Systematics of the ratio of electric quadrupole moments Q(21+) to the square root of the reduced transition probabilities B(E2;01+→21+) in even–even nuclei

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Systematics of the ratio of electric quadrupole moments Q(2+ 1 ) to the square root of the reduced transition + probabilities B(E2; 0+ 1 → 21 ) in even–even nuclei Y.Y. Sharon a , N. Benczer-Koller a,∗ , G.J. Kumbartzki a , L. Zamick a , R.F. Casten b,c

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a Department of Physics and Astronomy, Rutgers University, New Brunswick, NJ 08903, USA b Wright Laboratory, Yale University, New Haven, CT 06520, USA c FRIB, Michigan State University, East Lansing, MI 48824, USA

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Received 16 August 2018; received in revised form 20 September 2018; accepted 9 October 2018

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A survey √ of even–even nuclei in the nuclear chart showed regularities in the values of the ratio Q(2+ )/ (B(E2) for each nucleus for which independent measurements of both the electric quadrupole 1 + + + moment of the 2+ 1 state, Q(21 ), and the reduced transition probability B(E2; 01 → 21 ) exist. As predicted by the rotational model, this ratio was found to be close to unity for the deformed nuclei in the rare-earth region. For non-rotational nuclei the absolute value of this ratio is almost always considerably lower. This latter observation can be interpreted within different models for different classes of nuclei as illustrated by a set of schematic IBA model calculations. A general mixing model and simple geometrical models are also suitable for collective non-rotational nuclei. Near magic nuclei a mechanism within the seniority scheme is compatible with the observed small value of RQB . © 2018 Published by Elsevier B.V.

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Abstract

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Keywords: Electric quadrupole moment and reduced transition probability

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* Corresponding author.

E-mail address: [email protected] (N. Benczer-Koller). https://doi.org/10.1016/j.nuclphysa.2018.10.027 0375-9474/© 2018 Published by Elsevier B.V.

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1. Introduction

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Several methods exist to extract structural information on even–even nuclei. These include the energies of the low-lying yrast states and their ratios, the energies of intrinsic excitations, and a number of observables related to quadrupole transitions matrix elements. In the present paper the focus lies on the latter where two complementary approaches stand out. The first involves the electric quadrupole moment Q(2+ 1 ) and the second consists of a measurement of the + reduced E2 transition probability (BE2; 0+ been used to discuss the 1 → 21 ). Both have often √ )/ B(E2) across the nuclear effective deformation. A systematic inspection of the ratio Q(2+ 1 chart was carried out showing different patterns in deformed and non-deformed nuclei, which are interpreted here in terms of several appropriate models. While quadrupole moments are more difficult to measure, large numbers of measurements do indeed exist. The literature was surveyed using recent compilations of Q(2+ 1 ) data and + + (BE2; 0+ → 2 ) values from C to Pb. The Q(2 ) values included in the present investiga1 1 1 tions were obtained by Coulomb excitation with reorientation (CER), its precession (CERP), electron scattering (ES), muonic x-ray hyperfine structure (MU-x), Mössbauer effect (ME) or time-independent angular correlation (TDPAC). All these measurements of Q(2+ 1 ) are indepen+ dent of B(E2) measurements. Specifically, all Q(21 ) values were taken from the compilation + of N. Stone [1]. The B(E2; 0+ 1 → 21 ) values were taken from the compilation of B. Pritychenko et al. [2].

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2. Results

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RQB = − 

Q(2+ 1) B(E2; 0+ 1

→ 2+ 1)

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All currently available data, consistent with above criteria, are presented in Table A.1 in the Appendix. The nuclei, 94,96 Mo, 98,100,102,104 Ru, 130,134,136,138 Ba, 142 Ce, 144 Nd and 198,200,202 Hg, have double values of Q(2+ ), corresponding to the inability to determine whether 1 the analysis involved constructive or destructive interference. Both values have been kept in the evaluation. Out of the 120 nuclei considered, several have positive Q(2+ 1 ) values suggest+ ing oblate shapes. Most of the remaining nuclei have negative Q(21 ) values related to prolate shapes. The actinide region is not included in the present investigation due to the absence of measured Q(2+ 1 ) values beyond Pb in [1]. Since both Q and the B(E2) values reflect the deformation of the nucleus, it is interesting to look at their ratio. For this purpose the square root of the B(E2) value was used to form a dimensionless quantity, RQB ,

