THE TIME-DEPENDENT CLOSE-COUPLING METHOD FOR ELECTRON-IMPACT DIFFERENTIAL IONIZATION CROSS SECTIONS FOR ATOMS AND MOLECULES

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1 Intenatonal The Tme-Dependent cence Pess Close-Couplng IN: 9-59 Method fo Electon-Impact Dffeental Ionzaton Coss ectons fo Atoms... REVIEW ARTICE THE TIME-DEPENDENT COE-COUPING METHOD FOR EECTRON-IMPACT DIFFERENTIA IONIZATION CRO ECTION FOR ATOM AND MOECUE M.. PINDZOA F. ROBICHEAUX AND J. COGAN Depatment of Physcs Aubun Unvesty Aubun A UA Theoetcal Dvson os Alamos Natonal aboatoy os Alamos NM UA Abstact: We evew the applcaton of the tme-dependent close-couplng method to calculate enegy and angle dffeental coss sectons fo the electon-mpact onzaton of atoms and molecules. PAC Numbes: 4.8.Dp 4.8.Gs.. INTRODUCTION The tme-dependent close-couplng method was developed to study few-body dynamcs n atomc and molecula collson pocesses []. Applcatons nclude photon electon and poton collsons wth atoms molecules and the ons. In ths pape we evew the applcaton of the tme-dependent close-couplng method to calculate enegy and angle dffeental coss sectons fo the electon-mpact onzaton of atoms and molecules. The tme-dependent close-couplng (TDCC method was fst used used to calculate enegy and angle dffeental coss sectons fo the electon-mpact sngle onzaton of the hydogen [ ] and helum [4 5] atoms. Fo the electon-mpact sngle onzaton of the hydogen o helum atoms collson pobabltes ae found by the pojecton of a fully tme evolved sx-dmensonal coodnate space wavefuncton onto smple poducts of spatal functons epesentng two outgong electons n the feld of H + o He +. The sx-dmensonal wavefuncton fo two contnuum electons movng n a statonay sphecal Coulomb feld s epesented by an expanson of two-dmensonal (D adal functons n coupled sphecal hamoncs. The TDCC-D method along wth the conveged close-couplng [6] and the exteo complex scalng [7] methods povde a non-petubatve soluton of ths fully dffeental thee-body sphecal quantal poblem. The tme-dependent close-couplng method was ecently used to calculate enegy and angle dffeental coss sectons fo the electon-mpact double onzaton of the helum atom [8]. Fo the electon-mpact double onzaton of the helum atom collson pobabltes ae found by the pojecton of a fully tme evolved nne-dmensonal coodnate space wavefuncton onto antsymmetc poducts of spatal and spn functons epesentng thee outgong electons n the feld of He +. The nnedmensonal wavefuncton fo thee contnuum electons movng n a statonay sphecal Coulomb feld s epesented by an expanson of thee-dmensonal (D adal functons n coupled sphecal hamoncs. The TDCC-D method povdes a non-petubatve soluton of ths fully dffeental fou-body sphecal quantal poblem. The tme-dependent close-couplng method was also ecently used to calculate enegy and angle dffeental coss sectons fo the electon-mpact sngle onzaton of the hydogen molecule [9 ]. Fo the electon-mpact sngle onzaton of the hydogen molecule collson pobabltes ae found by the pojecton of a fully tme evolved sx-dmensonal coodnate space wavefuncton Intenatonal Revew of Atomc and Molecula Physcs ( Januay-June 9

