Triaxial Shapes in Even–Even Nuclei: A Theoretical Overview

Abstract

Triaxial shapes in even–even nuclei have been considered since the early days of the nuclear collective model. Although many theoretical approaches have been used over the years for their description, no effort appears to have been made for grouping them together and identifying regions on the nuclear chart where the appearance of triaxiality might be favored. In addition, over the last few years, discussion has started on the appearance of small triaxiality in nuclei considered so far as purely axial rotors. In the present work, we collect the predictions made by various theoretical approaches and show that pronounced triaxiality appears to be favored within specific stripes on the nuclear chart, with low triaxiality being present in the regions between these stripes, in agreement with parameter-free predictions made by the proxy-SU(3) approximation to the shell model, based on the Pauli principle and the short-range nature of the nucleon–nucleon interaction. The robustness of triaxiality within these stripes is supported by global calculations made in the framework of the Finite-Range Droplet Model (FRDM), which is based on completely different assumptions and possesses parameters fitted in order to reproduce fundamental nuclear properties.

1. Introduction

Atomic nuclei composed of an even number of protons and an even number of neutrons, called even–even nuclei, can exhibit a variety of properties and effects, like quadrupole, octupole (pear-like), and hexadecapole axially symmetric deformations, triaxial deformations, the existence of isomeric states, shape coexistence, and many others. Global calculations over large parts of the nuclear chart have been performed using various theoretical approaches, with their results compared to compilations of experimental data, and the regions of the nuclear chart in which these effects are favored having been pointed out. For example, quadrupole and hexadecapole deformations have been calculated in relativistic mean-field (RMF) theory [1], while relevant experimental data for the quadrupole deformation have been tabulated in Refs. [2,3]. Quadrupole deformations taking into account deviations from axial symmetry have been obtained by mean-field calculations involving the D1S Gogny interaction and taking into account departure from triaxiality [4], while parameter-independent predictions for the deformation variables have been obtained microscopically on a symmetry basis in Refs. [5,6]. Quadrupole, hexadecapole, and, in addition, octupole deformations have been calculated in the Finite-Range Droplet Model (FRDM) [7], while relevant experimental data for the octupole deformation have been tabulated in Refs. [8,9]. Regions in which octupole deformation appears have been reviewed in Refs. [10,11], while a microscopic derivation of octupole magic numbers has been recently given [12]. Shape isomers have been calculated within the Finite-Range Liquid-Drop Model (FRLDM) [13,14] and reviewed in Refs. [15,16,17]. Shape coexistence has been reviewed in Refs. [18,19,20,21,22,23], with islands on the nuclear chart on which shape coexistence can occur determined microscopically on a symmetry basis in a parameter-free way in Refs. [24,25].

Global calculations for triaxial nuclei, i.e., nuclei violating axial symmetry, have been performed within the Finite-Range Liquid-Drop Model [26,27], while stripes within which triaxiality is favored have been microscopically predicted recently in a parameter-free way in Ref. [28]. In parallel, it has been suggested by computationally demanding Monte Carlo Shell Model (MCSM) calculations [29,30,31], as well as by more accessible Triaxial Projected Shell Model (TPSM) calculations [32], that some amount of triaxiality is present in almost all nuclei across the nuclear chart.

It is the purpose of the present work to review existing theoretical predictions for triaxiality in even–even nuclei and detect regions of the nuclear chart in which triaxiality is favored. In Section 2, Section 3, Section 4, Section 5 and Section 6, the theoretical approaches used in the study of triaxiality are exposed, while in Appendix B, Appendix C, Appendix D, Appendix E, Appendix F, Appendix G and Appendix H, a detailed account of existing theoretical work for each series of isotopes in the region with $Z=24$–98 is separately given. Finally, in Section 7, Section 8 and Section 9, the findings of the present compilation are summarized and an outlook for further work is given.

2. Collective Model of Bohr and Mottelson

2.1. The Rigid Triaxial Rotor Model

In the collective model of Bohr and Mottelson [33,34,35,36], introduced in 1952 [33], nuclear properties are described in terms of the collective variables $\beta$ and $\gamma$, describing the departure from sphericity and the departure from axial symmetry, respectively. For the latter, the values $\gamma =0$ and $\gamma =60$° correspond to ellipsoids with prolate (rugby-ball-like) and oblate (pancake-like) axially symmetric shapes, while the intermediate values $0<\gamma <60$° correspond to triaxial shapes with three unequal semi-axes, with maximum triaxiality occurring at $\gamma =$ 30°.

Soon thereafter, two radically different special cases have been considered. On one hand, Wilets and Jean, in 1956, studied the $\gamma$-unstable case [37], in which the nuclear potential within the Bohr Hamiltonian depends only on $\beta$ and not on $\gamma$, allowing the exact separation of variables in the relevant wave function as a by-product. This limit has been found useful in the description of the spectra of several nuclei known at that time [38,39], later called $\gamma$-soft or $\gamma$-unstable nuclei, in which the value of $\gamma$ can vary without any energy expense.

On the other hand, Davydov and Filippov [40], in 1958, considered the case of rigid triaxiality, with the nuclear shape corresponding to an asymmetric top with fixed values of $\beta$ and $\gamma$, and studied spectra and transition probabilities within it [40,41]. This model has been called the Rigid Triaxial Rotor Model (RTRM) [42]. Soon thereafter, the cases of a constant value of $\gamma$ within the Bohr Hamiltonian [43], as well as of a $\gamma$ potential corresponding to small oscillations around a non-vanishing value ${\gamma }_0$ [44], were considered. The analytical solution for the triaxial rotor with $\gamma =30$° was given by Meyer-ter-Vehn [45] in 1975.

Several extensions of the RTRM have been introduced over the years, including the two-rotor model for the description of the scissors mode, formed by the separate collective motion of protons and neutrons [46,47], and the presentation of the RTRM as multiple Q-excitations built on the ground state [48], as well as the semiclassical versions of a cranked triaxial rotor [49] and a triaxial rotor model treated by a time-dependent variational principle [50], while for odd-A nuclei, the particle-rotor model has been introduced [45,51].

An interesting application of the RTRM regards the application of the analytical expressions of the RTRM for spectra [42,52,53,54] and/or transition rates [42,53] in order to extract values of $\gamma$ from the relevant data for several series of isotopes. The values extracted from the spectra and separately from the transition rates have been found to be consistent with each other and also different from zero, usually above 10°, serving as an early sign for the existence of some degree of triaxiality over extended regions of the nuclear chart (see Tables I and II of [53], Table II of [42], and Tables 1–3 of [54]).

In the RTRM, the three moments of inertia along the three axes are determined by the single parameter $\gamma$, representing an irrotational flow [55] within the nucleus. In 2004, this restriction was lifted by Wood et al. [56], which allowed for independent inertia and electric quadrupole tensors within the triaxial rotor model, thus obtaining the generalized triaxial rotor model (GTRM). The GTRM has been applied for the study of many nuclei [57,58,59,60,61,62,63] extracting the $\gamma$ values characterizing them (see, for example, Table 1 of [60]).

An early alternative description of triaxial nuclei in the Xe-Ba and Th regions has been provided within the framework of the Triaxial Rotation Vibration Model (TRVM) [64,65], which is an extension of the Rotation Vibration Model (RVM), in which the deviations of the shape coordinates around their static values are considered [66,67,68,69,70]. Triaxiality is obtained by taking into account the interaction between rotations and vibrations [64,65].

In recent years, the interpretation of $\beta$ and $\gamma$ collective bands as quadrupole vibrations has been questioned [71,72,73,74,75,76,77], with $\beta$ bands attributed to two-particle–two hole (2p2h) excitations [76], and $\gamma$ bands considered as a consequence of breaking of axial symmetry leading to triaxial shapes [75,77].

2.2. The Algebraic Collective Model

Extensive numerical calculations over several regions of the nuclear chart [78,79,80] have been performed in the framework of the Generalized Collective Model (GCM), introduced by the Frankfurt group in 1971 [81], in the potential of which all independent terms up to sixth order in $\beta$ are taken into account. A comprehensive review of work performed in the framework of the Bohr collective Hamiltonian has been given in Ref. [82].

A major step forward in numerical calculations related to the Bohr collective model has been taken through the introduction of the Algebraic Collective Model [83,84,85], triggered by the realization [86] that the Davidson potential [87], initially used for the description of rotation-vibration spectra of diatomic molecules, has a SU(1,1) × SO(5) structure [88,89], which can be applied to atomic nuclei, resulting in much faster convergence of the relevant numerical calculations, especially in the case of deformed nuclei, due to the optimal choice of the basis [90,91] in which the diagonalization of the Hamiltonian is performed.

The availability of a computer code for the ACM [92] greatly facilitated its application in the case of triaxial nuclei [93,94,95,96], in which the presence of a ${cos}^{2}3\gamma$ term has been found necessary.

2.3. Shape/Phase Transitions and Critical Point Symmetries

The introduction by Iachello of the critical point symmetries (CPSs) E(5) [97] in 2000 and X(5) [98] in 2001, related to shape-phase transitions (SPTs) [99,100,101] (also called quantum phase transitions (QPTs) [102]) from vibrational (near-spherical) nuclei to $\gamma$-unstable and to prolate axially deformed nuclei, respectively, have raised much interest on special solutions of the Bohr Hamiltonian [103,104], several of them related to triaxial shapes.

