aa r X i v : . [ nu c l - t h ] A ug Nuclear structure studies of double Gamow-Teller strength
N. Auerbach and Bui Minh Loc ∗ School of Physics and Astronomy,Tel Aviv University, Tel Aviv 69978, Israel (Dated: August 29, 2018)
Abstract
The double Gamow-Teller strength distributions in even- A Calcium isotopes were calculatedusing the nuclear shell model by applying the single Gamow-Teller operator two times sequentiallyon the ground state of the parent nucleus. The number of intermediate states actually contributingto the results was determined. The sum rules for the double Gamow-Teller operator in the fullcalculation were approximately fulfilled. In the case that the symmetry is restored approximatelyby introducing degeneracies of the f -levels, and the p -levels in the f p -model space, the agreementwith the sum rules was very close. ∗ [email protected] . INTRODUCTION The double charge-exchange (DCX) processes are a promising tool to study nuclear struc-ture in particular nucleon-nucleon correlations in nuclei. In the 1980s, the DCX reactionsusing pion beams that were produced in the three meson factories at LAMPF, TRIUMF, andSIN were performed. Studies at lower pion energies ( E ≤
50 MeV) have indeed producedclear signals of nucleon-nucleon correlations [1–5] and were successfully explained by thetheoretical studies [6, 7]. The pion DCX experiments discovered the double isobaric analogstates (DIAS) [8, 9]. At higher pion energies (
E >
300 MeV), the studies discovered thegiant dipole resonances (GDR) built on the IAS [10–13], and double giant dipole resonances(DGDR) [14–17] (See Refs. [10, 14] for definitions).At present, there is a renewed interest in DCX reactions, to a large extent due to theextensive studies of double beta-decay, both the decay in which two neutrinos are emitted(2 νββ ) and neutrinoless double beta (0 νββ ) decay. In DCX and ββ decay, two nucleons areinvolved. The pion, however, interacts weakly with states involving the spin and the pionDCX reactions do not excite the states involving the spin, such as the double Gamow-Teller(DGT) state. The DGT strength is the essential part of the ββ decay transitions. It wassuggested in the past that one could probe the DGT state and hopefully the 0 νββ decayusing DCX reactions with light ions [18, 19].In present days, DCX reactions are performed using light ions. There is a large programcalled NUMEN in Catania where reactions with O, and Ne have been done [20]. Thehope is that such studies might shed some light on the nature of the nuclear matrix elementof the double beta-decay and serve as a “calibration” for the size of this matrix element.These DCX studies might also provide new interesting information about nuclear structure.One of the outstanding resonances relevant to the 0 νββ decay is the double Gamow-Teller(DGT) resonance suggested in the past [18, 21]. At RIKEN, there is a DCX program usingion beams for the purpose of observing DGT states and other nuclear structure studies [22].At Osaka University, the new DCX reactions with light ions were used to excite the doublecharge exchange state and compare to the pion DCX reaction results. One additional peakappeared in the cross-section suggesting that it is a DGT resonance [23].The DGT strength distributions in even- A Neon isotopes was discussed in Ref. [24] andrecently the calculation of DGT strength for Ca was performed in Ref. [25]. In the present2aper, the DGT transition strength distributions in even- A Calcium isotopes are calculatedin the full f p -model space using the nuclear shell model code NuShellX@MSU [26, 27]. The single
Gamow-Teller operator is applied two times sequentially on the ground state of theparent nucleus to obtain the DGT strength. This method is different from the method usedin Refs. [24, 25].The properties of the DGT distribution are examined and limiting cases when the SU(4)holds or when the spin orbit-orbit coupling is put to zero are studied. DGT sum rules werederived in Refs. [24, 28, 29], and recently discussed in Ref. [30]. The DGT sum rules wereused here as a tool to asses whether in our numerical calculations most of the DGT strengthis found.
