Simultaneous γ -ray and electron spectroscopy of

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I. INTRODUCTION
The neutron-deficient mercury isotopes (Z = 80) around the neutron mid-shell at N = 104 constitute one of the most prominent examples of shape coexistence [1]. Laser spectroscopy studies in this region show dramatic changes of the charge radii between the neighboring isotopes [2,3]. This behavior, called shape staggering, indicates a large change of deformation between the measured ground and isomeric states [1]. The evolution of shape coexistence is demonstrated in the level energy systematics of the even-mass mercury isotopes that show two structures at low energies, one built on top of the ground state, interpreted as weakly oblate-deformed, and the other, built on top of the intruder 0 + 2 state, assumed to be prolate-deformed [1,4]. The excitation energies of the latter have a parabolic behavior as a function of neutron number, with the minimum at N = 102 in 182 Hg.
The presence of two coexisting bands is confirmed by other complementary experiments in this region. Lifetime measurements of the yrast-band members up to the 8 + state in even-mass 180−188 Hg isotopes have shown large E2 transition strengths, while they drop for the 2 + 1 state [5][6][7][8]. This behavior indicates a similar configuration of high-spin states and a mixing of two configurations in the 2 + 1 level. One should also note the decrease of the 4 + 1 → 2 + 1 transition strength from 180 Hg, where it is similar to the values between higher-spin members of the yrast cascade [5], to 188 Hg, where it is much closer to the B(E2; 2 + 1 → 0 + 1 ) [8]. This effect is interpreted as an evolution of the 4 + 1 state structure from prolate-to oblate-deformed shape [5].
The 186,188,190 Pb α-decay fine structure measurements reveal large hindrance factors for decays to the 0 + 2 states in 182,184,186 Hg, which is interpreted as an indication of a weak mixing between the 0 + states [9][10][11]. On the other hand, internal conversion coefficient (ICC) measurements between the first and the second 2 + states point to the existence of a large E0 component [12][13][14][15][16], which is interpreted as a fingerprint of mixing [1,17]. A Coulomb excitation (Coulex) study at ISOLDE [18,19] provided the monopole strengths between the lowest 2 + states and it confirmed strong mixing between these states.
While the existing experimental information points to a good qualitative description of shape coexistence in mercury isotopes, quantitative information is still lacking. Currently, the insufficient precision of the spectroscopic information, with uncertainties of the γ-branching ratios and the ICCs being as large as 30% [13], hinders * marek.m.stryjczyk@jyu.fi, he/him † These authors contributed equally the interpretation of the Coulex results [20]. Information on mixing of the 4 + and higher-spin states is also lacking. Different theoretical approaches have been tested in the region and while they are able to reproduce some of the observables, they point to contradicting conclusions, for instance regarding the intrinsic deformation of the 186 Hg ground state [3,21]. In order to increase the available amount of spectroscopic information and its precision, excited states in 182,184,186 Hg have been studied by means of the β decay of 182,184,186 Tl at the ISOLDE facility at CERN. The existence of isomers in the thallium isotopes with spin and parity 2 − , 4 − , 7 + and 10 − [22,23] enabled the population of excited states in the 182,184,186 Hg isotopes up to spin 12 while the simultaneous detection of γ rays and electrons allowed us to measure ICCs and, consequently, to identify transitions with E0 components.
The paper is organized as follows. In Sec. II the experimental setup is described. The analysis methods and information relevant for all three cases are presented at the beginning of Sec. III and the results for 182 Hg, 184 Hg and 186 Hg are provided in Secs. III A, III B and III C, respectively. In Sec. III D a method to extract mixing ratios for all three isotopes is presented together with the results. The discussion and the interpretation of the results as well as the comparison with the theoretical calculations are given in Sec. IV. In Sec. V, conclusions are drawn and an outlook is provided.

II. EXPERIMENTAL SETUP
Pure beams of 182,184,186 Tl were produced at the ISOLDE facility at CERN [24] in spallation of a thick UC x target by 1.4 GeV protons, delivered every 1.2 seconds or a multiple of this value by the Proton Synchrotron Booster (PSB). The produced nuclei diffused from the target material to a hot cavity, where the thallium isotopes were selectively ionized by the Resonance Ionization Laser Ion Source system [25] in a two-step ionization process. The first step excitation was performed via the 6p 2 P 1/2 → 6d 2 D 3/2 transition at 276.83 nm using a dye laser system and for the second step, the Nd:YAG laser at 532 nm was used. The ionized thallium isotopes were extracted from the ion source at 30 keV energy and mass-separated by the High Resolution Separator [24]. The beam was implanted into a movable tape at the center of the ISOLDE Decay Station (IDS) [26]. The tape was moved every 30 to 50 seconds, depending on the structure of the PSB supercycle, in order to remove daughter activities.
