resonances and continuum states in the breakup of halo nuclei

5
ELSEVI ER Nuclear Physics A738 (2001) 298 302 www.else\.ier.com/locate!npc R.esonances arid continuum states in the breakup of’ halo nuclei T. Myo "*, S. Aoyaina b, Ii. Kato ’, I<. Ikeda "Researcli Center for Nuclear Physics (RCNP), Osalta University, Ibaraki, 0sak;i 567-0047, Japaii "Infoririation Processing Center, Kitami Institute of Technology, Icitairii 090-8507, Japaii "Division of Physics, Graduate School of Science, Hokkaido University, Sapporo 060-08 10 ~ Japaii "RI-Be:~iii Science Laboratory-, RIKEN (Tlic Institute of Pliysical and Clieinical Researcli), Walto, Saitama 351-0198, Japan Coulomb breakup strengths of "Li are studied iii the complex scaliiig metliotl. In tlic: rrsults, we cannot find tlic dipole rcsonaiiccs and ~vt: decompose the tmnsitioii strengtlrs into tlic conti-ibiitions froin two-body "’"Li+n" and three-body "’Li-i-n+n" coiil~iiiuuiii states. The lon- ciiergy enliancemeiit in the breakup strength is produced by botli tlie tn-(1- and thrce-body continuum states, and the riilianceirient from the thrce-body coiitinuuiri states liwe a stroiig connection to the halo structurc: of "Li. 1. Introduction The "Li iiucleiis is kiion-n as a typical Borrotiiean system, which in(x:liaiiisiii is impoi,- taiit in t,lie forinat,ion of a lialo, but it is not yet fully imderstood. The large triat,t,cr radii1 of "Li siiggests a mixing of the (lslp)’ neutron coinponent in addition to tlic: (Opl12) orie. ..2iiotlier iiit cresting problem related to the lialo structure of "Li is a c1iar;rcteristic: excitation mode siich as the soft dipole resonance[1] ~ in diiclt the major conipoiicnt call he descxihed as ( lsIp)(0plp). Tlius, the behavior of the Is- and Op-orbits ol va1enc.o tieutroiis is very crncial to undrrstand tlie lialo struct,ure and tile excitd states in "Li. Measurements of tlie Coulomb healtup strength distributions of " Li have been per- formed by three groups at IVISU[2], RII<EN[3] and GS1[4]. The low energy en1iaric:emciit ol’ the strvrigth scciiis to indicate the existencc of the dipole resoiiaiice. altliough the sliapcs of dist,ributions in thee csperiinents at different incident energies do not coiiicide with eacli ot,lier. In atldit,ioii to this, the ineasiired invariant rriass spectriiiri of ’LLi+7r shows tlic low ciicrgy eii1iaricc.iricnt [4], which implies the csistence oC the lon,-lping 1 .s-orbit iii ’"Li. hi order to understand the observed enhanceinent of the "Li lirealiup strengt,li: therefore, we iniist coiisitic:r the coiitiiiuurri structure of the ’"Li+n binary component m(1 the coiitiiiuuin rcsponse of ’Li+n+n. iii addition to the three-hotiy rcsoiiaiice. 0375-94741s ~ see front matter i: 2001 Elsm ier B.V All riehts reuened. doi: 10.101 6~j.iitic~]~hysa.200~.~4.~49

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Page 1: Resonances and continuum states in the breakup of halo nuclei

ELSEVI ER Nuclear Physics A738 (2001) 298 302

www.else\.ier.com/locate!npc

R.esonances arid continuum states in the breakup of’ halo nuclei

T. Myo "*, S. Aoyaina b , Ii . Kato ’, I<. Ikeda

"Researcli Center for Nuclear Physics (RCNP), Osalta University, Ibaraki, 0sak;i 567-0047, Japaii

"Infoririation Processing Center, Kitami Institute of Technology, Icitairii 090-8507, Japaii