2 M.. Pndzola F. Robcheaux and J. Colgan onto smple poducts of spatal functons epesentng two outgong electons n the feld of H +. The sx dmensonal wavefuncton fo two contnuum electons movng n a statonay non-sphecal Coulomb feld s epesented by an expanson of fou-dmensonal (4D adal and angula functons n otaton functons. The TDCC-4D method povdes a non-petubatve soluton of ths fully dffeental thee-body non-sphecal quantal poblem. The emande of ths pape s oganzed as follows. The close-coupled equatons and vaous total and dffeental onzaton coss secton expessons ae pesented fo the TDCC-D method n ecton the TDCC-D method n ecton and the TDCC-4D method n ecton 4. elected total and dffeental coss secton esults fo the electon-mpact onzaton of H He H + and H ae pesented n ecton 5. In ecton 6 we conclude wth a summay and an outloo fo futue wo. Unless othewse stated all quanttes ae gven n atomc unts.. TIME-DEPENDENT COE-COUPING D METHOD The tme-dependent chodnge equaton fo electon scatteng fom an atom wth one actve electon s gven by: ψ ( t = H ( ψ ( t... ( whee the non-elatvstc Hamltonan s gven by: H ( = V ( + + V + Fo the hydogen atom: V ( = whle fo the helum atom:... (... ( V ( = 4ρ + VH a... (4 whee V H ( s the dect Hatee potental a s an adjustable paamete and ρ ( s the pobablty densty of the coe electon. The total sx-dmensonal electonc wavefuncton s expanded n coupled sphecal hamoncs fo each total obtal and spn angula momentum symmety: ( t ψ = whee P ( t ll Y ( ˆ ˆ l l... (5 l l ll Y( ( ˆ ˆ l l = Cm m Yl m Yl m ˆ ˆ... (6 m m ll l C mm m s a Clebsch-Godan coeffcent and Y ( ˆ lm s a sphecal hamonc. Upon substtuton of ψ ( t of Eq. (5 nto the tme-dependent chodnge equaton and pojecton onto the coupled sphecal hamoncs we obtan the tme-dependent close-coupled patal dffeental equatons fo each symmety: P whee ( t = Tl l ( Pl l ( t ll T l l ( = l l + W P t... (7 ll ll ll ( l + l + + V (... (8 and the couplng opeato s gven n tems of standad j and 6j symbols by: W ( = l + l + l + l + l + l + l l l l ( ( l l l l < + l l > l l... (9 An ntal condton fo the soluton of the tmedependent close-couplng equatons fo electon scatteng fom an atom wth one actve electon s gven by: ( P t = l l [ = P ( F ( s l l F ( P ( + δ δ ] δ δ s l l... ( 9 Intenatonal Revew of Atomc and Molecula Physcs ( Januay-June

3 The Tme-Dependent Close-Couplng Method fo Electon-Impact Dffeental Ionzaton Coss ectons fo Atoms... whee s the ntal lnea momentum F l ( s a adal wavepacet and P s ( s a bound adal obtal fo the gound state of the hydogen o helum atoms. Followng tme evoluton of the tme-dependent close-coupled patal dffeental equatons nto the asymptotc egon fo each symmety the total coss secton s gven by: σ = d d 4 ( ( P l l + + l l the enegy dffeental coss secton s gven by: d 4... ( α = d dδ α tan / ( ( P l l + + l l... ( and the enegy and angle dffeental coss secton s gven by: d d d α Ω Ω = d d tan / 4 δ α l ( + l + + s l l ( δ l + δ l e P Y ˆ ˆ. ( ( ( l l l l... ( In the coss secton expessons of Eqs ( ( α s the angle n the hypesphecal ( plane and ae the fnal lnea momentum and δ and δ ae scatteng phase shfts. The momentum space pojecton wavefuncton fo each symmety and l l coupled channel s gven by: Pl l = d d P l P l P t... (4 l l whee P l ( s a box nomalzed contnuum adal obtal calculated n the potentals of Eq. ( fo H + o Eq. (4 fo He +.. TIME-DEPENDENT COE-COUPING D METHOD The tme-dependent chodnge equaton fo electon scatteng fom an atom wth two actve electons s gven by: ψ ( t = H ( ψ ( t whee the non-elatvstc Hamltonan s gven by: H ( = V ( V ( Fo the helum atom: V ( =... ( V ( (6.... (7 The total nne-dmensonal electonc wavefuncton s expanded n coupled sphecal hamoncs fo each total obtal and spn angula momentum symmety: ( t ψ = whee Y ( ( ˆ ˆ ˆ l l l ll l ( P t l l l Y ( ( ˆ ˆ ˆ l l l... (8 ll l = C ( ˆ m mm CM m Yl m m m M m Y ˆ Y ˆ.... (9 lm lm Upon substtuton of ( t ψ of Eq. (8 nto the tme-dependent chodnge equaton and pojecton onto the coupled sphecal hamoncs we obtan the tmedependent close-coupled patal dffeental equatons fo each symmety: Intenatonal Revew of Atomc and Molecula Physcs ( Januay-June 9

