a of properties of low-lying states in neutron-deficient Hg isotopes : Coulomb excitation of 182Hg, 184Hg, 186Hg and 188Hg neutron-deficient

. The neutron-deﬁcient mercury isotopes serve as a classical example of shape coexistence, whereby at low energy near-degenerate nuclear states characterized by diﬀerent shapes appear. The electromagnetic structure of even-mass 182 – 188 Hg isotopes was studied using safe-energy Coulomb excitation of neutron-deﬁcient mercury beams delivered by the REX-ISOLDE facility at CERN. The population of 0 +1 , 2 , 2 +1 , 2 and 4 +1 states was observed in all nuclei under study. Reduced E 2 matrix elements coupling populated yrast and non-yrast states were extracted, including their relative signs. These are a sensitive probe of shape coexistence and may be used to validate nuclear models. The experimental results are discussed in terms of mixing of two diﬀerent conﬁgurations and are compared with three diﬀerent model calculations: the Beyond Mean Field model, the Interacting Boson Model with conﬁguration mixing and the General Bohr Hamiltonian. Partial agreement with experiment was observed, hinting to missing ingredients in the theoretical descriptions.


Introduction
The neutron-deficient mercury isotopes (Z = 80) serve as an illustrative example of shape coexistence [1,2], whereby at low excitation energies near-degenerate nuclear states are characterized by different shapes. The first observation of a dramatic change in the ground-state mean-square charge radii was observed through isotope shift measurements in 183 Hg and 185 Hg, when comparing to heaviermass mercury isotopes [3]. Since then a large amount of information has been collected for nuclei around the N = 104 midshell between N =8 2a n dN = 126 using different experimental techniques. This resulted, amongst others, in the observation of a large odd-even staggering in the isotope shifts in the mercury isotopes around 181−185 Hg [4], which has long been attributed to the intruder structure becoming the ground state in the oddmass isotopes and the observation of shape coexistence at low excitation energy in 185 Hg [5]. Recent results obtained from isotope-shift measurements extended the knowledge on the ground-state deformation systematics down to 177 Hg [6]. Further, lifetime measurements performed for 184 Hg and 186 Hg [7,8] suggested a sudden increase in deformation of the excited yrast states with the spin larger than two. Radioactive-decay studies identified coexisting bands in 184,186,188 Hg, assumed to be characterized by different deformations [9][10][11]. This phenomenon was observed in 182 Hg as well by means of in-beam spectroscopy studies [12].
The energy-level systematics of the even-even mercury isotopes ranging from A = 190 to A = 198 reveals a nearly constant excitation energy of the yrast states up to the spin I = 6 [13,14]. Mean-field calculations interpret these states as exhibiting a weakly-deformed oblate character [1, 15,16]. However, this regular pattern is distorted for the lighter mercury isotopes (N ≤ 106) through the intrusion of levels of a very collective rotational band of assumed prolate nature, which decreases in excitation energy reaching a minimum around mass A = 182, 184 [13,14]. In a shell-model picture, the energy evolution of the deformed states as a function of neutron number can be interpreted as arising from a proton pair excitation across the Z = 82 shell closure. This leads to extra valence proton pairs interacting with the valence neutrons through the attractive quadrupole protonneutron interaction [1]. By contrast, in a mean-field picture the difference in excitation energy between the oblate and prolate states results from the variation of the density of single-particle levels around the Fermi energy with deformation in the Nilsson diagram.
The energy of the 2 + 1 state, as well as the B(E2; 2 + 1 → 0 + 1 ) value in even-even mercury isotopes around N = 104 are relatively constant, which is commonly interpreted as a manifestation of a similar structure of these states. On the other hand, large conversion coefficients related to the substantial E0 components in the 2 + 2 → 2 + 1 transition, observed for 182,184,186 Hg [17,18], indicate a large degree of mixing. Indeed, as it was reported in ref. [19], the composition of the 2 + state changes significantly in the light mercury isotopes, which is reflected in large variations of mixing amplitudes extracted using the Variable Moment of Inertia (VMI) model. We performed Coulombexcitation (Coulex) studies using post-accelerated mercury beams and deduced E2 matrix elements between yrast and non-yrast states [20]. The results were interpreted within the two-state mixing model supporting the mixing of a weakly-deformed oblate-like structure with a more-deformed prolate-like structure.
In this paper we present the experimental details and the analysis procedure of the Coulomb-excitation studies of 182,184,186,188 Hg reported in ref. [20]. However, it should be noted that the results for 182,184 Hg reported in ref. [20] have been extracted using preliminary values of a number of γ-ray branching ratios and conversion coefficients determined in a β-decay study of 182,184 Tl populating excited states in 182,184 Hg. Some of these originaly used spectroscopic data, particularly total conversion coefficients α(2 + 2 → 2 + 1 )i n 182,184 Hg, turned out be erroneous and they were corrected in the subsequent, complete analysis of the same β-decay data set and published in ref. [17]. In the current paper, we performed a full re-analysis of the Coulomb-excitation data for 182,184 Hg using the corrected values of those spectroscopic data [17] which substantially differ from the preliminary values assumed in ref. [20]. Consequently, for these two nuclei, the data reported in the present paper replace the results from ref. [20].
The experimental technique, the production and postacceleration of mercury beams at REX-ISOLDE are presented in sect. 2. The data analysis and the extraction of the reduced matrix elements are presented in sects. 3,4 and in sect. 5, respectively. A comparison of experimental excitation energies and quadrupole moments with theoretical predictions based on three different models (the quadrupole collective General Bohr Hamiltonian model (GBH) [21], the interacting boson model with configuration mixing (IBM-CM) [22] and the beyond-meanfield model (BMF) [16]) is given in sect. 6. The experimental and theoretical monopole transition strengths, ρ 2 (E0; 2 + 2 → 2 + 1 ), are discussed in sect. 7. A summary and outlook are given in sect. 8.

Production and postacceleration of exotic, neutron-deficient mercury beams
The neutron-deficient mercury isotopes were produced through the spallation reaction induced by a 1.4 GeV proton beam, delivered from the Proton Synchrotron Booster at CERN, impinging on a molten lead target. The resulting products rapidly diffused out of the target, which was heated up to 600 • C, passed through the transfer line, and were ionized in a high-temperature plasma ion source. The 1 + ions were extracted from the ion source by applying a 30.2 kV electrostatic potential and were, subsequently, mass-separated by the General Purpose Separator (GPS). Possible beam contaminations from lead and gold isotopes were negligible since the temperature of the target container and the transfer line was kept around 600 • C: the vapor pressure of mercury (i.e., its evaporation rate out of the target container) is four orders of magnitude higher than that of lead, and twelve orders of magnitude higher than that of gold at this temperature.
The low-energetic singly-charged radioactive ion beam was then guided to the REX postaccelerator. The ions were injected into the REX Penning trap (REXTRAP) to cool and bunch the beam. The bunches were transmitted to the Electron Beam Ion Source (REXEBIS) afterwards, where they were brought to a higher charge state. The final charge states of the studied mercury isotopes are listed in table 1. In the experiment performed in 2007, the charge breeding time in EBIS was equal to 170 ms. This time is identical to the trapping time in REXTRAP. In consequence EBIS bunches the ions into the REX postaccelerator at a rate of ∼ 6 Hz. In 2008, the EBIS breeding time was set to 200 ms, corresponding to a bunching frequency of 5 Hz. In order to avoid additional stable beam contaminants in the radioactive ion beam (originating from e.g., the REXTRAP buffer gas and residual gases inside REXEBIS), the ions were separated according to their A/q ratio prior to injection in the REX linear accelerator (REX-LINAC) [23]. The highly-charged radioactive mercury ions were post-accelerated at the REX-LINAC to an energy of 2.85 MeV/A.