To detect the internal conversion electrons (ICE), the SPEDE spectrometer [27] was employed. In its heart there is a 24-fold segmented, 1-mm thick annular silicon detector cooled by circulating ethanol at about −20 o C. The SPEDE spectrometer was placed inside the IDS decay chamber at 16-mm distance in front of the tape, in the upstream direction of the beam. For the detection of β particles, a 0.5-mm thick 900 mm 2 silicon detector was mounted in the downstream direction. The γ radiation was detected by five High-Purity Germanium Clover detectors (HPGe). Four of them were placed in the upstream direction while the fifth one was placed in the downstream direction and it was used only for energy gating. Signals from the detectors were recorded using the Nutaq digital data acquisition system [28] with 100 MHz sampling frequency, running in a triggerless mode.
To calibrate the germanium detectors, an encapsulated 152 Eu source and a 138 Cs sample, produced on-line and implanted onto the tape, were used, while for the SPEDE spectrometer, the ICEs from the strong E2 transitions in 184,186 Hg and 138 Ba were utilized. More details regarding the setup calibration and its performance are reported in Refs. [29,30].

III. RESULTS
The β-decay schemes of 182,184,186 Tl were built using γ-γ, γ-electron and electron-electron coincidence spectra. The coincidence time window between any two signals was 300 ns. In all three measured cases, the beam was a mixture containing two or three β-decaying isomers of thallium in an unknown proportion. As a result, only γ-branching ratios for each excited state were extracted while apparent β feedings and log(ft) values were not determined. Tables with γ-branching ratios, γ-ray intensities normalized to the strongest 2 + 1 → 0 + 1 transition for each isotope and full decay schemes are provided in the Supplemental Material [31]. In the following sections, the information relevant for each isotope is presented.

A. Excited states in 182 Hg
The analysis of the coincidence data allowed us to confirm the decay scheme proposed by Rapisarda et al. in Ref. [13]. The only exception was the 1182-keV transition which was moved from the 2566-keV state to the 1794-keV state. Electron singles energy spectrum and a typical γ-ray energy spectrum with a gate on a γ ray are presented in Figs. 1 and 2. In total, 89 excited states and 193 transitions were identified in 182 Hg. Out of them, there were 57 new excited states and 136 new transitions. Six levels and eight transitions known from in-beam studies [33] were also observed. It should be noted that we observed a systematic shift of around 1 keV between the γ-ray energies reported in our work and Ref. [33]. A similar shift was observed in the previous β-decay study [13]. In addition, nine ICCs have been measured. A summary of the deduced levels with their de-exciting transitions is presented in Tables I and II [31]. The ICCs are given in Table I and a partial decay scheme is shown in Fig. 3.
The electron energy spectrum gated on the 723 and 773 keV γ rays feeding the 2 + 1 351-keV state is presented in Fig. 4. Two peaks are visible at 268 and 252 keV, which can be associated with the K-ICE from the 351-keV 2 + 1 → 0 + 1 transition and the de-excitation of the 335-keV 0 + 2 state, respectively. This observation proves the existence of a 16-keV 2 + 1 → 0 + 2 transition. The intensity ratio of these two peaks can be linked to the γ-ray intensity ratio  [31]. Levels and transitions known from the previous β-decay studies are plotted in black, shifted in the decay scheme in blue, known from other than β-decay studies in green and newly identified in red. Transitions not observed in this work for which the intensity limits have been determined are plotted with dashed lines. Spins, parities and proposed transition multipolarities are taken from this work and Ref. [32]. de-exciting the 351-keV state: where Ω K (335) and Ω tot (335) are the tabulated K and total electronic factors for the 335-keV E0 transitions, respectively, taken from Ref. [35], while α K (351) and α tot ( tions, respectively [34]. The extracted value can be converted into the ratio of the B(E2) transition strengths: Having this ratio and the B(E2; 2 + 1 → 0 + 1 ) = 0.33(2)e 2 b 2 value obtained in the Coulomb excitation studies [19], the absolute value of the matrix element | 0 + 2 E2 2 + 1 | = 2.2(3)eb was extracted. This result is in agreement with the [−2.2, 0.9] range given in Ref. [19] but only for the negative values. It is also in a good agreement with 0 + 2 E2 2 + 1 = −2.48eb from the two-state mixing calculations presented in Ref. [7] (see also Fig. 16 in Ref. [19]).
Although the sign of an individual reduced matrix element has no physical meaning and depends solely on the used convention, the sign of the interference term is an experimental observable. It is a product of three reduced matrix elements and it is important in the determination of the state's triaxiality using the quadrupole sum rule [36][37][38]. The combined analysis of this work and the results from Ref. [19] yields a sign of the 0 interference term to be negative.
