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Knockout Reactions: Analysis, Results and Applications

Arun K. Jain1∗

1Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai - 400 085, INDIA

Recent development of the incorporation of finite range nature of the interaction be-tween the incident particle and the struck particle has led to many startling revelationsin the knockout reaction analyses. It removed the huge inconsistencies obtained earlierin the conventional zero-range analyses. In the three body final state reaction of clusterknockout by incident nuclear particle where the residual nucleus not only bears testi-mony to the dynamics of the knocked out cluster before the event but also testifies thehappenings at the time of the hard collision event. In the α -cluster knockout from targetnucleus the gross interactions of the two α’s with the spectator residual nucleus are largeenough to differentiate between the strong peripheral bruising or the deep interpenetra-tion of the two α’s. Similar results were witnessed from the heavy cluster knockout in16O(12C, 212C)4He reaction performed at the Mumbai LINAC. Here the two 12C’s werefound to have a strong repulsion, where this range is fairly large ∼ 3.7 fm. The presentresults of the large finite range influence in the knockout reactions and their ability todetect change over from repulsive to attractive interaction enhances the possibility ofobserving the multiquark objects such as Dibaryon and Pentaquark, through nucleonknockout using protons and K+ mesons. Our approach for the knockout reactions pavesthe way to include finite range effects in atomic and molecular knockout physics as also inneutron multiplication calculations. The application of FR-DWIA on the heavy clusterknockout reaction opens up new avenues to use the heavy core knockout for the detailedinvestigation of heavy as well as Borromean halo nuclei.

Knockout reactions are direct reactionswhere the reaction time is small enough suchthat there is hardly any time for rearrange-ment, except for the rearrangement causedby the incident projectile. In knockout reac-tions simple kinematic energy momentum con-servation considerations, along with the as-sumption that the struck portion of the tar-get behaves as if it were free, lead to the ex-traction of the momentum distribution of theknocked out cluster from the target before itwas knocked out. For almost half a century itwas assumed in nuclear reaction physics thatthe short range nature of the nuclear forceswill justify the use of zero range (ZR) ap-proximation for the interaction responsible forknockout. To some extent it was justifiablein (p, 2p), reactions. This ZR-DWIA ap-proximation has the great advantage that theknockout interaction gets separated from the

∗Electronic address: [email protected]

whole matrix element in the form of an off-shell elastic scattering matrix element of theknockout partners. What remains now is thedistorted form factor representing the Fouriertransform of the bound wave function. TheZR-DWIA being the main reaction model, thesame had been applied for the analysis of clus-ter knockout reactions such as (p, pα), (p, pd),(α, 2α), (α, αd), (α, α3He), (α, α3H) etc. onlight and medium mass nuclei.

The absolute clustering spectroscopic fac-tors extracted from the cluster knockout re-actions with protons as projectiles have beenfound to be in reasonable agreement withthe nuclear structure calculations [1–5]. Inthe case of cluster knockout reactions us-ing α-particle as projectiles however, therearise orders of magnitude anomalies in theextracted cluster spectroscopic factors [6–11].So far it had been a puzzle that the ZR-DWIA formalism which seems to work nicelyfor the (p, 2p), (p, pα), (p, p3He) and otherproton induced knockout reactions fails mis-erably in predicting very low cross sections

Proceedings of the DAE Symp.on Nucl. Phys. 55 (2010) I7

Available online at www.sympnp.org/proceedings

for the (α, 2α),(α, αd), (α, α3He) and otherα- induced knockout reactions [10–12]. Ex-ceptions to these observations, however, wereseen for the (α, 2α) reactions on 9Be [13] and12C at ∼ 200 MeV, [14]. The small predictionsof absolute cross sections and hence large α -cluster spectroscopic factors from (α, 2α) reac-tions upto 140 MeV were ascribed to inducedα - clustering [6], or in terms of reduced opticaldistortion effects [15]. These ad hoc prescrip-tions can not, however, account for the goodfits of the ∼ 200 MeV data [13, 14].