Coulomb excitation of 182-188 Hg
The post-accelerated radioactive mercury beams were delivered to the Miniball set-up [24]. Coulomb excitation of 182,184,186,188 Hg was induced by 120 Sn, 107 Ag, and 112,114 Cd secondary targets with thicknesses of 2.3, 1.1, and 2 mg/cm 2 , respectively. The γ rays depopulating Coulomb-excited states in the ejectile and target nuclei were detected with the Miniball γ-ray spectrometer, which consists of eight clusters. Each of these clusters contains three HPGe crystals, which are electrically divided into six segments and a central electrode. During the experimental campaigns performed in 2007 and 2008, 23 and 18 out of 24 crystals were operational, respectively. To determine the full-energy γ-ray efficiency of the Miniball array over the range corresponding to observed γ-ray transitions in mercury isotopes, down to the Hg K α X-ray region (K α1 =7 0 .8 keV and K α2 =6 8 .9keV), 133 Ba and 152 Eu calibration sources were used. The absolute γ-ray efficiency of Miniball was 6.80(18)% and 5.30(14)% at 1.3 MeV in 2007 and 2008, respectively. The energies of scattered target recoils and mercury ejectiles were measured with a double-sided silicon strip detector (DSSSD) [25], placed inside the compact collision chamber at a distance of 32.5 mm behind the cadmium target and 33.5 mm behind the tin and silver targets. The angular range covered by the DSSSD in the laboratory frame measured with respect to the beam direction extended from 15.5 • to 51.6 • (for measurements performed with the cadmium targets) and from 15 • to 50.7 • (for measurements performed with the tin and silver targets), corresponding to the center-of-mass angular ranges for projectile and target shown in table 1. The DSSSD is subdivided into four quadrants with 16 annular and 12 radial strips per quadrant, which allowed for a measurement of the angular distribution of both mercury ejectiles and target recoils. Based on the different kinematics and requesting two particles to be present in each event (see sect. 3 B), the detected recoils and ejectiles were identified in the DSSSD.
The detected γ rays were emitted in flight causing a Doppler shift in their detected energy. This is due to the fact that: (i) the lifetimes of the excited nuclear states of the investigated mercury isotopes are typically several picoseconds, while the time of flight of the projectile or recoil particles from the target to the DSSSD is in the range of a few nanoseconds, and (ii) the target thicknesses used in the measurements were smaller than the range of the particles in the target. The angular segmentation of the DSSSD and Miniball detection set-ups was essential to perform a reliable, event-by-event Doppler correction of the γ-ray energy.
The beam energies were chosen such that the distance between collision partners was greater than 1.25(A Table 1. Experimental parameters of the measurements. The first four columns give the isotope of interest, its half-life, T 1/2 , the year of the experimental campaign and the selected charge state(s) for each mercury isotope. For each experiment the beam intensity, I Hg , measured at the secondary target, the beam energy, EHg, the center-of-mass angular range, θCM, corresponding to angular range covered by the DSSSD in the laboratory frame for different projectile and target nuclei are listed. These angular ranges correspond to one-and two-particle detection ranges (see text for details). The total measurement time, t exp, is given in the last column.

Isotope
T 3 Data selection

Timing conditions
Beams delivered by the REX-ISOLDE facility have specific timing properties which influences the way data are taken at Miniball. The beam delivered to the REX-LINAC is bunched. Triggered by the EBIS signal, the REX-LINAC is switched on for 800 µs. During this active time window the Miniball data acquisition system registers all the information from the γ-ray and particle detectors. The detected "in-beam" γ-ray spectra contain not only the prompt radiation following Coulomb excitation, but also γ rays originating from β-decay, natural background radiation and X rays from the accelerator cavities. They all contribute to the observed γ-ray background (random γ rays), while the particle background mostly originates from the elastic Rutherford scattering process. The un-conditioned in-beam γ-ray spectrum obtained during the 184 Hg-on-112 Cd experiment is presented in fig. 1(a). Coincidences between a particle and a γ ray are crucial to distinguish the Coulomb-excitation events from the background radiation. In order to select the Coulomb-excitation events, each γ ray arising directly from the collision was correlated to one or more coincident ejectile/recoil particles. The time difference between the detected γ rays and the particles in a time window of 4 µs in the experiment of 184 Hg on the 112 Cd target is shown in fig. 1(d). A time window of 300 ns width was defined to select the prompt particle-γ events. Random particle-γ coincidences, indicated in green in fig. 1(d), were attributed to the un-correlated particle-γ events. In order to subtract the random γ-ray spectrum from the prompt one, the former was scaled by the ratio of the widths of the respective time windows. The resulting scaled random γ-ray spectrum for 184 Hg is presented in fig. 1(b) in green. After this procedure only the transitions resulting from Coulomb excitation remain, as shown in fig. 1(c). The Doppler-broadened photopeaks of γ rays originating from the 2 + 1 → 0 + 1 ,4 + 1 → 2 + 1 ,2 + 2 → 0 +   Fig. 1. (Color online) Gamma-ray spectra from the 184 Hgon-112 Cd experiment illustrating the data processing: (a) Inbeam, un-conditioned γ-ray spectrum detected in the Miniball array during the EBIS pulse; (b) Gamma-ray spectrum collected in a prompt (black) and random (green) coincidence with detected ejectile/recoil particle(s) in the DSSSD; (c) Random-subtracted prompt γ-ray spectrum of 184 Hg. No Doppler-correction has been applied; (d) time difference between the detected γ ray and scattered particle with indicated prompt (black) and random (green) coincidence windows.
transitions in 184 Hg as well as the 2 + 1 → 0 + 1 transition in 112 Cd can be observed. A peak around 69 keV remains present, suggesting a production of X rays directly related to the collision. In the non-random-subtracted spectrum, a clear peak around 65 keV was present, arising mainly from the β + /EC decay of 184 Ir on top of the X-ray radiation from the accelerator cavities. The X rays from both these sources are not time-correlated with the collision and thus are not present in the random-subtracted spectrum. The remaining X rays are related to the mercury beam. This issue will be discussed in more detail in sect. 4.

Two-particle event selection
In order to identify the scattered mercury beam and target ions, events were selected demanding detection of exactly two particles in opposite quadrants of the DSSSD in coincidence with a γ ray registered in Miniball. The centre-ofmass angular range, where two particles are incident on the DSSSD, is reduced as compared to the centre-of-mass angular range covered by the particle detector presented in table 1, and extends from 76.8 • to 149 • (for measurements performed with the cadmium targets) and from 78.6 • to 150 • (for measurements performed with the tin and silver targets). The events of interest were chosen by requesting the absolute time difference between the detection of two particles to be ≤ 50 ns. Figure 2 presents the number of counts corresponding to the detected 182 Hg-beam and 112 Cd-target ions as a function of the energy and scattering angle in the laboratory frame. It shows a typical inverse-kinematics scattering pattern. The heavier-mass beam particles are detected at smaller angles in the laboratory frame of reference, while the recoiling target nuclei are scattered throughout the whole detection range of the DSSSD. The separation between the ejectile and recoil ions is significantly improved when requesting detection of exactly two particles scattered back-to-back in the centreof-mass frame (see bottom panel of fig. 2) compared to the spectrum obtained without this condition (top panel of fig. 2).

Data analysis 4.1 Mercury K X rays
In the background-subtracted γ-ray spectra intense K α,β X ray peaks are observed for all studied isotopes at energy of 69 and 80 keV in addition to the γ rays following the Coulomb excitation of target and projectile. As it will be discussced in the following sections, origins of these mercury X rays include internal conversion of observed γ rays and E0 transitions. An additional source is related with the heavy-ion induced K-vacancy creation processes. The cross-section for the emission of a K α X rays, originating from atomic processes, can be estimated from phenomenological approach, which gives the cross section as a function of beam energy, target proton number and ionization potential. Details of the method together with summary of all data concerning the observed X rays in Coulombexcitation experiments on isotopes in the light-lead region at ISOLDE are given in ref. [27]. For the analysis of the mercury data the Coulomb excitation of 188 Hg were used to scale the theoretical predictions of the expected K X-ray yield from the K-vacancy creation process. The only states populated in the Coulomb excitation experiment of 188 Hg were 2 + 1 at 413 keV and 4 + 1 at 1005 keV (see sect. 4.5). As no low-lying excited 0 + states or higher-lying 2 + or 4 + states, are observed in 188 Hg it was concluded that the only nuclear effect giving rise to mercury X rays, is internal conversion of observed γ ray transitions. After correcting for the latter, using known conversion coefficients [28], the remaining number of X rays was attributed to the heavy-ion induced K-vacancy creation process and used to rescale the calculated number of X rays predicted by theoretical formulas. A scaling factor of 0.037(5) results from a comparison of the number of observed and expected X rays. This factor is further useded to rescale the predicted amount of X rays originating from the heavy-ion induced K-vacancy creation process for lighter mercury isotopes. The contribution to the X-ray intensity from atomic effects observed in 182,184 Hg was deduced to be 13(3)% (for 182 Hg) and 14(4)% (for 184 Hg) of the total observed ones and the remaining excess of X rays indicate the presence of E0 deexcitations from the 0 + 2 and 2 + 2 states. The way how these two E0 transitions were distinguished using the γγ coincidences is presented in sects. 4.2 and 4.3.  Fig. 3. Two-particle gated and random-subtracted γ-ray spectra (2p − γ). The spectrum (a) is Doppler-corrected assuming emission from the 182 Hg ejectile, whereas (b) is Dopplercorrected assuming emission from the 112 Cd recoil. The population of the 2 + 1 state in 112 C da n do ft h e2 + 1 ,2 + 2 ,4 + 1 states in 182 Hg was observed. Intense K X-ray peaks are clearly visible in the spectrum (a) (see text for details). Figure 3(a) is adapted from ref. [20].

