Nuclear Shape-Phase Transitions and the Sextic Oscillator

Citation: Lévai, G.; Arias, J.M. Nuclear Shape-Phase Transitions and the Sextic Oscillator. Symmetry 2023, 15, 2059. https://doi.org/ 10.3390/sym15112059 Academic Editors: Peter Otto Hess and Charalampos Moustakidis Received: 14 October 2023 Revised: 28 October 2023 Accepted: 10 November 2023 Published: 14 November 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). symmetry S S Review Nuclear Shape-Phase Transitions and the Sextic Oscillator Géza Lévai 1,* and José M. Arias 2,3 1HUN-REN Institute for Nuclear Research (ATOMKI), P.O. Box 51, H-4001 Debrecen, Hungary 2Departamento de Física Atómica, Molecular y Nuclear, Facultad de Física, Universidad de Sevilla, Apartado 1065, 41080 Sevilla, Spain; [email protected] 3Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, Fuentenueva s/n, 18071 Granada, Spain *Correspondence: [email protected] Abstract: This review delves into the utilization of a sextic oscillator within the β degree of freedom of the Bohr Hamiltonian to elucidate critical-point solutions in nuclei, with a specific emphasis on the critical point associated with the β shape variable, governing transitions from spherical to deformed nuclei. To commence, an overview is presented for critical-point solutions E(5), X(5), X(3), Z(5), and Z(4). These symmetries, encapsulated in simple models, all model the β degree of freedom using an infinite square-well (ISW) potential. They are particularly useful for dissecting phase transitions from spherical to deformed nuclear shapes. The distinguishing factor among these models lies in their treatment of the γ degree of freedom. These models are rooted in a geometrical context, employing the Bohr Hamiltonian. The review then continues with the analysis of the same critical solutions but with the adoption of a sextic potential in place of the ISW potential within the β degree of freedom. The sextic oscillator, being quasi-exactly solvable (QES), allows for the derivation of exact solutions for the lower part of the energy spectrum. The outcomes of this analysis are examined in detail. Additionally, various versions of the sextic potential, while not exactly solvable, can still be tackled numerically, offering a means to establish benchmarks for criticality in the transitional path from spherical to deformed shapes. This review extends its scope to encompass related papers published in the field in the past 20 years, contributing to a comprehensive understanding of critical-point symmetries in nuclear physics. To facilitate this understanding, a map depicting the different regions of the nuclide chart where these models have been applied is provided, serving as a concise summary of their applications and implications in the realm of nuclear structure. Keywords: nuclear structure models and methods; collective models; Bohr Hamiltonian; quasiexactly solvable models; sextic potential PACS: 21.60.-n; 21.60.Ev; 21.60.Fw 1. Introduction Phase transitions are typically well-defined and associated with macroscopic systems [ 1 ]. However, in recent years, early indicators of phase transitions have been observed in mesoscopic systems [ 2 ], leading to the concept of quantum phase transitions [ 3 , 4 ]. In classical physics, phase transitions are traditionally classified into two categories: first-order and continuous phase transitions, based on the behavior of the order parameter at the critical point. At this juncture, the order parameter typically undergoes a discontinuous change from zero to a finite value in first-order transitions, whereas in continuous transitions, the change is smooth at the critical point. These same principles apply to quantum phase transitions. It is evident that the notion of a critical point is closely intertwined with the study of phase transitions. An intriguing facet of modern nuclear physics involves the exploration of the quantum shape-phase transition (QSPT) in atomic nuclei. In this context, QSPTs have been studied Symmetry 2023,15, 2059. https://doi.org/10.3390/sym15112059 https://www.mdpi.com/journal/symmetry Symmetry 2023,15, 2059 2 of 19 in nuclear physics with the introduction of shape variables. Different shape phases imply different spectral patterns (energy spectrum, electromagnetic transitions, etc.), and often different symmetries. The transitions from one shape phase to another thus correspond