Page 1

arXiv:nucl-th/0604072v2 13 Dec 2006

Collective-coupling analysis of spectra of mass-7 isobars:7He,7Li,

7Be,7B.

L. Canton(1),∗G. Pisent(1),†K. Amos(2),‡S.

Karataglidis(2,3),§J. P. Svenne(4),¶and D. van der Knijff(5)∗∗

(1)Istituto Nazionale di Fisica Nucleare, sezione di Padova,

e Dipartimento di Fisica dell’Universit` a di Padova,

via Marzolo 8, Padova I-35131, Italia

(2)School of Physics, University of Melbourne, Victoria 3010, Australia

(3)Department of Physics and Electronics,

Rhodes University, Grahamstown 6140, South Africa

(4)Department of Physics and Astronomy, University of Manitoba,

and Winnipeg Institute for Theoretical Physics,

Winnipeg, Manitoba, Canada R3T 2N2 and

(5)Advanced Research Computing, Information Division,

University of Melbourne, Victoria 3010, Australia

(Dated: today)

Abstract

A nucleon-nucleus interaction model has been applied to ascertain the underlying character of

the negative-parity spectra of four isobars of mass seven, from neutron– to proton–emitter driplines.

With one and the same nuclear potential defined by a simple coupled-channel model, a multichannel

algebraic scattering approach (MCAS) has been used to determine the bound and resonant spectra

of the four nuclides, of which7He and7B are particle unstable. Incorporation of Pauli blocking in

the model enables a description of all known spin-parity states of the mass-7 isobars. We have also

obtained spectra of similar quality by using a large space no-core shell model. Additionally, we

have studied7Li and7Be using a dicluster model. We have found a dicluster-model potential that

can reproduce the lowest four states of the two nuclei, as well as the relevant low-energy elastic

scattering cross sections. But, with this model, the rest of the energy spectra cannot be obtained.

PACS numbers: 24.10-i;25.40.Dn;25.40.Ny;28.20.Cz

∗Electronic address: luciano.canton@pd.infn.it

†Electronic address: pisent@pd.infn.it

‡Electronic address: amos@physics.unimelb.edu.au

§Electronic address: s.karataglidis@ru.ac.za

¶Electronic address: svenne@physics.umanitoba.ca

∗∗Electronic address: dirk@unimelb.edu.au

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I.INTRODUCTION

Currently there is much interest in the structure of, and reactions with, radioactive nuclei.

In particular, attention has been given to weakly-bound light nuclei, which may manifest

exotic structure. The very extended neutron matter distributions of6He and11Li that have

been called neutron halos are examples. They contrast with8He and9Li which have neutron

skins. Signatures of those neutron matter distributions have been noted in the cross sections

from elastic and inelastic scattering of the nuclei from hydrogen [1, 2]. As yet, relatively

few such experiments have been made (elastic and inelastic scattering of radioactive ion

beams (RIB) from a hydrogen target). We hope that will change in the near future, since

appropriate scattering theories to describe the events not only exist but also have been

implemented [2, 3]. Most of the existing RIB-hydrogen scattering data have been taken with

ions of medium energies, analyses of which are appropriately made using a g-folding model

of the optical potential for elastic scattering and a distorted wave approximation (DWA)

analysis of inelastic scattering data [4]. For low energies, however, a coupled-channel theory

is more relevant. Then the MCAS method [3] is a most appropriate way to proceed.

Low energy nucleus-nucleon scattering data exhibit resonances, analyses of which reflect

structure of the compound nucleus formed by the ion and proton target. Such was described

in a recent publication [5]. Also MCAS analyses of data from the scattering of low energy

14O ions from hydrogen [6] revealed that the spectrum should have a number of narrow

resonances at energies slightly higher than the observed two broad resonances that coincide

with the ground and first excited states of the proton-unstable15F. The centroids, widths,

and spin-parity assignments all are determined using the MCAS approach [5].

Herein we use the MCAS theory for the interactions of6He and of6Be with nucleons.

The spectrum of7Li is used to set that matrix of potentials. Thereafter with the interaction

held fixed, we predict the spectra of the other mass-7 isobars, of which7He and7B are

particle unstable. The MCAS method [3] appears particularly suited to this problem in that

it describes the bound and resonant spectra of even-odd light nuclei in terms of nucleon-

nucleus dynamics in the context of coupled-channel interactions. Even a simplistic collective

model, with geometric-type deformation for the structure of the nucleus, can give a very

reasonable description of the compound system [7]. That is so only if the non-local effects

of Pauli blocking are taken into account in the nucleon-nucleus model Hamiltonian. With

MCAS, using a collective model prescription for the input matrix of interaction potentials,

the Pauli exclusion principle can be taken into account by means of the Orthogonalizing

Pseudo-Potential (OPP) method [8]. Single-particle and collective-model aspects then can be

handled in an single approach for nuclear structure and scattering since both sub-threshold

and resonance states of the compound nucleus can be determined.