1. Entrance Channel Potential

There existed some uncertainties aboutthe entrance channel distorting potentials forknockout reactions. The entrance channel po-tential for the knockout reaction is strictly thepotential for the scattering of the incident par-ticle from the residual nucleus which is to beaveraged over the volume of the target nu-cleus [1]. Such a potential is not obtainablefrom any realistic experiment, therefore mostof the DWIA calculations use potentials whichreproduce the scattering data on the targetnucleus, A but with a scaling down factor [1]equal to the ratio of the mass numbers ofthe residual nucleus and the target nucleus,B

AVaA. This procedure had been adopted in

almost all the knockout DWIA calculations.Recently a single folding procedure for the

entrance channel optical potentials has beenworkedout [16]. For the A(a, ab)B knockoutreaction, the single folding model (SFM) effec-tive interaction of the incident particle, a withthe target nucleus, A is evaluated in termsof its interactions with the residual nucleus,B and the struck particle, b. These interac-tions are folded over the density distributionof the target, which can be approximated bythe square of the ground state inter-cluster ra-dial wave function.

VaA =

∫[tab(~rab−

B

A~R)+taA(~raA+

b

A~R)]ρ(r)d~R

As the projectile-ejectile, a-b interaction iscompletely accounted to all order’s by the cor-responding knockout t-matrix, is to be ne-glected while evaluating the entrance channel

FIG. 1: Various entrance channel optical poten-tial, (––––) Real folded, (· · · · ·) Imaginary Folded,

(– – –) RealB

AVaA prescription and (– · – · – ·)

ImaginaryB

AVaA Prescription. The 140 MeV re-

sults used (– · · – · · –) Real VaA and ( ) Imagi-nary VaA optical potentials. Fig (a) is for 77 MeV7Li(α, 2α)3H , (b) is for 140 MeV 12C(α, 2α)8Be,(c) is for 140 MeV 16O(α, 2α)12C and (d) is for200 MeV 12C(α, 2α)8Be reactions [16].

FIG. 2: The DWIA calculations using various en-trance channel potentials, (– – –) Warner et el ,(– · – · – ·) 0.7 times Warner et el , (· · · · ·) foldedVaA and (––––) 0.8 times folded VaA comparedwith 77 MeV 7Li(α, 2α)3H [16].



distorting potentials VEnt(r) for the knockoutreaction. On the other hand the effective in-teraction between a and B, taB(~raB) may beapproximated by the a-B optical potential,VaB(~raB). Therefore.

VEnt(~raA) =

∫VaB(~raA +

b

A~R)ρ(R)d~R

Using this procedure, the entrance chan-nel optical potentials have been evaluated [16]for various reactions such as 7Li(α, 2α)3H ,12C(α, 2α)8Be, 16O(α, 2α)12C and manyother α cluster knockout reactions at variousenergies (See Fig.1). In this figure all thefolded entrance channel potentials are seen tobe much different from the ones obtained from

either theB

Aprescription or the VaA approx-

imation. Both the real and imaginary foldedpotentials are seen to be deeper than the cor-

respondingB

AVaA potentials. The potentials

employed for the analyses of the 140 MeV databy Wang et al [6], however, are deeper thanthe folded potentials.

For the 77 MeV 7Li(α, 2α)3H reaction re-

sults for folding andB

Aprescription are com-

pared with the data in (Fig.2). It is seen inthis figure that while the shapes agree reason-ably well with each other the folding modelresults are smaller by almost a factor two inabsolute magnitude. In both the calculationsthe dip at the zero recoil momentum positionis seen to be still filled up in comparison to theexperimental results. It is also seen that whenthe entrance channel potential depths, both in

theB

Aas well as in the folding criterion, are

reduced by about 25 % the peak to valley ratioincreases sharply in better agreement with thedata. In the ℓ=0 knockout also the shape ofthe single peaked spectra do not change muchbut using the folding criterion the cross sec-tion is reduced by half in comparison to theB

Acriterion.

