Evidence against the wobbling nature of low-spin bands in $^{135}$Pr

The electromagnetic character of the $\Delta I=1$ transitions connecting the one- to zero-phonon and the two- to one-phonon wobbling bands should be dominated by an $E2$ component, due to the collective motion of the entire nuclear charge. In the present work it is shown, based on combined angular correlation and linear polarization measurements, that the mixing ratios of all analyzed connecting transitions between low-lying bands in $^{135}$Pr interpreted as zero-, one-, and two-phonon wobbling bands, have absolute values smaller than one. This indicates predominant $M1$ magnetic character, which is incompatible with the proposed wobbling nature. All experimental observables are instead in good agreement with quasiparticle-plus-triaxial-rotor model calculations, which describe the bands as resulting from a rapid re-alignment of the total angular momentum from the short to the intermediate nuclear axis.

PACS numbers: 21.10.Re, 21.60.Ev, 23.20.Lv,27.60.+j Wobbling motion was a topic of great interest in nuclear physics over the last years. This collective mode of excitation is considered as a clear fingerprint of the presence of stable triaxiality in a deformed rotating nucleus. Initially proposed by Bohr and Mottelson for even-even nuclei at high spin [1], the wobbling mode occurs when the three principal axes of the nuclear density distribution are unequal. Due to the triaxiality such nucleus rotates simultaneously around all of its three axes. It was demonstrated that the approximation of such rotation with wobbling motion, where the excited rotational bands are generated by the excitation of wobbling phonons, is a good approximation at high spin.
Experimentally, the wobbling motion has been identified through the observation of ∆I = 2 rotational bands with similar moments of inertia and excitation energies increasing with the number of wobbling oscillation quanta. The most important direct evidence of * Corresponding author: petrache@ijclab.in2p3.fr the wobbling mode is obtained from the examination of the connecting ∆I = 1 transitions between bands differing by one wobbling phonon, which, due to the fact that the wobbling motion involves the entire nuclear charge, should have predominately electric E2 character.
The pioneering experimental evidence for nuclear wobbling motion was reported in the triaxial superdeformed 161−167 Lu and 167 Ta odd-even nuclei two decades ago [2][3][4][5][6][7]. In all these nuclei the wobbling bands were assigned to the πi 13/2 intruder configuration with large deformation of ε 2 ≈ 0.4, and the spin of the odd nucleon parallel to that of the core.
Recently, a new type of wobbling motion in oddmass nuclei, called transverse wobbling, was proposed by Frauendorf and Dönau [8], in which the angular momentum of the odd nucleon is orthogonal to the angular momentum of the core. This is in contrast with the longitudinal coupling of the angular momenta of the odd nucleon and the core, accepted previously for the high-spin bands in the odd-mass Lu and Ta isotopes (see e. g. [9]). The new transverse wobbling mode was proposed for both the low-spin band in 135 Pr, and the high-spin bands of the odd-mass Lu isotopes [8]. This new idea opened an intense research activity which led to the publication of transverse and longitudinal wobbling bands in 135 Pr [10,11], 105 Pd [12], 130 Ba [13], 133 La [14], 127 Xe [15], 187 Au [16], 183 Au [17], and 136 Nd [18]. In particular the one-and two-phonon transverse wobbling bands in 135 Pr reported in Refs. [10,11] comprise previously observed negative-parity states without a proposed interpretation in Ref. [19]. The critical experimental evidence supporting the new wobbling interpretation is the measured magnitudes larger than one of the mixing ratios of three transitions connecting the one-phonon wobbling band to the yrast band, and of three transitions connecting the two-phonon to the one-phonon wobbling bands. It is well known from the literature that the extraction of the mixing ratios of ∆I = 1 transitions from angular distribution alone yields two solutions, |δ 1 | > 1 and |δ 2 | < 1, resulting from the χ 2 fit of the experimental points. This ambiguity can be solved by a linear polarization measurement. However, in the case of the wobbling bands of 135 Pr polarization results were carried out for only two out of the six linking transitions, and in addition, as pointed out in Ref. [20], the magnitude of the measured linear polarization was not used, but only its sign, to conclude on their predominant E2 character. Moreover, the polarization data for the two transitions linking the onephonon to the yrast band, presented in Ref. [10], are in contradiction with those of Ref. [21] published soon after Ref. [10], and performed using the same reaction and the same setup [21]. The last paper was followed by an erratum [22] which reports similar polarization results as those of Ref. [10], however some questions were raised with respect to the analysis, Ref. [23]. Considering the apparent uncertainty in the polarization data for the one-phonon band and the lack of any polarization data for the two-phonon band, doubts on the available experimental evidence for the proposed wobbling bands remain. Therefore, a further investigation of the mixing ratios and polarization asymmetries in 135 Pr appeared to us necessary to clarify the nature of the connecting transitions of the proposed wobbling bands.
