II-2. Nuclear Physics (Theory)

RIKEN Accel. Prog. Rep. 54 (2021)

RIKEN Accel. Prog. Rep. 54 (2021)

Isobaric analog state energy in deformed nuclei: A toy model†
X. Roca-Maza,∗1,∗2 H. Sagawa,∗3,∗4 and G. Colò∗1,∗2
The eﬀects of deformation on the energy of the isobaric analog state (IAS) are studied through microscopic
deformed Hartree-Fock-Bogolyubov calculations. A simple yet physical toy model is also presented to provide
guidance when predicting unknown IAS energies of deformed nuclei. The deformed HFB calculations are performed for several neutron-deﬁcient medium-mass andheavy nuclei to predict the IAS energies.
Isospin is one of the most important (approximate)symmetries in nuclei. The validity of isospin symmetry
has been established by the experimental observationof IASs by charge-exchange reactions. Recently, thesestates have been investigated extensively in connection
with the symmetry energy, particularly to determine theso-called slope parameter L. Thus far, theoretical studies
of IAS have mainly focused on spherical nuclei such as48
Ca, 90 Zr, and 208 Pb. However, a large number of deformed nuclei exist, but transparent information on theeﬀects of deformation on IAS is still lacking. In this paper, we derive a general formula for the eﬀects of deformation on the Coulomb direct contribution to the energyof the IAS and provide a simple albeit physical model.
Then, we study several neutron-deﬁcient medium-massand heavy nuclei, which are now planned to be studied
experimentally in RCNP, Osaka within the LUNESTAR
project.
The IAS energy EIAS can be deﬁned as the energy
diﬀerence between the analog state |A⟩ and the parent
state |0⟩ as follows:
EIAS = ⟨A|H|A⟩ − ⟨0|H|0⟩ =⟨0|T+ [H, T− ]|0⟩, (1)⟨0|T+ T− |0⟩∑A
where T± = i t± (i) are the isospin raising/lowering
operators. The direct Coulomb term of IAS energy foran axially symmetric nucleus can be evaluated as[β2n β2pCd,sphCd
EIAS = EIAS
1−4π]
(β2n − β2p )(β2n + β2p ) (β2n − β2p )2+,(2)+4π4π
where β2n(2p) is the quadrupole deformations for neutrons (protons). For β2n = β2p , this reduces to[]β2Cd,sphCd
1− 2 .
(3)= EIAS
EIAS
4π
From Eq. (3), one should expect that a larger quadrupole
deformation corresponds to a smaller IAS energy. Fora qualitative understanding of the eﬀect of deformation
†∗1
∗2
∗3
∗4

Condensed from the article in Phys. Rev. C102, 064303 (2020)
Dipartimento di Fisica, Università degli Studi di Milano
INFN, Sezione di Milano
RIKEN Nishina CenterCenter for Mathematics and Physics, the University of Aizu

Table 1. Results of deformed HFB calculations with SAMi
HFBEDF.1) All energies are in MeV. EIAS
is the sum of thedirect Coulomb, exchange Coulomb, and isospin mixing
contributions. The deformation eﬀect on IAS energy has
HFB≡
been estimated from the HFB calculations, ∆EIAS
HFB
HFB
EIAS
(β2n , β2p ) − EIAS
(β2n = 0, β2p = 0).
Nucl.

β2nβ2p

HFB
EIAS

102
46 Pd
106
48 Cd
112
50 Sn
120
52 Te
124
54 Xe
130
56 Ba
136
58 Ce
138
58 Ce
142
60 Nd
144
62 Sm
156
66 Dy
158
66 Dy
162
68 Er
164
68 Er
168
70 Yb
174
72 Hf
176
72 Hf180
74 W
184
76 Os
190
78 Pt
196
80 Hg

0.186
0.255
0.192
0.039
0.000
0.000
0.126
0.040
0.000
0.000
0.201
0.197
0.348
0.346
0.385
0.304
0.267
0.278
0.335
−0.147−0.180

0.174
0.256
0.199
0.042
0.000
0.000
0.158
0.047
0.000
0.000
0.225
0.217
0.378
0.377
0.413
0.319
0.276
0.299
0.355
−0.141−0.18713.34613.81014.08514.35314.74915.11415.35215.33115.50915.95016.75516.72316.97516.88117.17117.62917.53817.91118.22318.64918.906exp 2)
EIAS

14.01916.07516.861
16.77817.388

HFB∆EIAS−0.162−0.089−0.001−0.0010.000
0.000−0.081−0.0100.000
0.000−0.003−0.028−0.112−0.110−0.212−0.091−0.086−0.101−0.171−0.037−0.070on the IAS energy, Eq. (3) predicts, for very deformednuclei with β2 ≈ 0.8, a relative reduction in EIAS ofapproximately 5% with respect to the spherical nucleus.
Table 1 lists the results of deformed HFB calculations
for several neutron-deﬁcient nuclei as predicted by SAMiHFB
is the sum of the direct Coulomb, exchange
EDF. EIAS
Coulomb, and isospin mixing contributions. The deforHFB
mation eﬀect is also shown as ∆EIAS, and it varies from168
Yb at the maximum to zero for no de−212 keV information cases.
The deformed HFB results reproduce well the experimentally known IAS energies within an error of approximately100 keV and previde a good guide for the experimental search for new IAS states of these neutrondeﬁcient nuclei. In general, the deformation eﬀect issmall, but it might be important for the precise prediction of IAS energy within an accuracy of several tenthsof keV.
References

1) X. Roca-Maza, G. Colò, H. Sagawa, Phys. Rev. C 86,031306 (2012).
2) M. Antony, A. Pape, J. Britz, At. Data Nucl. Data Tables 66, 1 (1997).

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