A shape/phase transition in the $\gamma$ variable, leading from axial to triaxial nuclei, has been described by the critical point symmetry Y(5) [93,105], introduced by Iachello in 2003 [105], with ¹⁶⁴Dy and ^166,168Er suggested as possible candidates for it, while the shape/phase transition from prolate to oblate shapes [106] has been described in terms of the Z(5) [107] critical point symmetry, in which the $\gamma$ variable is confined in a steep harmonic oscillator potential around the value of maximal triaxiality, $\gamma =30$°, with the transition taking place in ^192−196Pt, suggested as a possible experimental manifestation. A $\gamma$-rigid version of Z(5), called Z(4), has also been introduced [108], in which $\gamma$ is fixed at 30°, with the transition taking place in ^128−132Xe, suggested as a possible experimental manifestation. In both Z(5) and Z(4), an infinite square well potential is used in the $\beta$ variable, in analogy to E(5) and X(5). Exact separation of the $\beta$ and $\gamma$ variables is obtained in Z(4), while in Z(5), an approximate separation of variables is achieved by assuming potentials of the form $u(\beta ,\gamma )=u_1\left(\beta \right)+u_2\left(\gamma \right)$, in analogy to the X(5) case [98,109]. A critical point symmetry called T(5) has also been introduced [110], containing X(5) and Z(5) as special cases. In the T(5) CPS, the approximate separation of variables is achieved, with an infinite square well potential used in $\beta$ and a steep harmonic oscillator potential centered around a specific value ${\gamma }_e$ being employed in $\gamma$. Furthermore, a special solution of the Bohr Hamiltonian called T(4) has been developed [111], which is a $\gamma$-rigid solution intermediate between Z(4) and X(4) [112], the latter being a solution intermediate between X(5) and X(3) [113], where X(3) is a $\gamma$-rigid version of X(5).

2.4. Special Solutions of the Bohr Hamiltonian

Several special solutions similar to Z(5) have been suggested, using in $\gamma$ the same steep harmonic oscillator centered around 30° used in Z(5), but replacing in $\beta$ the infinite square well potential with the Davidson potential [114], a hyperbolic Pöschl–Teller potential [115], an inverse square potential [116], or a finite well potential [117]. In addition, similar solutions achieving exact separation of variables by using potentials of the form $u(\beta ,\gamma )=u_1\left(\beta \right)+u_2\left(\gamma \right)/{\beta }^2$ have been considered, by using in $\gamma$ a steep harmonic oscillator potential around $\gamma =30$° and in $\beta$ the Coulomb potential [118,119], the Kratzer potential [118,119,120], the Davidson potential [119,121], the infinite square well potential [119], the Morse potential [122], the Hulthén potential [123], or the Killingbeck potential [124]. Furthermore, exact separation of variables has been used with a sextic potential in $\beta$ and a Mathieu potential in $\gamma$ [125,126,127,128,129], while a generalized Gneuss–Greiner potential has been used in Ref. [130].

In addition, several exactly separable special solutions similar to Z(4) [108] have been suggested, in which the infinite square well potential in the variable $\beta$ is replaced by the Davidson potential (Z(4)-D, [131]), or by a sextic potential (Z(4)-sextic, [132,133,134,135]). A special solution of the Bohr Hamiltonian providing a bridge between Z(4) [108] and X(4) [112] has also been developed [136].

In most of the above mentioned special solutions, a harmonic oscillator potential centered around $\gamma =30$° is used. A much sharper potential well, centered around $\gamma =30$°, is the potential $1/{sin}^2\left(3\gamma \right)$, the former potential corresponding to the second-order Taylor expansion of the latter [137].

2.5. Modifications of the Bohr Hamiltonian

A well known problem of the Bohr collective model [33,34] is that the nuclear moments of inertia are predicted to increase proportionally to ${\beta }^2$, while experimental evidence indicates that the increase should be much slower, especially for deformed nuclei [138]. One way to moderate the increase of the moment of inertia with increasing deformation $\beta$ is to allow the mass to depend on the deformation. The general formalism for quantum systems with mass depending on the coordinates has already been developed [139,140,141], with analytical solutions obtainable for several physical systems through the techniques of Supersymmetric Quantum Mechanics (SUSYQM) [142,143]. Deformation-dependent masses (DDM) have been introduced in the Bohr Hamiltonian with a Davidson potential [144,145], and a Kratzer potential [146]. Special solutions regarding triaxial nuclei using the DDM concept have been developed for the analog of the Z(4) solution using the infinite square well potential (Z(4)-DDM, [147]), the Davidson potential (Z(4)-DDM-D [148]), and the Kratzer potential (Z(4)-DDM-K, [149,150]). A special solution using the DDM concept has also been developed for the X(3) solution (X(3)-DDM, [147]).

Another attempt of modifying the moments of inertia in the Bohr Hamiltonian has been made by using the concept of minimal length [151,152], which comes from non-commutative geometry [153,154], leading to a generalized uncertainty principle [155,156] involving deformed canonical commutation relations. This concept has been applied in many fields, including string theory [157] and quantum gravity [158]. In relation to triaxial nuclei, the Z(4) special solution of the Bohr Hamiltonian has been modified by including the minimal length (Z(4)-ML [147]). A similar modification of the X(3) solution has also been worked out (X(3)-ML [147]).

It should be remarked that the connections between three different forms of unconventional Schrödinger equations involving position-dependent mass, deformed canonical commutation relations, or a curved space have been clarified in Ref. [139]. Therefore, the DDM and ML methods described in the last two paragraphs are not unrelated [159].

Another family of solutions of the Bohr Hamiltonian can be obtained by considering energy-dependent potentials, already used in other fields of physics [160,161]. Solutions of the Bohr Hamiltonian involving a coupling constant linearly dependent on the energy have been provided for the harmonic oscillator [162,163,164] and Coulomb [165] potentials. In relation to triaxiality, a solution for the Kratzer potential possessing a coupling constant linearly dependent on the energy has been provided [166].

A novel approach to SPTs has been introduced by Hammad in 2021 [167,168], making use of the fractional calculus [169,170,171,172]. Using fractional derivatives [169,170,171] in the Bohr equation, the fractional order of the derivative can be used in order to approach the critical point closer than what is allowed by derivatives of integer order, something useful since the neutron number, which is used as the control parameter in a series of even–even isotopes, is increasing by steps of 2. In addition, the use of conformable fractional derivatives [172] guarantees that the usual rules valid for derivatives in the usual calculus continue to be valid in the case of the fractional calculus as well. Analogues of the Z(5) CPS using conformable fractional derivatives have been considered for the Kratzer potential [173], as well as for the Morse [174], Tietz-Hua [174], and various multi-parameter exponential-type potentials [174] and a four inverse power terms potential [175].

3. The Nuclear Shell Model

The nuclear spherical shell model [176,177,178] was introduced in 1949 [179,180,181] in order to explain the extreme stability observed at the nuclear magic numbers 2, 8, 20, 28, 50, 82, 126,…, [182]. It is based on a three-dimensional isotropic harmonic oscillator (3D-HO), to which a spin-orbit interaction [181] is added, resulting in the modification of the 3D-HO magic numbers 2, 8, 20, 40, 70, 112, 168,…, [183,184,185] beyond the $sd$ shell.

While giving a satisfactory description of nuclear properties in the vicinity of the magic numbers, where nuclei of nearly-spherical shape are expected to occur, the spherical shell model fails to account for the large quadrupole moments observed in nuclei away from closed shells. This has been remedied through the introduction of the deformed shell model by Nilsson [186,187], in which nuclei are allowed to obtain axially symmetric shapes. The fact that spheroidal shapes offer greater stability than spherical shapes, being energetically preferable, was pointed out by Rainwater in 1950 [188].

Based on the spheroidal shapes introduced by Nilsson, the macroscopic–microscopic Finite-Range Droplet Model (FRDM) and Finite-Range Liquid-Drop Model (FRLDM) have been developed [7,26,27,189,190,191], allowing for the determination of the ground-state masses and deformation parameters of a large number of nuclei.

The concept of cranking of a spheroidal collective field about a fixed axis was introduced by Inglis in 1954 [192,193] and used in several early shell model calculations (see, for example, [194]), in some of which, the 3D-HO potential has been replaced by a Woods–Saxon potential [195]. An authoritative review of the development of the nuclear shell model can be found in Ref. [196]. The main difficulty associated with nuclear shell model calculations is the large size of the model space, which makes the computational needs extremely large.

A computationally tractable approach that has been widely used for studying triaxiality in several nuclei is the Triaxial Projected Shell Model (TPSM) [197,198], in which a triaxial Nilsson + BCS basis is used. The TPSM is a generalization of the axially symmetric Projected Shell Model (PSM) approach [199]. In PSM (TPSM), a Nilsson potential with axial (triaxial) deformation is used in order to generate the deformed single-particle states, while pairing correlations are treated by the usual BCS approximation. A three-dimensional angular momentum projection [138] is then carried out on the Nilsson+BCS quasiparticle states in order to obtain the many-body basis with states having good angular momentum, and subsequently, the Hamiltonian is diagonalized in the projected basis. TPSM calculations have been carried out for Ge [200], Se [200], Mo [201], Ru [201,202], Ce [203], Nd [203], Gd [204], Dy [205,206], and Er [205,207,208] series of isotopes, while detailed results for 30 triaxial nuclei ranging from Ge to U have been summarized in Refs. [32,209] (see Table 3 of [32]).

