Binding and excitation mechanisms of 6He, 10He and 11Li

NUCLEAR PHYSICS A ELSEVIER

Nuclear Physics A588 (1995) 29c--34c

Binding and excitation mechanisms of

6He, 1°He and

llLi

K. Kath", S. Aoyama", S. Mukai" and K. Ikeda b aDepartment of Physics, Hokkaido University, Sapporo 060, Japan bDepartment of Physics, Niigata University, Niigata 950-21, Japan

The binding mechanism and excitations of SHe, X°He and llLi neutron-rich nuclei are investigated by using core(4He, SHe or 9Li)+n+n three-body models. Both of the dineutron like correlation in addition to the mean field correlation, which are described by the hybrid-TV model, are shown to play a very important role in the binding of the Borromean three-body systems such as 4He+n+n and 9Li+n+n. Comparison between observed binding energies and theoretical ones calculated with a large model space and reliable core-n interactions are discussed. Excited three-body resonance states including the ground state of l°He are also discussed by solving them with the complex scaling method. 1. ~ T R O D U C T I O N As a typical example, the so-called halo structure in the nLi nucleus has been studied with much interest from both of experimental and theoretical sides.J1-2] The basic understanding of the halo structure in llLi has successfully been obtained within the 9Li+n+n three-body model. On the other hand, more quantitative description has been needed through the recent development of experimental studies. For the quantitative understanding, however, it is a serious problem that the 9Li+n+n model cannot reproduce the observed binding energy 250-4-80 keV. The experimental data of the 6He nucleus; the small two-neutron separation energy (0.975 MeV) and large r.m.s radius (2.33+0.0413]), also suggest that this nucleus has the exotic structure of the neutron halo or skin. In SHe, furthermore~ the excited 2 + state is observed as a resonance state. Since SHe is the simplest neutron rich nucleus and a typical Borromean three-body system[4] having no bound states in any binary sub-system, investigations of this nucleus are expected to provide a basic understanding on the weak binding mechanism and excitations of the Borromean three-body systems. In addition to 6He, new experimental observations on He-isotopes have recently been accumulated.J5] The 1°He nucleus has been found and its life time has been observed to be much shorter than the theoretical prediction.J6] Since 1°He is already beyond the neutron 0375-9474/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved. SSDI 0375-9474(95)00095-X

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K. Kato et al. / Nuclear Physics A588 (1995) 29c-34c

drip line, it is very interesting to investigate the binding mechanism and its excitation of this nucleus for the systematic understanding of neutron rich nuclei over the neutron drip line. To study those systems of He and Li isotopes, we carefully examine the core+n+n three-body models in a wide model space by taking account of two important correlations; one is a mean field correlation between core and valence neutrons and the other is an n-n correlation between valence neutrons. The former is taken into account by basis wave functions of the cluster orbital shell model (COSM)[7] in the V-type coordinates. The latter correlation has been discussed to overcome the small binding energy of the 9Li+n+n model calculated only within the COSM.[8] In order to take account of this correlation, a hybrid model combining the extended cluster model (ECM)[8] with the COSM has been proposed. Since the ECM basis functions are given by the T-type coordinates, we call the hybrid model of COSM and ECM as a hybrid-TV model. The ground state of 1°He and all excited states of 6He, 1°He, and 11Li nuclei appear as resonance states. To solve resonance states in three-body systems with correct boundary conditions, we have used the very powerful treatment of the complex scaling method[9] which has been successfully discussed in spectroscopic studies of the l°Li(gLi+n) system[10] and others. We can obtain resonance energies and widths by solving the complex scaled Hamiltonians of three-body systems by a famihar diagonalization procedure. In this paper, we report results of the binding energies of 4He+n+n, SHe+n+n and 9Li+n+n systems which are solved with the hybrid-TV model and a large basis space of the COSM and with the most reliable 4He-n, SHe-n and 9Li-n interactions. At the same time, we calculate resonance energies and widths of the excited states including an interesting excitation corresponding to the soft giant dipole state. From those results, we discuss the binding mechanism and excitations of SHe, l°He and 11Li neutron rich nuclei. 2. H Y B R I D - T V M O D E L OF C O R E + n + n

