arXiv:1407.4232v1 [nucl-th] 16 Jul 2014 uha obeki n ytm like so systems system and the the it alter cases break not these to does In as much valley. single stability a stan- the of removal to more closer of sit like behavior that systems the time. nuclear from short tightly-bound very different dard two-body a quite in resulting of apart is falls any the This it if and removed, unbound because is is system special particles are three corre- systems weakly-bound the that two These core plus a nucleus bound . normally stable parts, a They three to sponds of constitution. up peculiar made their are of because nuclei light ieppr fMiadvnIakr[],te0 the [4]), Isacker van and Mei of core’s papers nice the of presence binding or field. existing stabilizing mean of the thought be to in The can alone due but exist potential. form, not NN would bound that a a idealization via an interact is that dineutron correlated neutrons, a a clearly of is for pair It approximation well-working, cluster. dineutron but the a coarse, describes plus that core a model with two-body system studied a be with can explained systems and Borromean two-neutron of erties spectrum. continuous speci- the global the of which discarding ficity attributed, to ap- are state is of resonant properties approach continuum averaged single this the a in in linked continuum with whole state proximated is The resonant This a latter. of the given. presence is the unbound, to being (A+n) system nwiha xlnto ftesaiiyo ytm made systems of stability core, the a of of explanation an which in describe to systems. [1] bound coined three-body been special unbound has these an nuclei The yields Borromean states). also name bound core form cannot the neutrons (two of subsystem removal the viously h eoa fasnl eto ae i difference big a makes neutron single neither a because of removal of the case the is Different 6 systems. bound perfectly e ih ulicoet h eto rplns where drip-lines, neutron the to close nuclei light He, eti aybd ula ytm tn u among out stand systems nuclear many-body Certain nmn ntne ntels e er freapethe example (for years few last the in instances many In prop- the of many that proposed [3] Jonson and Hansen er g,Mga 2 rpsdaqaiaieargument qualitative a proposed [2] Migdal ago, Years arn ntecnium h uduoersos fteBor the of response quadrupole the continuum: the in Pairing A lstonurn ept h intermediate the despite neutrons two plus , ASnmes 11.v 11.y 26.60.Cs 21.10.Ky, 21.10.Gv, energy. numbers: g.s. PACS the above MeV latest 3.9 the 2 about narrow with at th the perspective found reproduces investigate into distribution We calculations strength quadrupole our composition. put we its and revealing accounts This state, interaction. ground contact-delta pairing simple a nabsso w-atcesae ul u fcontinuum of out built states two-particle of basis a in h rudsaeadlwligcniumsae of states continuum low-lying and state ground The 10 inor Li )Dpriet iFsc srnma“.aie”adINFN and “G.Galilei” Astronomia e Fisica di Dipartimento 1) 5 eeiti on om Ob- form. bound in exist He .Fortunato L. ninIsiueo ehooy ore 4 6,India 667, 247 Roorkee Technology, of Institute Indian 41 aor Ca i azl ,I311Pdv,Italy Padova, I-35131 8, Marzolo via 42 1 a or Ca, .Chatterjee R. , + )Dprmn fPhysics, of Department 