Volume92A, number4 PHYSICS LETTERS 8 Noveniber1982

CONTINUUM-DISTORTED-WAVE CAPTURE INTO THE nTH SHELL: 1, mDISTRIBUTIONS

D.S.F.CROTHERSand J.F.McCANNDepartmentofAppliedMathematicsand TheoreticalPhysics,The Queen’sUniversityof Belfast,BelfastBT7 iNN, Northern Ireland, UK

Received17 August 1982

Continuum-distorted-wave(CDW) intermediateenergycrosssectioncalculationsarereportedfor the process+ H(ls) —~C

5~(nlm)+ Ht For then ~3, 4, 5 and7 shellsthe I subleveldistributionis comparedwith other theoreticalcal-culations.TheCDW m subleveldistributionsarealso given, therebeingapparentlyno otherpublishedexperimentalortheoreticalvaluesavailablefor comparison.It is concludedthat OBK is extremelyunreliablein predictingm subleveldistri-butions.

Recentlythe continuum-distorted-wave(CDW)methodhasbeenused [1,21to describechargetrans- bnnam= I~ exp{—irB [q + (~c/v+~v)~1}fer processessuchas

X ~nn~m~B~rj31F1(iZ/v; 1;iUTB+jUTB), (6)

BZ++H(1s)_~B~~)+(n)+H+. (1)where n2 and mare parabolicquantumnumbers[3],

whereBZ+ is a fully strippedprojectileandZ,n and rH and rB are the positionvectorsof the electronrela-the impactvelocity u areessentiallyarbitrary.In par- tive to H+ and BZ+, respectively,~e is theresonanceticularit may beshown,usingparaboliccoordi- defectandthe ~~nn2~ arebound-statewavefunctions.nates[1], that theprior CDW crosssection for In order to obtainthecrosssectionfor

BZ++H(ls)_~B~~+(nn2m)+H~ (2) BZ++H(ls)_~B~i~(nlm)+H~, (7)

is givenby wherein standardspectroscopicnotationn, 1 andm

arethe sphericalquantumnumbers,the following re-2 CDW — 2 r ,~ 2 ciprocalunitary transfomiations[4] areemployed:2~rv Qnn2,,~—a0 j q uqIg0~2,,1~ , (3,

0 n-~1

whereq is thetransversecomponentof the changein ~ m(r) = l~I an21(n,m)~n lm(r), (8)linearmomentumof theheavy-particlerelativemo-

n—1—ImItion and g is given by

nn2m ‘~nimfr)= ~ a1,~2(n,m) ~nn2m(~ (9)

gnn2m= exp[7r(Z+ 1)/2v] F(1 — i/u)F(1 —iZ/u) n2~

X ab (4~ The coefficients are given [5] bynn2m’ ~ I

al~2(n,m)=(_l)’22C(kk1;Ml/2

2m), (10)a fdrH exp{ir11 [q + (z~c/v— ~v) ~3]}

where the Clebsch—Gordan coefficients, C, are de-X 1F1 (i/u; 1; ivrH + itrrH) V,.H~lOO(rH), (5) fined in ref. [6], the additional phase factor (— 1)’~2

maintains the convention of and the consistencybe-tweenref. [1] and ref. [3] and theparametersare

170 0 031-9163/82/0000--0000/502.75© 1982 North-Holland

Volume 92A, number4 PHYSICSLETTERS 8 November1982

given by n—i

= Q~i, (17)k~(n— 1), (11) 1=0

= ~(m + n1 — n2), (12) which maybe rearranged,usingthe unitarityof the := m — p1 =

3~(m— n1 + n2), (13) transformation,to give

n—I n—rn—in1=n—1—m—n2. (14) 2~v

2QWa~fqdq ~ ~m E Ig~ 12n

2m ‘

The Ccoefficientsarewidely tabulatedbutwechose 0 m0 112=0 (18)to use a subroutinefrom thewell-knownelectron— consistentwith ref. [1], includingthe definitionofatom collision R-matrix package[7]. ~m Expressions(15) and (16) involve a simplemodi-

It follows that fication to our earliercomputerprogram[1] for ex-n—rn—i 2 pression(18).

2~u2Q~~Wa~f q dq ~ a

1~2(n, m)g11~2~i - In table 1 we presentspecimenresultsforZ 6112=0 I andn = 7 showingthe probability distributionsover

0 (15) l(100Q111/Q11).Comparisonis madewith theweak-Wemay define couplingOBK theory [2,3,8—16].We merelymen-

tionin passingthatour OBK parabolic-coordinate

Qni Qnim’ (16) methodof calculationis compactin analogywith the

m=—lTable 1Percentagedistributionsof theCDW crosssectioninto thesubshells(n,1)100Q~~/Q~for the process:C

6~+ H(1s)—~C5~(n,1)+

1{F (wjth n 7), in the 013K andCDW approximations.ThemaximumcrosssectionQ~1,andthetotal crosssectionfor thenthshell aregivenin atomicunits (ag).

