Ⅱ-2. Nuclear Physics (Theory)

RIKEN Accel. Prog. Rep. 50 (2017)

Sum rule study for Double Gamow-Teller state†∗1,∗2 and T. Uesaka∗1H.Sagawa∗1,∗2and T. Uesaka∗1
H.Sagawa,

initial state |Ji Mi  by the simple notation |i. The sum
rule value for J = 0 excitations is evaluated as

Double charge exchange excitations (DCX) inducedby heavy ion beams at intermediate energies1,2) attract
a lot of interest with respect to new collective excitations such as the double Gamow-Teller giant resonance(DGTR) and double beta decay matrix elements. Anew research program based on a new DCX reaction(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 =250MeV/u to excite the spinisospin response. One notable advantage of this reaction is based on the fact that it is a (2p, 2n) type
DCX reaction, and one can use a neutron-rich target to excite DGT strength. Although many theoretical eﬀorts have been made to study double beta decays, DGT strengths corresponding to the double beta
decays are still too small to be identiﬁed in the reaction experiments. Minimum-biased theoretical prediction based on sum rules will provide a robust andglobal view of the DGTR and can be a good guideline for future experimental studies. In this paper, we
present useful formulas for analyzing DGT strength
using several sum rules for the double spin-isospin operator (σt− )2 . We also study DIAS excited throughthe double Fermi transition operator (t− )2 and double
GT operator (σt− )2 .
The sum rule for the single GT transitions is wellknown and proportional to the neutron excess,(J=0)= 2(N − Z)(N − Z + 1)
4
(N − Z)S+ − i|[iΣ̂ · (Ô− × Ô+ ) + Σ̂ · Σ̂]|i ,+3(5)where Σ̂ = α σ(α) and (Ô− × Ô+ ) is the vector product of two operators. The sum rule for the J π = 2+
ﬁnal states can be obtained in a similar way to thatfor the J = 0 states:(J=2)

fSum rule values for DGT transitions to J π = 0+ and2+ in Eqs. (5) and (6) are calculated for 10 nuclei,6He, 8 He, 14 C, 18 O , 20 O, 42 Ca, 44 Ca, 46 Ca, 48 Ca,
and 90 Zr, together with those for the DIAS states bythe Fermi and GT operators. In N >> Z nuclei, thesum rule values of DGT transitions are approximately
proportional to a factor (2J+1), i.e., the value for J =2 is ﬁve times larger than that for J = 0 in the same
nucleus. However, this proportionality is signiﬁcantly
modiﬁed in N ∼ Z nuclei. In the extreme, the sum rulefor J = 2 transitions is smaller than that for J = 0in the nuclei 6 He, 14 C, and 18 O with N = Z + 2 ,and the two values are almost equal in 42 Ca. In nuclei
N > Z +2, the J π = 2+ excitations dominate the DGTstrength because of the multipole factor (2J +1), morethan 0+ excitations. However, in nuclei with N ∼ Z,
the 0+ excitations become substantially strong, evenlarger than 2+ excitations in light nuclei with T = 1
such as 14 C and 18 O. The excitation to DIAS is alsostudied through (σt− )2 and t2− operators to investigate
the possibility of extracting the unit cross sections forDGT strength. We point out that the strength of DIAS
excitations by the (σt− )2 operator is competitive with
the DGT strength in the light T = 1 nuclei. This
characteristic feature may give a good opportunity toextract the unit cross section for DGT strength.

f(1)where the GT transition operators readσ(α)t± (α).
Ô± (GT) =(2)αThe DGT transition operator Ô± (GT)2 can be projected to good multipole states to be[Ô± (GT) × Ô± (GT)]Jμ , J = 0, 2.

(3)

The sum rule strength is expressed by the reduced matrix element
1JD±|Jf ||[Ô± × Ô± ]J ||Ji |2 ,(4)=2Ji + 1
JfReferenceswhere Ji and Jf are the angular momenta of the initial
and ﬁnal states, respectively. Hereafter, we denote the†∗1∗2(J=2)D−− D+= 10(N − Z)(N − Z − 2)102(N − Z)S+ + i|[iΣ̂ · (Ô− × Ô+ ) + Σ̂ · Σ̂]|i .
+3(6)S− − S+=|f |Ô− (GT)|i|2 −|f |Ô+ (GT)|i|2 ,= 3(N − Z),(J=0)

D−− D+= i| [Ô+ × Ô+ ](J=0) , [Ô− × Ô− ](J=0) |i,1) M. Takaki et al.: Proposal for experiment at RCNP,“Search for double Gamow Teller giant resonancesin 48 Ti via the heavy-ion double charge exchange
48
Ca(12 C, 12 Be(0+2 )) reaction” (2015).
2) F. Cappuzzello et al., Journal of Physics: Conference
Series 630, 012018 (2015).

Condensed from the article in Phys. Rev. C 94,064325(2016)
RIKEN Nishina Center
Center for Mathematics and Physics, University of Aizu

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