Page 1

arXiv:nucl-th/0202024v1 8 Feb 2002

Progress of Theoretical Physics Supplement

1

Collective Excitations and Pairing Effects in Drip-Line Nuclei

Continuum RPA in Coordinate-Space HFB

Masayuki Matsuo∗)

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

Japan

(Received )

We discuss novel features of a new continuum RPA formulated in the coordinate-space

Hartree-Fock-Bogoliubov framework. This continuum quasiparticle RPA takes into account

both the one- and two-particle escaping channels. The theory is tested with numerical calcu-

lations for monopole, dipole and quadrupole excitations in neutron-rich oxygen isotopes near

the drip-line. Effects of the particle-particle RPA correlation caused by the pairing interac-

tion are discussed in detail, and importance of the selfconsistent treatment is emphasized.

§1.Introduction

Collective excitation in unstable nuclei is one of the most attractive subjects

since the exotic structures in the ground state, such as halo, skin, and the presence

of loosely bound nucleons, may cause new features in the excitations, e.g. the low-

energy dipole mode that is being discussed extensively. The random phase approxi-

mation (RPA) or the linear response theory is one of the most powerful framework

to investigate such problems microscopically. Indeed the continuum RPA theory in

the coordinate-space representation1), 2)has played major roles so far since it can

describe the continuum states crucial for nuclei near drip-line.3)- 5)

The pairing correlation is another key feature of drip-line nuclei.6)- 8)To treat

the coupling of the continuum states as well as the density dependence of the

pairing correlation, the Hartree-Fock-Bogoliubov (HFB) theory formulated in the

coordinate-space representation6), 9)has been developed while the conventional BCS

approximation has inherent deficiency.

It is therefore important to combine the continuum RPA and the coordinate

space HFB in a consistent way in order to describe the excitations in unstable nuclei

near drip-line, especially when the pairing correlation play crucial roles. We have

recently shown that a new quasiparticle RPA (QRPA) satisfying this requirement is

indeed possible.10)In the present paper, we discuss characteristic features of the the-

ory and analyze excitations of near-drip-line nuclei, focusing on effects of the pairing,

by using numerical calculations performed for the monopole, dipole and quadrupole

excitations in neutron-rich oxygen isotopes. The previous continuum QRPA ap-

proaches employ the BCS approximation.11)- 14)Other QRPA approaches applied

to unstable nuclei neglect the escaping effects since some use the BCS quasiparti-

cle basis,15)- 17)and other adopt the coordinate-space HFB but use the discretized

canonical basis.18)The present formalism provides the first consistent continuum

∗)E-mail address: matsuo@nt.sc.niigata-u.ac.jp

typeset using PTPTEX.sty <ver.0.8>

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M. Matsuo

QRPA approach for the drip-line nuclei with pairing correlation.

§2.Continuum RPA in coordinate-space HFB

The Hartree-Fock-Bogoliubov theory describes the pairing correlation in terms

of quasiparticles and selfconsistent mean-fields including the pairing potential. To

correctly describe behavior of the quasiparticles in the surface and exterior regions

related to halo or skin, it is preferable to solve the HFB equation in the coordinate-

space representation6), 9)

H0φi(rσ) = Eiφi(rσ),

which determines the quasiparticle states and the associated two-component wave

functions

?

(2.1)

φi(rσ) ≡

ϕ1,i(rσ)

ϕ2,i(rσ)

?

.(2.2)

Here the HFB mean-field Hamiltonian is expressed in a 2 × 2 matrix form

?

˜h∗(r˜ σ,r′˜ σ′)

H0(rσ,r′σ′) ≡

h(rσ,r′σ′) − λδ(r − r′)δσσ′

˜h(rσ,r′σ′)

−h∗(r˜ σ,r′˜ σ′) + λδ(r − r′)δσσ′

?