4 M.. Pndzola F. Robcheaux and J. Colgan P whee ( t l l l ( T = l l l W W = T ( P ( t l l l l l l + l l l < j l l l W l l l l l l P t ( l + ( l l l l l l ( j l + + V l l + + δ δ = l l ( l l l l l l... (... ( ( l ( l ( l ( l ( ( < + l l > l l l l l l... ( l + δl l = ( l + ( l + ( l + ( l + ( ( ( + ( + < + l l l l l l l l l... ( and W l l l l l l = ( l l l δ l l > ( l + ( l + ( l + ( l + ( ( ( + ( + < + l l l l > l l l. l l... (4 An ntal condton fo the soluton of the tmedependent close-couplng equatons fo electon scatteng fom an atom wth two actve electons s gven by: ( P t = l l l whee ( = Pu ( F ( l δ δ δ δ δ... (5 l l l l l P l u s a coelated adal wavefuncton fo the two taget electons of the gound state of helum. Followng tme evoluton of the tme-dependent close-coupled patal dffeental equatons nto the asymptotc egon fo each symmety the total coss secton fo sngle onzaton s gven by: σ = d d ( ( + + P ( s... (6 l l sll the enegy dffeental coss secton fo sngle onzaton s gven by: d β = d d tan / δ β ( ( + + P sl l s... (7 l l 94 Intenatonal Revew of Atomc and Molecula Physcs ( Januay-June

5 The Tme-Dependent Close-Couplng Method fo Electon-Impact Dffeental Ionzaton Coss ectons fo Atoms... and the enegy and angle dffeental coss secton fo sngle onzaton s gven by: d d d β Ω Ω = d d tan / δ β ( + + l l ( l l + e δ + δ l l ˆ ˆ ( P s Y. sl l s l l... (8 In the coss secton expessons of Eqs (6 (8 β s the angle n the hypesphecal ( plane and ae the fnal lnea momentum and δ l and δ l ae scatteng phase shfts. The momentum space pojecton wave-functon fo each symmety and sl l coupled channel s gven by: ( P s sll 6 = l l l j Q j d d d P P P s l l P t sl l... (9 whee P l ( s a box nomalzed contnuum adal obtal calculated n the potental of Eq. (4 fo He + and j s summed ove the sx pemutatons of. The pobablty expanson coeffcents Q (j ae obtaned usng standad algebac educton methods. mlaly the total coss secton fo double onzaton s gven by: σ = d d d ( ( + + P l l l... ( l l l the enegy dffeental coss secton fo double onzaton s gven by: dαdβ = d d d ( tan ( / δ α ( tan / δ β + ( ( + + P l l l... ( l l l and the enegy and angle dffeental coss secton fo double onzaton s gven by: dαdβdω dω dω = d d d ( tan ( / δ α ( tan / δ β + ( + + l l l ( l l l + + e δ + δ + δ l l l l l l ( l l l ˆ ˆ ˆ ( P Y... ( In the coss secton expessons of Eqs ( ( α s the angle n the hypesphecal ( plane β s the angle n the hypesphecal ( + plane and ae the fnal lnea momentum and δ l δ l and δ l ae scatteng phase shfts. The momentum space pojecton wavefuncton fo each symmety and l l l coupled channel s gven by: ( P l l l 6 = l l l j Q j d d d P P P l l l Intenatonal Revew of Atomc and Molecula Physcs ( Januay-June 95

