Shape coexistence in neutron-deficient Hg isotopes studied via lifetime measurements in $^{184,186}$Hg and two-state mixing calculations

The neutron-deficient mercury isotopes, $^{184,186}$Hg, were studied with the Recoil Distance Doppler Shift (RDDS) method using the Gammasphere array and the K\"oln Plunger device. The Differential Decay Curve Method (DDCM) was employed to determine the lifetimes of the yrast states in $^{184,186}$Hg. An improvement on previously measured values of yrast states up to $8^{+}$ is presented as well as first values for the $9_{3}$ state in $^{184}$Hg and $10^{+}$ state in $^{186}$Hg. $B(E2)$ values are calculated and compared to a two-state mixing model which utilizes the variable moment of inertia (VMI) model, allowing for extraction of spin-dependent mixing strengths and amplitudes.


I. INTRODUCTION
Nuclei exhibiting different shapes at low energy have been of interest in nuclear structure ever since the discovery of a large jump in the mean-squared charge radius, associated with a dramatic change in shape between 187 Hg and 185 Hg observed in isotope shift measurements [1]. Calculations based on Strutinsky's shellcorrection method [2] interpreted this result as a transition from a weakly-deformed oblate to a more pronounced prolate-deformed shape. Further isotope shift measurements reveal that the weakly-deformed oblate character extends down to A = 182 in the even-mass Hg isotopes [3]. Calculations in these isotopes using the Nilsson-Strutinsky approach [4] predict two deformed minima, where the lowest-energy minimum corresponds to an oblate shape, β −0.15, and the second to a more deformed prolate shape with β 0. 27. In shell-model terms, these minima are associated with a proton zeroparticle-two-hole configuration, π(0p − 2h), and a twoproton excitation across the Z = 82 shell gap yielding a π(2p − 4h) configuration, respectively.
Spectroscopy of the even-mass mercury isotopes reveals a systematic trend of the intruding 0 + 2 band head (shown in Fig. 1) which minimizes in energy near the neutron mid-shell at N = 104. The excited states built upon these configurations become yrast above I π = 4 + 0.0 0.5  1. (Color online) Level energy systematics of even-mass mercury isotopes. Red circles refer to the assumed intruder states while blue squares refer to the assumed oblate states, guided by the results of the calculations in this work and that of Ref. [5]. The figure is an updated version of that in Ref. [6] using data taken from the NNDC database [7].
for A ≤ 186, and the 2 + levels become close enough in energy to mix strongly.
Large E0 components in 2 + 2 → 2 + 1 transitions indicate a large degree of mixing. An attempt to understand the mixing between the bands was made by measuring this E0 component in-beam in 180 Hg [8] and 186 Hg [9]. The conversion coefficient of this transition can also be measured following β decay, and there is an effort to provide  more experimental data in this region [10]. Mixing of the 0 + states can be quantified by comparison of ρ(E0) 2 values [11,12] which have been experimentally determined in 180 Hg [13], 184 Hg [14], and 188 Hg [15]. As described in a recent review on the topic [16], the microscopic shellmodel approach and the theoretical mean-field approach can both successfully reproduce the observed mixing. An analysis of α-decay hindrance factors [13,17] indicates a smaller prolate contribution to the ground-state band 0 + state.