(2.3)

where h includes the kinetic energy and the Hartree-Fock field in the particle-hole

(ph) channel and˜h is the selfconsistent pairing field in the particle-particle (pp)

channel. They are expressed in terms of the effective two-body interactions and the

normal and pair densities although we omit here their detailed expression. Prop-

erties of the static HFB equations and techniques to solve them are known.6), 9)

The quasiparticle excitation energy Eiis defined with respect to the Fermi energy

λ(< 0). The spectrum becomes continuous for Ei> |λ| and the quasiparticles above

the threshold energy |λ| can escape from the nucleus. This is a special feature we

have to take care of when we describe weakly bound systems with pairing correlation.

In order to describe the linear response of the system, we need to know the

motion of two quasiparticles propagating under the HFB mean-field Hamiltonian.

Assuming that the external field and the selfconsistent field are the local one-body

fields expressed in terms of the normal density ρ(r) =

pair densities ˜ ρ±(r) =1

2

?

response function for these operators. The unperturbed response function R0(ω) at

frequency ω, that neglects effect of the residual interaction, is easily derived from a

time-dependent extension of Eq.(2.1) as

?

σψ†(rσ)ψ(rσ) and the

, it is enough to consider

σ

?

ψ(r˜ σ)ψ(rσ) ± ψ†(rσ)ψ†(r˜ σ)

?

Rαβ

0(r,r′,ω) =1

2

?

+φ†

i

?

˜i(r′σ′)BG0(r′σ′,rσ,−Ei− ¯ hω − iǫ)Aφ˜i(rσ)

σσ′

?

φ†

˜i(rσ)AG0(rσ,r′σ′,−Ei+ ¯ hω + iǫ)Bφ˜i(r′σ′)

?

(2.4)

with use of the HFB Green function G0(E + iǫ) = (E + iǫ − H0)−1and the wave

functions φ˜i(rσ) of the quasiparticle states. Here φ˜i(rσ) is the one associated with

the negative energy quasiparticle state (with energy −Ei) conjugate to a positive

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Collective Excitations and Pairing Effects in Drip-Line Nuclei

3

energy state φi(rσ) with Ei. The index α,β and the symbol A,B refer to the three

kinds of densities ρ, ˜ ρ+and ˜ ρ−. Structure of Eq.(2.4) is similar to the familiar form

for unpaired systems1)except that we here use the quasiparticle wave functions

and the HFB Green function. However this expression is not satisfactory since the

summation?

effects accurately.

The continuum limit is introduced by first replacing the summation with a con-

tour integral of the HFB Green function G0(E) in the complex E plane. The contour

is chosen so that it encloses all the quasiparticle poles at −Eiof the negative energy

quasiparticle states (see, Fig.1). By doing this, the unperturbed response function

can be expressed as

iassumes that all the quasiparticle states belong to a discrete spectrum.

We have to take a continuum limit of this summation to deal with the continuum

Rαβ

0(r,r′,ω) =

1

4πi

?

CdE

?

σσ′

?TrAG0(rσ,r′σ′,E + ¯ hω + iǫ)BG0(r′σ′,rσ,E)

+TrAG0(rσ,r′σ′,E)BG0(r′σ′,rσ,E − ¯ hω − iǫ)?

(2.5)

in terms of products of the HFB Green function G0 and their integral. Now we

can implement the continuum quasiparticle spectrum by adopting the exact HFB

Green function which satisfies the proper boundary condition of outgoing wave for

the continuum states.9)The exact HFB Green function has poles on the real E axis

in the interval −|λ| < E < |λ|, corresponding to the discrete bound quasiparticle

states, and has the branch cuts, associated with the continuum states, along the real

axis for E > |λ| and E < −|λ| (Fig.1). We incorporate in this way the continuum

states within the framework.

ε’

ε’

i

-i

C

−λ

ImE

ReE

-Ecut0

λ

Fig. 1. The contour C in the integral representation of the response function. The crosses represent

the poles at E = ±Ei corresponding to the discrete bound quasiparticle states. The thick lines

are the branch cuts associated with the continuum states. The imaginary part ǫ′must satisfy

the condition 0 < ǫ′< ǫ.

