Relevant data is available at: http://www.astronomy.ohio-state.edu/~nahar/nahar_radiativeatomicdata/index.htmlThe contributions of higher partial waves to the elastic scattering amplitude are dominated by long-range interactions which fall off as r^-n as r → ∞. Closed-form expressions for the contributions of higher partial waves (2l > n - 3) to the scattering amplitude for various long-range interactions (n ranging from 3 to 8) are presented.This research has been supported, in part, by the Air Force Office of Scientific Research under Grant No. AFOSR-84-0143

Nahar, Sultana Nurun

Wadehra, J. M.

Physical Review A

KnowledgeBank at OSU

PHYSICAL REVIEW A VOLUME 36. NUMBER 3 AUGUST 1. 1987Contributions of higher partial waves to the elastic scattering amplitudefor various long-range interactionsJ. M. Wadehra and Sultana N. NaharDepartment ofPhysics and Astronomy, Wayne State University, Detroit, Michigan 48202(Received 29 January 1987)The contributions of higher partial waves to the elastic scattering amplitude are dominated bylong-range interactions which fall off as r -n as r--. 00. Closed-form expressions for the contributionsof higher partial waves (21) n' - 3) to the scattering amplitude for various long-range interactions (nranging from 3 to 8) are presented.where the transition matrix elements T/ are related tothe phase shifts 81 byWhen the interaction between a projectile and a targetis central, it is convenient to use the method of partialwaves to obtain the elastic scattering amplitude and thecorresponding differential and integrated cross sections.Each partial wave corresponds to a definite angularmomentum of the system. FOT a fixed incident energy ofthe projectile, the higher partial waves correspond tolarger impact parameters of the incoming projectile. Ifthe impact parameter is larger than the range of the in-teraction. the contribution of the corresponding partialwave to the scattering amplitude, for this particular ener-gy. is zero. Thus for short-range interactions betweenprojectile and target only a finite number of partial wavesmake contributions to the scattering amplitude. More-over, the number of contributing partial waves increasesas the impact energy increases. If the projectile-target in..teraction has a long-range tail, then, in principle, aninfinite number of partial waves will contribute to the am-plitude. In practice, the phase shifts of the first few (say,L) partial waves are computed exactly by solving for theasym ptotic radial part of the wave function and the phaseshifts of the higher partial waves (from L + 1 to infinity)are estimated by the Born approximation using only thelong-range interaction. (For some recent examples seeRefs. 1 and 2.) The purpose of this work is to provideclosed-form expressions for the contributions of the higherpartial waves to the scattering amplitude for variouslong-range interactions. As far as we are aware, such aclosed-form expression has been obtained previously onlyfor the 1/r4 potentialr' the results fOT other long-range in-teractions are new in the present work.Consider the elastic scattering of a projectile of mass I.tand energy E =fz2k 2/2p, by a central potential V(r) whichresults in a change in momentum from fik; to 17k! for theprojectile tk, = kf = k ). The momentum transferliq = -li( kf - k,.) is related to the scattering angle (J byq =2ksin((J/2). For later convenience we define a quanti-ty z as z=sin(O/2)=q/2k. The partial-wave expansionfor the elastic scattering amplitude can be written as(2)(3c)(3b)1 oc6/= k ~ (21 + 1 )T1BP/(cos6)I=L+lwhereHere IB(O) is the complete scattering amplitude in theBorn approximation, that is,If the quantities Ti for I > L are approximated by TIBwhich are obtained by using the Born approximation,then I( 6) can be rewritten as1 L1(8)=k~(21+1)T/P/(cos())+l1f, (3a)/=0Here ao=li2/me 2 is the Bohr radius, -e and m are thecharge and mass of an electron, and en are dimensionlesscoefficients. For certain values of n the two terms on theright-hand side of Eq. (3c) could be individually singularbut the difference, namely, J11 is always finite and non-singular. Substituting from (6) into (5) and using Eq.6.574.2 of Ref. 4 one obtains18(8)= - ~ f 00 dr V(r)r 2 sin(qr) (4a)~ 0 qr= - ~ f 00 dr V(r),2 i (21 + 1Ul(kr)p,(cos(J) (4b)n: 0 