to structural rearrangements in the nucleus. Identifying these is thus of primary importance in understanding certain aspects of nuclear structure. The Bohr Hamiltonian was developed to capture the dynamics of quadrupole-type surface excitations in nuclei [ 5 ]. It relies on two key variables: β≥ 0, which quantifies the deviation of the nuclear surface from a spherical shape, and γ∈[ 0, π/ 3 ] , which characterizes its deviation from axial symmetry. In addition, describing the orientation of the nucleus requires Euler angles, rendering the problem in a five-dimensional space. The kinetic term encompasses all five variables, whereas the potential depends solely on the intrinsic β and γ shape variables. The position of the potential minimum dictates the nuclear shape, which could be spherical, deformed prolate, deformed oblate, deformed triaxial, deformed γ -unstable, and so forth. Changes in a control parameter in the Hamiltonian can produce a transition from one specific form to another. The point of change is called a critical point and is characterized by a critical value of the appropriate control parameter. In the context of the Bohr Hamiltonian [ 5 ], a few decades ago, Iachello introduced the concept of critical-point symmetries concerning quantum systems like atomic nuclei [ 6 ]. Iachello formulated two critical-point symmetries within the geometric model. One of these symmetries, known as E(5), characterizes the critical point in the shape transition from spherical to deformed γ -unstable shapes [ 6 ] and is particularly relevant for describing continuous shape-phase transitions. Another symmetry proposed by Iachello, referred to as X(5), pertains to the critical point in the transition from spherical to axially deformed shapes [ 7 ] and serves as a benchmark for first-order phase transitions. In addition to these critical points, Iachello also discussed a phase transition in the angle variables [ 8 ] and particularized the phase transition from axial to triaxial shapes in nuclei. The corresponding critical point is known as Y(5). Soon after, Bonatsos and collaborators proposed a model for the critical point in the transition from oblate to prolate shapes, approximately separating variables in the Bohr Hamiltonian for γ=π/ 6 [ 9 ]. This critical point is called Z(5). Some other critical-point solutions, such as X(3) and Z(4), have been proposed and are discussed below. Another widely employed nuclear structure model for exploring shapes and shapephase transitions is the Interacting Boson Model (IBM) [ 10 ]. While the IBM was originally formulated within a second quantization formalism, it is possible to gain a geometric understanding of the model by leveraging the concept of coherent states and the intrinsic state formalism [ 11 – 13 ]. This approach unveils geometric shapes, akin to the Bohr model, which are associated with the various dynamical symmetries of the IBM. These correspond to special situations in which exact analytical solutions of the many-body problem can be obtained using group theory techniques. These dynamical symmetries are labeled as U(5), SU(3) ( SU(3) ), and O(6). With the introduction of shape variables, the U(5) limit corresponds to spherical shapes, the SU(3) ( SU(3) ) limit yields prolate (oblate) axially symmetric shapes, and the O(6) limit generates deformed γ -unstable shapes. This prompts the search for counterparts to the E(5) and X(5) symmetries within the IBM. In this context, the critical points are referred to as e( 5 ) and x( 5 ) , respectively, and they represent the transitional pathways from U(5) to O(6) and from U(5) to SU(3). Significant experimental and theoretical investigations have been conducted to explore quantum phase transitions, particularly for even-even nuclei within the IBM, as documented in various reviews [14–18]. The introduction of critical-point symmetries has sparked a quest to identify nuclei that align with the predictions of these models. Additionally, it has spurred investigations into potentials that yield exact solutions within the Bohr Hamiltonian framework (for a comprehensive review, please refer to Ref. [ 19 ]). Our contribution to these endeavors includes the proposal of the sextic oscillator potential for the β shape variable, making the potential energy surface independent of γ [ 20 – 23 ]. This potential exhibits remarkable flexibility: its characteristics