Having the continuum unperturbed response function R0(ω), we are then able

to take into account the RPA correlation caused by the residual interactions. The

RPA linear response equation for the transition densities at frequency ω reads

δρ(r,ω)

δ˜ ρ+(r,ω)

δ˜ ρ−(r,ω)

=

?

dr′

Rαβ

0(r,r′,ω)

κph(r′)δρ(r′,ω) + vext(r′)

κpp(r′)δ˜ ρ+(r′,ω)

−κpp(r′)δ˜ ρ−(r′,ω)

(2.6)

where κph(r) and κpp(r) are the residual interactions in the ph- and pp-channels.

The strength function for the external one-body field vext(r) is evaluated as f(ω) =

−1

πIm?drvext(r)∗δρ(r,ω).

Page 4

4

M. Matsuo

An important feature of the present continuum QRPA is that the linear response

equation (2.6) takes into account the pair transition density δ˜ ρ±as well as the normal

transition density δρ. Note that the three transition densities couple since the system

has pairing correlation. By solving Eq.(2.6), the RPA correlations responsible for

the excited states, represented by the ring diagram, are taken into account both

in the ph-channel (through δρ) and in the pp-channel (δ˜ ρ±). The particle-particle

RPA correlation may be called the dynamical pairing correlation to distinguish from

the pairing correlation already taken into account as the quasiparticles in the HFB

description of the ground state. We demonstrate in the following important roles

played by the dynamical pairing correlation for excitations of nuclei near drip-line.

Another novel aspect of the theory concerns with the continuum states. The

two-quasiparticle states appearing in the QRPA formalism are classified in three

groups; the first consists of two nucleons both occupying the bound discrete quasi-

particle states, the second with one particle in the bound states and the other in

the continuum states, and the last with two particles both in the continuum states.

The three configurations are incorporated through the product of two HFB Green

function G0appearing in Eq.(2.6) since G0describes exactly both the discrete and

continuum states. Thus the present linear response theory contains both the channel

of one-nucleon escaping and that of two-nucleon emission. The threshold energy for

the two-nucleon channel is twice the Fermi energy Eth,2= 2|λ|, while the thresh-

old for one-nucleon escape is Eth,1= |λ| + Ei,minwhere Ei,minis the lowest of the

quasiparticle energy Ei. See Ref.10) for details of the formalism.

§3.Numerical analysis for oxygen isotopes

3.1. Monopole: pairing selfconsistency

In the following, we present our numerical analysis performed for the neutron-

rich even-even oxygen isotopes including the neutron drip-line nucleus24O. The

adopted model assumes the Woods-Saxon potential for the single-particle potential.

As for the residual interaction in the ph-channel, the Skyrme-type density-dependent

delta force vph(r,r′) = (t0(1 + x0Pσ) + t3(1 + x3Pσ)ρ(r))δ(r − r′) is used.

model parameters for the Woods-Saxon potential and the Skyrme force are taken

the same as Shlomo-Bertsch.1)(Note that the adopted Woods-Saxon parameters are

slightly different from those in Ref.10).) Although this modeling of the potential and

the ph-interaction is not selfconsistent, an approximate selfconsistency is satisfied by

renormalizing the interaction strength t0and t1with an overall factor f so that the

dipole response has a zero-energy mode corresponding to the spurious center of mass

motion.1)

For the pairing interaction, we adopt the density-dependent delta force8), 19)

The

vpair(r,r′) =1

2V0(1 − Pσ)(1 − ρ(r)/ρ0)δ(r − r′), (3.1)

with ρ0= 0.16 fm−3. We use the same force to obtain the HFB pairing field for the

ground state and also to solve the linear response equation for the excitations. Thus

the calculation is selfconsistent in the pp-channel.

