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Study of the effects of nuclear and Coulomb interactions in the breakup of 19C on 208Pb

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2015 J. Phys. G: Nucl. Part. Phys. 42 015109

(http://iopscience.iop.org/0954-3899/42/1/015109)

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Study of the effects of nuclear and Coulombinteractions in the breakup of 19C on 208Pb

B Mukeru1, M L Lekala1 and A S Denikin2,3

1 Department of Physics, University of South Africa, Pretoria, South Africa2 Flerov Laboratory of Nuclear Reactions, Dubna, Russia3 Dubna International University, Dubna, Russia

E-mail: bmukeru@gmail.com, mukerb@unisa.ac.za, lekalml@unisa.ac.za anddenikin@jinr.ru

Received 4 September 2014, revised 29 October 2014Accepted for publication 7 November 2014Published 3 December 2014

AbstractA detailed partial wave analysis of the breakup of 19C on 208Pb at 67MeV/A isperformed to investigate the effects of the nuclear and Coulomb breakups. It isfirst shown that the breakup cross sections are dominated by p-waves, but allthe outgoing neutrons are not necessarily in the p-waves. The contributions ofthe other partial waves are important and account for the normalization of thebreakup cross section. The nuclear contribution is not negligible for anglesbelow °3 and in fact both nuclear and Coulomb breakups contribute equallybetween °2 and °3 . The incoherent difference of the full (coherent sum ofnuclear and Coulomb breakups) and nuclear breakup cross sections agree withthe data for low excitation energies. However, the full breakup cross sectionalone describes well the data for high excitation energies. We found that thesmall nuclear contribution does not directly imply small nuclear–Coulombinterferences, which was generally found to be destructive regardless whetherthe continuum–continuum couplings are included or not.

Keywords: breakup, interferences, couplings

(Some figures may appear in colour only in the online journal)

1. Introduction

Breakup reactions, in which the valence nucleon is removed from the projectile in itsinteraction with a target nucleus, plays a useful role in probing the structure of these nuclei[1]. Among the main focuses within this field, there are studies of the breakup cross sectionand the influence of the breakup on other reaction channels like fusion, elastic scattering, etc.Due to the complexity of the breakup process, the main cause of this breakup is not fully

Journal of Physics G: Nuclear and Particle Physics

J. Phys. G: Nucl. Part. Phys. 42 (2015) 015109 (11pp) doi:10.1088/0954-3899/42/1/015109

0954-3899/15/015109+11$33.00 © 2015 IOP Publishing Ltd Printed in the UK 1

established. In fact one important question when investigating a breakup process of weaklybound nuclei is what is the main interaction producing this breakup, the Coulomb or nuclearinteraction? Or what is the nature of their interference and how important are these? If theanswer to the first question cannot be predicted to some extend, it is rather difficult toanticipate any answer to the second.

The Coulomb breakup of 15C and 19C halo projectiles impinging on 208Pb target at 68, 35and 67MeV/u have been measured and analyzed by different groups, using differentapproaches [2–4, 6]. In [3, 4, 6], the Coulomb dissociation method, which is based on thefirst-order perturbation theory [7] was employed while in [2] the time-dependent Schrödingerequation was solved to investigate nuclear and Coulomb breakups of 19C. When using theCoulomb dissociation method to study Coulomb breakup reactions, one of the famous pro-cedure used is the scaling of the nuclear breakup cross section [3, 8, 9]. However, systematicstudies of Coulomb dissociation for loosely bound nuclei on a variety of targets, spanning arange of beam energies; have shown that the nuclear scaling is not always reliable andnuclear–Coulomb interferences can be large [8].

