Evidence of a higher nodal band α+44Ca cluster state in fusion reactions andα clustering in 48Ti

S. Ohkubo1

1Research Center for Nuclear Physics, Osaka University, Ibaraki, Osaka 567-0047, Japan(Dated: October 11, 2021)

In the nucleus 48Ti, whose structure is essential in evaluating the half-life of the neutrinolessdouble-β decay (0νββ) of 48Ca, the existence of α cluster structure is shown for the first time. Aunified description of scattering and structure is done for the α +44Ca system. By using a globalpotential, which reproduces experimental α +44Ca scattering over a wide range of incident energies,Eα=18 - 100 MeV, it is shown that the observed α +44Ca fusion excitation function at Eα=9 -18MeV is described well. The bump at Eα=10.2 MeV is found to be due to a resonance which is a 7−

state of the higher nodal band with the α +44Ca cluster structure in 48Ti. The local potential α+44Ca cluster model locates the ground band of 48Ti in agreement with experiment and reproducesthe enhanced B(E2) values in the ground band well. This shows that collectivity due to α clusteringin 48Ti should be taken into account in the evaluation of the nuclear matrix element in the 0νββdouble-β decay of 48Ca.

Alpha clustering is essential not only in the light 0p-shell and sd shell regions [1, 2] but also in the medium-weight fp shell region [3, 4] as evidenced typically in the44Ti region [5–15]. Recent evidence of the higher nodalband states with the α cluster structure in 52Ti [16], inwhich the intercluster relative motion is excited, suggeststhat α clustering may persist in nuclei in-between suchas 48Ti and 46Ti. In fact, the α spectroscopic factors of48Ti and 46Ti in the (6Li,d) α-transfer reactions [17, 18]are larger than that of 52Ti, which is the minimum inthe A=36-64 mass region [17]. This is also confirmed inthe (p, pα) reactions [19–21]. Ref. [22] reports that theexcess neutrons outside the core work as covalent bondingbetween the clusters [22].

The structure of 48Ti is crucial in determining the na-ture of neutrino, Dirac or Majorana particle in the mea-surements of neutrinoless double-β decay 0νββ of 48Ca[23]. The study of 0νββ [24], which violates lepton num-ber conservation, serves to solve the longstanding fun-damental questions beyond the standard model. The in-verse half-life of 0νββ of 48Ca(0+)→ 48Ti(0+) is given by[T 0ν

1/2]−1 = G0ν |< mββ > /me|2 |M0ν |2, where < mββ >

is the effective Majorana neutrino mass, me is the elec-tron mass, and G0ν ∼ 10−14 yr−1 is a phase-space factor.For the evaluation of the nuclear matrix element (NME)of the transition M0ν [25], it is essential to know theground state wave function of 48Ti accurately.

Several approaches such as the shell model [26–29], abinitio calculations [30, 31], quasi-particle random phaseapproximation (QRPA) [32–34], the projected Hartree-Fock Bogoliubov model [35], the generator coordinatemethod (GCM) [36, 37], the energy density functional[38, 39] and the interacting boson model [40] have beenreported. As for the α clustering aspects of 48Ti, a mi-croscopic α+44Ca cluster model calculation in the GCMwas done [41, 42]. However, no state corresponding tothe ground state of 48Ti was obtained. Experimentallyin the recent challenges [43, 44], no α cluster states havebeen observed in 48Ti. A discovery of a typical α cluster

state like the higher nodal band states observed in 52Ti[16], would shed light to the α clustering of the groundstate of 48Ti.

The purpose of this paper is to show that the existenceof a higher nodal band α+44Ca cluster structure in 44Tiis evidenced for the first time by investigating the ob-served α+44Ca fusion cross sections with a global Luneb-urg lens-like [45–47] potential that describes α +44Cascattering over a wide range of incident energies, Eα=18- 100 MeV. By using the local potential α+44Ca clustermodel the observed enhanced B(E2) values of the groundband of 48Ti are reproduced well. It is shown that the en-hanced B(E2) values are caused by α clustering in 48Ti.