To determine the magnitude and type of deformation of the two bands, their mixing strength, and to test the picture of shape coexistence, more precise data are required on the absolute transition strengths between the excited states of the nuclei in this mass region. Lifetimes of excited states in 180,182 Hg have been recently measured by this collaboration [18,19]. To extend this knowledge and address these missing data, lifetime measurements of excited states in 184,186 Hg have been performed. Partial level schemes of the nuclei studied in this work are given in Fig. 2. suppressed, high-purity Ge detectors [22] arranged into 17 rings of constant polar angle, θ, with respect to the beam. For this experiment, 100 detectors split into 16 rings were in use. The Köln plunger device was installed at the target position to allow for the Recoil Distance Doppler-Shift (RDDS) lifetime measurements. The distance of the 11 mg/cm 2 -thick Au stopper foil was varied with respect to the 0.6 mg/cm 2 thick Sm target within a range of 2-2000 µm and data were taken at 12 distances for 184 Hg (10 for 186 Hg). For the analysis of the data, γγ-coincidence matrices were built using GSSORT [23] and the ROOT framework [24]. The data from different HPGe detectors of Gammasphere were grouped by rings at similar angles. In the analysis of 184 Hg, the lifetimes are obtained by taking the weighted average of the lifetimes for different rings. Here, it was possible to obtain the velocity independently for each ring, utilizing the measured Doppler shift of the peaks. In the case of 186 Hg, where measurement times were shorter, the lifetimes were derived directly from the weighted average of the intensities whilst the velocity is obtained by weighting the velocities from the individual γγ-coincidence spectra for all combinations of ring pairs. Typical γγ-coincidence spectra obtained during the experiment are presented in Fig. 3, in which it is possible to see the variation of intensity of the fully Doppler-shifted component of the depopulating transition, I sh , with increasing distance. At the smallest distances the flight times of the nuclei are very short leading to few γ rays emitted in-flight. Consequently, when gating in the γ-γ matrix, a low number of events are observed, an effect present in Fig. 3 (a).
To account for the varying measurement time at different target-to-stopper distances, the intensities had to be normalized. For 184 Hg, the distances were normalized by employing a gate on the total intensity of the 340-keV, 6 + 1 → 4 + 1 transition and averaging the intensities of the 287-keV, 4 + 1 → 2 + 1 and 367-keV, 2 + 1 → 0 + 1 transitions at all angles. For 186 Hg, all possible γγ coincidences for the yrast transitions up to the 10 + 1 state, in spectra with θ < 90 • , were used for the normalization. The use of the coincidence differential decay curve method (DDCM) eradicates the influence of recoil de-orientation [25].
In the analysis of 184 Hg, lifetimes were obtained using the prescriptive coincidence DDCM [26,27]. While here we explain only the features of the method that are required for this analysis, the full details are contained in Refs. [26,27], to which the reader is referred for more information. The coincidence technique allows for the elimination of systematic errors usually introduced in RDDS-singles measurements by the unknown feeding history from states that lie higher in energy. The decay curves, as a function of distance, x, for this analysis were constructed using gates on the shifted component of the yrast transition directly feeding the state of interest, ensuring a significant simplification whereby the lifetime is determined using only the ratio of the unshifted (I us ) and the time derivative of the fully-shifted (I sh ) components of the depopulating transition.
Typical fits of continuously-connected second-order polynomials, performed with Napatau [28,29], are illustrated in Fig. 4. The lifetime is determined at every distance and should sit at a constant value. Deviations from this behavior indicate systematic effects, which can be identified easily with this method. The weighted average, τ av , is taken of the points inside of the sensitivity region, i.e. where the derivative of the decay curve is largest.
Lifetimes of the yrast states of 186 Hg up to the 10 + 1 state were similarly determined with the exception of the 4 + 1 state. In this case, the near doublet of the 4 + 1 → 2 + 1 and 2 + 1 → 0 + 1 transitions rendered it problematic to use the simple "gate from above" method. Instead, the method of "gating from below" [30] was used. The corresponding τ plot for the 4 + 1 state is found in Fig. 5. The intensities involving the 6 + 1 and 4 + 1 analysis were corrected for a contamination from the 15 − 2 → 13 − 2 transition, using a gate on the 13 − 2 → 11 − 2 transition.

III. RESULTS
The final weighted averages of the mean lifetimes of all states studied are shown in Table I along with the transition strengths, B(E2), and absolute transitional quadrupole moments, |Q t |, of the depopulating yrast transitions. Transition quadrupole moments, Q t , are related to the B(E2) values assuming a rotating quadrupole deformed nucleus using the rotational model: where I020|I 0 is a Clebsch-Gordan coefficient and In 184 Hg, the seven independent measurements of each of the lifetimes of the even-spin yrast states are presented in Fig. 6, as a function of the "ring" angle at which they  were determined in Gammasphere.
