Experimental Evidence for Transverse Wobbling in $^{105}$Pd

New rotational bands built on the $\nu$$(h_{11/2})$ configuration have been identified in $^{105}$Pd. Two bands built on this configuration show the characteristics of transverse wobbling: the $\Delta$$I$=1 transitions between them have a predominant E2 component and the wobbling energy decreases with increasing spin. The properties of the observed wobbling bands are in good agreement with theoretical results obtained using constrained triaxial covariant density functional theory and quantum particle rotor model calculations. This provides the first experimental evidence for transverse wobbling bands based on a one-neutron configuration, and also represents the first observation of wobbling motion in the $A$$\sim$100 mass region.

New rotational bands built on the ν(h 11/2 ) configuration have been identified in 105 Pd. Two bands built on this configuration show the characteristics of transverse wobbling: the ∆I=1 transitions between them have a predominant E2 component and the wobbling energy decreases with increasing spin. The properties of the observed wobbling bands are in good agreement with theoretical results obtained using constrained triaxial covariant density functional theory and quantum particle rotor model calculations. This provides the first experimental evidence for transverse wobbling bands based on a one-neutron configuration, and also represents the first observation of wobbling motion in the A∼100 mass region. Nuclear wobbling motion was initially discussed by Bohr and Mottelson [1]. This type of rotation is predicted to occur in triaxially deformed nuclei. The nucleus rotates around the principal axis having the largest moment of inertia and this axis executes harmonic oscillations about the space-fixed angular momentum vector. Its analog in classical mechanics is the motion of a free asymmetric top, while in quantal systems a corresponding example would be the rotation of molecules having different moments of inertia for the three principal axes. In nuclei, the expected energy spectra related to this motion are characterized by a series of rotational E2 bands corresponding to the different oscillation quanta (n). The signature quantum number of two consecutive bands is different, thus the yrast and yrare bands (corresponding to n=0 and n=1, respectively), look like signature partner bands with large signature splitting. The yrare band decays by ∆I=1 M1+E2 transitions to the yrast band. However, contrary to the case of signature partners, the multipole mixing ratios are very large, and the transitions have predominantly E2 character. Furthermore, the energy separation between the yrare and yrast bands, the wobbling energy, is expected to increase with increasing spin. Although Bohr and Mottelson predicted this motion for even-even nuclei where no intrinsic angular momentum is involved, the phenomenon in this simple form has not been experimentally documented to date.
The first experimental evidence for nuclear wobbling motion was reported in the odd-proton 163 Lu (Z=71) nucleus [2,3] and later in the 161 Lu, 165 Lu, 167 Lu nuclei [4][5][6], as well as in 167 Ta (Z=73) [7]. In these nuclei the wobbling mode is observed in the triaxial strongly deformed bands corresponding to the π(i 13/2 ) intruder configuration. Recently, wobbling motion was reported in 135 Pr (Z=59), where the wobbling bands have normal deformation and they are built on the π(h 11/2 ) configuration [8]. The expected different signature values and the predominant E2 character of the ∆I=1 transitions between the bands have been observed for all the above cases. However, the wobbling energy has been found to decrease with increasing spin contrary to theoretical expectations. Frauendorf and Dönau [9] interpreted this behavior as the consequence of the perpendicular orientation of the odd particle's angular momentum to the rotational axis, and they suggested to name the phenomenon as "transverse wobbling". This interpretation differs from that previously published for the Lu and Ta isotopes, and gener-ated great theoretical interest to clarify the situation using different models [10][11][12][13][14][15][16][17][18][19]. Very recently another type of the wobbling motion has been claimed in 133 La, the "longitudinal wobbling", where the wobbling energy was found to increase with increasing spin [20]. It is worth noting that all the wobbling bands observed so far correspond to a configuration of one proton coupled to the core. In this Letter, we report on experimental evidence for transverse wobbling motion in 105 Pd (Z=46, N =59). This is the first observation of transverse wobbling motion based on a one-neutron configuration, and also the first observation of wobbling motion in the A∼100 mass region.
High-spin states in 105 Pd were populated using the 96 Zr( 13 C,4n) reaction. The 13 C beam was provided by the Vivitron accelerator at IReS, Strasbourg. The beam impinged upon a stack of two self supporting metallic foil targets being enriched to 86% in 96 Zr, and each having a thickness of ∼0.6 mg/cm 2 . The emitted γ-rays were detected by the EUROBALL IV [21] spectrometer equipped with 15 Cluster detectors at backward angles and 24 Clover detectors at 90 • relative to the beam direction. Contaminants from the charged-particle reaction channels were eliminated using the highly efficient DIAMANT charged-particle detector array consisting of 88 CsI detectors [22,23] as an off-line veto. A total of ∼2 × 10 9 triple-and higher-fold coincidence events were obtained and stored onto magnetic tapes.
