When discussing the nuclear shape, the usual assumption is that the nucleus can be modelled as a liquid drop with the nuclear surface described as an infinite series of spherical harmonics R θ , ϕ = R 0 1 + ∑ l μ α l μ * Y l μ θ , ϕ , (1) where R 0 is the radius of the nuclear surface in the spherical configuration, Y l μ are the spherical harmonics of degree l , order μ , and α l μ are the (complex) time-dependent expansion parameters describing the deformation of the nuclear surface. The most important contribution to the departure from a spherical shape comes from quadrupole deformation, and the above infinite series is often reduced to the l = 2 term R θ , ϕ = R 0 1 + ∑ μ α 2 μ * Y 2 μ θ , ϕ . (2)

Equations 1, 2 describe the nuclear shape with an arbitrary orientation in space and can be transformed into the principal-axis frame using the Wigner rotation matrices, D ν μ J ( α , β , γ ) , via Equation 3 a 2 μ = ∑ ν D ν μ 2 α , β , γ α 2 ν , (3)

with the Euler angles ( α , β , γ ) chosen such that a 2 ± 1 = 0 . The commonly used deformation parameters β 2 and γ , which define the magnitude of the deformation and the deviations away from axiality, respectively, are defined in Equation 4 a 20 = β 2 ⁡ cos ⁡ γ , a 22 = 1 2 β 2 ⁡ sin ⁡ γ , (4)

with the restriction of 0 ≤ β and 0 ≤ γ ≤ π / 3 in order to not have the same quadrupole moments, and hence shapes, defined by a different set of coordinates.

In order to firmly identify shape-coexisting states, some key indicators are required. These indicators have been described in detail elsewhere (see, e.g., Ref. [1]), and those that can be extracted from β -decay studies will only be briefly outlined here.

As discussed above, locating states and determining their decays and spin-parities is a first requirement. Lifetime measurements of excited states provide an extremely important quantity since once determined, together with transition branching ratios, the reduced transition rate or B ( E 2 ) value can be found from Equation 5 B E 2 ↓ ; I i → I f = 9.527 × 1 0 6   B R E γ 5   A 4 / 3   t 1 / 2   1 + α δ 2 E 2 / M 1 1 + δ 2 E 2 / M 1 W . u . , (5) where B R is the total transition branching ratio (i.e., including both the γ -ray and conversion-electron fractions), E γ is given in keV, A is the mass number, t 1 / 2 is the level half-life in s , and α is the total conversion coefficient for the transition. In the case of mixed E 2 / M 1 transitions, the rate must be corrected for the transition mixing ratio δ ( E 2 / M 1 ) . The correction for the conversion coefficient can be neglected if it is very small, and its impact is much less than the experimental uncertainties on the level lifetimes or branching ratios. In this work, the Weisskopf units (W.u.) are used throughout. Care must be used if converting from B ( E 2 ↓ ) to B ( E 2 ↑ ) as the two quantities are related by Equation 6 B E 2 ↑ = 2 I upper + 1 2 I lower + 1 B E 2 ↓ , (6) where the notation on the level spins is obvious.

Once a B ( E 2 ) value has been determined, it can be related to an intrinsic quadrupole moment Q 0 via B E 2 ; I i K → I f K = 5 16 π   Q 0 2   ⟨ I i K 20 | I f K ⟩ 2   e 2 b 2 , (7) where the B ( E 2 ) value must be in units of e 2 b² and a K quantum number (the projection of the angular momentum onto the nuclear symmetry axis) is assumed. Extracting an intrinsic quadrupole moment via Equation 7 builds in an assumption of the nuclear shape as being axially symmetric (and hence K being a good quantum number). It should be noted that the value extracted in this way, often referred to as the rotational limit, gives an over-estimate of the actual quadrupole moment if the system possesses any softness or triaxiality. With Q 0 determined, it can be related to the deformation parameter β 2 via Equation 8 Q 0 = 3 5 π Z R 0 2   β 2 1 + 2 7 5 π β 2 + ⋯   ≈ 3 5 π Z R 0 2   β 2 1 + 0.36   β 2 . (8)

Ideally, it is better to extract the rotationally invariant ⟨ Q 2 ⟩ and ⟨ Q 3 ⁡ cos ⁡ 3 ⁡ δ ⟩ values that can be established using the Kumar–Cline sum rules [2–4]. In the principal-axis frame of the nucleus, the electromagnetic E 2 operator, M ̂ ′ ( E 2 ) , which is a rank-2 SO(3) tensor, has components that can be expressed using two parameters: M ′ E 2 , μ = 0 = Q ⁡ cos ⁡ δ M ′ E 2 , μ = ± 2 = 1 2 Q ⁡ sin ⁡ δ . (9)

The electromagnetic E 2 matrix elements M ( E 2 ) measured in the laboratory frame can be related to those in the principal-axis frame by making use of the invariant property of the electric quadrupole tensor under rotational (SO(3)) transformations. The products of the E 2 operators coupled to zero angular momentum are scalar quantities, and thus their expectation values can be expressed in terms of Q by − 1 2 I i 2 I i + 1 ∑ j 〈 I i ‖ M E 2 ‖ I j 〉 〈 I j ‖ M E 2 ‖ I i 〉 2 2 0 I i I i I j = 1 5 2 I i + 1 ∑ j 〈 I j ‖ M E 2 ‖ I i 〉 2 = 1 5 ⟨ Q 2 ⟩ , (10) where { } is a 6 j symbol. The sum formally extends over all states I j that can be reached from the state in question I i via a single E 2 transition; however, typically, only a few key states contribute to it. Products of three quadrupole tensors coupled to angular momentum zero can also be formed that are used to extract 2 35 ⟨ Q 3 ⁡ cos ⁡ 3 ⁡ δ ⟩ . However, this requires knowledge of not only the magnitudes but also the signs of the matrix elements which cannot be determined by β -decay.