can vary depending on the parameters, featuring either a spherical minimum, a deformed minimum, or both. Moreover, the transition between Symmetry 2023,15, 2059 3 of 19 these different shape phases can be controlled analytically. The sextic oscillator belongs to the category of quasi-exactly solvable (QES) potentials [ 24 , 25 ], implying that only the lowest few energy levels can be obtained exactly. Fortunately, these few lowest energy levels generally correspond to the set of nuclear states for which a sufficient amount of experimental spectroscopic information is available, allowing for a quantitative comparison of experiment and theory. We have also derived electromagnetic transition rates ( B(E 2 ) , B(E 0 ) ) for this potential in closed form and computed benchmark values associated with the transitions between different shape phases [21–23]. During the last 15 years, a systematic investigation of the quantum shape-phase transitions linked to the sextic potential has been developed. Many works have employed the sextic oscillator in the Bohr Hamiltonian with potentials dependent on the γ shape variable. Various nuclear shapes have been explored, including γ -stable triaxial [ 26 ], γ - stable prolate [ 27 ], γ -rigid triaxial [ 28 ], and γ -rigid prolate [ 29 ] nuclei (for an extensive review from 2015, see Ref. [ 30 ]). Numerical investigations for the sextic oscillator have also been conducted to describe double-well structures [ 31 ]. An examination of higher QES solutions of the sextic oscillator [ 32 ] concluded that solutions with up to two nodes appear to be the most suitable approximation for studies on phase transitions. Applications to different regions of the nuclear chart table have been reported with all these approaches. In this review, we present a summary of the most important results obtained in the past 20 years in the framework of the Bohr equation solved with a sextic potential in the β degree of freedom. This material is organized as follows. In Section 2, the Bohr Hamiltonian is revisited, and its solutions with a γ -independent sextic oscillator in the β variable are described. In Section 3, other various solutions of the Bohr Hamiltonian with a sextic potential in the β variable are discussed, including several ways of treating the angular variable γ. Some general conclusions are drawn in Section 4. 2. The Bohr Hamiltonian The Bohr Hamiltonian describes the collective motion of the nucleus as a fivedimensional system. The deformation variable β measures the deviation from the spherical shape, whereas γ accounts for the deviation from axial symmetry toward triaxial configurations. The remaining three variables are the three Euler angles ( θ1 , θ2 , θ3 ), which describe the spatial orientation of the deformed nucleus. After separating the angular variables, the Bohr Hamiltonian is written as follows [5]: H=−¯h2 2B"1 β4 ∂ ∂β β4∂ ∂β +1 β2sin 3γ ∂ ∂γ sin 3γ∂ ∂γ −1 4β2∑ k Q2 k sin2(γ−2 3πk)#+V(β,γ), (1) where B is the mass parameter and Qk(k= 1, 2, 3 ) are the components of the angular momentum in the intrinsic reference frame. It is worthwhile switching to reduced energies by rescaling the energy and potential terms as e=2B ¯h2E and v(β , γ) = 2B ¯h2V(β , γ) , respectively. Note that the deformation variable βis dimensionless, and so are v(β,γ)and e. The mass parameter B appears as a moment of inertia, and as such, it has the physical dimension mass ×length2. Its chosen value sets the energy scale [23]. In order to solve the two-dimensional partial differential Equation (1), one usually has to apply various assumptions and approximations. The problem can be simplified if we assume that the potential has a separable form: v(β,γ) = u(β) + w(γ) β2. (2) In this case, the terms with the γ variable can be separated into a second-order differential operator containing the differential terms and w(γ) . By solving the corresponding differential equation, the resulting eigenvalues then appear in the β -differential equation together with the β−2factor. Symmetry 2023,15, 2059 4 of 19 The electric quadrupole transitions are calculated using the first-order transition operator [5,6,33] T(E2)=tβD(2) µ,0(θi)cos γ+1 √2D(2) µ,2(θi) + D(2) µ,−2(θi)sin γ, (3) where D(j) m,m0(θi) are rotation Wigner D-matrices, and t is a scale parameter. In some situations, this operator may impose overly restrictive selection rules. These can be relaxed by adding a second-order term to (3), where t0is an additional parameter: T(E2)0=T(E2)+t0β2−D(2) µ,0(θi)cos(2γ) + 1 √2D(2) µ,2(θi) + D(2) µ,−2(θi)sin(2γ). (4) The B(E2) electric quadrupole transition rates can be calculated as the matrix elements of operators (3) or (4). 