Quadrupole and monopole transition properties of $0^+_2$ in Gd isotopes

The longstanding problem of characterization of the $0^+_2$ states in Gd isotopes is revisited by adopting the Nilsson$+$BCS mean field and the random-phase approximation. The interband electric quadrupole transition strengths varying almost two orders of magnitude are nicely reproduced at the same time as other observables. These results indicate that the $0^+_2$ states, in particular, those in lighter isotopes are well described as $\beta$ vibrations excited on top of deformed ground states without recourse to the shape-coexistence picture.


I. INTRODUCTION
The first 0 + excitation, denoted as 0 + 2 , is one of the fundamental excitations in atomic nuclei. It carries information about the nuclear shape and the pairing correlation. In medium and heavy nuclei, ground states of those which are off closed shell are more or less deformed.
Actually the latter has been widely confirmed in the nuclear chart. In contrast, properties of observed 0 + 2 are still controversial [2]. The most decisive observable is B(E2, 0 + 2 → 2 + 1 ). But its data often have relatively large error bars and they vary strongly with nucleon numbers. Not only as a particle-hole collective state, the shape vibration, but 0 + states can be excited as a particle-particle collective state, pairing vibration, via two-nucleon (2n) transfer reactions. In superfluid nuclei, their typical crosssection to 0 + 2 is estimated as 2% of that to 0 + 1 [3]. In the 70's, a lot of (p, t) and (t, p) experiments were done and found that transfer crosssections to 0 + 2 became comparable with those to 0 + 1 in transitional N = 88 − 90 nuclei, Ref. [4] for example of Gd. Their results were interpreted mainly in terms of the shape-coexistence picture [5,6], for example. On the other hand, a large-amplitude shape fluctuation encompassing two minima, if exist, was also conjectured [7].
Another observable that is known to be sensitive to shape deformation and coexistence is ρ 2 (E0), the reduced electric monopole transition strength measured through internal electron conversions [8,9]. The latter reference discussed a wide variety of medium and heavy nuclei based on available data of 0 + → 0 + , 2 + → 2 + , and 4 + → 4 + transitions. On the other hand, the E0 transition strength is one of indicators of the cluster structure in light nuclei [10]. Reference [11] compiles 0 + → 0 + transition data throughout the nuclear chart.
Theoretically the E0 strengths in medium-heavy nuclei have been systematically studied mainly in terms of the interacting boson approximation (IBA) model, [12], for example.
But characterization of 0 + 2 is still not decisive. This suggests that not only properties of 0 + 2 , such as the level energy, E2 and E0 transitions to the ground band, but also other information such as rotational band structure should be taken into account. Studies aiming at such a direction were pursued for 152 Sm, for example, [13][14][15]. In addition, the relation between the properties of the 0 + 2 in 154 Gd and the spectra of an adjacent odd-A nucleus was also argued trying to discriminate different pictures [16,17].

II. THE MODEL
We adopt a traditional mean field + random-phase approximation (RPA). The mean field is the Nilsson + BCS model, where standard notations for each quantity are understood. The l · s and l 2 terms are given by the singly-stretched coordinates. Their strengths are taken from Ref. [19]. The deformation of the oscillator potential is parameterized as where ω 0 is determined so as to conserve the nuclear volume. The cranking term −hω rot J x is also introduced when necessary. The residual pairing (P) plus isoscalar doubly-stretched quadrupole-quadrupole (Q-Q) interaction is given by are obtained from the spherical harmonics as Here ' ' indicates that the ground-state expectation values are subtracted, and the K = 0 terms mix into the mode under consideration when a rotation and/or γ deformation are introduced. The original P+Q-Q interaction was used to determine the parameters of Bohr's collective Hamiltonian [20] and applied to deformed Gd isotopes in Ref. [21]. The doublystretched Q-Q interaction was proposed to fulfill a shape selfconsistency in deformed nuclei and was shown to be effective in actual description of deformed nuclei [22,23], and further extended to rotating nuclei [24].