"Division of Physics, Graduate School of Science, Hokkaido University, Sapporo 060-08 10 ~ Japaii

"RI-Be:~iii Science Laboratory-, RIKEN (Tlic Institute of Pliysical and Clieinical Researcli), Walto, Saitama 351-0198, Japan

Coulomb breakup strengths of "Li are studied iii the complex scaliiig metliotl. In tlic: rrsults, we cannot find tlic dipole rcsonaiiccs and ~vt : decompose the tmnsitioii strengtlrs into tlic conti-ibiitions froin two-body "’"Li+n" and three-body "’Li-i-n+n" coiil~iiiuuiii states. The lon- ciiergy enliancemeiit in the breakup strength is produced by botli tlie tn-(1-

and thrce-body continuum states, and the riilianceirient from the thrce-body coiitinuuiri states liwe a stroiig connection to the halo structurc: of "Li.

1. Introduction

The "Li iiucleiis is kiion-n as a typical Borrotiiean system, which in(x:liaiiisiii is impoi,- taiit in t,lie forinat,ion of a lialo, but it is not yet fully imderstood. The large triat,t,cr radii1 of "Li siiggests a mixing of the (lslp)’ neutron coinponent in addition to tlic: (Opl12) orie. ..2iiotlier iiit cresting problem related to the lialo structure of "Li is a c1iar;rcteristic: excitation mode siich as the soft dipole resonance[1] ~ in diiclt the major conipoiicnt call

he descxihed as ( l s Ip) (0plp) . Tlius, the behavior of the Is- and Op-orbits ol va1enc.o tieutroiis is very crncial t o undrrstand tlie lialo struct,ure and tile exc i td states in "Li.

Measurements of tlie Coulomb heal tup strength distributions of " Li have been per- formed by three groups at IVISU[2], RII<EN[3] and GS1[4]. The low energy en1iaric:emciit ol’ the strvrigth scciiis to indicate the existencc of the dipole resoiiaiice. altliough the sliapcs of dist,ributions in t h e e csperiinents at different incident energies do not coiiicide with eacli ot,lier. In atldit,ioii to this , the ineasiired invariant rriass spectriiiri of ’LLi+7r shows tlic low ciicrgy eii1iaricc.iricnt [4], which implies the csistence oC the lon,-lping 1 .s-orbit iii ’"Li. hi order to understand the observed enhanceinent of the "Li lirealiup strengt,li: therefore, we iniist coiisitic:r the coiitiiiuurri structure of the ’"Li+n binary component m(1 the coiitiiiuuin rcsponse of ’Li+n+n. iii addition to the three-hotiy rcsoiiaiice.

0375-94741s ~ see front matter i: 2001 Elsm ier B.V All riehts reuened. doi: 10.101 6 ~ j . i i t i c ~ ] ~ h y s a . 2 0 0 ~ . ~ 4 . ~ 4 9

Page 2: Resonances and continuum states in the breakup of halo nuclei

T M j o et a1 /Nuclcas Phjsrcs A738 (2004) 298-302 299

For this purpose, we Iiave been developing the applicability of the coinplcx scaling rrietliod (CSM) [5], wliich is useful to find three-body resoiiarices beyond tlie two-body case[6,7]. For an unbound system, it is a big advantage of CSM to conduces to the separation riot only between resoiiances and continuum states but also between differciit kinds of continuum states starting from different thresholds[8]. We showed a successful results for the three-body Coulomb breakup reaction of 6He[8], where the 5He(3/2-)+n binary component doininates the El strength distribution. Iii this report, we apply method to investigate the Coulomb brealtup reaction of "Li[9].

