RIKEN Accel. Prog. Rep. 50 (2017)

Ⅱ-2. Nuclear Physics (Theory)

Eﬀective restoration of dipole sum rules within the renormalized
random-phase approximation†
N. Quang Hung,∗1 N. Dinh Dang,∗2 T. V. Nhan Hao,∗1,∗3 and L. Tan Phuc∗1IS†
∗1
∗2
∗3IV

5B(E1) (e 2 fm 4 )8(a) 48Ca

64
24(d) 48Ca

32
1

B(E1) (e 2 fm 4 )54(b) 52Ca

321
4B(E1) (e 2 fm 4 )B(E1) (10 2 e 2 fm 4 )B(E1) (10 2 e 2 fm 4 )B(E1) (10 2 e 2 fm 4 )The random-phase approximation (RPA) is one of
the simple theoretical methods, which is widely used
to describe the vibrational collective excitations of thenucleus. The RPA is derived basically based on thequasiboson approximation (QBA), which assumes thatthe operators of the particle-hole (ph) pairs have structure like ideal bosons, leading to the violation of thePauli principle owing to their fermion structure. The
application of the RPA to the light nuclear systems,
where the concepts of the mean ﬁeld and the QBA do
not properly work well, is therefore questionable. Oneof the typical methods to preserve the Pauli principle
between the RPA ph pairs is the renormalized RPA(phRRPA)1) . The latter includes the so-calledground
state correlation (GSC) factor Dph ≡ fh − fp with
fp and fh being occupation numbers of the particles
and the holes, respectively. Within the RPA, Dph isequal to 1, whereas within the phRRPA it is proportional to the RPA backward-going amplitudes (Yph )2
and is therefore always less than or equal to 1. However, the application of the phRRPA using this GSC
factor has been limited so far to the energy and B(E3)146value of the lowest 3−
Gd and 208 Pb only2) .
1 state in−
For the dipole excitation 1 , whose ﬁrst moment m1
known as the energy-weighted sum rule (EWSR) is themost important, no investigation was carried out. TheRPA fulﬁlls this EWSR sum rule. However, within thephRRPA, the matrix elements of the ph interactions
are reduced by the factor Dph , leading to the decreaseof the B(E1) values and therefore causing the violation of the EWSR. The goal of the present paper isto restore the EWSR violated within the conventional
phRRPA in an approximate and eﬀective way.
To restore the EWSR, we propose a simple RRPA
calculations taking into account, in addition to the phexcitations, the contribution of the pp and hh excitations in a perturbative way. The total isovector (IV)and isoscalar (IS) (compressional) dipole transition
probabilities are then given as B(E1) = B ph (E1) +B pp (E1) + B hh (E1), where B ss (E1) are the transition probabilities of the ss (ss = ph or pp or hh )transitions. The numerical calculations are carriedout using the self-consistent RPA code with Skyrme
SLy5 interaction3) for IV and IS dipole strength distributions in 48,52,58 Ca (Fig. 1) and 90,96,110 Zr isotopes. The results obtained show that the GSC beyond(c) 58Ca

3210102030
E (MeV)404(e) 52Ca

3215(f) 58Ca

43210102030
E (MeV)

40Fig. 1. Distributions of the IS and IV reduced transition
probabilities B(E1) for calcium obtained within theRPA, phRRPA, and RRPA. The solid and dashed vertical bars denote the B(E1) values obtained within theRPA and phRRPA, respectively. The dotted vertical
bars stand for the B pp (E1).

the RPA reduce the IS transition strengths, whereas
it slightly increases the total strength on the lowenergy region (the pygmy dipole resonance – PDR)
region and decreases the strength on the other side(the GDR region), leading to a signiﬁcant decrease ofthe EWSRs for both IS and IV modes obtained within
the phRRPA. This violation of the EWSR is then fully
recovered by taking into account the contribution of ppand hh excitations within the RRPA. This result reveals the reason why all the RPA extensions that do
not take into account the pp and hh excitations violate
the EWSRs. Consequently, the ratio of the energyweighted sum of strengths of the PDR to that of theGDR, which is almost zero in stable nuclei, increases
with the neutron number. As compared to the RPA
case, this ratio is in general signiﬁcantly larger within
the RRPA.
References1) K. Hara, Prog. Theor. Phys. 32, 88 (1964); K. Ikeda,T. Udagawa, and H. Yamaura, ibid. 33, 22 (1965); J.
Dukelsky and P. Schuck, Phys. Lett. B 464, 164 (1999).
2) F. Catara, N. Dinh Dang, and M. Sambataro, Nucl.
Phys. A 579, 1 (1994).
3) G. Colo, L. Cao, N. Van Giai, and L. Capelli, Comput.
Phys. Commun. 184, 142 (2013).

Condensed from the article in Phys. Rev. C 94, 064312(2016)
Institute of Research and Development, Duy Tan University
RIKEN Nishina Center
Texas A&M University-Commerce- 94 -