For the entrance channel optical potentialsit can therefore can be concluded that inthe conventional DWIA prediction of the ℓ=1

knockout spectra the discrepancy in the peakto dip cross section ratio arises mainly due tothe uncertainties in the choice of the entrancechannel potentials. The use of folded poten-tials are seen to change the DWIA predictionsfor absolute cross sections. Above all, the useof the cluster folding potentials appear to bemore consistent and aesthetically more satis-fying.

2. Finite Range t-Matrix

One of the basic approximations which hasbeen explicitly used in the conventional DWIAanalyses of the knockout reactions is thefactorization approximation (FA). Empiricaltests of this factorization approximation havebeen obtained in (α, 2α) reactions in variouskinematic conditions. Theoretically, however,it has been argued [17, 18] that the factoriza-tion may arise not only due to the short rangenature of the t-matrix effective interaction butmay also arise due to the constancy of theoptical distortion factors over the significantrange of the t-matrix effective interaction. Forexample in the extreme case of no optical dis-tortions or in other words the plane waves thefactorization would be exact even when thet-matrix effective interaction is of sufficientlylarge range. As the ratio of the PWIA to

the DWIA cross sections, (σPW

σDW)|(α,2α) in ta-

ble(II) of ref [1], is more than 3-orders of mag-nitude, which implies that the distortion ef-fects are too large. The saving grace howeveris that the experimental results are very closeto the PWIA, indicating thereby that the dis-tortion effects are very much over estimated inthe ZR-DWIA through the factorization ap-proximation.

The important finding in the comparisonof FR-DWIA and ZR-DWIA for (p, 2p) reac-tions [19–21] is to witness large differences inshapes and magnitudes of their energy sharingdistributions. While the p-p t-matrix effec-tive interaction was available [22, 23] for do-ing the FR-DWIA calculations for (p, 2p) re-actions the α-α t-matrix effective interactionwas not available for the FR-DWIA analysis of(α, 2α) reactions in the literature. We there-fore evaluated the α − α t-matrix effective in-



teraction [24] using the basic definition,

T± = V Ψ± (1)

where T , V ,Φ, and Ψ are the t-matrix effec-tive interaction, the realistic interaction, theplane wave scattering state and the realis-tic scattering state wave function with properasymptotic boundary condition. For (α, 2α)reaction the tαα-matrix is now to be evalu-ated using the α-α potential V which fits theα-α elastic scattering data. The α-α t-matrixeffective interaction, t+αα(E,~r) takes the formas,

t+αα(E,~r) = e−ıkzC(~r)Ψ+αα(~r)

≡∑

L=0,1,2,3...

tL(E, r)PL(r) (2)

Here it is to be noted that all the multipoleL-values (=0,1,2,3..), even as well as odd, con-tribute in the expansion of the above equation.Then,

tL(E, r) = (L+1

2)∑ℓ,m

Vℓ(r)ıℓ+3m(2ℓ+1)(2m+1)

eıσℓ

∫ +1

−1

P ∗

L(t)Pℓ(t)Pm(t)dtm(kr)uℓ(kr)

kr(3)

Where m(kr) is the spherical Bessel func-tion and uℓ(kr) is the radial wave function ob-tained from the solution of the Schrodingerequation for α-α scattering.

Using the α-α elastic scattering data onecan obtain the optical potentials which aremostly L-independent and have at the most6 to 9 free parameters which are constrainedwith in permissible limits due to specific rea-soning and logic. It so turned out that thereare many sets of optical potentials which fitthe elastic scattering. These sets have contin-uous as well as discrete ambiguities. Besidesthis uncertainty there are two distinct types ofoptical potentials used for α-α scattering, andsome other scattering systems, one of thesetypes are the all-through attractive potentialsand the other types are the L-dependent po-tential having a repulsive core [25]

It had been seen that a choice of the realpart of the α-α optical potential as a sum of