The level scheme of 105 Pd was constructed using the Radware analysis package [24] on the basis of the triplecoincidence relations, as well as energy and intensity balances of the observed γ rays. Several new rotational bands have been observed in 105 Pd. Among them there are negative-parity quadrupole bands with probable neutron h 11/2 configuration. Two of these bands have opposite signature than the previously known, yrast neutron h 11/2 band. Fig. 1 shows the yrast neutron h 11/2 band (band A) up to spin 43/2 and the two newly identified bands (bands B and C).
Linear polarizations and directional correlation from oriented states (DCO) ratios [25][26][27][28] were derived for the transitions of sufficient intensity. The observed values for the transitions relevant to the focus of the present Letter are compared in Fig. 2 with the values of different multipolarities and mixing ratios calculated for the experimental geometry. For the DCO ratio anlysis we used stretched E2 gating transitions, the attenuation coefficients of incomplete alignment were fitted to the known strong 1100 keV E2 and 1331 keV E1 transitions [29] in 105 Pd assuming pure stretched E2 and E1 multipolarities for them, respectively. Our analysis resulted a mixing ratio of -0.37 (8) for the 442 keV lowest inband M1+E2 transition in 105 Pd which reproduced well the -0.33 (13) value reported in Ref. [29]. The 1331 keV and 442 keV transitions are not shown in Fig. 1. Details of the experimental setup and data analysis, as well as the full level scheme, will be provided in a forthcoming publication [30].   Band A has been reported in Refs. [29,31] with spinparity values firmly assigned to the states up to spin 31/2 . Data from the present experiment confirm the previously reported values. The 17/2 − , 21/2 − and 25/2 − states of band B and the 21/2 − state of band C were reported in Ref. [29] as non-band levels. However, the levels belonging to bands B and C have been identified as rotational bands first in the present experiment. The DCO and linear polarization values derived for the 814 keV, 918 keV, 1089 keV and 1064 keV transitions agree well with stretched E2 multipolarity, confirming the E2-band character of band B. As it is seen in Fig. 2, the measured DCO and linear polarization values for the 991 keV, 1034 keV and 994 keV transitions are in good agreement with ∆I=1 M1+E2 multipolarity at large, δ=1.8(5), 2.3(3), 2.7(6) multipole mixing ratios, respectively. Thus, these transitions have predominantly E2 characters, however, they cannot be pure ∆I=2 E2 transitions because for such transitions the linear polarization values are expected to be between 0.65 and 0.7, like in the case of the 1100 keV gamma transition, contrarily to the measured negative values. Therefore, the observed DCO and polarization values allow only the 17/2 − , 21/2 − and 25/2 − spin-parity values for the initial states of the 991 keV, 1034 keV and 994 keV transitions, respectively. Strictly speaking, spins less by one or two units would also be allowed by the DCO and polarization data. However, it is very rare that levels of rotational bands decay to the same-spin or higher-spin states of another band, and in those cases they also decay to lower-spin levels of the other band. Existence of such transitions to lower spin states was excluded by the observed data in the present case.
The lowest-energy state of band C is fed by the 794 keV transition from the 25/2 − state of band B and decays by the 939 keV transition to the 17/2 − state of band B. As the M2, M3 and E3 transitions are not competitive with E2 and M1 transitions, the 794 keV and the 939 keV transitions can only be stretched E2 transitions. Thus the spin-parity of the lowest-energy state of band C can only be 21/2 − . Similarly, the second state of band C is linked to the 21/2 − and to the 29/2 − states of band B. Thus, its spin-parity can only be 25/2 − . This also confirms the E2-band character of band C. The adopted spinparity assignments for the four previously known levels of bands B and C are consistent with those reported in Ref. [29].
Band C decays to band A by the 1158 keV and 1159 keV transitions. Linear polarization value of -0.6(3) was derived for the sum of the two transitions. Unfortunately, no linear polarization values could be deduced separately for the two transitions because of their close energies. DCO value could not be derived even for the sum of the two transitions because their energies are close to that of the strong 1152 keV transition in band A. The fact that the 1158 keV, 1159 keV and 1152 keV transitions are all in coincidence with the intense gamma rays which could be used as coincidence gates, caused further difficulties in the analysis. The deduced linear polarization value agrees well with the multipolarity expected for the 1158 keV and 1159 keV transitions from the above spin-parity assignments for the band C states: namely that they are ∆I=1 M1+E2 transitions. However, it allows both small (0 ≤ δ ≤ 0.5) and large (1 ≤ δ ≤ 2.4) mixing ratios.