Assuming identical charge and mass distributions, one can relate the Q 2 parameter to the deformation parameter β 2 by [5] 〈 Q 2 〉 = q 0 2 〈 β 2 2 〉 , (11)

with q 0 = 3 4 π Z e R 0 2 and R 0 = 1.2   A 1 / 3 fm. The extraction of the ⟨ Q 2 ⟩ value depends only on squares of matrix elements, and thus any experimental technique that contributes spectroscopic data required to determine the B ( E 2 ; I i → I f ) values can be brought to bear. Once the lifetime of a level is available, the measurement of a γ -ray branching ratio provides the B ( E λ ) value for that transition. Thus, a β -decay measurement can be used to determine B ( E 2 ) values for previously unobserved transitions in this case. Care must be used, however, if the level lifetime is deduced from Coulomb excitation results, as is often the case in the evaluated data files. Here, depending on the details of the reaction and the analysis, the presence of a previously unobserved excitation pathway may modify the lifetime results extracted.

The electric monopole, or E 0 , transition plays a very important role in studies of shape coexistence since the magnitude depends on the difference in the root-mean-square values of the charge radii between the initial and final states. The first application of E 0 transition rates to shape coexistence appears in [6] in their study of shape coexistence in ¹¹⁶Sn. The operator for E 0 transitions is given in Equation 12 [7] M ̂ E 0 = ∑ i e i r i 2 , (12) where the sum extends over the A bodies in the nucleus with their charges e i and radial position r i . E 0 transitions are allowed only for Δ J = 0 transitions and are sensitive to the changes in the nuclear charge-squared radii. The usual quantity quoted when referring to E 0 transitions are the ρ 2 ( E 0 ) values, defined via Equation 13 [7] Γ E 0 = 1 τ E 0 = ρ 2 E 0 ∑ j Ω j Z , Δ E , (13) where Γ ( E 0 ) is the partial width for the decay, τ ( E 0 ) is the partial lifetime, and Ω ( Z , E e ) is the electronic factor that depends on the atomic number Z and the energy of the transition Δ E . The quantity ρ ( E 0 ) is defined by Equation 14 ρ E 0 = 1 e R 2 I f M E 0 I i (14) and carries all nuclear structure information. The expression typically used for the E 0 operator in Equation 12 takes the leading order term only of the operator that can be more generally written as in Equation 15 [8] M ̂ E 0 = ∑ i e i r i 2 − σ r i 4 + ⋯   . (15)

The value for the parameter σ depends on the derivatives of the electron wave functions at the origin and has a slight dependence on the electronic shell of the originating electron and the transition energy, and a stronger dependence on the nuclear charge distributions. In nearly all cases, σ < 0.1 and exceeding this value only in the heaviest nuclei (above Pb) and assuming an 1 / r charge distribution. For the case of constant charge distribution and Z < 100 , σ < 0.1 always. Recently [9], extensive beyond-mean-field calculations using the Gogny forces and the five-dimensional collective Hamiltonian explored the effect of using the expanded E 0 operator adopting σ = 0.1 . Figure 1 shows the results of their calculations. As expected, the addition of the r 4 term in the operator reduces the magnitude of the ρ 2 ( E 0 ) values, although this may be considered an over-estimate of the impact since the actual value of σ for these particular cases is expected to be much smaller.

Within a two-state mixing model, ρ 2 ( E 0 ) can be expressed as ρ 2 E 0 = a 2 b 2 Δ ⟨ r 2 ⟩ 2 Z 2 R 4 , (16) where a 2 and b 2 are the square of the mixing amplitudes of the two states, Δ ⟨ r 2 ⟩ is the difference in the mean-square charge radii, and R = 1.2 A 1 / 3 fm. In the collective variables of the Bohr model, the operator is given by Equation 17 (keeping the lowest-order term in Equation 15) [11] M ̂ E 0 = 3 Z 4 π 4 π 5 + β 2 + 5 5 21 π β 3 ⁡ cos ⁡ γ (17) so that in a two-level mixing solution with deformation parameters ( β 1 , γ 1 ) and ( β 2 , γ 2 ) , the E 0 strength is given by Equation 18 [7, 11] ρ 2 E 0 = 3 Z 4 π 2 a 2 1 − a 2 β 1 2 − β 2 2 + 5 5 21 π β 1 3 ⁡ cos γ 1 − β 2 3 ⁡ cos γ 2 2 . (18)

Although often the E 0 transitions are interpreted as occurring between shape-coexisting shapes, they can arise in any situation where there is a difference in the ( ⟨ r 2 ⟩ ) 2 values between two states. In the case of a spherical vibrator, for example, the nature of the E 0 operator gives rise to a selection rule that E 0 transitions are allowed for changes of the phonon number Δ N = 0 , ± 2 . For 0 + ( N = 2 ) → 0 + ( N = 0 ) , for example [7], ρ 2 E 0 = 2 5 3 4 π Z β rms 2 2 , (19) where β rms 2 in Equation 19 is the mean-square value of the amplitude of the surface vibration.

