Exploring shape coexistence in A < 190 nuclei through EM transition rate measurements

The nuclei having mass A ~ 190 exhibit some interesting structural phenomenon like shape coexistence and shape transition. The present work reports on some of the experimental evidence of shape coexistence in this mass region for some of the nuclei. To establish experimentally the shape of a nucleus and hence the presence of shape coexistence demands use of experimental tools such as coulomb excitation or lifetime measurements. Various possible measurable quantities and associated phenomena are discussed, together with details on experiments performed. Experiments performed to populate different isotopes in Tl, Pt and Hg nuclei. Several theoretical and experimental calculations suggest the presence of shape coexistence in different isotopes of Tl, Pt and Hg nuclei.


Introduction
The prolate -oblate shape coexistence has been reported in nuclei of many mass regions across the chart of nuclei, with Z ~ 82 and A ~ 190 tagged as the best region.In Hg nuclei having mass A = 185, huge isomer shift was observed, caused by the coexistence of slightly deformed oblate and strongly deformed prolate shapes [2].In this mass region in addition to shape coexistence, phenomena like shape transition and shape mixing have been observed [3 -5].For Pb nuclei the discovery of a spherical high-spin isomeric state and two deformed, high-K isomeric states in 188 Pb were instrumental in characterizing the presence of coexisting nuclear shapes [6].The three coexisting structures occur systematically in the even-even Pb isotopes corresponding to the spherical, oblate and prolate deformed shapes.On the other hand, for the Tl nuclei oblate -prolate shape transition as a function of increasing spin has been observed [3].The experimental studies done in neutron deficient Hg-nuclei suggest a prolate-oblate shape mixing from low to high spins in the nuclei with A < 190 isotopes.The mixing has been well reproduced in the theoretical studies done with the microscopic shell-model approach [7], the theoretical mean-field approach [7] and the phenomenological two-band mixing model [8,9] approach.These studies suggest that the degree of mixing between the two bands, decreases with increasing spin such that an almost pure excited energy states are expected above 4+ state in these even-even Hg nuclei.The present work is an attempt to make an understanding of the overall evolution of the shape coexistence picture in A < 190 nuclei through the measured electromagnetic transition rates in Pt, Hg and Tl nuclei.

Shape Coexistence across nuclear chart
The study of shape-coexisting states enables us to investigate the correlations of particles under different deformation conditions within the same nucleus.Across the nuclear chart, the shape coexistence has been defined in a variety of ways over the years.Earlier, the presence of minima in the potential energy surface of the nucleus suggested as shape coexistence [4].Later, shape coexistence understood as existence of two or more states at low excitation energy and having well defined and distinct properties.The experimental studies over the years have revealed the presence of shape coexistence in very many nuclei of different mass regions [7].Mostly, the shape coexistence has been observed in nuclei in which the proton and/or neutron number is near closed shell.In light nuclei with (Z, N) (4,8), (12,20), and (14, 28), 16 O and 40 Ca are historically one of the first ones to show coexistence in nuclei.
One of the important requirements of observing the shape coexistence in nuclei is the presence of their proton and/or neutron number near the closed shell.The other important condition is the presence of a strong shape driving intruder orbitals (e.g.low-Ω, πh9/2 or πi13/2 orbitals) near the fermi surface.As shown in Figure 1 (for Au nuclei), the location of deformation driving intruder πh9/2 orbital with respect to the Fermi surface is known to manifest shape coexistence phenomena in this mass region.For theses nuclei, the shape coexistence is generated due to the promotion of multiple particles across shell gaps.The energy required is provided by correlation energies due to both pairing and quadrupole-quadrupole interactions among the particles.The promotion of particles increases the number of valence particles and holes which then interact with the valence particles of other type to produce different shapes.