In exploring the α cluster structure in medium-weightmass region where the level density is high, a unifieddescription of α scattering including rainbow scattering,prerainbows, backward angle anomaly (BAA) or anoma-lous large angle scattering (ALAS) and the α clusterstructure in the bound and quasi-bound energy regionhas been very powerful [4–7, 14, 16]. In fact, the α clus-ter structure in the 44Ti region was successfully exploredfrom this viewpoint and the predicted α cluster K = 0−

band with the α+40Ca cluster structure [5–7], which is aparity-doublet partner of the ground band of 44Ti, wasobserved in experiment [8, 9]. Systematic theoretical andexperimental studies in the 44Ti region [10–15] confirmedthe existence of the α cluster in the beginning of the fp-shell.

The BAA in α particle scattering, which was firstlyobserved for α+40Ca [48, 49], was systematically investi-gated in comparison with the isotopes, 42Ca and 44Ca [50]at low energies, Eα=18-29 MeV. It was concluded [50]that α+ 44Ca scattering with no backward enhancement,which is described well by the so-called standard opticalwith a Wood-Saxon form factor, is normal. The normalbehavior of the angular distributions was attributed tostrong absorption due to the excess neutrons outside the40Ca core [41, 42, 50, 51] and little attention has beenpaid to the α+44Ca cluster structure of 48Ti in contrast

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FIG. 1. (Color online) Angular distributions in α+44Ca scat-tering at Eα=18 and 24.1 MeV calculated with the opticalpotentials in Table I (solid lines) are compared with the ex-perimental data (circles) [50]. The calculated angular distri-butions are decomposed into the internal-wave (dotted lines)and the barrier-wave (dashed lines) contributions. At 18 MeVthe solid line overlaps with the dotted line and the dashed lineis negligibly small and is not drawn.

TABLE I. The optical potential parameters used in Fig. 1 andthe volume integrals per nucleon pair, Jv, in unit of MeVfm3

for the real potentials. Eα, V , W and Ws are in units of MeVand rv, av, rw, aw, rs and as are in units of fm.

Eα Jv V rv av W rw aw Ws rs as

18 388 181 1.42 1.25 14.0 1.75 0.934 53.2 1.36 0.37824.1 356 166 1.42 1.25 10.6 1.75 0.934 17.0 1.36 0.378

to the α+40Ca cluster structure in 44Ti.I extend the global optical potential determined

in α+44Ca scattering over a wide range of ener-gies of Eα=24 -100 MeV [51] to the lower energiesEα=9 - 24 MeV. The optical potentials are given byU(r) = −V f2(r;Rv, av) + VCoul(r)− iWf2(r;Rw, aw)−i4asWs

ddrf

2(r,Rs, as) with f(r;Ri, ai) = 1/{1 +exp[(r −Ri)/ai]}. The Coulomb potential is assumedto be a uniformly charged sphere with a reduced radiusrc=1.3 fm. Firstly I show that the experimental angulardistributions in α+44Ca scattering at the lowest energyEα=18 MeV is reproduced well by the global potential.In Fig. 1 the calculated angular distributions at 18 MeVand 24.1 MeV are compared with the experimental data.The potential parameters V , W and Ws are searchedwith other parameters fixed as in Ref. [51], which are

FIG. 2. (Color online) Calculated excitation function of theα+ 44Ca fusion cross sections (solid lines) is compared withthe experimental data (filled circles) [55].

listed in Table I. The fits to the experimental data aremuch better than the ones calculated with the standardoptical model in Ref. [50] and the improvements are en-tirely due to the Luneburg lens-like shape of the real po-tential [45–47]. To see why BAA of the cross sections isabsent, the calculated angular distributions are decom-posed using the technique of Ref.[52] into the barrier-wave component reflected at the surface and the internal-wave component, which penetrates deep into the internalregion of the potential [53]. In Fig. 1, the barrier-wavesdominate and the internal-waves hardly contribute, notvisible at 18 MeV. The absence of the BAA in α+44Cascattering is due to no internal-waves contributions un-der strong absorption, and not due to the real part of thepotential since it is similar to the α+40Ca system as willbe shown in Fig. 6.