Originally, the MCAS method was developed and tested in studies with stable light

nuclei [3, 7, 9]. Subsequently the method was developed further to infer properties of a

light nucleus that is outside of the proton drip line [5]. Crucial to the outcome was the

introduction of a new concept which combines collectivity and single-particle dynamics,

namely Pauli hindrance. This accounts for the fact that the transition from Pauli-allowed to

Pauli-blocked single-particle orbits in shell-model structures is not a binary jump but might

change gradually with mass. The description in terms of the OPP entails the non-local

effects of nucleon state antisymmetrization, and appears perfectly suited parametrically to

describe a smooth transitional situation [5].

In the following section, features of the MCAS method and of the properties assumed

2

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for the target nuclei are discussed. This section also includes use of the MCAS method to

analyze the spectra of7Li and of7Be as dicluster systems of α+3H and α+3He respectively.

A dicluster-model potential is generated that describes bound and resonance states. The

elastic scattering cross sections of the cluster pair for energies to 14 MeV are presented and

discussed in that section as well. Thereafter, in Sec. III, the results of our calculations of

the coupled-channel problems of a nucleon with a mass-6 nucleus are given and compared

with the known spectra of the mass-7 isobars. Finally, in Sec. IV, we give conclusions that

may be drawn.

II.THE MCAS THEORY AND SOME STRUCTURE DETAILS

As all details of the MCAS theory have been presented previously [3], only salient features

are reviewed herein.

A.Application of MCAS to a nucleon-nucleus coupled-channel system

We use a simple collective-model prescription for the matrix of interaction potentials

between a proton and6He, similar to that used previously [3, 7, 8], taking just a quadrupole

deformation (deformation parameter β2) with a mix of central (0), spin-orbit (ls), l2(ll), and

spin-spin (Is) potential terms, and a Coulomb potential from a uniformly charged sphere of

radius Rc, viz.

Vcc′(r) = V0v(0)

cc′(r,β2) + Vlsv(ls)

+Vllv(ll)

cc′ (r,β2)

cc′(r,β2) + VIsv(Is)

cc′ (r,β2) + δcc′Vcoul(r,Rc) .

(1)

The channel index identifies the coupling of specific nucleon partial waves to specific target

states leading to each considered, and conserved, total spin-parity Jπof the compound

system, viz. c ≡??l1

states in these applications. It is important to note that the channel indices c incorporate

all relevant quantum numbers for a given single-particle state.

The model takes into account effects of core excitation and polarization by allowing

transitions from the ground state to the lowest two excited states which are assumed to

have collective nature. Therefore, to define the channel space, we assume that there are

three important states to consider in the spectrum of6He. They are the 0+ground, a

2+

first excited states of6He have been given those spin assignments [10], while the third is

that expected by a shell-model calculation [1]. Further, for simplicity, we consider transitions

between them to be effected by one and the same quadrupole operator though, in expansions,

we take the quadrupole deformation to second order [3]. The basic functional form of the

channel-coupling interactions is of Woods-Saxon type.

Starting from this local form in coordinate space, the full nuclear potential V contains,

in addition, the highly nonlocal OPP term. The final form is

2

?jI : J?, with parity (−1)lsince we consider only positive-parity target

1(1.78 MeV) first excited, and a 2+

2(5.6 MeV) second exited states. The ground and

Vcc′(r,r′) = Vcc′(r)δ(r − r′) + λcAc(r)Ac(r′)δcc′.

(2)

The function Ac(r) represents the normalized radial part of the single-particle bound state in

channel c, spanning the phase-space excluded by the Pauli principle. The OPP method for

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treating the effects of the Pauli-blocked states holds in the limit λc→ ∞, but it suffices to

set λc= 1000 MeV to get a stable spectrum where all forbidden states have been removed.

For Pauli-allowed states λc= 0. Pauli-hindered states are assumed when specific strengths

of λcare selected, greater than zero but much lower than 1 GeV and typically a few MeV.

Those strengths (λc) presently are treated as free parameters.

B.Aspects of structure

If we consider6He from the point of view of the simplest vibrational model, the two

excited states are assumed, respectively, to be a one-phonon and a two-phonon excitation

(both with L=2) from the ground. As a consequence, and allowing for the scale factor (of

2) that differentiates basic probability expressions for one-phonon couplings between the

states, this simple model predicts

B(E2;2+

B(E2;2+

2→ 2+

2→ g.s.) = 0 .

1) = 2 B(E2;2+

1→ g.s.)

Neither are close to most observed results as, empirically,

0.5 ≤R =B(E2;2+

B(E2;2+

2→ 2+

1→ g.s.)

1)

≤ 1.6 .