II. METHOD OF CALCULATION
The notion of a DGT was introduced in Refs. [18, 21]. First of all, the nuclear shell modelwave functions of the initial ground state, J = 1 + intermediate states, and J = 0 + , + finalstates were obtained using the shell model code NuShellX@MSU [26, 27] with the FPD6 [31]and KB3G [32] interactions, in the complete f p -model space. The maximum of the numberof intermediate states is 1000. Table I shows the total number of final states that is possiblein Ti isotopes. If the total number of final states is larger than 5000, the calculations weredone up to 5000 final states. As one will see later, that is enough to exhaust almost thetotal strength.After all wave functions were obtained, the single GT operator was applied two timessequentially. First, all transitions from the parent nucleus 0 + to all 1 + intermediate statesare calculated and then all transitions from 1 + intermediate states to each 0 + or 2 + in the TABLE I. The total number of final states in the f p -model space and f -model space including the f / and the f / orbits only. Ti Ti Ti Ti J πf + + + + + + + + f p -space 4 8 158 596 2343 9884 14177 61953 f -space 2 1 29 99 180 741 446 1899 FIG. 1. Illustration of the method of calculation described in the text. The notations 1 +max and0 +max are the highest states that can be reached in practice. operator is denoted as Y ± = A X i =1 σ t ± ( i ); t ± = t x ± it y , (1)with t − n = p and t + p = n where 2 t x and 2 t y are the Pauli isospin operators and σ is Paulispin operator. Then the single GT transition amplitude from the initial state | i i to the finalstate | f i is M ( GT ± ; i → f ) = h f || Y ± || i i√ J i + 1 , (2)and the GT transition strength given by B ( GT ± ) = | M ( GT ± ; i → f ) | (3)obeys the ”3( N − Z )” sum rule: X f B ( GT − ) − X f B ( GT + ) = S GT − − S GT + = 3( N − Z ) , (4)where the P f means summing over all eigenstates of J f T f . Because the f p -model spaceis limited, only the valence neutrons participate in the calculation for Calcium isotopes.Therefore, we have S GT + = 0. 4he dimensionless DGT transition amplitude is defined as M ( DGT ± ) = X n M ( GT ± ; i → n ) M ( GT ± ; n → f ) , (5)where n are the intermediate states. Note that this is a coherent sum. Finally, the DGTstrength is given by B ( DGT ± ) = | M ( DGT ± ) | , (6)or in more detail B (DGT − ; i → n → f ) = 12 J i + 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X n * f (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X i σ i t − ( i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n + * n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X j σ j t − ( j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i +(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (7)Note that B ( DGT − ; i → n → f ) depends on J f with J f = 0 and 2 only. The matrix elementin the case J f = 1 vanishes because the DGT operator changes sign under the interchangeof coordinates of two particles.The DGT sum rule for J f = 0 is given in Refs. [28, 29] and J f = 0 , S J f =0DGT = 6( N − Z )( N − Z + 1) − ,S J f =2DGT = 30( N − Z )( N − Z −
2) + 5∆ , (8)where ∆ = √ h | [ Y + × Y − ] (1) · Σ − Σ · [ Y − × Y + ] (1) | i , with Σ = P i σ ( i ). There is afactor of three difference between our work and the work in Refs. [24, 25] because the spinoperator is not projected in our calculation. The first terms of the sum rules depend onlyon N and Z , and the second terms (2∆ or 5∆) need to be calculated separately. The signof the second term makes the first one to be the upper limit for J f = 0 + and lower limit for J f = 2 + . III. RESULTS AND DISCUSSIONS
It is well-known that the single
GT strength is quenched (see Ref. [33]). In the shellmodel calculations, GT strength is fragmented. This is demonstrated in the case of Cain Fig. 2 as an example. The results were obtained with two different interactions that areoften used in the f p -model space FPD6 and KB3G interactions. Within the range of about17 MeV excitation energy, S GT − is approximately 24 exhausting the “3( N − Z )” sum rule5 IG. 2. The cumulative sum of the single Gamow-Teller strength B(GT) as a function of the numberof 1 + states (a) and excitation energies (b) of the intermediate nucleus Sc. The calculation usedFPD6 and KB3G interactions in the f p -model space. ( S GT + = 0 in our calculation). The cumulative sum of the single GT strength S(GT) as afunction of number of Sc states is shown in Fig. 2. One can expect that there are about500 intermediate 1 + states in Sc that actually contribute to the final results of the DGTstrength although the total number of 1 + states in this nucleus is many thousands.For the study of the DGT, first, we calculated the sum rule using it as a tool to asseswhether in our numerical calculations most of the DGT strength is found. The ∆ in Eq. (8) TABLE II. The properties of DGT transition using FPD6 and KB3G interactions in the complete f p − model space. S (DGT) is the DGT total strength with the FPD6 interaction. B is thetransition from the g.s. of the parent nucleus to the first J + state of the final nucleus. E (MeV)is the average energy (Eq. (9)).Initial nucleus Ca Ca Ca Ca J πf + + + + + + + S (DGT) 28.1 19.5 102.0 284.0 223.7 752.6 385.0Sum rule ≤ ≥ ≤ ≥ ≤ ≥ ≤ B (FPD6) 16.172 6.117 0.654 0.000 0.201 0.017 0.109 B (KB3G) 17.010 5.942 0.895 0.119 0.182 0.057 0.072 E (FPD6) 6.1 4.8 16.3 13.2 21.2 18.0 24.6 E (KB3G) 6.1 5.5 14.7 12.2 19.0 16.9 21.9 IG. 3. The cumulative sum of the double Gamow-Teller strength B (DGT; 0 + → + ) in Ca.FIG. 4. The cumulative sum of the double Gamow-Teller strength B (DGT; 0 + → + ) in Ca. can be obtained directly by subtracting from total sum the first term that depends onlyon N and Z (see Table II). In Ref. [29], ∆ was related to the magnetic dipole transition S ( M A isotopes including Calcium isotopes. Our results given in Table II are in agreementwith them (Note that there is a factor of three difference between our work and Ref. [30]).It means we exhaust all the DGT strength in the study. Obviously, the total DGT strengthdoes not depend on the choice of interaction.Because all the strengths are obtained, we can show not only the values of the total sumbut also the cumulative sums of the DGT. The cumulative sums are given in Fig. 3, and7 IG. 5. The cumulative sum of the double Gamow-Teller strength B (DGT; 0 + → + ) in Ca.