In [3] it was concluded that the shape of the angular distribution is not affected by thenuclear breakup effects below the grazing angle (∼ °2.7 ). Later in [2], it was shown that evenat °1.5 the nuclear effects are already important. But using a 19C binding energy of 0.53MeVas in [3], the results overestimated the data for low excitation energies (see figure 1(a) of [2]).In [4], the Coulomb breakup of 15C on 208Pb at 35MeV/u was analyzed. It was concludedthat i) the nuclear–Coulomb interferences are insignificant, (ii) the outgoing neutrons are all inthe p-waves and (iii) the breakup occurred in one step. However, a more accurate analysis ofthe effects of nuclear and Coulomb breakups requires a method capable of treating bothnuclear and Coulomb breakups at the same footing.

The continuum discretized coupled channel method (CDCC) [10–12] provides a non-perturbative approach in which to describe a breakup process, both Coulomb and nuclearbreakups are treated at the same footing. Multipole excitations are fully taken into account aswell as the final state interaction effects [13]. In fact, using this method in [5], all the threeconclusions of [4] were contradicted. The authors showed that although the p-waves aredominant (and is mainly E1 but would contain some nuclear contribution too), but all theoutgoing neutrons are not in the p-waves. They also stressed that the nuclear–Coulombinterferences are rather important. However, to have a broad picture of the importance andnature of the nuclear–Coulomb interferences, and to prove that all the outgoing neutrons arenot in the p-waves, a detailed partial wave analysis is required. On the light of these con-tractions, one may wonder as to whether this is a particularity of the 15C nucleus or it can begeneralized to other one neutron halo nuclei. An anticipated conclusion is not guaranteedgiven the different nuclear properties of these nuclei.

In this paper, CDCC calculations are performed for the 19C+208Pb breakup reaction. Thechoice of this nucleus is motivated by the availability of the experimental data, thus makingthe comparison easy. Moreover, both 15C and 19C are known to have similar ground stateconfiguration ( +2s1

2) [3, 14]. Our main objectives are (i) to perform a detailed partial waves

analysis, for a better understanding of the different partial waves contributions to the breakupcross section, (ii) to investigate the effects of both nuclear and Coulomb breakups on thebreakup cross sections, (iii) to analyze the importance and nature of nuclear–Coulombinterferences. The role of the continuum–continuum couplings (CCC) is also considered. Thepaper is organized as follows. In section 2, we recapitulate the main features of the CDCCmethod. In section 3, the results and discussions are presented, whereas section 4 summarizesour conclusions.

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2. Outline of CDCC method

In this section we highlight the main features of the CDCC method. More details can be foundin [10–12, 15, 16]. In general, we consider a projectile with a two-body structure ( = +p c v,)where the structureless valence nucleon (v) is loosely bound to the core (c), impinging on atarget t. For simplicity, no explicit target excitations are included other than the ones due to

−c t and −v t effective complex interactions. The internal Hamiltonian H0 of the projectilereads

μ= − +

Hr

V r2

d

d( ), (1)

cvcv0

2 2

2

where μcv is the reduced mass of the +c v system, and the interaction between the core andvalence nucleon is stimulated by the potential V r( )cv . In the CDCC method, the continuumwave functions of the projectile are sliced into bins (of index = …i i N, 1, 2, b, Nb being thenumber bins) of widths Δ = − −k k ki i i 1 and averaged over the relative momentum (k) toobtain [17–19]

∫φ ϕ=αα

α−

rW

r f k k( )1

( ) ( )d , (2)k

k

kℓi

i

1

where the functions ϕ r( )kℓj are continuum wave functions of the projectile, being radial parts

of eigenstates of H0 and are normalized according to

ϕ δ δ→ ∞ → +r F kr k G kr k( ) ( ) cos ( ) ( ) sin ( ), (3)kℓj

ℓ ℓj ℓ ℓj

with Fℓ and Gℓ being Coulomb functions [13, 20] and δ k( )ℓj are nuclear phase shifts. Inequation (2), α = i ℓ s j( , , , ) represents the relevant quantum numbers describing the states ofthe projectile, where the ground state corresponds to i = 0 and αW is a normalization factordefined as