(3)

Wave functions for6He have been obtained from a complete (0 + 2 + 4)?ω shell-model

calculation [1] in which the G-matrix interactions of Zheng et al. [11] were used. This no-core

model gave a spectrum with three low-lying states coinciding with the known 0+ground

and the two excited (resonant) states having centroids at 1.797 and 5.6 MeV. Of those the

1.797 MeV has been assigned 2+while the 5.6 MeV resonance is listed [10] with ambiguous

spin-parities of (0+,1−,2+). The shell model, as noted previously, anticipates 2+

states are radioactive with the 1.797 MeV state having a width of 113 keV. The width

of the 5.6 MeV state is uncertain. Other excited states are listed but lie much higher in

excitation [10], above 14 MeV.

Using bare charges and oscillator wave functions with an oscillator length of 1.8 fm, the

no-core shell model gave B(E2) values for γ-decays in6He of

2. All three

B(E2;2+

B(E2;2+

B(E2;2+

1→ g.s.) = 0.153 e2fm4

2→ 2+

2→ g.s.) = 0.036 e2fm4.

1) = 0.099 e2fm4

Thus this shell-model calculation of6He gives a ratio R = 0.647. This lies near the lower

limit of the empirical ratio range. Also the shell model predicts that the 2+

the ground with a significant probability (23.5% of that of the 2+

The isospin mirror,6Be, is particle unstable (decaying to an α and two protons) and from

the TUNL Data-Group project [10] it is thought to have a resonant 0+ground state and a

first excited one, possibly 2+, centered 1.7 MeV above. Nonetheless we take for6Be the same

shell-model spectroscopy of6He, under the assumption of charge symmetry of the nuclear

force. Thus we assume a second 2+excitation, centered around 5.6 MeV, also for the6Be

isotope. In the present analysis we treat all nuclear (target) states, either ground or excited,

2state decays to

1state).

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TABLE I: Parameter values of the (negative parity) p-6He interaction.

Strengths

V0

Vℓℓ

Vℓs

VIs

−36.817

R0fm;

2.8

−1.2346

a fm;

0.88917

14.9618

Rcfm;

2.0

0.8511

β2

0.7298

Geometry

as stable. In the MCAS scheme unstable states could be accommodated in the formalism.

We plan to do so in the near future. Of course the Coulomb interactions used in MCAS

calculations take into account the change from 2 to 4 protons and the OPP term refers to

states forbidden in the corresponding mirror system. The Coulomb radius was increased to

2.8 fm in this case as well.

As the shell-model calculations give for6He (and of6Be) comparable admixtures of pair

re-coupling and pair breaking in the two 2+states, in the collective model prescription of

the input matrix of interaction potentials, we have taken the two excitations to be equal

mixtures of first and second order terms in the quadrupole-deformation parameter. The

deformation β2is allowed to be a variable parameter, to be adjusted along with the other

parameters of the model interaction to best reproduce the known7Li spectrum.

We have also performed a shell-model calculation for7Li using the G-matrix shell-model

interaction of Zheng et al. [11]. The negative parity states were calculated within a complete

(0+2+4)?ω model space, as was previously published [12], while the positive parity states

were obtained using a (1 + 3 + 5)?ω model space. The only restriction in the latter was

the exclusion of the 5?ω 1p-1h components connecting the 0p shell to the 0i-1g-2d-3s shell.

The shell-model code OXBASH [13] was used for all the calculations from which states up

to, and including J =

comparison with those we obtain using MCAS.

Of course, there are many other models for the structure of these nuclei and of7Li in

particular; no-core shell models [14] other than that we have used, Green’s function Monte

Carlo studies [15], and cluster model investigations [16] to state a few. We also stress the

complementary nature of these methods. Cluster and shell-model studies are suitable, in

general, for analyses of different data, yet they can also give consistent descriptions of many

nuclear properties, as has been pointed out recently [17]. That consistency is evident when

one compares the electron scattering form factors for7Li deduced from a cluster model [16]

and from a no-core-shell-model calculation [12]. Those models provide equivalently good fits

to data to 2 fm−1.

Recently [18], a dicluster model of7Li was used to study electromagnetic properties and

break-up. The α-3H system is a single channel problem given the spectra of the two nuclei

involved. To reproduce the experimental energies of the four states in7Li considered, the

interaction strengths were adjusted in calculation of the p-wave and f-wave functions of

relative motion separately. With those wave functions a variety of data could be described

[18]; most of which data are sensitive primarily to the large radius properties of the wave

functions. In the following subsection, we develop a similar dicluster-model calculation that

reproduces the subset of mass-7 energy levels that significantly couples to the cluster channel.