SYSTEMS

The Hamiltonian of the core+n+n models for SHe, l°He and 11Li is given as 2 H = Y~+Ti - T ~

i=o

n! + Vcrt(~'~l) + Vcn('l~2 ) "4- Wnn('P) --~ Ay~ i=1

where Ti and Tc,~ are kinetic energy operators of the i-the cluster (i=0 (core), l, 2 (neutrons)) and of the center-of-mass of the total system, respectively. V~n(r/i) are potentials between core and the i-th neutron (i=1, 2) and details will be discussed in the next section. We employ the Minnesota force [11] with an exchange parameter u=0.96 as the two-neutron potential V=~(r). The last term of Eq.(1) is the pseudo-potential to project out the Pauli forbidden states fij (r/) of relative motion between the core nucleus and valence neutrons. We assume the harmonic oscillator shell model configuration for fij(rh) with a size parameter b0 fitting

Iflj(rli )

><

n n ¢:N /':N ~-¢~ )~'J ~ ~ : ) +.

j~j(r/i)h

n

®

(1) n

"o) 0

(,) COSM

ECM

V-type(COSM) and T-type (ECM) coordinates of the coreTn+n model.

Fig.l

K. Kato et al. / Nuclear Physics A588 (1995) 29c-34c

3 lc

the observed matter radius of the core nucleus and A=10 s (MeV). Using a variational method, we solve the SchrSdinger equation

H]qIjM > = E]~JM > •

(2)

To take account of both of the mean field correlation due to V~n(vh) and the n-n correlation due to V,,=(r), we employ the hybrid basis functions expressed by

qlgM = ~JM(V) + (~JM(T),

(3)

where q)ju(V) and ~jM(T) are COSM[7] (V-type) and ECM[8] (T-type) basis functions, respectively. Explicitly, they are given by b1

hi2

¢jM(V)

=

ili2 %,J,,'2,~2A^ [¢,:~, (~/1)¢,~j~(~h)] jM , E E i1,12 11,J1,12,j2

~JM(T)

=

Z ~-~ ,,L,I,S il ,i2 I,L,I,S

[[Ul ( r ) ~ L (R)]I[X1/2(1)X1/2(2)]S]JM

(4) ,

ill2 and ~1i2 where fi. is an antisymmetrization operator and Ctl,j~,l~j2 t,L,Z,s are variational bq bi2 efficients. As the basis sets of {¢,,j,(~/t), ¢hj2(~/2)} and {u~" (r), U~L~2(R)}, we use Gaussian functions with the size parameters {bil, bi2, flil, fli~} which are given by geometrical progression. To solve three-body resonance states, we apply the complex scaling method[9] to core+n+n systems. The complex scaling is defined by the following transformation: .3 V(O)~o(~) = exp(,~O)~o(exp(,O)z),

(5)

cothe the the

(6)

for any function ¢ ( z ) of a coordinate z ( = ,11, ~2, r or R ) , where 0 is a real value

parameter. Under this transformation, the SchrSredinger equation is given as

H(O)[~o > = E(O)I~o >,

H(O) = U(O)HU(O) -~,

Iff/o > = U(0)I~ > .

(7)

The ABC-theorem[9] indicates that the solutions of Eq.(7) have the following properties: 1) The resonance solutions are also described by square-integrable functions in addition to normalizable bound states. 2) The energies of bound states are not changed by scaling. 3) When we choose the scaling parameter 0 larger than the angle tan -I(F/2Er) corresponding to the resonance position (resonance energy E~ and width P), E~ and 1"/2 are obtained as real and imaginary parts of the complex eigenvalue E(O), e.g., E(O) = E, - iF~2. 4) The continuum spectra are obtained along lines on the complex energy plane, which start at the threshold energies of decays of the system into sub-systems and have an angle - 2 0 from the positive real axis. 3. B I N D I N G E N E R G I E S A N D E X C I T E D STATES OF 6He, 1°He A N D 1iLl

