departmentofphysics,facultyofscience,niigatauniversity

arX

iv:0

809.

1291

v1 [

nucl

-th]

8 S

ep 2

008

1

Di-neutron correlation in soft octupole excitations of neutron-richNi isotopes beyond N = 50

Yasuyoshi Serizawa1 ∗) and Masayuki Matsuo2 ∗∗)

1Graduate School of Science and Technology, Niigata University, Niigata 950-2181,

Japan2Department of Physics, Faculty of Science, Niigata University, Niigata 950-2181,

Japan

We investigate low-lying octupole response of neutron-rich Ni isotopes beyond the N=50shell closure using the Skyrme-Hartree-Fock-Bogoliubov mean-fields and the continuumquasi-particle random phase approximation. Performing detailed numerical analyses em-ploying the Skyrme parameter set SLy4 and a density-dependent delta interaction of themixed type, we show that a neutron mode emerges above the neutron separation energyas a consequence of the weak binding of neutrons and it exhibits strong influences of thedi-neutron correlation.

§1. Introduction

The di-neutron correlation, a spatial pair correlation with small correlationlength among neutrons, has been one of the central themes of the physics of two-neutron halo nuclei such as 11Li.1)–7) Although an affirmative experimental signatureof the neutron spatial correlation in 11Li is obtained rather recently,8) many theo-retical predictions have been accumulated, concerning not only two-neutron halonuclei1)–3), 9)–14) but also other neutron-rich systems including the surface area ofmedium- and heavy- mass neutron rich nuclei15), 16) and dilute neutron matter.17)–19)

Indeed the analysis of the spatial structure of the neutron Cooper pair in dilutematter17)–19) has revealed a mechanism that the di-neutron correlation may emergegenerically. Namely the spatial di-neutron correlation originates from a strong cou-pling feature of the neutron pair correlation, which becomes significant at low den-sities because of the strong momentum dependence of the attractive nuclear force in1S channel. Furthermore the induced pairing interaction caused by the exchange ofsurface phonons is claimed to give additional contribution to possible enhancementof the spatial correlation in finite nuclei.20)

The influence of the di-neutron correlation on nuclear structure could be mul-tifold. Namely it may influence not only the ground state but also various modesof excitations and dynamics. Attentions, however, have been focused so far mostlyon soft dipole excitation in two-neutron halo nuclei,1)–8), 12) and rather recently, inmedium- and heavy-mass neutron rich nuclei.15), 18) Concerning the latter case, wehave shown in our previous study15) that strong influence of the di-neutron corre-lation in the soft dipole excitation of proton semi-magic neutron-rich Ca and Ninuclei. Considering the possible generality of the di-neutron correlation, it is ex-

∗) E-mail: [email protected]∗∗) E-mail: [email protected]

typeset using PTPTEX.cls 〈Ver.0.9〉

http://arxiv.org/abs/0809.1291v1

2 Y. Serizawa and M. Matsuo

pected that this correlation may also emerge in other multipole modes of excitation.We anticipate it in multipole excitations with negative and natural parity becausethe di-neutron correlation involves a configuration mixing among single-particle or-bits with different parities. In the present paper, therefore, we examine low-lyingoctupole modes of excitation.

There exists only a few investigations of low-lying octupole excitations in neutron-rich nuclei while in stable nuclei the low-lying 3− state with a character of octupolesurface vibration is well established.21), 22) Experimental properties of the low-lyingoctupole modes in neutron-rich nuclei are known very little, with exceptions such as20O.23) On the theoretical side, it is predicted24)–26) that neutron halo nuclei exhibita neutron mode with large octupole strength in the continuum region above the neu-tron threshold, resulting from transitions from weakly bound orbits to continuumorbits, and thus having a completely different character with the surface vibration.Analyses based on the random phase approximation (RPA),27)–30) which can de-scribe both the continuum states and the collectivity, have shown coexistence of thecollective surface vibration and neutron continuum strength near the threshold inmedium mass neutron rich nuclei. Note however that models of Refs.27)–30) do notinclude the pair correlations, and analyses are limited to nuclei with doubly closedshell configurations, such as 60Ca, 28O and 68,78Ni. The quasi-particle random phaseapproximation including the pair correlation effect is employed in Ref.23) to describeoxygen isotopes 18−24O. There, however, the Hartree-Fock+BCS approximation isadopted instead of the Hartree-Fock-Bogoliubov scheme, and effects of the pair cor-relation are not discussed intensively. In contrast to these preceding works, we wouldlike to focus in the present paper on roles of the pair correlation and the possibilityof the di-neutron correlation especially in the low-lying octupole correlation. Forthis purpose we use the continuum quasi-particle random phase approximation (thecontinuum QRPA)15), 18), 31) which can describe both the continuum, the pairing,and the collectivities in neutron rich medium-mass nuclei. We choose nickel isotopes80−86Ni for numerical analysis, and perform detailed investigation of 84Ni which ischosen as an representative example. A preliminary report of this work is seen inRef.32)

§2. Skyrme-HFB plus continuum QRPA method

In the present study we assume the spherical symmetry of the ground state aswe analyze semi-magic nickel isotopes. We first construct the spherically symmetricground state and associated self-consistent mean-fields by means of the coordinate-space Skyrme-Hartree-Fock-Bogoliubov (Skyrme-HFB) method,33), 34) with which wecan describe properly the spatially extended wave functions of weakly bound quasi-particle orbits. We employ the Skyrme-force parameter set SLy4,35) which has beenextensively used for studies of the neutron-rich nuclei.36)–43) We shall also use an-other parameter set SkM∗43), 44) for comparison. In our previous study of the dipoleexcitation15) a Woods-Saxon potential was adopted, but here we perform fully self-consistent HFB calculations. Concerning the effective pair interaction, we use the

Di-neutron correlation in soft octupole excitations 3

density-dependent delta-type interaction (DDDI)2), 36), 43) which is given by

vpair(r, r′) =

1

2V0(1− Pσ)

(

1− ηρ(r)

ρ0

)

δ(r − r′). (2.1)

We choose the so-called mixed type pairing,38) i.e. η = 1/2, ρ0 = 0.16 fm−3, concern-ing the density dependence, and we determine the strength of the pair interactionV0 in the same way adopted in Ref.15) Namely we determined V0 = −315MeV fm3

for nickel isotopes so that the average pairing gap for neutrons is in overall agree-ment with the experimental odd-even mass difference evaluated with the three-pointformula45) (cf. Fig.1).

We describe excitation modes by means of the continuum QRPA. The contin-uum QRPA formalism is essentially the same as those in Refs.15), 31) except thatwe here formulate it on the basis of the Skyrme-Hartree-Fock-Bogoliubov mean-fields and different residual interactions. Concerning the residual interaction inthe pairing channel, we adopt the one derived from the functional derivatives ofthe pairing energy functional, as was in our previous study.15) Concerning theresidual interaction in the particle-hole channel, we use the same Skyrme inter-action employed in constructing the ground state, but we take a Landau-Migdalapproximation to the Skyrme interaction46)–48) in order to make the systematicnumerical calculations feasible. In practice, the particle-hole interaction is givenby vph(r − r

′) = {(F0/N0)(r) + (F ′0/N0)(r)τ · τ ′)} δ(r − r

′) where F0, F′0 and N0

are the Landau-Migdal parameters49)–51) and the associated normalization factor,evaluated for the Skyrme interaction. Their expressions are given, for example, inRefs.52), 53) The Landau-Migdal parameters are defined usually for symmetric nu-clear matter, but we treat the Fermi momentum kF in the expressions as a local

quantity kF (r) =(

3π2ρ(r)/2)1/3

related to the nucleon density ρ(r).53) Thus theforce strength is density dependent, and hence depends on the spatial coordinate r.