2) nmedium in ..of g.s. 209 11 bare Pb 210 ior Li 6 Pb, He 2 ajtSingh Jagjit , opetoo hmadcluaetedaoa pairing diagonal integral, the an calculate as and resonance, elements the them some matrix of adjust of energy two to the is couple to state procedure resid- bound standard of appropriate A role states mandatory. the continuum of is single-particle of treatment between estimation interaction proper the While ual a and simplicity. assumption, continuum of insightful sake the an the is for states this model decay- exponentially shell bound, ing, using by calculated been has ucsflpeoeooia oesand models phenomenological Successful Gaussian a or interaction delta interac- potential. contact pairing a often suitable most some tion, with notation, shorthand atce usd fadul ai oe o xml see example for two on core: have magic discussion doubly that the a nuclei of deeply-bound calcula- outside successful of particles the properties from this of taken for tion is paradigm calculations The in of seen type continuum. structures energy resonant low-lying the of the some for account ously lsdsel,teehv ensvrlsuis[]amdat aimed [6] studies several been have outside there two-particles shells, descriptions having closed this systems on nuclear Based standard the of nuclei. for these explanation contribution of convincing structure non-trivial level utterly a an furnishes give of that elements any interaction matrix play residual non-diagonal not and the does diagonal continuum The the role. where nucleus bound ttso orma uli uhas such bound nuclei, the of Borromean character of interac- stable states pairing the show explain delta to naturally contact tion, resid- paper a to this namely related of interactions, concepts ual purpose theoretical of the box extension is a an It how in diagonalization radius). functions a finite wave from of decaying example, exponentially for ba- a bound, (obtained, rather of but set continuum, true sis point the starting not a as calculations character. take for normally stable approaches their these of fact essential reason In are prime These the continuum. understand the effects to of incorporate presence to re- the fail to nuclear still due in they but behavior well, their fairly actions reasonable approximate a to to and structure their degree describe to used been have hs orma ytm r o e ul understood. fully yet not are systems Borromean These 6 p eaefudwti hl oe scheme, model shell a within found are He sae fteunbound the of -states o h orma hrce ftebound the of character Borromean the for + eoac,wieascn ie ekis peak wider second a while resonance, uduoersos ftesystem the of response quadrupole e xeietlrsls h calculated The results. experimental 1 .Vitturi A. 18 Szoed Padova, di -Sezione nHyestxbo 5,adeeply- a [5], textbook Heyde’s in O 1 5 oennucleus romean ences using nucleus, He R Ψ ′ 6 V eadsimultane- and He ( | binitio ab ~ r 1 − ~ r 2 6 models He | Ψin )Ψ 2 5He 2part. 6He |φ 2 p1/2(r,EC)| 0.005 0.0045 0.005 0.004 0.0045 0.0035 6.0 0.004 0.003 0.0035 0.0025 0.003 0.002 0.0025 0.0015 0.002 0.001 + − + 0.0015 (2 , 1 , 0 ) ± 0.0005 ( ) 0.001 0 1 0.0005 4.0 0 10 9 8 2 7 1 + 6 p 0 5 2 0 5 4 3 1 10 Ec (MeV) p p + + 15 3 2 2 1 , 2 + 20 2 2.0 1 2 2 r (fm) 25 p 1 p 3 + + 30 2 2 0 , 2 + |φ 2 p 3 2 p3/2(r,EC)| 2 0.025