Experiment on 182 Hg
Coulomb excitation of 182 Hg ions was induced by a 112 Cd secondary target. Either the projectile or the target nucleus can be excited in a collision. The two-particle-gated γ-ray spectra, random-subtracted and Doppler-corrected for mass A = 182 and A = 112, are shown in fig. 3(a) and (b), respectively. In the latter, a clear peak at the energy of 617 keV is visible corresponding to the 2 + 1 → 0 + 1 γ-ray transition in 112 Cd. Sharp peaks at energies of 261 keV, 352 keV, 548 keV correspond to the 4 + 1 → 2 + 1 , 2 + 1 → 0 + 1 and 2 + 2 → 0 + 1 γ-ray transitions in 182 Hg, respectively. Moreover, intense K α (69 keV) and K β (80 keV) X-ray peaks are clearly observed. The γ-ray and X-ray intensities are listed in table 2.
The observed K X rays are in prompt coincidence with two scattered particles and their Doppler correction is consistent with emission from the mercury projectile. As mentioned in sect. 4.1 they originate from: 1) the heavy-ion induced K-vacancy creation due to atomic processes taking place when the mercury beam passes through the target [27], 2) the internal conversion of the observed γ-ray transitions in 182 Hg, 3) the E0 de-excitation after Coulomb excitation.
Subtracting the number of X rays originating from the afore-mentioned first two sources, 7.2(6) × 10 2 counts corresponding to K α X rays remains in 182 Hg. Those were attributed to the E0 de-excitation of the 0 + 2 and 2 + 2 states. Since the mixing ratio δ( E2 M 1 ) for the 2 + 2 → 2 + 1 transition is unknown in the investigated mercury isotopes, the value of 1.85 was adopted for the analysis (see sect. 5.1 for details). Table 2. Measured γ-ray and K X-ray intensities (not efficiency corrected) for 182 Hg scattered on the 112 Cd target. The extracted intensities of the E02 + 2 → 2 + 1 and 0 + 2 → 0 + 1 transitions, corrected for the K α/Kβ X-ray branching ratio and fluorescence effect, are also shown. The angular range for simultaneous detection of two particles is given in the centreof-mass frame together with corresponding angular range for the target-nucleus (t) detection in laboratory frame. The intensities marked with asterisks were derived from the γγcoincidence analysis (as described in detail in the text).   to the Coulomb-excitation analysis. Level and γ-ray transition energies are given in keV. Widths of the arrows are proportional to the measured γ-ray intensities. The intensities of the 0 + 2 → 0 + 1 and 2 + 2 → 2 + 1 E0 transitions, deduced from the analysis of the K X-ray peaks (see text for details) and given in table 2, were also included in the analysis. The low-energy level scheme of 182 Hg is presented in fig. 4, showing the γ-ray transitions that were observed, and the states included in the Coulomb-excitation analysis. Note that for all figures displaying level schemes, the levels are organized into yrast and non-yrast parts. No attempt has been made to assign levels to rotational bands of states of similar intrinsic structure. The reason for this choice is the high degree of mixing of the lowest energy states in the investigated mercury isotopes and the absence of a firm nuclear-model independent interpretation regarding the type of deformation for specific states (see sect. 6). The 196 keV 2 + 2 → 2 + 1 γ-ray transition is not visible in fig. 3

Nucleus
Random-subtracted γγ-coincidence spectrum gated on the 2 + 1 → 0 + 1 transition at the energy of 352 keV in 182 Hg, demanding that at least one particle satisfies the kinematic condition. The γ rays were Doppler corrected for the 182 Hg ejectile.
The particle-gated γγ energy spectrum for 182 Hg is shown in fig. 5. Several γ-ray transitions, being in coincidence with the 2 + 1 → 0 + 1 γ ray, are visible in the spectrum: the 4 + 1 → 2 + 1 and 2 + 2 → 2 + 1 γ-ray transitions at energies of 261 keV and 196 keV, respectively, as well as K X rays. As the 4 + 1 → 2 + 1 transition is observed in both two-particle gated (singles) and particle-gated γγ (coincidences) spectra, the ratio of their intensities expressed as and equal to 0.26 (5) can be used to extract the intensity of the 2 + 2 → 2 + 1 transition in singles. The result is given in table 2 and details of the procedure are provided in ref. [29].
Moreover, in the γγ coincidence spectrum K α Xr a y s are clearly visible. From the detected 74(15) counts of K α X rays, 9(2) can be attributed to the K-vacancy creation due to atomic processes [27], i.e. process number 1. in the list given at the beginning of this section. The internal conversion of the observed 4 + 1 → 2 + 1 γ-ray transition in the coincidence spectrum is responsible for to 5(1) counts. After subtraction of these two sources of X rays 60(15) K α X rays remain. As they are in coincidence with the 2 + 1 → 0 + 1 γ-ray transition, they can be attributed to the conversion of the 2 + 2 → 2 + 1 transition. Further, using the R 4 + 1 →2 + 1 scaling factor, the number of counts arising from this conversion in the two-particle gated γ-ray spectrum can be found to be equal to 2.3(7) × 10 2 . Subtracting the latter from the total number of 7.2(6) × 10 2 K α Xr a y s , 4.9(9) × 10 2 counts remain. These were attributed to the E0 de-excitation of the 0 + 2 state. The final values were corrected for the K α and K β X-ray branching ratio equal to 3.6(1) [30], and the fluorescence effect [31]. The results are presented in table 2. . Two-particle gated, random-subtracted and Dopplercorrected for the 184 Hg ejectile γ-ray spectrum following the Coulomb excitation of 184 Hg induced by three different targets: 112 Cd (a), 107 Ag (b) and 120 Sn (c). The population of the 2 + 1 , 2 + 2 ,4 + 1 states in 184 Hg is clearly visible, as well as intense K Xray peaks. Additionally, in the experiment performed with the 120 Sn target, a weak 4 + 2 → 2 + 2 γ-ray transition (552 keV) and the 2 + 3 → 0 + 2 &2 + 3 → 2 + 1 doublet (around 610 keV) in 184 Hg were observed. Figure 6(c) is adapted from ref. [2].

Experiment on 184 Hg
Coulomb excitation of 184 Hg was performed using three different secondary targets: 112 Cd, 107 Ag and 120 Sn. Two-particle-gated γ-ray spectra, random-subtracted and Doppler-corrected for the ejectile are presented in fig. 6. The population of the 2 + 1 ,2 + 2 and 4 + 1 states in 184 Hg is clearly visible. Moreover, in the experiment with the 120 Sn target (Z = 50) the 4 + 2 and 2 + 3 states at 1086 keV and 983 keV, respectively, were excited, yielding the weak 552 keV 4 + 2 → 2 + 2 peak and a doublet of 2 + 3 → 0 + 2 and 2 + 3 → 2 + 1 γ-ray transitions. The low-energy level scheme of 184 Hg together with all observed γ-ray transitions in the Coulomb excitation experiment is shown in fig. 7. The extracted γ-ray and X-ray intensities are summarized in table 3.
Intense K α X-ray peaks are clearly visible in the twoparticle gated γ-ray spectra of 184 Hg. After subtracting those originating from the heavy-ion K-vacancy creation and from the internal conversion of the observed γ rays, a significant amount of 3.6(5) × 10 2 and 3.3(5) × 10 2 Xr a y s Widths of the arrows are proportional to the measured γ-ray yields. As in the case of 182 Hg, the 0 + 2 → 0 + 1 and 2 + 2 → 2 + 1 E0 transitions were deduced from the analysis of the K X-ray peaks (see text for details). Figure adapted from ref. [20].
remains in the spectra of 184 Hg collected with the 112 Cd and 120 Sn targets, respectively. The γγ coincidence analysis, analogous to that performed for 182 Hg, allowed these X rays to be attributed to the 2 + 2 → 2 + 1 and 0 + 2 → 0 + 1 E0 transitions in 184 Hg. The low level of statistics collected in the experiment with the 107 Ag target does not allow such an analysis to be performed. As an example, the γγ coincidence spectrum, gated on the 2 + 1 → 0 + 1 γ-ray transition in 184 Hg, for the experiment performed with the 120 Sn target is shown in fig. 8. The 286 keV 4 + 1 → 2 + 1 γray transition and the K α X rays are clearly visible. After correcting for the internal conversion and the K α -vacancy creation process, 55(17) K α X rays remain. For the experiment performed with the 112 Cd target the number of K α X rays deduced from the γγ coincidence analysis is equal to 29 (20). In both cases these numbers were attributed to the E0 component of the 2 + 2 → 2 + 1 transition in 184 Hg. Further, following the method described in sect. 4.2, the intensity of the E0 transitions between the excited 2 + 2 and 2 + 1 states as well as between the excited 0 + 2 state and the 0 + 1 ground state can be deduced for the two-particle-gated γray spectra. The R 4 + 1 →2 + 1 takes the value of 0.21 (8) for the 184 Hg + 112 Cd experiment and 0.22 (5) for 184 Hg + 120 Sn. The results are summarized in table 3. The final values were corrected for the K α /K β branching ratio and the fluorescence effect.

