Multiple s attering e(cid:27)e ts in quasi free s attering from halo nu lei: a test to Distorted Wave Impulse Approximation 1,2 1 1 1 1 R. Crespo, A. Deltuva, E. Cravo, M. Rodríguez-Gallardo, and A. C. Fonse a 1 Centro de Físi a Nu lear, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal 2 Departamento de Físi a, Instituto Superior Té ni o, Taguspark, Av. Prof. Cava o Silva, Taguspark, 2780-990 Porto Salvo, Oeiras, Portugal (Dated: February 4, 2008) 11 8 Full Faddeev-type al ulations are performed for Be breakup on proton target at 38.4, 100, 0 and 200 MeV/u in ident energies. The onvergen e of the multiple s attering expansion is inves- 0 tigated. The results are ompared with those of other frameworks like Distorted Wave Impulse 2 Approximationthatare based on anin omplete and trun atedmultiple s attering expansion. n a PACSnumbers: 24.50.+g J 9 2 I. INTRODUCTION aim of the present work. We onsider here the quasi free s attering of the A ] nu leus by a proton target leading to the breakup rea - h Interest in quasi free breakup of a Halo nu leus stems n−1 t mainly from the stru ture information about the wave tion11p(A, A n)p. We take as an example the breakup - of Be halo into a proton target at 38.4, 100, and 200 l fun tionofthestru kvalen eneutron. Qualitatively,the c quasifrees atteringisapro ess[1,2℄inwhi haparti le MeV/u. 11 u Re ently the elasti s attering of Be from a proton n kno ks a nu leon out of the nu leus (A) and no further n−1 target was analysed using the Faddeev/AGS multiple [ strongintera tiono ursbetweenthe(A )nu leusand s attering approa h [7℄. Its was shown that even at in- the two outgoing parti les. 1 termediate energiesof 200MeV/u oneneeds to takeinto v Assumingn−th1atthenu leus(A) anbewelldes ribedby a ount 3rd order multiple s attering ontributions that 9 aninert(A ) oreandavalen eneutron,thisrea tion arenotin ludedintheGlaubers atteringframework[8℄. 8 anbestudied usingthefew-bodyFaddeevorequivalent In this work we brie(cid:29)y review the Faddeev/AGS mul- 4 Alt, Grassberger, and Sandhas (AGS) s attering frame- 4 tiple s attering framework in se tion II. In se tion III work[3,4,5℄whi h anbeviewedasamultiples attering . wesumarizetheDWIAformalismandidentify theterms 1 seriesintermsofthetransitionamplitudesforea hinter- of the orresponding multiple s attering expansion. In 0 a ting pair. Both equations are onsistent with the or- se tion IV we show the al ulated (cid:28)vefold breakup ross 8 responding S hrödinger equation and therefore provide 0 se tion using the Faddeev/AGS equations and ompare anexa tdes riptionofthequantumthree-bodyproblem : with the results of the DWIA type. v but are more suitable for the numeri al solution. The i Faddeev equations areformulated for the omponents of X the wave fun tion, while the AGS equations are integral II. THE FADDEEV/AGS EQUATIONS r a equations for the transition operatorsthat lead more di- re tly to the s attering observables. In this se tion, for ompleteness we brie(cid:29)y des ribe Traditionallythisrea tionhasbeenanalysedusingthe the Faddeev/AGS multiple s attering framework whi h DistortedWaveImpulseApproximation(DWIA)s atter- treats all open hannels in the same footing. Let us ingframework[6℄. Inthisapproa honeassumesthatthe onsider 3 parti les (1,2,3) intera ting by means of two- in oming parti le ollides with thestru k nu leonasif it body potentials. In this se tion we use the odd man out was free, and therefore the ross se tion for the quasi notation appropriate for 3-body problems whi h means, free s attering is given by the produ t of the transition for example, that the