Quantum phase transitions and structural evolution in nuclei

Progress in Particle and Nuclear Physics 62 (2009) 183–209

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Quantum phase transitions and structural evolution in nuclei R.F. Casten A.W. Wright Nuclear Structure Laboratory, Yale University, P.O. Box 208124, New Haven, CT, 06520, USA

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Keywords: Evolution of structure in nuclei Collectivity Quantum phase-transitions Proton–neutron interaction Critical point symmetries Mapping of nuclei

a b s t r a c t This overview discusses the evolution of collectivity in atomic nuclei, with particular focus on the rapidly developing field of quantum phase transitions in the nuclear shape, and on trajectories of structural evolution in the N /Z plane. Particular stress is put on the interplay of a nucleon-based description of the driving mechanisms in the emergence of coherence and collectivity in nuclei with a more macroscopic perspective in terms of nuclear shapes and the symmetries and quantum numbers of the many-body system taken as a whole. © 2008 Elsevier B.V. All rights reserved.

Contents 1. 2. 3.

4.

Introduction............................................................................................................................................................................................. 183 Critical point symmetries ....................................................................................................................................................................... 189 Structural evolution ................................................................................................................................................................................ 201 3.1. Landau theory ............................................................................................................................................................................. 201 3.2. Evolutionary trajectories ............................................................................................................................................................ 202 Final comment......................................................................................................................................................................................... 207 Acknowledgements................................................................................................................................................................................. 207 References................................................................................................................................................................................................ 208

1. Introduction The purpose of this article is to discuss recent developments in the structure of medium mass and heavy nuclei focusing on regions of neutron and proton number where there are rapid changes in structure, and on structural evolution generally. Unless otherwise noted the discussion always refers to even-even nuclei and to low-lying collective modes. This paper is in no way intended as a thorough review of the literature. Rather, its focus is to give a picture of the physics involved and to discuss the main concepts and their applications to actual nuclei. One of the most pervasive features of atomic nuclei is the emergence of coherence and collective behavior out of the interactions among independent nucleons. In recent years this has been frequently phrased in terms of two broad, encompassing questions describing the field of nuclear structure physics. As stated in the recent 2007 U.S. Long Range Plan for Nuclear Science (URL: http://www.sc.doe.gov/np/nsac/docs/Nuclear-Science.Low-Res.pdf), these are:

• What is the nature of the nuclear force that binds protons and neutrons into stable nuclei and rare isotopes? • What is the origin of the simple patterns in complex nuclei? The first refers to a femtoscopic approach to understanding nuclear structure in terms of the orbits and interactions of the individual nucleons, subject to quantum mechanics, the Pauli principle and relevant conservation laws. The degrees of freedom are those of the individual nucleons and their multi-particle configurations. (Note: in this overview, we will limit our discussion of nuclei to the nucleonic level, ignoring sub-nucleonic degrees of freedom).

E-mail address: [email protected]. 0146-6410/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ppnp.2008.06.002

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Fig. 1. R4/2 values and 2-neutron separation energies in regions of heavy nuclei. The circle in the S2n plot highlights the discontinuity in S2n in a first order phase transitional region. This region is expanded on the right.

The second question above takes a perspective from a few fermis distance and looks at the nucleus as a whole. That this approach is useful is, in itself, already surprising. Nucleons orbit in the nucleus some 1021 times per second. Since a nucleon has a radius ∼ one fm and a nucleus has R ∼ 1.2A1/3 fm, the nucleons occupy on the order of 60% of the total nuclear volume. Instead of the utter chaos that might be expected, nuclei display astonishingly simple patterns and regularities. Some of these occur in excitation spectra (such as those associated with nearly perfect rotational bands), and some of them arise in looking + at data spanning sequences of nuclei. An example of the latter is the regular pattern of energy ratios R4/2 = E (4+ 1 )/E (21 ) seen on the top left in Fig. 1. This perspective looks at the ‘‘forest’’ instead of the ‘‘trees’’ and is a more macroscopic approach although ‘‘macroscopic’’ here still refers to distance scales of a few fermis. It invokes many-body degrees of freedom and many-body symmetries, quantum numbers of the many-body system, collective coordinates, and coherent motions. These two perspectives reflect the reality of mesoscopic systems such as the atomic nucleus, and are complementary, and mutually reinforcing. Together, they promise to provide a synergistic understanding of nuclear phenomena. Here, we will focus primarily on the second perspective, although the microscopic interactions that drive the emergence of collective and non-spherical shapes in nuclei will also be discussed. Our primary emphasis will be on aspects of collective behavior related to the recent intensely active area of quantum (zero energy), or ground state, phase transitional (QPT) behavior – that is, on the characterization of the equilibrium shapes and structure of nuclei – and to equilibrium structural evolution as a function of N and Z . Inherent to the idea of a phase transition is a control parameter, in terms of which a rapid change in structure occurs, and an order parameter describing the system which shows that change. Finite mesoscopic systems such as atomic nuclei present special problems in this regard and require care in choosing both quantities. Here, we are speaking of structural changes as a function of nucleon number but, clearly, in nuclei, this is a discrete integer quantity. As such one cannot even talk of the discontinuities in first or second derivatives of an order parameter with respect to the control parameter that are a common definition of first or second order phase transitions. Moreover, one cannot expect to see the sharp changes in the order parameter characteristic of infinite systems. This issue has been dealt with and we will not repeat such discussions here except to say that finite systems can and do exhibit behavior characteristic of quantum phase transitions, albeit somewhat muted relative to the infinite dimensional limit. Since we are dealing with shape changes in the quadrupole deformation,