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Collective Excitations and Pairing Effects in Drip-Line Nuclei

5

The pairing force strength V0 is fixed to a value V0 = 520 fm−3MeV which

gives an average neutron pairing gap ?∆? (see Ref.10)for its definition) reproduc-

ing the global trend ?∆? ≈ 12/√A MeV in18,20O. The calculated value is ?∆? =

2.37,2.83,2.84,2.89 MeV for neutron and zero gap for proton in18−24O. The model

parameters and the procedure of calculation is the same as those in Ref.10) except the

small difference in the Woods-Saxon parameters and the force renormalization fac-

tor, which is f = 0.689,0.704,0.750,0.775 for18−24O. Using the present parameters,

the peak energies of the giant quadrupole and dipole resonances are lifted by about

a few MeV (cf. Fig.3 compared with Fig.2 in Ref.10)), giving better description of

GR’s in16O. The calculation will be further improved if we use the Hartree-Fock po-

tential for the ph-part. We will not discuss here quantitative (dis)agreement with the

experiments,16), 23)- 26)but rather focus on qualitative aspects seen in the theoretical

analysis. The small imaginary part in the response function is fixed to ǫ = 0.2 MeV.

It has an effect to bring an additional width of 0.4 MeV in the calculated strength

function.

Let us first emphasize importance

of the selfconsistent treatment of the

pairing in the linear response equa-

tion bylookingintothe Nambu-

Goldstone mode associated with the

nucleon number conservation. Fig.2

shows the monopole strength function

for the neutron number operatorˆ N =

?dr?

since it is the conserved quantity. The

calculated strength function (solid line)

exhibits this feature, and there is essen-

tially no spurious excitation that could

have been caused by the nucleon num-

ber mixing in the HFB ground state. Note that the spurious strength forˆ N were

induced if we took into account the pairing correlation only in the static mean-field

and neglected the dynamical pairing correlation i.e. neglecting the pairing interac-

tion in the linear response equation (2.6) (see the dotted curve in Fig.2). In the

selfconsistent calculation the Nambu-Goldstone mode has an excitation energy very

close to zero. We can shift this excitation energy to exact zero by modifying the

pairing force strength V0in Eq.(2.6) just by less than 1%, indicating that a good

accuracy for the pairing selfconsistency is obtained in the actual calculation.

dB(N0)/dE [MeV-1]

E [MeV]

24O

No dynamical pairing

full pairing (selfconsistent)

0 10 203040

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

Fig. 2. The strength function for the neutron

number operator in24O.

σψ†(rσ)ψ(rσ). There should be

no response to the number operator

3.2. Quadrupole excitation

The strength functions for the proton, neutron, isoscalar, and isovector quadrupole

operators calculated for the drip-line nucleus24O are shown in Fig.3. The peak

around E≈ 18 MeV in the isoscalar strength and the broad distribution

Page 6

6

M. Matsuo

E [MeV]

Eth

dB(τ2)/dE [fm4/MeV]

neutron

proton

IS

IV

01020304050

0

50

100

150

200

IS

neutron

IV

proton

02468

0

200

400

600

800

1000

1200

Fig. 3. The strength functions for the isoscalar, isovector, proton and neutron quadrupole moments

in

indicated by an arrow.

24O. The threshold energy Eth,1n = Eth,2n = 2|λ| = 4.08 MeV for neutron emission is

dB(IS2)/dE [fm4/MeV]

E [MeV]

16O

051015202530

0

200

400

600

800

1000

1200

0

18O

051015202530

200

400

600

800

1000

1200

0

20O

051015 20 2530

200

400

600

800

1000

1200

0

22O

051015202530

200

400

600

800

1000

1200

0

24O

051015202530

200

400

600

800

1000

1200

Fig. 4. Isoscalar quadrupole strength function

for16−24O.

around E ≈ 25 − 40 MeV of the isovec-

tor strength correspond to the isoscalar

and isovector giant quadrupole reso-

nances, respectively, although isoscalar

and isovector characters are mutually

mixed. The most prominent feature is

presence of the intense low-lying state

at E = 3.6 MeV, which is very close to

the threshold energy Eth,1n= Eth,2n=

4.08 MeV for the one- and two-neutron

escape.This low-lying 2+state has

large neutron strength which is over-

whelming the proton strength.

energy weighted sum of the neutron

(and the isoscalar) strength in this state

amounts to 12 (13)% of the sum-rule

value. The neutron transition den-

sity δρ(r) has a large peak in the sur-

face region.Thus this low-lying state

has a characteristic of neutron sur-

face vibration; for which the dominance

of neutron strength marks significant

difference from the isoscalar low-lying

2+states in the stable nuclei.