Given the HFB mean-fields and the residual interactions, we solve the linearresponse equation

δρqL(r, ω)δρ+,qL(r, ω)δρ−,qL(r, ω)

=

∫

0dr′

Rαβ0,qL(r, r

′, ω)

∑

q′ κqq′

ph (r′)δρq′L(r

′, ω)/r′2 + vextqL (r′)

κpair(r′)δρ+,qL(r

′, ω)/r′2

−κpair(r′)δρ−,qL(r

′, ω)/r′2

.

(2.2)to describe the correlated multipole response of a nucleus against an external field.Here δρqL(r, ω), δρ+,qL(r, ω), and δρ−,qL(r, ω) are responses in the normal and abnor-mal densities with multi-polarity L, and κph and κpair are related to the interactionstrengths (F0/N0)(r), (F

′0/N0)(r) and V0(1− ηρ(r)/ρ0). Note that we construct the

response function Rαβ0,qL(r, r

′, ω) in terms of products of two single-quasi-particle HFBGreen’s function summed over not only the discrete quasi-particle states but also thecontinuum quasi-particle states.15), 31) This is possible because we use the exact HFBGreen’s function55) consisting of the regular and out-going wave solutions of the HFB


equation. We consider the external field

V ext(r) =∑

i

r3i Y30(ri) and∑

i

1± τ3,i2

r3i Y30(ri), (2.3)

for the isoscalar, neutron and proton strength functions dB(λL)/dE =∑

iB(λL; 0gs →3−i )δ(E−Ei) (λL =IS3, n3 and p3, respectively), which can be evaluated as dB(λL)/dE =− 7

π Im∫∑

q drvext,q(r)δρqL(r, ω) using the solution δρqL(r, ω) of Eq.(2.2).Numerical details are as follows. The HFB equation is solved using the radial

coordinate in a spherical box r = 0 − rmax. The Skyrme-HFB code is our orig-inal one, whose numerical procedures essentially follow those of Ref.33) We solvethe radial HFB equation using the Runge-Kutta method instead of the Numerovmethod adopted in Refs.33), 54) since we need a consistent evaluation of derivativesof the quasi-particle wave functions in constructing the Green’s function used in thecontinuum QRPA calculation. We have checked that results of our Skyrme-HFBcode agree with those produced by the code HFBRAD.54) The continuum QRPApart is based on our previous version15) employing the Woods-Saxon potential, buthere we replace the Woods-Saxon potential by the Skyrme-HFB mean-fields. Alsowe implement the Landau-Migdal approximation of the Skyrme interaction as theresidual interaction in the particle-hole channel. As the box size we choose rmax =22 fm, and an equidistant discretization with ∆r = 0.2 fm. Concerning the cut-offof the quasi-particle orbits, we set lmax = 17~ for the single-particle partial waves lj,and Emax = 60 MeV with respect to the quasi-particle energies. The cut-off energyis a standard choice adopted in many Skyrme HFB calculations,36), 37), 39), 42), 43) butslightly larger than that adopted in our previous Woods-Saxon calculation.15) Con-cerning the orbital angular momentum cut-off, we find that the ground state andthe octupole strength function have good convergence already at l = 12~, but toevaluate the transition densities we need larger l’s. Because we adopt the Landau-Migdal approximation, the consistency between the ground state and the excitedstates is partly broken. In order to minimize effects of the self-consistency breaking,we renormalize the strength of the particle-hole residual interaction as vph → f ×vphby a numerical factor f so that the spurious center-of-mass mode in the isoscalardipole excitation has zero excitation energy.

§3. Numerical analysis

3.1. Octupole strength functions in 84Ni

We shall first discuss a representative example 84Ni in order to clarify basicfeatures of low-lying octupole excitation.

In Fig. 2, we show the isoscalar octupole strength in 84Ni. Here the strengthfunction is calculated with smoothing parameter ǫ = 0.2 MeV, which is introducedin the linear response equation as the imaginary part of the excitation frequencyω + iǫ. This means that the strength function is folded with an Lorentzian functionwith FWHM = 2ǫ = 0.4 MeV. It is seen that there are essentially two groups ofstrength distribution; one around E = 26 - 32 MeV and the low-lying distributions


below E ∼ 10 MeV. We regard the high energy group as the 3-~ω high-frequencyvibrational mode.21) In the low-energy group, the sharp peak at E ≈ 4.15 MeVis most prominent, but another structure is also seen. We focus on the low-energygroup in the following discussion.

In Fig. 3 we show magnification of the isoscalar strength function in a low-lyingregion E = 0−8 MeV, together with the neutron and proton octupole strength func-tions. The smoothing parameter is chosen to a smaller value ǫ = 0.05 MeV in thiscase (and also in most of the following calculations unless mentioned explicitly). It isnow clear in Fig. 3 that the low-lying strength consists of two structures. Apart fromthe sharp peak at E = 4.16 MeV, there exists a broad bump which emerges abovethe one-neutron separation energy (the one- and two-neuron separation energies areE1n,2n = 1.86 and 2.37 MeV as indicated with arrows). Although the two structuresoverlap in the same energy region, their characters are clearly distinguished as fol-lows. Firstly, the broad bump does not form a well defined peak, and we considerit as a kind of continuum mode where fast neutron decay takes place. It carriesessentially only neutron strength. Secondly, the sharp peak has a small width eventhough it is located above the neutron separation energy. It can be regarded as anarrow resonance. It is also distinguished from the neutron mode by the fact that itcarries sizable proton strength.

In order to give more characterizations, we examined how the residual interac-tions influence these modes. Namely we performed three calculations where either orboth of the particle-hole and pairing residual interactions are switched off. Resultsare shown in Fig. 4. The sharp peak disappears when the particle-hole interactionis neglected, and it is influenced rather weakly by the pairing residual interaction.Thus the main origin of the sharp-peak mode is correlation due to the particle-holeresidual interaction. It suggests that this mode could be the surface vibrational modeconsisting of low-energy 1-~ω particle-hole transitions.21) On the contrary, the broadneutron mode exists even when the particle-hole residual interaction is neglected. Ithas different origin.