FIG. 3: Effective α-α t-matrix interaction, tL(r)vs r at 119.86 MeV for many L-values, (a) usingVℓ,α-α(r) with repulsive core and a longer rangeattraction, (b) using a purely attractive Vα-α(r).[26]

two attractive Woods-Saxon forms provide ex-cellent fits to the α-α scattering data over awide range of energies. Thus these two typesof optical potentials, widely differing in theirshape, their energy and angular momentumdependence, reproduce the elastic scatteringdata equally well. It is thus to be empha-sized that so far there has been no way tochoose one type of optical potential to theother type in their suitability for other appli-cations. The t-matrix effective interactions,tL(r) for repulsive core and all-through attrac-tive α-α interactions as seen in figs. 3(a) and3(b)respectively are drastically different fromtheir respective realistic interactions. More-over, the basic character of the tαα(r)’s, fromthe repulsive core α-α optical potential andfrom the all-through attractive α-α optical po-tential is drastically different. Most notable isthe vanishing of the effective interaction fromthe region of the repulsive core region of theoptical potential.

3. FR-DWIA Calculations

Having obtained the t-matrix for the α− αscattering using the optical potentials we arenow ready to formulate the transition ampli-tude, Tfi for the knockout reaction A(α, 2α)Bin the FR-DWIA formalism from the initialstate, i to the final state, f . It can be written[2, 17, 18],

d3σL,J

dΩ1dΩ2dE1= Fkin · SLJ

α ·∑∧

|T αL∧

fi (~kf , ~ki)|2(4)

where J and L (∧) are the total and orbital



(its azimuthal component) angular momentaof the bound α-particle in the target nucleus,Fkin is a kinematic factor and SLJ

α is thecluster spectroscopic factor. The conventionaltransition matrix element for the knockout re-action, T αL∧

fi (~kf , ~ki) using the finite range α-α

t-matrix effective interaction t12(~r12) is givenby [2, 17, 18]:

T αL∧

fi (~kf , ~ki) =

∫χ

(−)∗1 (~k1B, ~r1B)χ

(−)∗2 (~k2B , ~R2B)

t12(~r12)χ(+)0 (~k1A, ~r1A)ϕL∧(~R2B)d~r12d~R2B

(5)The distorted waves χ0, χ1 and χ2 of Eq.5

are evaluated using the optical potentials forthe α1-A, α1-B and α2-B. Finally all the rel-ative coordinates are expressed in terms of

~r12(≡ ~r) and ~R2B(≡ ~R). While using the ZR-DWIA the transition matrix element, Tfi ofEq. 5 was factorized into integrals over ~r and~R separately. The same is not possible whenone uses the full finite range t12(~r12) due tothe presence of optical distortions. This isbecause in the FR-DWIA formalism the cho-sen relative coordinates ~r and ~R get coupled

through the distorted waves χ(+)0 (~k1A, ~r1A)

and χ(−)∗1 (~k1B, ~r1B).

For the evaluation of T α,L,∧fi of Eq.5 the dis-

torted waves, χ(~k,~r) were expanded in termsof partial waves and then on the mesh of thespherical polar coordinates, r, θ, φ and R, Θ,

Φ the values of χ0, χ1, χ2, ϕL∧(~R) and t12(~r)were evaluated. The final result of Tfi is ob-tained by doing a 6-dimensional integration

over the mesh of ~r and ~R coordinates. Thecomputer code was checked by performing FR-plane wave impulse approximation (PWIA)calculations using the present 6-dimensionalintegration approach as well as through the3-dimensional integrations approach (becausein the plane wave case the 6-dimensional inte-gral of Eq. 5 factorizes into two 3-dimensional

integrals, one over ~r and the other over ~R).We performed the FR-DWIA analyses of

the typical (α,2α) data on 9Be at 197 [13] and

FIG. 4: Comparison of 9Be(α, 2α) data withthe FR-DWIA calculations using α-α interactionwhich is purely attractive(A) and having a repul-sive core (R+A), (a) for 197 MeV and (b) for 140MeV [26].

140 MeV [6], on 12C at 200 [14] and 140 MeV[6] and on 16O at 140 [6] and 90 MeV [? ].For these analyses, out of the many possiblet12(~r)’s, we used the α-α t-matrix effective in-teractions obtained from the two types of α-αoptical potentials, attractive with a repulsivecore, (R+A) on the one hand and all-throughattractive, (A) on the other hand.

In the calculations we employed the im-proved prescription for the entrance channelpotentials [16] where the folding model re-places the conventional (B

A ) prescription [1].