7

2, were obtained. We will present and discuss later the results in

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C.Application to the mass-7 dicluster systems

With the MCAS scheme, we have performed an equivalent dicluster calculation, identify-

ing the dicluster problem as a single-channel potential problem. We have ascertained a single

potential that gives a set of compound states in good agreement with some of those in the

spectrum of7Li [10]. The MCAS calculations were made without OPP. As a consequence,

there are two deeply bound spurious states which have to be neglected. This is not an

essential problem in single-channel studies as, by construct, they are orthogonal to the other

excited states. When one considers a coupled-channel problem, however, that is no longer

true and due care of the Pauli principle is needed to ensure that all determined states have no

spurious components [8]. Later we will incorporate a positive-parity interaction to analyze

scattering data. With that interaction, there are also spurious sub-threshold positive-parity

states in the evaluated7Li spectrum. There are three such states of spin-parity1

5

2

interaction we have determined. These are indeed spurious as there is no known positive

parity state in the spectrum below at least 11 MeV excitation [10].

We first discuss results obtained using an α-3H potential acting only in negative-parity

states. A standard Woods-Saxon form [3] was used with parameter values

2

+,3

2

+, and

+, having energies of −11.1, −6.8, and −9.0 MeV relative to the α+3H threshold for the

V0= −76.8 MeV

R0= 2.39 fm

Vll= 0.6 MeV

a = 0.68 fm

Vls= 1.7 MeV

Rc= 2.34 fm

(4)

Rcis the radius of a Coulomb sphere of charge. As the alpha-particle is treated solely as a

0+state in this model, VI·s= 0. For reference later, we define this (purely negative-parity)

interaction as Potential I. This was the form that we found best reproduced the known

energies of the four physical states of interest in7Li. Their values and resonance widths are

given in Table II. It is important to note that, with the dicluster model, there are no other

TABLE II: Spectral properties of7Li found using Potential I in the MCAS calculation of the α+3H

system. The energies (center of mass, MeV) are relative to the α-3H threshold. The widths, given

in brackets, are in keV.

Jπ

3

2

1

2

3

2

1

2

7

2

5

2

Exp.

spurious

spurious

−2.47

−1.99

2.18 (60)

4.13 (918)

Potential I

−29.4

−27.8

−2.47

−1.75

2.12 (83)

4.17 (834)

−

−

−

−

−

−

negative-parity states. But within the low-energy excitation range, a number of other states

are known experimentally [10].

1.

α scattering from3H

Cross sections at select center of mass scattering angles from the elastic scattering of

α-particles from a3H target for energies between 4 and 13.2 MeV have been measured and

6

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4

6

8101214

Elab(MeV)

0

100

200

300

θ = 54.7o

100

200

300

σel(E) (mb/sr)

θ = 125.3o

θ = 84.8o

100

200

300

FIG. 1: (Color online) Cross sections from3H(α,α)3H at the center of mass scattering angles listed.

Results differ by various amounts of positive-parity contributions as described in the text.

a phase-shift analysis made [19]. Three resonances were noted, with the phase-shift analysis

showing that the7

22

weaker5

2

were put at lab. energies of 5.2, 9.8, and ∼ 11.5 MeV which link to the known state values

in7Li at 4.65, 6.60 and 7.45 MeV excitation. The first two correspond to the 2.18 and 4.13

MeV states in respect to the α-3H threshold.

As there are no known positive-parity states in the spectrum of7Li, one can only rely upon

scattering data to assess the positive-parity α-3H interaction. In Fig. 1, a set of results are

given for three designated scattering angles at which data have been obtained [19]. Results

found using the Potential I interaction (no positive-parity interaction) are displayed by the

dot-dashed curves. The dashed curves depict results found with Potential I and assuming

that the negative- and positive-parity parameters are the same. The solid curves result when

that positive-parity central strength is reduced to −70 MeV, which interaction we identify

hereafter as Potential II.

At the scattering angle of 54.7◦the cross section found using Potential II reproduces

the data well at least to 10 MeV. It is interesting to observe that a decrease in strength of

the positive-parity interaction actually enhances the cross-section results to achieve a good

comparison with data. This is a signature of strong interference effects between partial-

wave contributions. Preference for Potential II is also confirmed by the results at the other

scattering angles.

−and first5

−states were built from the relative f-wave while the third, a

−, was built from the relative p-wave in the scattering. In sequence, their centroids

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TABLE III: Spectral properties found using the model Potential I in the MCAS method for the

7Be spectra from the α+3He system. The energies (center of mass, MeV) are relative to the α-3He

threshold. The widths are in keV.