The vast majority of measured ρ 2 ( E 0 ) values are between J π = 0 + states. For states with J ≠ 0 , E 0 transitions can occur, but their extraction requires a subtraction of the M 1 and E 2 contributions. The intensity of the K -conversion electron line, for example, has three contributions from the E 0 , M 1 , and E 2 components and is given by Equation 20 I K = I K E 0 + I K M 1 + I K E 2 (20)

and dividing by the γ -ray intensity, I γ , yields Equation 21 α K E 0 + M 1 + E 2 = I K E 0 + I K M 1 + I K E 2 I γ (21)

and thus Equation 22 gives the E0 contribution to α_K α K E 0 = α K E 0 + M 1 + E 2 − α K M 1 + δ 2 E 2 / M 1 α K E 2 1 + δ 2 E 2 / M 1 . (22)

An often used quantity is given in Equation 23 q K 2 E 0 / E 2 = I K E 0 I K E 2 (23)

and also using the standard E 2 to M 1 transition mixing ratio given in Equation 24 δ 2 E 2 / M 1 = I γ E 2 I γ M 1 (24)

results in Equation 25 α K E 0 + M 1 + E 2 = α K M 1 + δ 2 E 2 / M 1 1 + q K 2 E 0 / E 2 α K E 2 1 + δ 2 E 2 / M 1 , (25)

with α K ( M 1 ) and α K ( E 2 ) the K -conversion coefficients for M 1 and E 2 multipolarities of the transition at energy E γ , respectively. Although, in principle, the multipole mixing ratio δ ( E 2 / M 1 ) must be known, there are situations where this is not the case. For some combinations of Z and transition energy, the conversion coefficients α K ( E 2 ) and α K ( M 1 ) are nearly identical. Figure 2 demonstrates this for the Z = 50 Sn isotopes; if δ ( E 2 / M 1 ) is known, the accuracy of the extraction of α K ( E 0 ) can be improved, but unless very high statistics are obtained, lack of knowledge of δ may not be a limiting factor.

The region of nuclei surrounding the N = 20 Mg isotope ³²Mg has been the focus of many investigations for nearly five decades. The observation from mass measurements of an increase in the two-neutron separation energies appearing between ²⁹Na and ³¹Na was interpreted with the aid of Hartree–Fock calculations [12] that the Na isotopes appeared to have deformed ground states for ^31,33Na. Later measurements of the isotope shifts by laser spectroscopy indicated deformed ground state structures already setting in at ²⁸Na [13]. The presence of a low-lying 2 1 + state in the neighbouring ³²Mg was first found in β -decay studies of ³²Na [14]. The energy of the 2 1 + state was much lower than would be expected for a closed neutron shell at N = 20 , and given the trends observed in the masses of the Mg isotopes, it had already been speculated that the ground state of ³²Mg was deformed. This region was coined the “island of inversion” (see, e.g., Ref. [15]), i.e., a region where the deformed intruder configuration based on particle-hole excitations lies below the spherical normal configuration. The original island was suggested to be composed of the nine nuclei ^30-32Ne, ^31-33Na, and ^32-34Mg [15]. This region has since been expanded as additional data have been obtained, for example, the spin and magnetic moment of the ³¹Mg ground state are evidence of its intruder configuration [16]. As might be anticipated in a region with an inversion of configurations, shape coexistence should be manifested.

Measurements using the β - γ - γ -timing technique, following ³⁰Na decay by Mach et al. [17], found a lifetime of the 1788-keV 0 2 + state in ³⁰Mg of 3.9 (4) ns. This long lifetime was interpreted as resulting from the hindered nature of the transition from the purported intruder 0 2 + state to the 2 1 + state, where the latter is part of the normal shell model configuration [17]. From the intensity imbalance observed for the 1788-keV state, it was also hypothesized that an E 0 branch existed. This E 0 transition was later observed in a follow-up β -decay experiment with a value 1 0 3 × ρ 2 ( E 0 ) = 26.2 ( 75 ) extracted, which is consistent with rather weak mixing between the two configurations that have a large difference in their deformations [18].

The intruder nature of the ³¹Na ground state results in the preferential population by β -decay of the intruder states in the daughter ³¹Mg. This was taken advantage of in a measurement at the TRIUMF-ISAC facility [19] to extend the level scheme of ³¹Mg and measure the β - γ angular correlations of spin-polarized ³¹Na. Figure 3 displays the γ -ray spectra taken with HPGe detectors placed at 0 ° (left) and 180 ° (right) with respect to the polarisation axis with ³¹Na having a net positive (aligned along 0 ° ) or negative (aligned along 180 ° ) asymmetry. The degree of asymmetry can be extracted and is sensitive to the spin of the levels, and all positive-parity states below the neutron-separation energy (2.3 MeV) were firmly assigned using such data, as shown in Figure 4. A deformed band based on the 1 / 2 + ground state, as well as a band based on the 3 / 2 − 220 keV level, were assigned. The bands were interpreted [19] in terms of the Nilsson model and assigned as the ν 1 / 2 + [ 200 ] and ν 1 / 2 − [ 330 ] orbitals, respectively, which appear at the Fermi surface for β ≈ 0.3 − 0.4 . At slightly higher energy, the 673-keV 3 / 2 + state is assigned as the 0 p − 0 h (referring to excitations of the core) spherical state. These results place ³¹Mg firmly within the island of inversion.

The above method was also applied to study ³⁰Mg [20]. A number of additional levels and transitions were proposed, and angular correlations were performed. Using γ - γ angular correlations, they affirmed the spin 0 + of the 1788-keV level. The β - γ angular correlations provided eight further spin-parity assignments [20]. From their data, they proposed the level scheme, as shown in Figure 5, which includes grouping of levels into various configurations. A key finding was the assignment of the 3 1 + level that would be expected for a K = 2 γ band built on the deformed 0 2 + state. In a Davydov model interpretation, the location of the K = 2 state relative to the 0 2 + level was used to extract γ ≈ 24 ° [20]. Very strong feeding of levels near 5 MeV permitted observation of, or stringent limits to be assigned, for their decays to the ground state and 0 2 + bands. From the decay patterns, they could be assigned as having either a collective (deformed) or spherical characteristic, with the 1 + state considered a candidate for the M 1 “scissors” mode [20]. These results were consistent with a number of other studies (for a summary, see Ref. [1]), in which the ³⁰Mg ground state has a predominately normal configuration.