It is worth noting that the new lifetime for 4 + 1 state in 186 Hg is smaller than the previously measured value from Ref. [31]. The discrepancy in the first measurement is likely due to complications that the authors encountered in resolving the doublet and subsequent assumptions which were made.  Transitional quadrupole moments, |Q t |, for the evenspin yrast states are given in Fig. 7 for the mercury isotopes where 180 ≤ A ≤ 186. The 2 + states show no strong variation in |Q t | with mass number, whereas the collectivity of the 4 + states reduces with increasing mass number. This can be compared to the energy level systematics in Fig. 1 where the energy of the intruder states reaches a minimum at A = 182.

B. The 93 state in 184 Hg
After the even-spin, positive-parity yrast band in 184 Hg, the most populated band is the odd-spin rotational band built upon the I = 5 state, observed at 1.848 MeV, which becomes yrast at around 4 MeV. Analogous bands have been observed in the neighboring isotopes, specifically 178 Hg [34] and 180 Hg [35] and their structure discussed in terms of octupole correlations. Lifetime measurements of states in this band in 182 Hg have been performed and a consistency in the structure of these states has been extended to 184 Hg using the energy displacement of states differing by 3h [19]. The quadrupole moment measured here for the 9 3 → 7 3 transition in 184 Hg, |Q t | = 5.6(7) eb, is similar to that of the even-spin yrast band, |Q t | 7.7 eb, although it is smaller than those measured in the lighter isotopes. I. Properties of the states investigated in this study. The uncertainties presented on τav represent the 1σ statistical error and include an additional systematic uncertainty which accounts for the choice of the fitting function and relativistic effects [27], typically ≤ 3%. Gamma-ray energies (Eγ) and branching fractions (b.f.) of the depopulating γ-ray transitions (corrected for internal conversion), as well as spin and parity (I π ) values, are taken from Refs. [20,21]. In cases where only one depopulating transition is observed b.f. is assumed to be equal to unity. τprev values from Refs. [31][32][33] are shown for comparison. In order to shed light on the properties of the coexisting structures in these light mercury isotopes, phenomenological two-band mixing calculations have been carried out using the assumption of a spin-independent interaction between two rotational structures. In the calculations, the variable moment of inertia (VMI) model [36] was used to fit known level energies of rotational bands built upon the first two 0 + states, up to and including I π = 10 + and 4 + for yrast and non-yrast bands, respectively. Employing the method of Lane et al. [37], one can derive the wave-function amplitudes of the two configurations from the mixing strengths. These are shown in Table II along with the mixing strength, V , for each isotope and the band-head energies of the two configurations.
Similar mixing calculations have previously been performed for 180,182,184 Hg [38]. The present results for 180,182 Hg differ due to the inclusion of non-yrast states identified in recent studies [8,10], which provide additional constraints on the calculations. The calculation for 184 Hg presented in Table II is essentially identical to that in Ref. [38]. It places the I = 2 member of the intruder band just 5 keV above the corresponding state in the normal band before mixing. This leads to almost complete mixing between the states, producing a first-excited 2 + state which comprises 51% of the normal configuration and 49% of the intruder configuration. An alternative calculation in which the order of these states is reversed was also performed and yielded very similar parameters to those presented in Table II. In this scenario the unmixed states are nearly degenerate, so the degree of mixing is even greater. The resulting B(E2) values (presented later in Fig. 8) are not significantly different, so in what follows only the results from the former calculation will be discussed.
Low-lying levels in 184,186 Hg have also been interpreted assuming the mixing of spherical and deformed states [39], while an alpha-plus-rotor model has been used to extract spin-dependent interaction strengths and mixing amplitudes in 182,184,186 Hg [5]. The latter study predicted a contribution of the more-strongly deformed structure to the 2 + 1 state in 182 Hg of 76%, comparing very well to the value of 71% obtained in this work. We note here that this contribution drops to only 2.3% for the same state in 188 Hg. The corresponding isotones in the platinum nuclei were also interpreted recently using twoband mixing calculations and qualitatively similar conclusions were drawn regarding a strong degree of mixing for the low-spin states [38,40].
It is possible to determine the transitional quadrupole moment of the unperturbed I → I − 2 transitions in the normal (n) and intruder bands (i), Q I , using an average of the moment of inertia of the two states, J I , such that [36] where J I is the moment of inertia for a pure state with spin I, extracted from the fit. An evaluated value of the constant, fitted to data in the neutron-deficient A = 170 region, k = 45(2) eb keV −1/2 [41], was used in this study.