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an obvious order parameter would be the deformation β . However, that is not directly an observable and we will therefore use proxies for it such as the energy ratio R4/2 which exhibits (see below) sharp changes in rapid transitional regions [1]. Since rotational energies decrease rapidly when deformation sets in, a valuable alternative [2] is 1/E (2+ 1 ) which mirrors in its general topology the behavior of R4/2 which is < for near-closed shell nuclei, at or just above 2 for vibrational nuclides and ∼3.33 for well-deformed rotors. It is important to recognize one pervasive feature of collective behavior. Since it describes, by definition, coherence in the many-body system, it involves, microscopically, many configurations, whose number grows combinatorially with valence nucleon numbers. It is not unusual for there to be 1015 or more configurations in medium mass and heavy nuclei, in the valence space alone. Each has a specific seniority. The presence of v > 0 (in many cases, significantly greater than zero) components in the ground and excited states is a way of stating that particle-hole excitations abound. Therefore, since these will involve many different particles in many different orbits, an important consequence follows: Small changes in N and Z are not expected to change the collective structure substantially. Therefore a characteristic feature of normal collective behavior is a smooth evolution with N and Z . Much of our discussion will focus on cases where such gradual structural evolution is, in fact, replaced by rapid changes in structure with small changes in N and Z , and on the implications and origin of such phenomena. The idea of shape changes in nuclei as a function of N and Z has a long history. It has also long been recognized that there are regions of rapid structural change that presumably signal some special driving mechanism. It might be thought that such rapid changes go counter to what we just said. In a sense, they do, but they also reveal much about the origins of emergent collectivity. Again, Fig. 1 nicely illustrates this. The top left side (already referred to) shows the rapid rise in R4/2 between N = 88 and 90, where it increases from values of ∼2.3, typical of actual vibrational nuclei, to ∼3.0, the traditional borderline value separating spherical from deformed nuclei, to ∼3.33, the limiting value of the axial rotor paradigm. The top right side of Fig. 1 shows exactly the same data as on the left except, in this case, plotted against Z instead of N. Here we also see a rapid jump in R4/2 near N = 90, but this way of plotting the data reveals more. [This, by the way, is one of countless examples of how complementary physics emerges from plotting various observables in different ways, focusing on different perspectives.] Here, the plot of R4/2 against Z shows not only the existence of a shape transition, but also the change in curvature in the data for N = 88 and 90, concave to convex [2]. This, in turn, reveals a feature of the underlying shell structure that resolves the seeming contradiction between gradual and rapid changes in collective states. The general behavior of R4/2 is to minimize near closed shells (values <2 are typical of nuclei with just a couple of valence nucleons) and to maximize (near 3.33) for deformed nuclei near mid-shell. Fig. 1 (top right) shows that, for N ≤ 88, Z = 64 behaves as a typical closed shell while, for N ≥ 90, Z = 64 appears to be near mid-shell. Thus there is no contradiction in our understanding of the evolution of collectivity. Instead, the rapid change in structure reflects a change in shell structure, in the effective shell gaps. In this view, nuclei with N < 90 and Z ∼ 64 are quasi-magic and so the numbers of effective valence nucleons (protons) are very small. However, for N ≥ 90, Gd has ∼14 effective valence protons. Hence, the rapid change in structure between N = 88 and 90 does not, in effect, occur over a small change of two valence neutrons but is accompanied by a much larger change in effective valence proton number. Moreover, Fig. 1(right) thus shows that this change in proton shell structure occurs as a function of neutron number, suggesting the critical importance of the (valence) p–n interaction. Indeed, it is largely the competition between the spherical-driving pairing interaction and the deformation- and collectivity-driving p–n interaction that determines the evolution of structure in atomic nuclei and that plays an essential role in the appearance of quantum phase transitions. The importance of the p–n interaction has been stressed for decades, since the first suggestion by Goldhaber and de Shalit in 1953 [3]. Talmi has long emphasized this point [4] and it underlies the fundamental work of Federman and Pittel, Heyde, Nazarewicz, Otsuka, Dobacewski, Skalski, the macroscopic-microscopic method of Strutinsky, and others on the effects of the p–n interaction in altering shell structure itself [1,3–11]. In particular, the monopole component of the p–n interaction provides a ‘‘self-energy’’ term that alters the single particle energies of one type of nucleon as a function of the number of the other type [8]. The effects range from the dissolution of magicity for N = 20 in 32 Mg [12], to the rapid onset of deformation near A ∼ 100 and A ∼ 150 in terms of the Z = 40 and 64 subshell gaps, to the descent of intruder configurations in light Pb and Hg isotopes. These effects are seen in the need to invoke specific changes in the counting of valence nucleons in a couple of mass regions in order to obtain smooth systematics in the Np Nn scheme [11]. They are seen even in the systematics of rotational behavior in well deformed nuclei [13]. Of course, the effects of the p–n interaction depend on the orbits involved, being largest in highly overlapping orbits with similar shell model quantum numbers. In deformed nuclei they also depend on the nature and orbit inclinations of the deformed (Nilsson) orbits [14,15]. In recent years, numerous changes in shell structure have been found [16–19], especially in light nuclei, with the advent of experimental techniques to study nuclei far from the valley of stability. Finally, the influence of the valence p–n interaction has been codified in the above-mentioned Np Nn scheme [11] in which it is found that structure evolves smoothly with the product of the number of valence protons, Np , and the number of valence neutrons, Nn . We will return to this later in using the related concept of the P-factor [20] to predict where additional rapid changes in structure may occur. Interestingly, it is possible to extract empirical values for the p–n interaction using double differences of atomic masses. Specifically, the average p–n interaction of the last two neutrons with the last two protons is given by [21]

δ Vpn (Z , N ) = 1/4[{B(Z , N ) − B(Z , N − 2)} − {B(Z − 2, N ) − B(Z − 2, N − 2)}].

(1)

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Fig. 2. Average empirical interaction energies of the last two protons with the last two neutrons in the 208 Pb region. Based on Ref. [25].

This was first suggested by Garrett and Zhang almost 20 years ago [21] and investigated soon thereafter [22,23]. Following the 2003 Audi, Wapstra, and Thibault mass evaluation [24], a much larger number of measured masses became readily available and vetted, allowing a much greater number of these interactions to be extracted. Figs. 2 and 3 illustrate some of the empirical values of δ Vpn . Fig. 2 shows the Pb region and illustrates dramatically the dependence of δ Vpn on orbit [25]. When both protons and neutrons are both below their respective closed shells they both occupy low j, high n principal quantum numbers, highly overlapping orbits and hence show large δ Vpn values. When Z > 82 and N remains ≤126, the orbits are quite different (protons are in high j, low n orbits, neutrons in low j, high n) and the δ Vpn values are correspondingly low. When both protons and neutrons are above the 208 Pb doubly magic shell closures, the orbits are again similar and large δ Vpn values appear. It would be interesting, and a nice test of these ideas, if a δ Vpn value could be measured in the lower right quadrant. The first point needs only the mass of 208 Hg and such an experiment is planned [26]. Of course, away from closed shells, the wave functions of the ground states are so complex that single particle overlap arguments are too simplistic. However, one can still pursue this idea in the following way. Typical shells in medium mass and heavy nuclei generally have a characteristic pattern stemming from the shape of the mean field potential just alluded to which favors high orbital angular momentum orbits, and the spin orbit interaction which favors the highest j orbit in each major shell. As already noted, the normal parity orbit sequences in a major shell thus go from high j, low n to low j, high n. Thus, if both neutrons and protons are filling their respective shells to roughly similar levels within a major shell, the p–n interactions should be large while in the opposite case they should be small. These simple ideas are in fact verified experimentally, as shown in Fig. 3(top), taken from Ref. [27], where the largest values occur near the diagonal and smaller ones in the off-diagonal corners. That is, δ Vpn values are largest for cases of similar fractional filling of the respective proton and neutron shells. Again, mass measurements in the lower right quadrant would be an excellent test of these ideas. The importance of large p–n interactions in inducing emerging collectivity is most dramatically shown by comparing δ Vpn values to the systematics of R4/2 . It is convenient to plot R4/2 against Np Nn as shown in the lower panel of Fig. 3. Here, of course, R4/2 grows with the numbers of valence nucleons but note that the growth rates are higher for nuclei in the lower left quadrant of Fig. 3(top) where the p–n interactions are large, and smaller for nuclei in the upper left quadrant where δ Vpn interactions are considerably smaller. Despite decades of recognition of the importance of the p–n interaction this is, in fact, the first direct empirical correlation between the two [27]. Of course, there are many other observables besides R4/2 that can indicate rapid shape changes. Sharp increases in yrast B(E2) values, sharp drops in yrast energies, sharp increases in certain E0 transition matrix elements, and certain special degeneracies (see below) are examples. Here, though, we show one related to masses, namely two-nucleon separation energies. Clearly, a nucleus will take on a different shape than its neighbors only if it can gain energy in doing so. This