Experiment on 186 Hg
Similar to the experiments performed for 184 Hg, the 186 Hg ions were Coulomb excited by three different targets: 114 Cd, 107 Ag, 120 Sn. The γ-ray spectrum following Coulomb excitation of the 186 Hg beam on the 120 Sn target is presented in fig. 9. A sharp peak around 404 keV was identified as a doublet of the 2 + 1 → 0 + 1 and 4 + 1 → 2 + 1 γ-ray transitions at the energies of 405 keV and 403 keV, respectively. The 4 + 2 state at the energy of 1080 keV was weakly populated as well: the 675 keV 4 + 2 → 2 + 1 and 459 keV 4 + 2 → 2 + 2 γ-ray transitions were observed in the Table 3. Measured γ-ray and K X-ray intensities (not efficiency corrected) for 184 Hg scattered on the 112 Cd, 107 Ag and 120 Sn targets. The extracted intensities of the E02 + 2 → 2 + 1 and 0 + 2 → 0 + 1 transitions, corrected for the Kα/K β X-ray branching ratio and fluorescence effect, are also shown. The angular range for simultaneous detection of two particles is given in the centre-of-mass frame together with the corresponding angular range for the target-nucleus (t) detection in the laboratory frame. The intensities marked with asterisks were derived from the γγ analysis (details in the text).

Nucleus
Θ lab,t Transition Counts (7) doublet: experiments performed with the 107 Ag and 120 Sn targets. The low-energy level scheme of 186 Hg together with all observed γ-ray transitions in the Coulomb excitation experiment is shown in fig. 10. The intensity of the 4 + 1 → 2 + 1 γ-ray transition was deduced from the γγ analysis, analogous to those performed for 182,184 Hg. The coincident γ-ray spectrum, gated on the 2 + 1 → 0 + 1 and 4 + 1 → 2 + 1 doublet in 186 Hg, is presented in fig. 11. However, in this case the clear peak vis-  ible at the energy of 404 keV consists of two components: the 2 + 1 → 0 + 1 γ rays in coincidence with the 4 + 1 → 2 + 1 transition and the 4 + 1 → 2 + 1 γ rays in coincidence with the 2 + 1 → 0 + 1 γ-ray transition. Since both cases are just as likely to occur, half of the observed intensity should be attributed to the 4 + 1 → 2 + 1 transition. Details of the analysis are presented in ref. [29]. The extracted intensities are summarized in table 4. The 5/2 − 1 → 3/2 − 1 γ-ray transition from the excitation of the 107 Ag target nucleus was observed as well. The energy of this transition, 423 keV, is  Fig. 11. The random-subtracted γγ coincidence spectrum, obtained for 186 Hg, Coulomb-excited on the 120 Sn target, gated on the 2 + 1 → 0 + 1 and 4 + 1 → 2 + 1 doublet around 404 keV. The γ-ray energies were Doppler-corrected for the 186 Hg ejectile. Table 4. Measured γ-ray and K X-ray intensities (not efficiency-corrected) for 186 Hg scattered on the 114 Cd, 107 Ag and 120 Sn targets. The angular range for simultaneous detection of two particles is given in the centre-of-mass frame together with the corresponding angular range for the targetnucleus (t) detection in the laboratory frame. The photo-peak intensities marked with asterisks were derived from the γγ coincidence analysis (details are given in the text). Nucleus close to those of the 2 + 1 → 0 + 1 and 4 + 1 → 2 + 1 transitions in 186 Hg. As a result, a broad structure was observed in the spectrum Doppler-corrected for the 107 Ag recoil, which made the precise extraction of the 5/2 − 1 → 3/2 − 1 γ-ray intensity not possible [29]. However, the spectrum Dopplercorrected for 186 Hg, revealed a clear narrow peak on a broad background for the doublet of the 2 + 1 → 0 + 1 and 4 + 1 → 2 + 1 γ-ray transitions. This allowed determination of the intensity of these two transitions in an unambiguous way. The 2 + 2 state was not populated in the experiment. No 216 keV 2 + 2 → 2 + 1 γ-ray transition was visible in the twoparticle-gated γ-ray spectra. The 2 + 2 → 2 + 1 is a relatively highly converted transition with a total conversion coefficient, α tot (2 + 2 → 2 + 1 ), known to be equal 3.5(5) [32]. However, in the contrary to the case of 182,184 Hg, the number of K α Xra ysintheγγ coincidence spectrum is consistent with zero (see fig. 11) indicating that E0 de-excitation attributed to the 2 + 2 → 2 + 1 transition was not observed. Nevertheless, all known spectroscopic information concerning the 2 + 2 state was included in the analysis aiming extraction of the matrix elements using the GOSIA code (see sect. 5 for more details). Some X rays are visible in the two-particle-gated γray spectra, fig. 9. These remain after subtracting X rays originating from the K-vacancy creation and from known internal conversion of the observed γ-ray transitions. They are attributed to the E0(0 + 2 → 0 + 1 ) transition.

Experiment on 188 Hg
In the Coulomb excitation of 188 Hg induced by the 120 Sn, 114 Cd and 107 Ag targets, the 2 + 1 and 4 + 1 states in 188 Hg were populated (see fig. 12). The beam intensity of 10 5 pps yielded high statistics collected during 11.4 h and 15.9 h of data taking with the 120 Sn and 114 Cd targets, respectively. Significantly lower statistics were collected during the experiment with the 107 Ag target due to a much shorter data collection time compared to the measurements using the 120 Sn and 114 Cd targets (see table 1). As an example, the total Doppler-corrected and random-subtracted γ-ray spectrum obtained for 188 Hg, Coulomb-excited by the 120 Sn target, is presented in fig. 12. In the case of the experiment performed with the 114 Cd target, the intensity of the 592 keV 4 + 1 → 2 + 1 γ-ray transition in 188 Hg could not be extracted since it is contaminated by the Dopplerbroadened 2 + 1 → 0 + 1 transition in the 114 Cd target at the energy of 558 keV. This was not the case for the experiments performed with the 120 Sn and 107 Ag targets, where the 4 + 1 → 2 + 1 transition was clearly identified. Moreover, in the γγ coincident spectrum there is no indication of a population of higher-lying states, except for the 4 + 1 . Particularly, no statistically significant 411 keV 0 + 2 → 2 + 1 γ-ray transition was observed in coincidence with the 2 + 1 → 0 + 1 transition [29]. Weak K X rays are visible in the total γray spectrum in fig. 12. They partially originate from the internal conversion of the observed γ-ray transitions in 188 Hg. After subtracting this contribution the remaining number of X rays was attributed to the heavy-ion induced K vacancy creation due to atomic processes and used as a normalization for lighter mercury isotopes (see sect. 4.1 and ref. [27] for more details). The γ-ray and K X-ray intensities in 188 Hg extracted from the Coulomb-excitation experiments are given in table 5. Similar to 186 Hg + 107 Ag experiment, the unambiguous extraction of the 423 keV, 5/2 − 1 → 3/2 − 1 γ-ray transition in 107 Ag was not possible. The energy of this transition is close to the 413 keV, 2 + 1 → 0 + 1 transition in 188 Hg, which forms a broad structure in the spectrum Doppler-corrected for the 107 Ag recoil. However, as in the previous case, the spectrum Doppler-corrected for 188 Hg revealed a clear narrow peak for the 2 + 1 → 0 + 1 γ-ray transition on a broad background, allowing precise extraction of its intensity.