Large-scale shell model calculations have become feasible by the introduction of the Quantum Monte Carlo Diagonalization (QMCD) method [210,211,212,213], reviewed in [214,215]. Recent Monte Carlo Shell Model (MCSM) calculations have suggested rigid triaxiality in ¹⁶⁶Er [30], taking advantage of self-organization in quantum systems [29]. The prevalence of triaxial shapes in heavy nuclei has been recently suggested [31].

3.1. The SU(3) Symmetry

The shells of the 3D-HO used in the original spherical shell model are known to possess the symmetry U($(N+1)(N+2)/2$) [183,184,185], where N is the number of oscillator quanta. In each shell, an SU(3) subalgebra exists [216].

Elliott, in 1958, working in the $sd$ ($N=2$) shell with U(6) symmetry, managed [217,218,219,220,221] to build a bridge between the spherical shell model and the deformation described by the collective model. Initially [217], he classified the shell model states using the U(6)⊃SU(3)⊃SO(3) chain of subalgebras, in which nuclear bands occur within the irreducible representations (irreps) of SU(3), which are characterized by the Elliott quantum numbers $(\lambda ,\mu )$. Nuclear states are labeled by the angular momentum L (which characterizes the irreps of the angular momentum algebra SO(3)) and K, the “missing quantum number” [185,222] in the decomposition from SU(3) to SO(3). The quantum number K divides the states into rotational bands with increasing values of L appearing in each of them. Subsequently [218], he proved that all states belonging to a given band come from the same intrinsic state, obtaining simple expressions for the quadrupole moments, resembling those of a rotational model with permanent deformation. In different words, Elliott proved the existence of deformed states corresponding to quadrupole deformation within the spherical shell model. SU(3) has played, since then, a major role in nuclear structure, as summarized in the recent book by Kota [223].

It should be noticed at this point that a mapping of the invariants of SU(3) onto the invariants of the collective model provides a connection between the Elliott labels $\lambda$, $\mu$ and the collective variables $\beta$, $\gamma$, which read [224,225]

$$\gamma =arctan\left({\displaystyle \frac{\sqrt{3}(\mu +1)}{2\lambda +\mu +3}}\right),$$

(1)

$${\beta }^2\propto ({\lambda }^2+\lambda \mu +{\mu }^2+3\lambda +3\mu +3)\propto (C_2(\lambda ,\mu )+2),$$

(2)

where $C_2$ is the second order Casimir operator of SU(3) [185,222]

$$C_2(\lambda ,\mu )={\displaystyle \frac{2}{3}}({\lambda }^2+\lambda \mu +{\mu }^2+3\lambda +3\mu ).$$

(3)

The SU(3) symmetry is broken beyond the $sd$ shell by the spin–orbit interaction, which lowers in each shell the orbital with the highest eigenvalue of the total angular momentum j into the shell below. As a result, each shell, beyond the $sd$ one, ends up consisting of its remaining original orbitals, minus the defecting orbital that went to the shell below, plus the invading orbital (called the intruder orbital) that came from the shell above.

3.2. The Pseudo-SU(3) Symmetry

Approximate restoration of the SU(3) symmetry can be achieved in different ways. In the pseudo-SU(3) scheme [226,227,228,229,230,231,232,233], introduced in 1973 [228], a unitary transformation [234,235,236] is used in order to map the remaining original orbitals of each shell onto the full set of the orbitals of the shell below it, thus restoring the SU(3) symmetry for the remaining orbitals, which now live in the same shell with the intruder orbitals next to them. The intruder orbitals are not included in the recovered SU(3) symmetry and have to be taken into account separately, through shell model techniques [230,231].

3.3. The Proxy-SU(3) Symmetry

A different approximate restoration of the SU(3) symmetry is the proxy-SU(3) symmetry [5,6,237], introduced in 2017 [5,237]. In the proxy-SU(3) scheme, a unitary transformation [238] is used for the intruder levels, except the sublevel (accommodating two nucleons) that possesses the highest eigenvalue of the projection $j_m$ of the total angular momentum j. This unitary transformation [238] maps the rest of the intruder levels onto the defecting orbitals, which had passed into the shell below. As a result, SU(3) is restored in the shell under study, in which the orbitals obeying the SU(3) symmetry now live with the isolated orbital with the highest $j_m$, which, however, can in most cases be neglected, since from the standard Nilsson diagrams [239], one can see that it lies at the top of the shell; thus, it will be empty for most nuclei to be considered within this shell.

An important finding realized in the proxy-SU(3) framework is that the most important irreps, which lie lowest in energy and, therefore, accommodate the ground state band, also called most leading irreps in earlier literature [230,231], are not the ones possessing the highest eigenvalue of the second order Casimir of SU(3), given by Equation (3), but the highest-weight (hw) irreps [240], which are the most symmetric irreps allowed by the Pauli principle and the short-range nature of the nucleon–nucleon interaction. Although the two sets of irreps are identical in the lower half of the shell, they become different beyond midshell (see, for example, Table I of Ref. [5]). Although this fact has been known since the early days of the use of SU(3) in nuclear physics, as seen, for example, in Table 5 of Ref. [228] for the $pf$ shell, its consequences have been rather ignored. The dominance of the highest-weight irreps offers an explanation for the dominance of prolate over oblate shapes in the ground states of even–even nuclei, a long-standing problem considered unresolved until recently [241,242], predicts a prolate-to-oblate transition in the heavy rare earths around $N=114$ [5], and paves the way for the determination of certain regions on the nuclear chart in which islands of shape coexistence can occur [22,24,25], as corroborated through covariant density functional theory calculations [243,244] and experimental evidence [22,245].

In addition, the dominance of the highest-weight irreps predicts non-negligible values of the shape variable $\gamma$ over extended regions of the nuclear chart. This can be seen, for example, in Tables II and III of Ref. [5], in which it is clear that most nuclei possess $\mu$ values different from zero, resulting in non-zero values of $\gamma$ through Equation (1). These predictions are in good agreement with $\gamma$ values obtained from the data, as seen in Figures 5 and 6 of Ref. [5]. A detailed comparison of the proxy-SU(3) predictions for the collective variable $\gamma$ and its empirical values extracted from the experimental data has been given recently in Ref. [28].

It should be emphasized that these results are obtained in a completely parameter-free way, based only on the consequences of the Pauli principle and the short-range nature of the nucleon–nucleon interaction. The basic property leading to these results is the fact that the net result of the restrictions imposed by the Pauli principle on five (or more) nucleons (protons or neutrons) in a given shell, which can accommodate N nucleons, is not the same as the net result of the restrictions imposed by the Pauli principle on $N-5$ (or fewer) nucleons in the same shell, as seen, for example, in Table I of Ref. [5].

It should be noticed that the use of the highest-weight irreps within the pseudo-SU(3) scheme leads to predictions [246] compatible with those of the proxy-SU(3) approach. It is remarkable that two considerably different approximation schemes, involving different fractions of the valence nucleons, lead to very similar conclusions. This might be understood on the basis that both schemes are based on unitary transformations of some of the orbitals onto their pseudo- or proxy- counterparts.

It has been recently realized that the next-highest-weight (nhw) irreps within the proxy-SU(3) symmetry might have important contributions in some special cases [28]. Extensive results for the nhw irreps in several regions of the nuclear chart have been given in Ref. [247], in order to be used in future irrep-mixing calculations.

3.4. Regions of High Triaxiality Predicted by the Proxy-SU(3) Symmetry

The proxy-SU(3) symmetry predicts non-vanishing values of the collective variable $\gamma$ almost everywhere across the nuclear chart [5,28]. However, if one is restricted to $\gamma$ values between 15° and 45°, i.e., values close to the value $\gamma =30$°, which corresponds to maximum triaxiality, one sees that these occur within certain horizontal and vertical stripes on the nuclear chart, lying within the proton or neutron numbers 22–26, 34–48, 74–80, 116–124, and 172–182 [28]. Empirical information [28] supports the interpretation that strong triaxiality should appear preferably within these stripes. In Appendix B, Appendix C, Appendix D, Appendix E, Appendix F, Appendix G and Appendix H, it will be tested to what extent these predictions are in agreement with theoretical work on triaxiality accumulated over the years.

4. Algebraic Models Using Bosons

4.1. Interacting Boson Model-1

In the Interacting Boson Model (IBM), introduced by Arima and Iachello in 1975 [222,248,249,250,251], collective nuclear properties are described in terms of bosons, which correspond to correlated proton pairs and neutron pairs, counted from the nearest closed shells and bearing angular momentum 0 (s-bosons) and 2 (d-bosons). The model has an overall U(6) symmetry, possessing three different dynamical symmetries: U(5) [252], corresponding to vibrational (near-spherical) nuclei, SU(3) [253], describing axially symmetric deformed nuclei, and O(6) [254], appropriate for $\gamma$-unstable nuclei, which are soft towards triaxial deformation of their shape. Schematically, the three dynamical symmetries are placed at the vertices of the symmetry triangle of IBM-1 [255], also called the Casten triangle.