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Fig. 3. Top: Similar to Fig. 2 for the Z = 50–82, N = 82–126 region. Bottom: R4/2 values for the same region. Based on Ref. [27].

appears as a sudden increase in separation energies. Actually, what is seen is not so much an increase absolutely, but an increase relative to the trend with nucleon number. This is shown for the A ∼ 150 region in Fig. 1 (bottom). The sudden flattening in S2n at N ∼ 90 signals the spherical-deformed transition region. We note that this effect is primarily seen in first order phase transitions (see below) where the collective potential has two minima and the critical point occurs where they cross. In a second order phase transition, the evolution is smoother and a more delicate change in two nucleon separation energies, perhaps invisible on the scale of Fig. 1(bottom), is expected. This discussion highlights the complementarity of the two approaches to nuclear structure, the femtoscopic based on particle degrees of freedom and on the interactions of the constituent nucleons, and the collective or many-body perspective. We have already mentioned this but it bears emphasis. Collective models are inherently phenomenological— given some key observables they can often predict a raft of others. This allows the identification of collective degrees of freedom, of many-body symmetries, selection rules, and, often, analytic expressions for various observables and can reveal subtle effects (via small deviations from the collective paradigms) that would be invisible in the output of a microscopic calculation. However, with certain caveats, such approaches are not inherently predictive. In contrast, again with some caveats, nucleonic descriptions are predictive but, generally, as noted, it is more difficult to identify the essential collective coordinates or symmetries from such calculations. Only by combining these approaches in a symbiotic way can a true understanding of structure emerge, which is both predictive and descriptive. The twin caveats mentioned above are significant. Among collective models, some, such as the interacting boson approximation (IBA) model [28], have a microscopic aspect. This model (discussed later in this paper) is based on the ansatz of a bosonic pairing of nucleons and represents a (huge) truncation of the shell model. Therefore, even with constant

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Fig. 4. Sketches of energy surfaces as a function of β for first (left) and second (right) order phase transitions. The X (5) and E (5) potentials are also shown. The sketch at the left shows the energy surface at the critical point. That at the right shows the energy surfaces for nuclei before (Vmin (β) at β = 0), at (flat potential), and after (Vmin (β) at β > 0) the critical point.

parameters, the predictions of its Hamiltonian depend on the numbers of (valence) protons and neutrons. The caveat on the microscopic side is that models such as the shell model, often claimed to be predictive in the sense that, given predictions for one nucleus, predictions for the neighboring nucleus are parameter free, are not really so since the parameters are themselves phenomenological and depend on N, Z , and A. The study of QPT is a beautiful example of the complementarity of these two perspectives. We recall Fig. 1 (top right) which showed that the sudden change in macroscopic shape (witnessed by R4/2 ) is due to a breakdown of the single particle Z = 64 shell gap driven by the valence p–n residual nucleon-nucleon interaction in highly overlapping orbits. In terms of a microscopic picture, the p–n interaction lowers the energy of a configuration involving an ‘‘intruder’’ orbit on the other side of a subshell or shell gap. In the A ∼ 150 region [1] the proton 1h11/2 orbit energy is lowered when neutrons start occupying the highly overlapping 1h9/2 orbit near N ∼ 90. This obliterates the Z = 64 shell gap as shown so dramatically in Fig. 1, top right. The same thing happens in the A ∼ 100 region [5,6], where the fastest shape transition in medium mass and heavy nuclei occurs at N = 60. Here, neutron occupation of the 1g7/2 orbit lowers the energy of the proton 1g9/2 orbit, causing the Z = 40 subshell closure to vanish suddenly. In other regions, especially involving major shell gaps, the intruder state descends in energy but not far enough. This happens, for example, in the Cd nuclei and near Pb where well known deformed intruder configurations appear in the low lying spectrum but they do not cross the ground state, which remains essentially spherical [8]. From the collective perspective, on the other hand, one can think of QPT in terms of collective potentials in the coordinates of the many-body system, such as β , which characterizes the ellipsoidal nuclear deformation. A first order phase transition exhibits phase coexistence (a coexistence of spherical and deformed structures with competing minima in the collective potential V (β) against β ), and a discontinuity in the expectation value of β . The critical point of the phase transition occurs when one minimum in the potential crosses the other. This is illustrated in Fig. 4(left) where the thick curve shows a potential in a nucleus, as a function of β , with two minima, one for a spherical shape (β = 0) and one for a deformed shape. The curve shows the critical point where the minima are degenerate. For nuclei with fewer valence nucleons the deformed minimum would be higher while, for larger numbers of valence nucleons, the deformed minimum is the equilibrium configuration. The behavior of nuclei in the case of a first order phase transition will depend considerably on the height of the barrier between the two minima. It seems that these barriers are often quite low and so one expects considerable mixing of the two coexisting configurations. Second order phase transitions can also occur. These occur when deformation sets in smoothly with no development of a double minimum in the potential, without phase coexistence, and without a discontinuity in the expectation value of β . This is illustrated on the right in Fig. 4 which shows three curves, for different numbers of valence nucleons. One has a spherical minimum and another, after the phase transition, has a deformed minimum. The middle curve is at the critical point. There is no phase coexistence. Such transitions are expected in cases of γ -soft potentials for vibrator to γ -soft deformed structures. These preliminaries lead us now to a discussion of recent models for phase transitional behavior in nuclei. Historically such regions, with competing degrees of freedom, have long been considered the most difficult to treat theoretically and have long been a key testing ground for microscopic models. We will see that newly proposed models offer extremely simple approaches to phase transitional behavior in nuclei. This subject has recently been raised to new levels of interest because of two developments, one experimental and one theoretical. The former, driven largely by technological developments in γ -ray detectors, consists of the development of much more thorough, extensive and precise level schemes for particular transitional nuclei while the latter is driven by the development of a new class of theoretical descriptions, called critical point symmetries (CPS), that give parameter free (except for scale) predictions for nuclei at the critical points. We now turn to a discussion of these. In the process of presenting this discussion a variety of issues, some practical, others rather philosophical, will also be raised.