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(1)

The negative sign is used to make most RQB values positive since most quadrupole moments, including nearly all for deformed nuclei, which are predominantly prolate, are negative. As seen in Fig. 1a, where the measured RQB values are plotted, in nearly all cases, the ratio |RQB | lies between roughly 0.0 and 1.0, with a few outliers. This figure is the key result of this paper. + For reference to the structure, Fig. 1b shows the yrast energy ratio R42 = E(4+ 1 )/E(21 ). In + + deformed nuclei the yrast energies go as I (I + 1) and therefore E(41 )/E(21 ) = 10/3. This ratio takes on values in the range ∼2.0 to 2.4 for typical vibrational nuclei (the harmonic value is 2.0), and <2.0 for non- or weakly-collective nuclei near closed shells.

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Fig. 1. Fig. 1a shows the ratio RQB , as defined in Eq. (1), in terms of the measured electric quadrupole moments and reduced transition probabilities. The isotopes of each element, denoted by one color, are plotted as a function of increasing atomic number, Z, and for each Z, in order of increasing neutron number, N . The abscissa scale, labeled by mass number, A, is therefore non-linear but is indicated correctly at each of the tick marks. The dashed lines at 0.0, 0.5 + and 1.0 are visual guides only. Fig. 1b shows, for comparison, the energy ratios E(4+ 1 )/E(21 ). (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)

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Fig. 2. Histograms of RQB values binned for three different regions of nuclei, namely (a) all nuclei in the compilation of Table A.1, (b) the nuclei Ni to Ce and (c) the nuclei Nd to Os.

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3. Discussion

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There are some notable features in the behavior of RQB with atomic number. There is a systematic difference between rotational and non-rotational nuclei. For deformed and nearly deformed nuclei (mostly in the rare-earth region), RQB ∼ 1.0. In regions far from deformed nuclei RQB is smaller than unity. Both features are quite salient. These trends are highlighted in Fig. 2 which displays histograms of the frequency of RQB values containing all nuclei (Fig. 2a), the nuclei from Ni to Ce (Fig. 2b) and the nuclei from Nd to Os (Fig. 2c). It is clear that in the Ni to

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Ce region the predominance of the data lies in the region 0.3 < RQB < 0.7, while in the Nd to Os region most of data lie in the 0.8 < RQB < 1.0 space. Thus the RQB systematics reflects the underlying nuclear structures. 3.1. Rotational nuclei

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3K 2 − I (I + 1) Q(I, K) = · Qo (I + 1)(2I + 3)

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B(E2; (Ii , K) → (If , K)) = (5/16π) · Q2o · (Ii K20|If K)2

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and B(E2; (0, 0) → (2, 0)) = (5/16π) · Q2o

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Q(2, 0) = −(2/7) · Qo

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Q(2, 0) = −0.906 √ B(E2; (0, 0) → (2, 0))

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since the Clebsch–Gordan coefficient is equal to unity in this case. Eliminating Qo from the last two equations yields a ratio

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where the last term in parentheses on the right is a Clebsch–Gordan coefficient. For K = 0, I = 2, one obtains

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It is immediately visible that this value is very similar to RQB in Eq. (1) as seen in Fig. 1a for deformed nuclei and it is entirely reasonable that one obtains empirical values of RQB near unity for deformed nuclei. Note that a few non-rotational nuclei are also expected to have RQB values near unity. In the X(5) symmetry [3], the theoretical value of this ratio is 0.93, which is reflected in the large empirical values in X(5)-like nuclei such as 150 Nd, 152 Sm and 154 Gd. The above analysis for deformed nuclei assumes complete separation of rotational and intrinsic degrees of freedom including, implicitly, the assumption of good K values. However, in all known deformed nuclei there is K-band-mixing, primarily between the ground band and the γ band. One effect of band mixing is to modify the B(E2) values since their matrix elements now consist of two terms, a K = 0 → K = 0 unperturbed part and a K = 0 → K = 2 part. In principle, it would be interesting to analyze the deviations of RQB from 0.906 in terms of this band mixing. + However, the effects on the allowed matrix element M(E2; 0+ 1 → 21 ) values due to the coherent inclusion of a weak interband, ground state to γ bandhead, component are very small. Typical mixing amplitudes for the 2+ states are < 0.01 [4–6] and these multiply matrix elements that are also about a factor of five smaller than the intraband matrix elements. So the amplitudes of this