2 . Coupled chaniiel model of "Li

We describe "Li with an extended ’Li+n+n three-body model[lO]. Tlie Hariiiltoiiian of this model is given in the orthogonality condition model as follows:

3 2

H("Li) H("i) + xti - T G + c K n , i + + A P F Id)PF)(d)PFI , (1) i=l i=l

where H(’Li), t, and TG are the iiiteriial Harniltoiiiaii of ’Li, the kinetic energy of each cluster and tlie center-of-mass of the three-body system, respectively. T h e ’Li-n potential, K,,; is given by a folding-type one with MHN interaction[lO]. For the potential v,,L for two valence neutrons, Minnesota potential is used. The last term in Eq. (1) is a projectioii operator to remove the Paiili forbidden (PF) states from the ’Li-n relative motion, w h e r e the value of X ~ F is taken as 106 MeV. In Ref.[10], we discussed the effect of tail behaviour of the ’Li-n potential. T h e behavior of the s-wave state near the threshold is very sensitive to tlie tail part of the potential due to the spatial extension of the wave function. T h e tail potential also plays an important role in lowering the energy of the (Is1/2)’ compoiieiit with respect to that o f the (Opl,2)2 component in "Li. Then, \ye add a Yukawa-type t,ail potential to the original folding-type one. Tlie parameters in the potential are tleterininetl to reproduce tlie 1+ resonance at 0.42 MeV and tlie s-wave properties in l"Li[lO]. Tlic wave function of "Li is given as

Here, the ’Li nucleus is expressed by a superposition of the shell model configuration C,. We introduce tlie neutron pairing correlation in ’Li by employing C1 = (Opsp) ; aiid C2 = (Opsp):(Opl,2); for p-shell neutrons (N, is 2 ) , and solve a coupled-cliaiinel ’Li+n+n three-body problem[lO]. We describe tlie wave function xi (nn) of valence neutrons using the fiiiite number of the Gaussian trial functions for two relative coordinates. Tlleii we take a discrctized approximation for the spectrurn. We use the so-called MI-IN interaction to calculate the neutron pairing correlation in ’Li, which leads to the laz12 = 15 O/o of tlic pairing excitatioii[lO]. For ’’Li, the dynarnical coupling of pairing correlation providrs tlie so-called pairing-blocking effect. For I1Li, we adjust tlie (Op)’-( IS)^ pairing coupling between valence neutrons to reproduce tlie observed binding energy of "Li (0.31hIeV).

We prepare three types of the "Li wave function; P-1, P-2 and P-3, which have diifcrcnt ( l ~ ~ , ~ ) ’ probabilities in the ground state. Each wave function gives the ( 1 ~ ~ 1 ~ ) ~ proliability (matter radius) as 21.0% (3.33 fin), 29.4% (3.58 fm) arid 38.8% (3.85 fin), respect,ivcly. In Table 1, we list the s-orbit properties for three wave functions of I0Li.

Page 3: Resonances and continuum states in the breakup of halo nuclei

300 7: My0 et al. /Nuclear Physics A738 (2004) 298-302

Table 1 Results for the tliree types of wave functions of the present model; Scattering lengths a, and energies E of the 2- virtual state of "Li,

3 . Complex scaling method

We calculate tlie unbound states of "Li applying CSM to the ’Li+n+n system, where the relative coordinates E between ’Li and the two neutrons, a i d the inoineiita corre- sponding to tlie asymptotic channel a of ’Li+n+n and "Li(*)+n are trarisforrried as

+ [ e t a , k, + k, ( 3 )

20 is a rotation angle of tlie cuts in the Rierriann sheets of complex energies, and the continuum states are obtained on the Riemann cuts rotated down by 20. Hereafter TVC call these rotated contiiiuuin states as the contiiiuiirn ones. When we take a large 0, in addition to the three-body bound states (3BB), we obtain (i) discrete three-body resonances (3BR) (ii) two-body contiiiuuni states (2BC) of loLi( I+, 2+)+n, and (iii) tliree-body continuum states (3BC) of %+n+n, which are decomposed from tlie tlirce-body scatteriiig states.