In the simplest version of the model, called IBM-1, no distinction is made between bosons coming from proton pairs or neutron pairs. When this distinction is made, IBM-2 is obtained. In the standard form of IBM-1 and IBM-2, only one-body and two-body terms are taken into account in the Hamiltonian. However, higher-order terms exist and can be included if necessary.

The classical limit of IBM can be obtained through the use of coherent states [256,257,258]. In the classical limit, a potential energy surface (PES) corresponding to the IBM Hamiltonian is obtained, with its shape depending on the values of the free parameters appearing in the IBM Hamiltonian. Studying the classical limit of IBM-1, Van Isacker and Chen, in 1981 [259], proved that no triaxial shapes can be obtained within the standard IBM-1, in which only one-body and two-body terms appear in the Hamiltonian, while the addition of three-body (cubic) terms do give rise to triaxial shapes, with ¹⁰⁴Ru suggested as an experimental example [260].

The relation between the O(6) dynamical symmetry and triaxiality was clarified already in 1979 [261,262]. It has been shown that the O(6) limit of IBM-1 corresponds to the $\gamma$-unstable model of Wilets and Jean [37], if an infinite number of bosons is considered. While the RTRM with $\gamma =30$° satisfies the same selection rules, the predictions of the two models for spectra and $B\left(E2\right)$ transition rates are considerably different [261,262]. Experimental manifestations of the O(6) symmetry have been found in the Pt [263,264] and Xe-Ba [265] regions. It has also been found that deviations from the O(6) symmetry in both regions can be accounted for easily by adding cubic terms to the IBM-1 Hamiltonian [266]. Looking the other way around, one concludes that a small degree of triaxiality is present in the best experimental manifestations of the O(6) dynamical symmetry.

It has been further proved [267,268] within the O(6) limit of IBM-1 that the $\gamma$-unstable state can be generated from an intrinsic state with rigid triaxial deformation with $\gamma =30$°, thus demonstrating the formal equivalence between $\gamma$-instability and rigid triaxiality, especially in the ground state bands [269]. Despite this formal equivalence, detailed investigations have shown that most nuclei possess a $\gamma$-soft nature, as opposed to stable, triaxial ground-state shapes [270,271].

The quadrupole operator in IBM-1 reads [222]

$$Q_q={(d^{†}s+s^{†}\tilde{d})}_q^{\left(2\right)}+\chi {\left(d^{†}\tilde{d}\right)}_q^{\left(2\right)},$$

(4)

where $s^{†}$ (s) and $d^{†}$ ($\tilde{d})$ are the creation (annihilation) operators for the $L=0$ and $L=2$ bosons, respectively, while $\chi$ is a parameter that obtains the values $-\sqrt{7}/2$ and zero in the SU(3) and O(6) limits, respectively. It has been proved that the value of the collective variable $\gamma$ is connected to the choice of $\chi$ [272]. In more detail, the values of the collective variables $\beta$ and $\gamma$ can be estimated from the quadrupole shape invariants $QQ$, $QQQ$, and $QQQQ$ introduced by Kumar in 1972 [273] and widely used since then [274,275,276]. Recent studies [277] of the fluctuations of $\beta$ and $\gamma$, estimated through calculations of the quadrupole shape invariants within the configuration-interaction shell model, indicate that there is a certain degree of softness for $\beta$, while for $\gamma$, the fluctuations can become large, rendering the effective value of $\gamma$ not meaningful [277].

A mapping between the triaxial rotor and the SU(3) limit of IBM-1 has been established [278,279,280,281,282], in which the inclusion of higher-order terms of the type $LQL$ and $LQQL$ (where L is the angular momentum operator) in the IBM Hamiltonian is required. A similar mapping regarding the O(6) limit of IBM-1 has been recently established [283]. We conclude that in both the O(6) and SU(3) limits of IBM-1, higher-order terms are required in order to produce rigid triaxiality. This has to be distinguished from the effective triaxiality stemming from the softness of the relevant classical potential seen in the framework of the standard IBM-1 [284,285], for which the decrease of the effective $\gamma$ value with $\chi$ moving from the O(6) value $\chi =0$ to the SU(3) value $\chi =-\sqrt{7}/2$ has been established [285]. The same dependence of $\gamma$ on $\chi$ has been corroborated by Hartree calculations with angular momentum projection performed before variation [286].

The need for cubic terms in IBM-1 for the production of triaxiality has also been demonstrated through studies within the Algebraic Collective Model (ACM) [83,84,85] (see Section 2.2), which is an algebraic version of the Bohr collective model allowing for rapidly converging calculations [92,287]. It has been seen that the occurrence of triaxiality within the ACM requires the inclusion of terms proportional to ${cos}^2\left(3\gamma \right)$, which in IBM-1 correspond to terms quadratic in $QQQ$ [288,289].

The study of shape/phase transitions from axial to triaxial shapes within IBM-1 has demonstrated the need for a three-dimensional parameter space [290,291,292], which can be obtained only with the inclusion of higher-order terms, as opposed to the two-dimensional parameter space sufficing in the case of IBM-1, called the Casten triangle [255].

4.2. Interacting Boson Model-2

In the Interacting Boson Model-2 (IBM-2), a distinction is made between bosons corresponding to valence proton pairs and those corresponding to valence neutron pairs, counted from the nearest closed shells. Triaxial shapes occur in the SU(3) dynamical symmetry, when valence protons are holes and valence neutrons are particles, or vice versa [293,294,295,296,297,298].

In the SU(3) dynamical symmetry the parameter $\chi$ in the quadrupole operator of Equation (4) obtains the value $\chi =-\sqrt{7}/2$, which makes the Q operator a generator of SU(3). However, the operator Q with parameter value $\chi =+\sqrt{7}/2$ is also a generator of SU(3), called $\overline{\mathrm{SU}\left(3\right)}$ for distinction. SU(3) represents axially symmetric prolate (rugby-ball-like) shapes corresponding in the classical limit to $\gamma =0$, while $\overline{\mathrm{SU}\left(3\right)}$ represents axially symmetric oblate (pancake-like) shapes corresponding in the classical limit to $\gamma =60$°. Within IBM-2, SU(3) has been used for valence protons or neutrons outside closed shells and up to midshell, while $\overline{\mathrm{SU}\left(3\right)}$ has been used for valence protons or neutrons beyond midshell, which are counted from the closed shell above, i.e., they represent holes in their own shell. For N bosons corresponding to valence particles, the corresponding SU(3) irrep is $(2N,0)$, while for N bosons corresponding to valence holes, the corresponding SU(3) irrep is $(0,2N)$. Thus, in a nucleus with $N_p$ bosons corresponding to proton holes and $N_n$ bosons corresponding to neutron particles, the relevant irreps are $(0,2N_p)$ and $(2N_n,0)$, respectively, coupled into the most stretched irrep $(2N_n,2N_p)$, which corresponds to a triaxial shape, as one can see from Equation (1). The symmetry that results from the coupling of SU(3) and $\overline{\mathrm{SU}\left(3\right)}$ has been called SU(3)^∗ [293,294,295,296,297,298,299]. ${}_{44}{}^{104}\mathrm{Ru}_{60}$ [293,294,295,297] and ${}_{76}{}^{192}\mathrm{Os}_{116}$ [298] have been suggested as experimental manifestations of the SU(3)^∗ symmetry.

The presence of SU(3)^∗ in IBM-2 has as a consequence that the symmetry triangle of IBM-1 [255] is becoming a tetrahedron, with U(5), O(6), SU(3), and SU(3)^∗ at its vertices [300,301,302,303]. Shape/phase transitions within the parameter space of IBM-2 have been considered in detail in Refs. [304,305,306], while triaxial shapes have also been considered in Refs. [307,308]. Triaxiality in ^108−112Ru has been considered within IBM-2 in Ref. [309].

The assumptions made in relation to the SU(3) irreps used in SU(3)^∗ are supported microscopically by the findings of the proxy-SU(3) scheme, described in Section 3.3. In Table I of Ref. [5], one sees that in proxy-SU(3) prolate ($\lambda >\mu$), irreps occur in the lower half of each shell, while oblate ($\lambda <\mu$) irreps occur in the upper halves of shells. The main difference is that in IBM, the transition from prolate to oblate irreps occurs exactly at mid-shell, while in proxy-SU(3), it occurs further within the upper half of the shell. This kind of asymmetry has also been seen recently within an IBM model including higher-order interactions [310,311], called SU3-IBM for simplicity. The rich structure of the SU3-IBM model is expected to be the subject of intense investigations in the near future.

4.3. The Interacting Vector Boson Model

Within the Interacting Vector Boson Model (IVBM) [312,313,314], collective nuclear properties are described in terms of two vector bosons of angular momentum one, which can be considered as elementary excitations, using the Bargmann–Moshinsky basis [315,316,317]. Its overall symmetry is Sp(12,R), which is a noncompact algebra; thus, in general, the number of bosons is not conserved. However, it does contain a U(6) subalgebra, within which the boson number is conserved [314]. Within the IVBM, an SU(3)^∗ symmetry has been found [318], analogous to the one existing in IBM-2, and triaxial shapes, as well as shape/phase transitions involving triaxial shapes, have been considered within it [318], with ¹⁹²Os used as an example of experimental manifestation of this symmetry.

