Collective Excitations and Pairing Effects in Drip-Line Nuclei -- Continuum RPA in Coordinate-Space HFB --

Abstract

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

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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).

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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-
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