Reassigning the shapes of the 0+ states in the 186Pb nucleus

Across the physics disciplines, the 186Pb nucleus is the only known system, where the two first excited states, together with the ground state, form a triplet of zero-spin states assigned with prolate, oblate and spherical shapes. Here we report on a precision measurement where the properties of collective transitions in 186Pb were determined in a simultaneous in-beam γ-ray and electron spectroscopy experiment employing the recoil-decay tagging technique. The feeding of the 02+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${0}_{2}^{+}$$\end{document} state and the interband 22+→21+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${2}_{2}^{+}\to {2}_{1}^{+}$$\end{document} transition have been observed. We also present direct measurement of the energies of the electric monopole transitions from the excited 0+ states to the 0+ ground state. In contrast to the earlier understanding, the obtained reduced transition probability B(E2;21+→02+)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B(E2;{2}_{1}^{+}\to {0}_{2}^{+})$$\end{document} value of 190(80) W.u., the transitional quadrupole moment ∣Qt(21+→02+)∣=7.7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$| {Q}_{t}({2}_{1}^{+}\to {0}_{2}^{+})| =7.7$$\end{document}(33) eb and intensity balance arguments provide evidence to reassign the 02+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${0}_{2}^{+}$$\end{document} and 03+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${0}_{3}^{+}$$\end{document} states with predominantly prolate and oblate shape, respectively. Our work demonstrates a step-up in experimental sensitivity and paves the way for systematic studies of electric monopole transitions in this region. These electric monopole transitions probe the nuclear volume in a unique manner and provide unexploited input for development of the next-generation energy density functional models. The authors study an interesting phenomena of shape coexistence in 186Pb. In an elegant and well-documented experiment, they confirm the coexistence of the three 0+ states in the 186Pb nucleus and reassign the shapes associated with the excited 0+ states.

T he governing interactions in atomic nuclei give rise to diverse quantum phenomena such as single-particle motion, nucleon pairing and collectivity, and can result in a configuration that prefers a specific shape. At one extreme, competing shape-driving configurations can appear within a small energy range, a phenomenon commonly known as shape coexistence.
One of the richest regions of shape coexistence is formed by very neutron-deficient nuclei with proton number Z close to the magic number 82 and neutron number N close to 104 (midshell) 1,2 . Shape coexistence in this region was first established in the early 1970s in laser spectroscopy experiments that discovered a sudden change in the nuclear charge distribution of neutrondeficient Hg isotopes 3 . Over the years, a whole arsenal of spectroscopic techniques has been employed to study the competing structures in this region. For example, rotational bands have been investigated via in-beam γ-ray spectroscopy 4 , β-decay and αdecay fine structure measurements have probed the level energies of the band-head 0 + states [5][6][7] , whereas lifetime and Coulomb excitation experiments have shed light on the collectivity of transitions connecting the low-spin states [8][9][10][11] . The lifetimes of excited 0 + states have been measured for 190,192,194 Pb nuclei 12 .
The discovery of two low-lying excited 0 + states in the N = 104 neutron mid-shell nucleus 186 Pb in an α-decay fine-structure experiment established a feature in the atomic nucleus that remains unique across different physics domains 5 . While in its ground state the 186 Pb nucleus is spherical 13 , its first and second excited states were associated with predominantly π(2p − 2h) and π(4p − 4h) excitations across the Z = 82 shell gap and assigned with deformed oblate and prolate shape, respectively 5 . This triplet of 0 + states, which lie within an excitation energy range of 700 keV, represents a system where the perturbation of the ground state by less than half a per mille of its binding energy results in configuration changes that are associated with different shapes. However, despite all efforts, the linking transitions from rotational bands to the deformed band-head 0 + states have remained unobserved. Results from previous in-beam spectroscopy experiments on 186 Pb have been reported in refs. [14][15][16][17][18] . In contrast to ref. 5 , many theoretical works [18][19][20][21][22][23][24][25][26][27][28][29] have proposed opposite shape assignments for the 0 + states. The band-head 0 + states are the lowest excited states, which due to angular momentum conservation, can only de-excite via electric monopole (E0) transitions. Therefore, in-beam observation of conversion electrons is required.