The α2-B bound wave function, ϕL∧(~R) isgenerated as usual for the Woods-Saxon formwith Rbound=1.09 B

1

3 fm.

The results of the FR-DWIA computations,normalized to the data peak values, are pre-sented in Figs.4, 5 and 6. The curves ob-tained from the attractive, tαα(A)(~r) are muchcloser to the data. This arises because thetαα(A)(~r)’s peak close to r=0, which simulatesthe zero range behavior and hence the resultsare similar to the ZR-DWIA results [1]. Therepulsive core, (R+A) results are seen to be atmuch variance. This could arise due to the un-certainty in the choice of the repulsive core α-α potential parameters. Most important con-



TABLE I: Comparison of (α, 2 α) cross sections from FR-DWIA calculations and experimental dataon 9Be, 12C and 16O at various energies and spectroscopic factors (Sα) derived from the FR-DWIAcalculations and theory. Comparison of Bold face entries is emphasized [26].

Reaction Eα σα,2α(Peak)/Sr2MeV Sα

(MeV) (R+A) (A) Expt Ref. (R+A) (A) Theory Ref.9Be(α, 2α)5He 197 575 µb 26.4 µb 6.3 µb [13] 0.011 0.24 0.57 [4]

140 609 µb 19.1 µb 100 µb [6] 0.164 5.2312C(α, 2α)8Be 200 19.9 µb 552 nb 380 nb [14] 0.02 0.7 0.55, 0.29 [4, 5]

140 92 µb 2.5 µb 18.5 µb [6] 0.2 7.416O(α, 2α)12C 140 19.1 µb 0.51 µb 10.5 µb [6] 0.55 20.6 0.23, 0.3 [4, 5]

90 171 µb 14.3 µb 68 µb [? ] 0.4 4.75

FIG. 5: Same as Fig.4 but for12C(α, 2α) reaction,(a) for 200 MeV and (b) for 140 MeV [26].

clusion however, can be drawn by comparison(bold face entries) of the absolute peak crosssection values from the FR-DWIA calculationswith the data and the derived Sα-values withthe structure theory estimates [4, 5] in Table.I.

In Table. I it is seen that the absolutecross sections and Sα values for the ∼ 197-200MeV (α, 2α) reactions on 9Be and 12C usingthe purely attractive tαα(A)(~r) are in betteragreement with data in comparison to thatusing tαα(R+A)(~r) where the absolute crosssections are 20 to 35 times larger. For en-ergies at and below ∼ 140 MeV, both thetαα(A)(~r) and tαα(R+A)(~r) yield somewhat dis-torted shapes. Yet the peaks close to thezero recoil momentum position (normalized to

FIG. 6: Same as Fig.4 but for 16O(α, 2α) reaction,(a) for 140 MeV and (b) for 90 MeV [26].

the data peak values) yield Sα-values, seenin Table.1, much closer to the theoretical val-ues when tαα(R+A)(~r)’s are employed. On theother hand, the Sα-values obtained from thetαα(A)(~r)’s are 10 to 90 times too large as com-pared to the theoretical estimates [4, 5].

Differences of almost two orders of magni-tude are seen between the FR-DWIA predic-tions of the (α, 2α) reaction cross sectionsusing the repulsive core, (R+A) and purelyattractive, (A) α-α potentials. An obviousconclusion is that use of the conventional ZR-DWIA formalism and hence the factorizationapproximation for the analysis of (α, 2α) re-actions below ∼197 MeV was incorrect.

While, due to factorization, the FR-PWIA(α, 2 α) results for the (R+A) and (A) α-α



potentials match, the enhancement of the FR-DWIA results for the (R+A) case over that ofthe (A) case can be understood qualitatively.It is to be visualized that due to the α-α repul-sion the incident α-particle can knockout thebound α-cluster while remaining outside theabsorbing region. On the other hand whenthe α-α interaction is purely attractive thetαα(A)(~r) peaks at r=0 and hence the α1 has toenter the absorbing region to knock the boundα2 out. Thus the enhancement in the (R+A)case arises due to the reduction in the opticalabsorption as a result of the α-α repulsion.

