pA10
VARIATIONALSCHWINGERAPROACHTODIRECTEXCITATON OF HYDROGEN-LIKE(Li2+ (1s))TARGETTOTHESTATEn=2
BYPROTONIMPACTENERGIESFROM9keVto3MeV
Friha KHELFAOUI1,2,Boumediene LASRI1,2and Oukacha ABBES2
1DépartementdePhysique,UniversitéAboubekrBelkaïd–Tlemcen,B.P.119,Tlemcen,Algérie
2DépartementdePhysique,UniversitéMoulayTahar–Saïda,B.P.138,Ennasr,20000Saïda,Algérie
E-mail:
ABSTRACT:The1s՜2s,2p,3s,3pand3dexcitationcrosssectionsforhydrogen-like(Li2+)byprotonimpact havebeencalculatedinawideenergyrangefrom9keVto3MeV,usingSchwinger’svariationalprinciple withintheimpactparameterformalism.Thesecross sections are relevanttocontrolled nuclearfusionstudies. Theyarealsoimportantin fusion plasmaresearch[1].Thebehavior ofthe computedcrosssections arein excellentagreementwithavailabletheoreticalresults, obtained bytheclose-coupling method whichisthatof TCAO(Two-centreAtomicOrbital expansion) ofErmolaevandco-workers[1] andSCE (SingleCentre Expansion) ofHall andco-workers [2].
KEYWORDS: fusion plasma, excitationcross section, Schwingervariationalprinciple,atomic collision
Introduction:
Inner-shellprocessesinion-atom collisionshavebeenextensivelystudiedinthepast,both experimentallyandtheoretically, coveringawiderangeofimpactvelocities.Inthispaper,we haveperformedanapplicationofthefractional formoftheSchwingerprincipletostudy the direct excitations of the hydrogen-like by proton impact. This approach was applied to study thedirectexcitationof hydrogenatombyprotonimpactatenergies,from10keVto200keV [3].WeextendourvariationalSchwingerapproachtodirectexcitationofLithium ion(Li2+) target by proton impact at energies from9 keV to 3 MeV.
Thetotalcrosssectionsfortargetexcitation Li2+(1s) tothestatesLi2+( 2s,2p,n=2) are presentedinthisenergyrangeby theSchwinger’svariationalprinciplewhichisperformed
+
byexpandingthewavefunctions α
2p-1}.
and β
onfive-statesbasissetas{1s,2s,2p0,2p+1,
Wealsopresentseveralresultsobtainedfromotherstheoreticalmodelsassingleandtwo-
centre close-coupling approaches [1] [2].
Currentlythereareno experimentalresultsfor anyof theabovestated transitions. While an experimentalpreparationoftheLi2+(1s)withamergedprotonbeammaybefeasible,it appears unrealistic that an excited Li2+ion target could be prepared [2] .
Thesecrosssectionsareimportantin fusionplasmaresearch[4].Lithiumpelletsareusedto conditiontheplasmaconfinementwalls,i.e. toreduceedgeinfluxfromHandCimpurities whichreduceconfinementtimes.Lithiumpellets arealsousedindiagnosticprocedures,e.g. the measurement of hydrogen ion temperatures in the plasma core after neutral beam injection.Also,electronremovalfrom LiionsandchargetransfertoD+constitutesincreased radiative power lossesand decreased deuteron fuel densities [2].
Schwinger’s variational principle and close-coupling method applied to direct excitation: SeveraltheoreticalapproacheshavebeenmadetotheproblemofexcitationofLithiumion (Li2+)targetbyprotonimpact.AMErmolaevandMRCMcDowell(1987)usedtheclose- coupledatomicorbitalmodelwithintheimpactparameterformalism[1]intheenergyregion
17.5 keV to 3.0 MeV to calculate the cross section of the direct excitation:
H++ Li2+(1s)՜H++ Li2+(nl)*. n=2 (1) Two different TCAO bases have been used in the calculations:
1- A 32-state basis AO32 which have been applied to the problem of 1s՜2s, 2p excitations of
(Li2+)by proton impact varying the incident energy from17.5 keV to 3.0 MeV
2- A59-statebasisAO59whichhavebeenappliedtotheproblemof1s՜2s,2pand1s՜3s,
3p,3dexcitationsof(Li2+)byprotonimpactvaryingtheincidentenergyfrom17.5keVto
3.0 MeV.
K A Hall, J F Reading and A L Ford (1996) usedalso the close-coupled atomic orbitalmodel, basedonadevelopmentatasinglecenterwithin theimpactparameterformalism[2] buttheir theoreticalcrosssectionsof thereaction(1) havebeenobtainedbyperformingalargesingle FiniteHilbertBasisSet(FHBS) calculationincludingstateswith angularmomentum uptol=6, intheenergyregionfrom30keVto600keVThe resultsofthisapproachareinverygood agreement with the experiment results for the P + Hcollision[5] .