In the present work, results from an in-beam spectroscopic study of 186 Pb employing the SAGE spectrometer 30 are presented. SAGE is a state-of-the-art instrument that builds on years of endeavour put into research and development of γ-ray 31 and solenoidal electron 32,33 spectrometers. It allowed for the needed gain in sensitivity to probe the transitions between the low-spin states in 186 Pb and provided the first rigorous experimental evidence that supports the assignment of the 0 þ 2 state with predominantly prolate shape. The present work responds to an increasing demand, also recalled in the recent review article 34 , to advance our understanding on electric monopole transitions.

Results
Recoil-gated, α-tagged (RDT from now on) background subtracted prompt γ-ray and conversion-electron energy spectra are shown in Figs. 1 and 2, respectively. The most prominent γ-ray peaks have been labelled according to their transition energies. The region covering the non-yrast band, which is associated with an oblate shape, is expanded in the inset of Fig. 1. In the electronenergy spectrum in Fig. 2, the most prominent conversionelectron lines with corresponding K-, L-and M-components have been marked and labelled accordingly. The low-energy region below 150 keV is dominated by the δ-electron background.
Deconvolution of the electron energy spectrum has been performed and overlaid in Fig. 2. The blue line illustrates the total fit made for the electron energy spectrum. The dotted lines correspond to the calculated K-, L, and M-electron components of the most prominent E2 and M1 transitions, also including contaminant events. After subtracting the calculated components from the total fit, there remains an excess of electron counts, indicated by the red line in Fig. 2. This excess will be discussed below.
The high-statistics RDT γ − γ and γ − e − coincidence data allowed new transitions in 186 Pb to be established. The properties of the new transitions and transitions relevant to the present work are listed in Table 1. Based on coincidence relations and intensity balances, the new transitions have been placed in the level scheme as shown in Fig. 3. It is interesting to note that, despite the same target and beam combination as in refs. 17,18 , the slightly higher beam energy of the present work resulted in much higher relative feeding of the non-yrast states in 186 Pb.
De-excitation path passing through the 0 þ 2 state. An electron peak at 447 keV stands out in the RDT electron-energy spectrum, see Fig. 2. The peak cannot be associated with any previously known γ-ray transition in 186 Pb, suggesting the transition has a strong E0 component. If this is considered to be a K-conversion electron component, the corresponding transition energy would be 535 keV and the L-conversion electron component would have an energy of 520 keV. Indeed, an electron peak at that energy is observed in the RDT electron energy spectrum in Fig. 2. These are in a perfect agreement with the 0 þ 2 ! 0 þ 1 transition energy extracted in the α-decay of 190 Po 5 , confirming the earlier discovery of the 0 þ 2 state and the assignment of electron peaks in Fig. 2 with a transition energy of 535 keV.
The de-excitation path passing through the 0 þ 2 state at 535 keV is investigated in Fig. 4. An RDT γ-ray energy spectrum in coincidence with the 447 keV prompt electrons is presented in Fig. 4a. In addition to the known yrast-band transitions, a peak at 125.0(24) keV is observed. The energy matches with the level energy difference of 127(2) keV between the 2 þ 1 and 0 þ 2 states, thus the transition has been placed in the level scheme accordingly. The groups of counts appearing at around 160 and 350 keV could not be assigned with any transitions in 186 Pb. The total intensity value of I total ð2 þ 1 ! 0 þ 2 Þ ¼ 31ð10Þ was extracted from the weighted average of the 0 þ 2 ! 0 þ 1 K-electron events normalised with the K-electron components of the 2 þ 1 ! 0 þ 1 , 6 þ 1 ! 4 þ 1 and 8 þ 1 ! 6 þ 1 transitions observed in coincidence with the 4 þ 1 ! 2 þ 1 γ-ray transition. The corresponding spectrum is shown in Fig. 4b, where the presence of the K-conversion electrons of the 535 keV transition and peaks used for normalisation are evident. The spectrum also includes events at~113 keV that could correspond to the L-conversion electrons originating from the 2 þ 1 ! 0 þ 2 transition as expected from the high L-conversion coefficient of α L ≈ 1.4. However, they overlap with the δ-electron background preventing precise extraction of the energy and intensity. The γray intensity value of I γ ð2 þ 1 ! 