1=01 oc=k ~ (2/ + 1)T/BP, (cos(}) (4c)1=0and T/B is the Born approximation for T/, namely,TIB = - 2/1:: J OX>dr V(r)r 2j{(kr) • (5)" 0Now, since only the long-range part of the interactioncontributes to phase shifts of higher partial waves, it isreasonable to take for V(r), in Eq, (5), a general spherical-ly symmetric potential of the formV(r)=C,.a 8-le 2 /r" . (6)(1)1 ClC[(8)= k ~ (2/ + 1)T/P,(cos6) ,/=01458 @ 1987 The American Physical SocietyBRIEF REPORTS 1459(l Oa)where, as before, z = sin ((J12).the closed form expression for ~! in this case can be writ-ten asand the results of Table I to obtain the following closed-form expression for a!,61r.1J k 3a4C~f= r- 0 6mf z 3 L p/(cosO) ]X [-18+/;0(2/+5)(2/+3)(21-1)(2/-3) ,(12b)(12a)(lIb)[.!. - £ +2z 21n(z)3 6L (21 + 1)P,(cos8) ]-/;2 (/ +2)(1 + 1)/(/ -I) ,2uk 2a 3C~/= _ r- 0 53mwhere z =sin(8/2).(iv) n =6. A long-range potential of the form 1/r" ap-pears in various applications in atomic and molecularphysics. First, the nonrelativistic van der Waals interac-tion between two neutral atoms, each in its lowest energystate, behaves as 1/r 6 for separations much larger thanthe Bohr radius. 11 Next, the second term in the interac-tion energy between a charged projectile and a neutral po-larizable target behaves, in the adiabatic approximation, as11r 6. Third, the leading nonadiabatic correction to thedipole term in the interaction between a charged projectileand a neutral polarizable target behaves asymptotically as1I r 6• 12 In these cases one can useoc P,(cosO) z38 6 = /~o (21+5 )(21 +3)(21 -I )(2/- 3) = 18projectile in motion and a neutral polarizable target, the. leading correction term behaves'' asymptotically as 1/r s.The 1IrS retardation effect also appears when applied toan electron-ion system with the electron bound in a highRydberg state." 10 This term, which is an addition to the1/r 4 polarization potential, vanishes in the nonrelativisticlimit. Again using the results in Table I and00 (21 + 1)P,(cosO) 1 z2 2s, = /';2 (/ +2)(/ + 1)1 (/ -I) = 3 -"6 +2z In(z) •(11a)where z =sin((J12), we obtain the following closed-formexpression_ JLa oC 3 [ L (2J + 1)PI(cos8) ]al- m 1+21n(z)+,;t ut s i, . (9b)(ii) n =4. The case of l1r 4 potential, perhaps the mostwell known in atomic physics, arises as the leading term,in the adiabatic approximation, in interactions of acharged projectile (electron, ion, positron) with a neutralpolarizable target. Again using the relevant entry inTable I for T/B andQO P/(cos8) z8.= /;0 (21+3)(21-ll = -"2 '(7)the closed form expression for al becomes_ 21TJ.tka aC4 Is. ~ PJ (cosO) ]6.1- m 2 + /~o (21 +3)(21 -ll • (LOb)with z =sin«(J12). This expression has previously beenobtained by Thompson. 3(iii) n = 5. When retardation effects are taken into ac-count for interactions between a structureless chargedThe conditions of validity for this expression aren > 1, l>(n -3)/2 . (8)Table I shows explicit expressions for TIB for values of nfrom 3 to 8. The corresponding long-range interactions,along with their possible applications in atomic andmolecular physics, are now discussed individually.0) n = 3. A potential of the form 1I r 3 appears as theleading retardation term (that is, the effect of the finitenessof the speed of light) for electron-electron interactions.v''In a different context, when two identical neutral atoms,excited to different degrees, exchange a photon it leads 7 toa "resonance interaction" of the form 1I r 3• Using the ex-pression for TIB for n = 3 in Table I and the sumee (21 + 1)p/(cos(J)S3= ~ ut 1) =-1-21n(z) (9a)/=1 +r [l-!!. + 1. ]r [!!- _.!. ]en 1£. - n -2 2 2 2 2T/B=-T m V1T(kao) [n] [ 1 n]r - r 1+-+-2 2 2TABLE I. Explicit expressions for T'B for various long-range interactions.nCondition ofvalidity3456781/[1 (/ + 1)]21TI[ (21 +3)(21 + 1)( 2/ - 1)]21[3(/ +2)([ + 1)/ (/-1)]61TI[ (2/ + S)(21 + 3 )(21 + 1)(2/ - 1)(2/ - 3) ]8/[ 1SU + 3)( / + 2)( I + 1)1(1 - 1)( I - 2)]201T/[(21+ 7)(21 +5)(21 +3)(21 + 1)(2/-1 )(21-3)(2/ -5)]1';2:1I ~ 11~21~2/';2:3t z 31460 BRIEF REPORTS(v) n = 7. When retardation effects are taken into ac-count the interaction between two neutral atoms in theirlowest energy states behaves as 1I r 7 for extremely largeatom-atom separations.I°. 