Secondly I investigate the α+44Ca fusion cross sec-tions below Eα=18 MeV. In contrast to α+40Ca [5, 54–56], the α+44Ca fusion excitation function with no pro-nounced oscillations [55] has never been paid attention inthe last decades. However, one note in Fig. 2 that the ob-served excitation function is not monotonic with a bumpat around Eα=10.2 MeV where absorption is relativelysmall near the threshold. The origin of this bump, whichmay be a remnant of the fusion oscillations seen typi-cally for α+40Ca [54–56], 12C+12C [57], 16O+12C and16O+16O [58], seems to be related to the α+40Ca molec-ular resonance.

In calculating the fusion cross sections using the opti-cal potential, I followed the prescription used for α+40Cain Ref.[54] where the fusion cross section is defined as atotal reaction cross section due to the short-ranged imag-inary potential. The fusion cross sections at Eα< 18MeV are calculated using the optical potential at Eα=18MeV in Table I with rw=1.3 for the volume imaginarypotential and Ws=0 for the surface imaginary potential

3

FIG. 3. (Color online) Calculated excitation functions of theα+ 44Ca fusion cross sections (solid lines) are decomposedinto the partial wave contributions; even L (dotted lines) andodd L (dashed lines). The inset is for Eα=8.4-9.0 MeV.

due to direct reactions. The energy dependence of thestrength parameter, W , which was originally determinedat Eα>24 MeV in Ref. [51] is modified for Eα <18 MeVto decrease linearly toward the barrier top energy dueto the dispersion relation of the threshold anomaly [59],W=aEα + b for Eα ≥ 8.4 MeV with a=1.281 and b=-10.74.

In Fig. 2 the calculated fusion excitation function iscompared with the experimental data [55]. It is to benoted that the bump at Eα=10.2 MeV, which was notreproduced in Ref. [55], is reproduced well by the cal-culation. Also the calculated fusion cross sections donot decrease monotonically toward the threshold energyin accordance with the very smooth oscillatory behav-ior of the experimental data, which is a remnant of thepronounced oscillatory structure seen in the fusion exci-tation function for α+40Ca in the same energy region.The emergence of a bump at Eα=10.2 MeV is due to therelatively weak absorption near the barrier top energy,E=6.2 MeV, which corresponds to Eα=7.2 MeV.

In Fig. 3 the calculated fusion cross sections are decom-posed into the partial cross sections. One notes that thebump peak is caused by the partial fusion cross sectionswith L=7 (dashed line). The broad resonance-like be-havior of the L=6 and 8 partial fusion cross sections con-tributes to the non-monotonic behavior of the fusion ex-citation function. The calculated fusion excitation func-tion at Eα=8.4- 9 MeV is displayed in the inset. Althoughthe magnitude of the cross sections becomes smaller, onenotices the appearance of a bump at Eα=8.5-8.6 MeVwhere the absorption is much smaller than the bump atEα=10.2 MeV. It is found that this structure is causedby the L =5 partial cross sections (dashed line).

In order to reveal the origin of the bump structures inthe fusion excitation function, in Fig. 4 the phase shiftscalculated with V =181 MeV potential (referred as D181

FIG. 4. (Color online) The phase shifts for α+ 44Ca scatter-ing calculated with the D181 real potential. The vertical axisis δL-mLπ where mL is the number of the Pauli-forbiddenstates. See text. The resonances for odd L and even L corre-spond to N = 15 and N =16, respectively.

hereafter) at Eα=18 MeV by switching off the imaginarypotentials are displayed. One notes that the resonance atEα=10.3 MeV with a width Γc.m.=0.18 MeV is responsi-ble for the peak of the partial fusion cross section for L =7and therefore for the bump observed at around Eα=10.2MeV. The resonance at Eα=8.3 MeV with Γc.m.=0.22MeV is also responsible for the bump at Eα=8.3 MeV inthe inset of Fig. 3. There appear resonances for L =9 and11 at Eα=12.96 (Γc.m.=0.11 MeV) and 16.07 (Γc.m.=0.04MeV), respectively, however, the contributions to the fu-sion cross section are not seen clearly. The broad L =6and 8 resonances at around Eα=12 MeV and 16 MeV,respectively, are also the origin of the non-monotonic be-havior of the fusion excitation function.