0 þ 2 Þ ¼ 9ð3Þ was obtained using the   Spins and parities of the initial and final states (J π i and J π f ) are given in the first column. Transition energies (E J π i !J π f ), γ-ray intensities (I γ ) and K-and L-conversion electron energies (E K , E L ) and intensities (IK and IL) are listed in the other columns. Unless otherwise noted, the intensities are extracted from the recoil-gated, α-tagged γ-ray or electron singles energy spectra. Energies are given in the units of keV and intensities are normalized to the γ-ray intensity of the 2 þ 1 ! 0 þ 1 transition. a Extracted from level energy difference. b From recoil-gated, α-tagged 0 þ 2 ! 0 þ 1 K electrons with a gate on the 4 þ 1 ! 2 þ 1 γ-ray transition. c Value and uncertainty extracted from the deconvolution of the corresponding electron peak. d From recoil-gated, α-tagged γ rays in coincidence with the 2 þ 1 ! 0 þ 1 γ-ray transition. e Weighted average, see text for more details. f From γ-γ coincidence data. g From recoil-gated, α-tagged γ-ray energy spectrum with a gate on 424 keV γ rays.
value α total (E2) = 2.35(4) 35 . Such a small intensity is beyond the observational limit in the RDT γ-ray singles events.
Observation of the E0 transition de-exciting the 0 þ 3 state. A group of events is found in the RDT electron-energy spectrum around 574 keV (Fig. 2) where the K-conversion electron component of the E2ð2 þ 1 ! 0 þ 1 Þ transition is expected. Based on the observed γ-ray intensity and the calculated K-conversion coefficient 35 , the number of corresponding K-conversion events in the peak has been determined. As a result, the intensity of the peak at~574 keV cannot be solely explained by the 2 þ 1 ! 0 þ 1 transition or by other observed γ-ray transitions. It is also noteworthy, that the full width at half maximum resolution of the peak is 17 keV, which is larger than the expected resolution of 14 keV. Thus, it is considered the peak comprises two components, one that is assigned with the 2 þ 1 ! 0 þ 1 transition and the other to the excess events. The composition of the peak with the corresponding fits is visualised in Fig. 5. The centroid of the excess events is at 571 keV. If this was an L-conversion electron component of a transition, the corresponding more intense K-component should be observed at 498 keV, which is clearly not the case. The non-observation of a corresponding γray transition suggests a strong E0 component. The suggested energy of 659(4) keV is in excellent agreement with the 0 þ 3 ! 0 þ 1 transition [650 (20) keV] first observed in the α-decay of 190 Po 5 . Consequently, the 571 keV electron peak is assigned as the K-conversion electron component of the 0 (7). Unfortunately, transitions feeding the 0 þ 3 state are beyond the observational limit.
Discovery of the interband 2 þ 2 ! 2 þ 1 transition. The interband 2 þ 2 ! 2 þ 1 transition has not been observed previously. The missing γ-ray intensity analysis based on γ-γ coincidences, similar to that performed for 188 Pb which revealed considerable E0 components in the interband J → J transitions 36 , did not indicate any strong E0 components in 186 Pb 17,18 . However, this could be explained by insufficient γ-γ statistics as the intensity balances listed in Table 1 of ref. 18 do not rule out an additional transition feeding the 2 þ 1 state.    Further evidence for the highly-converted 2 þ 2 ! 2 þ 1 transition is presented in Fig. 6, where RDT γ-ray (a) and electron (b) energy spectra in coincidence with 662 keV γ rays stemming for the 2 þ 1 ! 0 þ 1 transition are shown. The γ-ray energy spectrum in Fig. 6a presents peaks associated with yrast-band transitions, whereas the γ-ray transition at 283 keV (see the inset) is absent. The upper limit for the γ-ray intensity of the 2 þ 2 ! 2 þ 1 transition was estimated to be I γ < 20. It is noteworthy, that apart from the 8 þ 2 ! 6 þ 2 transition at 424 keV with I γ = 61 (18), other non-yrast transitions are below the observational limit in Fig. 6a. In Fig. 6b, in addition to the two preceding yrast-band transitions, an electron peak emerges at 196(2) keV corresponding to the energy of the K-conversion electron component of the 283 keV 2 þ 2 ! 2 þ 1 transition. The intensity of the K-conversion electron component Table 1 is the weighted average of intensities extracted from RDT singles electrons (Fig. 2) and from RDT electrons with a gate on the 2 þ 1 ! 0 þ 1 γ-ray transition (Fig. 6). Based on obtained intensities, the lower limit of α K > 0.5 for the conversion coefficient can be extracted.
