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arXiv:nucl-th/0202024v1 8 Feb 2002

Progress of Theoretical Physics Supplement

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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

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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).

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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

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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-

tionbylookingintothe 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 10203040

-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