The K-, L-and M+ 1 -internal conversion coefficients of the 2 + 2 → 2 + 1 transition were determined from the γ-ray and electron energy spectra gated on the 526-, 576-, 748-  and 1171-keV γ rays (see Figs. 5 and 6). A fit to the L and M+ electrons is presented in Fig. 6. The sum of the extracted ICCs, which is equal to 7.6(7), is in a good agreement with the value of 7.2(13) reported in Ref. [13]. Employing the same gate, the K-ICC of the 548-keV 2 + 2 → 0 + 1 transition and the L-ICC of the 213-keV 2 + 2 → 0 + 2 transition were extracted. Both results are in excellent agreement with the theoretical value for E2 transitions [34].
The K-ICC of the 622-keV transition de-exciting the 973-keV state was obtained by gating on the 701-keV γ ray (see Fig. 7). Its value fixes a positive parity to the 973-keV state. The upper limit for the K-ICC of the 638-keV transition (α K < 0.029), extracted by employing the same gate, is consistent with a pure E2 character (α K (E2) = 0.012) and excludes an M 1 multipolarity (α K (M 1) = 0.040). Therefore, by combining both results, we propose the spin-parity assignment of 2 + for the 973-keV state. The energy gate set on the yrast 332-keV (6 + 1 → 4 + 1 ) transition allowed us to extract α K = 0.030(8) for the 586-keV γ ray. This value suggests a mixed E2/M 1 character, however, an E0 component cannot be excluded without an independent measurement of the δ mixing ratio. Based on this information, the de-excitation pattern and the level energy systematics (see Sec. IV A), we propose spin and parity of (6) + for the 1531-keV state.
The 211-keV transition de-exciting the 1719-keV state was observed only via ICEs (see Fig. 8 and the decay scheme in Supplemental Materials [31]). The lower limit of the K-ICC (α K > 0.9) was extracted from the γ-ray and electron energy spectra gated on the 1156-keV γ ray and it indicates an existence of an E0 component. This implies that both excited states, at 1719 and 1507 keV, have the same spin and parity.
The K-ICC of the 219-keV transition de-exciting the 1985 keV state was extracted by gating on the 576-keV γ ray (see the decay scheme in Supplemental Materials [31]). The value of 0.90 (21) is in 1σ agreement with a pure M 1 transition. By combining this information, the de-excitation of the 1985-keV state to the 4 + and 6 + states and the (5 − ) assignment of the 1766-keV level fed by the 219-keV transition, we propose (5 − ) spin-parity for the 1985-keV state. The 512-keV transition was observed in an electron energy spectrum gated on the 261-keV γ ray (see Fig.  9) and its placement was confirmed by matching energy as well as the presence of the 1218-keV γ ray feeding the 1124-keV level (see the decay scheme in Supplemental Materials [31]) in the γ-ray energy spectrum gated on the 261-keV line. Due to theoverlapping annihilation peak, the direct measurement of γ-ray intensity of the 512-keV transition could not be made. The branching ratio of 9.9(59) was determined by comparing the number of counts of the 1218-keV γ ray registered in coincidence with the 261-and 576-keV transitions. Due to large uncertainties, only the lower limit for the total ICC (> 0.65) of the 512-keV transition was extracted. Nevertheless, this value indicates the existence of a large E0 component in the 512-keV transition which allows us to firmly confirm the 4 + spin of the 1124-keV level.

B. Excited states in 184 Hg
Based on the coincidence analysis, we confirm the decay scheme reported in Ref. [13]. An electron singles energy spectrum and typical γ-γ and γ-electron spectra are presented in Figs. 10, 11 and 12, respectively. In total, 110 excited states and 178 transitions were assigned to 184 Hg. In particular, there were 126 new transitions and 85 new excited states. Four levels and 14 transitions previously observed in the in-beam studies [39] were also observed in this β-decay study. Furthermore, 12 ICCs were measured. The experimental results are summarized in Table II and in Supplemental Material in Tables  III and IV [31] while the partial decay scheme is presented in Fig. 13.
The level at 1872 keV from our study (see the decay  (14) 0.0755(9) 983.5 (1) 534 scheme in Supplemental Material [31]) has a 1 keV lower excitation energy compared to Ref. [39] and has a different de-excitation pattern. Thus, unlike the previous β-decay study [13], we propose that our 1872-keV level and the 1873-keV level from Ref. [39] are two different states.
The 1450-, 2036-, 2093-and 2309-keV γ rays have been placed in the decay scheme based on the energy sum arguments (see Supplemental Material [31]). These γ rays were not included in the determination of the energy of the excited states.