3.1. 6He We calculate the binding energy of SHe by using the hybrid-TV model for the

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K. Kato et al. / Nuclear Physics A588 (1995) 29c-34c

4He+n+n system. As the interaction between the 4He core and a valence neutron, we use the potential which is slightly modified from the local 4He-n potential constructed by Kanada et al.[13] to reproduce well the observed phase shifts including higher partial waves. To obtain well converged solutions, we increase the channel configurations (11,jl, 12, j2) step by step as seeing convergence of energies. In each step, we choose enough number and the optimal size parameters of Gaussian basis functions. Figure 2 shows the binding energy convergence for both cases of the hybrid-TV model and the COSM. In the COSM calculation with only the V-type basis functions, we increase angular momenta to l = 14 and j = 29/2 by starting with the (pl/2) 2 configuration. In addition to the V-type basis functions, the T-type functions of the I = L = 0 configuration are combined in the hybridTV calculations. 1.o We can see very rapid convergence in the 0,8 hybrid-TV model but slow convergence in ~ " 0.5 O.4 the COSM. This result indicates that the di~ o.2 neutron like correlation is essentially impor~ o.o tant in binding of 6He. The well-converged I.t --o.2 ~-o.4 binding energy 0.784 MeV is much improved from the previous calculations. However, a little shortage (0.191 MeV) remains still i 2 $ 4 5 6 T 8 g lOlllll$141511117111192011~t2~lll,~X~/'~tn30 in comparison with the experimental value Channel No. [n] 0.975 MeV. For excited 2 + states, we calculate the Fig.2 Energy convergence of the 4He+n+n resonance energies (Er) and widths (F) by ground states calculated by the hybrid-TV diagonalizing the complex scaled Hamilto- modeI(COSM+ECM) and the COSM. nian. As shown in Fig.3, we obtain the first 2 + state (Er=l.03 MeV and F=0.260 MeV) U.O 'o ' ' ' corresponding to the observed 2+ state (Er=0.822+0.025 MeV and F=0.113+0.020 56 MeV[14]). Although the calculated width is a little large due to a little higher resonance energy, the calculated excitation energy 1.81 , ~ - 3 . 0 MeV is in very good agreement with exper% o imental one (1.797+0.025 MeV[14]). Fur- g,,~ - 4 . 0 o o o thermore, our calculations predict the sec-5.0 0 . 0 1 . 0 2 . 0 3.0 4.0 5.0 ond 2+ resonance state(Er=2.7 MeV, F=4.8 Energy (MeV) MeV), a 1+ resonance state (Er=3.0 MeV, F=6.2 MeV) and the second 0 + resonance Fig.3 The 2+ eigenvalue distribution of the state (Er=4.2MeV, F=8.9 MeV). However, 4He+n+n complex scaled Hamiltonian. no 1- resonance state is obtained.

._-i.0;>_~.~~,+1, 0

, 00

3.2. 1°He The large binding energy (2.137 MeV[14]) of SHe is observed from the 6He+2n threshold and the SHe nucleus is considered to have a (Os)4(Op3/~) 4 closed shell configuration. Therefore, assuming that the SHe cluster is stable, we apply the SHe+n+n model to 1°He.

K. Kato et al. / Nuclear Physics A588 (1995) 29c-34c

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The SHe-n interaction is determined by fitting the binding energy energy(-1.13 MeV[14]) and resonance state at E~=3.8 MeV[14] of 9He. For the unbound ground state of mile, we calculate the resonance energy and width by applying the complex scaling method to the SHe+n+n model. As basis functions, we use the COSM wave functions. 0.0 The distribution of complex eigenvalues (J~=0 +) of H(#) is shown in Fig.4. The true three-body resonance is obtained at E~=2.14 MeV and F=1.63 MeV. This result well corresponds to the observed energy °<-" 1°He(0+) (1.2:1=0.3 MeV[6]) and width (<1.2 MeV[6]). We also obtain the excited 2 + resonance state at about 5 MeV excitation energy. , \°o \ --1.5 0.0 The present result calculated within the 1.0 2.0 3:0 4.0 Energy (MeV) COSM basis functions may be slightly improved by an extended calculation of the Fig.4 The 0+ eigenvalue distribution of the SHe+n+n complex scaled Hamiltonian. hybrid-TV model.