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state. Furthermore, in the rotational where Qo is the intrinsic quadrupole moment of the model the B(E2) value for the intraband E2 transition from the (Ii , K) state to the (If , K) state is given by

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An easy explanation can be given for the RQB values of the deformed nuclei. In the rotational model, in which one usually invokes a separation of rotational and intrinsic degrees of freedom, a given state is characterized by the projection K of its angular momentum I on its axis of symmetry and its quadrupole moment Q(I, K) is given by

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mixing are on the order of a few tenths of a percent compared to the allowed matrix element component – well, below the uncertainties in RQB . Another effect of the mixing in deformed nuclei is to decrease Q(2+ 1 ). From Eq. (2) it follows that Q(2, 2) = +(2/7) · Qo = −Q(2, 0). Again this effect is small since the Q(2,2) contribution is multiplied by a very small mixing amplitude. Suffice it to say that the RQB values shown for deformed nuclei are consistent with estimates based on the simple rotational results given in Eq. (6).

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3.2. Non-rotational nuclei

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Can one develop a similar understanding of the lower RQB values in non-rotational regions? The most obvious deviations from ∼unity in Fig. 1a are the negative values for Pt, Hg and Pb. This sign change relative to most values of RQB is due to positive quadrupole moments corresponding to oblate shapes near the end of the shell, as has been well documented in this region [7]. The generally smaller values of RQB , especially for the Ni to Nd range of isotopes displayed in Figs. 1a and 2b, can be qualitatively understood from simple geometrical models. For example, in the harmonic vibrator model, Q(2+ 1 ) = 0 and √ hence RQB is also zero since the B(E2) value However, Q is is finite. Actually both the values of Q and B(E2) reflect the deformation. √ linear in the deformation, and therefore senses its average value, while B(E2) is proportional to the root-mean-square deformation. In a deformed potential both should be similar and would yield an RQB value near unity. In a spherical or weakly-deformed potential RQB should decrease since the average deformation (reflecting zero-point motion) should approach zero faster than its root-mean-square value. To look at this effect more quantitatively, a set of simple IBA model [8] calculations was performed to see the behaviour of RQB for different patterns of structural evolution in Fig. 3a. The calculations used, for simplicity, 10 bosons and a Hamiltonian with constant QQ term, varying boson energy and internal parameter χ in the Q operator in the standard CQF formalism [9,10]. The calculations are schematic, but they illustrate the sensitivity of RQB to the nuclear structure and the different behavior of RQB for different evolutionary scenarios. The results are shown in + Fig. 3b where RQB is plotted against the yrast energy ratio E(4+ 1 )/E(21 ). The figure shows four trajectories which can be best described in terms of the standard structural triangle representing the space of IBA calculations shown in Fig. 3a. In this triangle the three symmetries of the IBA are located at the vertices and any contour along the legs or through the interior represents a path in the evolution of collective structure. Each path (trajectory) is defined by appropriate variations of the parameters in the IBA Hamiltonian. In Fig. 3b all the paths start at the SU(3) vertex for a rotor (RQB ∼ 0.906, energy ratio = 3.33). Trajectory A corresponds to the path from the symmetric rotor to a pure vibrator [U(5) vertex], where Q, and hence RQB , goes to zero at R42 = 2.0. It is interesting that the results even show the slight rise in the transitional region predicted by X(5) (see Section 3.1) resulting in a value of RQB = 0.93, somewhat larger than the rotor value. Note also that RQB does not significantly decrease until quite close to the vibrator limit along this trajectory. Trajectory B corresponds to an evolution from the symmetric rotor at SU(3) to a γ -soft rotor [O(6) vertex]. Here again, Q goes to zero at the terminus and so RQB curves down to zero at a faster rate and vanishes at R42 = 2.5, characteristic of O(6). Trajectory C goes from the symmetric rotor to a point along the leg from a vibrator to a γ -soft rotor. Along this leg of the triangle, R42 varies from 2.0 to 2.5 and RQB vanishes along the entire leg. The calculation illustrated corresponds to intersecting the leg at R42 ∼ 2.25.

