Volume71A, number1 PHYSICSLETTERS 16 April 1979

COLLISIONS BETWEEN ATOMIC SYSTEMS~

Victor FRANCOPhysicsDepartment,Brooklyn Collegeof theCity UniversityofNew York,Brooklyn, NY11210,USA

Received3 October1978

Revisedmanuscriptreceived29 November1978

We calculatethescatteringamplitudefor elasticandinelasticcollisionsbetweenarbitraryionsor neutralatomsby meansof anexpansionofa generalizationof theGlauberapproximationfor particle—atomcollisions.

The Glauberapproximation[1] wasintroducedto atomicphysicsin 1968 via anapplicationto scatteringofchargedparticlesby hydrogenatoms[2] . Thisapplicationhas led to numerousanalysesof scatteringby hydrogenby meansof that approximationor somevariant thereof[3,41.Theutility of theGlauberapproximationwassig.nificantly increasedwhena one.dimensionalintegralexpressionwasobtainedfor collisions of chargedparticleswith arbitraryatoms[5] . This led to manyapplicationsto scatteringof particlesby atomsor ions [3,6,71.Wenowextendthe applicability of the approximationto collisionsbetweencompositesystems,eachof which may be anion or a neutralatom,by employingan expansionof a generalizationof the Glauberapproximationfor particle—atomcollisions.

Considerthecollision of a projectileof nuclearchargeZ1 andN boundelectronswith a targetof nuclearchargeZ2 andM boundelectrons.Let u be theincidentvelocity of theprojectilerelativeto the target.Let the coordinateof the/thelectronof the projectile(target)relativeto its nucleusber~!(r1)with componentsz1andSj (z1 ands1)parallelandperpendicular,respectively,to v. Let R = b + ~denotethepositionof the projectilenucleusrelativeto thetargetnucleus,with b and~being,respectively,perpendicularandparallelto u. The Coulombinteractionbetweenthe projectileandthetargetis

{~},{r})=e2 (Z

1Z2/R_Z1E i~-Ri_1 -z

2E I~+RH+~ ~ I~_~+RI_I). (1)1=1 j—1 i=lj=1

Forsufficiently largeR this approachesVc(b,~) (Z1 — N)(Z2 M)e2/R.Notethat Vc = 0 if eitherthe projectile

or targetis neutral.Define a functionxby

x(b, {s~},{s})~_(1Iv)’f V(b,~,{,~},{~‘})d~. (2)

The correspondingfunctionx,~H(b,b—s) for collisionsbetweena chargedparticle(x) andneutralhydrogen(H) is[2]

x~(b,b —s) = (—2Z~e2/Ilu~)ln(Ib—si/b),

whereZxe and~ arethe chargeandrelativespeedof particlex. It follows from eqs.(1) and(2) that,apartfromthe contributionof Vc, x for collisionsbetweencompositesmaybe expressedentirely in termsof XpH andXeH

* This work wassupportedin partby theNational ScienceFoundation,theNational AeronauticsandSpaceAdministration,and

a CUNY PSC-BHEresearchaward.

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Volume 71A, number I PHYSICS LETTERS 16 April 1979

for collisionsof protons(p) and electrons(e) with hydrogenatoms.The contributionof V~maybe handledusinga techniqueemployedin Coulomb-nuclearstudies[81 andrecentlyappliedto atomic collisions [6] . The resultobtainedis

M N

2x(b, {~.},(S;}) = (2Z1 —N)~x~11(b,b—s1)+ (2Z2 -M) XpH(b,b +s1)

(3)MN

+�~~ [XeH(b+5,b+S)_Sj)+XeFI(bSi,b_Si+S))1 +2~~(b),

where~ is the contributionto x arising from thepoint Coulombinteraction V~.TheamplitudeFfi (q, k) for collisionsin which a composite,incident with momentum1~k,transfersmomentum

iiq to anothercompositeandtheentire systemmakesa transitionfrom initial stateito final statef is given,in theGlauberapproximation,by [8,9]

Ffi(q,k) = (ik/2ir) fei~1fIl ~—exp[i~(b,{s~,~s}} Ii>d2b, (4)

+ (ik/21r)f ~iq ~be~c(”) Ffi(b)d2bwhereeqs.(4) and (5) definea profile function Ffi(b), andJcöfi is theknown [1] point Coulombcontributiontothe scatteringamplitudearising from ~. Consequently

F~(b)= — (flexp[ix(b, {s~},{~})— i~~(b)]Ii> . (6)

OnceFfi(b) is obtained,the scatteringamplitudeFfi maybe calculatedfrom eq.(5).The profile function,FXH, for x-hydrogencollisionsis [21 F,~(b, b — s) = 1 exp[ixXH(b,b — s)I . We expand

Ffi of eq.(6) in termsof rPH andFeH. The first orderexpansionretainsin Ffi both the pH and eH profile functionsto first order;Thisprocedurestill accountsfor somemultiple collisionssincethe Glauberapproximationfor par-ticle—hydrogenatom scatteringtakesdoublescatteringinto account[2] . The result is f’~= FR) + ... ,wherethefirst order profile function,FR), is givenby

= (fIFU)Ii> (7)

2FW (2Z1 —N)EFPH(b,b —s1)+(2Z2—M)~FPH(b,b +s)

MN (8)

+~ ~ [FeH(b+S),b+S_Sj)+FeH(b_Si,bSi+S)]1=11=1

SinceFPH(b,b — s~)is a single-particleoperatoras far as thetargetelectronsare concerned,its matrix elementvanishesif, in th~final statef, more thanone electronin the targetis excitedor theprojectile is excited.A similarremarkappliesto FPH(b,b +s).The matrix elementsof the~eH’~vanish if more than one electronis excitedineither thetargetor projectile.