A study [21] of ³⁴Si following the β -decay of ³⁴Al observed the e − + e + pairs from a state at 2.719 (3) MeV that was assigned as the 0 2 + state. The half-life extracted was 19.3 (7) ns, leading to the 1 0 3 × ρ 2 ( E 0 ) = 13 ( 1 ) . The data leading to these results are presented in Figure 6. The 0 2 + state was interpreted as the deformed intruder state dominated by 2 p − 2 h components across the N = 20 neutron shell [21]. Weak mixing between the ground state and the 0 2 + state was deduced, with the intruder 0 + state having a deformation of β 2 = 0.29 ( 4 ) . A recent study of the decay of ³⁴Mg and ³⁴Al performed at ISOLDE [22] took advantage of the preferential feeding of the two β -decaying states in ³⁴Al. The β -decay of ³⁴Mg led to the population of the 1 + state only of ³⁴Al, which then preferentially fed 0 + and 2 + states in ³⁴Si. Conversely, extracting the beam of ³⁴Al strongly favoured the ³⁴Al 4 − state, resulting in the population of higher-spin states in the ³⁴Si daughter. This work was followed by a precise new measurement of the 2 1 + → 0 1 + / 2 1 + → 0 2 + branching ratio of 1779 (182) [22] vs. the previous result of 1380 (717) [21], resulting in B ( E 2 ; 2 1 + → 0 2 + ) = 7.2 ( 31 ) W.u. strongly favouring its assignment as the 2 + band member of the 0 2 + state. Unfortunately, the 4 + band member has not yet been identified. These data place shape coexistence in ³⁴Si on a firm footing.

Nuclei in the vicinity of N = 40 were first suggested to possess shape coexistence, following the observation in reaction studies [23, 24] of a 0 2 + level at the low excitation energy of 1.77(4) MeV in ⁶⁸Ni. The β -decay of ⁶⁸Co was studied [25], and with the superior energy resolution of HPGe γ -ray detectors and a Ge double-sided strip detector for conversion electrons, the 0 2 + state was determined to be located at an excitation energy of 1605 (3) keV. Figure 7 displays the data used to measure the energy of the 0 2 + state, as well as identifying the E 0 transition to the ground state. The measured E 0 strength of 1 0 3 × ρ 2 ( E 0 ) = 7.6 ( 4 ) was deduced and interpreted as resulting from shape coexistence, with the 0 2 + state as a deformed configuration involving both proton and neutron multiparticle-multihole configurations, and the spherical ground state. The excited states in ⁶⁸Ni have also been investigated in transfer reactions (see, e.g., Ref. [26]) and were used to support an interpretation of a predominantly ν ( g 9 / 2 ) 2 characteristic for the 0 2 + state but could not describe the magnitude of the 2 1 + cross section. An earlier suggestion of the similarity of the situation of neutron states in ⁶⁸Ni to the proton states in ⁹⁰Zr [27] – that the 0 1 + and 0 2 + states arise from the mixing of the ( g 9 / 2 ) 2 and ( p 1 / 2 ) 2 configurations—was qualitatively supported by the shell model calculations but discrepancies remained [26]. In a β -decay experiment [28] performed at the National Superconducting Cyclotron Laboratory (NSCL), the decays of ⁶⁸Fe and ⁷⁰Co were studied. The decay of ⁶⁸Fe populated selectively the low-spin β -decaying state of ⁶⁸Co, which preferentially fed the low-spin states of the ⁶⁸Ni daughter. A key observation was of the 430-keV 2 1 + → 0 2 + , shown in Figure 8, with a measured branching ratio of 0.12 (3)%. With the known lifetime of the 2 1 + state, a B ( E 2 ; 2 1 + → 0 2 + ) = 8.9 ( 28 ) W.u. was determined [28], which was nearly a factor of 3 greater than the B ( E 2 ; 2 1 + → 0 1 + ) value. This led the authors of [28] to conclude, through comparisons with both large-scale shell model and Monte Carlo shell model calculations, that the 0 2 + and 2 1 + states are the members of a shape-coexisting band, supporting some of the earlier interpretations [25]. The lifetime of the 0 3 + state was also measured that yielded a rather small B ( E 2 ; 0 3 + → 2 1 + ) = 2.4 ( 2 ) W.u.

The results from [25, 28, 29] lead to a different conclusion regarding the structure of ⁶⁸Ni than earlier interpretations [23, 24, 26, 27, 30], with the 0 2 + level as either the head of a deformed band or as a mixture of ν ( g 9 / 2 ) 2 + ν ( f p ) 2 configurations upon which seniority-type states can be built. The arguments are outlined in [1], and the conflicting interpretations are shown in Figure 9. Clearly, additional measurements that could elucidate the structure of ⁶⁸Ni are required, including new β -decay studies to confirm the existence of the 430-keV 2 1 + → 0 2 + transition.

The half-life of the 0 3 + level in ⁶⁶Ni was measured following the β -decay of ⁶⁶Co by Olaizola et al. [31]. The result, shown in Figure 10, of t 1 / 2 = 170 ( 7 ) ps is considerably longer than the half-life reported in a plunger measurement by Leoni et al. [32] of 134 (9) ps. However, using either half-life leads to a small B ( E 2 ; 0 2 + → 2 1 + ) value on the order of 0.1 W.u. Both the LSSM calculations and MCSM calculations indicate multiple shape coexistence in ⁶⁶Ni with spherical, prolate, and oblate configurations occurring for the first four 0 + states. The hindered nature of the transition was taken as evidence for substantially different shapes of the states, leading to shape isomerism. Figure 11 displays the energy level systematics for the even–even Ni isotopes, with the states coloured-coded for their presumed shapes.