Matrix elements determination
In order to determine the E2 matrix elements in 182,184,186,188 Hg, the Coulomb-excitation least-squares fitting code GOSIA [33,34] was used. The code constructs a standard χ 2 function built of measured γ-ray intensities and those calculated from a set of matrix elements between all relevant states. Additionally, known spectroscopic data, e.g., γ-ray branching ratios, multipole mixing coefficients and lifetimes, can be used as auxiliary data in the minimization procedure. These data enter the χ 2 function on an equal basis as γ-ray intensities observed in the Coulomb-excitation experiments.
The γ-ray branching ratios known from the β-decay studies of 182,184,186,188 Tl isotopes [17,[35][36][37] were used in the GOSIA analysis as additional data points serving as important constraints in the multidimensional χ 2 fit. Moreover, the E0 transitions were included as well, together with the known total conversion coefficients for the 2 + 2 → 2 + 1 transitions in 182,184 Hg [17] and 186 Hg [32], as described in more detail in sect. 5.2. Lifetimes of the yrast states in 182,184,186 Hg were measured independently using the RDDS method [19,38,39]. Lifetimes of excited states in 182 Hg were extracted in ref. [38] and in ref. [39]. Both publications report consistent results which were obtained by applying two different analysis procedures to the same data set. In the current Coulomb-excitation analysis we used lifetime values from ref. [39]. Lifetimes of the non-yrast 0 + 2 and 2 + 2 states in 186,188 Hg were taken from ref. [40].
Since lifetimes provide strong constraints in the multidimensional GOSIA fit, it was important to check the consistency between them and the Coulomb-excitation data. Thus, at the first stage of analysis no lifetime information was included and the Coulomb-excitation cross sections for the projectile were normalized to the known excitation cross sections for target nuclei. These calculations were performed with the GOSIA2 code [34], which is a modified version of the standard GOSIA code, capable of handling mutual excitation of target and projectile nuclei. The method is described in detail in ref. [41] and recently has been applied as well in refs. [42][43][44]. The analysis of the Coulomb-excitation data performed without using the known lifetimes of excited states in 182,184,186 Hg yielded results consistent within 1σ uncertainty with the lifetime values [29].
To exploit the dependence of the Coulomb-excitation probability on the scattering angle, and in this way gain the sensitivity to higher-order-effects, such as quadrupole moments or signs of the interference terms, the data for each mercury isotope were subdivided, depending on the experiment, into three or five subsets. They corresponded to different angular ranges of scattered particle. The influence of the scattering angle on multi-step excitation probability is illustrated in fig. 13, where an increase of the 4 + 1 → 2 + 1 γ-ray yield with respect to the 2 + 1 → 0 + 1 transition is observed for higher center-of-mass angles. The division of the data was a compromise between the number of independent data points for the γ-ray yields and the level of statistics obtained for the individual ranges. Due to the low statistics collected in the experiments with the 107 Ag target, no divisions were applied to these data. Instead, the total intensities were used in the analysis. The same applies to the 2 + 2 → 2 + 1 γ-ray and the E0i n 182,184 Hg. The current work presents the re-evaluated values, with respect to refs. [20,29], of matrix elements in 182,184 Hg, obtained using revised spectroscopic data char- acterizing the low-energy structure of these isotopes that have recently become available [17] and which differ from the values used in the previous analysis [20]. The change concerns mainly the values of the total conversion coefficients α tot (2 + 2 → 2 + 1 ) which is equal to 7.2(13) in 182 Hg and 14.2 (36) in 184 Hg, (instead of the values of 4.7(13) and 23(5) for 182 Hg and 184 Hg, respectively, that were used in the previous analysis), as well as the 2 + 2 → 0 + 2 /2 + 2 → 0 + 1 γray branching ratio in 184 Hg which changes from 0.082 (34) to 0.016 (9). The new analysis together with the obtained matrix elements is presented in detail in sect. 5.1.

Analysis of 182 Hg and 184 Hg
The level schemes of 182 Hg and 184 Hg, limited to wellknown states considered in the current analysis, are presented in figs. 4 and 7, respectively. The measured γ-ray yields, listed in tables 2 and 3, as well as the E0 transitions extracted between the pairs of 0 + and pairs of 2 + states in 182,184 Hg, were included in the GOSIA analysis. The method adopted to include the E0 decay in the GOSIA analysis is described in detail in sect. 5.2. In total, 19 [14] transitional and diagonal matrix elements were fitted to 40 [20] data points in 184 Hg [ 182 Hg].
The analysis of the Coulomb-excitation data brings information on the relative signs of transition matrix elements, as the latter may have a significant influence Table 7. Reduced matrix elements obtained in this work for 182 Hg and 184 Hg. The E2 matrix elements for 186,188 Hg are taken from ref. [20]. The relative signs of matrix elements were determined by analysing the influence of interference terms listed in table 6. The adopted sign convention enables a direct comparison with the two-state mixing model calculations presented in sect. 6 on the Coulomb-excitation cross sections. The absolute sign of an individual transitional matrix element has no physical meaning, since it depends on the arbitrary choice of the relative phases of wave functions of initial and final states. However, the sign of the product of the matrix elements-the so called interference term e.g., 0 + 1 E2 2 + 2 2 + 2 E2 2 + 1 2 + 1 E2 0 + 1 -is independent of the chosen convention and can be determined experimentally. The signs of three interference terms were determined for 182 Hg and 184 Hg and are listed in table 6. A convention adopted in the case of 182 Hg was that the signs of all transitional E2 matrix elements connecting the yrast states and the E2 matrix elements between the non-yrast states, as well as the sign of the 2 + 2 E2 2 + 1 matrix element, were fixed. The signs of all the remaining matrix elements were free, i.e., the signs of the interference terms could be determined in the GOSIA analysis. A similar approach was used for 184 Hg.
The relative signs of the matrix elements reported in the present work for 182,184 Hg were verified by performing the minimization procedure for all possible sign combinations of the interference terms. The obtained χ 2 values were compared. For example, when the 0 + 1 E2 2 + 1 2 + 1 E2 2 + 2 2 + 2 E2 0 + 1 product in 182 Hg is negative, it causes a four-fold increase in the total χ 2 value as compared to the positive sign of this term. Only in the case of the 2 + 1 E2 2 + 2 2 + 2 E2 0 + 2 0 + 2 E2 2 + 1 interference term in 182 Hg two equally deep χ 2 minima corresponding to the different signs of this product were found. The main difference between the sets of matrix elements for these two minima are the opposite signs of the 0 + 2 E2 2 + 1 matrix element. The signs of all other transitional E2 matrix elements are the same, and only slight differences in their magnitudes are observed. The uncertainties of the matrix elements reported in table 7 account for these small variations. The value of the 0 + 2 E2 2 + 1 matrix element in 182 Hg is given as a range.
The resulting set of reduced matrix elements in 182,184 Hg together with their relative signs is presented in table 7. The estimation of the statistical uncertainties of the matrix elements was performed in two steps. Firstly, the uncorrelated uncertainties were calculated. Then, all possible correlations between the matrix elements were taken into account. The final confidence interval, defined by the integral of probability distribution, is equal to 68.3% [33].
The differences between the re-evaluated values of the matrix elements and those published in ref. [20] mainly stem from the change in α tot (2 + 2 → 2 + 1 )andγ-ray branching ratios, as explained above. The values of the matrix elements related to the 2 + 2 and 0 + 2 states are most influenced by these changes. The reduced matrix elements obtained for 182,184 Hg reproduce all γ-ray branching ratios, total conversion coefficients α tot (2 + 2 → 2 + 1 ) and lifetimes within the 1σ uncertainty. Almost all experimental γ-ray yields were reproduced within 1σ uncertainty. The only exceptions are: (i) I(2 + 2 → 0 + 1 )i n 184 Hg for the experiment with the 107 Ag target reproduced within 1.3σ, (ii) I(4 + 1 → 2 + 1 )i n 184 Hg reproduced within 1.5σ in one of the experimental data sets with the 112 Cd target, (iii) I(E0, 2 + 2 → 2 + 1 )i n 182 Hg reproduced within 2σ. The diagonal matrix elements of the 2 + 1 ,2 + 2 and 4 + 1 states were included in the analysis as free parameters.
With the new, revised spectroscopic data, a careful analysis of the signs of the 2 + 1,2 diagonal matrix elements was performed for 182 Hg and 184 Hg. This analysis shows that changes in the value and/or sign of a given diagonal matrix element, for example from 1.1 eb to −1.6eb for 2 + 1 E2 2 + 1 in 184 Hg, do not significantly influence neither other values of the transitional E2 matrix elements, nor the quality of the χ 2 fit. Furthermore, because of the large uncertainty related to the E0 component of the 2 + 2 → 2 + 1 transition in 182,184 Hg we refrain from reporting any diagonal matrix elements for the 2 + states except for the 188 Hg. In the latter case, extraction of the 2 + 1 E2 2 + 1 matrix element is more straightforward, since only 2 + 1 and 4 + 1 states were populated in 188 Hg. For the comparison with the quadrupole sum rules results presented in fig. 3 of ref. [20], the Q 3 cos(3δ) invariant cannot be determined from the current results; consequently, no conclusion on the triaxiality of the 0 + states can be drawn. However, the values of the Q 2 invariants, analogous to those reported in ref. [20], can be extracted for the 0 + states from the E2 matrix elements presented in table 7. The Q 2 values for the 0 + 1 states in 182 Hg and 184 Hg are equal to 2.04(16) e 2 b 2 and 1.74(15) e 2 b 2 , respectively and are consistent with those published in ref. [20]. For the excited 0 + states the Q 2 =2.3(9) e 2 b 2 in 184 Hg and an upper limit of 7.1 e 2 b 2 can be given for 182 Hg. The latter are in a better agreement with equivalent values calculated with the beyond-mean-field and the interacting boson models as shown in fig. 3 in ref. [20]. It is worth to mention that while the Q 2 invariant for the ground state in even-even nuclei is almost completely determined by the 0 + 1 E2 2 + 1 matrix element [45], the situation becomes more complex for the higher-order invariant, i.e. Q 3 cos(3δ) , as discussed in refs. [43,45]. In order to extract the latter for the 0 + 1,2 states, knowledge on diagonal matrix elements of the excited 2 + states is particularly important as well as information about signs of all relevant interference terms.
The E2/M 1 mixing ratios, δ( E2 M 1 ), are not known for any transitions between the low-lying states in 182,184 Hg. In the current analysis the 2 + 3 → 2 + 1 and 4 + 2 → 4 + 1 transitions in 184 Hg were assumed to be of pure E2 character. In the case of the 2 + 2 → 2 + 1 transition in 182,184 Hg, a δ( E2 M 1 ) value of 1.85 was adopted, consistent with the known value in 202 Po [46]. An influence of the unknown E2/M 1 mixing ratio on the extracted E2 matrix elements was investigated. The E2/M 1 ratio was varied over several values between 0.5 and 5. For each value of the mixing ratio, δ( E2 M 1 ), a full minimization with the GOSIA code was performed. The solutions obtained with δ( E2 M 1 ) > 1c o rrespond to similar χ 2 values and no considerable change in other E2 matrix elements in 182,184 Hg was observed. A larger change in values of the matrix elements related to the 2 + 2 state is observed when δ( E2 M 1 ) < 1, as presented in fig. 14.