It is useful to evaluate integrated sums of octupole strengths associated with thetwo modes. It is, however, not easy to evaluate them separately since the sharp-peakmode and the broad neutron mode overlap in the same energy region. Neverthelesswe estimate them in the following manner. Concerning the broad neutron modewe integrate the strength functions in an energy region above the one-neutron sep-aration energy E1n < E < E1n + 1.5 MeV with an interval of 1.5 MeV, where theneutron strength dominates and the collective vibrational mode barely overlaps. Theboundary energies are E1n = 1.86 and E1n + 1.5 = 3.36 MeV in the case of 84Ni.Since the choice of the energy interval 1.5 MeV is rather arbitrary and probablysmall to cover the whole strength of the broad neutron mode, we expect that theremay be an underestimate by a factor of up to about 2. Concerning the strength ofthe surface vibrational mode, we define an energy interval E1 < E < E2 where thestrengths are integrated by noticing that this mode carries a proton strength, whichis a character clearly distinguishable from the broad neutron mode. In practice, wedefine the boundaries E1 and E2 at which the proton strength function is 1.0% ofthe value at the peak energy Epeak. In 84Ni, E1 and E2 are 3.50 and 5.00 MeV,


respectively, and there is no overlap between the two energy intervals. In Table I, welist the integrated isoscalar octupole strengths of the broad neutron mode and thesurface vibrational mode in 80−86Ni. The obtained isoscalar octupole strength for thebroad neutron mode is 5.85 × 104 fm6 in 84Ni. Note that the energy-weighted sumof this strength is 1.7 percent of the classical isoscalar octupole sum-rule value21), 56)

SEWcl = 147

4π~2

2m

(

N⟨

r4⟩

n+ Z

⟨

r4⟩

p

)

. The isoscalar strength of the surface vibrational

mode is 6.74×105 fm6 in the same nucleus, and it exhausts 29 percent of the classicalsum rule value. The strength of the broad neutron mode is smaller than that of thesurface vibrational mode by about a factor of ten.

3.2. Transition densities of low-lying octupole modes in 84Ni

The natures of the sharp-peak mode and the broad neutron mode become moreevident by looking into transition densities. We here analyze three kinds of transitiondensities

ρtriq(r) = 〈Φi|∑

σ

ψ†q(rσ)ψq(rσ) |Φ0〉 = Y ∗

LM (r)ρtriqL(r), (3.1)

P ppiq (r) = 〈Φi|ψ

†q(r ↑)ψ†

q(r ↓) |Φ0〉 = Y ∗LM (r)P pp

iqL(r), (3.2)

P hhiq (r) = 〈Φi|ψq(r ↓)ψq(r ↑) |Φ0〉 = Y ∗

LM (r)P hhiqL(r), (3.3)

where ρtriq(r) is the usual particle-hole transition density while P ppiq (r) and P hh

iq (r)are the transition densities for particle-pair and hole-pair, respectively, for eitherneutrons or protons (q = n, p). They are evaluated as

ρtriqL(r) = −C

πr2ImδρqL(r, ωi), (3.4)

P ppiqL(r) =

C

2πr2Im(δρ+,qL(r, ωi)− δρ−,qL(r, ωi)), (3.5)

P hhiqL(r) =

C

2πr2Im(δρ+,qL(r, ωi) + δρ−,qL(r, ωi)) (3.6)

in terms of the solutions of the linear response equation (2.2). Here C is a normaliza-tion constant, which is fixed so that the transition amplitudeMiqL =

∫

vextqL (r)ρtriqL(r)r2dr

gives the integrated isoscalar octupole strength of a mode under consideration bythe standard definition B(IS3) = 7|

∑

qMiqL|2.

In Fig. 5, we show the transition densities associated with the surface vibrationalmode. We evaluate them at E = 4.15 MeV, approximately at the peak energy. Itis seen in Fig. 5 that the particle-hole transition densities ρtriqL(r) of both neutronsand protons exhibit large and in-phase amplitudes at around the nuclear surface(the calculated matter r.m.s. radius of 84Ni is 4.28 fm). We thus confirm that themode is typical of the surface vibration in which neutrons and protons give coherentcontribution. It is seen also that the amplitude of the particle-hole transition den-sity ρtriqL(r) is significantly larger than those of the particle-pair transition density

P ppiqL(r) and the hole-pair transition density P hh

iqL(r) both for neutrons and protons.

(P ppiqL(r) = P hh

iqL(r) = 0 for protons is a trivial consequence of the zero proton pairing


gap ∆p = 0.) The dominance of the particle-hole amplitude is consistent with theobservation that the mode is generated by the particle-hole residual interaction (cf.Sec.3.1).

Figure 6 is the transition densities of the broad neutron mode, evaluated at E =3.0 MeV. They have characters different from those of the surface vibrational mode.We observe here two distinct features. Firstly, the particle-pair transition densityP ppiqL(r) of neutrons is significantly larger than the neutron particle-hole transition

density ρtriqL(r) in the nuclear exterior region. The ratio between the two transitiondensities is approximately a factor of 3 at r = 10 fm, and a factor of ∼ 10 atr = 15 fm. This indicates that the mode is characterized, especially in the externalregion, by motion of neutron pairs rather than by particle-hole excitations. In otherwords, the neutron pair correlation is the main character of this mode. Secondly,both the particle-hole and particle-pair transition densities of neutrons exhibit a verylong tail extending to r ∼> 20 fm, especially in the particle-pair transition density.This indicates that neutrons in weakly bound and continuum orbits participate informing this mode. In addition to these two features, we observe also that the protonamplitude of the particle-hole transition density is considerably smaller than that ofneutrons. This is in accordance with the dominance of the neutron strength alreadyobserved in the previous subsection.

We would like to emphasize that the residual pairing interaction playing rolein the QRPA equation brings the correlation to this mode, and that the pairingmean-field alone is not sufficient. This is seen in the difference between the solid anddashed lines in Fig. 7, where the dynamical pairing effect, i.e., the RPA correlationdue to the residual pairing interaction, is either included or neglected. It is seen thatthe dynamical pairing effect enhances the particle-pair transition density by a factorof ∼ 2. This indicates significant configuration mixing effect originating from theresidual pairing interaction. The important role of the pair correlation is seen evenwhen the particle-hole residual interaction is neglected (the dotted curve in Fig. 7).

The large dynamical pairing effect is analyzed in more details. In Fig. 8 weshow how the transition densities of the neutron mode changes if we include onlya part of neutron single-particle orbits with lower orbital angular momenta l, i.e.,including only orbits up to a smaller cut-off lcut on l. This is the same analysisthat we performed for the soft dipole mode.15) It is seen that the convergence ofthe particle-pair transition density P pp

iqL(r) of neutrons with respect to l is slow, andorbits with large l contribute coherently to produce the dynamical pairing effect.There is sizable contribution even around l ∼ lmax = 17. Note that the precisetreatment of the continuum states, guaranteed in the continuum QRPA approach, isessential to describe the correlation since high-l orbits with l > 4 are all continuumstates.

Based on the above two features, i.e., the large effect of the dynamical pairingand the high-l contribution, we can argue that a neutron pair moving in the neutronmode exhibits a spatial correlation at small relative distance between the pairedneutrons. The configuration mixing involving high-l orbits to a certain value, saylcorr, means a two-particle correlation with a small opening angle θcorr ∼ 1/lcorr, and


hence a large lcorr means a spatial correlation at small relative distance. These arethe same features seen in the case of the soft dipole excitation,15) and we concludethat the di-neutron correlation appears also in the neutron mode in the octupoleresponse.