The 367-keV 2 + 1 → 0 + 1 and the 608-keV 2 + 3 → 0 + 2 transitions are in a mutual coincidence (see Fig. 11) indicating the existence of the 9-keV 0 + 2 → 2 + 1 transition. To estimate its total intensity (I t (9)), a similar method as in the case of the 512-keV γ ray in 182 Hg was used. The number of counts in the 608-keV peak in the spectrum  gated on the 367-keV transition N Rg (608) was compared to the number of counts in the same peak in the γ-ray singles energy spectrum N Rs (608). The N Rg (608) value was corrected by the γ-gate detection efficiency γ (367), by the factor 3 4 to include the reduction of γ-detection efficiency in coincidence spectrum due to the fact that one out of four germanium detectors is being used for γ gating, and by the ICC of the gating transition α tot (367), leading to the following:  transition to the 375 keV transition, one can write: Having the ratio R and the mean lifetime of the 0 + 2 state (τ = 0.9(3) ns [41]), we were able to calculate ρ 2 (E0; 0 + 2 → 0 + 1 ) = 4.1(14) × 10 −3 , as well as The latter is in 2σ agreement with 1.3 +0.7 −0.5 e 2 b 2 from the Coulomb excitation studies [19].
The weak 119.2-keV 4 + 1 → 2 + 2 transition is very close to the strong 119.7-keV γ-ray originating from the decay of 184 Ir to 184 Os. The γ-ray intensity of the 119-keV transition N γ (119; Hg) was obtained by subtracting the contribution associated with the osmium line (N γ (119; Os)) from the total number of counts in the peak (N γ (119)). This contribution was calculated by scaling the number of counts in the strongest osmium peak at 264 keV (N γ (264; Os)) by the intensity ratios from Ref. [42]: By comparing the extracted value with the number of counts in the 287-keV 4 + 1 → 2 + 1 transition, an upper limit of the branching ratio for the 119-keV transition equal to 0.6 was obtained. The energy of this transition was calculated as the energy difference between the excited states.
A number of γ lines could be identified as doublet structures. There are two transitions with an energy around 1179 keV. The intensity of the 1445 keV → 367 keV transition was determined from the γ-γ coincidences, while for the 1179 keV → 0 keV transition it was determined as a difference between the intensity from the γ-ray singles energy spectrum and the intensity obtained from the coincidence data. The same method was also applied for pairs of transitions at 765 keV (1854 keV → 1089 keV from coincidence data, 1300 keV → 535 keV as a difference) and at 1082 keV (2495 keV → 1413 keV from coincidences, 1450 keV → 367 keV as a difference). In addition, the energy of the 1450 keV → 367 keV γ ray was determined as the energy difference between the excited states.
Based on the electron-electron coincidences gated on the 375-keV 0 + 2 → 0 + 1 transition (Fig. 14), a state at 1124 keV was identified. The lack of corresponding γ ray (α K > 1.256) indicates a strong E0 component in the 749-keV transition (Table II) and, thus, spin and parity of 0 + are attributed to the state. To determine the 749-keV transition's branching ratio, the number of K-ICE in the electron-electron spectrum was compared to the 589-keV γ-ray transition intensity after correcting them by detection efficiencies as well as a factor Ωtot(749) Ω K (749) = 1.2 to include ICEs from other atomic shells.
The observation of the 0 + 3 → 0 + 1 transition was beyond the observational limit. In addition, there is no known transition feeding the 1124-keV state, thus, an upper limit could not be deduced.
The L-and M+-ICCs for the 168-keV 2 + 2 → 2 + 1 transition were obtained from the γ-ray and electron energy spectra gated on the 367-keV 2 + 1 → 0 + 1 transition, see Fig. 15, whereas the K-ICE energy was below the detection threshold. However, by using the γ-imbalance method proposed in Ref. [13], the total ICC (α tot (168)) was extracted by comparing the number of 367-and 535-keV γ rays (I γ (367) and I γ (535), respectively) in the γray energy spectrum gated on the transitions feeding the where α tot (367) and α tot (159) are the total ICCs of the 367-and 159-keV transitions, respectively, calculated using BrIcc [34], Br γ (168) and Br γ (159) are the γ-branching ratios from this analysis (see Tab. III in Supplemental Material [31]) while Itot (9) Itot(9)+Itot(375) is the intensity ratio of the 9-keV transition, extracted in this work, see Eq. 3. The K-ICC was determined as a difference between the total and the L and M+ ICCs. The value obtained in our work (α tot = 12.8 (24)) is in good agreement with 14.2(36) reported in Ref. [13]. It should be noted that the main source of uncertainty comes from the precision of the Br γ (168) branching ratio.