13 This interaction is a replace-ment for and not an additive correction to the nonrela-tivistic van der Waals interaction. Using the appropriateentry from Table I andec (21+ 1)p/(cosO)S, ='';3 (l +3)(l +2)(1 + III (l -1)(/ - 2)(13a)the closed-form expression for ~f becomes8f..Lk 4o lJC7[-43 4 z2 1 z4tJ.f= 15m 240 z +s-SO+T1n(z)L (21 + 1 )PI(COSO) ]+'';3 (I + 3)(I +2)(/ + 1)1 (I - 1)(1 - 2) ,(13b)where z=sin(812).(vi) n = 8. In investigations of the Rydberg states ofhelium it is noted that terms up to 1II' 8 in the polariza-tion potential arising in the ion-core-electron interactioncontribute significantly to the energy levels. 14, IS With theanticipation that interaction terms behaving asymptotical-ly as 1/1'8 might also contribute significantly to scatteringprocesses, we provide the following closed-form expres-sion for at for this case,201TUksa6C [zS L P,(cos(}) ]i1f= r: 0 8 ~m 450+/~-(2-/-+-7-)-{2-/-+-S-X-2-/+-3-)(-2/---I-)(2-1-- 3-)-( 2-/---5-) , (14a)where z=sin(912). Now the sums S3, Ss, and S7 can beeasily evaluated by first doing the partial fractions of thesummand and then using the sums of Eq, (16). The sumsS4t 8 6 , and S8 are evaluated by first doing the partial frac-tions of the summand, then usingwith z = sine (J12), using the results in Table I and the sum00 PI(cose)Sa =';0 (21+7)(21 +5 )(21 +3 )(21-1 )(21 - 3)(2/- 5)(l4b)The closed-form expressions [Eqs. (9b), no», nn»,(12b), (13b), and (14a)] for tJ.f for various long-range po-tentials depend crucially on the evaluation of sums Sj,i = 3, ... ,8 [Eqs, (9a), (lOa), (11a), 02a), (13a), and(14b)]. These sums can be obtained as follows. On multi..plying both sides of the generating functionec P/(case> 4 3 2~ (/-2) =(-7z +6z +5z -4z)1=3-(6z 4 -6z 2+ 1)1n[z (1 +z)] ,(160(16g)(15)i t1PI(COS(J) = ( 1-2t cos(J+t 2 ) - 1/ 2/=0by t " (m = -3, -2, ~ 1,0,1,2,3) and integrating withrespect to t from 0 to 1, one can establish the followingsums:!Y:: Pj(cosO)~ =(20z s-10z4 _1Qz 3+ lOz2+6z _.!!)I =0 (I +4) 3 6+ (-20z 6+ 30z 4 -12z 2+ 1)X ln ( 1+ 1/ z) , (16a)00 PI (cose)~ =( -6z 3+3z 2+4z _1.)I =0 (I + 3) 2+ (6z 4 _6z 2 + 1)1n[( 1+z)/z] , (16b)IX PI (cose)~ (I 2) =(2z-1)+(-2z 2 + 1)ln [( 1+ z )l z ] , (16c)1=0 +00 PI (cos8)~ (1 1) =In[( 1+z)/z] , (l6d)1=0 +ee PI(cose)~ I =-In[z(l+z)], (16e)1=1sin8 dP,(cosf)[PI + 1(cos8)-P1_ 1(cos8)]/(21 + 1) = I (I + 1 )dO (17)[and similar lengthy relations for (PI + 2 - P, _ 2) I( 21+ 1)and (Pr+3-PI-3)/(2/ + 1)] and eventually utilizing thesums of Eq. (16). It is rather remarkable that after manypages of algebra the sums S 4, S6, and S 8 turn out to be assimple as in Eqs. (l Oa), (12a), and (14b), which suggeststhat there could be an easier and perhaps more generalprocedure (which obviously eluded us) of evaluating sumsof this kind.Finally we comment that in the present work the con-tributions of higher partial waves to the elastic scatteringamplitude for various long-range interactions are takeninto account via the first Born approximation. It is nowpossible to include, in principle, the additional correctionterms via the second Born approximation since an exactanalytical expression for the second Born contribution tothe transition matrix elements TI for any long-range in-teraction has been obtained recently by Wadehra.l"It is a pleasure to thank Professor Larry Spruch for ahelpful conversation. This research has been supported,in part, by the Air Force Office of Scientific Research un-der Grant No. AFOSR-84-0143.BRIEF REPORTS 1461IR. K. Nesbet and S. Gehman, Phys. Rev. A 33, 3815 (1986).2W. L. van Wyngaarden and H. R. J. Walters, J. Phys. B 19,1817 (1986).3D. G. Thompson, Proc. R. Soc. London Sere A 294, 160 (1966).41. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series,and Products (Academic, New York, 1965).sT. Fulton and P. Martin, Phys. Rev. 95, 811 (954).6L. Spruch, Phys. Today 39(11),37 (1986).'H. Margenau and W. W. Watson, Rev. Mod. Phys. 8, 22(1936).8J. Bemabeu and R. Tarrach, Ann. Phys. (N.Y.) 102, 323 (976).9E. J. Kelsey and L. Spruch, Phys. Rev. A 18, 15 (1978).tOLe Spruch and E. J. Kelsey, Phys. Rev. A 18, 845 0978}.11F. London, Z. Phys, 63, 245 (1930).12C. J. Kleinman, Y. Hahn, and L. Spruch, Phys. Rev. 165, 53(1968).13H. B. G. Casimir and D. Polder, Phys. Rev. 73, 360 (1948L14R. J. Drachman, Phys. Rev. A 26, 1228 (1982).lSC. K. Au, Phys. Rev. A 34, 3568 (1986).16J. M. Wadehra, J. Phys. B 19, L761 (1986).