From these FR-DWIA results it is obvi-ous that the α-α potential character changesdrastically at α-energies, Eα somewhere be-tween 140 and 200 MeV, corresponding toEα-α of 70 to 100 MeV. It can be qualita-tively understood in the Resonating GroupMethod, (RGM)-shell model picture (takingcare of Pauli’s exclusion principle). Here thefour neutrons(n) and four protons(p) of thetwo α-particles can exist in an overlapping po-sition if the two n’s and two p’s of one α-particle are in the lowest 1s1/2 shell modelstate and the other two n’s and two p’s ofthe other α in the next shell model state(1p3/2, which is situated around 21 MeV abovethe ground state of α-particle). The totalenergy of this overlapping system, Eα-α willthus be ∼4×21=84 MeV (corresponding toEα ∼2×84=168 MeV). Thus below Eα ∼168MeV, the two α’s would find it energeticallymore favorable to avoid their overlap with arepulsive core in their interaction. Above thisenergy, however the two α’s have no such re-striction and are free to have the usual attrac-tive force between them. This understandingof the change in the nature of the α-α inter-action is clearly validated by the present FR-DWIA analyses of the (α, 2α) data.

FIG. 7: Shell model (RGM) scheme of two α-particles when separated or overlapping at rela-tive energy of Eα-α [26].

Similar arguments with repulsive core,(R+A) interaction between α-d, α-t and α-3He are expected to remove the inconsisten-cies [10] in the (α, αd), (α, αt) and (α, α 3He)reactions in comparison to the correspondingknockout reactions using the proton projec-tiles.

4. Heavy Cluster Knockout

As the large anomalies of the (α, 2α) reac-tions may be understood in terms of the fi-nite size of the α-α knockout vertex. Basedon this finding, a heavy cluster knockout re-action, 16O(12C, 212C)4He has been concep-tualized and executed for the first time so asto reveal the true short distance behaviour ofthe 12C-12C knockout vertex.

This reaction is expected to differentiate be-tween the highly attractive (as obtained in thedouble folding model, (DFM) [27, 28] and L-dependent repulsive core (as obtained in theresonating group method, (RGM)) 12C-12Cinteraction [29, 30]. It was found impossi-ble to discriminate between these two kindsof 12C-12C interactions from the analysis ofthe 12C-12C elastic scattering data alone, seeFig. 8.

The beauty of the 16O(12C, 212C)4He re-action, as seen in Fig. 9, is that it differsfrom the 16O(α, 2α)12C [1, 6] reaction essen-tially in the knockout vertex (12C-12C vs α-α). The inter-cluster bound wave function dueto the α-12C bound state component of 16O)[1, 2, 14] as well as the optical distortions (α-12C optical potentials (except for some energydependence [31, 32] are the same. Due tothe larger size of the 12C-nucleus compared tothat of the α-particle (∼1.5 times) the finiterange effects in the (12C,212C) reaction cross

0.001

0.01

0.1

1

10

100

1000

10 20 30 40 50 60 70 80 90

dσ/d

Ω (

mb/

sr)

θcm

12C-12C Elastic Scattering at EC=106.8 MeVUsing Wieland’s Attractive (A) Pot.

Using repulsive core (R+A) Pot.

FIG. 8: 12C-12C elastic scattering using all-through attractive (A) optical potential Weilandet al and L-dependent (R+A) optical potential.



FIG. 9: First order knockout diagram for a)16O(α, 2α)12C, b) 16O(12C, 212C)4He reaction.

sections are expected to be more enhancedthan that in the (α, 2α) reaction [1, 6, 15].

The experiment was performed at Pelletron-LINAC facility at Mumbai using ∼2.5 pnA118.8 MeV 12C beam. A target with WO3 ofOxygen content equivalent of ∼ 36.2 µg/cm2

was used. The knockout of 12C from W canbe easily separated because of its large posi-tive Q-value.