Strength function for tlie operator 0, of rank X is expressed with the Green’s function. Q i ( [ ) is the initial wave function of "Li and we also apply CSM to tlie strength function.

wlirre the complex-scaled Green’s function GD(E, 6,") is given as

Ef m d [IJ! ( c ) (Gf (<) ) are the energy eigenvalues and eigenfunctioiis (bi-orthogonal eigeii- functions[8,9]) of the complex-scaled Harniltonian U ( Q ) , respectively. Finally, we obtain tlie decornposrd stiength function S X , ~ ( E ) for each final state v such as 3BC aiid 3BR,

4. Results

111 Fig. 1, me shorn the eigeiivdue distribiitioiis of tliree dipole excitcd stat,es ( l /2+ , 3/2+, 5/2 +) at 8 = 28". All eigenvaliies along t h e e lines of 2BC of ’"Li(l+, 2+)+n aiid 3RC of gLi+n+7~,. We cannot find any resonances with a sharp witlt,h at least (r /2ET < tail-’ 28). Since the wave function of s-wave valence neutron is spatially extended, dipole resoiiimces including s-wave component may teiid to decay easily.

Page 4: Resonances and continuum states in the breakup of halo nuclei

I: My0 et al. /Nuclear Physics A738 (2004) 298-302 301

0.0 1.0 2.0 0.0 1.0 2.0 0.0 1.0 2.0

Re(Energy) [MeV]

Figiire 1. Energy eigenvalues of tlie dipole excited states with 0=2S0 in CSM. Squares, triangles and circles indicate 2BC of ’’Li(l+, 2+)+n and 3BC of ’Li+!ri+n, respectively.

In Fig. 2(a), using the solutions of the contiiiuum spectra, we show the dipole strength functions. The strengths show the low energy enhancement whose height is sensitive to the ( 1 ~ ~ 1 ~ ) ~ probability. This enhancement is interpreted as a threshold effect coining from the continuurn states and reflects the halo structure of ’’Li. We also coinpare our results to the cxperirnental data of MSU and a disagreement of the shape is seen in thc strength. In Figs. 2(b) and 2(c), we derive the breakup cross sections using tlie equivalent plioton method with a Pb t,arget. We can see a good agrecmeiit with the data of RIKEN for the P-2 wave function with the

In Fig. 3, we separate the El transition strength of "Li into two-body and three-body continuum components for the three types of tlie (151p)’ probability in the ground state. The two-body contiriuuni component of l0Li( l+)+n shows a low energy enhancementj in each panel, whose peak positioii is just above tlie two-body thresliold (0.42 MeV) of I 0 l i i ( l+)+n. Aiiotlier two-body continuum component of ’’Li(2’)+n shows a broader structurc because a larger decay width of the 2+ state than that of the If state in l0Li broadens the strength. The low energy enhancement in two-body contiiiuuiri components is interpreted as a tliresliold effect of 2BC of the 1°Li(l+,2+)+(s-wave neutron) channels.

From Fig. 3, the contribution from 3BC of’ ’Li+n+n increases with the (1sl/2)2 prob- ability. The strength distuibutiori of 3BC has a peak at the low energy region (- 0.5 IIcV) a id slowly decreases with energies. With tlie increasing of’ the ( 151/2)2 probability in the "Li ground state, the strcngth of 3BC shows a sharper peak, and i ts magnitude increases markedly in comparison to the case of 2BC. Tliis indicates that tlie structurc of 3BC tleperitls strongly on the halo structure of the ground state. In fact, a large coil-

tribution of 3UC cannot he seen in tlie brealtup reactions of 6He having a 2.4% of the ( 1s1/2)2 probal-dity in the ground state[8]. This trend indicates that the mechanisms of’ the brealtup reactions of ’He and "Li are different, and this is caused by the different ( lsl/2)p probabilities in their ground states.

probability being around 30%.