2.1. Special Solvable Cases and Critical-Point Symmetries Several special cases have been discussed for which exact solutions can be obtained. These cases are based on further assumptions or approximations that simplify the mathematical formalism. Most of these are associated with various symmetries. Typically, these special cases have been introduced using the infinite square well as u(β) in (2), the flat bottom of which is considered an approximation of the shape-phase transition between two potential minima (see, e.g., Ref. [ 6 ] for details). Here, we briefly review these models, keeping in mind that they can also be implemented using the sextic oscillator as u(β) . In some of these models, the γ degree of freedom is omitted ( w(γ) = 0) or γ is restricted to a constant value, reducing the Bohr Hamiltonian to the Davydov–Chaban Hamiltonian [ 34 ]. These models are technically less involved, allowing for exact solutions. In others, w(γ) is approximated with some simple potential. E( 5 ) symmetry: A special case arises for γ -unstable potentials, which depend explicitly only on the βvariable, i.e., w(γ) = 0 in (2). Then, the β-dependence can be separated into an equation similar to the radial Schrödinger equation by using the substitution, Ψ(β,γ,θi) = β−2φ(β)Φ(γ,θi). (5) The β-differential equation is then −d2φ(β) dβ2+(τ+1)(τ+2) β2+u(β)φ(β) = eφ(β). (6) Here, τ is the seniority quantum number, which determines the allowed L angular momentum values (see Section 2.2). By taking the infinite square well as u(β) , this model is associated with the E( 5 ) symmetry describing the transition between the spherical and the deformed γ-unstable shape phases [6]. X( 5 ) symmetry: This model describes the transition from the spherical to axially deformed shape phases. Here, the collective Hamiltonian is the original Bohr Hamiltonian in five dimensions. This case corresponds to the situation in which the potential is separable, v(β , γ) = u(β) + e w(γ) , where the β2 factor in Equation (2) is approximated using its mean value hβ2i . In this situation, the differential equation is approximately separable in the variables β and γ . Then, a harmonic oscillator with a minimum at γ= 0 is used for the γ degree of freedom, whereas for the potential in β, an infinite square well is assumed [7]. Z( 5 ) symmetry: This model accounts for the prolate to oblate shape-phase transition. In this case, the variables in the Bohr Hamiltonian are approximately separated for γ=π/ 6. A harmonic oscillator potential with a minimum at γ=π/ 6 in the γ variable was proposed, whereas for the βvariable, an infinite square well was assumed [9]. Symmetry 2023,15, 2059 5 of 19 X( 3 ) symmetry: This is a γ -rigid (with γ= 0) version of the X( 5 ) model. It is called X( 3 ) because only three variables occur in this case, and the separation of variables is exact. The collective Hamiltonian (in reduced units) is given in this case by h=−1 β2 ∂ ∂β β2∂ ∂β −Q2 3β2+v(β,γ). (7) This is the case when γ is not considered as a variable but rather as a parameter with a given value ( γ -rigid case), such as γ= 0. For the β variable, an infinite square well is also assumed [35]. Z( 4 ) symmetry: This is another γ -rigid version but in this case, of Z(5). γ is considered as a parameter fixed to γ=π/ 6 (not a variable). In this situation, the Bohr equation is a function of four variables ( β and the Euler angles). The Bohr Hamiltonian splits into two equations: one for the βvariable and another for the Euler angles 1 β3 ∂ ∂β β3∂ ∂β −λ β2+ (e−u(β))φ(β) = 0 ; "1 4 3 ∑ k=1 Q2 k sin2(γ−2πk 3)−λ#Φ)θi) = 0, (8) where e and u are the reduced energies and potentials introduced previously, and θi (i = 1, 2, 3) are the Euler angles. For γ=π/ 6, the angular equation was solved long ago by Meyer-ter-Vehn [ 36 ], with λ=λL,α=L(L+ 1 )− 3 α2/ 4, where α are the eigenvalues of the projection of the angular momentum on the body-fixed axis and are even integers [ 36 ]. For the βvariable, an infinite square well is used here too [37]. All these critical-point solutions are summarized in Figure 1. βw βw βw βw βw β−degree of freedom γ−degree of freedom γ− independent γ = 0 0 γ = π/6 0 Critical Symmetry E(5) X(5) Z(5) X(3) γ− γ− rigid ISW ISW ISW ISW ISW HO rigid HO 0 0 0 0 0 γ = 0 γ = π/6 Z(4) Figure 1. Pictorial summary of the critical-point solutions in the βdegree of freedom. 