∫=α α−

W f k k( ) d . (4)k

k2 2

i

i

1

The bin energies are given by

∫εμ

=αα

α−

W

k f k k2

( ) d . (5)cv

k

k2

22 2

i

i

1

The form factor αf k( ) depends on the nature of the bins. For non-S-wave bins, one can use

=αf k( ) 1 for a nonresonant continuum in which case Δ=W k( )i i1 2 and ε μ= k̂ 2i i cv

2 2, with

= + +− −k k k k kˆ ( ) 3i i i i i2 2

12

1 . For S-wave bins, it is convenient to use =αf k k( ) as thisstabilizes the extraction of the three-body transition amplitude [21]. In this case the binenergies are ε μ Δ= − − k k k k( ) (10 ˆ )i i i cv i i

2 51

5 2. For resonant bins, we follow the description

of [13].Once the bins are constructed, the three-body Schrödinger equation

Ψ− =[ ]H E r R( , ) 0, (6)R JMCDCC

can be solved by first decomposing the CDCC wave function into radial and angular parts asfollows [13]

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∑Ψ χ Ω=α

α α ( )rR

Rr R r( , )1

( ) , (7)JML

LJ LJR

CDCC

,

where J is the total angular momentum and M its z-projection. The channel functionΩα r( , )LJ

R is defined by

Ω Φ Ω= ⊗α α ⎡⎣ ⎤⎦( ) ( )Yr r, i ˆ ( ) , (8)LJR

LL R

JM

with

Φ φ Ω= ⊗α α⎡⎣ ⎤⎦( )r Y Xrˆ ( ) ( ) i . (9)ℓ

ℓm r smjm

ℓ s

The substitution of equation (7) in equation (6) results into a set of coupled-channeldifferential equations for the coefficients χα R( )LJ , reading

∑μ

ε χ

χ

− − + + + −

− =

αα α α

α ααα α

≠ ′

′−′′

′′

⎡⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥

R

L L

RV R E R

V R R

2

d

d

( 1)( ) ( )

i ( ) ( ) 0, (10)

pt

LJ LJ

L L LL J L J

2 2

2 2

where μpt is the projectile-target reduced mass and αα′′V R( )LL J the potential matrix element,

coupling the ground state to continuum as well as continuum to continuum states of theprojectile and is given by

Ω Ω= +αα α α′′

′′ ( ) ( )V R U Ur r( ) , , , (11)LL J LJ

R ct ptL J

R

with Uvt and Uct being the core-target and nucleon-target phenomenological optical potentialsincluding nuclear and Coulomb components. They include absorption from all the channelswhich are not included in the model space. Equation (10) is solved with the usual scatteringboundary conditions at large R, given by

χ δ⟶ −α α α αα α α αα α−

′+

′⎡⎣ ⎤⎦( ) ( ) ( )R H K R H K R S K( )i

2, (12)LJ

where α±H are Coulomb–Hankel functions [13] and αα α′S K( ) is the S-matrix, with

=αμ ε+ α

K

E2 ( )pt

2. The breakup observables are calculated from the resulting S-matrix

following [12, 13, 21].

3. Results and discussion

Our results are presented and discussed in this section. We start by describing the projectilestructure and the CDCC inputs required to solve the coupled-equations (10).

3.1. Projectile structure and CDCC model space

In this paper, we adopt the ⊗+ +n sC(0 ) (2 )18 12

for the 19C ground state, with a binding energyof 0.53MeV as suggested in [3]. To obtain the ground state wave function, we use thepotential parameters for the 14C+n system, obtained from [22], where we only adjust thedepth of the central form factor to fit the binding energy. These parameters are listed intable 1. The same parameters are used to calculate the continuum wave functions. Thestructure of the continuum is presented in figure 1. One observes that the continuum is not

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structureless and exhibits clear resonances in the d-waves, =π + +j ,3

2

5

2. To obtain the

breakup observables for the reaction under investigation, we solved the coupled-equations (10) using the computer code FRESCO [13]. The input parameters employed, i.e.the different potential parameters and the CDCC model space parameters are summarized intable 1. These model space parameters were selected based on the convergence requirements.