Inthepresentwork,thecalculations,basedonthefractionalform oftheSchwingervariational principle,areperformed.Thevariationalapproachreportedhereuses methods thathavebeen described previously andso will be recalled [6] [7].
Let ψ+
and −
bethescatteringEigen-statesoftheHamiltoniansatisfyingoutgoingand
incomingwaveboundaryconditionsdefined,inacollisionwithoutrearrangement,bythe eikonal Lippmann-Schwinger equations:
+∞
+ ∫ + +
ψα(z)=α(z)+
dz′GT(z−z′)V(z′)ψα(z′)
−∞
(2a)
ψ−(z)=
β(z)+∫−∞
dz′
ψ−(z′)G−(z−z′)V(z′)
(2b)
whereVisthepotentialresponsiblefortheexcitationandobtainedbyomissionofthelong-
rangeprojectile-targetcoulombinteraction[8],i.e., ( −1
r r −1 ),
and β z
V =ZP R
− R− x
α(z) ( )
aretheinitialandfinalstatesofthetargetrespectively.Wherezisthecoordinatealongthe
straight trajectory of the projectile andG±
are target Green‘s operators.
The transition amplitude, for the impact parameterρr , may be written forα≠β, as:
r i (βV ψα )(ψβ V α)
aβα
(ρ)=−
v
(ψ− V
−VG+V
ψ+ )
(3)
wherethenotation ( ) indicatestheintegrationovertheelectroniccoordinatesaswellas
overzonlywhen GT
doesnotappear,andoverz′whenGT
ispresent,andvistheimpact
r
velocity.Itcanbeeasilyshownthattheexpression(3),forthetransitionamplitude
aβα(ρ) is
stationary under arbitrary variations on ψα
and onψβ
. Indeed, it is easy to show that:
δa (ρr)= 0 (4)
Untilfirstorderfor δψ+
and
δψ−
.Byexpanding ψ+
−
and β
ontwotruncatedbasissets
r
{i}and{ j}
respectively,onecalculatesapproximatetransitionamplitude
a%βα(ρ).Thetwo
basissetsarenotnecessarilyidenticalbuttheymusthavethesamefinitedimensionN.Then, usingthevariationalconditionδaβα(ρ) ,onegetstwoseparatefinitesetsoflinearequations
forthecoefficientsoftheexpansions:onefor ψ+
andonefor
− .Solvingthesesetsof
linearequationsprovideapproximatesolutions
ψ%+
and
ψ%−
forψ+
and
− ,respectively.
Finally,theinsertionoftheirsolutionsineq.(3)leadstothefollowingpracticalformofthe approximate transition amplitude:
N N
a~ (ρ)= ⎜−
⎟∑∑(βV i)(D−1) (j Vα)
(5)
r ⎛ i⎞
βα ⎝ v⎠
i=1
ij
j=1
Where(D−1)
elements:
is the element (i,j) ofthe matrixD−1, inverse of the matrix D defined by the
Dij =(i V−VGTV j)
(6)
Quantumtransition amplitude between initial and final states ofthe system(projectile-
r r
Target):α ⊗ kα
→ β ⊗kβ
is:
r +
Tβα=kβ⊗βVψα
=⎡kβ⊗β⎤V⎡ψ+E ⊗kα⎤
(a)
(b)
(7)
⎣ ⎦ ⎣ α ⎦
r r
=kβ,βVkα,ψ+E
r r
(c)
Where α ⊗
kα , β
+E
⊗ kβ
theinitialandfinalarestatesofthesystemtarget-projectile
respectively and ψα
is eikonal scattering state of the target.
Equations(7b,c)resultbytakingthefirstorderoftheexpansioninpowerof(1/ μ)
(μis
reducedmassofcollisionalsystem)forthescatteringwavebecauseourstudyisbasedonthe impactparameterformalism.