Jπ

3

2

1

2

3

2

1

2

7

2

5

2

Exp.

spurious

spurious

−1.59

−1.16

2.98 (175)

5.14 (1200)

Potential I

−28.0

−26.4

−1.53

−0.84

3.07 (180)

5.09 (1194)

−

−

−

−

−

−

2.The case of α +3He and7Be

Assuming charge symmetry and changing only the Coulomb-force details, we have used

Potential I to describe the states of7Be. We have also changed the Coulomb radius to 2.39

fm. That change has minimal impact. Comparison with known state energies [10] is given

in Table III. The energies of three of the four low lying states of7Be are found within 100

4

6

810 1214

Elab(MeV)

0

100

200

300

θ = 54.7o

100

200

σel(E) (mb/sr)

θ = 90.0o

θ = 125.2o

100

200

FIG. 2: (Color online) Cross sections from α(3He,3He)α at the center of mass scattering angles

listed. The curves are predictions made using the Potential II as the inter-nuclear interaction.

keV of the experimental values and the fourth, the

1

2

−state, within 320 keV. Even better

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are the widths for the two resonance levels being found to within 6 keV. Then we used the

MCAS results with Potential II to form elastic scattering cross sections at three center of

mass scattering angles (54.7◦,90.0◦,125.2◦) at which data has been taken [19]. The results

of those calculations are shown by the solid curves in Fig. 2. The comparisons with data are

very good, adding confirmation to our definition of the basic dicluster-interaction potential.

III. RESULTS OF NUCLEON+MASS-6 NUCLEI CALCULATIONS

Now we return to the description of A = 7 nuclei in terms of a nucleon plus mass-6

systems. The ground state energies of the four nuclei of interest,7He,7Li,7Be, and7B

lie at 0.445, −9.975, −10.676, and 2.21 MeV with respect to the relevant nucleon-emission

thresholds.

The known spectrum of7Li [10] consists exclusively of negative-parity states so that

provides no information on the even-parity interactions. In fact, all mass-7 isobars have only

negative-parity entries in their measured spectra up to the excitation energy studied. Since

the MCAS method also can define scattering, it is hoped that low–energy6He ion scattering

from hydrogen may soon be measured experimentally, not only to ascertain resonances but

also as the average scattering cross section will be sensitive to the positive-parity interaction

in the p+6He system, just as we have found using our dicluster-model potential. It is due

to this lack of experimental information that we restrict consideration hereafter to just the

effects of the odd-parity interaction.

That all the states are of negative parity may be understood in the simplest shell picture as

being dominantly a capture of a 0p shell nucleon upon the target states. To form a positive-

parity state one has to have capture in the 1s-0d shells which requires 1?ω additional energy.

For these nuclei, that would be 10 MeV or greater. The 0s shell is taken to be fully occupied

and so no capture can be made into it. Thus, in the MCAS approach for the mass-7 isobars,

the 0s shell is to be explicitly Pauli-blocked. This is achieved by the OPP containing a term

that prevents further proton- and neutron-occupancies in the 0s configurations of the cores.

All results that we present here have been produced with the potential parameter set

given in Table I. The components of the potentials are identified in the first line of this

table, and their strengths, in MeV, are given in the second. The third line identifies the

geometry parameters whose values are listed in the last line. Of the eight parameters of

the table, the effective nucleon-nucleus radius R0and the corresponding charge radius Rc

has been set consistently with shell-model information of6He, corrected due to the size of

the (impinging) proton. The remaining 6 parameters have been varied as fit parameters to

reproduce 6 out of the 11 known states of7Li, 8 of which are stable with respect to proton

emission.

The fit procedure gave a deformation parameter β2= 0.7298. This is a large deformation,

more than twice that inferred by using the shell-model B(E2) to determine a collective model

value. The diffusivity is larger than typically found with scattering from stable nuclei, but

that may simply reflect the neutron halo character of6He. The spectrum found from the

MCAS evaluations of the p+6He system is compared in Table IV with the experimentally

known spectrum of7Li [10]. Therein we also give the results obtained for the mirror case,

7Be, which has been treated in the MCAS formalism as an n+6Be system. The energies

are in MeV but the widths, shown in brackets, are in keV. The numbers in square brackets

are the corresponding experimental widths with respect to the t+α channel for7Li, and

3He +α channel for7Be. Also the decay into the channels n-6Li and p-6Li are included,

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TABLE IV: Experimental data and theoretical results for7Li and7Be states (Energies are in MeV,

widths are in keV). All energies are defined with thresholds, p+6He = 9.975 MeV with respect to

7Li ground state and n+6Be = 10.676 MeV with respect to7Be ground state.

Jπ

7Li

7Be

Exp.

−9.975

−9.497

−5.323 [69]

−3.371 [918]

−2.251 [80]

−1.225 [4712]

−0.885 [2752]

−0.405 [437]

−−

1.265 (260)

Theory

−9.975

−9.497

−5.323

−3.371

−0.321

−2.244

−0.885

−0.405

−−

0.704 (56)

1.796 (1570)

2.981 (990)

3.046 (750)

5.964 (230)

6.76 (2240)

Exp.