80Ni 82Ni 84Ni 86Ni

neutron mode

B(IS3) [104fm6] 1.61 4.23 5.85 16.11

SEW(IS3)/SEWcl [%] 0.7 1.6 1.7 3.4

surface vibrational mode

Epeak 5.56 4.78 4.16 3.84

B(IS3) [105fm6] 3.57 4.97 6.74 10.02

B(E3) [e2104fm6] 2.77 3.27 3.87 4.53

SEW(IS3)/SEWcl [%] 25.0 26.9 28.6 33.0

Mn/Mp 2.63 2.95 3.27 3.90

sum rule and separation energies

SEWcl [106fm6MeV] 7.92 8.78 9.77 11.36

E1n [MeV] 2.79 2.54 1.86 1.32

E2n [MeV] 3.84 3.26 2.37 1.37

Table I. The isoscalar and electric octupole strengths B(IS3) and B(E3) of the neutron mode and

the surface vibrational mode. The energy weighted sum SEW(IS3) of the isoscalar octupole

strength in unit of the classical sum-rule value SEWcl . is also listed. The neutron vs proton ratio

Mn/Mp of the transition amplitudes as well as the peak energy is also shown for the surface

vibrational mode. The values of SEWcl as well as the one- and two-neutron separation energies

S1n and S2n are also tabulated.

3.3. Isotopic dependence of the neutron mode

The strength of the neutron mode increases significantly as the system ap-proaches toward the neutron drip line. Figure 9 shows the strength functions cal-culated for 80,82,84,86Ni. The calculation is the same as that in §3.1. The calculatedone-neutron separation energy in these isotopes are E1n = 2.79, 2.54, 1.86, and 1.32MeV in the corresponding order, and shown in Fig.9 with an arrow. It is seen inall the nuclei that there exists above E1n a broad distribution of predominant neu-tron strength, which corresponds to the neutron mode. Clearly the magnitude ofthe strength increases monotonically with increasing N . This is also seen in theintegrated isoscalar strength B(IS3) of this mode, listed in Table I.

The transition densities of the neutron mode, evaluated at E = 4.0, 3.5, 3.0,and 2.5 MeV for 80,82,84,86Ni, respectively, are shown in Fig. 10. In the top panels wesee that the tail of the particle-hole transition density grows as the neutron numberincreases. It is known that such a long tail can be realized for particle-hole transitionsfrom weakly bound orbits, whose wave functions have long tail, to continuum orbitswith small kinetic energies. The long tail enhances the octupole strength as theoctupole operator has the radial form factor ∝ r3 giving a heavy weight at larger


distances. We thus see that the increase of strength with increasing N originatesfrom the effect of weak binding of neutrons.

Looking at the particle-pair transition densities shown in the bottom panels inFig. 10, we observe that the particle-pair transition density varies with N moredrastically than the particle-hole transition density. In 86Ni and also in 84Ni thistransition density does not show exponential decay in the outside of the nucleusr ≥ 10 fm, but it rather shows an oscillatory behaviour with its maximum amplitudeat around r ∼ 15 fm far outside the nucleus. This suggests significant emission of a

neutron-pair from the nucleus. This is of course related to the feature that the two-neutron separation energy E2n decreases from E2n = 3.84 MeV in 80Ni to a smallvalue 1.37 MeV in 86Ni. The di-neutron correlation in the neutron mode becomesmore significant as we approach the neutron drip-line.

3.4. Model dependence

Let us now investigate how our predictions on the neutron mode depends on themodel parameters. For this purpose, we shall compare the above results with thoseobtained with another Skyrme parameter set SkM∗,44) and those with a model adopt-ing a Woods-Saxon potential15) instead of the Skyrme-HFB self-consistent mean-fields. When calculating the octupole response for these models, we use the samemixed-type DDDI, but the interaction parameter V0 is adjusted separately to re-produce the average pairing gap of neutrons in stable nuclei as in the SLy4 case:V0 = −265 and −280MeV fm−3 for SkM∗ and WS, respectively. The obtained aver-age neutron pairing gap in 84Ni is 1.349, 0.797 and 0.569 MeV for the Woods-Saxon,SLy4 and SkM∗ models, respectively.

The octupole strength function in 84Ni obtained for the SkM∗ and Woods-Saxonmodels are shown and compared with that for SLy4 in Fig.11. It is seen that theneutron mode depends rather sensitively on the models. The Woods-Saxon modelproduces a significantly larger strength than SLy4 while it is smallest for SkM∗

among the three models. Figure 12 is a comparison of the transition densities of thesoft neutron mode of the Woods-Saxon model and that of the SLy4, both evaluatedat E = 3.0 MeV. (The transition densities for SkM∗ is not shown here as the strengthof the neutron mode itself is weak.) The Woods-Saxon model exhibits enhanced am-plitudes of the transition densities, especially of the particle-pair transition density,compared with the SLy4 model.

We can relate the model dependence to differences in the single-particle lev-els and the Fermi energy of neutrons. The neutron Fermi energy in 84Ni is λn =−0.722,−1.183 and −2.338 MeV for the Woods-Saxon, SLy4 and SkM∗ models,respectively. The neutron single-particle levels are shown in Fig.13. Clearly thelast neutrons are less (more) bound in the case of the Woods-Saxon model (theSkM∗ model). Thus the model dependence of the low-lying octupole strength canbe explained in terms of the weak-binding effect which increases the strength of theneutron mode.

We can speculate that the neutron mode depends also on the effective pair in-teraction. Let us examine explicitly dependence on the effective pair interaction.For this purpose we compare results obtained with the mixed type DDDI with an-


other calculation using the density-independent delta interaction, which is definedby Eq.(2.1) but with η = 0. (The same SLy4 is used as the Skyrme parameter set.)The strength V0 = −215MeV fm−3 is adjusted in the same way as for the mixedtype DDDI, i.e. to reproduce the average neutron paring gap in stable Ni isotopes.The density independent delta interaction is often called the volume pairing. Thevolume pairing produces average neutron pairing gap of 0.467 MeV in 84Ni, slightlysmaller than that in the mix-type DDDI. As far as the octupole strength function isconcerned, there is no significant dependence on the type of the pairing interactionas we see from the comparison in Fig.11. There is, however, large difference in theparticle-pair transition density P pp

iqL(r) of neutrons, as shown in Fig. 14. The am-

plitude of the particle-pair transition density P ppiqL(r) in the volume pairing case is

smaller by a factor of ∼ 2 than that of the mixed-type DDDI. It is seen also thatthe dynamical paring effect is significantly smaller in the volume pairing case. Theseresults indicate that the di-neutron correlation in the neutron mode is sensitive tothe choice of the effective paring interaction: effective pairing interactions such asthe mixed-type DDDI that has stronger interaction strength at low densities givestronger di-neutron correlation in the neutron mode. This feature is in agreementwith our previous finding for the soft dipole excitation.15)

3.5. Comparison with soft dipole excitation

The analyses in the preceding subsections revealed that the weak binding effectand the di-neutron correlation in the octupole neutron mode are similar to those seenin the soft dipole excitations.15) In this subsection let us make a more explicit andquantitative comparison between the dipole and octupole cases. For this purposewe calculate the electric dipole response using the same Skyrme-HFB + continuumQRPA model. (The calculations in our previous work15) is based on the Woods-Saxon model, and not suitable for direct comparison with the present calculations.)