Transition strengths are calculated as follows. For the initial state |i = |0 + 2 = X † |0 + 1 and the final state |f = |2 + 1 , where 2 + 1 is the first excited member of the ground-state band, the E2 transition strength is given by when the rotational effect, the difference between the intrinsic states of |0 + 1 and |2 + 1 , is ignored. Here I i 020|I f 0 = 1 for I i = 0, I f = 2, and [·, X † ] RPA denotes the transition amplitude associated with the RPA phonon X † . The rotational effect is taken into account by the method [25] based on the generalized intensity relation (GIR) [1]. The GIR indicates that the angular-momentum dependence of interband transition matrix elements fits into In the textbook of Bohr and Mottelson, M 1 and M 2 are obtained by fitting to the data. Such a fitting was also done for adjacent nuclei recently [15]. Reference [25] proposed a method to represent it in terms of intrinsic matrix elements given by the mean field and RPA. The concrete form for the present case is given by replacing where J is the moment of inertia of the ground-state band.
The non-dimensionalized E0 transition matrix element from |i = |0 + 2 = X † |0 + 1 to |f = |0 + 1 is given by with R = r 0 A 1/3 . Effective charges are not introduced, and Q (+) K and r 2 in these expressions are understood as their proton part multiplied by e. The X ratio [8] is defined by    The lightest two, 146 Gd and 148 Gd, are spherical, ǫ 2 = 0, as expected and they are omitted in TABLE I and the following calculation. Next one, 150 Gd, is almost spherical, ǫ 2 = 0.07. Experimentally the two-phonon triplet in terms of spectra of spherical nuclei splits to some extent and the 0 + among them was labeled as the quasi-β [27]. A boson expansion calculation included in that reference shows more developed rotational character.
Then we included 150 Gd in the following figures for the sake of comparison but obviously the N dependence is discontinuous to 152 Gd and heavier.
In the literature, the ground state of 152 Gd has been said to be spherical and this lead to the shape-coexistence interpretation of the 2n-transfer data [4], but the present result of the Strutinsky method disagrees. In the following we put some emphasis on this issue.

N dependence
Adopting the deformation ǫ 2 obtained above, the mean field plus RPA calculations are performed in three major shells N osc = 4 -6 for neutrons and N osc = 3 -5 for protons, which give phenomenologically appropriate results, as in Ref. [25]. Interaction strengths G n , G p , and κ  with the data. They vary almost two orders of magnitude. This steep variation is nicely reproduced by including the rotational effect. Its significance is shown in Fig. 1(b) by the ratio M 2 /M 1 . The effect is conspicuous in lighter isotopes because of its dependence on J determined from E 2 + 1 = 2 · (2 + 1)h 2 /2J in TABLE I. This is very contrastive to the γ vibration for which the corresponding 5 2 × B(E2, 2 + γ → 0 + 1 ) stays within 10 -20 W.u. (Fig. 5(b)). From this largeness of B(E2, 0 + 2 → 2 + 1 ), the 0 + 2 states in 152 Gd and 154 Gd have been thought of as typical β vibrations [2, 13, 28] but a different interpretation was also proposed as discussed later. The smallness in heavier isotopes, already presented in Ref. [25], will also be discussed later.
with the available data [11]. Note that Refs. [29,30] included a data point of ρ 2 (E0, 0 + 2 → 0 + 1 ) in their calculation for 158 Gd, but this is actually that of ρ 2 (E0, 2 + 2 → 2 + 1 ). See Refs. [9,31]. A recent large-scale calculation adopting the constrained Hartree-Fock-Bogoliubov theory with the Gogny D1S interaction [32] results in failure to reproduce the order of magnitudes of the observed ρ 2 (E0). Then we have to have recourse to more phenomenological models to discuss their actual isotope dependence. In the literature, the IBA model [33] and the geometrical coherent-state model [34] reproduce the data well. The present calculation gives similar results.