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2. Critical point symmetries We return now to Fig. 4 and the curves giving typical potential energy surfaces at the critical point. It is, in general, not possible to solve the Schroedinger equation analytically for such potentials. However, Iachello proposed a simplification [29,30] that provides analytic solutions trivially. That ansatz is shown by the infinite square wells also shown in the figure which approximate the outer boundary of the critical point potential by a vertical wall and which, in the case of a first order phase transition, ignores the small barrier between the two minimum. Thus the complex curve corresponding to either a first or second order phase transition is replaced simply by a square well in β . Of course, nuclei are three dimensional objects and require other coordinates for their description. A traditional approach is that of Bohr and Mottelson [31] who invoked a separation into intrinsic and laboratory degrees of freedom. Nuclei with quadrupole deformation are described by two intrinsic variables, β and γ . In addition to these, the orientation of the nucleus in space is described by the three Euler angles. Thus, overall, one has a 5-dimensional problem. One assumes that V (β, γ ) = V (β) + V (γ ) and, further, an infinite square well in β in the Bohr Hamiltonian. In the critical point symmetries first developed by Iachello, two potentials in γ are defined corresponding to two different CPS. In one, corresponding to a first order phase transition from a spherical vibrator to an axially deformed rotor, V (γ ) ∼ γ 2 , a harmonic oscillator. This gives the so-called X (5) CPS [30] via an approximate separation of variables. [In the CPS, the number in parenthesis refers to the number of effective dimensions. In the present case, and also in E (5) (see below), the other dimensions are the three Euler angles.] After assuming that V (β, γ ) = V (β) + V (γ ) and an infinite square well of width βw for V (β), the Bohr Hamiltonian gives the following Bessel equation in β :

ξ˜ 00 +

ξ˜ 0 z

 + 1−

v2 z2



ξ˜ = 0;

ξ˜ (βw ) = 0.

(2)

The solutions are Bessel functions the squares of whose roots give the eigenvalues of the 0+ states and of the sequences of states built on them. These solutions are parameter-free except for scale. The eigenvalues are given by

β,s,J =

(xs,J )2 βw2

(3)

where J is the angular momentum and xs,J is the sth zero of the Bessel function Jv (z ) where the order is given by

v=



J (J + 1) 3

+

9 4

1/2

.

Note that v is an irrational number. Here, z = β

ξs,J (β) = cs,J β −3/2 Jv (ks,J β)

(4) 2mEβ /h¯ . The eigenfunctions in β are given by

p

(5)

where ks,J = xs,J /βw . Before inspecting this solution a couple of remarks are useful. First, the separation of β and γ is approximate only. As this paper is not a thorough discussion of the formal aspects of this model and its approximations we will not delve into this here but simply refer the reader to the detailed discussion in [32] which is important to understanding the nature and significance of the approximations and which also brings in the effective centrifugal forces arising from β –γ coupling in the five-dimensional problem. The other CPS [29], called E (5), has a potential that is independent of γ , that is, it is γ -flat. It corresponds to a second order phase transition from a spherical vibrator to a γ -soft rotor (the Wilets–Jean model [33]). The solutions are again Bessel functions, in this case, of half-integer order. This solution contains the O(5) symmetry of γ -independent models. We will focus primarily on X (5), for several reasons. It is the more interesting case since most spherical-deformed transition regions evolve from spherical shapes to axial rotors. It has more independent predictions since, for the E (5) case, many E2 branching ratios are identical to those of the O(5) symmetry which applies to any structure along the entire trajectory from vibrator to γ -soft rotor. Moreover, in E (5), as a second order phase transition, the values of observable change more slowly [34]. Combined with the natural muting of phase transitional behavior due to the finite nature of the nuclear system, this means that such second order phase transitions are more difficult to identify although good candidates for E (5) have indeed been found in nuclei such as 134 Ba [35] and 102 Pd [36] and some Ru isotopes [37]. Turning then to X (5), the predictions for the yrast and yrare states are given in Fig. 5(left). It is important to stress that these are completely parameter-free except for scale in the energies and B(E2) values. However, the wave functions, relative B(E2) and other transition rate predictions are all completely independent of scale, and therefore parameter-free. The most obvious initial characteristic feature of X (5) is the R4/2 ratio of 2.91, intermediate between vibrator (∼2) and rotor (3.33) values. Consistent with this R4/2 value, the higher yrast states are also between the vibrator and rotor limits as shown in Fig. 6(left). Another key and unique prediction of X (5) stems from its linking of motion in the Euler angles to + intrinsic excitations, namely that the ratio E (0+ 2 )/E (21 ) has a well-defined value, 5.66. We note that, traditionally, in the

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Fig. 5. Comparison of X (5) to the data for 152 Sm. In this and subsequent similar figures the widths of the E2 transition arrows are roughly proportional to the B(E2) values and the numbers on or within these arrows are the B(E2) values in W.u. The call-out box on the right for the yrare B(E2) values is discussed in the text: The numbers in the left part of this box are the literature values prior to Ref. [42], those in the middle are from Refs. [42,43], while those on the right are from Ref. [44]. Based on Ref. [45]. See text.