slightly lower excitation energy and the

weaker strength of the low-lying 2+state

in comparison with the calculation in

The

(The

Page 7

Collective Excitations and Pairing Effects in Drip-Line Nuclei

7

E [MeV]

dB(IS2)/dE [fm4/MeV]

02468

0

250

500

750

1000

1250

1500

full pairing

No dynamical

pairing

no pairing

Energy weighted sum [fm4MeV]

E [MeV]

No dyanamical

pairing

full pairing

no pairing

sum-rule

01020304050

0

5000

10000

15000

20000

25000

Fig. 5. Isoscalar quadrupole strength function in24O (left) and its energy weighted sum (right).

The calculation with the full pairing effects (solid) is compared with the one with no pairing

effects (dotted), and the one where only the static HFB potential is taken into account and the

dynamical pairing effect is neglected (dashed).

Ref.10) are due to the different Woods-Saxon parameter.)

Figure 4 shows systematics of the isoscalar quadrupole strength for16−24O. The

isoscalar strength (and the neutron strength not shown here) of the low-lying 2+

state increases with the neutron number. The B(E2) value of this state, on the other

hand, is not very dependent of the neutron number; B(E2) = 17,18,20,18 e2fm4for

A = 18 − 24. Thus the neutron character of this vibration mode is enhanced as

the drip-line is approached. Significant amount of neutron strength in the interval

between the low-lying 2+state and the GQR, seen in the near-drip-line nuclei22,24O,

arises from the neutron continuum states. It is also seen that the width of the

isoscalar giant quadrupole resonance increases with the neutron number. This can

be naturally interpreted as an increase of the neutron escaping width.

Effects of neutron pairing on the quadrupole response are quite large.

pairing correlation increases drastically the collectivity of the low-lying 2+state,

as shown in Fig.5 (solid vs. dotted curves). The energy weighted isoscalar sum

S(IS2) =?E

be 730,1220,1680,2330 fm4MeV (and 43,47,54,62 e2fm4MeV ) for A = 18−24, while

by neglecting the pairing correlations these quantities become 73,132,277,646 fm4MeV

(and 3,4,7,14 e2fm4MeV), which are several times smaller. Here we emphasize that

the total pairing effect arises not only from the static HFB mean-field but also from

the dynamical RPA correlation induced by the pairing interaction. As Fig.5 (left)

indicates, the dynamical pairing correlation lowers the excitation energy of the 2+

state by ≈ 1 MeV (solid vs. dashed curves).

The dynamical pairing correlation has another important role as shown in Fig.5

(right), where accumulated energy weighted isoscalar sum?E

ted. Note that the energy weighted sum rule is satisfied quite accurately, but this

The

?dB(IS2)

dE

?

dE of the 2+state (and the E2 sum S(E2)) is calculated to

0E′?dB(IS2)

dE′

?

dE′is plot-

Page 8

8

M. Matsuo

dB(E1)/dE [e2 fm2/MeV]

E [MeV]

16O

10203040

0.0

0.2

0.4

0.6

0.8

0.0

18O

10203040

0.2

0.4

0.6

0.8

0.0

20O

10203040

0.2

0.4

0.6

0.8

0.0

22O

10203040

0.2

0.4

0.6

0.8

0.0

24O

10203040

0.2

0.4

0.6

0.8

Fig. 6. Electric dipole strength function for

16−24O. The arrows indicate the one- and

two-neutron thresholds.

r (fm)

δρ(r)

neutron

proton

22O

E=8.0 MeV

02468

-0.004

-0.002

0.000

0.002

0.004

Fig. 7. The transition density for the E1

response of neutron (solid) and proton

(dashed) at E = 8.0 MeV in22O, plotted

as a function of the radial coordinate r.

is achieved only by including selfconsis-

tently the dynamical pairing correlation

on top of the mean-field pairing effect.

If one neglects the dynamical pairing,

the selfconsistency in the linear response

equation is broken and the sum rule is

violated as seen for the dashed curve.