1 v(au)

1.0 2.0 4.0 6.0 8.0 10.0 14.14 20.00

0 CDW 0.85 2.76 2.84 13.47 25.43 34.92 47.60* 56.10*OBK 0.79 1.00 3.47 1.86 4.14 16.21 43.27 66.74*

1 CDW 5.62 3.82 24.24 40.25* 42.80* 40.43* 32.90 24.81OBK 4.11 4.38 5.23 19.57 48.54* 57.25* 47.88* 30.70

2 CDW 5.96 12.02 37.21* 30.21 20.98 15.14 9.86 8.02OBK 4.28 4.86 13.37 47.84* 37.57 23.18 8.30 2.48

3 CDW 10.62 29.45 24.30 10.86 6.29 4.91 4.53 4.91OBK 10.52 13.20 41.40* 25.23 8.83 3.17 0.53 0.08

4 CDW 27.13 31.75* 8.43 3.27 2.60 2.61 2.89 3.28OBK 10.88 8.45 28.87 5.05 8.76 0.19 0.02 0.00

5 CDW 33~43* 16.08 2.24 1.39 1.39 1.47 1.65 1.88OBK 18.31 37.34~ 7.09 0.43 0.04 0.01 0.00 0.00

6 CDW 16.39 4.09 0.74 0.55 052 0.52 0.56 0.98OBK 51.11* 30.78 0.57 0.01 0.00 0.00 0.00 0.00

Qn*l CDW 9.814~a) 1.376 3.67i~ 5955_5 2.751—6 2.167~ 5.134~ I.i87’-~OBK 9.897*2 1482fi 3.450_2 5.i79~ 2.O25~ 1.802—6 2.568~ 5.827_10

Q,~ CDW 2.936+2 4.333 9.868~ i.480~ 6.428—6 5.358~ 1.079_8 2.117_~0OBK 1.936~~ 3.968~~ 8.333—2 1.083~ 4.173~ 3.149—6 5.363~ 8.731~°

a)a±b=ax10~b•

171

Volume 92A, number4 PHYSICSLETTERS 8 November1982

aboveCOW method.Clearly thereis a similarity with eitherQOBK or QB2 In table 2 we compare our 1-bothour OBK distributionsand the secondBorn dis- probabilitydistribution of QCDW for n = 3, 4 and 5tribution [17] thoughthe detailsaresomewhatthf- at v = 3\f~/Sv0with theperturbed-stationary.stateferent.In particularour distributionsat v = 00 (PSS)valuesQ~S

5of ref. [19] at 27 keV/amu.The(whereu

0 = e2/h)and2uo are smoothby compari- similarity betweenthetwo distributionsandtheir

son.Moreover,exceptfor v = 8, 10 and 20 u0 the agreementwith theclassical-trajectoryMonte Carlo

maximumcrosssectionoccursat a lowervalueof!. (CTMC) distribution [19—21]at v = v0 appearsto beSuch aspectsare hardly surprisingsinceQ~

2exceeds satisfactory.Of coursetheabsolutevaluesof QCDWQCDW typically by a factor of 10 at u = 2vo andof 6 arethemselvesunreliableat sucha comparativelylow

at v = l0u~.TheBorn validity criterion(—Z/rB a per- velocity sincethewave functionsare notnormalisedturbation)is often interpreted[18] as [22]. Neverthelessit is reassuringthat thereis some

measureof agreementbetweenthesethreetheoriesv~max(1Z/n)v

0, (19) all of which describethe threecompetingCoulombwhichin thecontextof table 1 is only fulfilled at the interactionsin a non-perturbativemanner [231andhighervaluesof u. Wemaythussupposethat for the for completenesswe includevaluesfor QCDWN whichlowerv valuesof table 1, QCDW is morereliablethan are roughlya factor of 4 lower thanQCDW and in

Table2Valuesof 100Qnl/Qn andQ,~for tile process:C

6~+ H(Is)—* C5~(nl)+ H~,for n = 3,4.5. -

n5 10 l1 1=2 13 1=4 Q~(1O~6cni~)

PSS1191 1.3 5.4 10.5 23.8 59.0 12.29CDW 1.6 6.2 12.2 15.4 64.6 65.74CDWN 1.7 5.9 12.2 15.8 64.4 16.39OBK 2.8 16.1 30.9 30.2 20.0 304.62ElK 2.9 18.9 35.4 29.8 13.0 12.82CTMC 1 5 14 29 51 5.32Abramoveta!. [191 6 14 18 22 40 —~