Theoretically, a considerable debate about the validity of transverse wobbling motion in odd-mass nuclei is in course. The frozen approximation proposed in Ref. [8] appears unrealistic, because it leaves out the effect of the Coriolis force on the odd nucleon coupled transversely to the core [24][25][26]. Very recently, the wobbling interpretation of the low-lying yrare bands in odd-mass nuclei was questioned in Refs. [27,28]. It was shown that the transverse wobbling equations as given in Ref. [8] are not equivalent to the rotational Hamiltonian of the quasiparticle-plus-triaxial-rotor (QTR) model which describes three-dimensional rotation, and do not accurately describe the excitation of wobbling quanta. As the threedimensional rotation of a triaxial odd-mass nucleus represents in fact a precession of the total angular momentum around a certain axis, the low-lying yrare bands in triaxial nuclei were called Tilted Precession (TiP) bands [27].
The present work concentrates on accurate measurements of the mixing ratios of the transitions connecting low-lying states in 135 Pr, using both linear polarization and angular correlation analysis [29][30][31], in order to provide crucial evidence on the real nature of the low-lying excited bands in 135 Pr. The low-lying negative-parity bands in 135 Pr have been investigated using the 100 Mo( 40 Ar,1p4n) reaction at a bombarding energy of 152 MeV. The 40 Ar beam was provided by the K130 Cyclotron at the University of Jyväskylä, Finland. The target was a 0.5 mg/cm 2 thick self-supporting foil of enriched 100 Mo. Excited 135 Pr nuclei were produced with approximately 30% of the total cross section. Prompt γ-rays were detected by the JU-ROGAM II spectrometer comprising of 24 EUROGAM clover [32] and 15 EUROGAM phase one [33] Comptonsuppressed germanium detectors. Approximately 5.1 × 10 10 three and higher-fold γ-ray coincidence events were obtained and stored. The data were sorted into coincidence γ-γ matrices and γ-γ-γ cubes, and analyzed using the radware [34,35] and gaspware packages [36].
The partial level scheme of 135 Pr shown in Fig. 1 is constructed using the relative intensity (I γ ) balance and coincidence relationships of the γ-ray transitions. It is in agreement with Refs. [19,21], but in contrast with the grouping of the 827-, 764-and 1009-keV transitions as one band, which was interpretaed as two-phonon wobbling band in Ref. [11]. These three transitions do not correspond to increasing γ-ray energy, as expected for a rotational band. Therefore the second excited 19/2 − state in 135 Pr is assigned here as the band head of the new band 4, consisting of the 688-and 871-keV transitions which are much stronger than the 827-keV transition (see Fig. 2 (a) in the supplementary material [37]), and have the approximate I(I + 1) energy dependence of a rotational band. If one considers the 1009-keV transition reported in Ref. [11] but not observed in the present work, the 764-keV transition is the first transition of a possible new rotational band, band 5, built on the third excited 23/2 − state in 135 Pr, while the 827-keV transition links this band to band 4. The multipolarity and the mixing ratios of the γ-ray transitions were established based on two-point angular distribution (anisotropy) ratios R ac [30,38] and linear polarization measurements [39,40]. The complete experimental information on levels, γ-ray transitions, and experimental techniques are presented in the supplementary material [37].
Bands 1 and 2 reported in Ref. [19] were interpreted as the favored and unfavored signatures of the lowestenergy πh 11/2 orbital. In a more recent work, band 2 was extended to higher spin by one transition. In the same work, some previously observed transitions were assigned to a band, band 3 [11]. Based on angular distribution measurements, mixing ratios with magnitudes larger than one were derived for the 747-, 813-, and 755-keV transitions linking band 3 to band 1, and with magnitude smaller than one for the 594-keV transition linking band 2 to band 1 [10]. The positive values of the measured linear polarization for the 747-and 813-keV transitions were considered as confirming the larger-than-one values of the deduced mixing ratios. Based on these experimental results band 3 was interpreted as a one-phonon wobbling band, while band 2 was assigned as a signature partner of band 1. In another work [11], angular distribution and DCO-like measurements only were used to extract mixing ratios for the 450-, 551-, and 519-keV transitions, which were assigned as linking the two-and the one-phonon wobbling bands. The extracted largerthan-one magnitudes were considered as a deciding proof for the proposed wobbling phonon interpretation.