Shape coexistence in neutron deficient Pt, Hg and Tl nuclei with A < 190 via lifetime measurements:
Though spectroscopic measurements are useful for getting the signatures of shape coexistence in nuclei, deformation measurements via Coulomb excitation and lifetime measurements are needed for conformity.For this reason, a number of such measurement have been done and shape coexistence/shape mixing/shape transition has been established in many nuclei of this mass region.The Coulomb excitation studies did yield the deformation estimates with sign for even-mass 182−188 Hg [11,12] isotopes and clearly established contributions from two distinct weakly-deformed oblate and strongly-deformed prolate configurations in the observed low-lying states in 182−188 Hg nuclei.However, the Coulomb excitation measurements are challenging in this mass region as they require the prior and precise knowledge of certain input parameters (e.g.lifetimes, branching and mixing ratios, etc.) for the nucleus of interest, which is largely missing due to lack of data.On the other hand, the lifetime measurement studies provide deformation information in nuclei in a model independent way, a number of lifetime measurements [8, 13 -18] have been done to study the shape coexistence in the nuclei of this mass region.To contribute in such important efforts, our group has also performed few lifetime measurements in the nuclei of this mass region targeting nuclear shape coexistence using the experimental facilities available at the Inter University Accelerator Center (IUAC), Delhi [19 -22].In these studies, the RDDS lifetime measurement were done using the RDM plunger and Germanium detector array (GDA) set-up (shown in Figure 2) available at the IUAC, Delhi.In these studies, for 187 Tl nucleus, the Qt values were extracted and plotted as a function of the spin (shown in Figure 3) for the observed transitions in the h9/2 and the i13/2 bands.From the figure, it can be observed that the value of transition quadrupole moment for the h9/2 band decreases slightly from 2.6 to 2.1 eb as one moves from the spin 13/2 to 17/2 and then changes suddenly to 6.4 eb at spin 21/2.This increase in the value of the quadrupole moment is an indication of a major shape change taking place in the structure of 187 Tl.In the lifetime measurements done in 189 Tl nucleus, the transition quadrupole moments (Qt) for the πh9/2 and πi13/2 bands in 189 Tl are plotted as a function of spin in Figure 4.The observation of the oblate shape at low frequencies, oblateprolate shapes at mid rotational frequencies, and pure prolate shape at high frequencies in the prolate i13/2 band is a clear manifestation of the shape coexistence structure.Also, the results of TRS calculations for both the negative parity, positive signature (πh9/2), and the positive parity, positive signature (πi13/2) bands in 189 Tl at different rotational frequencies are shown in Figure 4. From the plots, it is observed that in both the bands, the 189 Tl nucleus shows oblate deformation in the ground.In 188 Pt, the results of the lifetime measurements reveal some interesting behavior of reduced transition probabilities (B(E2) values).To understand the behavior of B(E2) trend with increasing spin, experimental B(E2) values for different even isotopes of Pt were compared as shown in Figure 5. From comparison, it is observed that B(E2) values increase sharply at lower spins for all isotopes (A = 178-190) indicating a strong gain in collective behavior, while with the increasing mass number, B(E2) values at higher spins are more or less constant.Also, in order to get a better insight into the possible prolate-oblate shape transition in Pt, the average Qt values for the even-even Pt isotopes with increasing mass number has been shown in Figure 5.The plotted Qt values clearly demonstrated the changing nuclear shape in Pt nuclei.Beyond A = 182, average Qt value in Pt are found to decrease with increasing mass, with the lowest being Qt = 5.4 ± 0.6 eb for 188 Pt.This decreasing trend indicates the reducing axial prolate collectivity in Pt nuclei towards higher mass.In case of 187 Hg, as shown in Figure 6, the results of the lifetime measurements show that the B(E2) values are observed to increase with the spin of the decaying state, which is an indication of increasing collectivity in 187 Hg along the yrast configuration.As the spectroscopic studies indicate the ground state to be moderately prolate, the experimental B(E2) do indicate a moderate deformation near the ground state for this nucleus.However, the deformation increases with spin, indicating more and more collectivity of the yrast structure in this nucleus.The increasing nature of the measured B(E2) values with spin is also indicative of an interaction of the yrast band with other neighboring bands, thus changing its properties.As shown in Figure 6, the variation of PSM calculated B(E2) values do support the predicated shape transition in 187 Hg nucleus from small oblate at low spin to large prolate deformed structure at high spins in the yrast configuration.