intera tion between the pair (2,3) v amplitude betweenthe two ollidingparti lesmodulated 1 is denoted as . We assume that the system is non- by the wave fun tion of the nu leon inside the nu leus. relativisti and we write its total Hamiltonian as Furthermoretherelativemotionoftheparti lesintheen- H =H0+Xvγ , tran eand exit hannelsisdes ribed by distortedwaves. (1) γ Thisapproximationleads,inea horder,toanin omplete and trun ated multiple s attering expansion. H0 with the kineti energy operator and the intera tion v γ Inordertoobtaina urateinformationaboutthewave γ for the pair . The Hamiltonian an be rewritten as fun tion, one needs to investigate the e(cid:27)e t of higher or- H =H +Vα , α der res attering between the two oliding parti les and (2) the (An−1) nu leus in the al ulated kno kout observ- Hα α where is the Hamiltonian for hannel ables. In addition, the ontribution of the negle ted H =H +v , α 0 α terms in ea h order needs to be assessed. This is the (3) 2 Vα and represents the sum of intera tions external to α partition + Vα = Xvγ . (4) γ6=α α = 0 The partition orrVes0ponds to three free parti les in the ontinuum where is the sum of all pair inter- FIG.1: Singles atteringdiagramsforbreakupintheFaddeev a tions. The total transition amplitude whi h des ribes α β s attering framework. thes atteringfromtheinitialstate to the(cid:28)nalstate isgiveninthepostform(seeRefs.[5,7℄foraderivation) T+βα =Vβ + XT+βγG0tγ (5) waves orresponding to the relative motion of three free γ6=α parti lesthatmaybeexpressedinanyoftherelativeJa- G o−b1i variables. In the latter ase the ontribution of the 0 term is zero. with the transition amplitude The solution of the Faddeev/AGS equations an be t =v +v G t γ γ γ 0 γ (6) found by iteration leading to U0α = Xtγδ¯γα+XtγXG0δ¯γξtξδ¯ξα and the free resolvent γ γ ξ G =(E+i0−H )−1, 0 0 (7) + XtγXG0δ¯γξtξXG0δ¯ξηtηδ¯ηα E γ ξ η where is total energy of the three-parti le system. + ··· , Equivalently in the prior form we write (13) T−βα =Vα+XtγG0T−γα . (8) where the series is summed up by the Padé method [9℄. γ6=β The su essive terms of this series an be onsidered Uβα as (cid:28)rst order (single s attering), se ond order (double Onethende(cid:28)nestheoperators whi hareequivalent s attering) and so on in the transition operators. The to the transition amplitudes on the energy shell, su h breakup series up to third order is represented diagra- that mati allyinFigs. 1-3wheretheupperparti leistaken Uβα = δ¯ G−1+Tβα as parti le 1 s attering from the bound state of the pair βα α + (23). = δ¯ G−1+Tβα , βα β − (9) In our al ulations Eqs. (11 - 13) are solved exa tly afterpartialwavede ompositionanddis retizationofall δ¯ =1−δ βα βα momentumvariables. Sin e,inadditiontothenu learin- where . From Eqs. (5-8) Uβα =δ¯βαG−01+XUβγG0tγδ¯γα , (10) tinetrear at iotniobnebtweteweneeanlltthhereperpoatoirns,awnedin10 Blued,ewtehefoClloouwlotmhbe γ te hni al developments implemented in Refs. [10, 11℄ for α proton-deuteron and -deuteron elasti s attering1a1nd or breakup that were also used in Ref. [7℄ to study p- Be Uβα =δ¯βαG−01+Xδ¯βγtγG0Uγα . (11) elasti s attering. ThetreatmentoftheCoulombintera - γ tionintheframeworkofthree-bodyFaddeev/AGSequa- tionsisbasedonthemethodofs reeningoftheCoulomb intera tion plus renormalization[10, 11℄; it leadsto fully These are the well known three-body AGS equations [4℄ β =0 onverged results for the on-shell elasti , transfer and whi h for breakup ( in the (cid:28)nal state) be ome breakup observables that are independent of the hoi e U0α =G−01+XtγG0Uγα, (12) of s reening radius usedRin≥th1e0 al ulations, as long as γ it is su(cid:30) iently large ( fm for the observables onsidered in the present work). Uγα where isobtainedfromthesolutionofEq.