0.0 0.025 0.02 0.02 0.015 + 0 0.015 0.01 MeV 0.01 0.005 0.005 FIG. 1: (Color online) Left: experimental energy levels (reso- 0 nances) in 5He. Center: unperturbed energies of two particle 0 10 9 states built upon the scheme on the left. Right: experimen- 8 6 7 tal energy levels (bound ground state and resonances) of He. 0 6 5 5 10 4 Ec (MeV) Shades of gray and pink indicate widths. The experimental 15 3 20 2 r (fm) 25 1 energies are from Ref. [7] (in black) and Ref. [8] (in red). 30 Parentheses indicate uncertain spin-parity assignment.

showing how Borromean systems are bound due to the FIG. 2: (Color online) Calculated square of 5He continuum effect of pairing that brings the energy of the subsystem wave functions (top: p1/2, bottom: p3/2) as a function of below the threshold. The short-range radial variable and continuum energy. nature of the residual NN interaction between the oth- erwise unbound neutrons is what kills the oscillating tail of the continuum wave functions. But, returning to our proach, consisting in identifying the resonances through system, the energy of the unbound neutrons is not just a the phase shifts. In this case the width is connected to single energy, it is rather smeared on a continuous energy the first derivative of the phase shift. The poles of the range according to some distribution. How does the dif- S-matrix have been calculated with the Jost functions ferent energies, all present at the same time with different for WS + spin-orbit potentials. The result is that, for probability, combine into a single bound state? V0 = −41.2 MeV and Vls =6.5 MeV, one gets a p3/2 res- We start from the description of the unbound subsys- onance with real part 0.79 and imaginary part 0.49. The tem and we specialize our arguments to the lightest pro- p1/2 resonance comes at 1.27 MeV with a width of 1.62 totypical case of 6He. The subsystem 5He is unbound, it MeV. Both calculations give similar outcomes, with sim- exists only as a short-lived resonance that breaks up into ilar parameters, although the widths are not in perfect the α + n channel. The shell model predicts a bound, agreement. Therefore for the sake of easing the following completely filled, s state for the α core and an unbound calculations, we will use the simpler results coming from p doublet, further split by spin-orbit interaction. Ex- the first approach. + 6 perimentally the p3/2 and p1/2 resonances are found at Fig. (1) also shows the 0 ground state of He that is 0.789 MeV and 1.27 MeV above the neutron separation bound by 0.973 MeV and the 2+ narrow resonant state threshold [7]. Their width are quoted as 0.648 MeV and found at 1.797 MeV ±25 keV above the ground state. 5.57 MeV respectively (See Fig.1). Note that these val- According to standard databases another resonance is ues have been extracted from raw data within R-matrix found at about 5.6 ±0.3 MeV with uncertain spin-parity approach. The relative motion wave of the neutron with assignment (given as 2+, 1−, 0+). No other states are respect to the core is an unbound (EC > 0,k> 0), oscil- present up to 14 MeV. The widths of these resonances lating dipole (ℓ = 1) wave that must approximate a com- are 113 ±20 keV (narrow 2+) and 12.1 ±1.1 MeV (very bination of spherical Bessel functions at large distances broad) respectively. Recent experimental observations from the center. The continuum single-particle states of [8], trying to disentangle the complicated nature of the 5He can be reproduced fairly-well with a Woods-Saxon 6He continuum with the p(8He,t) reaction at SPIRAL (WS) potential of depth V0 = −42.6 MeV, r0 = 1.2 fm (GANIL), support the existence of two resonances above + and a =0.9 fm, with a spin-orbit coefficient of Vls =8.5 the neutron Sn: a2 state at 2.6(3) MeV (see Eq. 2-144 of Ref. [9]). These wavefunctions are MeV with Γ =1.6(4) MeV and a 1(+,−) state at 5.3(3) shown in Fig. 2. We have followed also a more refined ap- MeV with Γ =2(1) MeV. These states are shown in red 3