Furthermore, a Vector Boson Model with the SU(3) symmetry broken by the presence of a pairing interaction in the nuclear Hamiltonian has been introduced [319], in which the odd–even staggering within the $\gamma$-bands, which is a hallmark of triaxial behavior, as discussed in Section 6.2, can be satisfactorily described in terms of mixing of the $\gamma$-band with the ground state band [320,321]. In these considerations, the SU(3) irrep accommodating the ground state band and the $\gamma$ band (therefore, having $\mu \ge 2$) has been treated as a free parameter. It would be interesting to examine what the consequences will be of using [322] a microscopically derived SU(3) irrep, as the one occurring in the proxy-SU(3) scheme [5,6] within the VBM scheme.

4.4. The Coherent State Model

The Coherent State Model [323,324,325] is a boson model providing a description of the ground, $\gamma$, and $\beta$ bands based on the coherent state formalism. It has been used for the description of triaxial nuclei in Refs. [125,126,127,128,129].

5. Self-Consistent Mean-Field Methods

In the microscopic nuclear structure calculations based on the shell model, the nucleons are assumed to live in a specific single-particle potential, which represents the nuclear mean field. Then, configuration mixing is performed, involving all possible states in the realm of its application, called the shell model space. In the self-consistent mean-field methods, the opposite approach is used, namely, the nuclear mean-field is determined though a self-consistent calculation, fitting it to the nuclear structure data [326]. The simplest variational method for the determination of the wave functions of a many-body system is the Hartree–Fock (HF) method [138,327], first used in atomic and molecular physics. In the case of nuclei, the pairing interaction has to be included, leading to the Hartree–Fock–Bogoliubov (HFB) approach [138,327]. The inclusion of pairing is facilitated by the BCS approximation, which allows the formation of pairs of degenerate states connected through time reversal, as in the BCS theory of superconductivity in solids [328]. Three different choices have been made for the effective nucleon–nucleon interaction: the Gogny interaction [4,329,330] and the Skyrme interaction [331,332,333,334,335,336], used in a non-relativistic framework, and the relativistic mean-field (RMF) models [337,338,339,340,341,342,343,344,345,346,347], the last having the advantage of providing a natural explanation of the spin-orbit force. A compilation of various parametrizations of the effective interactions in use can be found in Table I of the review article [326]. Extensive codes have been published, facilitating the use of the Skyrme forces [348,349] and the RMF [340,350] all over the nuclear chart.

The self-consistent mean-field approach used in nuclear structures [351,352] strongly resembles the density functional theory used for the description of many-electron systems [353,354]. The main difference is that in electronic systems, one can derive, very accurately, the energy density functionals by ab initio methods from the theory of electron gas, while in nuclear structures, the effective energy density functionals are motivated by ab initio theory [355], but they contain free parameters fitted to the nuclear structure data.

In early theoretical work on triaxial nuclei, modified harmonic oscillators [356,357] and generalized forms [358] of the Nilsson model have been used. In addition, an early microscopic–macroscopic approach for the calculation of nuclear potential energy surfaces is the Nilsson–Strutinsky model. In phenomenological models, like the liquid drop model of vibrations and rotations (see Appendix 6A of [36]), the distribution of nucleons in phase space is supposed to be homogeneous, which is not the case in the nuclear shell model, in which the distribution becomes inhomogeneous. Strutinsky’s method ([359], see also section 2.9 of [138]) allows for calculations of the shell-model corrections to the liquid drop energy of the nucleus, starting from a modified Nilsson’s level scheme [186,187,360]. The corrections depend on the occupation number and on the deformation. The Nilsson-Strutinsky model has been used for the description of triaxial nuclei for example in [361,362,363,364].

The self-consistent mean-field approaches determine the ground state properties of nuclei. A method allowing the calculation of excitation spectra and B(E2) transition rates has been introduced by Otsuka and Nomura [365,366,367,368,369,370,371]. The parameters of the Interacting Boson Model (IBM) [222,248,249,250,251] are determined by mapping the total energy surface obtained from the relativistic mean-field approach onto the total energy surface determined by IBM. One is then free to use the IBM Hamiltonian with these microscopically derived parameters for the calculation of spectra and electromagnetic transition rates. This method has already been applied to some regions of the nuclear chart exhibiting triaxiality [371,372,373]. A similar method has been recently used in the non-relativistic mean field framework employing the Skyrme interaction [374,375].

Triaxiality in various nuclei has been considered within the HF method in Refs. [376,377,378,379,380,381,382,383,384], and within HFB in Refs. [385,386,387,388,389,390,391]. Studies of triaxility in various nuclei using the Gogny force can be found in Refs. [392,393,394,395,396,397,398,399], while the Skyrme force has been used in studying triaxiality in specific nuclei in Refs. [201,374,400,401,402,403,404,405]. Studies of triaxiality in specific nuclei using the relativistic mean field approach can be found in Refs. [371,372,373,406,407,408,409,410,411,412,413,414,415]. These will be discussed in the relevant sections regarding the corresponding series of isotopes.

6. Empirical Signatures of Triaxiality

6.1. The Shape Parameter $\gamma$

Within the Rigid Triaxial Rotor Model [40,41], described in Section 2.1, the value of the shape parameter $\gamma$ can be obtained directly from the ratio of the energy of the second $2^{+}$ state, $2_2^{+}$, over the energy of the first $2^{+}$ state, $2_1^{+}$

$$R={\displaystyle \frac{E\left(2_2^{+}\right)}{E\left(2_1^{+}\right)}}$$

(5)

through the expression [28,255]

$$\gamma ={\displaystyle \frac{1}{3}}{sin}^{-1}\left({\displaystyle \frac{3}{R+1}}\sqrt{{\displaystyle \frac{R}{2}}}\right).$$

(6)

One can also obtain $\gamma$ from the ratio of transition rates (branching ratio)

$$R_2={\displaystyle \frac{B(E2;2_2^{+}\to 2_1^{+})}{B(E2;2_2^{+}\to 0_1^{+})}}$$

(7)

through the expression [28,255]

$$R_2={\displaystyle \frac{20}{7}}{\displaystyle \frac{\frac{{sin}^{2}3\gamma }{9-8{sin}^{2}3\gamma }}{1-\frac{3-2{sin}^{2}3\gamma }{\sqrt{9-8{sin}^{2}3\gamma }}}}.$$

(8)

The ratios R and $R_2$ are shown in Figure 3 of Ref. [28]. The ratio R starts from the value 2 at $\gamma =30$° and raises towards infinity for $\gamma =0$, while the ratio $R_2$ shows the opposite behavior, starting from 1.43 at $\gamma =0$ and raising towards infinity at $\gamma =30$°. At $\gamma =15$°, one has $R=6.85$ and $R_2=2.71$. Nuclei with $R<6.85$ and $R_2>2.71$ are expected to have $\gamma >15$°, while nuclei with $R>6.85$ and $R_2<2.71$ are expected to have $\gamma <15$°.

However, atomic nuclei are not rigid bodies; therefore, the collective variables $\beta$ and $\gamma$ do not obtain a fixed value for each nucleus. In addition, $\beta$ and $\gamma$ are not directly measurable quantities. It is, therefore, desirable to try to estimate the shape of the nucleus through model-independent quantities that would be direct observables. It turns out that the quadrupole shape invariants introduced by Kumar in 1972 [273,274] can serve this purpose.

The quadrupole shape invariants [273,274,275,276] are the expectation values of higher powers of the quadrupole operator, calculated in a given state, which can be taken to be the ground state. The second-order quadrupole shape invariant, $q_2$, which corresponds to the matrix elements of $Q·Q$, turns out to be proportional to ${\beta }^2$, while the third-order invariant, $q_3$, which corresponds to the matrix element of ${\left(QQQ\right)}^0$, turns out to be proportional to ${\beta }^{3}cos\left(3\gamma \right)$. Therefore, $q_2$ can serve in estimating the value of $\beta$, while the ratio $q_3/q_2^{3/2}$ can serve for estimating the value of $cos\left(3\gamma \right)$. nth order invariants of the form $q_n/q_2^{n/2}$ can also be used [276].

In the framework of the IBM, one can easily see that the collective variable $\gamma$ is connected [272] to the value of the parameter $\chi$, appearing in the expression for the quadrupole operator given in Equation (4). The question of equivalence between $\gamma$-instability and rigid triaxility within the O(6) dynamical symmetry of the IBM has been posed [267], concluding that the ground states of the two cases are equivalent, but the excited states are not [269].

Recent calculations [277] in the framework of the configuration-interaction shell model, allowing for the calculation of higher-order quadrupole invariants and, therefore, of the fluctuations of the shape variables $\beta$ and $\gamma$, show that $\beta$ has a non-negligible degree of softness, while $\gamma$ is usually characterized by large fluctuations, rendering its effective value not meaningful [277]. A by-product of this method is that doubly magic nuclei cannot be considered as spherical, because the notion of a well-defined shape does not apply to them [277].

6.2. Rigid Triaxiality vs. $\gamma$-Softness

The relative displacement of the levels with odd angular momentum L relative to their neighbors with even L within the $\gamma$ band, called the odd-even staggering, has long been considered as an indicator of triaxiality [416,417].