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φ2M+ 2

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U = −βψ(2+ 1)

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To further understand how lower RQB values can arise in non-deformed or weakly deformed nuclei, a simple mixing calculation that can qualitatively account for values of RQB between zero and the rotor value is presented below. Away from well-deformed nuclei, the first two 2+ states can still mix, such as in soft or vibrational nuclei. In the former, dynamically deformed shapes result from zero point motion in a finite potential. In the latter, their parentages are, respectively, the one- and two-phonon states. In still other regions intruder states appear in the low-lying spectrum, as shown for instance in references [11–14]. A simple mixing model of 2+ states analogous to that often invoked in deformed nuclei, but more general, can be considered, to understand why RQB is typically less than one for nuclei + away from the rotational regions. In most cases, the 2+ 2 level lies often closer to the 21 level than in deformed nuclei and these two states can mix more readily. In this case, the wave functions for these, mixed, 2+ states can be written as U φ2M+ = αψ(2+ 1) 1

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3.3. The influence of mixing in nearly spherical or weakly deformed nuclei

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Trajectory D is more complex. It corresponds to a trajectory starting at the symmetric rotor, arcing into the middle of the triangle and then dropping to the triangle leg from the vibrator to rotor [U(5) to SU(3)]. As one enters the triangle RQB drops at first. But, noting that RQB decreases very slowly along trajectory A, in trajectory D, RQB actually increases back up nearly to 0.9 when the trajectory hits the vibrator to rotor leg. The message from these illustrative calculations, or an infinity of others that could have been shown, is that RQB has a complex behavior throughout the span of collective structures, that RQB can be very sensitive to structure, and that it can be used, in conjunction with other observables such as R42 , to help pinpoint the structure of a given nucleus.

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Fig. 3. Results for a simple IBA model calculation for RQB . The arrows in Fig. 3a depict four trajectories, A, B, C, D for which the RQB values were calculated. The resulting RQB are shown in Fig. 3b.

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where α 2 + β 2 = 1 and U and M denote the unmixed and mixed wave functions, respectively. The nature of the unperturbed basis states is left undefined for now.

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The second term inside the bracket involves a transition from the second 2+ state to the 0+ ! ground state. In most models for non-deformed nuclei this transition will be hindered compared to the yrast transition. For example, in the pure harmonic vibrator or in γ -soft nuclei, it is forbidden. Moreover, the mixing amplitude β is expected to be small compared to the amplitude α of the unperturbed transition. Hence, this second term in Eq. (8) can be neglected, leading to

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+ M + U + 2 2 B(E2; 0+ 1 → 21 ) ∼ α < ψ(21 ) ||r Y2 ||ψ(01 ) >

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+ M + M M 2 Q(2+ 1 ) = 0.759 < φ( 21 ) ||r Y2 ||φ( 21 ) >

+ U U 2 = 0.759 < αψ(2+ 1 ) + βψ(22 ) ||r Y2 ||

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> √ √ where 0.759 = 16π/5 · (2220|22) · [1/ 2J + 1] and the factor in the middle of this product is the appropriate Clebsch–Gordan coefficient (0.5345). The direct terms involve the unperturbed + quadrupole moments of the unmixed 2+ 1 and 22 states which can be taken as quite small for weakly deformed nuclei. For this heuristic analysis, these terms were ignored and the focus was set on the cross terms. The quadrupole moment of the physical (mixed) 2+ 1 state then becomes, with Ji = 2

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U + βψ(2+ 2)

+ U + U M 2 Q(2+ 1 ) ∼ 0.759 · 2 · αβ < ψ( 21 ) ||r Y2 ||ψ( 22 ) >   + U ∼ 1.52αβ B(E2; 2+ 2 → 21 ) · 2Ji + 1  + U ∼ 3.40αβ B(E2; 2+ 2 → 21 )

Hence for non-deformed nuclei

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+ M This B(E2; 0+ is thus equal to the unperturbed value except for the (likely small) 1 → 21 ) deviations of α from unity. The quadrupole moment Q of the 2+ 1 state in this mixing model needs to be calculated from the diagonal matrix element of the E2 operator in the mixed 2+ 1 state,

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+ + U + U 2 2 2 [< αψ(2+ 1 ) ||r Y2 ||(ψ(01 ) + βψ(22 ) ||r Y2 ||(ψ(01 ) >]