∆ 2.0 1.0 0.5 0.2 0.1_____ E in Fig. (1). Several theories are listed in Ref. [8] and 20 ______J=0 ______compared with the available experimental information. ______2 ______G=g ∆E Most of them show the same set of levels that we have ______g ______constructed on p orbitals. The somewhat puzzling nature ______6 15 ______of the excited states of He has been discussed recently in ______Ref. [10] where it is concluded that the spin and parity ______1e+05 ______+ ______of the 5.3 MeV state is most probably 0 , in contrast ______10 ______with the analysis proposed with the experimental data. ______The crudest model with two non-interacting parti- ______10000 ______5 ______cles in the above single-particle levels of He produces ______Energy (MeV) ______5 positive-parity states when two neutrons are placed ______5 ______1000 2 ______in the p3/2 and p1/2 unbound orbits. Namely the p3/2 ______configuration couples to J = 0, 2 (these states can nat- ______urally be assumed as the main components of the two ______0 100 6 _ _ _ _ _ 0 20 40 60 80 100 lowest states of He), the p3/2p1/2 configuration couples N 2 to J = 1, 2 and the p1/2 configuration couples only to 5 10 20 50 100 N J = 0. The unperturbed energies of these configura- tions are 1.578, 2.06 and 2.54 MeV respectively, as indi- cated in the second column of Fig.(1). Ab initio theories FIG. 3: (Color online) Left: Eigenspectrum of the interacting [8, 11] (at the level of 12 ~ω) find the sequence of levels two-particle case for J = 0 for increasing basis dimensions, N. − (0+, 2+, 2+, 1+, 0+) with a third 0+ lowering rapidly as The coefficient of the δ contact matrix, G, has been adjusted the basis is increased (see Fig.1 of cited Ref. [11]), con- each time to reproduce the g.s. energy (right). The actual strength of the pairing interaction, g, is obtained by correcting firming our simpler scheme. A few relevant statements with the energy spacing ∆E and it is practically a constant. can now be done: if the 1− attribution of part of the strength is confirmed, then its nature cannot be associ- ated with 2 neutrons sitting in p orbitals. One might won- der whether dipole strength should be present: certainly goes from 0.1 fm to 100.0 fm with the potential discussed a highly collective dipole mode built at high excitation above (Notice that this amount to 2.4 Gb of data for each energy as a coherent superposition of p-h states must component). With these wavefunctions, using the mid- exist, but its nature in 6He calls for promotion of one point method with an energy spacing of 2.0, 1.0, 0.5, 0.2 neutron from the p to the sd shell. The low-energy tail and 0.1 MeV, corresponding to block basis dimensions of of this giant dipole resonance indeed is expected to come N =5, 10, 20, 50 and 100 respectively, we formed the two down till zero and therefore it might mix significantly particle states and calculated the matrix elements of the with other states and make the spin-parity assignment pairing interaction (∼ 4Gb of data for the largest case). very difficult. This strength might also come from other This has been diagonalized with standard routines and configurations, such as α + (2n) cluster configurations, it has given the eigenvalues shown in Fig. (3) for the that mix with the α + n + n configuration. J = 0 case. The coefficient of the δ−contact matrix, G, The five states discussed above are not discrete, but has been adjusted to reproduce the correct ground state rather depend on the energies of the two continuum or- energy each time. The actual pairing interaction g is ob- bitals tained by correcting with a factor that depends on the aforementioned spacing between energy states and it is (j) φℓ,j,m(~r, EC )= φℓ,j (r, EC )[Yℓmℓ (Ω) × χ1/2,ms ]m . practically a constant, except for the smallest basis. The biggest adopted basis size gives a fairly dense continuum These can be combined into a tensor product two-particle in the region of interest. wave function, The radial part of the S = 0 g.s. wavefunction ob- (J) tained from the diagonalization in the largest basis is ψJM (~r1, ~r2) = [φℓ1,j1,m1 (~r1, EC 1) × φℓ2,j2,m2 (~r2, EC 2)] M displayed in the upper part of Fig.(4). Due to symmetry that must be discretized and used as a basis for calcu- reasons and to the fact that ℓ1 = ℓ2 = 1, there is no lations. It is clear that one needs at least to introduce S = 1 component for a δ−interaction (see Ref. [14], ch. the residual interaction between continuum states, a task 20). In fact, in this case, we can write the two-particle Y + that requires careful numerical implementation because wavefunction as Ψ(~r1, ~r2) = Ψ(r1, r2) JM (Ω1, Ω2)χS=0. one deals with large datasets. We take an attractive pair- It is symmetric with respect to the exchange of coordi- ing contact delta interaction, −gδ(~r1 − ~r2) for simplicity, nates of the two identical neutrons. It shows a certain although, as it is well-known, density dependent interac- degree of collectivity, taking contributions of comparable tions might be more appropriate [12]. We have calculated magnitude (though not all of the same sign) from several the continuum single-particle wavefunctions, with ener- basis states, while in contrast the remaining unbound gies from 0.0 to 10.0 MeV, normalized to a delta similarly states usually are made up of a few major components. to Ref. [13], for the p-states of 5He on a radial grid that The surface plot shows the exponential behavior typical 4

Ψ 4 (r1,r2) 0.4 20 + 2 0.4 0.3 1 /MeV) 10 0.3 4 3 0.2 fm

0.2 2 0 0.1 0.1 0 5 (e C 0 0 2 g.s.