The observed three bands show the features of a pair of wobbling bands with oscillation quanta zero and one (bands A and B, respectively) and the signature partner band of band A (band C). Indeed, the multipolarities of the lowest-lying linking transitions between bands B and A are M1+E2 with large, δ=1.8(5), 2.3(3), 2.7(6) multipole mixing ratios for the 991 keV, 1034 keV and 994 keV transitions, respectively. These mixing ratios mean around 80% (calculated as δ 2 /(1 + δ 2 )) E2 content, which is expected for the wobbling band, but not expected for the signature partner. We note that the 991 keV, 1034 keV and 994 keV transitions were also reported in Ref. [29] and δ=0.46(10) as well as δ=0.62 (18) were derived for the 991 keV and 1034 keV transitions, respectively, from angular distribution measurement. While the present DCO results also allow δ=0.59 (20) and 0.40 (6) values for the two transitions, respectively, the linear polarization data disagree with these smaller mixing ratios, but strongly support the larger δ=1.8(5) and 2.3(3) values.
Band C is a candidate for the signature partner of band A. The two bands have the same parity and similar alignments [30] but opposite signature. Furthermore, band C decays to band A by the 1158 keV and 1159 keV transitions. Although the mixing ratios of these transitions could not be derived unambiguously, the possible smaller mixing ratio value deduced from the present experiment is in a good agreement with this scenario. In Ref. [29] a mixing ratio of δ=1.3(9) was reported for the 1158 keV transition. Due to the large uncertainty this value can allow a rather small mixing ratio, thus it can be in agreement with the signature partner interpretation, too.
The difference between the mixing ratio values measured for the linking M1+E2 transitions between the wobbling bands in 135 Pr and in 105 Pd is their opposite signs. While the sign is positive in 105 Pd, it is negative in 135 Pr. The sign of the mixing ratio value is determined by the sign of the M1 matrix element assuming that the quadrupole deformation is of same type. The sign of the M1 matrix element is proportional with the (g p − g R ) factor [1], where g R is the rotational gyromagnetic factor. Its value is approximately Z/A, which is ∼0.4 for both nuclei. However, g p , the gyromagnetic factor of the odd particle, is different for the protons and the neutrons moving in high-j intruder (j = l + 1/2) orbitals. It has a large positive value (>1) for protons, while it has a negative sign for neutrons. Thus, the sign of the (g p − g R ) factor is opposite for high-j protons and neutrons [32].
In order to explore the nature of the observed rotational band structures in 105 Pd, they have been studied by the constrained triaxial covariant density functional theory (CDFT) [33,34] as well as the quantum particle rotor model (PRM) [9,19,[35][36][37][38]. The configurationfixed CDFT calculations [33,34] with the effective interaction PC-PK1 [39] reveal that the ν(1h 11/2 ) 1 configuration has a triaxial shape of β = 0.27 and γ = 25 • , which fulfills the conditions required for the presence of wobbling bands. With the configuration and deformation parameters obtained, it is straightforward to perform PRM [9,19,37,38] calculations in order to study the energy spectra and electromagnetic transition probabilities for the observed rotational sequences in 105 Pd. In the PRM calculations, the neutron particle is described by a single-j shell Hamiltonian [40] and the pairing effect is included using the standard BCS quasiparticle approximation with the empirical pairing gap ∆ = 12/ √ A = 1.17 MeV and the Fermi surface located at the beginning of the h 11/2 subshell. The triaxial rotor is parametrized by three angular-momentum-dependent moments of inertia J i = a i 1 + bI(I + 1) [41,42] (known as the ab formula) to take into account the soft character of the potential energy surface revealed by CDFT calculations [43].
Here, i = m, s, l denotes the medium, short, and long axes, respectively, and the corresponding parameters a m,s,l = 5.89, 3.74, 1.27 2 /MeV and b = 0.023 −2 are determined by fitting to the energy spectra of bands A and B.
The calculated rotational frequency ( ω(I) = [E(I) − E(I − 2)]/2) and energy spectra as functions of spin I for bands A (solid line), B (short dash line), and C (short dash-dot line), in comparison with those of the experimental data, are shown in Fig. 3. It is seen that the PRM calculations can reproduce bands A and B well. For band C, the energies are overestimated by about 500 keV. A similar problem is also seen in Ref. [8] for 135 Pr.