The region of shape coexistence has been extended steadily in the vicinity of ⁶⁸Ni. Pauwels et al. [33] used the β -decay of ⁶⁷Fe to propose that ⁶⁷Co had a spherical ( 7 / 2 − ) ground state and a deformed ( 1 / 2 − ) first excited state at 492 keV interpreted as a proton 1p-2h intruder configuration with a Nilsson configuration π 1 / 2 − [ 321 ] . A similar interpretation was also suggested for the ( 1 / 2 − ) state at 1,095 keV in ⁶⁵Co combining information from deep-inelastic scattering reactions and β -decay of ⁶⁵Fe [34]. The decay of ⁶⁶Fe to ⁶⁶Co was studied by Liddick et al. [35], and they identified the ⁶⁶Co ground state as a proton intruder, with positive parity based on the strong β -feeding from the ⁶⁶Fe 0 + ground state. The β -decay of the mass 66 chain, ⁶⁶Mn-⁶⁶Fe-⁶⁶Co-⁶⁶Ni, was also investigated by Stryjczyk et al. [36] who employed MCSM calculations to understand the structures of the states in the daughter nuclei. The strong β -feeding from the deformed ⁶⁶Mn ground state to the 1 + state at 2874 keV in ⁶⁶Fe was taken as evidence of the latter’s deformed characteristic. The ⁶⁶Fe ground state, on the other hand, had a rather diffuse shape in the MCSM calculations, and the 1 + ground state of ⁶⁶Co was weakly deformed, with the well-deformed configuration identified as a 1 2 + state observed at 982 keV, in contrast to the interpretation of [35].

Very recently, the N = 40 island of inversion was extended above Fe isotopes [37]. The β -decay of ⁷⁴Cu was studied with the GRIFFIN spectrometer at TRIUMF-ISAC, building on an extensive decay scheme that was developed at the Holifield radioactive beam facility using three clover HPGe detectors [38]. With the GRIFFIN array, which comprised 12 HPGe clover detectors, γ - γ angular correlations were performed that resulted in a significant revision of the ⁷⁴Zn level scheme. As shown in Figure 12, the 2099-keV level which had been favoured to have ( 4 + ) [38], was firmly assigned as 3 + . Furthermore, additional γ -ray transitions were observed that were key to assigning band structures; the 359-keV 2 3 + → 0 2 + transition and the 730-keV 2 3 + → 4 1 + transition unambiguously lead to a 2 + assignment for the level at 2148 keV. With these assignments, a modified level scheme, as shown in Figure 13, was constructed, where the transitions are labelled with their relative B ( E 2 ) values normalized to 1 for the highest energy transition from each level. The large relative values for the in-band transitions are consistent with the assigned structure of a “ K π = 2 + ” band and a K π = 0 + band. The results were interpreted with the aid of large-scale shell model calculations, as well as beyond-mean-field calculations that indicated that the ground state band had, on average, a greater 2 p − 2 h content and a slightly more deformed and triaxial nature than the 0 2 + band [37].

Finally, there have been two different β -decay experiments investigating the structure of ⁸⁰Ge [39, 40]. The first [39] was an experiment performed at the ALTO facility and used the decay of ⁸⁰Ga to study both the γ rays and conversion electrons emitted. A weak peak in the conversion-electron spectrum at 628 keV was observed that was attributed to an E 0 transition in ⁸⁰Ge and placed as feeding the ⁸⁰Ge ground state. This result implied the existence of the 0 2 + state at only 639 keV, a dramatic lowering of the 0 2 + from 1,547 keV in ⁷⁸Ge, and shape coexistence in ⁸⁰Ge [39]. However, in a β -decay experiment performed at the TRIUMF-ISAC facility with the GRIFFIN array, no such transition was observed in spite of the increase in statistics [40]. This null result ruled out shape coexistence occurring at very low excitation energies in ⁸⁰Ge.

It had been suggested for some time that shape coexistence occurs in the N = 60 mass region [41], and the observation [42] in the β -decay of ¹⁰⁰Y of the 0 2 + level in ¹⁰⁰Zr at the low excitation energy of 331 keV supported this. This discovery was quickly followed by a measurement of the 0 2 + lifetime [43] that determined t 1 / 2 = 3.37 ( 30 ) ns. A lifetime measurement [44] using the β - γ - γ fast timing technique reported the lifetime of the 2 1 + level to be 0.55 (2) ns, yielding B E 2 ; 2 1 + → 0 1 + = 80 ( 3 ) W.u., and also found [45] a considerably longer lifetime for the 0 2 + level of 5.60(15) ns, leading to 1 0 3 × ρ 2 ( E 0 ) = 92 ( 17 ) and B ( E 2 ; 0 2 + → 2 1 + ) = 13.3 ( 10 ) W. u. These data were then used in a two-state-mixing calculation to determine the deformation parameters for the deformed ground-state band of β D = 0.34 ( 1 ) , and for the spherical configuration β S = 0.16 ( 2 ) . The ¹⁰⁰Zr lifetimes were in excellent agreement with measurements [46] performed in parallel using a γ - γ coincidence technique involving a small-crystal Ge detector and a BaF₂ detector of 0.40 (8) ns ( 2 1 + ) and 5.36 (23) ns ( 0 2 + ) . At around the same time, studies of the β -decay of the deformed ⁹⁷Rb [47] suggested the presence of shape coexistence in the N = 59 ⁹⁷Sr isotope. Rotational bands based on the ν 3 / 2 [ 422 ] and ν 3 / 2 [ 541 ] Nilsson orbitals were suggested at excitation energies of 585 keV and 648 keV, respectively, with lower-lying levels considered to be spherical. This was confirmed shortly thereafter [48] via lifetime measurements using the β - γ - γ technique. The lifetimes extracted within the suggested ν 3 / 2 [ 422 ] band, an example of which is shown for the 5 / 2 + → 3 / 2 + transition in Figure 14, established that the in-band transitions were indeed enhanced and corresponded to an intrinsic quadrupole moment of | Q 0 | = 3.5 ( 4 ) e b.