From the same gate on the 367-keV 2 + 1 → 0 + 1 transition, the K-ICC of the 617-keV 2 + 3 → 2 + 1 transition was determined. The extracted value α K = 0.066(6) indicates the existence of an E0 component and allows us to confirm the spin and parity of 2 + for the 984-keV state proposed in the previous work [13].
The K-ICC of the 644-keV 2 + 4 → 2 + 2 transition was obtained from the spectra gated on the 535-keV γ ray, see Fig. 17. The extracted value α K = 0.100 (14) indicates the existence of an E0 component which allows us to confirm the 2 + assignment of the 1179-keV state proposed in Ref. [13]. From the same gate, the lower limit for the K-ICC of the 449-keV transition was extracted (α K > 0.355) and the result supports the 2 + assignment of the 984-keV level.
The ICEs from the 552-(4 + 2 → 2 + 2 ) and 554-keV ((3) + 1 → 2 + 2 ) transitions create one unresolved peak at 470 keV in the electron energy spectrum, as presented in  previously proposed spin (3) [13]. The K-ICC of the 646-keV transition was determined from the spectra gated on the yrast 287-keV 4 + → 2 + γ ray. Although the obtained value, α K = 0.072 (13), has a relatively large uncertainty, it is more than 2σ larger than the coefficient of a pure M 1 transition (α K (M 1) = 0.0386(6)), which indicates the existence of an E0 component. As a result, we were able to firmly establish the spin and parity of 4 + for the 1300-keV state.
The 214-keV transition de-exciting the 1300-keV state has been observed solely via ICEs (see Fig. 18). The limit for the K-ICC (see Tab. II), which was extracted from the spectra gated on the 720-keV γ rays, implies an E0 transition. This conclusion also confirms our 4 + assignment to the 1300-keV state. The ICC of the 556-keV 6 + 2 → 6 + 1 transition was obtained from the γ-ray and electron energy spectra gated on the 340-keV γ ray and points out to a mixed E2/M 1 multipolarity. However, as in the case of the 586-keV transition in 182 Hg, the existence of an E0 component cannot be excluded without an independent measurement of the δ mixing ratio.

C. Excited states in 186 Hg
Based on the coincidence analysis, we confirmed most of the decay scheme reported in the latest evaluation [43] and substantially extended it. Typical spectra are presented in Figs. 19 and 20 while portions of the γ-ray and electron singles energy spectra are presented in Fig. 7 in Ref. [44]. In total, 102 excited states and 156 transitions were associated with 186 Hg, including 91 new transitions and 68 new levels. Nine states and 17 transitions known from the in-beam studies [43] have been also observed in this β-decay study. The summary of the measured γ rays with the branching ratios is presented in Tables V and VI in Supplemental Material [31] and the extracted ICCs are summarized in Table III. The partial decay scheme is presented in Fig. 21.
Compared to the previous β-decay studies [15,43], three previously unplaced transitions, 413, 726 and 1273 keV, were put in the decay scheme based on the γ-γ coincidence data. It should be noted that the placement of the 413-keV γ ray is in agreement with the in-beam studies [45]. We were not able to confirm the existence of two excited states at 1966 and 2056 keV, which are reported in the evaluation [43]. The former was supposed to de-excite via the emission of a 288-keV γ ray, which has not been observed, while the latter was proposed to decay by emitting a 1248-keV γ ray. In our analysis, this transition is in coincidence only with the Hg x rays and the 511-keV annihilation peak (see Fig. 22). Based on these coincidences and the fact it has the same energy as the new 1248-keV state established in the γ-γ coincidence analysis, we propose it de-excites this level to the ground state.
The 353-keV transition de-exciting the state at 1434 keV was observed only via ICEs, see Fig. 25. The limit for the K-ICC (α K > 1.54) was extracted from the spectra gated on the 675-keV 4 + 2 → 2 + 1 γ ray and it points out to a presence of a strong E0 component. As a result, the previously proposed (3 + ) assignment of the 1434-keV state [46] was changed to 4 + . By employing the same γray energy gate at the 675-keV transition, the K-ICC of the 597-keV 6 + 2 → 4 + 2 transition was extracted and the E2 multipolarity of this transition was confirmed. It should be noted that the spin-parity assignments reported in the ENSDF evaluation [43] for the 1660.0-, 1868.9-, 2138.8-and 2428.4-keV states (see Supplemental Material for the full decay scheme [31]) were based on the same theoretical calculations as for the 1434-keV state [46]. Since the assignment was incorrect for one state, we do not adopt them for other levels. By gating on the 403-keV 4 + 1 → 2 + 1 transition, K-ICC of the 272-(4 + 2 → 4 + 1 ) and 626-keV (4 + 4 → 4 + 1 ) transitions were extracted. In spite of large uncertainty, related mostly to the limited γ-ray statistics, it is firmly established that the 272-keV transition has an E0 component while in the case of the 626-keV line the value indicates a mixed E2/M 1 multipolarity. However, similarly to the 6 + 2 → 6 + 1 transition in 182,184 Hg, the E0 component cannot be excluded without an independent measurement of the δ mixing ratio.