2.2. The Sextic Oscillator as u(β) The solutions of the general sextic oscillator u6(r) = A−2r−2+A0+A2r2+A4r4+A6r6, (9) Symmetry 2023,15, 2059 6 of 19 cannot be written in an exact form. However, some of its special subsets admit partial exact solutions. In this case, the Ai coefficients satisfy certain constraints. This happens for the quasi-exactly solvable (QES) subset of the sextic oscillator [ 24 ], both as a radial ( r∈[ 0, ∞) , A−26= 0) and as a one-dimensional ( x∈(−∞ , ∞) , A−2= 0) problem. Choosing the former with r=β∈[ 0, ∞) and incorporating the β−2 term from Equation (6) using A−2= (τ+ 1 )(τ+ 2 ) , the normalizable solutions are written in terms of an infinite power series of β2 multiplied by a suitable pre-factor containing exponential forms of β2 and β4 . Substituting it into the Schrödinger equation and collecting the terms with the same powers β2i , its coefficients are found to satisfy an infinite three-term recurrence relation. However, the recursion can be terminated by a suitable choice of the parameters, leading to a polynomial form, P(M)(β2), of the solutions: φ(β)∼βτ+2exp−a 4β4−b 2β2P(M)(β2), (10) where a and b are model parameters defining the β potential (see below). In this way, the first M+ 1 normalizable solutions can be obtained in closed form. Note that for A6> 0, the sextic oscillator tends to infinity asymptotically, so it has an infinite number of bound states. The termination condition leads to constraints on the coefficients of the potential terms, so eventually, the QES sextic oscillator potential takes the form u(β) = uπ(β) = (b2−4acπ)β2+2abβ4+a2β6+uπ 0. (11) Here, a and b are the parameters appearing in the exponential term of φ(β) in Equation (10). The a= 0 choice recovers the harmonic oscillator. cπ is a constant that is determined by τand Mas 2cπ=τ+2M+7 2. (12) It is worth noting that all these conditions are necessary to obtain a QES potential with state-independent exact analytical solutions. Relation (12) implies that the normalizable solutions of the QES sextic oscillator are composed of φ(β) functions (10) with all combinations of τ and M for which τ+ 2 M=const . It is obvious that cπ differs for τ -even ( c+ ) and τ -odd ( c− ) combinations, which also introduces a slight difference in the coefficient of the quadratic potential component. This effect, which proved to be marginal in the applications in [ 20 – 23 ], was handled by prescribing that the minima of the two potentials coincide. This can be achieved by applying different constants in potential (11). For M= 2, these constants are: u+ 0=0, (13) u− 0=           (b2−15a)(β+ 0)2−(b2−17a)(β− 0)2 +2ab[(β+ 0)4−(β− 0)4] + a2[(β+ 0)6−(β− 0)6]; if b<(15a)1 2 −(b2−17a)(β− 0)2−2ab(β− 0)4−a2(β+ 0)6; if (15a)1 2<b<(17a)1 2 0 ; if b>(17a)1 2 (14) where βπ 0=1 3a[−2b±(b2+12acπ)1 2]. (15) See Refs. [20–23] for the detailed calculations. The parameter space (a , b) is divided into three characteristic domains, each corresponding to a specific potential shape in Figure 2. On the right of the parabola a=b2/( 4 cπ) , the coefficients of both the β2 and β4 potential terms are positive, so the u(β) potential shows a monotonously increasing trend with β , and it has a minimum at β0= 0. Above the parabola, the coefficient of the quadratic term is negative, so the potential has a local maximum at β0= 0 and a minimum at β1> 0. Finally, on the left of the parabola, the coefficients of the quadratic and quartic terms are positive and negative, respectively, re- Symmetry 2023,15, 2059 7 of 19 sulting in a local minimum at β0= 0, a local maximum at β1> 0, and another minimum at β2>β1 . It can be proven that the latter minimum is always deeper than that located at β0= 0. Potentials corresponding to parameters lying on the parabola lack the quadratic term, so they have a relatively flat bottom near β= 0. This situation is close to the infinite square well, which has been considered as the model of phase transition from the spherical to the axially deformed minimum in γ -unstable nuclei with E( 5 ) symmetry. It has been shown [ 20 , 21 ] that various benchmark numbers (ratios of energy eigenvalues or electric quadrupole transition rates) are constant along the parabola. V Η.