−10.676

−10.246

−6.106 [175]

−3.946 [1200]

−3.466 [400]

−−

Theory

−11.046

−10.680

−6.409

−4.497

−1.597

−−

−2.116

−1.704

−3.346

−0.539

0.727 (699)

1.995 (231)

2.009 (203)

4.904 (150)

5.78 (1650)

3

2

1

2

7

2

5

2

5

2

3

2

1

2

7

2

3

2

3

2

1

2

3

2

5

2

5

2

7

2

−

−

−

−

−

−

−

−

−1.406 [?]

−0.776 [1800]

0.334 (320)

−

−

−

−

3.7 (800) ?a

4.7 (700) ?a

−

−

−

6.5 (6500) ?b

aFor these states spin and parity are unknown [10].

bSpin-parity of this state has been assigned as1

2

−[10].

respectively, when these channels are open. Above the relevant zero-energy thresholds the

experimental widths are indicated in round brackets and are compared with values calculated

with MCAS. As the theoretical widths refer only to the specific nucleon decay channels, they

differ somewhat in significance with respect to the experimental ones which include other

break-up contributions. Our evaluation produced 12 states to 15 MeV excitation in7Li.

The lowest 9 match states match known spin-parity states in the spectrum. while the next

three calculated levels are in the energy region in which two resonant states of undetermined

spin-parity are known. The matched states agree quite well in energies save for a crossing

of the3

22

error, calculated over the 8 states in7Li below the proton emission threshold, is

−|2with the5

−|2states. A measure of the quality of result is that the mean square

µ =1

N

??

(Eth− Eexp)2= 0.2728 MeV.

(5)

By including the resonances, assuming the assignments, the mean square error remains

good, namely µ = 0.2966 MeV. Also the agreement between widths for the resonances is

satisfactory given that no adjustments have been made to get a better fit to them. But the

experimental widths have various components. The ground state in7Li is stable and the

first excited decays only electromagnetically. The next two can decay also by emission of a

triton or an α particle as the threshold for that decay is 2.467 MeV above the7Li ground

state value. The next states in the spectrum can also decay by neutron emission as the

n+6Li threshold lies 7.25 MeV above7Li ground.

10

Page 11

00.20.4

β2

0.6

0.8

-10

-8

-6

-4

-2

0

2

4

E (MeV)

3_

1_

7_

5_

3_

7_

1_

5_

3_

1_

3_

5_

FIG. 3: (Color online) The calculated energy spectrum of7Li as β2varies.

To interpret the structure of the evaluated spectra, we follow the procedure used previ-

ously [7, 8] of allowing the deformation to decrease to zero. The variation of the spectrum

is shown in Fig. 3. Each state is identified on the right by twice its spin and its par-

ity. The dots show the degenerate spectrum values when β2 = VIs = 0. States cluster

as the deformation vanishes with the residual separation of groups due to the effect of the

(target)spin-(proton)spin interaction VIsbeing finite. Evidently, the individual states retain

an element of their zero deformation grouping for all of them track smoothly with β2and

do not cross any member of another group. Some states within each group do cross (inter-

change their order in the spectrum as β2decreases) when β2lies in the range 0.2 to 0.35.

Nevertheless, the mixing of basis states is evident from the rapid divergence of the members

of the two primary groups with increasing β2.

Setting that interaction strength VIs= 0 at zero deformation gives the spectrum of degen-

erate states shown in the leftmost column of Fig. 4. The states are identified by their value

of (2×spin)-parity. The mapping of our results against the experimental spectrum discussed

above is clear on comparing the central (theoretical) against the empirical (rightmost) spec-

tra. Comparing now with the zero deformation results, it is clear that the dominant term

in the ground state specification is the coupling of a 0p 3

target. Of course with deformation non-zero, the actual description is a linear combination

of all such basis3

2

coupling a 0p 3

unperturbed spectrum is 1.78 MeV. The next state with spin-parity1

quartet of states built primarily from the coupling of a 0p 3

−state which resolves to the unique coupling

of a 0p 1

MeV) in that limit lies 5.36 MeV above the ground state value and that is the spin-orbit

splitting of the single proton states in this model. Finally there is the doublet of states

formed by coupling a 0p 1

of these assignments comes from tracking the widths of resonances found as β2decreases.

2proton to the ground state of the

−states. The next four states in the spectrum dominantly are formed by

2proton to the first excited 2+state of the target as the energy gap in the

2

−is a member of the

2proton to the second excited 2+

state of the target. Then there is another

2proton with the ground state when the deformation vanishes. Its energy (0.73

1

2

2proton to the first excited 2+state of the target. Confirmation

11

Page 12

-10

-5

0

5

Unperturbed Theoretical Experimental

3_

1_, 3_, 5_, 7_

1_

3_, 5_

1_, 3_, 5_, 7_

3_

5_

3_

5_

3_

1_

1_

1_

7_

7_

1_

3_

5_

5_

7_

3_

5_

1_

3_

?