-0.1 )/dE +

10 2

8 → 0 6 + 2 4 r (fm) 2 4 2 1 2 6 2 r (fm) 8 1 10 0 dB(E2; g.s.

0 2 0 2 4 6 8 10 0.04 2 (p ) ~97.2% E (MeV) (p1/2) ~2.8% 3/2 C 0.02 i C 0 FIG. 5: (Color online) Quadrupole strength distribution with -0.02 respect to the break up threshold. The total strength (black) 2 -0.04 is split into the contribution of the (p3/2) and (p3/2p1/2) 0 5000 10000 15000 20000 components, in blue and red respectively. The insert shows | i > the full curve for the total strength.

FIG. 4: (Color online) Ground state wavefunction (S = 0) for N =100 as a function of the coordinates of the two neutrons and corresponding contour plot (upper part). Decomposition and obtain eigenvalues, that are all unbound, and the of the g.s. into the J=0 basis (lower part) as a function of an corresponding eigenvectors. arbitrary basis state label: the basis is divided in two blocks, We compute the B(E2) values (Fig.5) to these eigen- 4 × (0) 4 × (0) 10 [p1/2 p1/2] components and then 10 [p3/2 p3/2] values and adjust the strength of the pairing matrix to components. The ordering in each block is established by the get the energy centroid of the first peak at about the right sequential energies of each pair of continuum s.p. states, i.e. position (E = 0.76 MeV, Γ ∼ 0.2 MeV). The width is a (EC1 ,EC2 ) = (0.1, 0.1), (0.1, 0.2), . . . ,(0.1, 10.0), (0.2, 0.1), (0.2, 0.2), . . . (10.0, 10.0). bit larger then the experimental value. We also obtain a second peak at about E = 2.91 MeV with an asym- metric width at half maximum of Γ ∼ 1.8 MeV. While 2 the first peak is mainly due to (p3/2) components, the of a bound state, despite being the sum of many products second peak is clearly identified as arising mainly from of oscillating wavefunctions. One can see from the bot- (p3/2p1/2) components. Measuring energies from the g.s., tom part of the figure that the square of the amplitudes . 2 the second peak is found at about 3 88 MeV. A notewor- of the (p3/2) components are dominant summing up to thy feature of this peak is that it is found at an energy 97.2%. In principle also the s-continuum should be intro- 2 higher than the corresponding unperturbed two-particle duced in the picture, because the (s1/2) configuration of state, despite the attractive nature of pairing: this is course couples to J = 0.We did not introduce it in the 5 a consequence of the asymmetric long tail in energy of calculation because the p-resonances dominate the He the p resonance in 5He. The total integrated strength spectrum and because the numerical computations are 1/2 amounts to about 8.8 e2fm4, of which about 3/4 is in the already very demanding. Following Ref. [15], however, first peak. This value can be compared with the value the contribution of p2 in 6He is estimated to be the most of 9.7471 e2fm4 obtained in Ref. [18]. To the best of relevant with a percentage of about 83%. In the lower our knowledge Fig. 6 of the cited paper is the only pub- part of the Fig.(4) one can also see that the chosen cut lished theoretical prediction for E2 strength distribution in energy is appropriate because the states with a label in 6He and with some little differences, we essentially approaching 104 and 2·104 become progressively less and confirm that result. According to our calculations the less important. second peak does not match with the recently identified Notice that, in our approach, there is no information 2+ strength at 2.6 MeV [8]. Possibly other components, on the angular correlation, that has nevertheless been ex- like s and d continuum states of 5He, when taken into ac- tensively investigated by various authors: it corresponds count theoretically might affect the quadrupole response C˜2 to the ℓ ;00(θ12) of Ref. [4]. of 6He. We plan to thoroughly investigate this aspect. While most theoretical studies have focused on dipole Several theories disagree on the predictions for the low- strength [16, 17], we have performed a set of calculations lying continuum states of 6He and the available experi- for quadrupole transitions. After constructing a basis of mental information cannot be considered as completely (2) the same size made up of two parts, namely [p3/2×p3/2] free from model assumptions. For example in Ref.[8] one (2) and [p3/2 × p1/2] , we diagonalize the pairing matrix might wonder that the procedure of extracting the posi- 5 tion and width of the secondary 2+ state from the shoul- pairing interaction. Other similar studies have used ar- der of the primary 2+ resonance peak is not free from tificially bound p-states or have used a box to discretize arbitrariness in the choice of background. This is all the the continuum. We analyze the E2 response showing more true when dealing with exotic beams with low in- where we expect two resonances to occur. The stark tensity and when several competing processes might cre- mismatch between theory and experiments on the posi- ate a background, as in the present case. The choice of tion of the higher resonance calls for further work, be- Breit-Wigner parameterization is another factor might cause our understanding of drip-line Borromean systems also influence the outcomes. Certainly a lager body of passes through the proper description of the lightest and experiments is needed in order to unravel the structure foremost example of them, 6He. of low-lying resonances of 6He. We would like to thank J.A.Lay and P.Descouvemont We have shown how the bound Borromean ground for useful suggestions. J.Singh gratefully acknowl- state of 6He emerges from the coupling of two unbound edges the finanacial support from Fondazione Cassa di p-waves in the 5He continuum, due to the presence of the Risparmio di Padova e Rovigo (CARIPARO).