The calculated electric dipole strength function dB(E1)/dE is shown in Fig.15.In the same energy region as the octupole neutron mode (i.e. just above the one-neutron separation energy), there emerges a broad distribution of the E1 strengthcorresponding to the soft dipole excitation. It is seen also that the monotonic increaseof the strength with N is in parallel with that of the octupole neutron mode. If weevaluate the B(E1) strength integrating within an energy interval [E1n, E1n+5MeV]above the one-neutron separation energy (as done in Ref.15)), the obtained E1strength is B(E1; 0gs → 1−) = 0.83, 1.59, 2.23, 3.33 e2fm2 for 80,82,84,86Ni, respec-tively. The energy weighted sum in the same energy interval is 1.9, 3.3, 4.0 4.8%of the classical (TRK) sum rule value for 80,82,84,86Ni. The E1 strength and theenergy weighted sum in a smaller energy interval [E1n, E1n+1.5MeV] are 0.11, 0.24,0.35, 0.87 e2fm2 and 0.15, 0.30, 0.36 0.68% for the same isotopes. If we comparethe strengths of the octupole neutron mode and the soft dipole mode in terms ofthe fraction of the energy weighted strength to the classical sum rule, 1.7% for theoctupole neutron mode and 4.0% for the soft dipole in 84Ni (0.4% if the same energyinterval [E1n, E1n + 1.5MeV] is used) are comparable.

In Fig. 16, we show the transition densities of the soft dipole mode. The tran-sition densities are evaluated at the same excitation energy E = 3.0 MeV and in


the same energy interval [E1n, E1n + 1.5 MeV] as is done for the octupole neutronmode. Comparing the transition densities of the octupole (Fig.8) and dipole (Fig.16)modes, both are similar in that the particle-pair transition density P pp

iqL(r) is mostdominant, and also that the dynamical pairing effect and coherent contributions ofthe high-l orbits play significant role to enhance the particle-pair transition density.If we compare the absolute magnitudes of the transition densities, it is seen that themagnitude of the particle-pair and particle-hole transition densities of the octupoleneutron mode is smaller than those of the soft dipole mode by a factor of ∼ 2 ataround r = 10 fm where the particle-pair amplitude is largest. Apart from thisdifference, the significance of the di-neutron correlation is comparable in both cases.

The most noticeable difference between the octupole and dipole response is thatin the octupole response the neutron mode overlaps with the surface vibrationalmode present in the same energy region, and the strength is overwhelmed by thelatter. In contrast, the soft dipole excitation in the dipole response is well separatedfrom the other mode of excitation, the giant dipole resonance. This may make itmore difficult to identify experimentally the octupole neutron mode.

3.6. Surface vibrational mode

Let us mention briefly the octupole surface vibrational mode, corresponding tothe sharp peaks around E ≈ 5.6 − 3.8 MeV in 80−86Ni. The peak energies as wellas the isoscalar and electric octupole strengths associated with the surface vibra-tional mode are listed in Table I. The most noticeable feature is that the isoscalarstrength increases steeply with increasing N from 80 to 86 approximately by a factorof three. We also see that the neutron vs. proton ratio Mn/Mp of the transitionamplitudes increases more steeply than the the nominal ratio N/Z. The doubleratio (Mn/Mp)/(N/Z) increases from 1.41 to 1.88 with increasing N . These indi-cate that the enhanced collectivity of the surface vibrational mode is due mainlyto neutron contributions. Note that the ratio of the energy weighted sum of theisoscalar strength to the classical sum rule value SEW

cl stays constant around 25-33%. This is partly because the increase of the strength is compensated by the decreaseof the excitation energy (see Table I), and partly because the sum rule value SEW

clitself increases with N due to increasing radial expectation value

⟨

r4⟩

nof neutrons.

This suggests that the increase of the isoscalar and neutron strength of the surfacevibrational mode is regarded as a kind of softening caused by the weak binding ofneutrons.

The enhanced collectivity of the octupole vibrational mode in neutron-rich nucleiclose to the drip-line is pointed out in Ref.,27) which however analyzed a doubly-closed-shell nucleus 60Ca using the Skyrme-HF plus continuum RPA without thepair correlation. Our results suggest that the enhanced collectivity is generally seenin nuclei near the neutron drip line. A large deviation of the Mn/Mp ratio from thenominal ratio N/Z is pointed out in a Skyrme-BCS+ QRPA calculation23) for theneutron-rich oxygen isotope 24O, but with much smaller deviation in less neutron-rich isotopes 18−22O. We refer also to Ref.,57) which discuss enhanced collectivitydue to the weak binding effects in the case of the low-lying quadrupole vibrationalmode.


§4. Conclusions

We have investigated the low-lying octupole excitations of the neutron-rich Niisotopes beyond the N = 50 shell closure using the continuum QRPA based on theSkyrme Hartree-Fock-Bogoliubov mean-fields. In addition to the surface vibrationalmode of the 1-~ω character, a broad strength distribution of predominantly neutroncomponent appears just above the neutron separation energy. This broad neutronmode exhibits the following distinctive features. (i) The transition amplitude forneutron pair density is significantly larger than that of the usual particle-hole transi-tion density of neutrons. The neutron pair correlation is therefore the most essentialaspect characterizing the neutron mode. (ii) The large neutron-pair transition den-sity originates from a coherent contribution of the neutron high-l orbits, which pointsto the presence of the spatial correlation among neutrons involved in this mode. (iii)The transition densities of the neutron mode display a long tail extending in theoutside of the nucleus. The strength of this mode increases monotonically with in-creasing N from 52 (80Ni) to 54, 56 · · · . Both indicate that the mode originatesfrom the weak binding of neutrons. From these features we conclude that the spatialdi-neutron correlation shows up also in the neutron mode in nuclei near the neu-tron drip-line, similarly to the soft dipole mode. This supports our expectation thatsoft modes having the di-neutron character may emerge generically in medium massneutron-rich nuclei near the drip-line. It is interesting to investigate this generalityin more details, e.g. by looking into modes with other multipolarities. Such ananalysis is in progress, and will be reported elsewhere.

§5. Acknowledgments

This work was supported by the Grant-in-Aid for Scientific Research (Nos.17540244,20540259) from the Japan Society for the Promotion of Science, and also by theJSPS Core-to-Core Program, International Research Network for Exotic Femto Sys-tems(EFES). The numerical calculations were carried out on SX5 at Research Centerfor Nuclear Physics in Osaka University, and SX8 at Yukawa Institute for TheoreticalPhysics in Kyoto University.

References

1) P. G. Hansen and B. Jonson, Europhys. Lett. 4 (1987), 409.2) G. F. Bertsch, H. Esbensen, Ann. Phys. 209 (1991), 327; H. Esbensen, G. F. Bertsch,

Nucl. Phys. A 542 (1992), 310.3) K. Ikeda, INS Report JHP-7 (1988); Nucl. Phys. A 538 (1992), 355c.4) D. Sackett, K. Ieki, A. Galonsky, C. A. Bertulani, H. Esbensen, J. J. Kruse, W. G. Lynch,

D. J. Morrissey, N. A. Orr, B. M. Sherrill, H. Schulz, A. Sustich, J. A. Winger, F. Deak,

A. Horvath, and A. Kiss, Z. Seres, J. J. Kolata, R. E. Warner, and D. L. Humphrey, Phys.Rev. C 48 (1993), 118.

5) S. Shimoura, T. Nakamura, M. Ishihara, N. Inabe, T. Kobayashi, T. Kubo, R. H. Siemssen,I. Tanihata, and Y. Watanabe, Phys. Lett. B 348 (1995), 29 .