Contributions of higher partial waves to the elastic scattering amplitude for various long-range interactions

J. M. Wadehra

Sultana N. Nahar

Crossref

The contributions of higher partial waves to the elastic scattering amplitude are dominated by long-range interactions which fall off as r^(−n) as r→∞. Closed-form expressions for the contributions of higher partial waves (2l\u3en-3) to the scattering amplitude for various long-range interactions (n ranging from 3 to 8) are presented

Wadehra, J M

Nahar, Sultana N

Digital Commons@Wayne State University

Wayne State UniversityPhysics and Astronomy Faculty ResearchPublications Physics and Astronomy8-1-1987Contributions of higher partial waves to the elasticscattering amplitude for various long rangeinteractionsJ M. WadehraWayne State University, ad5541@wayne.eduSultana N. NaharWayne State UniversityThis Article is brought to you for free and open access by the Physics and Astronomy at DigitalCommons@WayneState. It has been accepted forinclusion in Physics and Astronomy Faculty Research Publications by an authorized administrator of DigitalCommons@WayneState.Recommended CitationWadehra JM, Nahar SN. Contributions of higher partial waves to the elastic scattering amplitude for various long-range interactions.Phys. Rev. A. 1987;36(3):1458-1461. doi: 10.1103/PhysRevA.36.1458Available at: http://digitalcommons.wayne.edu/phy_astro_frp/100PHYSICAL REVIEW A VOLUME 36, NUMBER 3 AUGUST 1, 1987Contributions of higher partial waves to the elastic scattering amplitudefor various long-range interactionsJ. M. Wadehra and Sultana N. NaharDepartment of Physics and Astronomy, Wayne State University, Detroit, Michigan 48202(Received 29 January 1987)The contributions of higher partial waves to the elastic scattering amplitude are dominated bylong-range interactions which fall off as r " as r ~ ao. Closed-form expressions for the contributionsof higher partial waves (2l ~ n —3) to the scattering amplitude for various long-range interactions (nranging from 3 to 8) are presented.When the interaction between a projectile and a targetis central, it is convenient to use the method of partialwaves to obtain the elastic scattering amplitude and thecorresponding differential and integrated cross sections.Each partial wave corresponds to a definite angularmomentum of the system. For a fixed incident energy ofthe projectile, the higher partial waves correspond tolarger impact parameters of the incoming projectile. Ifthe impact parameter is larger than the range of the in-teraction, the contribution of the corresponding partialwave to the scattering amplitude, for this particular ener-gy, is zero. Thus for short-range interactions betweenprojectile and target only a finite number of partial wavesmake contributions to the scattering amplitude. More-over, the number of contributing partial waves increasesas the impact energy increases. If the projectile-target in-teraction has a long-range tail, then, in principle, aninfinite number of partial waves will contribute to the am-plitude. In practice, the phase shifts of the first few (say,L) partial waves are computed exactly by solving for theasymptotic radial part of the wave function and the phaseshifts of the higher partial waves (from L +1 to infinity)are estimated by the Born approximation using only thelong-range interaction. (For some recent examples seeRefs. 