(11)with α,β,γ =(1,2,3) U. βTαhes atteringamplitudesarethema- trix elements of al ulated between initial and (cid:28)nal III. THE DWIA/PWIA states that are eigenstates of the orresponding hannel H (H ) A(a,ab)B α β Hamiltonian with the same energy eigenvalue. Let us onsider the rea tion where an in i- a For elasti s attering and transfer those states are made dent parti le kno ks out a nu leon or a bound lus- α(β) b A upbytheboundstatewavefun tionforpair timesa ter in the target nu leus resulting in three parti les α(β) α(β) (a,b,B) relativeplanewavebetweenparti le andpair . in the (cid:28)nal state. This rea tion has been ana- For breakup the (cid:28)nal state is a produ t of two plane lyzed within the DWIA as des ribed for example in the 3 V V aB bB The potentials and are taken to be the opti al a+B b+B potentials whi hdes ribethe and s attering + + ǫ ǫ aB bB atenergies and respe tively. Thereforeonewrites T ∼hη(−)η(−)|t |φ η(+)i. AB aB bB ab Bb aA (19) If we now introdu e plane waves in the exit hannel FIG. 2: Double s attering diagrams for breakup in the Fad- T ∼hχ χ |(1+t G )(1+t G )t |φ η(+)i, deevs attering framework. AB aB bB aB 0 bB 0 ab Bb aA (20) we then obtain the trun ated multiple s attering series t +t G t +t G t +t G t G t . + + ab aB 0 ab bB 0 ab aB 0 bB 0 ab (21) The (cid:28)rst order term is the single s attering represented diagramati allyin Fig. 4. This is the only term retained + + in Plane Wave Impulse Approximation (PWIA) al u- lations. The se ond and third order terms in Eq. (21) represented in Fig. 5 are due to distortion in the exit hannel. The relative importan e of these terms to the + + s attering anbe assessed by performing Faddeev multi- ple s attering al ulations. Thedistortionintheentran e hannelismoredi(cid:30) ult to bridge with few-body al ulations. In order to esti- matethis distortionin the ase ofa lightemited parti le FIG.3: Triples atteringdiagramsforbreakupintheFaddeev one ould say for instan e, that s attering framework. η(+) ∼(1+G t )χ . aA 0 aB aA (22) workof Chantand Roos[6℄. Thisis an approximatefor- Thisleadstoadditionalmultiples attering ontributions malism and its validity in the appli ation to radioa tive t G t +t G t G t +t G t G t +O(t4). ab 0 aB aB 0 ab 0 aB bB 0 ab 0 aB beam studies needs to be assessed. In the work of ref- eren e [6℄ it is assumed that the proje tile stru ks the (23) eje ted parti le freely and that the transition amplitude is given by These terms are represented in Fig. 6. We note that the T =hη(−)|t |φ η(+)i, distortion on the entran e and exit hannels ontribute AB Bab ab Bb aA (14) to a trun ated and in omplete multiple s attering series φ (Bb) at ea h order. Bb where is the bound state wave fun tion, and η(+) η(−) We on entrate on the study of the multiple s atter- aA and Babdes ribetherelativemotionoftheparti les ing expansion. In addition, we want to investigate if the intheentran eaVndexit hannels,respe tively,negle ting single s attering term be omes the dominant ontribu- ab ηth(+e)intera tion . Therefore in the entran e hannel tion astheenergyin reases. Itwasshownin theworkof aA satis(cid:28)es Chant and Roos [6℄ that, ontrary to what is expe ted, (T +V −V −ǫ )η(+) =0, the DWIA results do not onverge to the PWIA results aA aA ab aA aA (15) as the proje tile energy in reases. The importan e of ǫ T in luding the distortionin omparisonwith the full Fad- aA aA where is the relative kineti energy and the rel- deev multiple s attering series is also estimated in the ative kineti energVy op−eraVtor.