6) M. Zinser, F. Humbert, T. Nilsson, W. Schwab, H. Simon, T. Aumann, M. J. G. Borge,L. V. Chulkov, J. Cub, Th. W. Elze, H. Emling, H. Geissel, D. Guillemaud-Mueller,P. G. Hansen, R. Holzmann, H. Irnich, B. Jonson, J. V. Kratz, R. Kulessa, Y. Leifels,H. Lenske, A. Magel, A. C. Mueller, G. Munzenberg, F. Nickel, G. Nyman, A. Richter,


K. Riisager, C. Scheidenberger, G. Schrieder, K. Stelzer, J. Stroth, A. Surowiec, O. Teng-blad, E. Wajda, and E. Zude, Nucl. Phys. A 619 (1997), 151.

7) K. Ieki, A. Galonsky, D. Sackett, J. J. Kruse, W. G. Lynch, D. J. Morrissey, N. A. Orr,

B. M. Sherrill, J. A. Winger, F. Deak, A. Horvath, A. Kiss, Z. Seres, J. J. Kolata,R. E. Warner, D. L. Humphrey, Phys. Rev. C 54 (1996), 1589.

8) T. Nakamura, A. M. Vinodkumar, T. Sugimoto, N. Aoi, H. Baba, D. Bazin, N. Fukuda,T. Gomi, H. Hasegawa, N. Imai, M. Ishihara, T. Kobayashi, Y. Kondo, T. Kubo, M.Miura, T. Motobayashi, H. Otsu, A. Saito, H. Sakurai, S. Shimoura, K. Watanabe, Y.X. Watanabe, T. Yakushiji, Y. Yanagisawa, and K. Yoneda, Phys. Rev. Lett. 96 (2006),252502.

9) M. V. Zhukov, B. V. Danilin, D. V. Fedorov, J. M. Bang, I. J. Thompson, and J. S. Vaagen,Phys. Rep. 231 (1993), 151.

10) F. Barranco, P. F. Bortignon, R. A. Broglia, G. Colo, and E. Vigezzi, Eur. Phys. J. A 11

(2001), 385.11) T. Myo, S. Aoyama, K. Kato, and K. Ikeda, Phys. Lett. B 576 (2003), 281.12) K. Hagino and H. Sagawa, Phys. Rev. C 72 (2005), 044321.13) K. Hagino, H. Sagawa, J. Carbonell, and P. Schuck, Phys. Rev. Lett. 99 (2007), 022506.14) K. Hagino, N. Takahashi and H. Sagawa, Phys. Rev. C 77 (2008), 054317.15) M. Matsuo, K. Mizuyama, and Y. Serizawa, Phys. Rev. C 71 (2005), 064326.16) N. Pillet, N. Sandulescu and P. Schuck, Phys. Rev. C 76(2007), 024310.17) M. Matsuo, Phys. Rev. C 73 (2006), 044309.18) M. Matsuo, Y. Serizawa, K. Mizuyama, Nucl. Phys. A 788 (2007), 307c.19) J. Margueron, H. Sagawa, K. Hagino, Phys.Rev. C 77 (2008), 054309.20) A. Pastore, F. Barranco, R. A. Broglia and E. Vigezzi, arXiv:0801.1385.21) A. Bohr and B. R. Mottelson, Nuclear Structure vol. II (Benjamin, 1975).22) R. H. Spear, At. Data Nucl. Data Tables 42 (1989), 55.23) E. Khan, Y. Blumenfeld, Nguyen Van Giai, T. Suomijarvi, N. Alamanos, F. Auger,

G. Colo, N. Frascaria, A. Gillibert, T. Glasmacher, M. Godwin, K. W. Kemper,V. Lapoux, I. Lhenry, F. Marechal, D. J. Morrissey, A. Musumarra, N. A. Orr, S. Ottini-Hustache, P. Piattelli, E. C. Pollacco, P. Roussel-Chomaz, J. C. Roynette, D. Santonocito,J. E. Sauvestre, J. A. Scarpaci, and C. Volpe, Phys. Lett. B 490 (2000), 45.

24) S. A. Fayans, Phys. Lett. B 267 (1991), 443.25) H. Sagawa, N. Van Giai, N. Takigawa, M. Ishihara, and K. Yazaki, Z. Phys. A 351 (1995),

385.26) F. Catara, C. H. Dasso, and A. Vitturi, Nucl. Phys. A 602 (1996), 181.27) I. Hamamoto, H. Sagawa, and X. Z. Zhang, Phys. Rev. C 64 (2001), 024313.28) X. R. Zhou, E. G. Zhao, B. G. Dong, X. Z. Zhang, and G. L. Long, Nucl. Phys. A 723

(2003), 375.29) M. Yokoyama, Prog. Theor. Phys. Suppl. 142 (2001), 325.30) P. Papakonstantinou, J. Wambach, E. Mavrommatis, and V. Yu. Ponomarev, Phys. Lett.

B 604 (2004), 157.31) M. Matsuo, Nucl. Phys. A 696 (2001), 371; Prog. Theor. Phys. Suppl. 146 (2002), 110.32) Y. Serizawa and M. Matsuo, Soryushiron Kenkyu (Kyoto) 112 (2005), B93; M. Matsuo,

Y. Serizawa, and K. Mizuyama, J. of Phys. Conf. Ser. 20 (2005), 113.33) J. Dobaczewski, H. Flocard, and J. Treiner, Nucl. Phys. A 422 (1984), 103.34) J. Dobaczewski, W. Nazarewicz, T. R. Werner, J. F. Berger, C. R. Chinn, and J. Decharge,

Phys. Rev. C 53 (1996), 2809.35) E. Chabanat, P. Bonche, P. Haensel, J. Meyer, and R. Schaeffer, Nucl. Phys. A 635 (1998),

231; Nucl. Phys. A 643 (1998), 441.36) J. Dobaczewski, W. Nazarewicz, and P.-G. Reinhard, Nucl. Phys. A 693 (2001), 361.37) J. Dobaczewski, W. Nazarewicz, and M. V. Stoitsov, Eur. Phys. J. A 15 (2002), 21.38) J. Dobaczewski and W. Nazarewicz, Prog. Theor. Phys. Suppl. 146 (2002), 70.39) M. V. Stoitsov, J. Dobaczewski, W. Nazarewicz, S. Pittel, and D. J. Dean, Phys. Rev. C

68 (2003), 054312; http://www.fuw.edu.pl/∼dobaczew/thodri/thodri.html40) S. Mizutori, J. Dobaczewski, G. A. Lalazissis, W. Nazarewicz, and P.-G. Reinhard, Phys.