Ο. β βmin = 0 V β βmin > 0 V β βmin > 0 βmin = 0 a = b /15 2 b a a = 0 V β Figure 2. Sextic model space (a , b) and the corresponding energy surfaces u(β) (11) in the different regions for M = 2, which corresponds to c+= 15 / 4 and c−= 17 / 4. The parabola a=b2/ 15 separates spherical from deformed shapes for b> 0 and deformed shapes from a coexistence region with two minima: one spherical and another deformed ( b< 0). The figure was taken from Ref. [ 22 ] and is licensed under CC BY-SA 4.0. Before proceeding, we note that the sextic oscillator can also be formally obtained through a suitable transformation from the bi-confluent Heun equation. In particular, the polynomial solutions of this equation coincide with the QES solutions of the sextic oscillator (see [38] for details on the relation between the two approaches). Once the potential u(β) in Equation (11) has been selected, i.e., a , b , and cπ are fixed, the energy eigenvalues and eigenstates can be determined. For this, the eigenvalues of the (M+ 1 )×(M+ 1 ) matrix (obtained from the termination of the three-term recurrence relation) have to be diagonalized. This amounts to finding the roots of an (M+ 1 ) ’th-degree algebraic equation. This has to be carried out for each M and τ satisfying prescription (12). This task can be completed numerically; however, for M= 0, 1, and 2, the roots can be obtained in closed analytic form. In the first publications proposing the application of the sextic oscillator in the Bohr Hamiltonian [ 20 , 21 ], solutions up to M= 1 were considered, and the energy eigenvalues and eigenstates were determined in closed analytical form. Later, the formalism was extended up to M= 2 [ 22 ], which extended the basis to 30 states (from 10), with nodal excitation up to ξ= 3 (from 2) and angular momentum up to L= 10 (from 6). For this, the roots of a cubic algebraic equation had to be calculated in the standard way. Here, we briefly recall the essential formulas for the energy eigenvalues and refer to Ref. [ 22 ] for any further expressions (coefficients appearing in the polynomial solutions, integrals appearing in the matrix elements, etc.). The energy eigenvalues are determined for τ -even and τ -odd states separately. According to Equation (12), the choice τ+ 2 M= 4, which allows combinations such as τ= 0, M= 2; τ= 2, M= 1; and τ= 4, M= 0, corresponds to c+= 15 / 4. Similarly, τ+ 2 M= 5 includes combinations such as τ= 1, M= 2; τ= 3, M= 1; and τ= 5, M= 0, corresponding to c−= 17 / 4 [ 20 ]. This difference between c+ and c− also defines the difference between the τ -even and τ -odd potentials as 2 aβ2 . The energy eigenvalues are displayed in Table 1, where the various λ(M) i quantities are obtained from the solution of the (M+1)’th-order algebraic equations. Symmetry 2023,15, 2059 8 of 19 Table 1. The energy eigenvalues of the sextic oscillator adapted to the Bohr Hamiltonian [ 22 ]. The constant terms uπ 0 in Equations (13) and (14) that depend on the even or odd value of τ are subtracted for simplicity. The λ(M) i and ˜ λ(M) i quantities are displayed in Equations (16) and (17). The fifth column displays the angular momenta Lcontained in the given (ξ,τ)multiplet. ξ τ Meξ,τ−uπ 0L 1 0 2 (λ(2) 2;b>0, λ(2) 1;b<0, 0 1 2 1 9b+λ(1) −2, 4 1 4 0 5b+λ(2) 32, 4, 5, 6, 8 2 0 2 5b+λ(2) 30 2 2 1 9b+λ(1) +2, 4 3 0 2 5b+(λ(2) 1;b>0, λ(2) 2;b<0, 0 1 1 2 7b+(˜ λ(2) 2;b>0, ˜ λ(2) 1;b<0, 2 1 3 1 11b+˜ λ(1) −0, 3, 4, 6 1 5 0 15b2, 4, 5, 6, 7, 8, 10 2 1 2 7b+˜ λ(2) 32 2 3 1 11b+˜ λ(1) +0, 3, 4, 6 3 1 2 7b+(˜ λ(2) 1;b>0, ˜ λ(2) 2;b<0, 2 When introducing s= ( 2 τ+ 5 )/ 4, the quantities shown in Table 1become the following [22]: λ(1) ±=2b±2(b2+18a)1 2, e λ(1) ±=2b±2(b2+22a)1 2, (16) λ(2) i=Λ(2) i(5/4),s=5/4 for τ=0 e λ(2) i=Λ(2) i(7/4),s=7/4 for τ=1 (17) where Λ(2) 1(s) = 4b−2rcos(φ 3), Λ(2) 2(s) = 4b+2rcos(π 3−φ 3), Λ(2) 3(s) = 4b+2rcos(π 3+φ 3), cos(φ) = −64ab r3, r≡r(s) = ±16 3(b2+2a(4s+1))1 2, (18) and the prescription sgn(r) = −sgn(b)has to be satisfied. The (ξ , τ) multiplets contain one or more physical states with a given L angular momentum value. The rules determining the allowed values follow from the O( 5 )⊃O( 3 ) decomposition: Construct τ= 3 n∆+ν , where n∆= 0, 1, . . . . Then, the allowed L values Symmetry 2023,15, 2059 9 of 19 are L=ν , ν+ 1, . . . , 2 ν− 2, 2 ν (note that the value L= 2 ν− 1 is missing). Table 1displays the allowed