∼InVsta(n~rdar)d DWIA al u- present work. We refer as DWIA-full a multiple s at- aA ab aB aA lations one takes . For the exit tering al ulation that in ludes the multiple s attering hannel one assumes as aηg(−oo)d approximation to write terms in Eqs. (21) and (23). The al ulations based on the 3-body wave fun tion Bab as a fa torized produ t these equationsfollowexa tly the samepro edureasex- η(−) ∼η(−)η(−), plainedbeforeforthesolutionoftheFaddeev/AGSequa- Bab aB bB (16) tions(11)through(13), both in termsofinitialand (cid:28)nal states as well as treatment of Coulomb, so that one may where obtain a onsistent omparison between di(cid:27)erent terms (TaB +VaB −ǫaB)ηa(+B) =0, (17) in the multiple s attering representation of DWIA and Faddeev/AGS amplitudes. and Wenotethatfurtherapproximationsareusuallymade (T +V −ǫ )η(+) =0. in standard appli ations of the DWIA when evaluating bB bB bB bB (18) the transition amplitude, namely: 4 1.80×104 2) 38.4 MeV θn=15o SDSS V sr θ10Be=0o MS e FfrIaGm.ew4:orSki.ngle s attering diagram in the DWIA s attering mb/M 1.20×104 θn=30o S ( θn=45o d 2 Ω d 6.00×103 1 + Ω d σ/ 5 d 0.00 0 30 60 90 120 S (MeV) FIG.5: Diagramsduetothedistortionintheexit hannelin theDWIAs attering framework. F11IG. 7: (Color online) Cross se tion for the breakup Be(p,pn) at 38.4 MeV. • The potential approximationin the entran e han- VaA−Vab ∼VaB(~raA) 8.00×103 nel . • 2) 100 MeV θn=15o SDSS The fa torizationapproximation, whi h is only ex- sr MS a t in PWIA. eV 6.00×103 θ10Be=0o M • On-shell approximation of the transition ampli- mb/ θ=30o tude. dS (24.00×103 θn=45o n Ω We referto this asDWIA-standard. An attempt to esti- d 1 mate the e(cid:27)e t of su h approximations was done for ex- Ωd 2.00×103 ample in the work of Ref. [12℄ for (p,2p) rea tions. The σ/ 5 Faddeev al ulationsandtheDWIA-fulldonotmakeuse d ofsu happroximationsinthesummationofthemultiple 0.00 50 100 150 200 250 300 s attering expansion, and therefore we are not address- S (MeV) ingherethevalidityofsu happroximationswhi harere- sponsibleforadditionalshort omingsofDWIA-standard vis-a-vis the exa t al ulations. F11IG. 8: (Color online) Cross se tion for the breakup Be(p,pn) at 100 MeV. IV. RESULTS asde1s0 ribed in [7℄. Forthe potential betweenthe proton We now pro eed to solve Faddeev/AGS equations (11 and Be ore we use a phemenologi al opti al model -1113) and to al ulate the observables for the proton- with parameters taken from the Watson global opti al Be breakup. As mentioned above the treatment of the potential parametrization[7, 15℄. The energy dependent Coulomb intera tion, as well as the al ulational te h- parameters of the opti al potent1ia1l are taken at the pro- nique, is taken over from Refs. [10, 11℄. 10 ton laboratory energy of the p- Be rea tion in inverse 1F0irstwedes ribethepairintera tionsp-n, p- Beand kinemati s. n- Be. Forp-nwetaketherealisti nu leon-nu leonCD Inthe solutionof the Faddeevequationswe in lude n- Bonnpotential[1130℄. Theintera tionbetweenthevalen e pℓ pa≤rti8al w1a0ves up to rℓel≤ati3ve total a10ngular momLe≤ntu1m0 np neutronand the Be oreisassumedto beL-dependent , n- Be up to , and p- Be up to . Three-body total angular momentum is in luded up to 100. As a (cid:28)rst result we show indF5iσg/s.dΩ7(cid:21)9dΩthdeS(cid:28)vefold dif- 1 2 ferentialbreakup rossse tion versusthe + + S E /A = lab ar length at 38.4 MeV, 100 MeV, and 200 MeV respe tively. The kinemati al on(cid:28)gurations are (θ ,φ ) i i hara terized by the polar and azimuthal angles of the two dete ted parti les labeled 110and 2 in Fig. 10. We assume those parti les to be the Be ore (1) and FIG.6: Diagramsduetothedistortioninthein oming han- the neutron (2). We take thr(eθe ,qφua)si=fre(e0◦s, 0a◦t)tering nel in the DWIAs attering framework. 