Electron-conversion component of the 4 þ 2 ! 4 þ 1 transition. The γ-e − coincidence statistics were not sufficient to obtain information about the conversion-electron component of the interband 4 þ 2 ! 4 þ 1 transition by gating on the feeding transition. Since the 4 þ 2 ! 4 þ 1 transition energy overlaps with the 8 þ 1 ! 6 þ 1 transition energy, gating with a transition below the 4 þ 1 state included both transitions and resulted in too few statistics for deconvolution analysis. However, RDT γ-ray and electron singles data were sufficient for the deconvolution of the electron energy spectrum based on the extracted γ-ray intensities. This is illustrated in Fig. 7, where a peak at the corresponding K-conversion electron energy has been divided into its components. The properties of these five components, as identified in Fig. 7, are based on the observed E2 γ-ray transitions. A comparison of the sum of the calculated intensities of the electron components to the measured electron energy spectrum reveals a clear excess of electrons. In order to match with the experimental observations, an additional electron excess component (red line in Fig. 7) with free fit parameters has been included, leading to a measured intensity of I = 12 (8). If the electron excess was of L-electron origin, the corresponding K-electron line at 252 keV should be observed, which is not the case. An α K = 0.32(30) conversion coefficient was extracted for the 4 þ 2 ! 4 þ 1 transition.

Discussion
The α K > 0.5 value obtained for the 2 þ 2 ! 2 þ 1 transition is higher than expected for an M1, E2 or mixed E2/M1 transition (α K,M1 = 0.409, α K,E2 = 0.075), suggesting the presence of an E0 component. This is only possible for a ΔK = 0 transition between states having the same spin and parity. Consequently, the level at 945 keV can be firmly assigned as the J π i ¼ 2 þ 2 state. Using similar arguments for the 4 þ 2 ! 4 þ 1 transition, the tentative assignment of the ð4 þ 2 Þ state at 1337 keV in refs. 17,18 can be now fixed. Based on these assignments and the angular distribution information in refs. 17,18 , levels at 1738 and 2162 keV can be also firmly assigned as J π i ¼ 6 þ 2 and J π i ¼ 8 þ 2 states, respectively. Accordingly, spins and parities of the non-yrast band up to the 8 þ 2 state are fixed. It also follows from the ΔK = 0 criteria that the M1 components for the interband transitions are ruled out.
The E0 transition probability is defined as where ρ 2 is the transition monopole strength and Ω(E0) is the electronic factor 37,38 . For the monopole strength, the following relation can be extracted: where I K (E0) is the K-conversion electron intensity of the E0 transition, I γ (E2) and W γ (E2) are the γ-ray intensity and transition rate of the competing E2 transition, respectively, and Ω K (E0) is the electronic factor for the K-electron conversion of the E0 transition.