Lvalues for the given (ξ,τ)multiplets. In what follows, the energy scale is redefined such that E= 0 corresponds to the ground-state energy. The rescaled energy eigenvalues are denoted as Eξ,τ=eξ,τ−e1,0 . (19) With this model, one can obtain parameter-independent values (up to a scale factor) for energies and transition probabilities at the critical point between spherical and deformed γ - unstable shapes. These benchmark numbers are summarized in Figure 3, as taken from [ 22 ] for M=2. (M, ) τ ξ = 2 (M, ) τ ξ = 3 (M, ) τ ξ,τ R ξ,τ R ξ,τ R R = ξ,τ E Eξ,τ 1,1 * * 0 2 4,2 6,4,3,0 8,6,5,4,2 10,8,7,6,5,4,2 1 0 2.61 3.75 6.91 ξ = 1 0 2 4,2 6,4,3,0 (2,0) (2,1) (1,2) (1,3) 3.02 4.29 6.25 7.61 (2,0) (2,1) (1,2) (1,3) (0,4) (0,5) 0 2 (2,0) (2,1) 6.59 8.06 5.66 LL L ξ = 3 ξ = 2 ξ = 1 1 1 2 3 Mτ 2 2 0 1 ∆ =1 ∆ =0 L 0 0 4 510 8 7 6 5 4 2 Mτ 2 2 0 12 0 ∆ =0 6 4 3 4 0 0 ∆ =0 ∆ =1 Mτ 2 2 1 1 0 1 2 36 8 6 5 4 2 2 0 4 2 0 2 1 1.87 2.53 3.23 3.74 1.32 1.20 34 1.03 1.69 2.20 0.77 0.90 2.78 1.13 1.05 2.00 2 2.00 2.53 1.87 1.80 1.71 2.84 1.33 1.21 1.81 2.57 1.53 1.53 2.56 0.45 2.53 2.53 1.58 2.78 1.89 1.18 Figure 3. Benchmark numbers for energies normalized to the excitation energy of the first 2 + ( left ) and B(E 2 ) values normalized to B(E 2; 2 + 1→ 0 + 1) = 100 ( right ) characterizing the transition from the spherical to the deformed domain within the sextic oscillator model. These quantities are fixed for any point of the critical parabola separating the two domains. The figures were taken from Ref. [ 22 ] and are licensed under CC BY-SA 4.0. 3. Applications of the Sextic Oscillator in the Bohr Hamiltonian Due to its flexible shape, the sextic oscillator seems to be a suitable candidate to serve as the β -potential u(β) in the Bohr Hamiltonian. It is the lowest-order polynomial potential that admits two minima. With a potential minimum at β= 0, it can describe spherical nuclear shapes, whereas with a minimum at β> 0, it can account for a nucleus with a deformed shape. Furthermore, the transition between the two shape phases can be implemented smoothly within an exact analytical framework simply by varying the potential parameters a and b in (11). These features have been employed in a variety of models describing a wide range of nuclear shapes and transitions between them. Several review papers have discussed solvable models within the framework of the Bohr Hamiltonian, paying special attention to the shape-phase transitions and critical-point solutions related to them [ 19 , 30 , 39 ]. In the present review, we focus on models in which the sextic oscillator was considered as the β potential and report on recent developments in the field, placing them in the context of earlier results. 3.1. w(γ) = 0 The first application of the sextic oscillator was proposed in 2004 in Ref. [ 20 ] to describe the transition between spherical and γ -unstable nuclear shapes. In this case, the γ shape variable does not play a role. Based on the corresponding critical-point symmetry, this approach can be called E( 5 ) -sextic. The formalism was first developed for solutions with M= 0 and 1, encompassing ten states altogether: three 0 + , three 2 + , one 3 + , two 4 + , and one 6 + levels. These model states can typically be assigned to the first four levels of the ground-state band Kπ= 0 + 1 , the first three levels of the quasiγ band Kπ= 2 + , and the bandheads of two further Kπ=0+bands. The formalism of the sextic oscillator was developed in Ref. [ 20 ], where the energy eigenvalues, wavefunctions, and B(E 2 ) transitions were derived. The evolution of the Symmetry 2023,15, 2059 16 of 19 4. Summary and Conclusions In summary, we have provided an overview of recent research into the phase transitional behavior exhibited by the equilibrium shapes of atomic nuclei. Our exploration is based on the Bohr equation and its solutions using simple potentials. We revised and briefly proposed critical-point solutions (CPSs) linked to the use of an infinite square well for the β variable and various selections for the γ degree of freedom: E(5), X(5), Z(5), X(3), and Z(4). These simple CPSs shed light on the structural properties at the phase transitional point. Then, extensions and alternatives to CPSs by substituting the infinite square well in β with a sextic potential were revised, offering valuable insights into their relevance and accuracy. These studies outline the broader