Combining knowledge of the wave-function amplitudes with the intrinsic quadrupole moments of the pure states, Q I , it is possible to extract the B(E2; I → I − 2) values of the mixed states [37]. The relative sign of the intrinsic quadrupole moments of the two configurations must be assumed to be positive or negative and was found to be best reproduce the data when positive. This feature has been noted in previous calculations [37,42] and is at odds with what is expected from the rotational model when The results of the calculations for the yrast sequences are compared in Fig. 8 with those extracted from the lifetimes measured in this work. Good agreement is found for the majority of yrast transitions, even for cases where the states are strongly mixed. The observed discrepancies for I ≥ 6 in 186 Hg may indicate a breakdown of the simple two-band picture as other structures begin to influence the yrast states [43,44]. For the nuclei presented in Fig. 8, the experimental and calculated in this work (points) plotted as a function of spin and compared to those extracted from the mixing calculations (solid black line, dashed line represents the uncertainty in the constant, k). For reference, the intra-band B(E2) values calculated for the pure unperturbed normal (red) and intruder (blue) bands are also shown. Data for 180,182 Hg are taken from Ref. [18] and the point at I = 10h in 184 Hg is taken from Ref. [33]. B(E2; 2 + 1 → 0 + 1 ) values are similar in each isotope, while the B(E2; 4 + 1 → 2 + 1 ) varies and, in each case, is not consistent with a transition within either of the pure bands. The low B(E2; 2 + 1 → 0 + 1 ) value is interpreted as being due to a transition between the weakly-deformed oblate 0 + ground state and a more strongly-deformed prolate 2 + state [18]. However, the admixture of the normal and intruder configurations for the 0 + and 2 + states is unique in each isotope and a similar B(E2) value is not necessarily indicative of similar structures across the mass range.
In Fig. 9, calculated B(E2) values are plotted as a func- tion of the energy difference between the pure, unmixed 0 + band heads. The parameters used in the calculations are those of 184 Hg, but since they are not too dissimilar for all of the isotopes, the curves can be considered representative of the mercury isotopes around the neutron mid-shell. The four isotopes compared have ∆E 0 values in the range of 250-550 keV, marked by the vertical lines in Fig. 9. One clearly observes that above 200 keV the B(E2; 2 + 1 → 0 + 1 ) values remain constant, even though the square of the mixing amplitude of the I = 2 states, α 2 2 , drops from 0.80 at 200 keV down to 0.015 at 1 MeV. In contrast, the B(E2; 4 + 1 → 2 + 1 ) values vary significantly in the range of interest and are much more sensitive to α 2 , while the B(E2; 6 + 1 → 4 + 1 ) values become more sensitive at larger band-head energy differences.

V. SUMMARY AND CONCLUSIONS
Lifetimes of excited states have been measured by employing the RDDS technique. Yrast states up to I π = 8 + in 184 Hg and I π = 10 + in 186 Hg have been studied. Dis-tinct differences in deformation for the assigned normal and intruder bands could be shown. The lifetime of the 4 + 1 state in 186 Hg was found to be shorter than the previously measured value. Lifetimes of the 9 3 state in 184 Hg and the 10 + state in 186 Hg have been measured for the first time. The other lifetimes are consistent with previous measurements, while the uncertainty could be reduced significantly. These more precise lifetime values, with those of Ref. [18], have been a vital input to the analysis of Coulomb excitation experiments at the REX-ISOLDE facility [45,46].
Rotational bands built upon the first two I π = 0 + states have been considered in terms of a two-state mixing model and the mixing amplitudes of the two configurations extracted as a function of spin. It is observed that, while the ground state remains composed of predominantly one configuration, namely the assumed weakly-deformed normal structure, in all of the evenmass mercury isotopes considered (180 ≤ A ≤ 188), the first excited 2 + state changes dramatically in its composition. This is in contrast to a naïve interpretation emanating from the systematics of the 2 + level energies and B(E2; 2 + → 0 + ) values, which are both strikingly similar across the mass range.