Odd-even staggering in $\gamma$ bands is studied using the quantity [416,417]

$$S\left(L\right)={\displaystyle \frac{E\left(L_{\gamma }^{+}\right)+E\left[{(L-2)}_{\gamma }^{+}\right]-2E\left[{(L-1)}_{\gamma }^{+}\right]}{E\left(2_1^{+}\right)}},$$

(9)

which measures the displacement of the $(L-1)$ level in relation to the average of its neighbors, L and $(L-2)$, normalized to the energy of the first excited state of the nucleus.

In Figure 1a, the odd–even staggering occurring in the $\gamma$-rigid Davydov model [40,41] for various values of $\gamma$ is shown. It is observed that, irrespectively of the value of $\gamma$, minima occur at odd values of the angular momentum L.

In Figure 1b, the odd–even staggering occurring in several $\gamma$-soft models, ranging from the vibrator with U(5) symmetry to the O(6) limit of the interacting boson model (IBM) [222], and including the critical point symmetry E(5) [97] and the E(5)-${\beta }^6$ model [418], in which the infinite square well potential in the deformation variable $\beta$, used in E(5), is replaced by a ${\beta }^6$ potential. It is observed that for all models, minima occur at even values of L, which is the opposite behavior in comparison to what is seen in the case of the rigid triaxial models in Figure 1a. This behavior can be easily attributed to the O(5) symmetry underlying the U(5) and O(6) dynamical symmetries of IBM [222], which implies that the levels of the $\gamma$-band are grouped by the degeneracies imposed by O(5) [419,420] as 2, (3,4), (5,6),…, (see Table I and Figure 3 of Ref. [418] for details), while in the rigid triaxial rotor of Davydov [40,41], the relevant grouping is (2,3), (4,5), (6,7),….

In addition to the opposite sign of staggering, in Figure 1, it is seen that the size of the odd–even staggering in the rigid triaxial cases is an order of magnitude higher than the staggering in the $\gamma$-soft cases.

It should be noticed that although very few nuclei exhibit odd–even staggering corresponding to rigid triaxial shapes (see Figure 4 of Ref. [417]), many nuclei exhibit odd–even staggering corresponding to soft triaxiality [421].

In corroboration of these observations, consistent modeling [422] of several observables of heavy nuclei reveals that for the triaxiality parameter $\gamma$, the zero value does not occur even in deformed nuclei, but rather, the $\gamma$ values exhibit a variance of around 8° [422], while acquiring values of 10° or higher (see, for example, Figure 5 of Ref. [5]).

Another quantity exhibiting different behavior in the $\gamma$-rigid and $\gamma$-soft cases is the ratio [423,424]

$$R\left(L\right)={\displaystyle \frac{E\left(L_{\gamma }^{+}\right)-E\left(L_g^{+}\right)}{E\left(2_{\gamma }^{+}\right)-E\left(2_g^{+}\right)}},$$

(10)

based on the energy differences between the $\gamma$-band and the ground state band (g) at the same angular momentum L. In Figure 2a, one sees that in the rigid triaxial case, the ratios R(L) increase rapidly and gradually bending upwards, while in Figure 2b, one observes that in the $\gamma$-soft cases, the ratios R(L) increase much more slowly and linearly. The qualitative differences between these two cases can be clearly seen in Figure 3.

It should be noticed that the rigid triaxial rotor of Davydov [40,41] at $\gamma =30$° has the same selection rules for electric quadrupole transitions as the quadrupole vibrator model and the $\gamma$-unstable model of Wilets and Jean [37]. Therefore, the selection rules of electric quadrupole transitions cannot help in distinguishing rigid triaxial behavior from the $\gamma$-soft one (see p. 191 of Ref. [255]).

7. Global Systematics of Triaxiality

After examining in Appendix B, Appendix C, Appendix D, Appendix E, Appendix F, Appendix G and Appendix H separately each isotopic chain in the region $Z=24$–98, and drawing partial conclusions in various specific regions, we are now going to try to build the general picture regarding areas on the nuclear chart in which triaxiality is favored.

We are going to consider first the predictions made within the proxy-SU(3) approximation to the shell model. A review of the proxy-SU(3) symmetry scheme and its connection to the Nilsson model and to the spherical shell model has been given in Ref. [6], while the group theoretical details regarding the highest weight irreducible representations of SU(3) and their dominance have been given in Refs. [240,425].

In Figure 4, the proxy-SU(3) predictions, taken from Equation (1), the deformation variable $\gamma$ for all experimentally known [426] nuclei with $18\le Z\le 80$ and $18\le N\le 124$ is shown [28]. The horizontal and vertical stripes covering the nucleon numbers 22–26, 34–48, 74–80, 116–124, and 172–182, within which substantial triaxiality is expected to occur [28], are also shown. We remark that non-zero values of $\gamma$ occur almost everywhere across the nuclear chart, in agreement with recent suggestions in the framework of the Monte Carlo Shell Model [29,30,31] and the Triaxial Projected Shell Model [32] that a non-zero degree of triaxiality appears everywhere on the nuclear chart, including regions of prolate deformed nuclei considered as ideal examples of axial deformation [29,30,31]. In addition, recent studies of consistent modeling of several observables in heavy nuclei has also demonstrated the need for broken axial symmetry, with the variance of the triaxiality parameter $\gamma$ centered in the range of $\gamma \sim 8$° [422].

Figure 5 is derived from Figure 4 by eliminating nuclei with low triaxiality, i.e., nuclei with $\gamma =0$–15°, being close to prolate ($\gamma =0$) shapes, and nuclei with $\gamma =45$–60°, being close to oblate ($\gamma$ = 60°) shapes, and showing only nuclei with high triaxiality, i.e., nuclei with $\gamma$ = 15–45°, being close to maximally triaxial ($\gamma =30$°) shapes. We see that a ”staircase” pattern appears, falling within the stripes in which triaxiality is expected to be favored according to the proxy-SU(3) predictions [28].

At this point, it is worth examining to which extent these proxy-SU(3) predictions agree with available experimental data [426].

In Figure 6, all nuclei with known [426] $2_1^{+}$ and $2_2^{+}$ levels are shown, divided into two groups, those with $R<6.85$ (expected to have $\gamma >15$°, as discussed in Section 6.1) and those with $R>6.85$ (expected to have $\gamma <15$°). We see that many nuclei belong to the first group. However, not all of them are expected to exhibit strong triaxiality, since $R=2$ also corresponds to the simple vibrator with U(5) symmetry [222].

In order to confront this problem, all nuclei up to $Z=80$ with experimentally known $2_1^{+}$ and $2_2^{+}$ levels, as well as $B(E2;2_2^{+}\to 0_1^{+})$ and $B(E2;2_2^{+}\to 2_1^{+})$ transition rates, have been collected in Table I of Ref. [28]. The experimental $R_{4/2}=E\left(4_1^{+}\right)/E\left(2_1^{+}\right)$ ratio, a well-known [255] indicator of collectivity, is also shown, in order to facilitate the recognition of nuclei being close to the simple vibrator value of $R_{4/2}=2$, mentioned above.

In Figure 7, the nuclei of Table I of Ref. [28] are subdivided into two groups, those with $R<6.85$ and $R_2>2.71$ (expected to have $\gamma >15$°, as discussed in Section 6.1) and those with $R>6.85$ and $R_2<2.71$ (expected to have $\gamma <15$°). We see that by taking into account the value of the branching ratio, the number of candidates for substantial triaxiality is drastically reduced, while most of the remaining candidates are aligned along the regions (indicated in Figure 5 by green stripes) for which oblate SU(3) irreps are predicted by the proxy-SU(3) symmetry, in agreement with the situation appearing in Figure 2.

The reasoning behind Figure 5 and Figure 7 assumes that large values of the collective variable $\gamma$ imply large triaxiality. Thinking in terms of potential energy surfaces (PES), this means a minimum in the PES close to 30°. But the question remains if this minimum is deep, implying robust triaxiality, or shallow, suggesting $\gamma$-softness.

One way to address this question is to consider the decrease in energy due to triaxiality occurring within the macroscopic–microscopic approach of the Finite-Range Droplet Model (FRDM) or Finite-Range Liquid-Drop Model (FRLDM) [26,27]. It is plausible that larger reduction in energy due to triaxiality would mean more robust triaxiality.

Nuclei for which the FRDM predicts a decrease in energy due to triaxiality equal or larger than 0.01 MeV, taken from Ref. [27], are shown in Figure 8. We see that the “staircase” structure regarding experimentally known nuclei seen in Figure 5 is reproduced, thus corroborating the assumption that large values of $\gamma$ imply robust triaxiality. It should be noticed that these two predictions come from completely different models of completely different nature. The FRDM predictions come from a macroscopic–microscopic model with parameters fitted in order to reproduce a great variety of nuclear observables over the whole nuclear chart, while the proxy-SU(3) predictions come from symmetry arguments, based on the Pauli principle and the short-range nature of the nucleon–nucleon interaction [238,240], in a completely parameter-free way. In addition, in Figure 8, we see regions of nuclei not appearing in Figure 5, since they correspond to yet experimentally unknown nuclei with $Z=42$–46, $N=96$–110 and $Z=52$–62, $N=120$–122. These additional regions also fall within the stripes of favored triaxiality predicted by proxy-SU(3).