The summed energy spectrum is seen in Fig.10 where one can easily identify the two promi-nent broad peaks marked by arrows separatedby ∼ 4 MeV. The higher (E1+E2) peak corre-sponds to the two correlated 12C’s emitted intheir ground state, and the second peak corre-sponds to the decayed 4.44 MeV excited stateof one 12C in coincidence with the other 12Cin its ground state. The energy sharing spec-

trum ( d3σdΩ1dΩ2dE1

vs E1 ) is shown in Fig. 11.

The broad peak in this spectrum (akin to theℓ=0 knockout) occurred at E1 ≃57 MeV withrecoil momentum, p3 ≃41 MeV/c. The un-certainty in the absolute cross section is esti-mated to be around 30 %

A conventional ZR-DWIA estimate of thepresent reaction was performed. Comparedto the experimental peak cross section valueof ∼ 125 ± 50 µb/ Sr2MeV the two afore-mentioned ZR-DWIA prescriptions (B/A andSFM) provide the values of 3.23 and 0.56 µb /Sr2 MeV respectively (Spectroscopic factors36.7 and 222 respectively). The plane waveimpulse approximation (PWIA) provided thisvalue to be 290 µb /Sr2 MeV, with a spectro-scopic factor, SF ∼ 0.43 much closer to thevalue of 0.23 , from structure theoretical esti-mates [4, 5].

For the present case of 16O(12C, 212C)4Hereaction only the attractive optical potentials(see Fig. 12(1)) was available to fit the 12C-12C elastic scattering. Using a repulsive coreradius of 3.65 fm a 12C-12C L-dependent op-tical potential of short range repulsion and

0

2

4

6

8

10

12

14

16

18

20

80 85 90 95 100 105 110 115 120

Coi

ncid

ent C

ount

s

E1+E2 (MeV)

12C4.4 MeV

12Cg.s.

FIG. 10: Summed energy spectrum for the 118.8MeV 16O(12C, 212C)4He reaction, the peaks cor-responding to both 12C′s in their ground statesand one 12C in ground state and another in firstexcited state (4.44 MeV) are identified.

0

50

100

150

200

45 50 55 60 65

182.4 125.4 66.0 3.2 -72.8

d3 σ/dΩ

1 Ω

2 d

E1

(µb/

sr2 M

eV)

(a) ZR-DWIA

P3 (MeV/c)

DataFolding*222

B/A*38.7PWIA*0.43

0

50

100

150

200

45 50 55 60 65

d3 σ/dΩ

1 Ω

2 d

E1

(µb/

sr2 M

eV)

E1 (MeV)

(b) FR-DWIA

DataRep. Core (R+A) 3.65 fm * 0.9

Att. Pot. (A) Wieland * 10

0

50

100

150

200

45 50 55 60 65

d3 σ/dΩ

1 Ω

2 d

E1

(µb/

sr2 M

eV)

E1 (MeV)

(b) FR-DWIA

DataRep. Core (R+A) 3.65 fm * 0.9

Att. Pot. (A) Wieland * 10

FIG. 11: Energy sharing spectrum for the 118.8MeV 16O(12C, 212Cg.s)

4He, compared with calcu-lations (normalized to the peak) (a)(·· ·· ··) PWIA,and ZR-DWIA results using entrance channel op-tical potential, (· · ··) B/A prescription and (—-)SFM prescription, (b) FR-DWIA results (—–) us-ing 3.65 fm repulsive core, (R+A) 12C-12C poten-tial and (·· ··) all through attractive, (A) 12C-12Coptical potential of Weiland et al. [27]t

FIG. 12: 12C-12C optical potential VC−C(r) 1)all through attractive (A), 2) with L-dependentrepulsive core (R+A).



longer range attraction (R+A) was obtainedby fitting the various phase shifts from the all-through attractive, (A) optical potential.