Although the information on the transition rates is lacking, it is intriguing to use estimates to evaluate the monopole strengths of the interband transitions. A BðE2; 2 þ 1 ! 0 þ 1 Þ ¼ 6 W.u. has been extracted from the lifetime of the 2 þ 1 state 8,9 , while BðE2; 2 þ 2 ! 0 þ 1 Þ ¼ 3:9 W.u. has been measured for 184 Hg in a Coulomb excitation experiment 10 . In both cases, transitions are between states of different intrinsic configurations. Assuming BðE2; 2 þ 2 ! 0 þ 1 Þ ¼ 5 W.u., a ρ 2 monopole strength value of 100(60) × 10 −3 can be extracted for the E0 component of the 2 þ 2 ! 2 þ 1 transition. Similarly, ρ 2 = 40(30) × 10 −3 can be obtained from the present data for the 4 þ 2 ! 4 þ 1 transition assuming BðE2; 4 þ 2 ! 2 þ 2 Þ ¼ 100 W.u., a typical value for an oblate band 8,9 . A large monopole strength is a fingerprint of mixing between two coexisting states with different deformation 39 . Since the spherical J π states are expected to lie well above the energy of the prolate and oblate J π states, they can be omitted in the following mixing calculations. According to the two-level mixing model, the monopole strength can be expressed as: where Z is the atomic number, a and b are the mixing amplitudes with the relation a 2 + b 2 = 1 and β 2,i is the quadrupole deformation parameter of the J þ i state. Quadrupole deformation values of |β 2 | = 0.29(5) and |β 2 | = 0.17 (3) have been extracted for the prolate and oblate bands, respectively, from lifetime measurements performed in this region 8,9 . Adopting these values, the amount of mixing needed to produce the monopole strength estimated for the 2 þ 2 ! 2 þ 1 transition is of the order of 10% (a 2 = 0.9), while it is only~4% (a 2 = 0.96) for the 4 þ 2 ! 4 þ 1 transition. It is noteworthy, that the presence of a possible M1 component in the interband transitions would reduce the value obtained for the monopole strength, further decreasing the amount of mixing. Accordingly, small prolate-oblate mixing can be expected for the yrast 2 þ 1 and 4 þ 1 states, which is in line with the results obtained in the lifetime measurements 8,9 . Calculations performed in the IBM framework predict more pronounced mixing 40 .
The intra-band transition intensities of the non-yrast band (oblate) were extracted to be around 10% of the corresponding yrast-band (prolate) transitions and with decreasing spin the majority of the de-excitation of the non-yrast states ends up to the yrast band. The feeding of the excited 0 + states is weak as the feeding transitions have to compete against higher energy transitions to the 0 + ground state. In the present work, the first and second excited 0 + states were fed at the level of~2.2% and~1.1% of the total intensity feeding the ground state, respectively.
The only observed feeding of the 0 þ 2 state is via the 2 þ 1 ! 0 þ values under the assumption of a rotating quadrupole-deformed nucleus are 7.7(33) eb and 9.0(5) eb, respectively, suggesting that the 0 þ 2 state has a predominant prolate component. The de-excitation of the 0 þ 2 state with I total ð0 þ 2 ! 0 þ 1 Þ ¼ 26ð6Þ extracted from the corresponding K-electron peak in the singles RDT electron energy spectrum of Fig. 2 is fully covered by the feeding transition with the intensity of I total ð2 þ 1 ! 0 þ 2 Þ ¼ 31ð10Þ extracted from the electron energy spectrum of Fig. 4 gated with 261 keV γ rays. Nevertheless, it is interesting to estimate the potential side-feeding within the given margins. As discussed earlier, the 2 þ 2 state can be considered almost pure oblate state. Provided the 0 þ 2 state were predominantly oblate, the BðE2; 2 þ 2 ! 0 þ 2 Þ value could be estimated to be~50 W.u. Assuming BðE2; 2 þ 2 ! 0 þ 1 Þ ¼ 5 W.u., an intensity value of I total ð2 þ 2 ! 0 þ 2 Þ ¼ 19ð2Þ can be obtained. Such a large sidefeeding does not fit within the given margins, suggesting the 0 þ 2 state does not have a notable oblate admixture.
The feeding of the 0 þ 3 state was beyond the observational limit. Taking BðE2; 2 þ 2 ! 0 þ 1 Þ ¼ 5 W.u. as above and assuming the feeding of the 0 þ 3 state is solely from a predominantly oblate 2 þ 2 state, a reduced transition probability of can be obtained. This shows that the 2 þ 2 ! 0 þ 3 transition can be a collective transition within a band.