A different picture appears for the FRDM predictions beyond $Z=82$, depicted in Figure 9, in which most of the nuclei with a decrease in energy due to triaxiality equal or larger than 0.01 MeV are lying outside the stripes for favored triaxiality predicted by the proxy-SU(3) symmetry. In particular, this happens in the regions with $Z=82$–94, $N=154$–160, and $Z=88$–92, $N=108$–114, as well as in $Z=96$–108, $N=130$–142.

In Figure 10, we have gathered the partial conclusions reached in Appendix B, Appendix C, Appendix D, Appendix E, Appendix F and Appendix G of the present review up to $Z=86$, $N=124$. It is seen that nuclei for which substantial evidence for triaxiality exists fall within the stripes in which triaxiality is predicted to be favored by the proxy SU(3) symmetry, shown in Figure 5. Agreement of these nuclei with Figure 8, in which nuclei with FRDM predictions of the decrease in energy due to triaxiality equal or larger than 0.01 MeV are depicted, is also seen.

Different conclusions are drawn from Figure 11, in which the partial conclusions reached in Appendix F, Appendix G and Appendix H of the present review above $Z=70$, $N=108$ are depicted. In this case, the partial conclusions of the present review indicate the presence of triaxiality within the region $Z=88$–98, $N=132$–154. There is neither overlap between these nuclei and the nuclei predicted by FRDM to exhibit decrease in energy due to triaxiality equal or larger than 0.01 MeV, shown in Figure 9, nor overlap with the stripes in which triaxiality is favored by the proxy-SU(3) symmetry.

In summary, most of the theoretical work within various models considered in this review predicts substantial triaxiality for nuclei in the region up to $Z=82$, $N=126$ lying within the stripes favoring triaxiality according to the proxy-SU(3) symmetry, exhibiting, in addition, a considerable (larger than 0.01 MeV) reduction in energy due to triaxiality according to the FRDM model. In contrast, for nuclei beyond $Z=82$, $N=126$, theoretical work within various models considered in this review predicts triaxiality in the region $Z=88$–98, $N=132$–154, in apparent disagreement with the predictions of both the proxy-SU(3) symmetry and the FRDM model.

In a first attempt to resolve this puzzle, we have extended Table 1 of Ref. [28] into the actinide region, including all nuclei for which the first two $2^{+}$ states, as well as the B(E2) transition rate among them, is known [426]. As already discussed in Section 6.1, nuclei with $R>6.85$ and $R_2<2.71$ are expected to have $\gamma <15$°. This is indeed the case for all nuclei shown in

Table 1. Nuclei with experimentally known $2_1^{+}$ and $2_2^{+}$ levels, as well as known $B(E2;2_2^{+}\to 0_1^{+})$ and $B(E2;2_2^{+}\to 2_1^{+})$ transition rates. Data have been taken from Ref. [426]. In nuclei in which $\beta$- and $\gamma$-bands are assigned in Ref. [426], the $2^{+}$ state of the $\gamma$-band is chosen as the $2_2^{+}$. Energies are given in MeV, while B(E2) transition rates are given in W.u. The ratios R and $R_2$ are calculated from Equations (5) and (7), respectively. The ratio $R_{4/2}$, a well-known [255] indicator of collectivity, is also shown, along with the $\gamma$ value obtained from the ratio R (called ${\gamma }_R$), from the ratio $R_2$ (called ${\gamma }_{R_2}$), and from the proxy-SU(3) symmetry using Equation (1) (called ${\gamma }_{hw}$), using the SU(3) irreps given in Table 2. See Section 7 for further discussion.

Nucleus	$2_1^{+}$	$2_2^{+}$	R	$2_2^{+}\to 0_1^{+}$	$2_2^{+}\to 2_1^{+}$	${\mathit{R}}_2$	${\mathit{R}}_{4/2}$	${\mathit{\gamma }}_{\mathit{R}}$	${\mathit{\gamma }}_{{\mathit{R}}_2}$	${\mathit{\gamma }}_{\mathbf{hw}}$
	MeV	MeV		W.u.	W.u.
${}_{90}{}^{230}\mathrm{Th}_{140}$	0.053	0.781	14.68	2.9 $\left({\displaystyle \genfrac{}{}{0pt}{}{+6}{-4}}\right)$	5.5 $\left({\displaystyle \genfrac{}{}{0pt}{}{+10}{-8}}\right)$	1.90	3.271	10.4	10.4	6.8
${}_{90}{}^{232}\mathrm{Th}_{142}$	0.049	0.785	15.91	2.9 (4)	7.2 (7)	2.48	3.284	10.0	14.1	7.7
${}_{92}{}^{234}\mathrm{U}_{142}$	0.043	0.927	21.31	2.9 (5)	4.9 (8)	1.69	3.296	8.7	8.1	7.3
${}_{92}{}^{238}\mathrm{U}_{146}$	0.045	1.060	23.61	3.04 (18)	5.3 (4)	1.74	3.303	8.3	8.8	6.5
${}_{98}{}^{250}\mathrm{Cf}_{152}$	0.043	1.032	24.15	2.3 (3)	3.7 (4)	1.61	3.321	8.2	6.9	10.4

, in agreement with $\gamma$ values shown for actinides in the original Davydov papers [40,41] (see Table 3 of [40] and Table 5 of [41]), as well as to $\gamma$ values provided by the extended Thomas–Fermi plus Strutinsky integral method [377] (see Tables II and III of [377]), discussed in Appendix H. In contrast, the Triaxial Projected Shell Model (TPSM) [32,198,209] appears to be providing, in the actinide region $\gamma$ values systematically higher than these of the aforementioned approaches (see Table 2 of [198], Table 2 of [209], and Table 3 of [32]). It should also be noticed that a method exists for extracting the value of $\gamma$ from the experimental $R_{4/2}$ ratio [52]. This is convenient, since the $R_{4/2}$ ratio is known experimentally in many more nuclei than the R ratio of Equation (5), but has the handicap of providing systematically much higher values of $\gamma$ than these provided by the R and $R_2$ (Equation (7)) ratios (see Table I of [52] for a long list of examples). The results provided by the TPSM and through the use of the $R_{4/2}$ ratio have also been taken into account in Appendix H, thus blurring the whole picture. It is clear that further microscopic work is needed in order to clarify this issue.

Finally, in Figure 12, Figure 13, Figure 14 and Figure 15, we provide, for quick qualitative reference, the proxy-SU(3) predictions in several parts of the nuclear chart, obtained by using the numerical results for the highest weight (hw) and next highest weight (nhw) irreps given in Tables 2–6 of Ref. [247]. The values shown are the ones corresponding to the hw irrep, except in the cases in which $\mu =0$ occurs for the proton hw irrep and/or the neutron hw irrep, in which case the values depicted are the average of the hw and nhw cases, in a simple schematic approximation, as described in Ref. [28]. The following observations can be made.

(a) In Figure 12, the protons lie within the region 34–48, in which triaxiality is favored [28]. If the neutrons also lie in regions favoring triaxiality (34–48 and 74–80), the values of $\gamma$ raise above 30°.

(b) A similar situation appears in Figure 15. In W-Hg the protons lie within the region 74–80, in which triaxiality is favored. If the neutrons also lie in the region 116–124, in which triaxiality is favored, the values of $\gamma$ raise above 30°.

(c) In Figure 13, the protons lie outside the regions in which triaxiality is favored (22–26, 34–48, 74–80, and 116–124) [28]. If the neutrons lie in the region 74–80, in which triaxiality is favored, the values of $\gamma$ raise above 15°.

(d) A similar situation appears in Figure 14. The protons lie outside the regions in which triaxiality is favored. If the neutrons lie in the region 116–124, in which triaxiality is favored, the values of $\gamma$ raise above 15°.

In short, we conclude that $\gamma$ values above 30° are reached when both the protons and the neutrons of a given nucleus lie within regions in which triaxiality is favored, while for $\gamma$ values above 15° to be achieved, it is necessary for either the protons or the neutrons of a given nucleus to lie within regions in which triaxiality is favored.

It should be mentioned that predictions for the collective variable $\gamma$ have also been derived [246,428] using the pseudo-SU(3) symmetry [228,229,230,231,232]. It is remarkable that although the pseudo-SU(3) and proxy-SU(3) schemes are based on different approximations to the shell model, discussed in Section 3.2 and Section 3.3, respectively, their predictions for the $\beta$ and $\gamma$ collective variables are very similar, once the appropriate h.w. irreps are used in each of them (see Figures 4 and 8 of [246], as well as Figures 3 and 4 of [428], for comparisons between the pseudo-SU(3) and proxy-SU(3) predictions). In addition, detailed predictions for the values of the $\beta$ and $\gamma$ collective variables have been given using the D1S Gogny interaction [4]. Comparisons between the D1S Gogny and proxy-SU(3) predictions have been carried out in Refs. [246,429,430,431,432]. It is seen that in most cases, the proxy-SU(3) predictions are lying close to the ones provided by the D1S Gogny calculations, lying within the variance determined in the latter (see Figures 1 and 2 of [429], Figures 1 and 2 of [430], Figures 2, 3, 8 and 9 of [431], Figures 9 and 11 of [432], and Figures 3 and 7 of [246]).