For band A, the rotational frequency is almost constant from spin I = 27/2 to 39/2, which presents an upbend phenomenon and is understood by the gradual alignment of a h 11/2 neutron pair. Such an alignment process can be reproduced by the PRM calculations due to the use of angular-momentum-dependent moments of inertia. After the upbending, the configuration becomes a threequasiparticle configuration ν(1h 11/2 ) 3 , whose quadrupole deformation parameters are β = 0.29 and γ = 10 • from the CDFT calculations, and the data can be reproduced by the PRM (dash line) with the moments of inertia taken as irrotational flow type J k = J 0 sin 2 (γ − 2kπ/3) with J 0 = 20 2 /MeV.
For band B, the experimental rotational frequency has a discontinuity between I = 29/2 and 33/2, which is understood by the alignment of a proton g 9/2 pair given that its alignment is 2 smaller than that of band A in the region I ≥ 39/2. Hence, the unpaired valence nucleon configuration for band B at I ≥ 33/2 is assigned as π(1g 9/2 ) −2 ⊗ ν(1h 11/2 ) 1 , whose deformation parameters are β = 0.25 and γ = 28 • according to the CDFT calculations. With this configuration and J 0 = 21 2 /MeV, the corresponding experimental rotational frequencies and energies can be well reproduced as shown in Fig. 3 (short dotted line), and thus supports the configuration assignment.
With the successful reproduction of the energy spectra of bands A and B, the wobbling energy E wob (as defined in Ref. [8]) can also be reproduced by the PRM calculations, as shown in Fig. 3(c). In agreement with the experimental observation, the calculated wobbling energy decreases with spin until I = 29/2, which presents the characteristic of a transverse wobbler. Note that the increasing energy difference between bands A and B in the region I ≥ 33/2 cannot be interpreted as evidence of a longitudinal wobbler [9], since their configurations are different as discussed above.
In Table I, the experimental and theoretical mixing ratios δ as well as the transition probability ratios B(M 1) out /B(E2) in and B(E2) out /B(E2) in for the transitions from band B to A in 105 Pd are listed. It is known that B(E2) out /B(E2) in is proportional to tan 2 γ [1,13]. It is found that the PRM results are in good agreement with the data. Thus, the microscopic input of the triaxial deformation parameter from the CDFT calculation is appropriate.
The mixing ratios δ and B(M 1) out /B(E2) in are proportional to Q 0 /g eff and (g eff /Q 0 ) 2 , respectively, with Q 0 the intrinsic quadrupole moment and g eff = g νh 11/2 − g R the effective gyromagnetic factor. It was found that in the PRM calculations, the B(M 1) out values would be overestimated by about a factor of 3−10 [8,9]. This is due to the scissors mode which is mixed with the wobbling motion and cannot be considered in the PRM calculations [44]. With the successful reproduction of the energy spectra and electromagnetic transitions in 105 Pd, the angular momentum geometries of bands A and B are examined in the PRM [19]. Indeed, it is found that the neutron angular momenta in bands A and B are similar and mainly align along the s-axis in the region I ≤ 25/2. The wave functions of band A are symmetric and peaked at the s-axis below I = 23/2, which indicates a principal axis rotation around the s-axis. At I = 29/2, the rotational mode of band A changes to a planar rotation. With onephonon excitation, the wave functions of band B are antisymmetric and have a node around the s-axis. These features are the same as in the case of 135 Pr [19], thus further confirming the transverse wobbling interpretation for bands A and B of 105 Pd in the region I ≤ 29/2.
In summary, we have studied nuclear transverse wobbling in 105 Pd, where the wobbling bands are based on the ν(h 11/2 ) one-neutron configuration. The predomi-nant E2 character of the ∆I=1 M1+E2 transitions between the wobbling bands is confirmed by the precise measurement of DCO values and linear polarization data. The transverse wobbling nature of these bands conforms well to results from calculations using constrained triaxial covariant density functional theory and the quantum particle rotor model. This observation provides the first experimental evidence for transverse wobbling bands based on a one-neutron configuration, and is also the first observation of wobbling motion in the A∼100 mass region.
The authors thank Professor C. M. Petrache for stimulating and useful discussions. This work was supported by the National Research, Development and Innovation Fund of Hungary, financed under the K18 funding scheme with project nos. K128947 and K124810. This work was also supported by the European Regional Development