To conclude, the B(E2) value shows that the 2 þ 1 ! 0 þ 2 transition is a collective transition, whereas the transition quadrupole moment, albeit with large uncertainty, suggests a transition within the prolate band rather than a transition between different configurations. The intensity balances between the low-spin states imply small mixing between the excited 0 + states. In particular, the oblate admixture in the 0 þ 2 state has been found to be small. Consequently, the 0 þ 2 state is assigned as the band-head of the yrast band and associated with predominantly prolate shape. Accordingly, the 0 þ 3 state is considered to be the band-head of the non-yrast band associated with predominantly oblate shape. In Fig. 8, level energy systematics in the neutron-deficient even-mass Pb nuclei has been plotted including the new assignments of the 0 + states. The parabolic behaviour of the intruder states is now more evident at the minima that take place at N = 104 and N = 106 for prolate and oblate configurations, respectively. This is also well in-line with the very recent study on the odd-mass 187 Pb nucleus with N = 105 which shows that the prolate minimum is below the oblate one in energy 41 .
Many theoretical calculations predict the prolate minimum in 186 Pb to be lower in energy than the oblate one (see refs. [18][19][20][21][22][23][24][25][26][27][28][29], although opposite results have also been obtained 5,42,43 . Moreover, the lowest oblate minimum in energy is predicted at N = 106 by several calculations 20,22,24,[26][27][28][29] 40 . Experimental results obtained on mixing of the 0 + states are only available for the oblate and spherical states in the heavier 190,192,194 Pb isotopes 45 . However, the fact that the assignments of the excited 0 + states presented here are in contrast to what was proposed by Andreyev et al. warrants further discussion. Their arguments were based on (i) the level energy systematics of the oblate 0 + states, (ii) the extrapolation from the high-spin members of the prolate band and (iii) reduced 190 Po α-decay widths 5 . The competition between the intruding oblate and prolate minima in the neutron-deficient Pb isotopes is most striking at the N = 104 mid-shell. In the light of shape staggering between different isotopes 46 and configuration mixing 45 observed in the region, it is questionable whether the level energy systematics can be used as a valid argument for association of 0 + states with specific shape, particularly when direct measurements of shapes, such as the spectroscopic quadrupole moments, are lacking. Secondly, extrapolating the band-head energy from higher spin members of the yrast band is not straightforward in the presence of configuration mixing, particularly when all states involved in mixing are not known. Finally, there is no firm experimental evidence regarding the shape of the ground state in 190 Po and the existing theoretical predictions are conflicting. Beyond mean-field calculations present the ground state as predominantly oblate 29 , whereas particlecore model calculations suggest predominantly prolate shape 47,48 . While the ground state is a mixture of spherical, oblate and prolate configurations with unknown admixtures, reduced α-decay widths cannot be unambiguously used as a tool to assign the configuration of the 0 þ 2 state in 186 Pb. It is noteworthy, that many experimental findings providing supporting arguments for the proposed new assignments, such as lifetime measurements of prolate intruder states in 186 Pb 8,9 and in-beam γ-ray spectroscopy of 186 Pb and 190 Po 17,18,48 nuclei, have been published after ref. 5 .

Conclusions
The N = 104 neutron mid-shell nucleus 186 Pb has been studied in a simultaneous in-beam γ-ray and electron spectroscopy experiment employing the SAGE spectrometer and the recoil-decay tagging technique at the Accelerator Laboratory of the University of Jyväskylä. The nucleus of interest is unique as it possesses a triplet of 0 + states within 700 keV of the ground state.
The existence of the excited 0 + states has been confirmed and their level energies have been determined with higher precision compared to earlier work through direct observation of the 0 þ 2 ! 0 þ 1 and 0 þ 3 ! 0 þ 1 E0 transitions. The observation of the 2 þ 1 ! 0 þ 2 transition allowed the collectivity of the transition feeding the band-head 0 þ 2 state to be assessed. The BðE2; 2 þ 1 ! 0 þ 2 Þ value obtained suggests the 0 þ 2 state has a large prolate admixture. To the best of our knowledge, this marks the first observation of a transition feeding an excited predominantly prolate 0 + state in Pb nuclei.