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+ M B(E2; 0+ 1 → 21 ) = [1/(2Ji + 1)]·

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+ M + The value of B(E2; 0+ 1 → 21 ) can be expressed in this mixing model (the 01 state is considered unmixed),

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+ U B(E2; 2+ 2 → 21 ) + U B(E2; 0+ 1 → 21 )

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The right side can be evaluated in a number of models. As an example, take the harmonic + + + quadrupole vibrator model, in which B(E2; 2+ 2 → 21 )/B(E2; 01 → 21 ) = 0.4. In the IBA model this result is modified by finite nucleon number effects and is typically about 0.32. Exper+ imentally, in most nuclei with A > 100 and 2.0 < E(4+ 1 )/E(21 ) < 3.0 – that is, medium-mass and heavy, collective, non-deformed nuclei – this ratio ranges from ∼ 0.1 to 0.4. Using, for the sake of this discussion, a value of 0.4 for the B(E2) ratio gives

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The quantity on the left is RQB . For RQB = 0.5, typical of many non-deformed nuclei, as seen in Fig. 1, one obtains β ∼ 0.23. Hence a simple mixing picture is able to give RQB values significantly smaller than unity in non-deformed nuclei. If mixing amplitudes were provided by other observables, RQB could be predicted and tested for consistency against the measured RQB data. Conversely, the mixing amplitude can be determined from the observed value of RQB .

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3.4. Nuclei near closed shells

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Finally, one can look at nuclei very close to magic in terms of the seniority scheme. In the present work the ground state has seniority zero and the 2+ 1 state has seniority 2. The dependence of the matrix element of the E2 operator on particle number is different for seniority-conserving cases such as Q(2+ 1 ) and seniority-changing situations such as + + B(E2; 01 → 21 ). For the extreme simple case of valence particles of one type occupying a single j orbit of configuration |j n >, the quadrupole moment goes approximately as (1-2f) where  √ + f is the fractional filling of the j shell, and the B(E2; 0+ 1 → 21 ) is proportional to f (1 − f ). The former decreases as the shell fills and, indeed, crosses from prolate to oblate at mid-shell. The latter has a weaker dependence, being rather flat over much of a shell, while decreasing to zero at shell boundaries. Note that, in the same j shell, the quadrupole moment of holes is opposite in sign to that of particles. Of course, real nuclei are not describable as single j configurations, or even pure shell model configurations of any sort, but this general behavior persists, even into deformed nuclei. Indeed, small deviations from the parabolic behavior of B(E2) values in the light Sn nuclei have been an area of active study in recent years ([15] and references therein). Generalization to a multi-j situation gives intrinsic quadrupole moments that are generally prolate, turning oblate only at the extreme ends of major shells.

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√ In summary, in this paper new systematics of the ratio Q/ B(E2) are presented. In the rotational nuclei of the rare-earth region RQB ∼ 0.9 as expected for deformed nuclei where there is a good separation of rotational and intrinsic degrees of freedom and where the standard rotational model relations among B(E2) values and quadrupole moments apply. Away from the rotational region, RQB < 1. This result can be qualitatively accounted for in terms of a simple mixing model and other simple models appropriate to the regions in question. Within an IBA framework, one can plot calculated values of RQB against R42 for any point in the triangle, specifying the IBA wave function. Several examples were shown, illustrating that measured values of these two quantities in a given nucleus can shed light on its structure. It remains intriguing why |RQB | >1 just below and above the rotational region, especially for 148,150 Nd and 200,202 Hg, as well as for the light isotopes 20,22 Ne and 24,26 Mg. Generally, it would be interesting to see if more microscopic and sophisticated models can account in detail for the systematics of RQB that have been presented here.

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Acknowledgements

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The authors thank S. Yeager, Xiaofei Yu for their contributions to the data collection and M. Caprio, F. Iachello, E.A. McCutchan and S.J.Q Robinson for discussions. Y.Y.S. acknowledges a 2017 Stockton University sabbatical grant. The authors appreciate the support of the National Science Foundation and the U.S. Department of Energy.