3.2. Energy distribution cross sections

In this section we investigate the different nuclear and Coulomb contributions to the breakupcross sections. However, we look first at different partial waves contributions. The results arepresented in figure 2. As expected, it is clear that the p-waves are largely dominant. We didnot plot the g-waves since they were found to be insignificant. Regarding the nuclear andCoulomb breakups, we present in figure 3(a), the Coulomb, nuclear breakup cross sections aswell as their coherent sum (full). One can notice that the Coulomb breakup cross section issystematically larger than the nuclear one as it could be expected for heavier systems.However, it is even larger than the coherent sum; rising the issue of destructive Coulomb–

Figure 1. 18C+n phase shifts and resonances structures for different partial waves.

Table 1. Core-target, neutron-target optical potential parameters and core-neutronpotential parameters and CDCC model space. The optical potential parameters aretaken from [2]. The central part of the core-neutron potential has the Woods–Saxonform, while the spin–orbit has the Thomas one.

V (MeV) RV (fm) a (fm)V W (MeV) RW (fm) a (fm)W

18C+208Pb

200.0 5.39 0.9 76.2 6.58 0.38

n+208Pb 29.48 6.93 0.75 13.18 7.47 0.58

18C +n V0 (MeV) R0 (fm) a (fm)0 V (MeV fm )SO2 RSO (fm) a (fm)SO

58.02 2.651 0.6 23.761 2.651 0.6

CDCC ℓmax εmax λmax rmax Rmax Lmax

4 9 MeV 5 60 fm 1000 fm 9000ℏ

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nuclear interferences. To compare our results with the data of [3], we subtract incoherently thenuclear cross section from the coherent sum. Our results are presented in figure 3(b). From thefigure, one observes that this incoherent difference (represented by F–N in the figure) is ingood agreement with the data for low excitation energies (up to 1MeV). However, forexcitation energies above 1MeV; the data are well fitted by the coherent sum.

3.3. Partial waves integrated breakup cross sections and interferences

To investigate quantitatively the Coulomb and nuclear breakup contributions as well asdifferent partial waves, we performed a numerical integration of the corresponding partialdifferential breakup cross sections. The interesting breakup properties are summarized intable 2. We present the results for the different partial waves, where σ ℓ

CN is the full breakupcross section (coherent sum) and σ σ+ℓ ℓ

C N stands for the incoherent sum of the nuclear andCoulomb breakup cross sections. As already mentioned, the p-waves are dominant for the fulland Coulomb breakups, whereas for the nuclear breakup, the d-waves are leading. The table

Figure 2. Different partial waves contributions to the energy distribution cross section.

Figure 3. Energy distributions cross sections of the 19C+208Pb breakup reaction. F–N isthe difference between the coherent sum Coulomb+nuclear (full) and the nuclear.

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also shows that above the d-waves, both Coulomb and nuclear breakups contribute almostequally. In fact it can be observed that for the g-waves, although insignificant; the nuclearcross section takes over the Coulomb one. The sixth row represents the sums of all the partialwaves, defined as

∑σ σ==

ˆ . (13)x

ℓ

ℓ

xℓ

0

max

It can be observed that the incoherent integrated difference of full and nuclear cross sections(i.e. σ σ−CN N) agrees quite well with the experimental data on Coulomb dissociation asmeasured in [3]. The effect and nature of the interferences can be analyzed using thefollowing relation

σ σ σ σ σ σ σ σ= + + ⇒ = − +( ). (14)CN C N I I CN C N

If σ < 0I , then we have destructive interferences, otherwise they are constructive. Using theabove relation, we obtained the seventh column. One notices that the interferences aredestructive in all the partial waves. Still there is another way to investigate the nature of theseinterferences, that is to use the ratio σ σ σ−( )ℓ ℓ ℓ

CN N C. It has been previously shown for looselybound light projectiles that, this ratio is always less than one [23]. Our results show that thisratio remains less than one for all the partial waves, thus reflecting once again the destructive

Table 2. Different partial waves contributions to the integrated breakup cross sections.The numerical integration is performed up to εmax = 8 MeV. The experimental valuefor the total Coulomb dissociation cross section is 1.190±0.119b [3]. All the crosssections are expressed in barns.