Fig. 6. Left: Comparison of yrast energies in 152 Sm with rotor, vibrator, and X (5) models. Based on Ref. [45]. Right: E-GOS plots [38] of the same normalized to 0.5 for J = 2+ . The horizontal line labeled R4/2 = 3.0 is a special case of such a structure, using the AHV (Eq. (12)).

conceptual context developed by Bohr and Mottelson (as distinct from the Bohr Hamiltonian itself), rotational and intrinsic degrees of freedom are separately parameterized. The prediction of a specific value for the ratio of yrast and intrinsic energies is fixed in X (5) and, like all X (5) predictions, cannot be varied and therefore serves as a sensitive test of the model. As a further consequence of linking these two energies scales, X (5) predicts a near degeneracy of the yrast 6+ level and the 0+ 2 + level, namely E6/0 ≡ E (6+ 1 )/E (02 ) = 0.96. The occurrence of degeneracies as a signature for rapid shape changes was alluded to earlier and here we see the most obvious case. This near degeneracy of the 6+ yrast state and the first excited 0+ state was noticed almost immediately when the X (5) scheme was proposed and thought to be a curiosity that might be useful as a signature for X (5). It has since been realized to be of more fundamental significance. This will be discussed further below. It is interesting to discuss the yrast and yrare energies in a slightly different way, using the recently developed technique of E-GOS plots (This acronym will be defined below). The concept of E-GOS plots [38] was proposed several years ago as a very efficient way to not only help identify the structure of a given nucleus but also to discern changes in that structure as a function of spin. We briefly introduce the idea by first looking at two extreme models, the symmetric rotor and the harmonic vibrator. In the rigid rotor, the energies of the yrast states are E (J ) = AJ (J + 1)

(6)

where A is the inertial parameter. γ -ray energies within a band (say, the yrast band) are given by Eγ (J ) = E (J ) − E (J − 2) = A[4J − 2]

(7)

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Fig. 7. Similar to Fig. 6 for the yrare energies in 152 Sm.

and so the ratio of Eγ to J (E-Gamma Over Spin or E-GOS) is given by



Eγ /J = A 4 −

2 J



.

(8)

In units of A, this evolves from 3 for J = 2 up to 4 for high J, and so is a gradually increasing and asymptotic function of J. Now consider the yrast energies of a harmonic vibrator E (J ) = nE (2+ 1 )

(9)

where n is the phonon number. Then Eγ = constant = E (2+ 1 )

(10)

and Eγ /J = E (2+ 1 )/J

(11)

which decreases hyperbolically from E (21 )/2 to zero. An E-GOS plot is thus simply Eγ /J plotted against J. Since it involves level energy differences rather than the energies themselves, it has a derivative character and is quite sensitive to structure. For intermediate structures the slopes in E-GOS plots range between the rotor and vibrator extremes. One particular case of interest is R4/2 = 3.0 which traditionally marks the boundary where axial rotation begins to set in. A very general phenomenological model is that of the Anharmonic Vibrator (AHV). In this model, the yrast energies are given by +

E (J ) = nE (2+ 1 )+

n(n − 1) 2

4

(12)

+ + + where 4 = E (4+ 1 ) − 2E (21 ) is the anharmonicity of the 4 state, that is, its deviation in energy from twice the 2 energy, and n = J /2. n is the phonon number in a vibrational nucleus. This formula is very general. For 4 = 0, Eq. (12) gives the + harmonic vibrator. For 4 = (4/3)E (2+ 1 ), it gives the rigid rotor expression. For 4 = E (21 ), it gives R4/2 = 3.0. In this case, E (J )/J = constant and thus the E-GOS plot is flat. So, interestingly, the phase transitional point (R4/2 ∼ 3) roughly serves to section E-GOS plots into the two classes of increasing and decreasing with J, so that nuclei on the vibrator side of the phase transition are downsloping while those to the rotor side are upsloping. Of course, the AHV is only one example of a model that parameterizes R4/2 so this point is qualitative only. Fig. 6(right) shows E-GOS plots for a vibrator, a rotor, R4/2 (AHV) = 3.00, and X (5). Note that X (5) is just slightly decreasing in this plot, compatible with its R4/2 = 2.91 which is just under the horizontal benchmark value of 3.0. Likewise, the energies of the excited states of the yrare 0+ 2 sequence are intermediate between the vibrator and rotor as shown on the left in Fig. 7. Another key feature of X (5) are the intra and interband B(E2) values. These are also shown in Fig. 5 and are also intermediate between the vibrator and the rotor. We will come back to this soon when we compare X (5) with the data. First, though, we need to identify nuclei that could be candidates for X (5). While historically, events proceeded differently (the nucleus that became the first manifestation for X (5) was recognized [39–42] as being at the critical point before X (5) was developed) we will approach it in a more pedagogical way. As we have discussed, the emergence of collective behavior in the complex nuclear many-body system is driven by the p–n interaction, which competes primarily with the like-nucleon pairing interaction. A measure of that competition is simply the number of p–n interactions divided by the number of pairing interactions. This assumes that each pairing interaction is the same as every other, and similarly for the p–n interaction. This is a very crude approximation but we will use it since it has been shown to be remarkably successful. The total number of p–n interactions is simply Np Nn where Np and Nn are the numbers of valence nucleons counted to the nearest closed shells since every valence proton interacts with every valence

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Fig. 8. Contours of P ∼ 5 denoting the locus of possible candidates for X (5) structure.

neutron. The number of pairing interactions is Np + Nn since each nucleon only interacts via the pairing interaction with a single, identical, nucleon in the equivalent time reversed orbit. Thus, we consider the so-called P-factor [20] P = Np Nn /(Np + Nn ).

(13)