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Appendix A. Data Table

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36 Ar 40 Ar 42 Ca

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50 Cr

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52 Cr 54 Cr 54 Fe

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58 Ni

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60 Ni

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62 Ni

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64 Ni 64 Zn 66 Zn

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68 Zn

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70 Zn

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70 Ge

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72 Ge

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74 Ge

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76 Ge 74 Se

DP

18

0.0040 0.0043 0.0333 0.0230 0.0438 0.0314 0.0927 0.0208 0.0296 0.0208 0.0301 0.0332 0.0951 0.0467 0.0951 0.0662 0.0275 0.1052 0.0622 0.0853 0.0608 0.0981 0.1220 0.0650 0.0916 0.0889 0.0674 0.1494 0.1370 0.1199 0.1510 0.1790 0.2087 0.3060 0.2735 0.3570

0.0002 0.0004 0.0016 0.0004 0.0009 0.0007 0.0005 0.0006 0.0007 0.0012 0.0016 0.0017 0.0025 0.0025 0.0025 0.0029 0.0016 0.0032 0.0025 0.0042 0.0031 0.0020 0.0060 0.0012 0.0016 0.0030 0.0032 0.0040 0.0033 0.0021 0.0080 0.0030 0.0030 0.0150 0.0030 0.0200

TE

18 O

0.03 0.01 0.03 0.04 0.03 0.02 0.03 0.06 0.02 0.03 0.06 0.04 0.08 0.07 0.06 0.01 0.16 0.07 0.02 0.08 0.14 0.03 0.05 0.06 0.02 0.12 0.20 0.02 0.13 0.16 0.03 0.06 0.06 0.02 0.06 0.07

δB(E2) [e2 b2 ]

EC

17

0.06 −0.04 −0.23 −0.19 −0.29 −0.21 0.16 −0.05 −0.16 0.04 0.11 0.01 −0.19 −0.14 −0.21 −0.18 0.08 −0.36 −0.08 −0.21 −0.05 −0.23 −0.27 −0.10 −0.10 0.05 0.40 −0.14 −0.08 −0.11 −0.24 0.03 −0.13 −0.19 −0.19 −0.36

RR

12 C

CO

16

B(E2) [e2 b2 ]

UN

15

δQ(2+ 1) [b]

6

9

14

Q(2+ 1) [b]

5

8

13

12

4

7

Table A.1 + + + + + Measured values of Q(2+ 1 ) [1], B(E2; 01 → 21 ) [2] and their uncertainties δQ(21 ) and δB(E2; 01 → 21 ) which were used in the calculation of RQB and δRQB . The electric charge unit e is subsumed in the definition of Q in Ref. [1].

11

3

RQB

δRQB

−0.952 0.549 1.260 1.253 1.387 1.186 −0.526 0.347 0.930 −0.277 −0.634 −0.055 0.614 0.648 0.681 0.688 −0.482 1.110 0.321 0.719 0.203 0.734 0.773 0.392 0.330 −0.168 −1.541 0.362 0.219 0.306 0.618 −0.071 0.285 0.343 0.363 0.603

0.476 0.137 0.166 0.264 0.145 0.114 0.099 0.416 0.117 0.208 0.346 0.220 0.260 0.324 0.195 0.036 0.965 0.218 0.081 0.275 0.568 0.097 0.146 0.235 0.066 0.402 0.772 0.053 0.351 0.462 0.082 0.142 0.132 0.042 0.115 0.125

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHA AID:21382 /FLA

82 Se

8

94 Kr

9

94 Mo

10 11

96 Mo

12

98 Mo

13

100 Mo

14

96 Ru

15

98 Ru

16 17 18 19 20

100 Ru 102 Ru 104 Ru

21 22

102 Pd

23

104 Pd

24 25

106 Pd 108 Pd 110 Pd

26

106 Cd

27

108 Cd

28

110 Cd

29

112 Cd

30

114 Cd

31 32

116 Cd 112 Sn 116 Sn

33

118 Sn

34

120 Sn

35

122 Sn

36

124 Sn

37

126 Sn

38 39

122 Te 124 Te 126 Te

40

128 Te

41

130 Te

42

130 Ba

43 44 45 46 47

134 Ba 136 Ba

RQB

δRQB

−0.34 −0.26 −0.31 −0.22 −0.50 −0.13 0.01 −0.20 0.04 −0.26 −0.25 −0.15 −0.21 −0.01 −0.44 −0.27 −0.63 −0.34 −0.78 −0.20 −0.20 −0.46 −0.51 −0.58 −0.47 −0.28 −0.45 −0.40 −0.37 −0.35 −0.42 −0.09 −0.17 −0.14 0.02 −0.13 0.03 0.00 −0.57 −0.45 −0.23 −0.22 −0.12 −1.02 −0.09 −0.26 0.15 −0.19 0.07