The natures of the level structures in the N < 60 Zr isotopes were greatly speculated upon given the suggestion of shape coexistence in ⁹⁸Zr [41] and also the similarity of their structures with those of the Sr isotopes. New 0 + states were assigned [49] in ⁹⁸Zr in an experiment using mass-separated beams at the ILL to extract ⁹⁸Rb ions. The decay sequence, ⁹⁸Rb → 98 Sr → 98 Y, resulted in only the low-spin β -decaying state in ⁹⁸Y being populated from the ⁹⁸Sr spin 0 + parent. New E 0 transitions of 423 keV and 564 keV were observed in the decay of ⁹⁸Y. These were placed as a cascade of E 0 transitions from levels at 1859 keV and 1436 keV in ⁹⁸Zr, establishing them as the 0 4 + and 0 3 + states, respectively. Additionally, γ - γ angular correlation measurements [49] performed with mass-separated beams of fission products ⁹⁸Sr and ⁹⁸Y at the Jülich research reactor proved conclusively that the states were 0 + . The half-lives of these states were also measured, assuming that they are fed directly in the β -decay, with t 1 / 2 = 0.69 ( 10 ) ns ( 0 3 + ) and 0.24(10) ns ( 0 4 + ) . The data permitted the extraction of the transition rates, and a two-state mixing model was assumed for the large 1 0 3 × ρ 2 ( E 0 ) = 290 ( 80 ) value determined for the 0 3 + → 0 2 + transition. With an assumption of maximal mixing and taking one configuration to be spherical, the deformed configuration must have | β | > 0.21 [49]. It should be noted that while the presence of large E 0 strengths imply significant differences in root-mean-square values of the deformation β , the structure of the 0 + states in ⁹⁸Zr remains contested. For example, [50] interpreted the 0 3 + state as the head of a well-deformed band, with the 2 2 + , 4 1 + , and 6 1 + states as band members. In contrast, [51] interpreted the 0 3 + , 2 2 + , and 4 1 + states as a two-quasiphonon triplet of weakly deformed states and 6 1 + as a three-quasiphonon state. The conversion electrons and E 0 strengths in ⁹⁸Sr were re-investigated in [52], performing a two-state mixing analysis using the ρ 2 ( E 0 ) values, the B ( E 2 ) values for the 2 1 + → 0 1 + , 0 2 + → 2 1 + , and 4 1 + → 2 1 + transitions, and the energies of the states, and found a solution that minimized the total χ 2 value that resulted in the unperturbed configurations to have β D = 0.38 ( 1 ) and β S = − 0.23 ( 2 ) , i.e., that the “spherical” configuration is actually an oblate structure. The Coulomb excitation results of [53, 54] that deduced a near-zero quadrupole moment for the 2 2 + state Q s = + 0.0 2 − 0.12 + 0.13 eb, however, is consistent with a spherical state rather than one with oblate deformation.

In ⁹⁹Zr, data from a series [55–58] of ⁹⁹Y β -decay measurements performed at Jyväskylä, ISOLDE, and at the ILL were used to suggest rotational bands based on the ν 3 / 2 [ 541 ] and ν 3 / 2 [ 422 ] Nilsson orbitals with band heads at 575.5 keV and 724.3 keV, respectively [55]. A short time later [55], using a superior setup at Jyväskylä, part of the evidence used for the deformed interpretation, namely an enhanced transition connecting the 5 / 2 − and 3 / 2 − band members, was found to be questionable, and the transition connecting the 7 / 2 − and 5 / 2 − band members was concluded not to exist. However, later experiments have observed the 58-keV 5 / 2 − → 3 / 2 − transition and reaffirmed the existence of the 3 / 2 [ 541 ] band (see, e.g., Refs. [57, 58] that includes discussions of other structures in ⁹⁹Zr).

The nature of states in the Zr isotopes with N < 58 was probed in a ⁹⁴Y β -decay measurement [59] that discovered the 2 2 + → 0 2 + transition in ⁹⁴Zr, as shown in Figure 15. The extracted branching ratio of 0.150(6)% was combined with the level lifetime from DSAM measurements, following the ( n , n ′ γ ) reaction resulting in B ( E 2 ; 2 2 + → 0 2 + ) = 19 ( 2 ) W.u. This was the first firm determination of the shape coexistence scenario based on absolute B ( E 2 ) values for the even–even Zr isotopes.

The systematics of selected states in the Zr and Sr isotopic chains are shown in Figure 16; the states assigned to the spherical ground states for N < 60 and the deformed ground states for N ≥ 60 and 0 2 + and 0 3 + , are shown. The results of a new experiment [60] performed with Gammasphere studying the β -decay of ¹⁰⁰Y discovered the 0 4 + and 0 5 + states at 1294.5 and 1774.0 keV, respectively, in ¹⁰⁰Zr, and also observed a 366.8 keV transition placed as the 2 3 + → 0 3 + transition, supporting the interpretation of a rotational band proposed in [61]. New results are also reported [62] for ⁹⁸Zr from a β -decay experiment performed at the TRIUMF-ISAC facility with the 8 π spectrometer. From γ - γ angular correlations, the 0 5 + and 0 6 + states were found at 2418 keV and 2749 keV, respectively, and many additional levels were identified as 2 + states. Further studies are required to be able to identify the existence of possible bands built on these new 0 + states.