Since, as seen earlier in this section, the proxy-SU(3) symmetry seems to predict $\gamma$ values consistent with empirical data, it would have been interesting to be able to compare detailed proxy-SU(3) predictions to existing data for spectra and B(E2) transition rates, especially in nuclei in which the presence of triaxiality is experimentally confirmed. However, calculations of spectra [433] and B(E2)s [429,430,434] within the proxy-SU(3) scheme are still under development. A by-pass is provided by detailed calculations of spectra and B(E2)s within the IBM-1 framework, in which the free parameters of IBM-1 are determined through self-consistent mean-field calculations using a Skyrme energy density functional, by forcing the potential energy surfaces of IBM-1 to agree with the microscopically derived ones. Detailed spectra and B(E2) transition rates have been provided for ^162−184Hf [374,435], ^168−186W [374,435], and ^160−180Er [375], proving that the addition of triaxiality, as predicted by the proxy-SU(3) symmetry, largely improves the agreement of the calculated IBM-1 spectra and B(E2)s to the data. Extension of these calculations to neighboring series of isotopes, starting with the Yb ones, is in progress [436]. It should be remembered, as discussed in Appendix E.6, that recent experimental work [437] on ¹⁵⁴Sm, one of the deformed nuclei showing non-vanishing triaxiality according to Monte Carlo Shell Model calculations [31], determines the collective parameter values to be $\beta =0.2925\left(25\right)$ and $\gamma =5.0\left(15\right)$°, in close agreement with the proxy-SU(3) predictions [247] of $\beta =0.296$ and $\gamma =4.31$° (see Table 3 of [247]), as well as with the Monte Carlo Shell Model predictions of $\beta =0.28$ and $\gamma =3.7$° [31] (see also Table III of [437]). Detailed calculations of spectra and B(E2)s employing the proxy-SU(3) hw irreps in the framework of the Vector Boson Model have also been started [322], providing an alternative by-pass.

8. Open Questions

In the present review, attention has been focused on nuclei exhibiting empirical and/or theoretical values of the collective variable $\gamma$ around 30° (15° $<\gamma <$ 45°), for which, in addition, the theoretical reduction [26,27] in the potential energy of the ground state due to triaxiality is significant (larger than 0.01 MeV). As seen in Figure 5, Figure 8, and Figure 10, there is indeed a strong correlation between values of $\gamma$ around 30° and substantial reduction in the potential energy of the ground state due to triaxiality. It is, therefore, safe to refer to these nuclei as triaxial.

What has not been addressed in detail, and should probably attract more attention in future work, is the distinction between rigid triaxiality and $\gamma$-softness. In the former case, a potential exhibiting a deep minimum at a $\gamma$ value around 30° is expected to occur, while in the latter case, the potential in $\gamma$ is expected to have a shallow minimum, allowing $\gamma$ to obtain different values at minimal expense in energy.

The empirical feature most widely used for making a distinction between rigid triaxiality and $\gamma$-softness is the odd–even staggering within the $\gamma$-band [416,417], as described in Section 6.2. However, the very nature of the $\gamma$ band, as well as of the $\beta$ band, has been, in recent years, a point of dispute [71,72,73,74,75,76,77]. While in the framework of the collective model of Bohr and Mottelson, these are considered as bands built on vibrational modes of the $\gamma$ and $\beta$ degrees of freedom, respectively [36], recent work suggests a two-particle–two-hole (2p2h) nature for the $K=0$ ($\beta$) band [76] and breaking of the axial symmetry, i.e., triaxiality, for the $K=2$ ($\gamma$) band [75,77].

Many $K=0$ bands have been seen experimentally in several nuclei [438,439,440,441,442,443,444,445], for which various different theoretical interpretations have been suggested [446,447,448,449,450,451,452,453,454,455,456], making the identification of the microscopic mechanism underlying the first excited $K=0$ band in each nucleus a tricky point [457]. $K=2$ bands tend to have moments of inertia more similar to those of the ground state bands, but still, corrections to this simple picture are required. The need for different mass coefficients in the ground state, $\gamma$, and $\beta$ bands has been demonstrated in Refs. [458,459,460,461,462,463].

Recent calculations in the framework of the Monte Carlo Shell Model [29,30,31] and the Triaxial Projected Shell Model [32] favor the interpretation of the $K=2$ bands as due to deviation from axial symmetry, which can be small in certain regions of the nuclear chart, but large in others, as already depicted in Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11.

The question of the transition from $\gamma$-softness to rigid triaxial deformation should also be addressed. Is it signifying a shape/phase transition, and of which order? $\gamma$-softness is expected to occur from nearly spherical to moderately deformed nuclei, characterized by the O(5) symmetry, which guarantees seniority [419,420] to be a good quantum number, thus producing odd–even staggering of $\gamma$-soft type in the $\gamma$ bands [417] (see also Table I and Figure 3 of [418]). Indeed, the E(5) critical point symmetry [97] characterizing the second-order shape/phase transition from spherical to $\gamma$-soft nuclei does have an O(5) subalgebra, guaranteeing the presence of seniority as a good quantum number. The Y(5) shape/phase transition [105] in the angle variable $\gamma$ from axially to triaxially deformed nuclei is of second order, while the X(5) shape/phase transition [98] from spherical to prolate ($\gamma =0$) axially deformed nuclei is of first order. Introducing non-zero equilibrium $\gamma$ values in X(5), the T(5) critical point symmetry [110] is obtained. However, X(5) and T(5) represent special approximate solutions of the Bohr Hamiltonian with a yet unknown algebraic structure.

The apparent disagreement for nuclei beyond $Z=82$, $N=126$ of the theoretical work within various models considered in this review with the predictions of both the proxy-SU(3) symmetry and the FRDM model should be further investigated, with an eye on its microscopic origin. Beyond mean-field calculations performed recently [464] using the Gogny interaction and taking triaxiality into account for the first time, a predominance of triaxial shapes in transitional superheavy nuclei is suggested, using the ${}_{114}{}^{288-296}\mathrm{Fl}_{174-182}$ isotopes as a testground. Extension of the proxy-SU(3) calculations, like the ones reported in Ref. [247], to the regions of the actinides and the superheavy nuclei are also called for.

On the other hand, the present study has been limited to nuclei above $Z=22$. Triaxiality is indeed seen in lighter nuclei, with clustering [465] playing a major role in this region. A detailed study of triaxiality in light nuclei might be an interesting project.

Finally, the present study has been focused on triaxiality in the ground state band and the $\gamma$-band of even–even nuclei. Higher-lying triaxial bands can occur in odd–odd nuclei by coupling two particles to a triaxial rotor, as first carried out in the framework of the tilted axis cranking theory [466], based on experimental evidence for the odd–odd nucleus ${}_{59}{}^{134}\mathrm{Pr}_{75}$ presented in Ref. [467]. The role of the rotating mean field of triaxial nuclei in breaking the chiral symmetry has been understood in Ref. [468], with the relevant theoretical developments reviewed in Ref. [469] and the relevant experimental findings for chiral doublet bands in odd–odd and odd nuclei collected in Ref. [470]. Evidence for chiral bands in even–even nuclei has been recently observed in the nucleus ${}_{60}{}^{136}\mathrm{Nd}_{76}$ [471], with theoretical approaches with the Particle Rotor Model [472] and the Triaxial Projected Shell Model [473] supporting the existence of five possible chiral doublets in it. For chiral bands, a separate study would be required.

9. Conclusions and Outlook

The main conclusions of the present study are summarized here.

(a) Triaxiality in even–even nuclei appears to be present all over the nuclear chart, albeit in a non-uniform way.

(b) Values of the collective variable $\gamma$ close to maximal triaxiality (30°) appear within specific stripes on the nuclear chart (22–26, 34–48, 74–80, 116–124, and 172–182), predicted by the proxy-SU(3) symmetry in a parameter-free way, based on the Pauli principle and the short-range nature of the nucleon–nucleon interaction.

(c) Calculations within the Finite-Range Droplet Model (FRDM), which is based on completely different assumptions and uses free parameters fitted in order to reproduce basic nuclear data, show that the reduction of the potential energy of the ground state due to triaxiality shows non-negligible values (above 0.01 MeV) roughly within the above mentioned stripes, thus correlating $\gamma$ values close to maximal triaxiality (30°) with deep minima of the potential energy.

(d) The existing few results for $\gamma$ values coming from computationally demanding Monte Carlo Shell Model calculations offer support to the above conclusions.

(e) Calculations performed within a variety of theoretical approaches support the above findings up to $Z=82$, $N=126$. Further work is needed in heavier nuclei, in which discrepancies from the above picture might start to appear.

(f) On the microscopic front, values of $\gamma$ close to 30° (maximal triaxiality) can be reached in nuclei in which the protons or the neutrons fall within the above-mentioned nucleon intervals favoring triaxiality. Values of $\gamma$ close to 60° (oblate shapes) can be reached in nuclei in which both the protons and the neutrons fall within the above-mentioned nucleon intervals favoring triaxiality.

Points demanding further investigation include the distinction between rigid triaxiality and $\gamma$-softness, as well as the nature of the transition from one kind of deformation to the other. Relatively scarce information exists on triaxiality in the actinides and in the superheavy elements, calling for extension of the above-mentioned studies in these regions.