Part waves σ ℓCN σ ℓ

C σ ℓN σ σ+ℓ ℓ

C N σ σ−ℓ ℓCN N σ ℓ

I σ σ σ−( )ℓ ℓ ℓCN N C

s 0.1459 0.1696 0.0515 0.2211 0.0944 −0.0752 0.56p 1.0503 1.3668 0.0514 1.4182 0.9989 −0.3679 0.73d 0.1743 0.1586 0.0722 0.2308 0.1021 −0.0565 0.64f 0.0377 0.0304 0.0290 0.0594 0.0087 −0.0217 0.29g 0.0295 0.0123 0.0196 0.0319 0.0099 −0.0024 0.81σ̂x 1.4377 1.7377 0.2237 1.9614 1.214 −0.5237 0.62

Figure 4. Angular distributions of the coherent and incoherent sums of Coulomb andnuclear and Coulomb and nuclear breakup cross sections.

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nature of the nuclear–Coulomb interferences. The table also shows that the integrated breakupcross section due to interferences is much larger than the nuclear breakup cross section.

3.4. Angular distribution breakup cross sections

Motivated by the results of [2, 3], we also performed the angular distribution calculations.Our results are presented in figure 4(a). It can be seen that for angles between °1 and °3 , thenuclear contribution is not negligible. In fact the figure shows that between °2 and °3 , bothCoulomb and nuclear breakups contribute equally. The nuclear contribution is rather negli-gible beyond °4 . For a better understanding of the nuclear effects, we plot in figure 4(b) theangular distributions of the nuclear–Coulomb interferences. This figure shows clearly that theoscillatory behavior of the breakup cross section is a result of these interferences, which areconstructive and destructive for forward angles (below °2 ), and exclusively destructive forangles above °2 .

The nuclear geometry is also a crucial ingredient when analyzing breakup reactions,which determines the range of the nuclear forces. It is believed that for halo projectiles, thenuclear effects go even beyond the projectile-target relative distance. To analyze these effects,we first present the angular momentum distribution breakup cross sections in figure 5(a). Theresults show that the nuclear breakup builds it contribution within ⩽ ⩽ L200 600 , with amaximum around = L 300 , where it is even greater than the Coulomb contribution.Moreover, a nuclear absorption for small angular momenta is noticed, where the Coulombcontribution prevails and its extension to large angular momenta can be attributed to its longrange behavior. To get more insight into the effect of the nuclear forces, we determine thegrazing impact parameter (bgr), which is related to the grazing angle (θgr) through [24, 25]

θ= =

( )b

Z Z

EL Kb

e

2 tan 2and . (15)

t p

ptgr

2

grgr gr

Using the grazing angle θ = °2.8gr , we obtained a grazing impact parameter bgr = 12.43 fm,corresponding to a grazing angular momentum Lgr = 389ℏ. The impact parameter distributionbreakup cross sections are presented in figure 5(b). As we can see, the nuclear breakup effectsextend much beyond the grazing impact parameter as it could be expected due to the halonature of the 19C projectile.

Figure 5. Angular momentum and impact parameter distribution breakup crosssections. σ σ= Kb L.