The stable Cd isotopes have been at the forefront of nuclear structure studies of shape coexistence for over 40 years. Early on [63], the even–even Cd nuclei were interpreted as having a level structure expected for spherical vibrators; however, an extra 2 + state was discovered in the vicinity of the two-phonon triplet in ¹¹⁴Cd in an ( n , γ ) reaction [64]. An additional 0 + state, also close in energy to the two-phonon triplet, was soon discovered in ¹¹³Cd( d , p ) ¹¹⁴Cd reaction, as well as an extra 0 + and 2 + pair of states in the ¹¹¹Cd( d , p )¹¹²Cd reaction [65]. Although their natures were speculated upon for some years, it was not until 1977 in an experiment studying the β -decay of the ¹¹⁰In 7 + ground state that the rotational-like band based on the 1473-keV 0 2 + level was finally elucidated [66]. This was achieved by having sufficient sensitivity for weak, low-energy transitions enabled through the use of Ge(Li) γ -ray detectors with their superior energy resolution over previous investigations using NaI detectors [67]. Using results that had nearly simultaneously been obtained from the two-proton-transfer reaction (³ H e , n ) [68], the rotational band was interpreted as being a 2 p − 4 h proton excitation, with the promotion of two g 9 / 2 protons into either g 7 / 2 or d 5 / 2 orbitals above the Z = 50 closed shell. It was also postulated at that time that a similar deformed rotational band occurred in ¹¹⁴Cd. Shortly thereafter, important information was obtained from conversion electron spectroscopy of ¹¹²Cd and ¹¹⁴Cd [69]. The data for ¹¹²Cd were obtained following the β -decay of ¹¹²In, whereas that for ¹¹⁴Cd were deduced from data obtained from a neutron capture reaction. With lifetimes deduced from in-beam studies [69], 1 0 3 × ρ 2 ( E 0 ) values were determined for the decay of the 0 2 + and 0 3 + states. The values for the 0 2 + → 0 1 + transitions, 37(11) and 30(8) for ¹¹²Cd and ¹¹⁴Cd, respectively, can be contrasted with the values of 0.48(11) and 1.7(2) for the 0 3 + → 0 1 + transition. For the 0 3 + → 0 2 + transition, the value in ¹¹²Cd, 8.1(19), is significantly larger than that for ¹¹⁴Cd at 0.41(9). Generally, these values were in line with the expectations of the vibrational model, where for Δ N = 2 E 0 , transitions are allowed [7]. Alternatively, the ρ 2 ( E 0 ) values are also consistent with a shape coexistence scenario.

The Cd isotopes were systematically studied by the Jyväskylä group, as reported in [70], that included light-ion fusion–evaporation reactions and also the β -decay of ^106,108,110In. The In activities were produced through ( p , n ) reactions on foils of ^106,108,110Cd with their decays studied by both γ -ray spectroscopy and conversion-electron spectroscopy. Although only singles measurements were performed, a number of conversion coefficients were extracted that included E 0 transitions. The e − and γ -ray spectra for the decay of ¹⁰⁸In are shown in Figure 17, demonstrating the very high signal-to-background obtained. Specifically important were the observations of strong E 0 transitions such as that at 1913 keV for ¹⁰⁸Cd, as shown in Figure 17. In this study, the shape-coexisting intruder bands were assigned from ¹⁰⁶Cd through ¹²⁰Cd [70]. Figure 18 shows the current state of the excitation energy systematics where the presumed nature (outlined below) of the states is reflected in the colour coding [1].

A major step forward in the understanding of the structure of the Cd isotopes occurred with the combination of lifetimes determined from analysis of Doppler-shift data from ( n , n ′ γ ) reaction and results from β -decay measurements performed at the TRIUMF-ISAC facility with the 8 π spectrometer [72–74]. These β -decay studies reached a very high level of sensitivity for the observation of weak, low-energy γ -ray branches between states at relatively high excitation energy. It was shown conclusively that the strong-mixing scenario, proposed decades earlier to explain the decay pattern of the 0 2 + and 0 3 + states [75], led to serious discrepancies for ¹¹⁰Cd [74]. The data, with some of the key coincidence spectra shown in Figure 19, further permitted the assignment of rotational bands built on excited 0 + states and the assignment of “ K = 2 ” bands, as shown in Figure 20 for ¹¹²Cd. The γ -ray transitions that were newly observed in those studies are highlighted in red. It is remarkable that despite how well studied both ¹¹⁰Cd and ¹¹²Cd were, many new transitions, and even states, were found. From a comparison of the experimental results to beyond-mean-field calculations, it was suggested that ^110,112Cd possessed multiple deformed shapes ranging from prolate, triaxial, and oblate [72, 73]. This is a major shift in the interpretation of the Cd isotopes, which had long been considered excellent examples—in fact often cited in textbooks — of spherical vibrational systems. A series of Coulomb excitation experiments have been conducted with the aim of providing definitive answers to the shapes of excited states in ¹¹⁰Cd, early results of which have been published in [76].

A series of conversion electron measurements were performed by [77–79], following the β -decay of In parents. The measurements took advantage of Cd ( p , n ) reactions to produce the In activities and obtained very high signal-to-background ratios that enabled the observation, or set upper limits, of many E 0 branches including those for 2 2,3 + → 2 1 + transitions. The observations were interpreted taking into account configuration mixing within the proton–neutron interacting boson model (IBM-2) framework. A reasonable reproduction of the E 0 strengths was obtained [78, 79], although the 2 3 + states in ^110,112,114Cd were interpreted as having a mixed-symmetry characteristic rather than of intruder origin. A conversion electron study, following the β + /EC-decay of ¹¹⁰In, extracted a new E 0 branch from the 4 + member of the intruder 0 2 + band [80]. The data, combined with that from [10], indicate that the shape-coexisting states continue to experience mixing with increasing spin in the bands.