The FR-DWIA calculations for the energy

sharing distributions,d3σ

dΩ1dΩ2dE1vs E1 of

the 16O(12C, 212C)4He reaction are comparedin Fig. 12(b). The calculations (normal-ized to the peak) are seen to be nicely re-producing the shape of the experimental dis-tribution for the repulsive core, (R+A) 12C-12C t-matrix while use of the attractive, (A)t-matrix shifts the distribution. The ab-solute cross sections for the all-through at-tractive,(A) t-matrix and with the repulsivecore, (R+A) are 12.5 µb/Sr2 MeV and 136.7µb/Sr2MeV respectively. The corresponding

spectroscopic factors of 10 and 0.9 respectivelyare to be compared with structure theoreticalestimate of 0.23. The reasonable (R+A)-fit(within the uncertainties of the experimentalabsolute cross section as well as those associ-ated with the theoretical estimates) leads usto the conclusion that the 12C-12C interactionaround 105 MeV should have a repulsive core.This finding is quite contrary to the conclu-sion arrived at by the proponents of the doublefolding model prescription for 12C-12C inter-action around 100-140 MeV incident energy.

Our finding of repulsion in the 12C-12C in-teraction (below r ∼ 3.65 fm) to explain the16O(12C, 212C)4He reaction data is in conflictwith the findings of large attraction at thesedistances from the double folding model. Crit-icism of the double folding model arises fromthe use of unjustified nucleon-nucleon (N-N)M3Y- effective interaction as also from thefailure in accounting for the proper antisym-metrization of 12C-12C microscopic wave func-tion in the double folding model. The M3Y-effective interaction peaks at r=0 while it hasbeen clearly shown by us [24] that any effectiveinteraction between particles having a shortrange repulsion, as is the case with nucleons,the effective interaction has to vanish in theregion of strong repulsion. Even the use ofdensity dependent (N-N) effective interaction[33] has the drawback that it does not predictthe transition from repulsive core to strong at-

traction at some definite energy as seen in the(α, 2α) case.

5. Conclusions

It is a clear message from our present workthat heavy ion optical potentials in this en-ergy region have highly repulsive core and theattractive potentials obtained from the doublefolding model are incorrect and unsuitable fordescribing the absolute cross sections of theknockout reactions. Subsequently it reflectson the reliability as well as the density andenergy dependence of the effective N-N inter-action. The heavy cluster knockout reactionof the 16O(12C, 212C)4He type can thereforeprove to be a nice tool for determining thetrue nature of the short range component ofthe heavy ion interaction as also the applica-bility of the RGM formalism to find the properheavy ion potentials.

The 16O(12C, 212C)4He reaction, per-formed at 118.8 MeV. It is first of its kindwhere exclusive measurements of the heavycluster knockout are performed and will thusbe even more informative in the case ofheavy cluster knockout from heavy nucleias also from the Halo nuclei, for example9Li knockout from the 11Li beam in the9Be(11Li,9 Li9Be)2n reaction and α knockoutfrom 6He in the 4He(6He, 2α)2n reaction.

Other heavy cluster knockout reaction suchas 24Mg(12C, 212C)12C, 32S(16O, 216O)16Oetc can be used to study di-nuclear molecularstructure of heavier nuclei as also for studyingthe halo nuclei such as 11Li, 6He etc. Heavycluster knockout may also be invoked to checkthe application of the effective N-N interactionin the reaction dynamics.

Extreme sensitivity of the cluster knockoutreactions to the short range behavior of thecolliding partners opens up the possibility ofprobing this aspect of the particles involvedat the knockout vertex. In (p, 2p) reactionsfor example, one should be able to see the be-havior of the nucleon-nucleon interaction atshort distances or otherwise, if there is a pos-sibility of dibaryon formation at some energythen one should be able to decipher it fromthe FR-DWIA analyses of the (p, 2p) reactions



[20]. Similarly one can visualize observing the∆-resonance in (π, πp) reaction [34] and thepentaquark, Θ+ [35] in (K+, K+n) reactiondue to enhanced distortion effects. In heavyion knockout reactions also one can investi-gate the short range behavior of the heavyions involved at the knockout vertex which israther difficult to observe in the elastic scat-tering. The present results and conclusionsmay be very instructive in studies involving(e, 2e) reactions on atoms, knockout of atomsfrom molecules, (n, 2n) reactions for neutronmultiplication and in many other disciplinesinvolving direct knockout reactions.

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