Summary
The shape coexistence phenomenon was observed all across the nuclear chart.In heavy nuclei, Z ∼ 82, there is a competition between energy gaps and residual interaction which manifests the occurrence of the shape coexistence in these nuclei.The nuclear deformation can be measured directly from coulomb excitation or indirectly from the lifetime measurement.To understand the shape coexistence, experiments were carried out to populate different isotopes in Tl, Pt and Hg.The experimental Qt values were compared with the theoretical predictions and found out to be in agreement.However, in order to firmly establish prolate-oblate shape coexistence in Hg isotopes, more odd-A isotopes with A~ 185 need to be probed with lifetime measurements.

Figure 1 .
Figure1.Systematics of the 1h9 /2 proton intruder states in the odd-mass Au isotopes (left hand side) and in the odd-mass Tl isotopes (right hand side) (Adopted from[10]).3.Shape coexistence in neutron deficient Pt, Hg and Tl nuclei with A < 190 via lifetime measurements:Though spectroscopic measurements are useful for getting the signatures of shape coexistence in nuclei, deformation measurements via Coulomb excitation and lifetime measurements are needed for conformity.For this reason, a number of such measurement have been done and shape coexistence/shape mixing/shape transition has been established in many nuclei of this mass region.The Coulomb excitation studies did yield the deformation estimates with sign for even-mass 182−188 Hg[11,12] isotopes and clearly established contributions from two distinct weakly-deformed oblate and strongly-deformed prolate configurations in the observed low-lying states in 182−188 Hg nuclei.However, the Coulomb excitation measurements are challenging in this mass region as they require the prior and precise knowledge of certain input parameters (e.g.lifetimes, branching and mixing ratios, etc.) for the nucleus of interest, which is largely missing due to lack of data.On the other hand, the lifetime measurement studies provide deformation information in nuclei in a model independent way, a number of lifetime measurements[8, 13 -18]  have been done to study the shape coexistence in the nuclei of this mass region.To contribute in such important efforts, our group has also performed few lifetime measurements in the nuclei of this mass region targeting nuclear shape coexistence using the experimental facilities available at the Inter University Accelerator Center (IUAC), Delhi[19 -22].In these studies, the RDDS lifetime measurement were done using the RDM plunger and Germanium detector array (GDA) set-up (shown in Figure2) available at the IUAC, Delhi.In these studies, for 187 Tl nucleus, the Qt values were extracted and plotted as a function of the spin (shown in Figure3) for the observed transitions in the h9/2 and the i13/2 bands.From the figure, it can be observed that the value of transition quadrupole moment for the h9/2 band decreases slightly from 2.6 to 2.1 eb as one moves from the spin 13/2 to 17/2 and then changes suddenly to 6.4 eb at spin 21/2.This increase in the value of the quadrupole moment is an indication of a major shape change taking place in the structure of 187 Tl.In the lifetime measurements done in 189 Tl nucleus, the transition quadrupole moments (Qt) for the πh9/2 and πi13/2 bands in189 Tl are plotted as a function of spin in Figure4.The observation of the oblate shape at low frequencies, oblateprolate shapes at mid rotational frequencies, and pure prolate shape at high frequencies in the prolate i13/2 band is a clear manifestation of the shape coexistence structure.Also, the results of TRS calculations for both the negative parity, positive signature (πh9/2), and the positive parity, positive signature (πi13/2) bands in189 Tl at different rotational frequencies are shown in Figure4.From the plots, it is observed that in both the bands, the 189 Tl nucleus shows oblate deformation in the ground.

Figure 2 .
Figure 2. Left hand side showing picture and detailed schematics of RDM plunger set-up available at IUAC, Delhi.Right hand side showing picture of GDA set-up used in the experiment.(Taken from [23])

Figure 3 .
Figure 3. Left hand side, plot showing variation of observed transition quadrupole moment (Qt) with increasing spin.Right hand side, plot of TRS calculated β2 and γ deformation parameters as a function of rotational frequency.Data taken from [19].

Figure 4 .
Figure 4. Left hand side, plot showing variation of observed transition quadrupole moment (Qt) with increasing spin.Right hand side, plot of TRS calculated β2 and γ deformation parameters as a function of rotational frequency.Data taken from [20].

Figure 5 .
Figure 5. Left hand side, the comparison of B(E2) values for even-A 182,184,186,188 Pt isotopes with increase in spin.Right hand side, the comparison of average experimental Qt values for even-mass Pt isotopes (A = 182-188) with mass number.Data taken from [21].