3.3. Dipole excitation

The dipole excitation is very inter-

esting since the microscopic structure

of the low-energy E1 mode, called of-

ten the soft dipole mode or the pygmy

dipole resonance, is currently debated

3), 7), 20)- 22), 26)while a large effect of the

pairing correlation on this mode is also

pointed in the case of the halo nucleus

11Li.8)We here calculate the electric

dipole strength by using the operator

Dµ= eZ

A

?

explicitly removed. The result is shown

in Fig.6.As the neutron number in-

creases the E1 strength in the low energy

region (e.g. E < 15 MeV) develops sig-

nificantly; the energy weighted sum be-

low 15 MeV is S(E1) = 7,11,16,21% of

the TRK sum-rule value for A = 18−24.

The transition density δρ(r) (shown in

Fig.7) indicates that only neutrons are

moving in the exterior region, whereas

in the surface region neutrons and pro-

tons move coherently in the direction

opposite to the outside neutrons. Thus

the low-energy structure has at least

partly the character of the pygmy res-

onance or the soft dipole mode.20), 7)

The low-energy strength may also be re-

lated to the so called threshold strength

since the strength increases sharply just

above the one-neutron threshold Eth,1n

(see Fig.6).In fact, it cannot be de-

scribed well by discretizing the neutron

continuum states with use of, e.g.

spherical box boundary condition with

nrY1µ(n) − eN

A

?

prY1µ(p),

from which the center of mass motion is

a

Page 9

Collective Excitations and Pairing Effects in Drip-Line Nuclei

9

E [MeV]

n 2n

full pairing

no dynamical pairing

no pairing

dB(E1)/dE [e2 fm2/MeV]

22O

102030 40

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Fig. 8. Effects of the pairing correlations on the dipole response in22O. The solid curve is obtained

with full pairing effects included. For the dotted curve no pairing effect is included whereas in

the calculation shown by the dotted curve only the static HFB potential is taken into account

and the dynamical pairing effect is neglected. The TRK sum rule is satisfied for the solid and

dotted curves, but not for the dashed curve.

a box radius R = 20 fm since this is a structure consisting of genuine continuum

states. Note however that the low energy strength does not arise purely from unper-

turbed neutron continuum states, as we see below.

Neutron pairing effect on the E1 strength is strong as shown in Fig.8, where

different contributions of the effects are also examined. The total pairing effect

enhances the low-energy E1 strength near the threshold E ∼ 8MeV. It is also seen

that the static mean-field pairing increases the low-energy strength with respect to

the unpaired response (dashed vs. dotted curves). This effect may be attributed

to partial filling, due to the pairing, of the neutron 2s1

on the threshold strength. However, more significant is the enhancement caused

by the dynamical pairing correlation (solid vs. dashed curves). This indicates that

the enhanced low-energy strength is not simply the threshold strength associated

with the unperturbed quasi-neutron states occupying the continuum and the weakly

bound orbits, but rather it is due to the pairing RPA correlation that mixes different

two-quasineutron states. A similar effect is seen in Ref.8) in their analysis of11Li

using a two-neutron continuum shell model.

2orbit which is influential

§4. Conclusions

The new linear response theory formulated in the coordinate-space HFB enables

us to describe the collective excitations in nuclei near drip-line where the pairing

correlation and the coupling to continuum states are important. The dynamical

RPA correlations in the excited states are taken into account both in the ph- and

pp-channels in a way consistent with the description of the static HFB mean-field.

Page 10

10

M. Matsuo

The continuum states in the one- and two-particle escaping channels are included.

Since the theory satisfies the pairing selfconsistency both in the HFB ground state

and in the linear response equation, there is no spurious excitation of nucleon number

and the energy weighted sum rule is guaranteed with good accuracy. The dynamical

pairing correlation in the pp-channel causes strong enhancement in the strength of

the low-lying quadrupole neutron vibration and of the low-energy dipole excitation

as demonstrated with the numerical analysis for oxygen isotopes near the neutron

drip-line.

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