The next aspect is the isotope dependence. Preceding the data for N = 92, Ref. [12] argued that the rise from N = 88 to 90 as well as in other isotope chains is a signal of the spherical-deformed shape phase transition and consequently ρ 2 (E0) would stay large in heavier isotopes. Unfortunately this has not been proved to apply. In the present calculation, the maximum occurs at N = 88 not 90. Figure 3 compares the calculated X ratios to the data. The reason why Refs. [35,36] included a data point of 158 Gd is the same as above. This figure indicates that the isotope dependence is predominantly determined by the denominator. The present calculation reproduces the rising trend but it is quantitatively weaker. This comes from the result that B(E2) in heavier isotopes looks to be larger than the data. This point will be discussed  Fig. 1(a). Data are taken from Ref. [11].
later. The discontinuity between N = 86 and 88 seen in Figs. 1(a) and 2 disappears because both the denominator and numerator vary to a similar extent.
[18] suggests a moderate deformation and actually in the present calculation, the rotational-model expression gives In this nucleus, high-spin states of the 0 + 1 and 0 + 2 bands were studied in the 2000s [37,38]. These works show smooth behavior of these bands starting from the bandhead with gradual stretching. Moreover, a g factor measurement of 2 + 1 -6 + 1 also supports rotational character of the low-spin members of the ground-state band [39]. Actually the present calculation gives a smooth behavior as a function of the rotational frequency, for example, g = µx Jx = 0.41 at hω rot = E 2 + 1 2 = 0.172 MeV, which is very close to the collective value, g R = Z A . Next, an implication of the conspicuous magnitude of B(E2, 0 + 2 → 2 + 1 ) is mentioned. As first discussed by Kumar [40], not only in-band but also interband B(E2) brings information about the deformation. Based on this, the model-independent effective deformation, β eff , is examined and compared with the IBA model [41]. According to this work, the square of the effective deformation of the 0 + 2 state is given by The summation is expected to be almost saturated with j = 1 and 2. In the present mean field plus RPA model, the j = 2 term gives the static deformation of the 0 + 2 state while the j = 1 term gives the zero-point amplitude of the β vibration. Those converted by from the B(E2) values in Fig. 1(a) are compared with corresponding static deformation, Fig. 4. Here the subscript designates the isoscalar quadrupole moment.
The RPA is a small-amplitude approximation. It is not obvious from the ratio of β 0 = 0.073 (no-rot) to β s = 0.207 whether 152 Gd is situated within the applicability of the RPA. In order to look into this, we compare the interband/in-band ratio of B(E2) to the case of the wobbling that is another example of strong interband E2 transitions previously accounted for in terms of the RPA. In the present case, the calculated ratio of the j = 1 (interband) and j = 2 (in-band) terms, , amounts to 0.12 (no-rot) and 0.52 (GIR). The wobbling excitations in the triaxially super/strongly deformed states in Lu isotopes were observed [42][43][44], calculated in terms of the RPA [45][46][47][48][49], and in other models [50][51][52]. Their ratios are B(E2,I→I−1) B(E2,I→I−2) ∼ 0.2. The associated fluctuation, the wobbling angle θ, is about 0.44 radian, for example, which fulfills a criterion of validity of the small amplitude approximation, tan θ ≃ θ [48]. In comparison of the present ratio, 0.12 that is directly given by the RPA, with that of the wobbling case, 0.2, we consider that the RPA is applicable to the β vibration in 152 Gd. Then, the effective value, β 0 = 0.154 (GIR), looks to indicate that, even if there exists some difference between the equilibrium deformations of 0 + 1 and 0 + 2 that is ignored in the present model, it would be of little relevance as conjectured in Ref. [7]. 2 ) 2 , respectively, in the present case as discussed in the case of 154 Gd below. Consequently, the pair transfer crosssection is also expected to be enhanced without recourse to the shape coexistence. Actually it is shown in Ref. [53] that the 154 Gd (p, t) 152 Gd crosssection is stronger for 0 + 2 than for 0 + 1 . This fact does not contradict the RPA result.