We have seen (Figs. 2 and 3) that, in heavy nuclei, the p–n interaction is roughly 250 keV. The pairing interaction is about 1 MeV. Thus we would expect deformation to set in when the integrated p–n interaction begins to dominate the pairing interaction, or around P ∼ 4. It turns out empirically that this zeroth order estimate is quite good and that P ∼ 5 is a slightly better estimate. In Fig. 8 we show a portion of the nuclear chart with a contour (dark boxes) showing the locus of P ∼ 5 nuclei. These nuclei would therefore be candidates for a shape transitional region. Consider the left edge of the contour in the rare earth region which is essentially vertical along the N = 90 isotones, 150 Nd, 152 Sm, 154 Gd, 156 Dy (after which it curves ‘‘northeastward’’). Such nuclei might be ideal candidates for X (5) since it has long been known that these nuclei are indeed transitional and that the deformed nuclei in the first half of the rare earth region are more or less axially symmetric. (They have γ values of ∼10◦ which corresponds to K admixtures in, say, the 4+ states, of ∼0.03). Moreover, these nuclei have transitional R4/2 values ∼3.0. Historically, sensitive studies [39] of the 152 Sm level scheme led to a suggestion that this nucleus gave evidence for a first order phase transition [40,41]. Following this, Iachello developed the X (5) CPS [30]. We now turn to a comparison of X (5) with the data for 152 Sm. Although many nuclei have been studied in the context of X (5), we will focus on 152 Sm because it was the first candidate identified and remains the most thoroughly studied [39–45]. Moreover, this allows us to delve in more detail into the comparison with X (5) for a given case rather than to merely survey similar results in several nuclei. We will more briefly discuss a couple of other candidates later. Fig. 5(right) shows the experimental 152 Sm level scheme based on Ref. [45] which pointed to 152 Sm as an excellent manifestation of the evident physics embodied in X (5). The agreement with X (5) is striking when it is realized that X (5) corresponds to such a crude model, an infinite square well, and that it is parameter-free. We first look at the yrast and yrare energies. These are shown in Figs. 6 and 7, respectively, in comparison with the vibrator, the rotor, and X (5). These figures show the same results in two formats, normal plots (left) of E (J ) vs. J, and in EGOS plots (right). Clearly, the agreement of X (5) with the data for both band spacings is excellent. Note that, by construction, these plots show relative band energies: clearly from Fig. 5 there is a serious discrepancy for the absolute spacings in the 0+ 2 sequence: we will return to this shortly. + + One of the most striking confirmations that X (5) is a reasonable description is the near degeneracy of the 61 and 02 levels + mentioned above and the closeness of the predicted energy ratio E (0+ 2 )/E (21 ) to the data. As noted above, the latter is 5.66 152 in X (5): it is 5.62 in Sm. This is a unique prediction of X (5) and stems from its inherent linking of inertial behavior (quasirotational behavior) and intrinsic excitations. In many collective models the scales of these have to be put in separately by hand. A further prediction of X (5) is the structure of the excited 0+ sequence. X (5) predicts that it has a slightly smaller R4/2 value and slightly smaller intrasequence B(E2) values. Fig. 5 shows that R4/2 in 152 Sm is indeed slightly lower than for the ground state sequence. As for the yrare B(E2) values there is an interesting historical development (see Fig. 5). Prior to + + + the recent studies of 152 Sm, the accepted B(E2 : 2+ 2 → 02 ) and B(E2 : 42 → 22 ) values were 520 (170) and ≈330 W.u.! Although these are clearly wrong, it took new measurements [42,43] to obtain the values 107 (27) and 204 (38) W.u., considerably lower than previous results and quite in line with the X (5) predictions. The latter is based on a differential plunger lifetime measurement and is therefore likely to be reliable. Later, additional measurements [44], suggested slightly

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+ + Fig. 9. Relative B(E2) values from the 0+ 2 -sequence to the yrast states, for X (5) and several N = 90 nuclei, where the B(E2 : 02 → 21 ) values have been normalized to each other. Based on Ref. [45].

revised numbers, also shown in Fig. 5. Both these later sets are substantially less than the early ones and in good agreement with X (5), the former more so. The intersequence transitions provide a number of tests of X (5), Inspection of Fig. 5 shows first that the X (5) predictions, relative to the intraband B(E2) values, are substantially larger than observed. However, this observation hides an important aspect of these numbers. The relative inter-sequence transitions agree almost exactly with X (5) with the single exception + of the 4+ 2 –61 transition. This is shown in Fig. 9 (for several nuclei) where the experimental inter-sequence B(E2) values are + compared with X (5) by normalizing both sets to the same B(E2 : 0+ 2 → 21 ) value. A similar discussion applies to the yrare energies. At first glance in Fig. 5, there seems to be a gross mis-prediction by X (5). More careful inspection, however, shows that it is the scale of the X (5) energies, that is, the inertial parameter, which differs from the data not the relative energies. The latter, in fact, agree almost exactly with X (5) as was shown in Fig. 7. This discussion highlights an important point in comparing models with the data. Often, comparison of absolute and relative values of various observable gives different and complementary information. Here, the relative values of both yrare energies and inter-sequence B(E2) values are well reproduced by X (5). By allowing us to identify that the discrepancy lies in the absolute values and thus with the scale, we can narrow down its possible physical origins. In fact, it is rather clear where the problems lie. First, consider the overprediction of the inter-sequence B(E2) values. Note that R4/2 for 152 Sm is 3.01, slightly to the rotor side of X (5). Of course, there is no a priori reason why any given nucleus should happen to ‘‘land’’ at exactly the phase transitional point and it appears that 152 Sm is a bit past the critical point. We can see this very nicely in the E-GOS plot of Fig. 6 where the experimental values are slightly above the X (5) values. How does this relate to the B(E2) values? It is well known that inter-sequence B(E2) values decrease in going from the vibrator (where many they are allowed transitions comparable to ‘‘intra-band’’ values) to the rotor where they are quite weak (for example, in the SU (3) limit of the IBA [28], they vanish by the SU (3) selection rules). Recognizing that most observables change most rapidly just around the phase transition region [34], where, indeed, their second derivations cross zero, even a small shift of a structure from the critical point towards the rotor could lead to considerably diminished inter-sequence B(E2) values. The situation with the scale of energies in the 0+ 2 sequence reflects another aspect of X (5). The square well approximation embodied in X (5) clearly is too drastic—real nuclei do not show such sharply changing potentials. Caprio [46] has carried out calculations with a sloped outer wall. The effect is easy to see as sketched in Fig. 10. Here the longer wave lengths at higher energies with a sloped outer wall lead to lower energies and to a reduction in spacings of the yrare sequence. Caprio has shown [46] that a specific value of the slope angle gives reasonable agreement with the data. [It is also worth mentioning here that, besides the approximation of a separation of β and γ degrees of freedom in solving the Bohr Hamiltonian (see Ref. [32]), X (5) also assumes an infinite well. Clearly, this is a very rough approximation. However, finite well calculations have been carried out by Caprio [47] as well and show only minor changes relative to X (5).] Thus we see that the reasons for discrepancies in scales for the yrare energies and the 0+ 2 -ground band B(E2) values are different. For the energies, the explanation seems to be the limitations of the extreme vertical outer wall assumed in X (5) while, for the B(E2) values, the explanation lies in the fact that 152 Sm is slightly to the rotor side of X (5). This, finally, highlights another point relevant to any simple model. Simple models, by definition, are simple, and, undoubtedly, too simple. When they succeed at some level, they are almost immediately followed by models that incorporate additional degrees of freedom or complications. The classic example of this is the energy expression for a symmetric rotor, E (J ) ∼ J (J + 1), which was found to describe reasonably well the yrast energies of many well-deformed nuclei. Such early success validates the core concept of the rotor model but also allows the recognition that deviations occur and led to models incorporating extensions to the rotor formula. This is illustrated in Fig. 11 where we show the energies of 164 Er, a well-deformed rotor, compared to the rigid rotor [J (J + 1)] expression. Of course, for the low lying levels, the agreement is excellent but, one also sees deviations. These are small but non-zero. The key point here is that without the paradigm provided by the original formula, the deviations would not have been noticeable. Once these deviations are recognized, one

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Fig. 10. Illustration of the effects of a sloped outer wall instead of a square well in the context of X (5). In the sloped well case, the wave length (λ2 ) within which a wave function is confined for a state at higher energy is larger, lowering its energy relative to that for the square well (λ1 ).