1 and 2.) The purpose of this work is to provideclosed-form expressions for the contributions of the higherpartial waves to the scattering amplitude for variouslong-range interactions. As far as we are aware, such aclosed-form expression has been obtained previously onlyfor the 1/r potential; the results for other long-range in-teractions are new in the present work.Consider the elastic scattering of a projectile of mass p,and energy E =Pi k /2p by a central potential V(r) whichresults in a change in momentum from Ak; to RcI for theprojectile (k; =k/ =k ). The momentum transferAq = ttt'(k/ —k; ) is related to the scattering angle 0 byq =2ksin(0/2). For later convenience we define a quanti-ty z as z =sin(0/2)=q/2k. The partial-wave expansionfor the elastic scattering amplitude can be written asoof(8)= —g (21+ 1)TtPt (cos0),k 1=0where the transition matrix elements T1 are related tothe phase shifts 61 byTt =exp(ifit)sin(5t) . (2)If the quantities T1 for l &L are approximated by T1Bwhich are obtained by using the Born approximation,then f(0) can be rewritten as1 Lf (0)=—g (21 + 1)TtPt(cos0)+ bf,k 1=0 (3a)whereoobf=—g (2l + 1)TtttPt(cos8)1=L+1L=fz(8) ——g (2l + 1)Tttt Pt(cos0} .k1=o(3b)(3c)Here fB(8) is the complete scattering amplitude in theBorn approximation, that is,f, (0)= —'", f "d. v(.)""""o qr (4a)f dr V(r}r g (2l + 1)jt (kr)pt(cos0) (4b)$2 0 1=0oo= —g (2l + 1)Tttt Pt (cos8)k1=o (4c}and T1B is the Born approximation for T1, namely,TlB =2P k oo dr V(r)r jt (kr) .2 ~ 22 0Here ao ——R /me is the Bohr radius, —e and m are thecharge and mass of an electron, and C„are dimensionlesscoe%cients. For certain values of n the two terms on theright-hand side of Eq. (3c) could be individually singularbut the difference, namely, Af is always finite and non-singular. Substituting from (6) into (5) and using Eq.6.574.2 of Ref. 4 one obtainsNow, since only the long-range part of the interactioncontributes to phase shifts of higher partial waves, it isreasonable to take for V(r), in Eq. (5), a general spherical-ly symmetric potential of the formV(r)=C„ao 'e /r" .36 1458 1987 The American Physical Society36 BRIEF REPORTS 1459n —2T(~ = — ~&7r(kao)"—2 mnr—21 nI I+ —+—2 2n 3 nr I ——+ —r ———2 2 2 2(7)projectile in motion and a neutral polarizable target, theleading correction term behaves asymptotically as 1/r ~The 1/r retardation effect also appears when applied toan electron-ion system with the electron bound in a highRydberg state. ' This term, which is an addition to the1/r polarization potential, vanishes in the nonrelativisticlimit. Again using the results in Table I andThe conditions of validity for this expression aren & 1, 1& (n —3)/2 . (8)(2!+ 1)P~(cosg)~(1+2)(l+1)l(l —1) + 2z ln(z),1 z3 6Table I shows explicit expressions for Tlq for values of nfrom 3 to 8. The corresponding long-range interactions,along with their possible applications in atomic andmolecular physics, are now discussed individually.(i) n =3. A potential of the form 1/r appears as theleading retardation term (that is, the effect of the finitenessof the speed of light) for electron-electron interactions. 