The two-level mixing calculations performed in the present work indicate that the 2 + and 4 + states are close to pure states with predominant components of~90% and~96%, respectively.
These findings call for a Coulomb excitation experiment to be performed. The SPEDE spectrometer 49 in conjunction with the Miniball spectrometer 50 with radioactive ion beams from HIE-ISOLDE 51 could provide information on diagonal and transitional matrix elements in 186 Pb. Together with the data obtained in the present work, the monopole strength for interband transitions could be determined. Moreover, the intrinsic configuration of intruder states in this region could be probed in transferreaction experiments, e.g., at the ISOLDE Solenoidal Spectrometer 52 . The recoil-shadow method 53 has been proposed 54 to measure the lifetimes of the 0 + states and to extract the monopole strength of the E0(0 + → 0 + ) transitions. In addition, simultaneous in-beam γ-ray and electron spectroscopy could shed light on the configuration assignments of the 0 + band head states in the 188 Pb nucleus.
Results obtained in the present work provide new insight into the nucleus at the heart of triple shape coexistence. It forms the basis for a systematic study of electric monopole transitions in the region and beyond with in-beam spectroscopic methods which have only now become possible.

Methods
The experiment was performed at the Accelerator Laboratory of the University of Jyväskylä. The use of SAGE+RITU+GREAT 30,55,56 instrumentation, described below, allowed for collection of simultaneous in-beam conversion-electron and γ-ray data employing the recoil-decay tagging method 57,58 . Nuclei of interest were produced via the 106 Pd( 83 Kr,3n) 186 Pb reaction with a beam energy of 365 MeV. The 83 Kr beam impinged on a 1-mg cm −2 thick target with intensities varying between 4 and 5 particle nA. During 108 h of beam on target, 6.34 × 10 5 α decays were recorded and correlated with 186 Pb nuclei.
Prompt γ rays and conversion electrons were observed with the SAGE spectrometer. SAGE consisted of 10 EUROGAM Phase I and 24 EUROGAM Clover-type Compton-suppressed germanium detectors 59 around the target for the detection of γ rays, and an annular segmented silicon detector placed upstream of the target for detection of conversion electrons. Transportation of conversion electrons was made by means of a solenoid coil that was operated with a current of 800 A. The δelectron flux arising from atomic collisions in the target was suppressed by employing a high-voltage barrier at −35 kV. The full width at half maximum resolution values of 2.9 keV and 4.5 keV for 261 keV and 662 keV γ rays, respectively, and 11 keV and 14 keV for 173 keV and 447 keV conversion electrons, respectively, were determined from RDT γ-ray and electron singles data.
The RITU gas-filled separator was employed to separate fusion evaporation residues (referred to as recoils in the present work) from the primary beam particles. The GREAT spectrometer, deployed at the focal plane of RITU, was used to identify recoils implanted in the double-sided silicon strip detector (DSSD) based on the deposited energy and characteristic α-decay energy. The maximum searching time between recoil implantation and the 186 Pb α decay within the same DSSD pixel was 15 s which is equal to approximately three times the half-life of 186 Pb (t 1/2 = 4.83(3)s) 60 . The shallow implantation depth of recoils allowed for 45% of α particles to escape from the DSSD without full energy deposition. Approximately 19% of the escaped α particles were detected by the PIN diode box of GREAT  measure the energy-loss of the recoils and scattered beam particles in a gas medium and to extract the time-of-flight information between the MWPC and the DSSD. Data were collected using the fully digital Total Data Readout (TDR) acquisition system 61 . TDR operates without a common hardware trigger and events are timestamped with a global 100 MHz clock. The GRAIN software package 62 was used for temporal and spatial correlation of events. The final analysis was completed using the ROOT framework 63 .
Due to the long α-decay searching time, random correlations with 186 Tl, 187 Tl, 184 Hg and 186 Hg nuclei gave rise to contaminant events. The false correlation level was~6% of all recoils. The contaminant nuclei are well known and their contribution was taken into account in the analysis.

Data availability
The data obtained in the present work and the corresponding metadata are available from https://doi.org/10.23729/a6444894-5fe7-4683-a9d5-a8ff1e28208b.