oftheothertwo,theformercanbeestimatedthroughanorder-of-magnitudecomparisonwiththeE1M11s2p3P0–1s21S0two-photontransitioninhelium-likeheavyions.Westartedfromarecentevaluation[40],whichgives3.14×109s 1forRa.Inordertoobtainagooddependenceontransitionenergyandradialmatrixelements,weevaluatedtheE1M1matrixelements(followingEq.(2)ofRef.[40])

The multiconfiguration Dirac-Hartree-Fock (MCDHF) model has been employed to calculate the transition rates between the nine lowest levels of radium. The dominant rates were then used to evaluate the radiative lifetimes. The decay of the metastable $ 7s7p

4Willbeinsertedbytheeditor

forHe-likeRa(usingonly1s2p3P1asanintermediatestate)andforneutralRa(usingonlythe7s7p3P1).Thenweintegratedoverphotonenergieswiththe ve-pointGauss-Legendreformula.Finally,weevaluatedtheratioofthesetwovaluestoscaletheHe-likeE1M1rate.Thisgivesanorderofmagnitudeestimateof9.6×10 3s 1.Suchlifetime(≈100s)ofthe7s7p3Pstatewouldbecomparabletothelifetimesofnucleiofseveralradiumisotopes(itwouldbe0signi cantlyshorteronlyincomparisonwiththenucleiofthemoststableradiumisotopesspanningthemassrange223-229).Thiswouldbethecaseofspin-zeroisotopesofradium.

Fortheisotopesofradiumwithanonzerovalueofnuclearspin,thehyper ne-inducedtransitionmustalsobeconsidered.Weestimatedthe7s7p3Psimple0–7s21S3

three-statemodel,inwhichthewavefunctionsofthehyper necomponents0transitionratewithaoftheupperP0statearedescribedbyasymmetry-adaptedcon guration-state-functionexpansionoftheform

|7s7p3P0IF HFS=c0|7s7p3P0IF +c1|7s7p1P1IF +c2|7s7p3P1IF (3)runningovertheappropriatehyper necomponentsIFofthe1Ptransitionrate(ins 1)maybeapproximatelyexpressed1and3Pas

1states.Thehyper ne-inducedA(7s7p3P0o

→7s21S0)=

2.02613×1018

I(I+AHFM1S(1P1,3

1)

P0)

AHFM1S(3I(I+1)

P1,3

P0)

The multiconfiguration Dirac-Hartree-Fock (MCDHF) model has been employed to calculate the transition rates between the nine lowest levels of radium. The dominant rates were then used to evaluate the radiative lifetimes. The decay of the metastable $ 7s7p

Willbeinsertedbytheeditor5

1

s(if thetwo-photontransitionisswitchedo thee ectiveHamilto-A3P0 1S0=0.02935 3

1

nianmethodyieldsAP0 S0=0.0197s 1).Intheprocessofthiscalculation,wefoundoutbycomparisonwiththeresultsofmdfgmecode[44]thatnon-orthogonalitybetweenspinorbitalsplaysveryimportantroleintheevaluationofnon-diagonalhyper nematrixelements(neglect-ingitmayevenleadtoasignchange).Weemployedanewcode,developedbyoneofus(P.J.)toevaluatecorrelatedo -diagonalhyper nematrixelementsfromGRASPwavefunctions,takingintoaccountthee ectofnon-orthogonalitybetweenspinorbitals.

(1)o

Theaccuracyislimitedbytheelectricdipolematrixelements 7s7p3P1 Q1 7s21S0 and

(1)oS31

7s7p1P1 Q1 7s21S0 aswellasbytheo -diagonalmagneticdipoleconstantsAHFM1(P1,P1),

(1)33o213HFS1S3

S0 weAHFM1(P1,P0)andAM1(P1,P0).Incaseofthematrixelement 7s7pP1 Q1 7s31

mayassumethe5%relativeaccuracyfortheP1 S0transitionratefromexperiment[5].TheaccuracyoftheA(1P1 1S0)rateismoredi culttoestimate(seeref.[25]).Thisisthestrongest,’allowed’transitionintheradiumspectrum,but(asmentionedabove)thecontributionofits

(1)o

matrixelement 7s7p1P1 Q1 7s21S0 tothetotalvalueofthecalculatedrateofthehyper ne-(1)oinducedtransition3P0 1S0is3.7timessmallerthanthatofthe 7s7p3P1 Q1 7s21S0 matrixelement,soevenarelativelylargeerrorbarwouldbequenched.Wemayveryconservativelytaketheentiredi erencebetweentheBabushkinandCoulombgauge nalvaluesfromtable1inreference[25]astheerrorlimit,obtainingarelativeaccuracy25%fortheA(1P1 1S0)transitionprobability.Theaccuracyofthediagonalmagneticdipoleconstantswasestimatedtobe6%(see[26]).Sinceano -diagonalconstantdependsonbothstatesratherthanonone,we,againquiteconservatively,doubledthe’diagonal’errorlimitestimateandassumed12%asthecontributionsoftheo -diagonalhyper neconstantstotheerrorbar.Eventually,theaboveprocedureyieldsthelifetimeofthemetastable3P0stateτ=34(15)sof22388Raisotope.Thelifetimeisbasedonboththe2-photonandhyper ne-inducedchannels.Theerrorbardoesnotincludethe2-photoncontribution.

4.2The7s7p3P1state

Figure1presentsthetransitionprobabilityA(3P1 1S0)asafunctionofthesizeofthemul-ti …… 此处隐藏：6125字，全部文档内容请下载后查看。喜欢就下载吧 ……