?

7_

E (MeV)

FIG. 4: (Color online) The calculated energy spectrum of7Li compared to the experimentally

known one. The left column is the zero deformation and VIs= 0 result. Spin-parities to the top

most experimental levels have not been assigned

For those states that are in, or move into, the continuum with decreasing β2, their centroids

and widths are given in Table V. In this, the subscript n designates the rank of the state

in the theoretical spectrum, while the letter ‘m’ in the brackets, which otherwise contains

the width, signifies a value less than 1 eV. Usually it is much less than that. The quartet

TABLE V: Centroids (MeV) and widths (keV) of resonance states in7Li as β2decreases.

β2for7Li

Jπ|n0.7

1

2

??2

7

2

??2

5

2

??2

3

2

??3

1

2

??3

3

2

??4

5

2

??3

0.6 0.5 0.40.30.2 0.1

0.18 (1)

1.21 (m)

0.80 (m)

0.65 (m)

0.0

0.49 (m)

1.33 (m)

0.80 (m)

0.59 (m)

0.23 (m)

0.41 (m)

0.69 (30)

0.53 (m)

0.65 (m)

0.69 (18)

0.81 (m)

0.82 (m)

0.68 (10)

1.04 (m)

0.89 (m)

0.67 (4)

0.11 (m)

0.70 (42)0.71 (54)

1.76 (1510) 1.63 (1303) 1.49 (1180) 1.32 (858) 1.15 (652) 0.97 (652) 0.80 (334) 0.72 (418)

2.96 (900)2.90 (848) 2.83 (780) 2.76 (680) 2.69 (652) 2.65 (600) 2.63 (580) 2.66 (610)

3.04 (740)2.99 (700) 2.92 (646) 2.84 (584) 2.75 (526) 2.66 (480) 2.59 (440) 2.53 (398)

of states, all having widths less than 1 eV, coincide with the zero deformation and VIs= 0

degenerate set having an energy of 0.96 MeV. They are built upon a single particle bound

in the continuum. The

2

degenerate value of 2.52 MeV, are built upon single particle resonances in the continuum

1

−state that tends to 0.724 MeV, and the doublet that tends to a

12

Page 13

TABLE VI: Experimental data and theoretical results for7He and7B states. All energies are in

MeV and relate to thresholds of −0.445 MeV for n+6He and of −2.21 MeV for p+6Be.

Jπ

7He

7B

Exp.Theory

0.43 (100)

1.70 (30)

2.79 (4100)

3.55 (200)

6.24 (1900)

Exp. Theory

2.10 (190)

3.01 (110)

5.40 (7200)

5.35 (340)

3

2

7

2

1

2

5

2

3

2

−

0.445 (150)

−−

1.0 (750) ?a

3.35 (1990)

6.24 (4000) ?b

2.21 (1400)

−

−

−

−

aObserved very recently and interpreted as a1

bSpin-parity of this state is unknown

2

−state.

whose inherent width is 580 keV.

The same nucleon-nucleus matrix of interactions, but with no Coulomb terms, was used

to evaluate the spectrum of the isospin mirror system n+6Be. Again in making comparison

with the experimental spectra for7Be, one must bear in mind the threshold energies of

reactions,3He + α of 1.586 MeV, p+6Li of 5.806 MeV, and n+6Be at 10.676 MeV. Our

predicted spectrum of7Be is compared with the known values also in Table IV and again

the round brackets around the calculated widths are solely for neutron emission. The five

lowest lying states in the known spectrum compare reasonably with the MCAS values. The

calculations give more states than are known to date above an excitation energy of ∼ 8.5

MeV in7Be, and there are a few crossings. But the result is a limited one in that it is

predicated upon charge symmetry and the simple collective model prescription. Studies

using other model prescriptions are in progress.

The spectrum of7He, so far as it is known experimentally, has three resonant states with

only the ground being quite narrow. There is a claim [20] of a fourth resonance

1 MeV above threshold which we include in Table VI. Our calculation puts it higher in

energy, as do shell-model calculations. Additionally we find a narrow

MeV excitation that has not been observed. In the MCAS calculations of the7He nucleus,

one must introduce a Pauli-hindrance effect on the 0p shells. This effect produces a ground

state that is unbound with respect to neutron emission. Specifically, one must invoke an

hindrance of both the 0p 3

of an exotic and non-compact structure (6He) being used as a basis in the channel coupling.

A similar discussion applies also for a proton coupled to6Be states. However, only the

ground state of7B is known and it is encouraging that the MCAS calculation has found

that resonance energy accurate to 5%. As shown in Tab. VI, we also predict three more

resonances, two of which have widths sufficiently narrow to be detected in experiments.