5 . Summary

111 suiriinary, x7e investigate the Coulomb breakup reaction of "Li employing tlie coin- plex scding method. From the results, we cannot find any dipole resonaiices with ii

sharp .ividth. The observed low energy enhaiicement in the dipole strength comes froin the continuum states of I1Li, where the two-body coiitinuum states and the three-body

Page 5: Resonances and continuum states in the breakup of halo nuclei

302 7: M y ) et al. /Nuclear P1ij:cic.y A738 (2004) 29X-302

7 1.6 a, 1.4

3 1.2 E 1.0

w- a, 0.6

0.6

7 0 2

Y

g 0 4

0.0 %

: ’.. p.3 ...... I ,

’-, ... MSU -

. , .. : .%..

;. ... .&.. c ’... ... . ...

-o.... i’

0 0 5 1 1 5 2 2.5 3 Energy [MeV]

18

0 0.5 1 1.5 2 2.5 3 3.5 4 Energy [MeV]

- 600 > 9 600

Y E 400 w

. D

z 200

u n

p.3 ........ GSI t)-

0 1 2 3 4 5 6 Energy [MeV]

0.8

0.6

0.4

0.2

y 0.0 a, 2 1.2

N; 1.0

0.8

2 0.6

W 0.4

E 0.2

g 0.0 7

1.6 1.4 1.2 1 .o 0.8 0.6 0.4 0.2 0.0

’~i+n+n Total -- - -I ’ 1 P-2

’~i+n+n -- loLi(, +)+n ....... ’0Li(2+)+n

>..: ....................... :~~ _.,, Iy

----___ .Ji-,-, -- [J,*’.. .. ... ..... ..... ... k ..-. - .--

0 0.5 1 1.5 2 2.5 3 3.5 4

Energy [MeV]

Figure 2. Dipole strengths and cross sec- tioiis of "Li in coiriparison to the exper- iineiital dat,a;(a)[2], (b)[3] and (c)[4].

Figure 3. Decomposition of tlie traiisi- tioii streiigth of IILi with the three types of the wave functions.

coiitiiiuiim oiies give comparable contributions. This result also means that the breakup mccliaiiism of "Li is different from that of GHe. Furthermore, it is found that the ( 1 ~ ~ p ) ~ - coinpoileiit, in the "Li grouiid state is responsible for the increases of the contribution of the three-hody coiitinuuni states niid the lorn energy enhariceirient in the streiigth.

As the origin o l the low energy eiiliaiiceiiieiit in the strength, the threshold clfect is consitleretl. Since the thee-body coiitiiiuum states include virtual states iii ’OLi, it, is iiit,crestiiig to iiivestigate tlie effect of virtual states oii the breakup reaction.

REFERENCES

1. I<. Ilteda; Nucl. Phys. A538(1992)355c. 2. I(. Ielti e i ul., Phys. Rev. Lett. 70(1993)730. 3. S. Shiiiioura et al., Pliys. Lett,. B348(1995)29. 4. 11. Ziiiser et al., Nucl. Phys. A619(1997)151. 5. ,J. Aguilar and J.M. Combcs, Coininuii. Math. Pliys. 22(1971)269.

E. Balslev aiitl J.M. Combes, Coiiiinuii. Math. Pliys. 22(1971)280. 6. S. Aoyama, I<. IGttij, ’I?. My0 m d I<. Ikeda, Prog. Tlieor. Phys. 107(2002)543. 7. E. Garrido, D. V. Fedorov aiid A. S. Jensen, Nucl. Pliys. A708(2002)277. 8. T. hlyo, S. Aoyania, I<. KatG and I<. Ilteda, Pliys. Rev. C 63(2001)054313. 9. T. I I p , S. Aoyama, I<. KatG and I<. Ikeda, Phys. Lett. B576(2003)281. 10. T. Myo, S. Aoyama, I<. I<ato and I<. Ikeda: Prog. Thcor. Phys. 108(2002)133.