Fig. 11. Illustration for 164 Er of the effects of adding a higher order term, BJ 2 (J + 1)2 , to the rotor expression for the yrast levels. The simplest rotor expression, E ∼ J (J + 1), deviates from the observed energies for high spins, albeit by a small amount. However, these deviations reveal new physics as embodied in the higher order terms whose predictions are much improved.

tries to account for them by embellishing the model with additional degrees of freedom. In this case, one of the earliest attempts involved the addition of higher order terms, E (J ) ∼ AJ (J + 1) + BJ 2 (J + 1)2 + · · ·. Clearly, better agreement will be obtained, as seen in the figure, and then the issue becomes one of whether the additional parameters (extra complication) are worth the improvement in comparison with the data. Alternately, one can say that the deviations provide evidence for additional physics beyond the simple model and allow one to probe the physical origin of such effects. To some extent, how one feels about this is a matter of taste. In any case, the simple description gives the essential physics and allows one to spot deviations from it that reveal subtle physics that would not be otherwise evident. For the case of the rotor one finds that small deviations on the order of a few keV point to evidence for centrifugal and Coriolis effects in nuclei. The same kind of historical development applies in the case of X (5) and it is left to the reader to make such a judgment as to the relative merits of a simpler, parameter-efficient description compared to more accurate descriptions such as a sloped outer wall that requires additional parameterization. The question naturally arises whether other models can do as well as X (5) in reproducing the data on transitional nuclei such as 152 Sm. Actually, as our discussion has intimated, this is not the correct question. The correct question also brings in the number of parameters in these models as well. Historically, many models have been applied to 152 Sm including

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Fig. 12. Illustration of a band-mixing interpretation of 152 Sm as an alternative to X (5). To achieve a reasonable (albeit not perfect) fit to the 152 Sm data necessitates using unperturbed quasi-bands that have non-rotational energies and requires 12 parameters (or 11 except for scale). The mixing matrix elements are 240 keV (0+ ), 216 keV (2+ ), and 180 keV (4+ ). From Ref. [42].

the traditional rotor model with parameterized intrinsic excitations, rotation-vibration models, the IBA, the pairing plus quadrupole model, and others. A fair discussion of all these in comparison with X (5) and the data is a complex task, precisely because of the issue of parameters, some of which are often hidden in some models that globally fit many nuclei. Such a discussion also needs to involve a fair assessment of what constitutes agreement or disagreement. This will vary from observable to observable (e.g., in-band energies, intrinsic energies, intraband and interband B(E2) values) and raises the issue of assessing uncertainties as well. Considering experimental uncertainties alone [44] is misleading since no one expects these models to work at, say, the 100 eV level. One needs to incorporate an estimate of the theoretical uncertainties, that is, the expected quality of model predictions. This is a task well beyond the present paper and has not yet been carried out. Instead, our goal here has been to assess whether X (5) provides the essential physics, which we have seen that it does. Of course, good agreement with the data for 152 Sm can indeed be obtained with other geometric models but only by utilizing a large number of parameters. This has been discussed extensively in Ref. [42] where it was shown that mixing of two traditional rotor bands (ground and β bands) or three-band mixing with a γ -band cannot reproduce the data but that, by starting with two unperturbed R4/2 values ∼3.27 and 2.61, respectively, and mixing matrix elements of ∼200 keV, one could reproduce the data. However, the essential point is the large number of parameters and the non-rotor unperturbed starting point which, not being paradigmatic, needs to be explicitly parameterized. It is useful to see how this comes about since there has been much confusion in the literature as to whether the 152 Sm results can be understood from the traditional picture of deformed nuclei. The ideas are illustrated in Fig. 12. We first note that both the yrast and the excited 0+ sequence of (yrare) states in X (5) and in nuclei such as 152 Sm have R4/2 values intermediate between vibrator and rotor. Hence, it is impossible to start with a rotor model, as some have claimed possible, with each band having its own R4/2 = 3.33, and obtain the experimental or X (5) levels √ by mixing. Mixing repels levels of the same spin and the mixing matrix elements for K = 0 bands increase with spin as J (J + 1). Therefore, the lower band would have a reduced perturbed R4/2 value after mixing (strong mixing could therefore reproduce an R4/2 ∼ 3.0) but the excited band would have a perturbed R4/2 value above 3.33. One is thus forced to start with unperturbed R4/2 values in which that for the ground band is larger than the perturbed value while that for the excited 0+ 2 band is less than the perturbed value. That implies that one is no longer invoking a rotor model at all but a model with extreme shape coexistence, in which the bands do not correspond to any of the traditional paradigms. That implies that each of the unperturbed energies is a free parameter. For states up through just the 4+ level this entails 5 parameters + + + + [E (2+ 1 ), E (41 ), E (02 ), E (22 ), and E (42 )]! One could, of course, invoke a model here to reduce the number of parameters, such as the anharmonic vibrator, but this is, in effect, just another model assumption. Of course, the strength of the interaction needs to be parameterized as well in order to produce the final energies. Since one is not starting any longer from a rotor model, there is no pre-ordained spin dependence for these matrix elements. Thus one has three additional parameters, one for each spin up to 4+ . Finally, to fit the B(E2) values, one needs a separate parameter for each unperturbed value. If one wants to fit, say, as in Fig. 12, four intraband and four interband B(E2) values, there are an additional 4 parameters representing the unperturbed intraband transition strengths. Totalling all these gives us 12 parameters! Clearly, with that, a good fit can be obtained as illustrated in Fig. 12. However, the process is quite arbitrary, it is inefficient, there is no guarantee that a unique solution is obtained, and it does not start from any known solution to the Bohr Hamiltonian, even if intruder states are invoked. Moreover, additional parameters would be required to fit states with J > 4+ . This exercise therefore serves as little more than as an existence proof of a solution. Of course, the number of

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Fig. 13. Left: Potentials V (β) ∼ β 2n against β , showing the approach to an X (5) potential for large n. Right: Yrast energies predicted with these potentials [using the same γ -dependence as in X (5)] along with the X (5) predictions. From Ref. [58].