How these resonances are understood as the highly ex-cited states with the α+44Ca cluster structure in 48Ti?In Fig. 5(a) the energy levels calculated with the D181potential is displayed. The number of the Pauli-forbiddenredundant states in the α+44Ca cluster model [41] ismL = (12− L)/2 for even L ≤ 12 and mL = (13− L)/2for odd L ≤ 13. mL = 0 for L ≥ 14. According tothe generalized Levinson theorem [60], the phase shiftδL in which the existence of the Pauli-forbidden statesare taken into account should satisfy δL =mLπ at Eα=0and tends asymptotically to δL =0 at Eα=∞. Shownin Fig. 4 are the phase shifts, δL-mLπ, to make easy tosee the band structure of N = 15 and N = 16 whereN = 2n+L with n being the number of nodes in therelative wave function with the α+44Ca molecular struc-ture. Surprisingly the calculated lowest Pauli-allowedN =12 band, which satisfies the Wildermuth conditiondue to the Pauli principle, falls in correspondence to theexperimental ground band of 48Ti. One finds that theL = 7 resonance that contributes to the bump of the fu-sion cross section at Eα=10.2 MeV is a member state ofthe higher nodal N = 15 band with the α+44Ca clusterstructure, in which the intercluster relative motion is onemore excited compared with the N = 13 band. This 7−

of N = 15 is a second example of the higher nodal band

4

FIG. 5. Energy levels of 48Ti with respect to the α thresh-old calculated with the potentials (a) V=181 MeV and (b)V=171 MeV are compared with (c) the experimental groundstate band [67]. The energy levels are calculated in the boundstate approximation except the N=15 band resonances. Thehorizontal dotted lines indicate the incident energy Eα=8.4 -18 MeV of the fusion excitation functions in Fig. 3.

with negative parity in addition to 44Ti [13, 14], whichgives strong support to the persistence of the α clusterstructures in 48Ti. It is highly desired to observe the 5−

state theoretically predicted at Eα=8.3 MeV as well asthe 3− and 1− states of the N = 15 band in a preciseexperiment such as sub-barrier fusion reactions [61, 62]and transfer reactions etc, although the α-strengths aswell as the N = 13 and 14 bands, may be fragmented asin 40Ca and 44Ti [11–14].

The reason why the 7− state of N = 15 is observed inFig. 3 is related to the considerably high α threshold en-ergy of 48Ti, 9.45 MeV, due to a non- α nucleus. Becauseof this, although the excitation energy Ex=18.8 MeV ofthe 7− state is high, the energy from the α thresholdbecomes relatively small. The N = 14 band, which isa higher nodal band of the ground band N = 12, startsjust above the α threshold. Its observation in additionto the known analog higher nodal bands in 20Ne [63–66],40Ca [11, 12], 44Ti [11, 13, 14] and 52Ti [16, 44] wouldalso reinforce the α cluster structure in 48Ti.

As for the N = 12 ground band, the D181 potentialwith Jv=388 MeVfm3 determined at Eα=18 MeV locatesthe ground state 0+, which overbinds, -4.27 MeV, com-pared with the experimental ground state. When dis-cussing the ground band deep below the threshold, aswas the case in 52Ti [16], the potential must be read-justed by taking into account that the volume integralof the real potential decreases toward the threshold dueto the threshold anomaly [59]. With a decreased poten-

TABLE II. Calculated intercluster rms radii (fm) and B(E2)values (W.u.) for the J → (J − 2) transitions of ground bandof 48Ti. The B(E2) values calculated with the D171 potential(cal-1) and the L-dependent V (MeV) (cal-2) are comparedwith the experimental data [73].

Jπ < R2 >1/2 B(E2) V B(E2)

cal-1 cal-1 exp.[73] cal-2 cal-20+ 4.41 171.02+ 4.37 13.6 15.0 169.4 13.84+ 4.33 17.9 18.4 167.7 18.8

tial strength, V=171 MeV (D171) with Jv=367 MeVfm3,the calculated ground band in Fig. 5(b) falls well incorrespondence with the experimental ground band inFig. 5(c). In Table II calculated rms intercluster radiiand B(E2) values are given. The B(E2) values are cal-culated with the D171 potential and the L-dependentV tuned to reproduce the experimental excitation en-ergy of the ground band becaese the L-dependence of Vhas been known widely, for example, in 20Ne [68], 44Ti[5, 6], 94Mo [69, 70], 212Po [69, 71]and 46,50Cr [72]. Asmall effective charge ∆e = 0.1e is introduced for pro-tons and neutrons. The experimental B(E2) values [73]are reproduced well. This small effective charge seemsreasonable considering that core excitations are impor-tant in the 44Ti region [13, 74] and that the observedB(E2 : 2+1 (1.157MeV) → g.s.) of 44Ca, 10.9 W.u., islarge. The rms charge radius < r2 >1/2=3.71 fm ofthe ground state calculated using the experimental val-ues < r2 >