The upper limit for the K-ICC of the 242-keV (8 − 1 ) → 8 + 2 transition de-exciting the (8 − 1 ) K isomer (T 1/2 = 82(5) µs [43]) was obtained from the spectra  gated on the 811-keV γ ray (see Fig. 26). The result allows us to firmly establish an E1 multipolarity and leads to a positive parity assignment for the 1976-keV state. Since this state belongs to the band built on top of the 1229-keV state [47][48][49], we propose a positive parity for all the band members. This result resolves a discrepancy regarding the spin and parity of the 1229-keV state, pointed out in the previous ENSDF evaluation [49], and is in agreement with the 4 + assignment proposed in the most recent evaluation [43].

D. Multipole mixing ratios
The determination of K-, L-and M+-ICCs for the 2 + 2 → 2 + 1 transitions in all three isotopes allowed us to determine the q 2 K (E0/E2) and δ(E2/M 1) mixing ratios [35]. The experimental ICC of the E0 + M 1 + E2 transition from the i atomic shell (i = K, L, ...) can be expressed as [35]: where α i (M 1) and α i (E2) are the calculated ICCs for pure M 1 and E2 transitions, respectively, while δ 2 and q 2 i are the aforementioned mixing ratios. The q 2 i values for different atomic shells i and j are linked with the following relation [35]: where Ω i (E0), Ω j (E0) are the theoretical electronic factors for E0 transitions. By having two or more ICCs, the likelihood function χ 2 can be written as: where s α exp i being the uncertainty of the experimental ICC α exp i . Free parameters were restricted to q 2 K < 1000 and |δ| < 10 by setting priors. The posterior density functions (pdf) were obtained using the Markov Chain  Monte Carlo method [29]. A pdf for 182 Hg is shown in Fig. 27. Values reported in Table IV are the medians and 16th and 84th percentiles of the marginalized pdf or, in cases where only limits are provided, the 5th percentiles.
The extracted δ mixing ratio limits are in line with δ = 1.85 used in Ref. [19] to determine ρ 2 (E0; 2 + 2 → 2 + 1 ) in 182,184 Hg. The q 2 K values from our work and from Ref. [50] are in agreement for 184,186 Hg but not for 182 Hg, where the literature value of q 2 K = 28 +7 −8 is more than 3σ away from our result. This indicate a stronger contribution of the E0 component in the 2 + 2 → 2 + 1 transition.
The extracted mixing ratios, together with the 2 + 2 states lifetimes, can be used to reevaluate the ρ 2 (E0; 2 + 2 → 2 + 1 ) values. However, we note that the known lifetimes are extracted from the Coulomb excitation study [19] and they depend on the spectroscopic input from the previous experiments. The new branching ratios and conversion coefficients from this work will lead to a different set of matrix elements in the Coulex analysis and, as a result, different lifetimes and monopole strengths.

A. Spin and parity assignments
In the previous sections, the spin and parity of a number of states was determined on the basis of the measured ICCs. The analysis of the de-excitation paths allows us to assign spins and parities of several low-lying states. The details are discussed below.
182 Hg, 1507 keV, J π = 3 − ,4 + : this state de-excites solely to the 2 + states and it is fed from the (5 − ) state. Since none of the discussed states exhibits isomeric properties, the only considered transition multipolarities are E1, M 1 and E2. That leads to two possible spins, 3 − and 4 + . This assignment allows us to propose the same spins for the 1719 keV level, as they are connected by a transition with an E0 component.
182 Hg, 1531 keV, J π = (6) + : the ICC between this and the 6 + 1 states indicates an E2/M 1 character and, thus, a positive parity, while the decay to the 4 + and 6 + states and the similar energies of the 6 + 2 states in 180,184 Hg (1504 keV [12] and 1550 keV, respectively) suggest a tentative spin assignment of (6). Since this state was proposed in Ref. [33] to be a bandhead of band 7 (see Fig.  3 of [33]), with levels being connected by E2 transitions, we propose that the states belonging to this band, including the 1942-keV state observed in our work, have spins and parities from (8) + to (16) + .
