II-2. Nuclear Physics (Theory)

RIKEN Accel. Prog. Rep. 54 (2021)

RIKEN Accel. Prog. Rep. 54 (2021)

Double charge-exchange phonon states†
X. Roca-Maza,∗1,∗2 H. Sagawa,∗3,∗4 and G. Colò∗1,∗2
The possibility of inducing double charge-exchange(DCX) excitations by means of heavy-ion beams at
intermediate energies has recently fostered interest in
new collective excitations such as double isobaric analog states (DIAS) and double Gamow-Teller giant resonances (DGTR). A research program based on a newreaction, (12 C, 12 Be(0+2 )), is planned at the RIKEN
RIBF facility with high-intensity heavy-ion beams at
the optimal energy of Elab = 250 MeV/nucleon to excite the spin-isospin response.1) A big advantage ofthis reaction is based on the fact that it is a (2p, 2n)type DCX reaction, and one can use a neutron-richtarget to excite DGTR strength. In this report, wepresent some formulas to evaluate diﬀerent combinations of the average excitation energies of DIAS andDGTR by ∑using commutator relations for the douA
ble isospin i,j=1 t− (i)t− (j) and spin-isospin operator∑Ai,j=1 σ(i)t− (i)σ(j)t− (j). Here, t = τ /2, and σ andτ denote the Pauli matrices in spin and isospin space,respectively. Speciﬁcally, we present formulas to estimate EDIAS − 2EIAS from the most relevant isospin
symmetry breaking (ISB) terms in the nuclear Hamiltonian and EDGTR − EDIAS − 2(EGTR − EIAS ) from a
simple albeit realistic Hamiltonian including separable
residual interactions.
The expectation value for the energy of the DIAS isdeﬁned asEDIAS ≡ ⟨DIAS|H|DIAS⟩ − ⟨0|H|0⟩ ,in other words, there is no isospin mixing, andT+ |0⟩ = 0. Remembering that the EIAS is writtenas, within the same approximation (i.e., no isospin
mixing in the ground state), EIAS = ⟨IAS|H|IAS⟩ −⟨0|H|0⟩ = ⟨0|[T+ , [H, T− ]]|0⟩/⟨0|T+ T− |0⟩, one caneventually write
EDIAS = 2EIAS +The energy of the double GT state can be deﬁned in
an analogous way to Eq. (3) as
EDGTR =

T− |IAS⟩⟨IAS|T+ T− |IAS⟩1/2⟨0|[T+2 , [H, T−2 ]]|0⟩,⟨0|T+2 T−2 |0⟩⟨0|[O+ , [O+ , [[H, O− ], O− ]]]|0⟩.
2(N − Z)(N − Z − 1)
(6)
Double GT and IAS average excitation energies havebeen determined for the ﬁrst time using double andquartic commutator relations.
In order to provide semi-quantitative theoretical estimates, we haveadopted two approximations. First, an independent
particle picture has been assumed. We have also provided expressions in which, by simplifying further, theneutron and proton distributions have been taken as
hard spheres. This simpliﬁcation has turned out to bevery useful to capture the main terms dominating thecalculated quantities.
In conclusion, within our approach, double IAS andGT resonance energies in neutron-rich nuclei are dominated by the same physics as their single counterparts because the main contribution is from 2EIASand 2EGTR , respectively. Hence, the eﬀect of twobody Coulomb interaction has a decisive eﬀect on theaverage energy EDIAS , while the spin-orbit and residual isospin and spin-isospin interactions play a substantial role in the average energy EDGTR − EDIAS .
More speciﬁcally, we have found that the corrections
due to quartic commutators follow the approximate
laws EDIAS − 2EIAS ≈ 23 A−1/3 MeV (even the isospin
mixing eﬀects are accounted in EIAS ) and EDGTR −EDIAS − 2(EGTR − EIAS ) ≈ 16A−1 MeV.

(1)

(2)

(3)

assuming that the ground state has good isospin;
†
∗1
∗2
∗3
∗4(5)

EDGTR = 2EGT +is the deﬁnition of the DIAS state in terms ofthe IAS that, in turn, can be written as |IAS⟩ ≡∑AT |0⟩/⟨0|T+ T− |0⟩1/2 , where T+ = i t+ (i) and T− =∑−A
i t− (i) are the isospin raising and lowering operators, respectively.
Starting from Eq. (1) and the deﬁnitions of DIASand IAS previously given, one may write the excitationenergy of DIAS as
EDIAS =

22⟨0|[O+, [H, O−]]|0⟩,22⟨0|O+ O− |0⟩

where the GT transition operators are O± =∑A
i σz (i)t± (i). After some algebraic manpulations, we
can rewrite the energy of DGTR (5) as

where |0⟩ represents the ground state, and|DIAS⟩ ≡⟨0|[T+ , [T+ , [[H, T− ], T− ]]]|0⟩. (4)2(N − Z)(N − Z − 1)

Reference

Condensed from the article in Phys. Rev. C. 101, 014320(2020)
Dipartimento di Fisica, Università degli Studi di Milano
INFN, Sezione di Milano
RIKEN Nishina Center
Center for Mathematics and Physics, the University of Aizu

1) M. Takaki et al., Proposal for Nuclear Physics Experiment at RI Beam Factory, “Search for double GamowTeller giant resonances in ββ-decay nuclei via the heavyion double charge exchange 48 Ca(12 C, 12 Be(0+2 )) reaction.”- 39 -1