landscape of possibilities in understanding structural evolution within atomic nuclei. This exploration opens up the prospect of a fresh perspective on the intricate journey of nuclear structure evolution, marking a promising avenue for future research in the field. Figure 7displays the region of the nuclide chart where models based on the sextic oscillator have been applied in the Bohr Hamiltonian. Only even-even nuclei are shown, indicating the models in terms of which the given nucleus has been discussed (using both exact and numerical methods) in any of the works reviewed here. The E( 5 ) -sextic model, describing the transition from the spherical to the γ -unstable shape phase, is applied on both sides of the Z= 50 shell closure and to nuclei near the lower side of the Z= 82 shell closure. The spherical phase appears close to the N= 50 shell closure for nuclei with Z< 50, the N= 82 shell closure for Z> 50, and the N= 126 shell closure for Z< 82. The X( 5 ) model accounting for the transition from spherical to axially deformed shape phase in regions further away from shell closures has been applied to nuclei with N> 82 and Z< 82. The γ -rigid version of this model, X( 3 ) , has been applied more extensively in this region and also on both sides of the Z= 50 shell closure. Frequently, the same nuclei have been discussed in terms of the X( 3 ) and E( 5 ) frameworks. The Z( 5 ) model, describing the transition between prolate and oblate shape phases, has been applied to a few nuclei further away from the shell closures with Z< 82 and N< 126. Its γ -rigid limit has been applied close to this region and also near the domain near the shell closures with Z> 50 and N< 82. There are a few nuclei that have been discussed in terms of three models simultaneously ( 128 Xe, 130 Xe, 192 Pt, 194 Pt, and 196 Pt: E( 5 ) , X( 3 ) , and Z( 4 ) ; 188 Os and 190 Os: E(5); and X(5)and Z(5)). Transitions between nuclear shapes through critical points are also displayed in Figure 7in the models reviewed here. A simple pattern (diagonal lines formed by triangles) seems to emerge for the critical-point nuclei in the E( 5 ) -sextic model on both sides of the Z= 50 shell closure. These nuclei separate the region of spherical nuclei close to shell closures from that of gamma-unstable nuclei somewhat further away. Regions associated with the other models are not extensive enough to identify similar systematics. In conclusion, it is crucial to underscore that the incorporation of a sextic potential into the Bohr–Mottelson model provides a unique avenue for investigating the evolution of nuclear shapes across the entire nuclide chart. Within this framework, quasi-exact solutions hold substantial significance, serving as a pivotal milestone and reference point for any prospective numerical solutions aimed at comprehensively characterizing the broader landscape of the sextic potential’s influence on nuclear shapes. Symmetry 2023,15, 2059 17 of 19 Figure 7. Even-even nuclei in terms of various models utilizing the sextic oscillator as u(β) . The color and position of the dots inside the boxes identify the models: E( 5 ) -sextic: green dot in the middle; X( 5 ) -sextic: red dot in the lower-left corner; Z( 5 ) -sextic: black dot in the lower-right corner; X( 3 ) -sextic: pink dot in the upper-left corner; Z( 4 ) -sextic: blue dot in the upper-right corner. Nuclei proposed as critical are marked with a triangle (instead of a dot). Author Contributions: All authors contributed equally to this review article. They have read and agreed to the published version of the manuscript. Symmetry 2023,15, 2059 18 of 19 Funding: This work was partially supported by the Consejería de Economía, Conocimiento, Empresas y Universidad de la Junta de Andalucía (Spain) under Group FQM-160, under Projects No. P20-01247 and No. US-1380840, and also through Projects No. PID2019-104002GB-C22, No. PID2020-114687GB- I00, and No. PID2022-136228NB-C22 funded by MCIN/AEI/10.13039/50110001103 and “ERDF A way of making Europe”. This work was supported by the National Research, Development and Innovation Office (NKFIH, Hungary), Grant No. K128729. Data Availability Statement: Not applicable. Conflicts of Interest: The authors declare no conflict of interest. References 1. Landau, L.D.; Lifshitz, E.M. Statistical Physics; Pergamon Press: Oxford, UK, 1969. 2. 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