182 Hg, 1985, 2037, 2342, 2418 and 2448 keV, J π = (5 − ): there are significant differences in the decay pattern of these states in 182 Hg compared to states at similar excitation energy in 184,186 Hg -in the latter nuclei the excited states de-excite by emission of no more than four different γ rays while in the former, five or more de-excitation paths exist. All these states feed the 4 + and 6 + states and do not feed the 2 + and 8 + states, which indicates spin 5. In addition, in the β-decay of 180 Tl(4 − ) to 180 Hg [12], similar states at 1797 and 2348 keV were observed and both of them had low log(ft) values, which suggests an allowed decay and, consequently, a negative parity. The measurement of the magnetic dipole moments by means of laser spectroscopy suggested a similarity in the structure of 180 Tl(4 − ) ground state and the low-spin 182 Tl(4 − ) isomer [23]. Although in our study we cannot extract log(ft) values, based on the presented arguments we tentatively propose spin (5 − ) to the 1985, 2037, 2342, 2418 and 2448 keV levels.

B. Comparison with the theoretical models
The experimental results were compared to calculations from two theoretical models available in the literature: Interacting Boson Model with Configuration Mixing (IBM-CM) which employs the D1M parametrization of the Gogny energy density functional (IBM Gogny) [51] and the Beyond Mean-Field based model (BMF) which uses the SLy6 parametrization of the Skyrme interaction [52]. Furthermore, additional calculations have been performed within the IBM-CM approach with the phenomenological parametrization (IBM Phen) [53], the General Bohr Hamiltonian (GBH) method [19,38,54,55] as well as the symmetry-conserving configuration mixing (SCCM) model [8,56,57].
The first information regarding the structure of 182,184,186 Hg can be retrieved by analyzing the potential energy surfaces (PES) as a function of deformation. In case of SCCM, the curve obtained with the particlenumber variation after projection (PN-VAP) method [58] points to a complex structure, with a global oblate minimum at β 2 ≈ −0.15, two normal-deformed (ND) prolate minima at β 2 ≈ 0.1 and 0.25 and one super-deformed (SD) prolate minimum at β 2 ≈ 0.6 ( Fig. 28). Furthermore, there is one additional minimum in 184 Hg at β 2 ≈ 0.45 and in 182 Hg at β 2 ≈ −0.35. A projection of PN-VAP wave functions onto angular momentum creates a particle-number and angular momentum projection (PNAMP) set whose structure remains rather unchanged for J = 0, with the global ND oblate minimum at β 2 ≈ −0.17 and a prolate minimum at β 2 ≈ 0.3 at almost identical energy. One exception is an appearance of a shallow ND oblate minimum at β 2 ≈ −0.35 in both 184 Hg and 186 Hg. These results are consistent with the recent laser spectroscopy study which determined the ground state |β 2 | value to be about 0.2 [2,3].
Comparisons of the experimental energies of excited states with the theoretical predictions are presented in Fig. 29. The best agreement is obtained with IBM Phen but it should be kept in mind that this model was fitted to the experimental data. The only significant discrepancy can be observed for the energy of the 2 + 4 state in 184 Hg. At the same time, the IBM Gogny calculations reproduce rather poorly the excitation energies with the exception of 186 Hg. It might be related to the fact that for 182,184 Hg these calculations predict strongly deformed ground-state bands and weakly deformed bands built on top of the 0 + 2 states [51] which contradicts the experimental findings [2,3]. On the other hand, for 186 Hg the ground-state band is predicted to be weakly oblate-deformed [51].
The results from GBH, SCCM and BMF show that the energy differences between the calculated states belonging to the same band are systematically larger than the experimental values, but this is a known deficiency of these calculations [19]. A very poor reproduction of the third 0 + and 2 + states in SCCM and BMF might be related to the restriction to only axial deformations.
The relative B(E2) values were derived from the mea-   Fig. 3 for the decay scheme). Symbol "-" indicates that the particular ratio was not calculated in a given model.   Fig. 13 for the decay scheme). Symbol "-" indicates that the particular ratio was not calculated in a given model.  In particular, the B(E2;6 + 2 →4 + 2 ) B(E2;6 + 2 →4 + 1 ) ratio is overestimated in all three nuclei and the largest discrepancy, of an order of magnitude, is observed in 184 Hg. In addition, the B(E2;8 + 2 →6 + 2 ) B(E2;8 + 2 →6 + 1 ) ratio in 186 Hg is overestimated by three orders of magnitude. This discrepancy is related to a very small B(E2; 8 + 2 → 6 + 1 ) value predicted by the model. The reproduction of the B(E2) ratios by IBM Gogny, GBH, SCCM and BMF is in general poor. For many values, the theoretical models do not reproduce the order of magnitude of the observable. However, a comparison of the known experimental B(E2) values with the theory (see Table 8 in Ref. [19] and Table VII in Supplemental Material [31]) indicates that while the intra-band transitions are reproduced rather well, the main issue is the correct predictions of the inter-band transition strength, which can differ up to two orders of magnitude. A similar pattern can be observed in 188 Hg [8].