(2) 154 Gd The 0 + 2 state at E = 681 keV in this nucleus is another candidate of typical β vibrations [2]. In contrast, the 0 + 3 state at E = 1182 keV is thought to have a smaller deformation [54] and to be a pairing isomer [28,55]. Higher-lying states above 1 MeV were also investigated [56]. Although a possibility of interpreting the 2 + state at E = 1531 keV as the β ⊗ γ double excitation assuming that the 0 + 2 is a β vibration is reserved, the authors of this reference suggest that the 0 + 2 state has a shape different from that of the ground state rather than is the β vibration on top of it based on the non-existence of the two-phonon β vibrational state. A similar argument was done also for 152 Sm [57,58], but the non-existence of the two-phonon β vibrational state does not necessarily mean that of the one phonon.
Reference [16] further proceeds in this direction; the 0 + 2 is also a pairing isomer with a smaller deformation although transition properties are not considered. Microscopically the main component of the 0 + 2 is (ν[505] 11 2 ) 2 in this scenario. An important consequence of their argument is that this scenario leads to the non-existence of the 0 + 2 ⊗ ν[505] 11 2 band in the adjacent odd nuclei, 153 Gd and 155 Gd, because of the blocking effect. In an accompanying paper [17], the authors studied 155 Gd and concluded that the 0 + 2 ⊗ν[521] 3 2 and the γ⊗ν[505] 11 2 bands exist but the 0 + 2 ⊗ ν[505] 11 2 does not. Note here that the γ ⊗ ν[505] 11 2 is an unusually high-K band. That was also observed already in Ref. [59]; the spin assignments of these two works differ by one unit from each other. Here we examine the results of RPA calculations on 1) the ground state of 154 Gd, 2) the done on their ground states specified by blocking an appropriate quasiparticle state [60] obtained by the calculation for 154 Gd. The difference between 153 Gd and 155 Gd is specified by the chemical potential that gives the correct particle number. The interaction strengths G n , G p , and κ 2 ) 2 , respectively. In contrast, the only large forward amplitude is that of (ν[505] 11 2 ) 2 , |ψ| = 0.995. This proves that the main origin of the collectivity is different from the main forward component. In the case of the [521] 3 2 of 155 Gd, 2), collectivity is reduced by blocking one of prolate orbitals but the resulting β vibration is still collective enough. In the case of the [505] 11 2 of 155 Gd, 3), the wave function is changed dramatically by loosing the main forward component. Consequently the K = 0 strength is pushed up to higher energies but still distinguishable from other non-collective states.
We also confirmed that the γ vibration is almost not affected because there are no neutron quasiparticle states that constitute K = 2 pairs with [505] 11 2 . These results prove that the characteristics of the spectra of 154 Gd and 155 Gd can be accounted for in terms of the β vibration. However, it should be noted that the isomerism of 0 + 3 and the difference in the crosssections of (p, t) and (t, p) transfers to 0 + 2 are out of the scope of the present calculation that does not contain the quadrupole pairing.