0=4 10 1=1 1=2 1=3 Q4(I0~

6cni2)

PSS1191 2.3 9.1 30.5 58.0 20.26CDW 4.5 11.3 36.3 47.9 30.85CDWN 4.1 11.0 36.9 48.0 7.38OBK 17.7 32.1 30.5 19.7 86.58ElK 24.0 38.2 28.1 9.8 1.47CTMC [191 2 7 25 10.3 17.6Abramoveta!.1191 7 20 25 48 —

UDWA 1271 4.1 14.5 33.4 47.9 13.8

n3 /0 11 /2 Q3(10

16cin2)

PSS[191 8.5 28.9 62.6 1.96CDW 20.8 23.2 56.0 4.38CDWN 17.4 20.3 62.3 0.90OBK 3.6 37.0 59.4 4.46ElK 4.6 36.3 59.1 0.04CTMC [191 6 32 62 6.23Abramovcc al. [191 11 29 60

172

Volume 92A, number4 PHYSICSLETTERS 8 November1982

Table 3Percentagedistributionsof theCDWand OBK crosssectionsinto the (n, 1, m) subshells,100Q~l~mi/Qnl,whereQ~ijmi= ~rnQnlm(em = 1 form = 0, and2 form * 0) for theprocess:C

6~+ H(1s)-~C5~(n1m)+ H~with n = 7. v = 5 X i08 cm ~ (= 2.29 au).

I m10 mHl m~2 mH3 mj4 m~5 mH6 ,~i=oQ~limjQnl(a~)

CDW OBK

6 CDW 20.53 34.30 19.04 10.34 10.18 4.85 0.75 4.215—2 2.765OBK 21.78 38.56 25.09 11.01 3.04 0.48 0.03

5 CDW 31.56 42.57 16.14 6.38 2.82 0.52 1.802_i 5.940OBK 20.31 38.79 27.16 11.14 2.39 0.21

4 CDW 41.45 42.34 12.40 3.35 0.46 4339_i 2.378OBK 11.34 33.70 37.86 15.11 2.00

3 CDW 55.68 35.87 7.60 0.84 5.118_i 1.252OBK 46.01 33.43 16.52 4.05

2 CDW 78.64 18.49 2.87 2.701—i 1.333OBK 46.56 47.51 5.93

1 CDW 82.85 17.15 6.343—2 3.659~OBK 69.61 30.39

0 CDW 100.00 4.381-2 2.559’OBK 100.00

= ~fl~i Qnl = 1.5453 1.4290~

Table 4The same as table 3, v = 10.00 au (=21.87 x 10~cms~).

1 im~0 m~1 rn12 m~—3 m14 m1—5 m~6 mi_OQilImVrQnl(t~)

CDW OBK

6 CDW 10.06 4.81 23.03 2.64 24.42 28.34 6.70 2.801~ i.661~2OBK 31.02 44.27 18.93 4.85 0.83 0.10 0.01

5 CDW 1.88 25.84 6.46 20,31 35.53 9.99 7.881~ 1.633~°OBK 35.64 45.62 15.31 3.03 0.38 0.02

4 CDW 11.94 16.99 16.36 41.22 13.49 1.3978 6.001~OBK 42.53 44.38 11.44 1.54 0.11

3 CDW 22.61 22.00 40.79 14.60 2.631—8 9.9888OBK 52.34 40.44 6.77 0.45

2 CDW 39.65 49.73 10.62 8.111_8 7.297~OBK 66.06 31.45 2.49

1 CDW 73.56 26.44 2.167~ 1,802—6OBK 83.29 16.71

0 CDW 100.00 1.871~ 5.104~OBK 100.00 — ______

5.3583~ 3.149-6

173

Volume92A.number4 PHYSICSLETTERS 8 November1982

which thecontinuumdistortedwave functionsare by a grant from theScienceand EngineeringResearch

normalized[22] in the united-atomlimit (detailswill Council. Oneof us(JMcC) thanksthedepartmentof

be publishedelsewhere[24]). Sufficeit to say that Educationfor NorthernIreland for theawardof athere is little difference between the QCD\V and post-graduatestudentship.We thank LouisDuhé forQCDWN distributions. MoreoverPSS,CTMC and a preprintof ref. [17].CDW all recognisethat the electronicdensitydistribu-tion cannotadjustto the rapid rotation of the inter- References

nuclearaxis [25,26] which is reflected in strongnon-adiabaticrotationalcoupling. It will alsobe noted [1 I).S.F.Crothers,J. Phys.B14 (1981)1035.

from table 2 that whereasPSS, CTMC, COW,CDWN. [2] D.S.F.CrothersandN.R. Todd.J. Phys.1313 (1980)

UDWA[271and strong-rotationalcoupling theories 2277.[3] K. Omidvar, Phys.Rev. 153 (1967)12!.

all predict that the/ = n — I stateis predominantly [4] V. Rojansky,Phys.Rev.33 (1929) 1.populated[28], irrespectiveof thevalueof n, our [5] D. Park, Z. Phys.159 (1960) 155.