To further understand the poor reproduction of the B(E2) ratios, the SCCM model Collective Wave Func-    tions (CWF) (see Fig. 30) can be analyzed. The CWF which are the weights of the intrinsic deformations in each calculated state reveal that in all three nuclei, each band has a rather constant deformation parameter. They also show that the overlap between the oblateand prolate-deformed states is very small which can be linked to a small mixing between states exhibiting different deformation. As a result, the predicted inter-band B(E2) values are too low. It should be noted that the exploratory studies of the SCCM model performed for 188 Hg indicated that this behavior might be related to an absence of triaxial degrees of freedom [8].
Underestimation of the inter-band transition strength by the IBM Gogny calculations was linked to the energy difference between the prolate and oblate minima on the Potential Energy Surfaces [51]. For 182,184 Hg this difference is large, therefore despite the availability of the triaxial degrees of freedom, the mixing between two configurations is hindered. At the same time, for 186 Hg the mixing strength was determined to be too strong for the low-lying states which might explain a systematic overestimation of the measured B(E2) ratios.
The monopole strength ρ 2 (E0) is directly proportional to the changes in the mean-square charge radii [17] and, consequently, carries important information to assess shape changes. In the case of 184 Hg, we were able to reevaluate ρ 2 (E0; 0 + 2 → 0 + 1 ) × 10 3 = 4.1 (14) by combining the intensity ratio with the known lifetime. For the 0 + 2 → 0 + 1 transitions in 182,186 Hg and the 2 + 2 → 2 + 1 transitions in all three isotopes the monopole strength is known from the literature [50]. The comparison between the experimental values and the theoretical models is presented in Table VIII.
Unlike the case of the B(E2) ratios and the excitation energies, the IBM Phen predictions for the monopole strenght differ by up to one order of magnitude from the experimental data. The IBM Gogny calculations predict correctly the monopole strength between the 0 + states in 184 Hg, however, for other analyzed cases it is underestimate by up to two orders of magnitude.
The results of GBH and BMF calculations are of the same order of magnitude. However, the monopole strength is overestimated between the 0 + states and underestimated between the 2 + states. One explanation of this effect might be an incorrect estimation of mixing between the low-spin states, as suggested in Ref. [52]. The SCCM calculations are able to correctly reproduce the monopole strength in 184 Hg but the predictions for 182,186 Hg are too low compared to the experimental values.
It should be noted that in all discussed cases the large relative uncertainties hinder more quantitative assessment of different theoretical approaches. In addition, we bring attention to the fact that the experimental monopole strengths between the 0 + states in 182,186 Hg might be incorrect. In case of 182 Hg, ρ 2 (E0; 0 + 2 → 0 + 1 ) was extracted in a model-dependent way. The same approach applied to 184 Hg leads to the two-orders-ofmagnitude higher value than the experimental result [19]. For 186 Hg the method used to extract the lifetime of the 0 + 2 state [59] suffers from unaccounted systematic effects. As shown in Ref. [60] and discussed in details in Sec. VD therein, the same method applied to the lifetime extraction of the 2 + 2 state in 188 Hg resulted in one-order-ofmagnitude difference compared to the fast-timing experiment [60].

V. CONCLUSIONS AND OUTLOOK
A spectroscopic study of 182,184,186 Hg has been performed at the ISOLDE Decay Station at the ISOLDE facility at CERN. The excited states were populated in the β decay of 182,184,186 Tl isotopes produced in the spallation of a UC x target. The collected data allowed us to confirm the existing decay schemes and to add to them a large number of new transitions and excited states. Internal conversion coefficients were measured for 23 transitions, out of which 12 had an E0 component. In 182 Hg, a B(E2) ratio from our study combined with the results from the Coulomb excitation study allowed us to extract the sign of one interference term and to extend the systematic comparison of matrix elements with the two-state mixing model. By using electron-electron coincidences, a 0 + 3 state was identified in 184 Hg. The experimental results were compared with theoretical calculations. All models described qualitatively the structure of the analyzed nuclei and pointed to the coexistence of oblate-and prolate-deformed structures. However, the quantitative description is still lacking as none of the discussed approaches was able to predict correctly all the observables. A relatively good reproduction of the data was obtained in the phenomenological Interacting Boson Model with Configuration Mixing and the microscopical symmetry-conserving configuration mixing model. In particular, the latter was able to correctly reproduce the order of magnitude of the monopole strengths in 184 Hg.
The results presented in this work provide an important complementary spectroscopic input for future Coulomb excitation experiments [20,61]. They also indicate that the future experiments should focus on life-time measurements, in particular for the low-lying yrare states, and the angular correlation to better characterize E0 transitions and, consequently, shape coexistence in these nuclei.