Using the energy conservation, quantumtransition amplitude can be written:
Z ⎜⎛Z −1
r 2r
iηv.ρr
2i P⎝ T ⎠ r
Tβα(η)=iv∫dρe ρ
v aβα(ρ)
(8)
Where
ZP andZT
arethechargeofprojectileandthetargetnucleusrespectivelyandηristhe
transverseimpulsiontransfer(ηr.vr=0 ).Fromthisrelationwecanseethatthephasefactor
2iZP ⎛⎜Z −1⎞⎟
ρ v isreintroduced,itrepresentsthepotentialcontributionbetweentheprojectile
and the whole target inter-aggregate contribution [6].
For an excitation process, the differentialcross section is given by the relation:
dσβα =
μ kα
βα
(ηr)2
(9)
dΩ 4π2 k
whereΩ indicates the solid angle. Using the case oflittle longitudinal impulsion transferkα ≈1,
kβ
the total cross section will be:
σβα
+∞
=2π dρρ
0
a (ρr)2
(10)
Inthepresentcalculations,thetotalcrosssectionhasbeenevaluatedbysubstitutingthe expressionoftheapproximatetransitionamplitude(5)in(10)andomissionoftheintegration
overtherangeofimpactparameters[ρ0,+∞[inourcaseρ0=11.2au. Therefore,thetotalcross
section may be approximated as:
σβα
=2π
ρ0
∫0 dρρ
a%βα
(ρr)2
(11)
Inthecode,apiecewiseSimpsonintegrationismade.Anautomaticproceduremanagesto
+
ensureagivenaccuracyoftheoutcome.ThetargetoperatorGT
hasbeenexpandedonthe
wholediscretespectrum.Thecontributionofthecontinuum hasbeentakenintoaccountusing an analyticalcontinuation which consists to evaluate the art closeto ionization threshold.
Inourstudy,wearedealingwithtargetexcitationfromthegroundstate.Hence,thestates i
and j ,intheexpression(5),arechosentobeintheset{υ}
ofEigen-statesofthetarget,
which are solutions of the eikonal Schrodinger equation of the target:
⎛ ∂ ⎞
(12)
⎜ −iv +H T ⎟ υ(z) =0
⎝ ∂z ⎠
Further, they are restricted to a subset which contains the lowest target states, includingα and
β.Using thesteps of our calculations, we find theresults of the following approximations:
% r − i (βV α)
1-The first Born approximation (Born-I):aβα(ρ)is replaced by v
in (11).
r − i
βV −VG+V α
2-The second Born approximation (Born-II):a%βα(ρ)is replaced by (
T )in (11).
r i (βV α)(βV α)
3-Schwinger-Born-approximation:a%βα(ρ)is replaced by− v
(βV
−VG+V
α)in (11).
4-Schwinger55approximation(Schw55):ψ+
and ψβ
areexpandedontheabove-mentioned
five-statebasissetas{1s,2s,2p0,2p+1,2p-1}.Thisapproximationisusedtostudythe excitation ofLi2+(1s)to the states 2s, 2p.
Results and discussions:
The present 1s՜2s excitation total cross sections, represented by Figure 1, show that the Schw55results,whichrefertotheapproachof Schwingerwith5states,haveapeaklocated around 75 keV. But above 800 keV this procedure, even in the Figure 2 and Figure 3, represented the total cross sections for excitation to the state 2p and to the level n=2 respectively, joinwithan almostsimilarresultsobtainedusingdifferenttheoreticalapproaches: TCAO-32, TCAO-59 of Ermolaev et al [1] and FHBS of Hall et al [2].
TheapproachFHBS(FiniteHilbertBasisSet) givesresultsslightly higherthanthoseof Ermolaevetal[1].AlthoughthecrosssectionsarisingfromTCAO-32,TCAO-59andFHBS aregenerallyaboveourSchw55results,theirbehaviorissimilartothebehaviorofcross sectionsobtainedbyourtheoreticalprocedure.Thetwotheoriesprovidealsouptoproton energy around 75 keV.
thebehaviorofourresults,fortheexcitation1s՜2poftheapproximationSchw-B,presented in Figure 2 proved in very good agreement with the behavior of the results of both approaches TCAO-32andTCAO-59.Whilethosearisingfrom Schw55areslightlylowerintheenergy range 30-300 keV.