1 1 on(cid:28)gurations: ( onf 1) with and 5 2MeV sr) 65..0000××110033 20θ100B eM=0eoV θn=15o SDMSSS 2eV sr) 1.80×104 38θ1.04Be =M0oeV θn=15o SSSS ((nnpp )+ pcore) ΩΩddS (mb/12432...000000×××111000333 θn=45o θn=30o ΩddS (mb/M1216..2000××110043 θn=45oθn=30o d Ω 5σ/ 1.00×103 σ/d d 5 d 0.00 200 250 300 350 400 450 500 0.00 0 30 60 90 120 150 S (MeV) S (MeV) F11IG. 9: (Color online) Cross se tion for the breakup F11IG. 11: (Color online) Cross se tion for the breakup Be(p,pn) at 200 MeV. Be(p,pn) at 38.4 MeV. y 8.00×103 θ=15o 2sr) 100 MeV n SSSS ((nnpp )+ pcore) p ϕ12 pp eV 6.00×103 θ10Be=0o 1 M ϕ1 ϕ2 mb/ θn=30o θ1 p2 x S ( 4.00×103 θn=45o θ2 Ωd2 d 1 Ωd 2.00×103 z σ/ p 5d 3 0.00 50 100 150 200 250 300 S (MeV) FIG. 10: (Color online) Kinemati angles for breakup. F11IG. 12: (Color online) Cross se tion for the breakup (θ ,φ ) = (45◦,180◦) (θ ,φ ) = (0◦,0◦) 2 2 1 1 Be(p,pn) at 100 MeV. (θ ,φ ) = (30◦,;1(8 0o◦n)f 2) with (θ ,φ ) = 2 2 1 1 a(0n◦d,0◦) (θ ,φ ) = (15;◦,(1 8o0n◦f)3) with S 2 2 and . The ar length E i is related to the energies of the two deteS ted par- produ es well the full al ulation at 200 MeV/u energy. S = dS ti les in the standard way [14℄ as R0 with ThereforetheFaddeev/AGSmultiples atteringseriesfor dS = (dE2 +dE2)1/2 1 2 . In ea h on(cid:28)guration, the solid breakup has onverged at se ond order level at this en- ergy. lines represent the full multiple s attering (MS) results obtained from the solution of Faddeev/AGS equations. Nowwe ompareFaddeev/AGSandDWIA-fullresults The single s atteringapproximation(SS) represented by for the breakup observables. We begin by analyzing thedashedlinesis al ulatedtakingintoa ountthepro- the single s attering term in Figs. 11(cid:21)13. The breakup ton s attering from the stru k nu leon and the s atter- rossse tion al ulatedtakingintoa ountbothproton- ing from the valen e ore as represented by diagrams in neutron and proton- ore ontributions is represented by Fig. 1. As follows from Fig. 7 the SS aproximationover- the solid line. The ir les orrespond to the PWIA, i.e., estimates the exa t result in any on(cid:28)guration at 38.4 without the proton- ores attering ontribution. At 38.4 MeV/u. At 200 MeV/u, the SS already provides a good MeV/u this ontribution is very small in the on(cid:28)gura- representation of the full al ulation in the ase of onf tions1and2,butissizablewhentheneutronisemmited 1, but in reasingly overestimates MS in onf 2 or onf 3 in the forwardregion. Therefore, even at thesingle s at- where the neutron is emitted at smaller angles. Thus, teringlevel, al ulationsbe omeinna urateinsome on- ontrary to what is predi ted by Chant and Roos [6℄, if (cid:28)gurations at low energies. At high energies the proton- the neutron is not dete ted at very forward angles, the ores attering ontribution isnegligiblein all on(cid:28)gura- single s attering approa hes the full al ulation as the tions. beam energy is in reased. The double s attering (DS) Sin e, as shown in Figs. 7(cid:21)9, the DS results are very approximation,representedbythe ir les,and al ulated losetothefullMSresults,we omparenextinFigs.14(cid:21) taking into a ount the diagrams of Figs. 1 and 2, re- 16 the predi tions of Faddeev/AGS and the DWIA-full 6 6.00×103 8.00×103 θ=15o θ=30o 2b/MeV sr) 54..0000××110033 2θ0100Be =M0oeV θ=30no SSSS ((nnpp )+ pcore) 2mb/MeV sr) 6.00×103 3θ810.B4e= M0oeV θn=45o n θn=SDD1SWW5+oIIDAAS --F ddaiidssd