Fig. 4 for Ca, Fig. 5, and Fig. 6 for Ca, and Fig. 7 for Ca. In these figures, the solidlines are the shell model calculations described in Section II using the FPD6 interaction.They are denoted as “FPD6 with LS”. The results using the KB3G interaction are alsoshown as dotted lines. The long-dash lines are the calculation with the FPD6 interactionin the SU(4) limit. The SU(4) limit in our work is restored approximately by making the f / and f / ; p / and p / degenerate following Ref. [34]. It means there is no spin-orbitcoupling and therefore they were denoted as “FPD6 without LS”. We want to show that inthe SU(4) limit, the cumulative sums approach the horizontal lines (denoted as the “SU(4)limit”) that represent the values of the terms that depend only on N and Z in Eq. (8).It is in agreement with the fact that ∆ vanishes in the SU(4) limit according to Ref. [35].In the cases of Ca (Fig. 3, and Fig. 4), the sum rules are totally exhausted because allintermediate states and all final states can be taken into account. In the cases of Ca, and Ca (Figs. 5–7), the cumulative sums are still increasing. For the case of the DGT to the2 + in Ca, we choose not to do the calculation because the total number of final states aretoo large. The result is not convergent using the standard NUSHELLX@MSU code [36].Most of the sum rule is satisfied, and therefore the entire distributions of DGT strengthnow can be discussed. We remind that Ref. [24] showed the entire DGT distributions foreven- A Neon and recently Ref. [25] showed the result for Ca for the first time. For thelightest nucleus, Ca, the DGT distributions with FPD6 and KB3G interactions are shownin Table III and IV. The difference between the results of the two interactions is not large.8
IG. 6. The cumulative sum of the double Gamow-Teller strength B (DGT; 0 + → + ) in Ca.
In the case B (DGT; 0 + → + ) of Ca, the transition from the g.s. of the parent nucleus tothe first J + state of the final nucleus ( B ) is large because the g.s. of Ti is the DIAS ofthe g.s. of Ca. Moreover, in the SU(4) limit the g.s. of Ti absorbs all the DGT strength(36) following Refs. [28, 29].The DGT distributions are drawn in Figs. 8–12. They contain inserts which show theDGT strength in the low-lying states of , , Ti. B is a very tiny fraction of the totalstrength. For example, the strength in the ground state of Ti is only 3 × − of the totalstrength (see Table II). This strength enters in the calculation of the ββ decay. In Ref. [25], FIG. 7. The cumulative sum of the double Gamow-Teller strength B (DGT; 0 + → + ) in Ca. IG. 8. B (DGT; 0 + → + ) for Ca as the function of the excitation energy of the final nucleus Ti (a). The DGT transitions to low-lying states are also shown (b). The strengths were smoothedby using Lorentzian averaging with the width of 1 MeV. it is pointed out that a very good linear correlation between the DGT transition to theground state of the final nucleus and the 0 µββ decay matrix element exists.When the strengths are spread by using the same Lorentzian averaging with the widthof 1 MeV to simulate the experimental energy resolution, the results show that the DGTdistributions are not single-peaked. The distributions have at least two peaks and in somenuclei as many as four major peaks. We should remind that the single GT resonances haveat least two peaks [37].