Measurements of the β -decay of Ag isotopes extending into the neutron-rich Cd region were pursued at ORNL using the CARDS array [81] that consisted of three or four clover-type HPGe detectors in close geometry and the BESCA Si(Li) detector for conversion electrons, replacing one of the clover Ge detectors. A new β -decaying isomer was discovered in the ¹¹⁶Ag parent [82]. The key for its observation was the outstanding energy resolution achieved for the BESCA detector for conversion electrons. The β -decay of three states in ¹¹⁶Ag, the 0 − ground state, the newly found 48-keV 3 + state, and the 129-keV 6 − state, populated a wide spin range of levels in the ¹¹⁶Cd daughter. Despite the sensitivity achieved, the placement of the key 262-keV 2 3 + → 0 3 + γ ray, which has been observed in one experiment only [83] with a reported branching ratio of 0.5% and used to establish the intruder band, could not be confirmed [84]. Rather, the β -decay measurement [84] reported a 2 σ upper limit of 0.6%. In ¹²⁰Cd, the data obtained [85] from the decay of the ¹²⁰Ag, which includes the ( 0 − , 1 − ) ground state and the 4 ( + ) and 7 ( − ) isomeric states, did not observe the decay of the previously assigned 0 2 + at 1388.9 keV but did observe the decay of the higher-lying 0 3 + state at 1744.9 keV. Furthermore, a previously unobserved γ ray was observed in coincidence with the 505.6-keV 2 1 + → 0 1 + γ ray, as shown in Figure 21, but without any additional coincidences. These facts were used to remove the 1388.9-keV state from the level scheme and to establish a new level at 1136.0 keV that was assigned as the 0 2 + state. This new assignment is reflected in the energy systematics shown in Figure 18. If this placement is confirmed, it would imply a dramatic drop in energy for the configuration that, in the lighter Cd isotopes, is suggested to be based on an oblate shape [72, 73].

Studies using the CARDS array were also performed for the decays of ^124,126Ag [86]. From their data for ¹²⁴Ag decay, they assigned a state at 1573.5 keV as the 0 2 + state, using similar arguments as for the ¹²⁰Cd 0 2 + state above. This may be the head of the “oblate” band; however, no higher-lying band members were assigned. A second excited 0 + state was suggested at 1924.8 keV. These new states would suggest that the 0 2 + state in ¹²²Cd, a nucleus which has not been thoroughly studied since the early 1990s [87], remains undetected since it is currently assigned as a level at 1705 keV, and such a dramatic rise and fall of the excitation energy of the 0 2 + state between ¹²⁰Cd and ¹²⁴Cd is unexpected.

New β -decay measurements have been initiated using the GRIFFIN spectrometer at the TRIUMF-ISAC facility that will include the study of ^104,106In decay and ^{112,116,118,120}Ag decay. The neutron-rich Ag isotopes, in particular, have multiple β -decaying states that can be separated somewhat via selective laser ionisation. The decay of ^118,120Ag will also be studied at the Jyväskylä facility that will offer the advantage of using the Penning trap mass spectrometer JYFLTRAP to achieve highly purified isomeric beams [88].

The shape-coexisting states in the mid-shell Sn isotopes were first discovered through a series of ( α , 2 n γ ) reactions by [89] and have mostly been investigated through a variety of reaction studies. Of particular interest has been the two-proton-transfer studies [68] that observed large populations of the 0 2 + states in ^114,116,118Sn, as well as in α -particle transfer reactions ^122,124Te( d , 6 Li)^118,120Sn [90], that reveal the microscopic natures of the 0 2 + states having important proton-pair contributions. The energy level systematics is shown in Figure 22 for the even–even Sn isotopes. Of particular interest is that both the 0 2 + and 0 3 + states display the characteristic parabolic-shaped trend expected for shape-coexisting structures with the minimum at the neutron mid-shell. The systematics plotted here use the 0 2 + level as the intruder band head based on its strong population in the proton-transfer reactions.

The structure of the deformed intruder bands in ^116,118Sn has recently been investigated with β -decay at the TRIUMF-ISAC facility. In ¹¹⁶Sn, the decay of ^116m1In was used to seek weak, low-energy transitions from the low-spin excited states, especially the 0 + and 2 + states. The collected data enabled the direct observation of the 85-keV 2 2 + → 0 3 + transition that previously had its intensity indirectly inferred. This is shown in Figure 23. Due to the large Compton-scattering background that was present, the authors of [91] performed a careful investigation of its possible impact to contribute to the signal at 85 keV. As shown in Figure 23, the 85-keV peak is clearly due to a coincidence with the 734-keV 0 3 + → 2 1 + γ -ray transition. The measured branching ratio for the 85-keV transition results in a B ( E 2 ) value of 100(8) W.u. that is a factor of 2.2(3) greater than the 355-keV 2 2 + → 0 2 + transition, leading to the suggestion that the 0 3 + level should be identified as the intruder band head. This is shown in Figure 24. Interestingly, upper limits were established [92] and led to negligible E 0 components in the 2 2 + → 2 1 + and 2 3 + → 2 1 + transitions, implying that the mixing of the intruder and spherical 2 + states must be very small.

A recent study [93] of the neighbouring nucleus ¹¹⁸Sn via the β − decay of ¹¹⁸In was performed with the GRIFFIN spectrometer at the TRIUMF-ISAC facility. The measurement collected a very large statistical sample that resulted in significant revisions of relative intensities for decay of some levels, most notably for the 284-keV transition from the 2 2 + level that populates the 0 2 + state. This particular transition was shown to be part of a triplet of γ rays at 285 keV, and its branching ratio for decay from the 2 2 + level was refined from 2.6(2)% down to 1.33(6)%, reducing B ( E 2 ; 2 2 + → 0 2 + ) to 21(4) W.u. from its previously adopted 39(7) W.u [93]. However, the intruder band in ¹¹⁸Sn remained as previously assigned with the 0 2 + state as its band head.