(3) 156 Gd  (4) 158 Gd Figure 5(a) also indicates that the observed B(E2, 0 + 2 → 2 + 1 ) in 158 Gd looks evidently smaller than expected from the systematics. Actually this is one of the curious properties that have this nucleus extensively studied but have not been resolved yet. Since an early study [31], the 0 + 3 state at E = 1452 keV has been known to be more collective than the 0 + 2 state at E = 1196 keV. Although ρ 2 (E0, 0 + 2 → 0 + 1 ) has not been reported up to now, ρ 2 (E0, 2 + 2 → 2 + 1 ) = (0.72±0.21)×10 −3 and ρ 2 (E0, 2 + 3 → 2 + 1 ) = (25±4)×10 −3 were reported in that work and reevaluated as ≤ 0.8×10 −3 and (17±3)×10 −3 , respectively, in Ref. [9] for the rotational-band members. The quadrupole transition strengths were measured much later in Ref. [61] as B(E2, 0 + 2 → 2 + 1 ) = 1.1 W.u. and B(E2, 0 + 3 → 2 + 1 ) = 2.1 W.u., see also Ref. [62]. In addition to the fact that the latter is larger, both of them are smaller than expected for ordinary β vibrations. Later a large number of 0 + states were reported [63]. Moreover, in Ref. [62], B(E2, 0 + n → 2 + 1 ) were measured for a lot of states up to 0 + 10 . This result proves that the E2 strengths are strongly fragmented and pushed up to higher energies; the largest one is B(E2, 0 + 8 → 2 + 1 ) = 7.7 +1.5 −0.7 W.u. References [64,65] suggest a contribution of two-phonon octupole vibration to producing large number of 0 + states based on the geometrical collective model and the IBA model. In the projected shell model [66], the excited energies and the number of 0 + states are accounted for by two-and four-quasiparticle states but associated B(E2) are much smaller than observed. Lo Iudice et al. [29] and Gerçeklioglu [30] performed RPA calculations. The former includes the quadrupole pairing interaction. The resulting number of 0 + states is less than observed in the RPA calculation but quasiparticle-phonon couplings with octupole two-phonon states improve the result. The latter includes the spin-quadrupole interaction. The number of 0 + states is reproduced without an octupoleoctupole interaction. Both calculations, however, failed to account for the character of the 0 + 2 and 0 + 3 states. Our RPA result for the distribution of excited 0 + states is presented in  0 . This figure shows that the overall distribution is reproduced quite well without an octupole-octupole interaction but the obtained 0 + 3 is not collective. None of higher states have B(E2) strengths larger than 1 W.u.
One of possible origins of quadrupole collectivity at high energies conjectured in Ref. [62] is the two-phonon γ vibration. The K = 0 two-phonon γ vibration is known only in 166 Er [67] although the K = 4 ones are known more as briefly reviewed in Ref. [68]. The quasiparticlephonon coupling model calculation in Ref. [29] looks to include such a type of excitation but reported B(E2) are much smaller.
(5) 160 Gd Very recently an upper limit of B(E2, 0 + 2 → 2 + 1 ) was reported [69]. The calculated value is slightly larger than the reported upper limit as shown in Fig. 5

(a). But it is open whether
there is a problem similar to 158 Gd.

IV. CONCLUSIONS
The long-debating problem of the characterization of the 0 + 2 states in Gd isotopes has been revisited. The model adopted is a traditional mean field plus RPA. The doubly-stretched quadrupole-quadrupole interaction is used. The rotational effect on the transition strengths are accounted for by that on the intrinsic matrix elements based on the generalized intensity relation. Calculations have been done paying attention to properties of rotational bands.
The most decisive property to characterize the 0 + 2 states is B(E2, 0 + 2 → 2 + 1 ). Its steep N dependence ranging two orders of magnitude is nicely reproduced. In particular, those in lighter isotopes, 152 Gd and 154 Gd have been shown to be understandable as β-vibrational excitations on top of deformed ground states as early thought [2]. To this end, an implication of the strengths of B(E2) and rotational properties for the former, and the relation to the spectra of the adjacent odd-A nucleus for the latter have been investigated. Consequently the present calculation supports the picture of Ref. [14]. The monopole transition strength, ρ 2 (E2, 0 + 2 → 0 + 1 ), is also thought to be sensitive to the shape deformation/coexistence. The available data have been reproduced fairly well within the present model but data are still too scarce to utilize for discriminating different theoretical pictures.
Looking at relatively weak B(E2) in heavier isotopes more closely, however, a disagreement remains in 158 Gd; a strong fragmentation of B(E2) strengths to higher energies is not accounted for in the present model as well as preceding works.