However, the χ 2 fit of the angular distribution data usually yields two solutions for the mixing ratio, |δ 1 | > 1 and |δ 2 | < 1. For instance, Fig. 2 shows the angular distribution data for the 747-, 813-and 450-keV transitions from Refs. [10,11], together with the fitted curves yielding |δ 1 | > 1 (the blue dashed line [10,11]) and |δ 2 | < 1 (the red solid line [20]). Therefore, both solutions δ 1 and δ 2 are possible. In order to select the correct one, linear polarization measurements are needed. However, out of the six linking transitions in 135 Pr for which the δ 1 solution was selected, linear polarization data are available for only two, the 747-and the 813-keV transitions. Furthermore, the magnitude of the measured linear polarization was not used to select the correct solution of δ, but it was assumed that the positive sign of the polarization confirms the solution with larger magnitude. However, the positive sign does not necessarily select the |δ| > 1 values [40], as also pointed out in Ref. [20].
In the present work, new mixing ratios for the 747-, 813-keV, and 450-keV transitions are determined from complementary measurements of linear polarization P and angular distribution ratio R ac , see Figs. 2(b), 2(d) and Table I. As illustrated in Fig. 2, the P − R ac analysis allows the extraction of the mixing ratios δ for the 747-, 813-and 450-keV transitions of -0.47 +9 −22 , -0.37 +10 −14 , and -0.31 +10 −13 , respectively. The smaller-than-one absolute mixing ratios of the three transitions are in contradiction with the larger-than-one mixing ratios reported in Ref. [10,11], and indicate predominant magnetic character at the level of 82%, 88% and 91% for the 747-, 813-, and 450-keV transitions, respectively. The higher lying 755-keV connecting transition is too weak to allow the extraction of polarization asymmetry (as in a previous work [10]). However its R ac ratio is very similar to that of the 813-keV transition, suggesting similar solutions for δ.
The predominant magnetic character of the 747-and 813-keV transitions connecting bands 3 and 1 is therefore in distinct disagreement with the predominant electric character deduced in Ref. [10] and incompatible with the proposed one-phonon wobbling interpretation of band 3. As an additional test, let us assume that band 3 is a one-phonon wobbling excitation with respect to band 1. In this case, the B(E2; 747) reduced transition probability to band 1 would be large, while B(E2; 526) to band 2 would be negligible, as it would correspond to a decay to the signature partner band 2, which is forbidden. The presently measured ratio B(E2; 747)/B(E2; 526) = 0.7 +0.7 −0.5 indicates similar strengths of the two transitions, and is therefore in disagreement with the wobbling interpretation of band 3. The predominant magnetic character of the 450-keV transition is also inconsistent with the previously proposed interpretation [11], which described it as a link between a two-and a one-phonon wobbling bands. Therefore the second 19/2 − state in 135 Pr does not have wobbling nature. TABLE I. The experimental polarization value P , angular correlation ratios Rac, mixing ratios δ, and ratios of out-of-band and in-band reduced transition probabilities of the connecting transitions between bands 3 and 1, and between bands 4 and 3.
Eγ Prior to this work, the negative-parity states in 135 Pr have been investigated using different models: the quasiparticle-plus-triaxial rotor model, tilted axis cranking (TAC) mean-field calculations, the triaxial projected shell model, and constrained triaxial covariant density functional theory, as well as the particle rotor model [8,10,11]. Band 1 was assigned to the πh 11/2 config- uration, band 2 was considered as the signature partner of band 1, and bands 3 and the levels of band 5 together with the 19/2 − level of band 4 were interpreted as onephonon and two-phonon transverse wobbling bands, respectively. The experimental results described above and the theoretical calculations reported in the following do not support the proposed transverse wobbling nature of these bands. In order to explore the nature of the five bands discussed in the present work, we performed quasiparticleplus-triaxial-rotor calculations [41] with standard parameters for the Nilsson potential and pairing, adopting shape parameters of ǫ 2 = 0.16 and γ = 26 • . Standard irrotational-flow moments of inertia were employed, supported by empirical evaluations on several triaxial nuclei [42], while their spin dependence was described by Harris parameters of J 0 = 12.5h 2 MeV −1 and J 1 = 12.5 h 4 MeV −3 . In variance with the QTR calculations of Refs. [10,11], we did not use the frozen approximation of the particle angular momentum, and did not modify the relative magnitude of the irrotational-flow moments of inertia, as in Ref. [8], in particular we did not increase the moment of inertia along the short axis. The single-particle degrees of freedom were considered by including 10 negative-parity orbitals near the proton Fermi surface, allowing effects such as Coriolis re-alignment of the valence nucleon, as well as single-particle excitations. The proton Fermi level was set at 44 MeV. The B(M 1) reduced transition probabilities were derived using gfactors of g s,ef f = 0.6g s,f ree , and g core = Z/A. The calculated excitation energies of the five lowest-energy negative-parity bands are shown in Fig. 3(a) in comparison with the experimental data.