With the same nucleon-nucleus interaction, fixed by properties of7Li, and only modified

by a Coulomb field, we describe spectra of two unbound nuclei7He and7B. The calculations

of these differ from those for the isospin mirrors,7Li and7Be, in regard to the OPP terms.

While Pauli blocking of the 0s 1

ones have additional OPP terms responsible for Pauli hindrance in the 0p 3

The parameter λcwas set as 17.8 MeV for the 0p 3

considered, while for the 0p 1

1

2

−at ≃

7

2

−resonance at 1.7

2and 0p 1

2shells. With this system, such effects might be a reflection

2shell is common to all four nuclides, the two particle-unstable

2and 0p 1

2shells.

2shell in each of the three target states

2shell, λcwas set as 36.0 MeV for the 0+g.s., but as 5.8 MeV

13

Page 14

MCAS Experimental Shell model

3_

3_

1_

5_

1_

3_

1_

1_

7_

5_

7_

1_

3_

5_

5_

7_

5_

3_

?

7_

3_

3_

1_

7_

5_

1_

3_

7_

5_

3_

1_

5_

3_

0

5

10

15

?

3_

E (MeV)

FIG. 5: (Color online) The calculated energy spectra of7Li compared to the experimentally known

one. The left and right columns are the MCAS and the no-core-shell-model results respectively.

The known spectrum [10] is given in the middle column.

for the two excited 2+states. The same hindrance effects were used for both7He and7B.

Finally we compare the MCAS results for7Li with both the known spectrum and with that

determined from our no-core-shell-model calculations. Those spectra to 15 MeV excitation

are shown in Fig. 5 and each state identified by twice its spin and its parity. In both

calculated spectra there is a high lying triplet of states, the1

with a doublet of resonances of unknown spin-parity in the known spectrum [10]. All lower

excitation states have defined spin-parity and are paired to ones in both calculated spectra.

The shell-model results reproduce the known spectrum [10] quite well and that calculation

also found the lowest energy positive parity state to be a1

The sequencing of the states in the spectrum given by the shell-model calculation is also

very good with one minor cross over of the7

2

occurring with the

2

between states in the MCAS and shell-model spectra with the MCAS spectra (of states not

used in determination of the Vcc′(r)) being slightly compressed while the shell-model spectra

is slightly expanded in energies in comparison to the known states. This, we believe could

be evidence of the need for more collectivity in the shell-model description and a softening

of that given by MCAS.

2

−|3,3

2

−|4, and5

2

−|3that coincide

2

+state at 33.7 MeV excitation.

−|2and1

2

−|2states, and a larger cross over only

5

−|3at ∼ 15 MeV excitation. As noted there is a 1:1 correspondence

14

Page 15

IV.CONCLUSIONS

We have used a collective-model prescription of a three-state (0+ground, 2+

spectrum for the isospin-mirror nuclei,6He and6Be, in forming the coupling interactions

with an extra nucleon to yield the bound and resonant spectra of the mass-7 isobars. We

used only a quadrupole deformation but chose parameters consistent with the mass-6 tar-

gets having extended (halo) nucleon distributions. We have also used a dicluster-model

potential to assess states in7Li and7Be that can have strong coupling in the cluster-cluster

channels,3H+α and3He+α. The first four levels of the nuclei are well reproduced as are

the widths of the f-wave resonances. With this interaction, the low-energy elastic scatter-

ing cross sections of the clusters are also well reproduced. However, the dicluster model

gives no other state in the spectra, while a number of other states have been observed. In

contrast, a complete reproduction of the bound and resonance levels for all mass-7 isobars

was found from the coupled-channel solutions of the nucleon-mass-6 systems when a single,

fixed, nucleon-nucleus interaction was used. Specifically we found very good reproductions

of the spectra of the stable isobars,7Li (from p+6He) and7Be (from n+6Be). The other two

isobars,7He (from n+6He) and7B (from p+6Be) are, as known [10], particle unstable. The

MCAS predictions for their (resonant) ground states is consistent with the available data

once Pauli-hindrance in the 0p shells is invoked. Other yet to be discerned resonances are

predicted, suggesting a complex scenario of low-lying odd-parity resonances.

1, and 2+

2)

Acknowledgments

This research was supported by the Italian MIUR-PRIN Project “Struttura Nucleare e

Reazioni Nucleari” and by the Natural Sciences and Engineering Research Council (NSERC),

Canada. K. A. and D. v.d. K. gratefully acknowledge the support and hospitality of the

I.N.F.N. (section Padova) and the University of Padova during visits in which this research

was developed, as do L. C. and J. P. S. for that given by the University of Melbourne during

their visits.

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[2] S. Stepantsov et al., Phys. Lett. 542, 35 (2002).

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15