0.07 0.09 0.07 0.07 0.30 0.08 0.08 0.08 0.08 0.09 0.07 0.08 0.08 0.09 0.04 0.07 0.04 0.03 0.07 0.12 0.15 0.11 0.07 0.04 0.03 0.08 0.08 0.04 0.04 0.01 0.04 0.10 0.04 0.10 0.07 0.10 0.13 0.02 0.05 0.05 0.05 0.05 0.05 0.15 0.15 0.12 0.12 0.06 0.07

0.4320 0.3430 0.2521 0.1830 0.2470 0.2072 0.2072 0.2775 0.2775 0.2695 0.5300 0.2379 0.4010 0.4010 0.4927 0.4927 0.6320 0.6320 0.8260 0.8260 0.4600 0.5290 0.6600 0.7640 0.8650 0.4070 0.4190 0.4260 0.5010 0.5360 0.5800 0.2320 0.2062 0.2070 0.1975 0.1887 0.1622 0.1269 0.6500 0.5600 0.4738 0.3800 0.2960 1.1380 1.1380 0.6650 0.6650 0.4130 0.4130

0.0160 0.0120 0.0082 0.0100 0.0280 0.0074 0.0074 0.0059 0.0059 0.0057 0.0270 0.0065 0.0130 0.0130 0.0041 0.0041 0.0120 0.0120 0.0170 0.0170 0.0230 0.0150 0.1700 0.0200 0.0230 0.0120 0.0140 0.0210 0.0220 0.0250 0.0260 0.0110 0.0050 0.0040 0.0024 0.0045 0.0040 0.0073 0.0300 0.0280 0.0093 0.0071 0.0100 0.0460 0.0460 0.0190 0.0190 0.0110 0.0110

0.517 0.444 0.617 0.514 1.006 0.286 −0.022 0.380 −0.076 0.501 0.343 0.308 0.332 0.016 0.627 0.385 0.792 0.428 0.858 0.220 0.295 0.632 0.628 0.664 0.505 0.439 0.695 0.613 0.523 0.475 0.551 0.187 0.374 0.308 −0.045 0.299 −0.074 −0.000 0.707 0.601 0.334 0.357 0.221 0.956 0.084 0.319 −0.184 0.296 −0.109

0.111 0.156 0.142 0.166 0.609 0.176 0.176 0.153 0.152 0.174 0.100 0.164 0.128 0.142 0.060 0.100 0.066 0.044 0.095 0.133 0.222 0.156 0.156 0.066 0.050 0.128 0.130 0.076 0.069 0.041 0.069 0.208 0.089 0.220 0.158 0.230 0.323 0.056 0.087 0.084 0.074 0.082 0.093 0.174 0.141 0.149 0.148 0.095 0.109

3 4

RO OF

80 Se

7

δB(E2) [e2 b2 ]

DP

78 Se

B(E2) [e2 b2 ]

TE

6

76 Se

δQ(2+ 1) [b]

EC

5

2

Q(2+ 1) [b]

RR

4

Isotope

CO

3

1

Table A.1 (Continued)

2

UN

1

[m1+; v1.289; Prn:11/10/2018; 15:07] P.10 (1-12)

Y.Y. Sharon et al. / Nuclear Physics A ••• (••••) •••–•••

10

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHA AID:21382 /FLA

[m1+; v1.289; Prn:11/10/2018; 15:07] P.11 (1-12)

