arXiv:nucl-th/0202024v1 8 Feb 2002
Progress of Theoretical Physics Supplement
Collective Excitations and Pairing Effects in Drip-Line Nuclei
Continuum RPA in Coordinate-Space HFB
Graduate School of Science and Technology, Niigata University, Niigata, 950-2181,
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.
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
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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
Here the HFB mean-field Hamiltonian is expressed in a 2 × 2 matrix form
˜h∗(r˜ σ,r′˜ σ′)
h(rσ,r′σ′) − λδ(r − r′)δσσ′
−h∗(r˜ σ,r′˜ σ′) + λδ(r − r′)δσσ′
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
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
ψ(r˜ σ)ψ(rσ) ± ψ†(rσ)ψ†(r˜ σ)
˜i(r′σ′)BG0(r′σ′,rσ,−Ei− ¯ hω − iǫ)Aφ˜i(rσ)
˜i(rσ)AG0(rσ,r′σ′,−Ei+ ¯ hω + iǫ)Bφ˜i(r′σ′)
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
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
1) S. Shlomo, G. Bertsch, Nucl. Phys. A243 (1975), 507.
2) G.F. Bertsch, S.F. Tsai, Phys. Rep. 18 (1975), 125.
3) I. Hamamoto, H. Sagawa, X.Z. Zhang, Phys. Rev. C57 (1998), R1064; Nucl. Phys. A648
I. Hamamoto, in this proceedigs.
4) S.A. Fayans, Phys. Lett. B267 (1991), 443.
5) F. Ghielmetti, G.Col` o, E. Vigezzi, P.F. Bortignon, R.A. Broglia, Phys. Rev. C54 (1996),
6) J. Dobaczewski, H. Flocard, J. Treiner, Nucl. Phys. A422 (1984), 103.
J. Dobaczewski, W. Nazarewicz, T.R. Werner, J.F. Berger, C.R. Chinn, J. Decharg´ e, Phys.
Rev. C53 (1996), 2809.
S. Mizutori, J. Dobaczewski, G.A. Lalazissis, W. Nazarewicz, P.-G. Reinhard, Phys. Rev.
C61 (2000), 044326.
J. Dobaczewski, in this proceedings.
7) P.G. Hansen, B. Jonson, Europhys. Lett. 4 (1987), 409.
8) G.F. Bertsch, H. Esbensen, Ann. Phys. 209 (1991), 327.
H. Esbensen, G.F. Bertsch, Nucl. Phys. A542 (1992), 310.
9) S.T. Belyaev, A.V. Smirnov, S.V. Tolokonnikov, S.A. Fayans, Sov. J. Nucl. Phys. 45
10) M. Matsuo, Nucl. Phys. A696 (2001), 371.
11) A.P. Platonov, E.E. Saperstein, Nucl. Phys. A486 (1988), 63.
12) I.N. Borzov, S.A. Fayans, E. Kr¨ omer, D. Zawischa, Z. Phys. A355 (1996), 117.
13) S. Kamerdzhiev, R.J. Liotta, E. Litvinova, V. Tselyaev, Phys. Rev. C58 (1998), 172.
14) K. Hagino, H. Sagawa, Nucl. Phys. A695 (2001), 82.
15) E. Khan, Nguyen Van Giai, Phys. Lett. B472 (2000), 253.
16) E. Khan, et al., Phys. Lett. B490 (2000), 45.
17) G. Col` o, P.F.Bortignon, Nucl. Phys. A696 (2001), 427.
18) J. Engel, M. Bender, J. Dobaczewski, W. Nazarewicz, R. Surman, Phys. Rev. C60 (1999),
M. Bender, J. Dobaczewski, J. Engel, W. Nazarewicz, preprint nucl-th/0112056.
19) J. Terasaki, P.-H. Heenen, P. Bonche, J. Dobaczewski, H. Flocard, Nucl. Phys. A593
20) Y. Suzuki, K. Ikeda, H. Sato, Prog. Theor. Phys. 83 (1987), 180.
21) F. Catara, E.G. Lanza, M.A. Nagarajan, A. Vitturi, Nucl. Phys. A624 (1997), 449.
22) D. Vretenar, N. Paar, P. Ring, G.A. Lalazissis, Nucl. Phys. A692 (2001), 496.
P. Ring, in this proceedings.
23) J.K. Jewell, et al., Phys. Lett. B454 (1999), 181.
24) P.G. Thirolf, et al., Phys. Lett. B485 (2000), 16.
25) F. Azaiez, Phys. Scr. T88 (2000), 118.
M. Belleguic, et al., Nucl. Phys. A682 (2001), 136c.
26) A. Leistenschneider, et al., Phys. Rev. Lett. 86 (2001), 5442.