eeexxeiivtt+in dS (m 3.00×103 θ=45o n ΩdS (24.00×103 Ω2 n d Ωdd12.00×103 σΩ/d12.00×103 σ/ 1.00×103 5d 5 d 0.00 0 30 60 90 120 0.00 S (MeV) 200 250 300 350 400 450 500 S (MeV) F11IG. 14: (Color online) Cross se tion for the breakup F11IG. 13: (Color online) Cross se tion for the breakup Be(p,pn) at 38.4 MeV. Be(p,pn) at 200 MeV. 6.00×103 θ=15o 2sr) 5.00×103 100 MeV n SDDSWW+IIDAAS --F ddaiidssd eeexxeiivtt+income approa h in luding in both al ulations the orrespond- mb/MeV 4.00×103 θ10Be=0o θn=30o ing multiple s attering terms up to se ond order. As S ( 3.00×103 θn=45o d explainedbeforetheDWIA-fullapproa hprovidesanin- Ω2 omplete series in all orders, thought in (cid:28)rst order the Ωd12.00×103 e(cid:27)e ts of the missing term are negleagible at high ener- d gies as shown in Figs. 11(cid:21)13. 5σd/ 1.00×103 0.00 The Faddeev/AGS (solid line) results in lude all the 50 100 150 200 250 300 S (MeV) single and double s attering terms shown in Figs 1(cid:21)2. TheDWIA-fullassumesthatthe ontributionofthes at- tering between the proton and the neutron is dominant F11IG. 15: (Color online) Cross se tion for the breakup and therefore the single s attering ont1r0ibution due to Be(p,pn) at 100 MeV. the s attering of the proton from the Be ore is ne- gle ted. The ir les in lude the (cid:28)rst and se ond order diagramsoriginatedfromthedistortionsintheexit han- V. CONCLUSIONS nel representedin Fig. 5, and the dashed urvein ludes, 11 in addition, the se ond order diagrams originated from We have al ulated the breakup ross se tion for Be the distortion in the in oming hannel represented in breakupfromprotonsat38.4,100and200MeV/u. Quasi Fig. 6. The DWIA-full results to se ond order (dashed frees attering onditionsare onsideredin di(cid:27)erent on- line) provideapoorapproximationto the Faddeev/AGS (cid:28)gurations for the emitted neutron. The Faddeev/AGS at38.4MeVinany on(cid:28)guration. Astheenergyin reases DWIA-full approa hes the exa t result. However one 5.00×103 should keep in mind that DWIA-standard al ulations θ=15o assume further approximations as dis ussed in Se t. III. 2eV sr) 4.00×103 2θ0100B eM=0eoV n SDDSWW+IIDAAS --F ddaiidssd eeexxeiivtt+income M In Figs. 9 and 11(cid:21)13 we show that in the high en- ergyregimePWIAapproa hestheexa tresultatleastin mb/ 3.00×103 θn=30o esonmeregysurietgaibmlee,birneaokrduepr toong(cid:28)egtuararteiloinabs.leTphreerdeif otrioe,ninoftthhies ΩdS (22.00×103 θn=45o d breakupobservablesinquasifrees attering,weadvo ate Ω1 that a better approa h is to work with the PWIA s at- σ/d 1.00×103 tering framework within suitably hosen on(cid:28)gurations 5d rather than in DWIA where un ontrolable approxima- 0.00 250 300 350 400 450 500 tionsaremade. Onthe ontrary,in PWIA thefa toriza- S (MeV) tion is exa t and the transition amplitude for the s at- tering between the in ident and kno ked parti le is on theenergyshellintheweakbindinglimitofthekno ked F11IG. 16: (Color online) Cross se tion for the breakup Be(p,pn) at 200 MeV. parti le. 7 formalismisusedtoshowthatattheseenergiesthemul- ontrolableapproximationssu hasthefa torizationand tiple s attering expansion has onverged at the se ond on-shellapproximations,oneadvo atesthatatthesehigh order. Wehavealsoshownthatathighin identenergies energies it is adviseable to use the exa t approa h or in- the single s attering (PWIA) approa hes the full al u- stead the PWIA in suitable on(cid:28)gurations. lation in some suitable on(cid:28)gurations, namely when the A knowledgements: The work of R. Crespo is neutron is not emitted at forward angles. 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