Five rotational bands lying at very similar excitation energies, in particular at high spin, are observed experimentally. The calculated bands do appear at similar excitation energies at low spins, but their excitation energies differ significantly from each other at high spin. This overestimation at high spin can be caused by several reasons, for instance by the interaction between the observed one-and three-quasiparticle bands, which results in lowering of the excitation energies of the experimentally observed one-quasiparticle bands, or by the model assumption that all bands are described with the same set of moments of inertia.
The QTR model predicts rotational bands that result from a mixture of collective and single-particle excitations. The three lowest-energy orbitals from the h 11/2 sub-shell, labelled here as #14, #15 and #16, have dominant contributions in the wave functions of the negativeparity states. The largest contributions to the wave functions of the 11/2 − and 15/2 − states of the yrast band 1 correspond to the #14 orbital and to projection of the total angular momentum on the short axis of K s = 11/2 and 15/2, respectively. The 19/2 − state of band 1 has largest contribution from orbital #15, while for the higher-spin states the dominant contribution is from orbital #16, which reflects the fact that the singleparticle alignment along the intermediate axis is largest. Thus, the yrast band corresponds to a rapid re-alignment of the single-particle angular momentum from the short to the intermediate axis. In parallel, the total angular momentum also changes its orientation from the short to the intermediate axis.
The calculated bands 2 and 3 have the same signature, but different nature. Band 2 corresponds to a dominant single-particle contribution of orbital #15 for the 13/2 − state and of orbital #16 for all other states, reflecting thus an alignment of the single-particle orbital angular momentum along the intermediate axis. Band 3 corresponds to dominant contribution from orbital #14 for the 13/2 − , 17/2 − , and 21/2 − states, and from orbital #15 for the higher-spin states.
Bands 4 and 5 have the same signature as band 1. The experimentally observed 19/2 − and 23/2 − band heads of these bands correspond in the calculations to a dominant contribution from orbital #14 with K s = 19/2 and 23/2, respectively, having thus full alignment of the total angular momentum along the short axis. Therefore the negative-parity states that correspond to largest contribution from fully aligned along the short axis total angular momentum are the 11/2 − and 15/2 − states of band 1, the 19/2 − state of band 4 and the 23/2 − state of band 5. This highlights the QTR prediction that the alignment along the short axis is not favorable at high spin, where the total angular momentum is instead aligned along the intermediate axis.
One can evaluate the re-alignment of the angular mo-  [10] for |δ| > 1 (blue diamond) and |δ| < 1 (green square) are shown. (c) Calculated projection of the singleparticle angular momentum along the direction of the total angular momentum j || =< I.j > /|I| for the five bands. (d) Transition probability ratios obtained from this work (filledcircle) for the 813-keV (21/2 − →19/2 − ) out-of-band and 726-keV (21/2 − →17/2 − ) in-band transitions compared with the present QTR calculations (red line), and to the results of Matta et al. [10] for |δ1| > 1 (blue diamond) and |δ1| < 1 (green square). menta by examining the expectation value of the projection of the single-particle angular momentum along the direction of the total angular momentum, j || =< I.j > /|I|, see Fig. 3. For bands 1, 2, 3, and 5 the magnitude of j || is almost constant, showing that the relative orientation of the two angular momenta remain almost unchanged as a function of spin. Therefore they re-align from the short to the intermediate axis simultaneously. The value of j || is very close to the maximum of j ||,max = 5.5, suggesting that the two angular momenta are aligned with respect to each other.
The present experimental data on the mixing ratios of the linking transitions between bands 3 and 1, and on the corresponding B(E2) out /B(E2) in reduced transition probability ratios, are compared with the present calculations in Fig. 3. There is a very good agreement for both quantities. It should be noted that the selected mixing ratios with magnitude |δ 1 | > 1 in Ref. [10], out of the two possible solutions for δ, is in disagreement with the present experimental results and calculations. However, the selection of the second solution, |δ 2 | < 1, from the data of Ref. [10] would be in good agreement with the calculations, as shown in Fig. 3(d).
In summary, the present work reports new data on 135 Pr, in particular the mixing ratios deduced from a combined polarization and angular distribution measurements, for three linking transitions between the previously proposed one-and two-phonon wobbling bands and the yrast band. The deduced absolute values of the mixing ratios are all smaller than one, suggesting predominant M 1 character, which excludes the one-and twophonon wobbling nature. The results of extensive QTR calculations are in good agreement with the experimental data, and indicate that the bands are tilted precession bands based on the νh 11/2 configuration.