6 w h e e P M.. Pndzola F. Robcheaux and J. Colgan P t... ( l l l l ( s a box nomalzed contnuum adal obtal calculated n the potental of Eq. (7 fo He TIME-DEPENDENT COE-COUPING 4D METHOD The tme-dependent chodnge equaton fo electon scatteng fom a homonuclea datomc molecule wth one actve electon s gven by: ψ ( t = H ( ψ ( t... (4 whee the non-elatvstc Hamltonan s gven by: H ( = V ( + θ + V ( θ (5 Fo the hydogen molecula on: V ( θ = + cosθ 4 R R + + cos θ 4 R R whle fo the hydogen molecule: V ( θ = + cosθ 4 R R... (6 + VH θ + R + R cos θ 4 θ 4 a ρ (... (7 whee R s the ntenuclea dstance V H ( θ s the dect Hatee potental a s an adjustable paamete and ρ ( θ s the pobablty densty of the coe electon. The total sx-dmensonal electonc wavefuncton s expanded n smple poducts of otatonal functons fo each M total pojecton onto the ntenuclea axs and spn angula momentum symmety: ( t Ψ = whee m Φ φ = lm m m ( θ θ P t sn θ sn θ m m Φ φ Φ φ... (8 m m m e φ... (9 and M = m + m. Upon substtuton of ψ ( t of Eq. (8 nto the tme-dependent chodnge equaton and pojecton onto the smple poducts of otatonal functons we obtan the tme-dependent close-coupled patal dffeental equatons fo each M symmety: ( lm Pm m θ θ t whee ( θ θ T mm l M = Tm m ( θ θ Pm m ( θ θ t m m M m m m m ( + W θ θ l M m m and K ( ( P θ θ t... (4 m sn θ = ( K ( + K ( θ + + V ( θ... (4 K θ ae netc enegy opeatos. The couplng opeato s gven by: M mm mm W θ θ = ( ( < + > ( q q ( + q! P! q ( cos θ ( cos m θ φ φ Φ φ q P d d 96 Intenatonal Revew of Atomc and Molecula Physcs ( Januay-June

7 The Tme-Dependent Close-Couplng Method fo Electon-Impact Dffeental Ionzaton Coss ectons fo Atoms... q whee ( cos P q φ φ e Φ φ Φ φ Φ φ m m m... (4 θ s an assocated egende functon. An ntal condton fo the soluton of the tmedependent close-couplng equatons fo electon scatteng fom a homonuclea datomc molecule wth one actve electon s gven by: ( θ θ = P t lm mm whee P s θ θ δ = P ( F ( s lm m ( F ( δ + θ m M lm ( P s θ δ m M δ m... (4 θ s a bound obtal fo the gound state of the H + o H molecules. Followng tme evoluton of the tme-dependent close-coupled patal dffeental equatons nto the asymptotc egon fo each M symmety the total coss secton s gven by: σ = d d ( + 4 l M l M P (... (44 l m l m l m l m the enegy dffeental coss secton s gven by: d α = d d tan / 4 δ α Y ˆ Y ˆ δ (46 l m lm m m M I n t h e c o s s s e c t o n e x p e s s o n s o f E q s ( 4 4 ( 4 6 α s the angle n the hypesphecal ( plane and ae the fnal lnea momentum θ and φ ae angles the ncomng electon maes wth the ntenuclea (z axs and δ and δ l ae scatteng phase shfts. The momentum l space pojecton wavefuncton fo each M symmety and m m coupled channel s gven by: l M * Pl m l m = d dθ d dθp l m θ ( P θ * l m P θ θ t... (47 l M m m whee P lm ( θ s a box nomalzed contnuum obtal calculated n the potentals of Eq. (6 fo H + o Eq. (7 fo H EECTED REUT The TDCC-D method was fst appled to the electonmpact sngle onzaton of the hydogen atom. The total coss secton calculated [] usng Eq. ( s compaed wth expement [] n Fg.. Pevous conveged closecouplng calculatons [] had also acheved excellent ageement wth expement. ( P l M l m l m + l M l m l m... (45 and the enegy and angle dffeental coss secton s gven by: d d d α Ω Ω = d d tan / 4 δ α l * ( Yl ( M + θ φ l M l + l ( δ l + δ l l M e P l m l m l m l m Fg. : Total electon-mpact sngle onzaton coss secton fo hydogen squae boxes: TDCC-D calculatons [] sold ccles: expement [] (. Mb =. 8 cm The TDCC-D method was then appled to the electon-mpact sngle onzaton of the helum atom. The Intenatonal Revew of Atomc and Molecula Physcs ( Januay-June 97