Rev. C 61 (2000), 044326.41) K. Bennaceur, J. Dobaczewski, and M. P loszajczak, Phys. Lett. B 496 (2000), 154.42) E. Teran, V. E. Oberacker, and A. S. Umar, Phys. Rev. C 67 (2003), 064314.

http://arxiv.org/abs/0801.1385

http://www.fuw.edu.pl/~dobaczew/thodri/thodri.html


43) M. Bender, P.-H. Heenen, and P.-G. Reinhard, Rev. Mod. Phys. 75 (2003) 121.44) J. Bartel, P. Quentin, M. Brack, C. Guet, and H. B. Hakansson, Nucl. Phys. A 386 (1982),

79.45) W. Satula, J. Dobaczewski, and W. Nazarewicz, Phys. Rev. Lett. 81 (1998), 3599.46) E. Khan, N. Sandulescu, M. Grasso, and Nguyen Van Giai, Phys. Rev. C 66 (2002),

024309.47) E. Khan, N. Sandulescu, and Nguyen Van Giai, Phys. Rev. C 71 (2005), 042801(R).48) N. Paar, D. Vretenar, E. Khan, G. Colo, Rep. Prog. Phys. 70 (2007), 691.49) A. B. Migdal, Theory of Finite Fermi Systems and Application to Atomic Nuclei (John

Wiley & Sons, 1967)50) J. W. Negele and H. Orland, Quantum Many-Particle Systems (Addison-Wesley, 1987)51) S. -O. Backman, A. D. Jackson, and J. Speth, Phys. Lett. 56B (1975), 209.52) Nguyen Van Giai and H. Sagawa, Phys. Lett. 106B (1981), 379.53) M. Bender, J. Dobaczewski, J. Engel, and W. Nazarewicz, Phys. Rev. C 65 (2002), 054322.54) K. Bennaceur, J. Dobaczewski, Comp. Phys. Comm. 168 (2005), 96.55) S. T. Belyaev, A. V. Smirnov, S. V. Tolokonnikov, S. A. Fayans, Sov. J. Nucl. Phys. 45

(1987), 783.56) P. Ring and P. Schuck, The Nuclear Many-Body Problem, (Springer-Verlag, 1980).57) M. Yamagami, Phys. Rev. C 72 (2005), 064308.


0.0

0.5

1.0

1.5

56 64 72 80 88

pair

ing

gap

[MeV

]

Mass NumberFig. 1. The average neutron pairing gap of nickel isotopes. The solid line shows the values obtained

in the present HFB calculation using SLy4 and the mixed-type DDDI. The dashed line is the

experimental value extracted from the nuclear masses using the three-point formula.


0

2

4

6

8

10

0 10 20 30

stre

ngth

[10

5 fm

6 / M

eV]

E [MeV]

84Ni isoscalar

full

unperturbed

0

2

4

6

8

10

0 10 20 30

stre

ngth

[10

5 fm

6 / M

eV]

E [MeV]

84Ni isoscalar

full

unperturbed

Fig. 2. The isoscalar octupole strength dB(IS3)/dE, plotted with the solid curve, obtained for84Ni using SLy4 and the mixed-type DDDI. The smoothing parameter is ǫ = 0.2 MeV. The

unperturbed strength is also shown with the dotted curve. The arrows indicate the one- and

two-neutron separation energies.

0

2

4

6

8

10

0 2 4 6 8

stre

ngth

[10

5 fm

6 / M

eV]

E [MeV]

0

2

4

6

8

10

0 2 4 6 8

stre

ngth

[10

5 fm

6 / M

eV]

E [MeV]

isoscalarneutronproton

0

10

20

30

3 4 5

Fig. 3. The low-energy part of calculated octupole strengths in 84Ni. The isoscalar, neutron and

proton strengths are plotted with the solid, dashed and dotted curves, respectively. The smooth-

ing parameter is ǫ = 0.05 MeV. See also the caption of Fig.2.


0

2

4

6

8

10

0 2 4 6 8

stre

ngth

[10

5 fm

6 / M

eV]

E [MeV]

0

2

4

6

8

10

0 2 4 6 8

stre

ngth

[10

5 fm

6 / M

eV]

E [MeV]

isoscalarfullno dyn. pairingpairing onlyunperturbed

Fig. 4. The effects of the residual interactions on the low-energy part of the isoscalar octupole

strength in 84Ni, shown in Fig. 3. The solid curve is the full calculation while the dashed

and dotted curves are the results neglecting the residual pairing and particle-hole interactions,

respectively. The dot-dashed curve is the unperturbed strength where both of the residual

interactions are neglected.


0.0

0.2

0.4

0.6

0 5 10 15 20

r2P

pp(r

), r2

Phh

(r) [

fm−1

]

r [fm]

neutron Ppp

neutron Phh

proton Ppp,hh

0.0

0.2

0.4

0.6

0 5 10 15 20

r2P

pp(r

), r2

Phh

(r) [

fm−1

]

r [fm]

neutron Ppp

neutron Phh

proton Ppp,hh

0.0

0.2

0.4

0.6

0 5 10 15 20

r2P

pp(r

), r2

Phh

(r) [

fm−1

]

r [fm]

neutron Ppp

neutron Phh

proton Ppp,hh

0.0

0.2

0.4

0.6

r2ρt

r (r) [

fm−1

]

84Ni E=4.15MeV

neutronproton

0.0

0.2

0.4

0.6

r2ρt

r (r) [

fm−1

]

84Ni E=4.15MeV

neutronproton

Fig. 5. The transition densities evaluated at E = 4.15 MeV for the octupole response of 84Ni (cf.

Fig. 4). In the upper panel shown is the particle-hole transition density r2ρtriqL(r) weighted

with the volume element r2. The neutron and proton amplitudes are plotted with the solid

and dashed curves, respectively. The particle-pair transition density r2P pp

iqL(r) and the hole-

pair transition density r2P hhiqL(r) of neutrons are shown in the bottom panel with the solid and

dotted curves, respectively. P pp

iqL(r) and P hhiqL(r) for protons (the dashed curves) are exactly

zero.


-0.01

0.00

0.01

0.02

0 5 10 15 20

r2P

pp(r

),r2 P

hh(r

) [fm

−1]

r [fm]

neutron Ppp

neutron Phh

proton Ppp,hh

-0.01

0.00

0.01

0.02

0 5 10 15 20

r2P

pp(r

),r2 P

hh(r

) [fm

−1]

r [fm]

neutron Ppp

neutron Phh

proton Ppp,hh

-0.01

0.00

0.01

0.02

0 5 10 15 20

r2P

pp(r

),r2 P

hh(r

) [fm

−1]

r [fm]

neutron Ppp

neutron Phh

proton Ppp,hh

-0.01

0.00

0.01

0.02

0.03

r2ρt

r (r) [

fm−1

]

84Ni E=3.0MeV

neutronproton

-0.01

0.00

0.01

0.02

0.03

r2ρt

r (r) [

fm−1

]

84Ni E=3.0MeV

neutronproton

Fig. 6. The same as Fig. 5 but for the transition densities evaluated at E = 3.0 MeV.