OBK and eikonalcalculations(basedon refs. [29, 161 E.U. CondonandG.H. Shortley,The theoryof atomic30]) fail to do soforn = 4 and 5. Onceagainthis is spectra(CambridpeUP., London.1951).

hardly surprisingsincethe initial interaction between [7] K.A. Bcrrington,PG. Burke,J.J. Chang,AT. Chivers,W.D. Rohb and K.T. Taylor, ComputerPhYS. Comm. 8

electron and projectile is not merely a perturbation.(1974) 149.

So far asabsolutevaluesare concernedevenCDWN [8] M.R,C.McDowell andJ.P.Coleman.Introductionto

must be suspectsince it fails to peakat n = 4, unlike the theoryof ion—atomcollisions (North-Holland,

PSSand CTMC. Neitheris this surprisingat such a Amsterdam,1970).

low velocity whererefined orthogonal[22] and 19] J.E. Golden,J.H. McGuirc and K. Onsidvar,Phys.Rev.Al 8 (1978)2373.close-couplingeffectsmustbe important.

[10] N. Toshiioa, J. Phys.Soc.Japan46 (1979) 927.In tables3 and 4 wepresentspecimenCOW and [11] N. Toshima, J. !‘hys. Soc. Japan46 (1979)1295.

OBK resultsforZ = 6 andti = 7 showingthe probabil- [12] V.P. Shevelko.J. Phys. B13 (1980) L319.ity distributionsoverm (100~,~Qmim/Qni)with i~ [13] J. Burgdorfer,J. Phys.B14 (1981)1019.

chosenastheaxis of quantization.For~ = [141 J. Burgdorfer,Phys.Rev. A24 (1981) 1756.thereare someinterestingoscillationsasm varieswith [151 E.G.Berezhko,V.V. Sizov andN.M.Kabachnik,J. Phys.1314(1981)2635.I fixed (table 4). Thereareapparentlyno other theo- 16] T.S. Ho, D. Umberger,R.L. Day. M. Lieberand FT.

reticalor experimentalresultsavailablefor cOmpari- Chan,Phys.Rev. A24 (1981)705.

son(sinceunfortunatelyUDWA [27] hasapparently [17] Li. Duhc.i andiS. Briggs, J. Phys.B14 (1981)4595.

only beenappliedto Ne10~,and sinceCIS [31] has [18] LI). LandauandE.M. Lifshitz, Quantummechanics(PcrgamonPress,Oxford, 1977).apparentlyonly beenappliedat asymptoticallyhigh 19] TA. Green,E.J. ShipseyandJ.C. llrowne, Phys.Rev.

velocitiesand eventhen for only Z I and~ ~ 3) A25 (1982) 1364.

and no empiricalselectionrule would appearto ap- [201A. Salop,J. Phys.B12 (1979)919.

ply. It will also be notedthat thereare considerable [21] RE. Olson,Phys.Rev. A24 (1981) 1726.discrepanciesbetweenthetwo theories,and it is no- (22] D.S.F.Crothers,J. Phys.B15 (1982)2061.

table that the m = 0 statedoesnot predominatein [23] D.S.F.Crothers,Adv. At. Mo!. Phys.17 (198!) 55.[24] D.S.1-.Crothers,PhysicaScripta (1983),to bepub-

general [27,28]. Althoughthis is mainly dueto the lished.

doublingeffect form ~ 0 at the lowervelocity (table [25] D,R. Batesand D. Sprevak,J. Phys.B3 (1970)1483.

3), at thehighervelocity (table4) this is notthe [26] V.A. Abramov, FE. BaryshnikovandV.S. Lisitsa,case.Whetherthis is a defectof COW or whetherthe 5~)V.Phys.JETPLett. 27 (1978)464.

UDWA trend [27] is more typical must remainan [27] H. Ryufuku andT. Watanabe,P1i’s. Rev. A20 (1979)1828.

openquestion. [281 R.K. JanevandL.P. Presnyakov,Phys.Rep.70(1981)

Resultsfor othervaluesofZ. v and n are available 1.

upon requestfrom the authors. [29] F.T. Chan and J. Eichler, Phys.Rev. A20 (1979) 1841.[30] T.S. Ho, M. LieberandFT. Chan,Phys.Rev. A24

(1981)2925.The calculationswerecarriedout usinga link to [31] 1). Belkii. J. Phys.BlO (1977) 3491.

the CRAY-i at theDaresburyLaboratoryprovided

174