1/2α =1.676 fm and < r2 >

1/244Ca=3.518 fm [75] is

to be compared with the experimental value 3.59 fm[75].The calculated intercluster distance of the ground stateis about 85% of the sum of the experimental rms chargeradii of the two clusters, which is only slightly smallcompared to 87% for the ground state of 44Ti [6]. ForJ = 0+, the overlaps of the six deeply bound forbid-den states supported by the potential with the harmonicoscillator wave functions with ~ω=10.41 MeV, which isnearly equal to ~ω=10.5 MeV used in the ab initio cal-culations of 48Ti in Ref. [76], are 1.000, 1.000, 0.999,0.999, 0.996 and 0.977 for the oscillator quanta NHO=0,2, · · · , 10, respectively. This means that the obtainedground state wave function is orthogonal to the Pauli-forbidden states with NHO ≤ 10 in the resonating groupmethod mimicking Saito’s orthogonality condition model[77]. The probability with quanta NHO component in theground state wave function is 78.0, 11.3, 4.2, 1.8 and 1.0% for NHO = 12, 14, · · · , 20, respectively. The dominantNHO = 12 shell-model like component, 78%, is largerthan that of 44Ti in Ref. [6]. The existence of the signif-icant amount of higher NHO ≥ 14 components, however,means that the ground state has α clustering if mod-est. The 2+ and 4+ states have the similar character.This means that α clustering, i.e. sizable four-particle

5

FIG. 6. (Color online) The D181 α+44Ca potential at Eα =18MeV (solid line) is compared with the Luneburg lens potential[45–47] with R0=5.68 fm and V0=173 MeV (points) and theenergy-independent equivalent local potential for L =0 in theGCM calculation [42] (medium dashed line). For comparisonthe α+40Ca potential used in the α cluster calculation of 44Ti[6] (long dashed line) and the α+48Ca potential at Eα =18MeV [16] (dash-dotted line) are also displayed.

excitations from the fp-shell to the higher major shells,contributes not only to the large B(E2) values but alsoto the 0νββ decay half-life of 48Ca, making longer thanthe shell model within the NHO = 12 model space.

In Fig. 6 the D181 potential is compared with a Luneb-

urg lens [47] potential, which is a truncated harmonic os-cillator potential [45, 46] given by V (r) = V0

(r2/R2

0 − 1)

for r ≤ R0 and V (r) = 0 for r > R0. The potentialresembles the Luneburg lens potential in the internal re-gion, which explains why the potential embeds the deeplybound Pauli-forbidden states with N <12 and locatesthe Pauli-allowed N = 12 band in correspondence to theexperimental ground band. The energy dependence ofJv=388 MeVfm3 at 18 MeV and 356 MeVfm3 at 24.1MeV are consistently in line with Jv=340 MeVfm3 atEα=29 MeV in Ref. [78]. The D181 potential in Fig. 6is reasonable being in-between the potentials for α+40Caand α+48Ca. The equivalent local potential of the mi-croscopic GCM cluster model calculation[41, 42] belongsto a shallower potential family, which explains why theN =12 band appears above the α threshold energy andthere appears no state corresponding to the ground stateof 48Ti.

To summarize, the α+44Ca fusion excitation functionat Eα=9-18 was reproduced well by a global Luneb-urg lens-like potential, which fits α+44Ca scattering atEα=18-100 MeV. The existence of a 7− state with theα+44Ca cluster structure of the higher nodalN =15 bandin 48Ti was confirmed for the first time at Ex = 18.8MeV in the bump of the fusion excitation function. Thefirst Pauli-allowed N =12 band with the α+44Ca clusterstructure falls below the α threshold in correspondencewell with the experimental ground band of 48Ti. Theexperimental B(E2) values of the ground band were re-produced well by calculations and the enhancement isfound to be α clustering.

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