[1] M.V. Zhukov, B.V. Danilin, D.V. Fedorov, J.M. Bang, [10] H.T.Fortune, Phys.Rev. C 89, 014326 (2014). I.J. Thompson and J.S. Vaagen, Phys. Rep. 231, 151- [11] S.Baroni, P.Navr´atil and S.Quaglioni, Phys.Rev.Lett. 199 (1993). 110, 022505 (2013). [2] A.B. Migdal, Sov. J. Nucl. Phys. 16, 238 (1973). [12] G.F.Bertsch and H.Esbensen, Ann.Phys. 209, 327-363 [3] P.G. Hansen and B. Jonson, Europhys. Lett. 4, 409 (1991). (1987). [13] N.Austern, C.M.Vincent and J.P.Farrell Jr., Ann.Phys. [4] P.Mei and P.van Isacker, Ann.Phys.(N.Y.) 327, 1162- 114, 93-122 (1978). 1181 (2012). [14] A.de Shalit and I.Talmi, Nuclear Shell Theory, (Aca- [5] K.Heyde, Basic ideas and concepts in , demic Press Inc. 1963). 3rd Ed., (IOP Publishing Ltd, Bristol, UK 2004). [15] K.Hagino and H.Sagawa, Phys.Rev. C 72, 044321 (2005) [6] K.Hagino, A.Vitturi, F.P´erez-Bernal and H.Sagawa, J. [16] S.Aoyama, T.Myo, K.Kato and K.Ikeda, Prog. Theor. Phys. G: Nucl. Part. Phys. 38, 015105 (2011); A. Vitturi Phys. 116 (1), 1-35 (2006). and F.P´erez-Bernal, Nucl. Phys. A 834, 428c (2010). [17] P. Descouvemont, E.Pinilla and D.Baye, Prog. Theor. [7] TUNL, Nuclear Data Evaluation, Phys. Supplement 196, 1-15 (2012). http://www.tunl.duke.edu/NuclData/General_Tables/5he.shtml[18] J.A.Lay, A.M.Moro, J.M.Arias and J.G´omez-Camacho, [8] X.Mougeot et al., Phys. Lett. B 718, 441-446 (2012). Phys.Rev. C 82, 024605 (2010). [9] A. Bohr and B.R. Mottelson, , Vol. I (World Scientific Publishing Co. Pte. Ltd. 1998)