~~

3.3. 11Li

The basic understanding of the halo structure in this nucleus has successfully been obtained within the 9Li+n+n model.[15] However, every calculation of the 9Li+n+n model cannot reproduce the binding energy quantitatively. To understand the binding mechanism, it is necessary to examine carefully both of the model space and the interactions between clusters. We here calculate the binding energy of the 9Li+n+n model by using the improved 9Li-n interaction[10] which has been determined on the basis of the new data[16] for l°Li and a large model apace of the Hybrid-TV model basis functions. In Fig.5, we show the binding energy con100 vergence in both cases of the COSM and the 50 hybrid-TV model basis functions. Although >~ _: ",,~. COSM Bases the large improvement is obtained by ex~-100 tending the model space from the COSM to :', h0 - 1 5 0 the hybrid-TV model and by using the im~ -2oo proved 9Li-n interaction, the observed bind-~50 COSM + ECM ing energy (250+80 keV) cannot be repro~o0 i i i i i 2 3 4 $ 6 duced. To reproduce the observed binding -35~ Channel no. In] energy, we have to strengthen the 9Li-n interaction by 2.75 %. Furthermore, we can Fig.5 Energy convergence of the 9Li+n+n see that the contribution from s-waves be- ground states calculated by the COSM(circles: tween 9Li and neurons is not so large (,,,50 without s-wave, crosses: with s-wave) and the hybrid-TV model. keV). To estimate effects of the 9Li-core excitation, which are not included in the 9Li+n+n model, we investigate effects of l p - l h and 2p-2h excitations in the 9Li-core. For this purpose, we first perform a coupled channel calculation of (9Li+n)+(SLi*+n) where lplh and 2p-2h configurations are taken for 9Li*. The obtained result indicates that these l p - l h excitations have small contributions. However, it is very interesting to estimate contribution from the 2p-2h excitation to the biding energy in 11Li.

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4. S U M M A R Y

We have performed the 4He+n+n, SHe+n+n and 9Li+n+n three-body model calculations for neutron rich nuclei 6He, l°He and 11Li. In those calculations, we emphasis two important correlations in the binding mechanism described by the hybrid-TV model combining COSM with ECM. To obtain more quantitative conclusion, we have adopted the most reliable 4He-n, SHe-n and 9Li-n interactions based on the recent experimental data for SHe, 9He and l°Li. The results and main discussions are summarized as follows: 1) The 4Heq-n+n system has been investigated in detail as the typical example of the core+n+n model. The binding energy 0.784 MeV is obtained and largely improved from previous calculations, though small amount 0.191 MeV remain still for the experimental value 0.975 MeV. The first excite 2 + resonance is well reproduced by this model. This is a genuine three-body resonance state in the Borromean system. In addition to the observed 2 + state, the model calculation predicts other 2+, 1+ and 0 + resonance states at 3,-,5 MeV excitation energies. On the other hand, the 1- state corresponding to the soft dipole mode is not found as a resonance state in the energy region lower tan 10 MeV and with the width smaller than 10 MeV. 2) The ground state of 1°He has been calculated as a resonance state of the SHe+n+n system. The observed short life lime is well explained. The excited 2 + resonance state is predicted at the excitation energy higher than about 5 MeV. 3) The binding energy of 1iLl has been investigated with the 9Li+n+n hybrid-TV model, where the improved 9Li-n potential has been used on the basis of the new experimental data of l°Li. The obtained binding energy is still short by about 1 MeV from the observed one. To explain this lack of the binding energy, we have discussed effects of the 9Li core excitation. REFERENCES

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