8 c o n v e g e d c l o s e - c o u p l n g [ 6 ] a n d R-matx e l e c t o n - m p a c t s n g l e o n z a t o n o f H M.. Pndzola F. Robcheaux and J. Colgan total coss secton calculated [4] usng Eq. ( s compaed wth expement [5] n Fg.. Pevous wth pseudostates [7] calculatons had also acheved excellent ageement wth expement. The enegy and dffeental coss secton calculated [4] usng Eq. ( s compaed wth expement [8] n Fg.. The dffeental coss secton s at an ncdent enegy of E = 44.6 ev equal enegy shang wth E = E =. ev co-plana wth φ = φ = and wth θ = 47 Good ageement s found between theoy and expement. The TDCC-D method was fst appled to the electon-mpact double onzaton of the helum atom. The total coss secton calculated [9 ] usng Eq. ( s compaed wth expement [] n Fg. 4. Excellent ageement s found between theoy and expement. The TDCC-4D method was fst appled to the +. The total coss secton calculated [] usng Eq. (44 s compaed wth expement [4] n Fg. 6. Excellent ageement s found between theoy and expement. Fg. : Enegy and angle dffeental electon-mpact sngle onzaton coss secton fo helum. old lne: TDCC-D calculatons [4] sold ccles: Expement [8] (. Kb =. cm. Fg. : Total electon-mpact sngle onzaton coss secton fo helum. quae boxes: TDCC-D calculatons [4] sold ccles: expement [5] (. Mb =. 8 cm The enegy and dffeental coss secton calculated [8] usng Eq. ( s compaed wth expement [] n Fg. 5. The dffeental coss secton s at an ncdent enegy of E = 9 ev equal enegy shang wth E = E = E = 9. ev co-plana wth φ = φ = φ = and wth θ = 45 and θ = 5. Good elatve ageement s found between theoy and expement although the measuement magntudes have been dvded by a facto of 5. Fg. 4: Total electon-mpact double onzaton coss secton fo helum. quae boxes: TDCC-D calculatons [] sold ccles: Expement [] (. Kb =. cm The TDCC-4D method was then appled to the electon-mpact sngle onzaton of H. The total coss secton calculated [5] usng Eq. (44 s compaed wth expement [6] n Fg. 7. Pevous R-matx wth pseudo-states [7] calculatons had also acheved excellent ageement wth expement nea the onzaton theshold. 98 Intenatonal Revew of Atomc and Molecula Physcs ( Januay-June

9 The Tme-Dependent Close-Couplng Method fo Electon-Impact Dffeental Ionzaton Coss ectons fo Atoms... The enegy and angle dffeental coss secton calculated [9] usng Eq. (46 s compaed wth expement [9] n Fg. 8. The dffeental coss secton s at an ncdent enegy of E = 5.4 ev equal enegy shang wth E = E =. ev co-pependcula wth θ = θ = 9 and wth φ = φ = ξ. Good elatve ageement s found between theoy and expement whee expement s nomalzed to theoy at ξ = 9 and theoy s aveaged ove all molecula oentatons. Fg. 7: Total electon-mpact sngle onzaton coss secton fo H. quae boxes: TDCC-4D calculatons [5] sold ccles: expement [6] (. Mb =. 8 cm Fg. 5: Enegy and angle dffeental electon-mpact double onzaton coss secton fo helum. old lne: TDCC-D calculatons [8] sold ccles: Expement [] (. b =. 4 cm Fg. 8: Enegy and angle dffeental electon-mpact sngle onzaton coss secton fo H. old lne: TDCC-4D calculatons [9] sold ccles: Expement [9] (. Kb =. cm. Fg. 6: Total electon-mpact sngle onzaton coss secton fo H +. quae boxes: TDCC-4D calculatons [] sold ccles: Expement [4] (. Mb =. 