Implementation of the E0 decay into the GOSIA analysis
The analysis of the intense K X-ray peaks measured for 182,184 Hg revealed that the 2 + 2 → 2 + 1 transitions are strongly converted. Furthermore, the intensities of the 0 + 2 → 0 + 1 and 2 + 2 → 2 + 1 E0 transitions in 182,184 Hg were deduced. Moreover, the total conversion coefficient of the 2 + 2 → 2 + 1 transition, α tot (2 + 2 → 2 + 1 ), measured in β/EC decay of 182,184 Tl [17] was extracted for 182 Hg and 184 Hg. Such data are crucial for the Coulomb-excitation analysis, as transitions under investigation contain large E0 components, which need to be taken into account when extracting matrix elements. As it is not currently possible to declare the E0 transitions directly in the GOSIA input files, an indirect method was applied, which has also been tested before e.g., in refs. [43,44]. The E0 decay path of the 0 + 2 state was simulated in the fit by an M 1t r a nsition via a virtual 1 + state, introduced in addition to the known level schemes of 182,184 Hg. The extra 1 + 1 states were placed below the 0 + 2 state, at 259 keV and 306 keV excitation energy in 182 Hg and 184 Hg, respectively, and connected to the 0 + 2 state by a 69 keV M 1 transition. The choice of the excitation energy for the virtual 1 + state is arbitrary. However, it was checked that changing this excitation energy does not influence the final results. The virtual 0 + 2 → 1 + 1 M 1 transition is utilised to enable the E0 decay path of the 0 + 2 state. The 0 + 2 M 1 1 + 1 matrix element was introduced in the GOSIA input file, together with the 1 + 1 M 1 0 + 1 matrix element enabling depopulation of the 1 + 1 state. In a similar way the E0 component of the 2 + 2 → 2 + 1 transition was taken into account. A second additional 1 + 2 state was included in the level schemes of 182 Hg and 184 Hg, placed between the first two 2 + states at the energy of 479 keV and 465 keV, respectively. The M 1 matrix elements connecting the 2 + 1,2 states with the 1 + 2 states were introduced as well. Since low-energy Coulomb excitation proceeds predominantly via E2 (and E3) transitions, the introduction of these additional matrix elements does not influence the calculated excitation pattern. In contrast, the M 1 transitions strongly influence the de-excitation process.
The branching ratio, which represents in the analysis the α tot (2 + 2 → 2 + 1 ), is interpreted as the ratio of the I(E0; 2 + 2 → 2 + 1 ) intensity and the 2 + 2 → 2 + 1 γ-ray intensity of mixed E2/M 1 multipolarity, This can be further expressed by the total conversion coefficient α tot (2 + 2 → 2 + 1 ), As the experimental E0 intensity I E0 (2 + 2 → 2 + 1 )i s known for 182 Hg and 184 Hg, it needs to be taken into account as well. Such data were included in the analysis assuming that the E0 de-excitation of the 2 + 2 proceeds via the 2 + 2 → 1 + 2 transition As the E0 components are represented in the GOSIA analysis by M 1 γ-ray transitions, the experimental intensity I E0 (2 + 2 → 2 + 1 ) needs to be corrected for internal conversion as follows: Similarly for the I E0 (0 + 2 → 0 + 1 ) intensity: Correcting for the experimental intensity I E0 (2 + 2 → 2 + 1 ) given by eq. (5) and expressing the I E2 γ I E2 γ +I M 1 γ term in eq. (3) by the E2/M 1 mixing ratio δ, one obtains The value of the branching ratio given by eq. (7) was introduced in the GOSIA input file as an additional data point; in this way the total conversion coefficient α tot (2 + 2 → 2 + 1 ) was included in the fit. As described in sect. 5, the value of δ( E2 M 1 )=1.85 was adopted. The 2 + 2 M 1 1 + 2 matrix elements in 182,184 Hg were fitted in such a way that the best reproduction of the experimental E02 + 2 → 2 + 1 intensities and BR( values defined by eq. (7) was achieved. Similar, the 0 + 2 M 1 1 + 1 matrix elements were fitted to reproduce the experimentally determined E00 + 2 → 0 + 1 intensities. The use of the M 1 multipolarity to represent the E0 decay paths is an arbitrary choice. Other possibilities for the virtual transitions simulating E0 decay, e.g., M 2transitions, were also tested and no influence on the final solution was observed [29].
6 Results and comparison with three models 6