To evaluateFR) we define thetransition densitymatrix for the target,p~r1),by

p~(ri)=f~p!’*(rj rM)~,11~(rl rM)d3r

2...d3rM,

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Volume 71A, number 1 PHYSICS LETTERS 16 April 1979

where i~i~andI,l/f arethe initial and final statewave functionsof the target.We assumethat p~may beexpressedas

p~(~)zrEdkTrnkTe_~kTrYf~T*Ø~)y~T(,), (9)

with a similarexpressionfor p~(r’)for the projectile(with P replacingthe indexT). Our resultscanbe trivially ex-tendedto includethe moregeneralform in which the subscriptsITmT and11~,m?1.arereplacedby lkTmkT andlkTmkT, respectively.However,since suchformsare notusuallyemployedin describingatoms,we shall explicitlypresentthe solution for wave functionssatisfyingeq.(9).

Let jp, f~,‘1~~T denoteinitial (i) and final (I) statesof the projectile (P) and target(T). Let 1 21 + 1 for anyangularmomentumquantumnumber,~e e

2/hu,~ (Z1 _iV)(Z2_M)fle, ~ T m4 — mT,~

7l~P— mp,~ “~T”M~ IC)l~~I,andM~ I’~1t~+ ‘MT’ ~ After considerablecalculation,the 3(M+JV)+ 2-dimensionalintegral for the scatteringamplitudeF~)(q)reducesto

F~)(q) f~(q)6ç1+ ~t~i~(zi — 41V)Mf~’(q)+ ~fTiF~2 — ~MWf~(q)

= (10)

+ +MNik f J~~(qb)[y~(b) + ‘y~’~(b)j(kb)2~bdb,

f~(q)= i(2k/q)1 +2l~q_3[F(1+in~)/F(1—i~~)]E dkT(_1)nkT(lTl~mTm~(a/aa~y~kT+1

X ~ i~~ + (_i)m~ E (4 /i)”2i*(1Tl~l; mT’ - m+) (11)

X c(1.~1-I~-1,00)YrT*(.Lir cbq)(2/qy(a/ao~~)!1kT_1+1

(a~T—1 —i~—(l+MT)/2,—1, —l —i~—(l—MT)/2

33k q2 —1 —i~—(l—MT)/2,0,—i —ii~ — (l+MT)/2

f~(q)= (—i)1P~i~f~(T-~P), (12)

= 2ir [F(i + ~~1e)Ifl~fle)] X,k,4L d~J4kTi’~— l(iPi;1Tr.!~/jL)lI2o,4l,L;mp,— m~)

C(lpl~L;00)C(l1.1~1;mT,—m~)C(lTl’~r1;00)2~~L!(_2/a,~TYZkT+ 3y~iP*(4~cbq)YrT*(41T,cbq)

X ~‘ G24L~—MT/2,(1+3)12, 1+1/2, 1 —l/2,MT/2 (13)y2 ~ + MT/2, 1 + 112, ~ — MT/2,(n~+ 3)/2,(~kT+ 4)/2 )

1 (_a/a~~)n~— L +1(a~+y2)—~’— 1~M~(1-’Y)dy,

7~(b)= ~ ~e-P). (14)

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Volume 71A, number 1 PHYSICSLETTERS 16 April 1979

In eqs.(11) and (13), G~is a Meijer G-function [10] .The notationY~T(T P) meansthe indicesT and Pareinterchangedeverywherein thefunction‘y~”.If at leastone of theatomicsystemsis neutral(as in ion—atomoratom—atomcollisions)our resultssimplify further to give a closedform expressionforFR~.

The reductionof the scatteringamplitudefrom anintegralof dimension3(M+N) + 2 to a two-dimensionalintegral for ion—ion collisionsandto closedform for ion—atomand atom—atomcollisionsmakespractical,for thefirst time, a ratherwide variety of applicationsof the Glauberapproximationto collisionsbetweencompositeatomicsystems.

References

[1] R.J.Glauber,in: Lecturesin theoreticalphysics,Vol. 1, cds. W.E. Brittin et al. (Interscience,New York, 1959)p. 315.

121 V. Franco,Phys.Rev. Lett. 20 (1968)709.[3] E. Gerjuoyand B.K. Thomas,Rep.Prog. Phys.37 (1974) 1345 andreferencestherein.[4] G. FosterandW. Williamson Jr.,Phys.Rev.AI3 (1976)2023;

J.N. Gauand J. Macek,Phys.Rev. AI2 (1975)1760;T. lshiharaandJ.C.Y.Chen,Phys.Rev. A12 (1975)370.

[5] V. Franco,Phys.Rev. Lctt. 26 (1971) 1088.[6] B.K. ThomasandV. Franco,Phys.Rev.A13 (1976)2004.[7] T. IshiharaandJ.C.Y.Chen,J. Phys.B8 (1975)L417;

II. Narumi andA. Tsuji, Frog. Theor.Phys.53(1975)671.[8] V. Francoand G.K. Varma,Phys.Rev. C12 (1975) 225.[9] V. Franco,Phys.Rev. 175 (1968) 1376.

[10] A. Erdelyi Ct al., Higher transcendentalfunctions,Vol. 1 (McGraw-Hill, New York, 1953) p. 206.

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