-0.01

0.00

0.01

0.02

0 5 10 15 20

r2P

pp(r

) [fm

−1]

r [fm]

fullpairing onlyno dyn. pairing

-0.01

0.00

0.01

0.02

0 5 10 15 20

r2P

pp(r

) [fm

−1]

r [fm]

fullpairing onlyno dyn. pairing

-0.01

0.00

0.01

0.02

0.03

r2ρt

r (r) [

fm−1

]

84Ni

neutronE=3.0MeV

Fig. 7. The neutron transition densities r2ρtriqL(r) and r2P pp

iqL(r) of the neutron mode, evaluated at

E = 3.00 MeV. Here the effects of the residual interactions are shown. The dotted curve is the

case where the residual particle-hole interaction is neglected while the dashed curve is the case

where the dynamical pairing effect, i.e. the residual pairing interaction, is neglected. The full

calculation is also shown with the solid curve for comparison. See also the caption of Fig. 5.


-0.01

0.00

0.01

0.02

0 5 10 15 20

r2P

pp(r

) [fm

−1]

r [fm]

lcut=7,9,...,17

-0.01

0.00

0.01

0.02

0.03

r2ρt

r (r) [

fm−1

]

84Ni

neutronE=3.0MeV

Fig. 8. The dependence of the neutron transition densities of the neutron mode, evaluated at E =

3.0 MeV, on different values of lcut = 7, 9, · · · , 17~. The upper and lower panels are for the

particle-hole and particle-pair transition densities, r2ρtriqL(r) and r2P pp

iqL(r), respectively. The

transition densities obtained neglecting the dynamical pairing effect are also plotted with the

thin solid curve for comparison.

0

4

8

12

0 2 4 6

stre

ngth

[10

5 fm

6 /M

eV]

80Ni

0

4

8

12

0 2 4 6

stre

ngth

[10

5 fm

6 /M

eV]

80Ni

isoscalarneutronproton

0 2 4 6E [MeV]

82Ni

0 2 4 6

84Ni

0 2 4 6 8

86Ni

Fig. 9. The octupole strength functions obtained for 80−86Ni. See also the caption of Fig. 3


-0.01

0.00

0.01

0.02

0.03

0.04

0 5 10 15 20

r2P

pp(r

) [fm

−1] full

no dyn. pairing

-0.01

0.00

0.01

0.02

0.03

0.04

0 5 10 15 20

r2P

pp(r

) [fm

−1] full

no dyn. pairing

-0.01

0.00

0.01

0.02

0.03

0.04

r2ρt

r (r) [

fm−1

] 80NiE=4.0MeV

0 5 10 15 20

82NiE=3.5MeV

0 5 10 15 20

r [fm]

84NiE=3.0MeV

0 5 10 15 20

86NiE=2.5MeV

Fig. 10. The neutron transition densities of the neutron mode obtained for 80−86Ni. The upper

and lower panels are for the particle-hole and particle-pair transition densities, r2ρtriqL(r) and

r2P pp

iqL(r), respectively. We calculate the transition densities in each nucleus at the excitation

energy listed in the Figure. The transition densities obtained neglecting the dynamical pairing

effect are also plotted with the dashed curves for comparison.

0

5

10

0 2 4 6

stre

ngth

[10

5 fm

6 /M

eV]

E [MeV]

Woods-Saxonmix

0 2 4 6 8

SkM*mix

0

5

10

0 2 4 6

stre

ngth

[10

5 fm

6 /M

eV]

E [MeV]

mixSLy4

0 2 4 6 8

volumeSLy4

Fig. 11. The octupole strength functions in 84Ni, obtained with the Woods-Saxon model (left

top) and the Skyrme HFB model with SkM∗ (right top). The mixed-type DDDI is used in

both cases. The bottom panels show the strength functions for the Skyrme parameter set SLy4

but with different pairing interaction: the mixed-type DDDI (bottom left) and the volume-type

density-independent delta interaction (bottom right). See the caption of Fig. 3 for the definition

of curves.


0.00

0.01

0.02

0.03

0.04

0 5 10 15 20

r2P

pp(r

) [fm

−1]

r [fm]

0.00

0.01

0.02

0.03

0.04

0 5 10 15 20

r2P

pp(r

) [fm

−1]

r [fm]

0.00

0.01

0.02

0.03

0.04

r2ρt

r (r) [

fm−1

]

thick: WSthin: SLy4

0.00

0.01

0.02

0.03

0.04

r2ρt

r (r) [

fm−1

]

thick: WSthin: SLy4

Fig. 12. The comparison of the neutron transition densities r2ρtriqL(r) and r2P pp

iqL(r) of the neutron

mode in 84Ni obtained with the Woods-Saxon model (the thick curves) vs. those with the Skyrme

HFB with SLy4 (the thin curves). The mixed-type DDDI is used and the excitation energy is

E = 3.0 MeV in both cases. The dashed curves in the bottom panel represent the results

obtained neglecting the dynamical pairing effect.

-8

-6

-4

-2

0

neut

ron

s.p.

ener

gy [

MeV

]

2d3/2

3s1/22d5/2

1g9/2

WS SLy4 SkM*

Fig. 13. The neutron single-particle energies for 84Ni in the Woods-Saxon model and the Skyrme

Hartree-Fock model with SLy4 and SkM∗. The dashed line indicates the neutron Fermi energy.


0.00

0.01

0.02

0.03

0 5 10 15 20

r2P

pp(r

) [fm

−1]

r [fm]

0.00

0.01

0.02

0.03

0 5 10 15 20

r2P

pp(r

) [fm

−1]

r [fm]

0.00

0.01

0.02

0.03

r2ρt

r (r) [

fm−1

]

thick: volumethin: mix

0.00

0.01

0.02

0.03

r2ρt

r (r) [

fm−1

]

thick: volumethin: mix

Fig. 14. The dependence of the neutron transition densities r2ρtriqL(r) and r2P pp

iqL(r) of the neutron

mode on the pairing interaction. We compare results obtained with the mixed-type DDDI (the

thin curves) and the volume-type density-independent delta interaction (the thick curves). The

excitation energy is E = 3.0 MeV in both cases. The Skyrme parameter set SLy4 is used.

0

4

8

12

0 5 10 15 20 25

stre

ngth

[10

5 fm

6 /M

eV]

E [MeV]

isoscalarocutpole

0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25

0

2

4

6

8

stre

ngth

[e2

fm

2 /M

eV]

80Nielectricdipole

82Ni 84Ni 86Ni

Fig. 15. The electric dipole strength function dB(E1)/dE in 80−86Ni, shown in the top panels,

obtained with the Skyrme parameter set SLy4 and the mixed-type DDDI, compared with the

isoscalar octupole strength function dB(IS3)/dE shown in the bottom panels. The smoothing

parameter is ǫ = 0.2 MeV.


-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

0 5 10 15 20

r2P

pp(r

) [fm

−1]

r [fm]

lcut=5,7,...,17

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

r2ρt

r (r) [

fm−1

]

84Ni dipole

E=3.0MeV

neutron

Fig. 16. The same as Fig.8 but for the soft dipole excitation. Here the angular momentum cut-

off is lcut = 5, 7, 9, · · · 17~. The transition densities are evaluated at E = 3.0 MeV using the

mixed-type DDDI and SLy4.

departmentofphysics,facultyofscience,niigatauniversity

Documents