Y.Y. Sharon et al. / Nuclear Physics A ••• (••••) •••–•••

7

142 Ce

8 9

144 Nd

10

146 Nd

11

148 Nd

12

150 Nd

13

148 Sm

14

150 Sm

15 16

152 Sm 154 Sm 154 Gd

17

156 Gd

18

158 Gd

19

160 Gd

20

164 Dy

21

166 Er

22 23 24

170 Er 170 Yb 172 Yb 174 Yb

25

176 Yb

26

176 Hf

27

178 Hf

28

180 Hf

29 30 31

180 W 182 W 184 W 186 W

32

184 Os

33

186 Os

34

188 Os

35

190 Os

36 37 38

192 Os 192 Pt 194 Pt 196 Pt

39

198 Pt

40

198 Hg

41 42 43 44

200 Hg 202 Hg

45

204 Hg

46

204 Pb

47

206 Pb

−0.14 0.08 −0.50 −0.16 −0.37 −0.15 −0.28 −0.78 −1.46 −2.00 −1.00 −1.30 −1.67 −1.87 −1.82 −1.93 −2.01 −2.08 −2.08 −1.90 −1.90 −2.18 −2.22 −2.18 −2.28 −2.10 −2.02 −2.00 −2.10 −2.10 −1.90 −1.60 −2.70 −1.63 −1.46 −1.18 −0.96 0.60 0.48 0.62 0.42 0.68 0.84 0.96 1.11 0.87 1.01 0.40 0.23 0.05

0.06 0.06 0.30 0.05 0.05 0.06 0.06 0.09 0.13 0.05 0.30 0.20 0.02 0.04 0.04 0.04 0.04 0.04 0.15 0.40 0.20 0.03 0.04 0.05 0.06 0.02 0.02 0.02 0.40 0.40 0.20 0.30 1.20 0.04 0.04 0.03 0.03 0.20 0.14 0.08 0.12 0.12 0.12 0.11 0.11 0.13 0.13 0.20 0.09 0.09

0.2300 0.2300 0.4840 0.4572 0.4572 0.5040 0.5040 0.7480 1.3380 2.7070 0.7130 1.3470 3.4611 4.3450 3.8720 4.7000 5.0900 5.1830 5.6160 5.7230 5.8380 5.7200 6.0900 5.8500 5.1890 5.4200 4.7360 4.6470 4.1500 4.1230 3.7060 3.5000 3.2140 3.0640 2.5000 2.3450 2.0300 1.9400 1.6310 1.4010 1.0720 0.9612 0.9612 0.8550 0.8550 0.6150 0.6150 0.4240 0.1587 0.0989

RQB

0.0110 0.0110 0.0380 0.0050 0.0050 0.0150 0.0150 0.0220 0.0300 0.0300 0.0350 0.0260 0.0021 0.0044 0.0016 0.1100 0.1100 0.0130 0.0680 0.0450 0.0810 0.7000 0.1500 0.1600 0.0890 0.1700 0.0630 0.0030 0.1400 0.0420 0.0350 0.0380 0.0790 0.0720 0.0360 0.0900 0.1000 0.0650 0.0680 0.0680 0.0500 0.0070 0.0070 0.0280 0.0280 0.0210 0.0210 0.0210 0.0069 0.0028

0.292 −0.167 0.719 0.237 0.547 0.211 0.394 0.902 1.262 1.216 1.184 1.120 0.896 0.897 0.925 0.890 0.891 0.914 0.878 0.794 0.786 0.912 0.900 0.901 1.001 0.902 0.928 0.928 1.031 1.034 0.987 0.855 1.506 0.931 0.923 0.771 0.674 −0.431 −0.376 −0.524 −0.406 −0.694 −0.857 −1.038 −1.200 −1.109 −1.288 −0.614 −0.577 −0.159

δRQB

2 3

RO OF

140 Ba

[b]

δB(E2) [e2 b2 ]

DP

6

[b]

B(E2) [e2 b2 ]

TE

5

δQ(2+ 1)

EC

138 Ba

Q(2+ 1)

RR

Isotope

3 4

1

Table A.1 (Continued)

CO

2

UN

1

11

0.126 0.125 0.437 0.074 0.076 0.086 0.088 0.124 0.157 0.110 0.372 0.195 0.022 0.035 0.027 0.149 0.149 0.055 0.131 0.187 0.139 0.382 0.175 0.182 0.152 0.186 0.117 0.027 0.275 0.224 0.139 0.181 0.702 0.127 0.091 0.117 0.109 0.154 0.120 0.096 0.124 0.126 0.128 0.147 0.156 0.184 0.190 0.310 0.227 0.286

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHA AID:21382 /FLA

12

1

[m1+; v1.289; Prn:11/10/2018; 15:07] P.12 (1-12)

Y.Y. Sharon et al. / Nuclear Physics A ••• (••••) •••–•••

References

1 2

2

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

RO OF

9

DP

8

TE

7

EC

6

RR

5

CO

4

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UN

3

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47