TABLE III. B (DGT; 0 + → + ) for Ca using FPD6 and KB3G interactions in the complete f p -model space. E ex (MeV) is the excited energy of Ti.FPD6 KB3G E ex B (DGT) S (DGT) E ex B (DGT) S (DGT)0.0 16.172 16.172 0.0 17.010 17.0106.0 0.442 16.614 5.7 0.281 17.29110.9 0.782 17.396 11.3 0.120 17.41114.9 10.692 28.088 15.4 11.085 28.496 ABLE IV. The same as Table III, but for B (DGT; 0 + → + )FPD6 KB3G E ex B (DGT) S (DGT) E ex B (DGT) S (DGT)0.0 6.117 6.117 0.0 5.942 5.9432.3 1.536 7.653 2.6 0.520 6.4635.1 0.125 7.778 5.2 0.009 6.4726.6 5.523 13.301 7.2 1.355 7.8277.3 4.916 18.217 7.8 9.679 17.5069.7 0.071 18.288 10.1 0.188 17.69411.6 0.039 18.327 11.9 0.017 17.71114.2 1.148 19.475 14.2 1.047 18.757FIG. 9. The same as Fig. 8, but for B (DGT; 0 + → + ). Figs. 13–15 show the dependence of the DGT distributions on the number of intermediatestates. We can see that about 100 intermediate states actually contribute to final resultsin the cases of Ca (Figs. 13–14). Although the total number of intermediate states inheavier isotopes, including Ca is many thousands, about 500 intermediate states actuallycontribute to final results (Fig. 15). The sum rule is useful to determine this number (Weremind that the number of intermediate states involved in the calculations for 0 νββ decay is11
IG. 10. B (DGT; 0 + → + ) in Ca.FIG. 11. B (DGT; 0 + → + ) in Ca. smaller (see Ref. [38])). Fig. 16 and Fig. 17 show the DGT transitions to J πf = 0 + togetherwith the transition to J πf = 2 + in Ca and Ca. As one can see the DGT transitions to J πf = 2 + are larger than the transitions to J πf = 0 + .The centroid or average energies of the DGT distributions defined by E = P f E f B f ( DGT − ) P f B f ( DGT − ) (9)are given in Table II. In Ti, with the FPD6 interaction for example, the average energyfor the J = 0 + is E = 21 . J = 2 + it is lower, E = 18 . Tithe average energy J = 0 + is E = 24 . IG. 12. B (DGT; 0 + → + ) in Ca.FIG. 13. The dependence on the number of intermediate states of B (DGT; 0 + → + ) in Ca. Thenumbers in parentheses are the corresponding total strengths. average energy of the Ca DGT giant resonance and 0 µββ decay nuclear matrix elementwas pointed out. The authors conclude that the uncertainties due to the nuclear interactionin the calculation of the DGT distribution in Ca are relatively under control. In our work,Figs. 18–20 show the DGT distributions are calculated with FPD6 and KB3G interactions.We see that the distributions and the average energies (see Table II) using FPD6 and KB3Gare similar. Our calculated distribution for Ca is in agreement with the recent result usingthe same KB3G interaction but a different method (when the factor of three is taken intoaccount). As one can see in Fig. 20 the DGT giant resonance in Ca is at the energy around13
IG. 14. The same as Fig. 13, but for B (DGT; 0 + → + ) in Ca.FIG. 15. The same as Fig. 13, but for B (DGT; 0 + → + ) in Ca. Fe( B, Li) were presented. In this reaction, several resonances were excited inagreement with the pion DCX studies. In addition, there is a peak at 25 MeV excitation,that the authors indicate that it could be the DGT resonance.In addition, the calculations in the f -model space (including the f / and the f / orbitsonly) using the same Hamiltonian are given in Fig. 21. For Ca, there are two 0 + DGTstates at the excitation energies 0 . . .
178 and13 . + DGT state at 0 . . IG. 16. B (DGT; 0 + → + ) and B ( DGT ; 0 + → + ) in Ca.FIG. 17. The same as Fig. 16 but for Ca.FIG. 18. B (DGT; 0 + → + ; 2 + ) in Ca using FPD6 and KB3G interactions. f -model space, weobtained exactly the sum rules even for the case of the DGT to 2 + in Ca because thecalculation can be done without any limitation as now the total number of possible finalstates is strongly reduced (see Table II). The DGT distributions in the f -model space aremuch more concentrated. The analytical calculation in the limited f -model space helps usknow where the DGT strengths concentrate. FIG. 19. The same as Fig. 18, but for Ca.FIG. 20. B (DGT; 0 + → + ) in Ca using FPD6 and KB3G interactions. The strengths are spreadwith the width 1 MeV and divided by the factor of 3 for comparing to Ref .[25]. IG. 21. The DGT distributions in the f -model space (including the f / and the f / orbits only)in Calcium isotopes using the FPD6 interaction. IV. CONCLUSIONS
The general features and trends of the DGT sum rules in even- A Calcium isotopes aredescribed using the numerical results. The properties of the entire distributions of theDGT are discussed. By studying the stronger DGT transitions, in particular, the DGTgiant resonance experimentally and theoretically, the calculations of ββ -decay nuclear matrixelements can be calibrated to some extent. There is no doubt that the pion DCX is a sensitiveprobing tool of nuclear structure. Nowadays the ion DCX reactions have been discussedmainly in the context of 0 νββ , however, the ion DCX reaction itself is a new probing toolof nuclear structure, in particular of spin degrees of freedom. The DGT resonance is justone example. Because two nucleons participate in the DCX reactions with pions or heavy17ons, one can expect that the nucleon-nucleon interaction and correlations can be probed,especially for the nucleus that is far from the stability regime. ACKNOWLEDGMENTS
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