8 cm 6. UMMARY The tme-dependent close-couplng method has been successfully appled to calculate electon-mpact dffeental onzaton coss sectons fo atoms and molecules. In ths pape we evewed the fomulaton of the TDCC-D method fo the sngle onzaton of atoms the TDCC-D method fo the double onzaton of atoms and the TDCC-4D method fo the sngle onzaton of datomc molecules. Each TDCC method goes beyond petubaton theoy to acheve an exact numecal descpton of a scatteed electon movng n the feld of one o two ejected electons and the feld of the onzed coe of the atom o molecule. Few-body quantal effects Intenatonal Revew of Atomc and Molecula Physcs ( Januay-June 99

10 M.. Pndzola F. Robcheaux and J. Colgan ae stongest fo cetan enegy and angle dffeental coss sectons as fo example when the scatteed and ejected electons move off wth the same enegy and close to the same angula decton. In the futue we plan to apply the TDCC methods to calculate dffeental coss sectons fo the electon-mpact sngle onzaton of seveal dffeent alal atoms the electon-mpact double onzaton of Be and the electon-mpact sngle onzaton of + and. ACKNOWEDGMENT Ths wo was suppoted n pat by gants fom the U Depatment of Enegy and the U Natonal cence Foundaton. Computatonal wo was caed out at the Natonal Enegy Reseach centfc Computng Cente n Oaland Calfona and at the Natonal Cente and Insttute fo Computatonal cences n Oa Rdge Tennessee. REFERENCE [] M.. Pndzola F. Robcheaux.D. och J.C. Beengut T. Topcu J. Colgan M. Foste D.C. Gffn C.P. Ballance D.R. chultz T. Mnam N.R. Badnell M.C. Wtthoeft D.R. Plante D.M. Mtn J.A. udlow and U. Kleman J. Phys. B 4 R9 (7. [] J. Colgan M.. Pndzola F. Robcheaux D.C. Gffn and M. Baetschy Phys. Rev. A (. [] J. Colgan and M.. Pndzola Phys. Rev. A 74 7 (6. [4] J. Colgan M.. Pndzola G. Chldes and M.A. Khaoo Phys. Rev. A 7 47 (6. [5] J. Colgan M. Foste M.. Pndzola I. Bay A.T. telbovcs and D.V. Fusa J. Phys. B 4 45 (9. [6] I. Bay D.V. Fusa A.. Khefets and A.T. telbovcs J. Phys. B 5 R7 (. [7] C.W. McCudy M. Baetschy and T.N. Rescgno J. Phys. B 7 R7 (4. [8] M.. Pndzola F. Robcheaux and J. Colgan J. Phys. B 4 5 (8. [9] J. Colgan M.. Pndzola F. Robcheaux C. Kase A.J. Muay and D.H. Madson Phys. Rev. etts. (8. [] J. Colgan O. Al-Hagan D.H. Madson C. Kase A.J. Muay and M.. Pndzola Phys. Rev. A (9. [] M.. Pndzola and F. Robcheaux Phys. Rev. A 54 4 (996. [] M.B. hah D.. Ellott and H.B. Glbody J. Phys. B 5 (987. [] I. Bay and A.T. telbovcs Phys. Rev. etts (99. [4] M.. Pndzola and F. Robcheaux Phys. Rev. A (. [5] R.K. Montague M.F.A. Hason and A.C.H. mth J. Phys. B 7 95 (984. [6] D.V. Fusa and I. Bay Phys. Rev. A 5 79 (995. [7] E.T. Hudson K. Batschat M.P. cott P.G. Bue and V.M. Bue J. Phys. B 9 55 (996. [8]. Roual J. Rode B. Rouvellou H. Ehadt A. Pochat I. Bay and D.V. Fusa J. Phys. B 7 (998. [9] M.. Pndzola F. Robcheaux J. Colgan M.C. Wtthoeft and J.A. udlow Phys. Rev. A 7 75 (4. [] M.. Pndzola F. Robcheaux and J. Colgan Phys. Rev. A (7. [] M.B. hah D.. Ellott P. McCallon and H.B. Glbody J. Phys. B 75 (988. [] M. Du A. Don J. Ullch.P. Cao A. Czasch A.. Khefets J.R. Goltz and J.. Bggs Phys. Rev. etts (7. [] M.. Pndzola F. Robcheaux and J. Colgan J. Phys. B 8 85 (5. [4] B. Peat and K.T. Dolde J. Phys. B 6 49 (97. [5] M.. Pndzola F. Robcheaux.D. och and J. Colgan Phys. Rev. A (6. [6] H.C. taub P. Renault B.G. ndsay K.A. mth and R.F. tebbngs Phys. Rev. A (996. [7] J.D. Gofnel and J. Tennyson J. Phys. B 8 67 (5. Intenatonal Revew of Atomc and Molecula Physcs ( Januay-June