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3.5. Role of the CCC

By CCC we mean couplings between two continuum states. The removal of these couplingsin the coupling matrix, retains only couplings to and from the ground state. The effects of theCCC have been investigated in different works for the breakup of 8B on light and mediumtargets for angular distributions [26–29]. It was concluded that these effects reduce theangular distribution breakup cross section. However they have not yet been fully establishedfor other projectiles. On the other hand, it not clear how the CCC influence the Coulomb–nuclear contribution. To this end, we investigate the role of these CCC for the reaction underconsideration. We first look at the energy distribution breakup cross sections. Our results arepresented in figure 6. From this figure one can observe that the Coulomb breakup is dominantfor low excitation energies (⩽1MeV), whereas the nuclear breakup dominates the rest of thespectrum. As in the previous discussion, we consider the contributions of the different partialwaves. The idea is to check the nature of the nuclear–Coulomb interferences in each partialwave. We present in table 3 the integrated partial breakup cross sections. The results showthat the nuclear breakup is more dominant in all partial waves. In fact apart from the p-waves,the Coulomb contribution is negligible in all the other partial waves. Moreover, looking at thecross sections due to the interferences or the ratio σ σ σ−( )CN C N, we may conclude that thenuclear–Coulomb interferences remain destructive in all the partial waves, except the p-waveswhere they are constructive. Coming to the angular distributions, figure 7(a) shows that for

Figure 6. Full, Coulomb and nuclear energy distribution breakup cross sections. Thesecalculations are made without continuum–continuum couplings.

Table 3. Integrated partial breakup partial cross sections (in barns). The continuum–

continuum couplings are excluded.

Part waves σ ℓCN σ ℓ

C σ ℓN σ σ+ℓ ℓ

C N σ ℓI σ σ σ−( )ℓ ℓ ℓ

CN C N

s 0.9320 0.0029 1.6964 1.6993 −0.7644 0.55p 3.9195 1.8055 2.0217 3.8272 0.0923 1.05d 0.7358 0.0065 0.9639 0.9639 −0.2346 0.76f 0.2286 0.0002 0.2763 0.2763 −0.0479 0.83g 0.0936 0.0000 0.1058 0.1058 −0.0122 0.89σ̂x 5.9095 1.8151 5.06411 6.8792 −0.9697 0.81

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angles less than °1 ; the nuclear contribution is negligible. However it suddenly increasesabove °1 , where it becomes dominant; crosses the Coulomb cross section at °2 and peaks at °3before dropping again. The angular distribution of the interferences is shown in figure 7(b).From this figure one observes that, the effects of the interferences are maximum where thenuclear and Coulomb breakups contribute equally (i.e. at °2 ). This figure serves to show onceagain that the oscillatory behavior of the full breakup cross section is a result of the nuclear–Coulomb interferences. Recently, the effect of these interferences on the elastic scatteringcross section was investigated for halo projectiles and it was found that this effect is sig-nificant [30]. Considering figures 4(a) and 7(a), we find that the effects of the ccc are toreduce the breakup cross sections in the vicinity of the grazing angle. The results as presentedin tables 2 and 3 demonstrate that the CCC reduces the total breakup cross section by a factor4 approximately.

4. Conclusion

We have investigated in more details the breakup of 19C impinging on 208Pb. We found thatthe breakup cross sections are dominated by the p-waves, but all the outgoing neutrons are notnecessarily in the p-waves. We showed that this dominance is independent of the CCC. Theincoherent difference of the full and nuclear breakup cross sections in the presence of theCCC, results in a fair description of the data for low excitation energy, whereas the fullbreakup cross section alone fits the data for high excitation energies, showing that the nuclearcontribution cannot be simply disregarded. The breakup cross section due to the nuclear–Coulomb interferences is much larger than the nuclear breakup cross section, thus provingtheir importance. Removing the CCC, the cross sections are more dominated by the nuclearbreakup, where the interferences in the p-waves become constructive. But the general pictureshows that the the CCC donʼt modify the nature of the interferences.

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Figure 7. Non continuum–continuum couplings angular distribution cross sections.

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