InFigure3,onereportstotalcrosssectionsforexcitationtotheleveln=2whichareobtained by summingexcitation cross sections to 2s and 2p states. It is essential to note that the theoreticalresultsprovidedbythefirstandsecondBornapproximationsgreatlyoverestimate thetotalcrosssectionsfortheexcitationtothe leveln=2inthefieldoflowenergy.Butthey give the same pace as other theoretical results for high energies.
IntheTCAO-32,TCAO-59,Schw-B,Schw55results presentedhere,wehave smallvaluesof theexcitationtotalcrosssectionsathighimpactvelocitieswherewehavetherequisiteenergy toremovetheelectron.Thentheionizationprocessisthemostdominantinthisenergyrange. Atlowandintermediateenergiestheresultsdifferduetotheexistenceofmultiplescattering, but they have the same behavior.The lack of experimental results cannot rule on the validity of each theoretical calculation presentedhere.
100.0
10.0
Ermolaev etal(TCAO-32) Ermolaev etal(TCAO-59) Hall etal(SCE)
Born-I Born-II
Schwinger-Born (Shw-Born) Schw55
100.00
10.00
1.00
1.0
0.10
Ermolaev et al.(TCAO-32) Ermolaevet al (TCAO-59) Halletal(SCE)
Borb-I Born-II
Schwinger-Born(Schw-Born)
Sch55
0.1
1 10 100 1000 10000
Impact energy E(keV)
0.01
1 10 100 1000 10000
ImpactenergyE(keV)
Figure1:crosssectionsfortheexcitation tothestate2sofLi2+byprotonimpacts
Figure2:crosssectionsfortheexcitation thestate2p ofLi2+byprotonimpacts.
100.0
10.0
1.0
Ermolaev etal(TCAO-32) Ermolaev etal(TCAO-59) Hall etal(SCE)
Born-I Born-II
Schwinger-Born (Schw-Born)
Schw55
0.1
1 10 100 1000 10000
Impact energyE(keV)
Figure3:crosssectionsfortheexcitationto thelevel n=2ofLi2+byprotonimpacts.
Conclusion:
WehavesuccessfullyappliedtheSchwingervariationalmethodtostudytheexcitationofthe ionLi2+(1s)totheLi2+(n=2)state byimpactofprotonswithenergiesrangingfrom9keVto3
Mev.Thedirectexcitationcrosssections,deducedfrom thisnewapproach,showverygood convergenceof the variationalapproachwhen one increasesthenumberof thetargetstateson whichthescatteringstatesareexpanded.Goodresultsareobtainedwhenthescatteringstates
+ and
β aredevelopedonthebasisof5states,comparedwiththoseofErmolaevetal[1]
and those of Hall et al [2].
At higher energies, the varioustotal cross sections prove to be comparable. This deduction shouldbesupported,ofcourse,bythenewclose-couplingcalculationswhichareconstantly used bases of atomic orbitals increasingly large.
Inthiscase,fortheexcitationtheionLi2+(1s)byabareion,itshouldbepossibletoimprove thepresentvariationalapproachbyintroducingatleastthecapturegroundstate1sinthe
expansion of bothψ+
andψ− .
Finally,thepresent-dayvariationalprocedureappearstobepowerfultoolstoinvestigatethe excitation process in atomic collisions at intermediate impact velocities.
References:
[1]A. M. Ermolaev and M. R. C. McDowell; J. Phys. B: At. Mol. Phys. 20 L379-L383 (1987) [2] K. A. Hall, J. F. Reading and A. L. Ford; Phys. B:At Mol. Opt Phys. 29 pp1979-
1994(1996)
[3]B.Lasri,M.Bouamoud,R.Gayet,NuclearInstrumentsandMethodsinPhysicsResearch
B 251, pp66–72 (2006)
[4]RA.Phaneuv,RKJaneuv,Atomic ad Plasma-Materialinteraction Processes in Controlled Thermonuclear Fusion ed R K Janev and H W Drawin(Amsterdam:Elsevier) pp371–80 (1993)
[5] A M Ermolaev, J. Phys. B: At. Mol. Opt. Phys. 23 (1990) L45-L50.
[6]M.Bouamoud,ThèsedeDoctoratd’EtatesSciences, Universitéde BordeauxI,1988 (unpublished).
[7]R.GayetandM.Bouamoud,NuclearInstrumentsandMethodsinPhysicsResearch
B 4 .515-522 (1989)
[8] K. Janev and A. Salin, Ann. Phys (N.Y.) 73,136(1972)