.1 Theoretical tools
In the following section the experimental and theoretical results concerning excitation energies, reduced transition probabilities and spectroscopic quadrupole moments are compared and discussed within the framework of three different models: i) a quadrupole collective model based on the General Bohr Hamiltonian (GBH) [21], ii) a beyondmean-field model (BMF) [16], and iii) an interactingboson model with configuration mixing (IBM-CM) [22]. An effective Skyrme interaction is employed in both the GHB and BMF approaches used here, and both methods are based on a set of BCS-type self-consistent mean-field states that cover a wide range of quadrupole deformations.
A BMF calculation consists of several consecutive steps. First, a set of deformed mean-field states with different axial quadrupole moments is constructed by solving the Hartree-Fock plus Bardeen-Cooper-Schrieffe (HF+BCS) equations with suitably chosen constraints. Each of these states is then projected on the proton and neutron numbers of interest and the targeted angular momenta. In the final step, the resulting symmetry-restored states of the same I are mixed in the framework of the generator-coordinate method (GCM). The projection not only provides a spectrum of states with different angular momenta, but it also restores the selection rules for transitions between them. The GCM describes the shape fluctuations of the collective states and also yields a set of orthogonal states, so that their properties can be directly compared with experiment. Since there is no truncation of the model space in BMF models like the 2hω truncation in the shell model space, there is no need to introduce effective charges and the matrix elements of the E0a n d E2 operators are calculated with bare charges. At each step of the calculation, the SLy6 parametrization of the Skyrme interaction is used in connection with a surfacetype pairing interaction. For a more detailed description of the calculations, we refer to ref. [16].
A general theory of the collective quadrupole model employing the general Bohr Hamiltonian can be found in ref. [21], while a detailed report on an application of the model is presented in ref. [45]. Here we briefly recall some of its main points. The β and γ dynamical variables of the model, which describe the deformation of a nucleus in the so-called intrinsic frame, are directly related to a quadrupole tensor of the nuclear-mass distribution (however, with no specific shape assumed). All quadrupole degrees of freedom, including nonaxiality and rotations, are treated on an equal footing. The General Bohr Hamiltonian is determined by seven functions: the potential energy and six inertial functions. These seven functions are calculated from the microscopic mean-field theory using the Adiabatic Time-Dependent Hartree-Fock-Bogoliubov (ATDHFB) approach. The E2 electromagnetic transitions are described by a collective operator directly related to the nuclear charge distribution. The computational details are the same as in ref. [45]. Mean-field configurations for given deformations are obtained through constrained HFB calculations using the SLy4 variant of the Skyrme force and the seniority (constant G) force as a pairing interaction. Neither the BMF nor the GBH model contain any free parameter, the value of which would be fixed by comparison with the properties of the excited states considered here.
The GBH can be regarded as a computation friendly approximation of a symmetry-restored GCM. Its space of dynamical variables includes the full β − γ plane and thus triaxial shapes. The use of ATDHFB masses in the GBH incorporates some effects that in a projected GCM would require consideration of time-reversal-breaking cranked states, which up to now has never been done in a systematic way. The SLy4 interaction used in the GBH is in many respects very similar to the SLy6 interaction used in the BMF calculations; they are fitted with the same protocol, but with a different recipe to correct for the center-of-mass motion. In consequence, SLy6 has a smaller surface-energy coefficient than SLy4, leading to deformation-energy surfaces that are slightly softer [47]. This tends to reduce the excitation energy of coexisting shapes of different deformations, which is one of the possible reasons of some of the differences between the results obtained with the BMF and GBH approaches reported below.
The interacting-boson model [48] is a leading algebraic model approach, making use of the U(6) symmetry of an interacting system built from L =0a n dL = 2 (s and d) bosons. The IBM is a symmetry-dictated truncation of the nuclear shell model where the bosons represent pairs of fermions. The number of bosons can be related to the number of valence protons and neutrons present in the corresponding shell-model space N = nπ+nν 2 .
An extended version of the IBM model, which can be applied when approaching closed shells, is presented in refs. [49,50]. In this version of the model one includes additional bosons, which are related with possible mparticle n-hole excitations. Consequently, the full model space also contains a part built from N + 2 bosons (for m =2a n dn = 2). The interaction amongst the N and N + 2 parts within the full model space gives rise to what is called the IBM-CM model.
Extensive use has been made of the interacting boson model with configuration mixing ([1] and references therein) in various regions of the nuclear chart, with particular attention to the isotopic chains in the Pb region. This approach gives the possibility to describe modes of excitations that exhibit different collective characters. The application to the neutron-deficient Hg nuclei was discussed in detail in ref. [22], where both the construction of the Hamiltonian, the E2 operator as well as an extensive discussion of the results have been presented at length.

Energy spectra and electric quadrupole properties
Experimental and theoretical results concerning the electromagnetic structure of 182,184,186,188 Hg nuclei are compared in fig. 15. Energies, B(E2) reduced transition probabilities and spectroscopic quadrupole moments are given for the yrast and the first excited states with even spin I. The quadrupole properties are summarized in table 8. The experimental values for the yrast transition probabilities above the 4 + 1 state in 182,184,186 Hg are taken from refs. [19] and [38]. Those for the transitions between the yrast and the first excited 0 + and 2 + states in 188 Hg are taken from ref. [51].
In the BMF calculations, the wave function for each state is obtained by mixing the deformed mean-field wave functions projected on the same angular momentum and particle numbers. Looking at fig. 9 of ref. [16], one can see that the BMF I = 0 states have particularly complicated structures. The only case where the ground state is dominated by the projected prolate configurations is 182 Hg. Even in this case, the first excited 0 + state involves projected oblate and prolate configurations with similar weights. At higher spin, the shape mixing decreases and starting at I = 6 (sometimes I = 4) either projected oblate or prolate configurations dominate the BMF wave functions. This is reflected in fig. 15 by the fact that the spectroscopic quadrupole moments do not vary much with increasing spin for the highest spin states. The same can be noticed for GBH results.
Both BMF and GBH energy spectra are too spread out compared to the experimental data. A well-known deficiency of these models is that they systematically underestimate moments of inertia. This is generally attributed to the conservation of time-reversal invariance imposed on the mean-field states. This prevents the reduction of pairing and the alignment of single-particle states to be taken into account when performing the projection on finite angular momenta.
The spectroscopic quadrupole moments of the yrast 2 + states are negative for the BMF and GBH approaches pointing to predominantly prolate states. The only exception is the BMF result for 188 Hg, where the spectroscopic quadrupole moment is positive and the calculation slightly overestimates the experimental value. The wave function of this 2 + state is predominantly composed of projected oblate mean-field states, as are those of the heavier Hg isotopes (see fig. 9 and fig. 17 of ref. [16]). The B(E2) values are systematically overestimated, both in BMF and GBH calculations. This deficiency can have several causes. The most probable ones are either too large weights of projected deformed prolate wave functions or an overestimation of the deformation already at the mean-field level. As discussed in ref. [16] these problems are ultimately linked to the wrong relative position of the single-particle levels at sphericity. Unfortunately, these positions cannot be improved by an obvious change in the parametrizations of the EDFs. , BMF [16] and GBH [21,52] predictions. Reduced E2 transition probabilities (arrows) and spectroscopic quadrupole moments (loops) are given in e 2 b 2 and eb units, respectively. The experimental values are taken from table 8 and refs. [19,38,51]. In each spectrum the left part presents the yrast levels, while the non-yrast states are displayed on the right side. Blue (red) is used for transitions and moments connecting the yrast (non-yrast) states. Transitions between yrast and non-yrast states are marked in green. Good reproduction of the experimental data can be noticed when comparing to the results of the IBM-CM model. Note that in the IBM-CM approach seven parameters per isotope, plus two other parameters (the latter two fixed for the whole isotopic chain) were obtained through a least-squares fit to the available experimental information. In order to extract the IBM parameters the measured energies up to the 8 + 1 level, including the yrast and the non-yrast 0 + 2 ,2 + 1,2,3,4 ,4 + 1,2,3 ,6 + 1,2 and 3 + 1,2 states, jointly with known measured B(E2) values involving these states, were used. For the case of 182,184,186 Hg the experimental B(E2) values between 2 + 2 and 2 + 1 ,0 + 1,2 and 4 + 1 , as well as B(E2; 10 + 1 → 8 + 1 ), were not included in the fit of the IBM parameters (for more details see ref. [22], sects. III B and III C, tables I and III therein). In this respect, it can be expected that the IBM-CM model reproduces the experimental data better as compared to the BMF and GBH, for which no parameters are fitted to the properties of the Hg nuclei. A geometric interpretation of the IBM-CM can be obtained using the intrinsic state formalism. This provides a way to extract the mean-field energy corresponding to the IBM-CM Hamiltonian. Moreover, quadrupole deformation variables β and γ could be extracted for the Hg nuclei from the quadratic and cubic quadrupole invariants (see ref. [22], sects. V.B and V.C for a more detailed description). A good agreement between the IBM-CM model calculations and the experimental results supports the description of the nuclear structure in the mercury isotopes as resulting from two coexisting configurations characterized by different deformations. An alternative procedure to extract the IBM-CM parameters can be used, in which the parameters are derived starting from a self-consistent mean-field calculation. This has been realised by Nomura et al., using the Gogny D1M force (see refs. [53,54] and references therein for a detailed description of the method used, as well as its application to the Hg nuclei).