'In a different context, when two identical neutral atoms,excited to different degrees, exchange a photon it leads toa "resonance interaction" of the form 1/r . Using the ex-pression for T~~ for n =3 in Table I and the sum(2l + 1)PI(cosg)S3= g 1 (1 + 1) = —1 —21n(z) (9a)where z =sin(g/2), we obtain the following closed-formexpressionpaoC3 (21+ 1)PI(cosg)bf=m1+21n(z)+ g l(l+1) (9b)(ii) n =4. The case of 1/r potential, perhaps the mostwell known in atomic physics, arises as the leading term,in the adiabatic approximation, in interactions of acharged projectile (electron, ion, positron) with a neutralpolarizable target. Again using the relevant entry inTable I for Tl~ and2pk aoC~ 1 zbf=- +2z'ln(z)3m 3 6(2l + 1)Pi(cosg ), (l +2)(l +1)l(l —1) (1 lb)where z =sin(g/2).(iu) n =6. A long-range potential of the form 1/r ap-pears in various applications in atomic and molecularphysics. First, the nonrelativistic van der Waals interac-tion between two neutral atoms, each in its lowest energystate, behaves as 1/r for separations much larger thanthe Bohr radius. " Next, the second term in the interac-tion energy between a charged projectile and a neutral po-larizable target behaves, in the adiabatic approximation, as1/r . Third, the leading nonadiabatic correction to thedipole term in the interaction between a charged projectileand a neutral polarizable target behaves asymptotically as1/r . ' In these cases one can useS6—oo Pt(cosg)o (21+5)(2l +3)(21—l)(21 —3)z318(12a)(1 la)the closed form expression for Af in this case can be writ-ten asS4 ——Pi (cosg)l~o (21+3)(21—1) 2 ' (10a) and the results of Table I to obtain the following closed-form expression for l5f,the closed form expression for b,f becomes2rrpka pC4 z I- PI (cosg )m 2 I o (2l +3)(21 —1) (lob)with z =sin(g/2). This expression has previously beenobtained by Thompson.(iii) n =5. When retardation effects are taken into ac-count for interactions between a structureless charged6~pk aoC6z3 L PI (cosg)x 18 I o (2l+5)(2l+3)(21 —1)(21 —3)where, as before, z =sin(g/2).(12b)TABLE I. Explicit expressions for TI~ for various long-range interactions.m (ka{))TIBPCq1/[1 (I + 1)]2n/[(21 +3)(2l + 1)(21—1)]2/[3(l +2)(l + 1)l (1 —1)]6~/[(21 + 5)(21 + 3)(2l + 1)(2l —1)(2l —3)]8/[15(l +3)(l +2)(l + 1)l (1 —1)(l —2)]20m /[(2! +7)(21+5)(2l +3)(2l + 1)(2l —1)(2l —3)(2l —5)]Condition ofvalidity1&11&11&21&21&31&31460 BRIEF REPORTS 36S7 ——oc' (21 +. 1 )Pt (cosO), (I +3)(l +2)(l + 1)l (I —1)(l —2)43 4 z 1240 5 8042ln(z), (13a)the closed-form expression for bf becomes(U) n =7. When retardation e6'ects are taken into ac-count the interaction between two neutral atoms in theirlowest energy states behaves as 1/r for extremely largeatom-atom separations. ' ' This interaction is a replace-ment for and not an additive correction to the nonrela-tivistic van der Waals interaction. Using the appropriateentry from Table I and8V'k + oc7 —43~z15m 2401 z+ ln(z)2'(2I + 1 )P~ (cosO)+3 (&+3)(I +2)(1 +1)1(l—1)(l —2)(13b)where z = sin(O/2).(Ui) n =8. In investigations of the Rydberg states ofhelium it is noted that terms up to 1/r in the polariza-tion potential arising in the ion-core —electron interactioncontribute significantly to the energy levels. ' ' With theanticipation that interaction terms behaving asymptotical-ly as 1/r might also contribute significantly to scatteringprocesses, we provide the following closed-form expres-sion for bf for this case,20m@k a OC8 5 L P~(cosO)450, (2I +7)(2l +5)(2l +3)(21 —1)(21 —3)(2l —5) (14a)Ss=oo Pi(cosO)(2l +7)(21+5)(2l + 3)(2l —1)(2l —3)(2l —5)5450 (14b)with z =sin(O/2), using the results in Table I and the sum - P, (cosO)= (2z —2z)+ (2z —1)ln[z (1+z)],(I —1)P~(cosO)=( —7z +6z'+5z' —4z)(1 —2)—(6z —6z +1)ln[z(1+z)],(16f)(16g)The closed-form expressions [Eqs. (9b), (10b), ( 1 lb),(12b), (13b), and (14a)] for bf for various long-range po-tentials depend crucially on the evaluation of sums S;,i =3, . . . , 8 [Eqs. (9a), (10a), (1 la), (12a), (13a), and(14b)]. These sums can be obtained as follows. On multi-plying both sides of the generating functionwhere z =sin(O/2). Now the sums S3, Sq, and Sq can beeasily evaluated by first doing the partial fractions of thesummand and then using the sums of Eq. (16). The sumsS4, S6, and Sq are evaluated by first doing the partial frac-tions of the summand, then usingg t 'Pi (cosO) = (1 —2t cosO+ t )1=0(15) sinO dPI (cosO)[P&+,(cosO) PI &(cosO)]/—(21+ 1)= I I+1 dO (17)X ln( 1+ 1/z ),Pi(cosO)(~+3) =( —6z'+ 3z'+4z ——', )+ (6z —6z + 1)ln[(1+.z)/z],Pi(cosO), , (1+2) =(2z—1)+(—2z + 1)ln[(1+z)lz],- PI(cosO), (I + I)= ln[( 1 +z) /z],P~(cosO)= —ln[z ( 1+z)],1=1 l(16a)(16b)(16c)(16d)(16e)by t (m = —3, —2, —1,0, 1,2, 3) and integrating withrespect to t from 0 to 1, one can establish the followingsums:- P&(cosO)= (20z' —10z' ——", z'+ 10z'+ 6z ——", ), (1+4)+ ( —20z'+ 30z' —12z'+ 1)[and similar lengthy relations for (P~+2 P& q)/(2!+1)—and (P(+3 Pt 3)/(21+1)] and eventually utilizing thesums of Eq. (16). It is rather remarkable that after manypages of algebra the sums S4, S6, and S8 turn out to be assimple as in Eqs. (10a), (12a), and (14b), which suggeststhat there could be an easier and perhaps more generalprocedure (which obviously eluded us) of evaluating sumsof this kind.Finally we comment that in the present work the con-tributions of higher partial waves to the elastic scatteringamplitude for various long-range interactions are takeninto account via the first Born approximation. It is nowpossible to include, in principle, the additional correctionterms via the second Born approximation since an exactanalytical expression for the second Born contribution tothe transition matrix elements T~ for any long-range in-teraction has been obtained recently by Wadehra. 'It is a pleasure to thank Professor Larry Spruch for ahelpful conversation. This research has been supported,in part, by the Air Force Office of Scientific Research un-der Grant No. AFOSR-84-0143.BRIEF REPORTS 1461'R. K. Nesbet and S. Geltman, Phys. Rev. A 33, 3815 (1986).2%'. L. van %'yngaarden and H. R. J. %'alters, J. Phys. B 19,1817 (1986).~D. G. Thompson, Proc. R. Soc. London Ser. A 294, 160 (1966).4I. S. Cxradshteyn and I. M. Ryzhik, Table of Integrals, Series,and Products (Academic, New York, 1965).~T. Fulton and P. Martin, Phys. Rev. 95, 811 (1954).L. Spruch, Phys. Today 39(11),37 {1986).7H. Margenau and W. W. Watson, Rev. Mod. Phys. 8, 22(1936).8J. Bernabeu and R. Tarrach, Ann. Phys. (N.Y.) 102, 323 (1976).9E. J. Kelsey and L. Spruch, Phys. Rev. A 18, 15 (1978).' L. Spruch and E. J. Kelsey, Phys. Rev. A 18, 845 (1978).'F. London, Z. Phys. 63, 245 (1930).'2C. J. Kleinman, Y. Hahn, and L. Spruch, Phys. Rev. 165, 53(1968).I3H. B. G. Casimir and D. Polder, Phys. Rev. 73, 360 (1948).R. J. Drachman, Phys. Rev. A 26, 1228 (1982).'5C. K. Au, Phys. Rev. A 34, 3568 (1986).6J M %adehra J Phys 8 19 L761 {1986)

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Contributions of higher partial waves to the elastic scattering amplitude for various long range interactions

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Contributions of higher partial waves to the elastic scattering amplitude for various long-range interactions

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