Comparison to the two-state mixing calculations
As reported in ref. [20], the electromagnetic properties of even-even neutron-deficient mercury isotopes can be described in terms of mixing of two structures, which coexist at low-excitation energy. Matrix elements as well as signs of their products (interference terms), extracted from Coulomb-excitation measurements, can be compared to those resulting from the two-state mixing model. Within this phenomenological approach, following the notation introduced in ref. [55] and elaborated in ref. [56], the observed physical states can be written as linear combinations of two unmixed structures -structure I and structure II-with specific mixing amplitudes. The latter, taken from ref. [19], were derived from the fit of the known higher-lying level energies in the rotational bands, built upon the first two 0 + states, using the variable moment of inertia model [57]. States with spin I>4a r e weakly mixed and manifest a rotational-like character, whereas a stronger mixing was deduced for states with spin I = 2, reaching the maximum of mixing for 184 Hg.  For 182-188 Hg the Coulomb-excitation data could be well reproduced by mixing between less-deformed oblate-like and more deformed prolate-like configurations [20], with their quadrupole properties assumed to be constant for the four isotopes considered. In fig. 16 a comparison between the experimental matrix elements and those resulting from the two-state mixing model is shown.
This figure is analogous to that presented in ref. [20] however, it contains the re-evaluated matrix elements for 182 Hg and 184 Hg. Experimental results, i.e., magnitudes of matrix elements and signs of their products, are well reproduced by the two-state mixing model. Moreover, a significantly better agreement between the experimentally determined 2 + 1 E2 0 + 2 and 4 + 1 E2 2 + 2 matrix elements and those calculated within the twostate mixing model is now obtained for 184 Hg as compared to the results presented in ref. [20]. This is directly related to the experimentally extracted signs of the interference terms: 2 + 1 E2 4 + 1 4 + 1 E2 2 + 2 2 + 2 E2 2 + 1 and 2 + 1 E2 0 + 2 0 + 2 E2 2 + 2 2 + 2 E2 2 + 1 , which are both negative in 184 Hg. The signs of these terms are opposite to those published in ref. [20].
The unperturbed quadrupole moments of the two 2 + states, Q s (2 + ) TSM , belonging to two different unmixed configurations, were deduced from the experimental 2 + 2 E2 2 + 1 matrix elements extracted for the mercury isotopes using the two-state mixing model (TSM). This yielded values of quadrupole moments for the pure 2 + states equal to 136 e fm 2 and −303 e fm 2 [20]. A comparison of the spectroscopic quadrupole moments (Q s ) for the 2 + 1 and 2 + 2 states in 182,184,186,188 Hg is shown in fig. 17. These Q s values are calculated with the BMF, IBM-CM and GBH models and presented together with the unmixed values: the Q s (2 + ) TSM ones and those obtained with the IBM-CM model, Q s (2 + ) IBM pure . The latter corre-    spond to the same Hamiltonian as the one used to obtain the results shown in fig. 15, column IBM-CM, but removing the interaction term acting between the 2h − 0p and 4h − 2p proton configurations. The BMF calculations coherently predict that the dominantly prolate configuration is the lowest in energy up to N = 106. A crossing between oblate and prolate configurations, corresponding to the positive and negative values of the quadrupole moments, respectively, takes place between N = 106 and N = 108. For the IBM-CM this transition happens at N = 104, while no change of the structure of the 2 + states is observed for the GBH. In the two-state mixing model the unperturbed values of Q s are assumed to be the same for all four mercury isotopes. This assumption is consistent with the IBM-CM calculations, predicting indeed almost constant values of unperturbed Q s (see fig. 17). The same constant trend for the calculated Q s values also appears for the BMF and GBH calculations. The calculated Q s values of the oblate 2 + states coincide with the pure ones from the two-state mixing model, while the absolute value for the prolate 2 + states is underestimated in BMF and GBH.
To summarize, though the excitation energy of the 2 + 1 state and the energy difference E(2 + 2 ) − E(2 + 1 ) are almost constant for even-even 180-188 Hg (as seen in the energy systematics in refs. [13,14]), and the same is true for the B(E2; 2 + 1 → 0 + 1 )a n dB(E2; 2 + 2 → 0 + 1 ) values as depicted in fig. 18(a), the underlying nuclear structure undergoes a dramatic change. As shown in the evolution of the IBM-CM wave functions in fig. 18(b), the composition of the 2 + 1 states changes from a rather pure structure-I character (the so-called regular configuration) for 186,188 Hg to a state dominated by structure II (the intruder configuration) in 182 Hg. This substantial change in nuclear structure is also supported by the evolution of the IBM-CM Q s values presented in fig. 17 or that of the theoretical and experimental B(E2; 2 + 2 → 2 + 1 ) values, as demonstrated in fig. 18(a). In conclusion, similar energies of states in an isotopic chain and similar transition probabilities do not always reveal a similar structure -the underlying mixing configuration can be somehow concealed as stated in refs. [22], [20] and [19].

Monopole transition strength
The large total conversion coefficients for the 2 + 2 → 2 + 1 transition in 182,184 Hg are a significant indicator of a strong mixing between the configurations having different shapes [60]. Combining the extracted B(E2; 2 + 2 → 2 + 1 ) values from the Coulomb-excitation experiment with a measured total conversion coefficients α tot (2 + 2 → 2 + 1 ) [17], the E0 transition strengths can be determined for 182 Hg and 184 Hg.
In fig. 19, the E0 strengths of the 2 + 2 → 2 + 1 transition deduced from the experimental data are compared to the two-state mixing model predictions, the BMF model [16], the GBH model [52] and the IBM-CM model [63]. While the experimental value for 186 Hg is in agreement with all four calculations, the value for 182 Hg deviates from the predictions of both the BMF and GBH model calculations.
It is only consistent, within 1σ, with the two-state mixing approach and the IBM-CM model. For 184 Hg none of the calculations is in agreement with the experimental result. Both BMF and GBH models predict similar magnitudes of the ρ 2 (E0; 2 + 2 → 2 + 1 ) especially around neutron midshell N = 104. Interesting to note is the rising trend of the BMF and GBH E0 strengths towards the lighter mercury isotopes, which is very different from the IBM-CM predictions. The IBM-CM calculations indicate that the largest values of the ρ 2 (E0; 2 + 2 → 2 + 1 ) occur around neutron number N = 104. Both IBM-CM and GBH models predict a drop of the ρ 2 (E0) strengths for more neutrondeficient Hg isotopes (from N =98toN = 96). The origin of this behavior is currently not known.

Summary and outlook
Multi-step Coulomb-excitation experiments with postaccelerated radioactive ion beams of neutron-deficient, eveneven 182,184,186,188 Hg isotopes were performed at the REX-ISOLDE facility at CERN. As a result, sets of E2 matrix elements were extracted between low-energy 0 + ,2 + and 4 + states populated in the experiments. The Coulombexcitation data for 182,184 Hg were re-evaluated since new, revised spectroscopic information, α tot (2 + 2 → 2 + 1 )a n dγray branching ratios, have become available [17]. Complementary to our previous work [20], a systematic comparison of experimental results, i.e., level energies and reduced quadrupole transition probabilities, with theoretical predictions, is shown. The results of calculations using the GBH and BMF models are, to a certain extent, in agreement with the experimental data. In the yrast bands of 182,184,186 Hg the experimental B(E2) transition probabilities exhibit a very smooth behavior for states with spin J ≥ 4, and this trend, as well as the absolute B(E2) values, are fairly well reproduced by the GBH and BMF models. According to these models these states are of a prolate nature and lie lower in energy as compared to the oblate ones. A stronger mixing was deduced for states with I = 2, reaching a maximum for N = 104 [19]. For the low-lying 2 + and 0 + states the comparison with theory is less successful, partly due to the fact that the excitation energies of the different configurations are not correctly reproduced and their relative positions are reversed. In the case of the BMF calculations inclusion of the triaxial degree of freedom may be necessary in the description of the low-energy electromagnetic structure of the neutron mid-shell mercury isotopes.
Properties of the lowest-lying states of even-even 182-188 Hg were also interpreted within a two-state mixing model. It is interesting to note how well the experimental results can be reproduced within this simple approach supporting the underlying assumption of two unperturbed different configurations that mix when states with equal spin and parity are close in energy. The results clearly show that the low-energy electromagnetic structure of 182-188 Hg isotopes can be described in terms of mixing of two rotational configurations which coexist at low excitation energy. Mixing between a weakly deformed oblate-like band and a more deformed prolate-like band gains importance when going towards neutron midshell N = 104.
Because of the limited beam energy, only low-lying states could be studied at REX-ISOLDE. With the higher beam energy, up to 5 MeV/A, possible nowadays at HIE-ISOLDE, Coulomb excitation of neutron-deficient mercury isotopes can provide much richer information thanks to the higher multi-step excitation cross sections and increased sensitivity to the diagonal matrix elements. Thus our knowledge of higher-lying non-yrast states can be extended. Moreover, the quadrupole sum rules method can then be used to extract the shape invariants in a given state independently of the nuclear-structure models [64,65].
In order to draw firm conclusions from Coulombexcitation experiments with exotic beams, complementary spectroscopic data are crucial [2]. Recently performed β/EC decay of neutron-deficient even-even Tl isotopes at HIE-ISOLDE intend to provide these data i.e., precisely measure γ-ray branching ratios, conversion coefficients and mixing ratios for the low-lying (yrast and non-yrast) states in Hg isotopes [66]. Moreover, future Coulomb excitation experiments will also benefit from the use of the electron spectrometer SPEDE [67] which will provide direct information on intensities of conversion electrons, being of great importance for the nuclei in the N = 104 region [68].
Additional spectroscopic information for the higherlying collective states can be obtained using unsafe Coulomb excitation. Beam energies available at HIE-ISOLDE up to 10 MeV/A will enable few-nucleon transfer reaction experiments to probe nuclear states in the light lead mass region. Furthermore, future Coulomb-excitation experiments of light odd-mass mercury isotopes aim to study the shape-coexisting isomers present in mercury nuclei. This, combined with one-neutron transfer reactions, will shed light on the underlying single-particle nature of these states. All these efforts open new possibilities to characterize shape coexistence in the mercury region [69].