parameters can be reduced – say to ∼6–8 – if further models are used to link different observables but this is then, again, arbitrary. What is clearly called for is a simple model that embodies the key physics. With experimental R4/2 values of 3.01 and 2.69 there is little value in attempting a rotor description. In effect, in one simple step, X (5) produces exactly such a model, which well reproduces (albeit not perfectly of course) the data, and that is precisely its appeal. In this context, the quality of agreement achieved (even for spins well above 4+ ) by the parameter-free X (5) solution is astounding. Often, when simple models are invoked, the focus tends to be on discrepancies, and this is, indeed, both important and fruitful but it misses the main point that it is remarkable that such models work at all. When they do, one concludes that they contain the core of the essential physics. It is also hardly surprising that disagreements do surface and one of the goals of studying transitional nuclei is to identify the physical origins of those disagreements (as we have done). The success of X (5) has spawned considerable new theoretical effort to develop new generations of simple models. Many approaches have been developed [48–62], mainly by Bonatsos and collaborators, that simulate critical point nuclei, or the evolution of structure near the critical point. We will not discuss these in detail here—they have been reviewed recently [63, 64]. However, it is worth illustrating the ideas with one particular example, and to mention two others. Consider a harmonic oscillator in β , V (β) ∼ β 2 . Depending on the potential in γ , it gives a kind of anharmonic vibrator spectrum. The square well ansatz of X (5) or E (5) can be arbitrarily well approximated by a potential V ∼ β x , for large enough x, as shown in Fig. 13(left). Thus one might expect that a sequence of potentials of the form V ∼ β 2n with n = 1, 2, 3, 4, . . . could simulate a transition region from the vibrator to the critical point. In Fig. 13(right), we show the results [58] for the yrast levels for such a sequence of potentials, where we use the same γ -dependence as in X (5). We see that the levels approach the X (5) limit for large n. In fact, even for n = 4 the agreement with X (5) is reasonable. Another potential that can be easily solved is the Davidson potential defined by [65]. VDavidson ∼ β 2 + β04 /β 2

(14)

where β0 is a parameter. For β0 = 0 we recover the harmonic oscillator while for large β0 the potential goes asymptotically to zero. Fig. 14 shows examples of the Davidson potential for three values of β0 . None of these is very close to X (5) or E (5) and, indeed, their spectra are quite different than X (5) or E (5) but they are useful in simulating the entire range of structures from a near vibrator to a deformed rotor. The application of these models to transitional nuclei in an exactly separable form was considered in detail in Refs. [54,55,61]. Another model is the so-called Confined β Soft (or CBS) model [62]. This starts with the X (5) ansatz of a square well but with an inner wall that can be parametrically moved from β = 0 to some βmax . As β → βmax , the potential becomes more deformed and approaches the rigid rotor. This model thus spans the entire region from X (5) to the rotor and has had impressive success. A complementary version of this model for the E (5) to γ -soft rotor transition was considered in Ref. [52]. Thus far, the focus has been on the β degree of freedom. However, all these models specify a potential in γ and we now turn to a brief discussion of this. Fig. 15 [66] shows the X (5) spectrum extended to include an excitation analogous to the γ vibration and compares the energies and decay properties of this sequence of states with the data on 152 Sm. Note that this involves two additional parameters, the excitation energy of the γ band and the scale of γ to ground band B(E2) values. (This scale is separate from that of the X (5) B(E2) values discussed for the 0+ -based sequences. The intra-γ band B(E2) values, though, do have the same normalization as used for other intra-band transitions.) Overall, the agreement is quite good. The predicted γ -band energies reproduce the experimental ones well, far better, for example, than does the rotor model. This is shown in Fig. 16. The small energy staggering between adjacent odd- and

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Fig. 14. Similar to Fig. 13(left) except for three Davidson potentials with different values of β0 in Eq. (14). Based on Refs. [61,64].

+ 152 Fig. 15. Comparison of X (5) with 152 Sm for the properties of the γ -band. From Ref. [66]. The call-out box for the 2+ Sm is discussed γ –22 transition in in the text: the left number is from Ref. [42], the right one is from Ref. [44]. The X (5) prediction of 0.20 W.u. agrees with the latter.

even-spin states is almost exactly the same as in the data, while the spacing between such level pairs is slightly greater than observed. The X (5) intraband B(E2) values within the γ -band are consistent with the data but the latter are only very poorly known. Additional measurements of these in-band B(E2) values would be a valuable test of the γ degree of freedom in X (5). For the interband B(E2) values to the ground band the agreement is quite good, although, again, it would be useful to reduce the experimental uncertainties. The transitions from the γ -band to the 0+ 2 sequence of states are particularly significant and present a interesting historical tale. Most of these B(E2) values are very small and so are the X (5) predictions. However, Fig. 15 shows one very + + glaring discrepancy, namely in the 2+ γ –22 transition. As with the other γ to 02 band B(E2) values, X (5) predicts a very small value. However, while for other γ –0+ 2 transitions, the measured B(E2) values are also indeed small, for this case only, the measurements of Ref. [42] gave a collective value of 27 W.u. It was pointed out in Ref. [66] that this represented a glaring discrepancy with X (5). Interestingly, subsequent measurements by Kulp et al. [44] showed that this transition is almost completely M1, in contrast to the substantial E2 component extracted in Ref. [42]. With this new result of Ref. [44], the agreement of X (5) with the data is excellent, as seen in Fig. 15.

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Fig. 16. Comparison of γ -band energies in 152 Sm with X (5) and the rotor. From Ref. [66].

Fig. 17. Comparison of X (5) with the level scheme for 150 Nd. From Ref. [67].

We have focused on 152 Sm for several reasons. It was the first candidate for the X (5) CPS. Following initial comparisons, the community’s interest in this topic sparked a number of further measurements, further delineating the comparison with the CPS. By now, a large number of other nuclei have been discussed as possible X (5) candidates. The first group, and the most discussed to date, are other N = 90 nuclei, such as 150 Nd [67], 154 Gd [68–71], and 156 Dy [72,73]. These have been discussed in other reviews [63,64]. Nevertheless, it is worth mentioning a few key points. Along with 152 Sm, probably the best candidate for X (5) is 150 Nd. Its level scheme for the yrast and yrare levels, taken from Ref. [67], is shown in Fig. 17. Generally speaking the level of agreement with X (5) is comparable to that for 152 Sm, although the details are a bit different. 152 The most obvious difference is that the scales of the 0+ Sm 2 to ground B(E2) values are somewhat closer to X (5) than in but the relative values are in less good agreement, as seen in Fig. 9. Fig. 9 also includes the comparison of intersequence B(E2) values in 154 Gd. In 156 Dy, Ref. [72] found quite good agreement between X (5) and the data although measurements of level lifetimes [69,73] found that the yrast B(E2) values approach the rotor values with increasing spin. Turning to other regions besides the N = 90 nuclei, the question arises of how to identify appropriate candidates. Are there systematic ways of searching for other candidates for X (5)? One is to use the locus of P ∼ 5 in Fig. 8. We need to be cautious, however, understanding that P ∼ 5 only suggests regions of nuclei where the p–n interaction begins to dominate the pairing interaction so that a spherical-deformed shape transition occurs, as typified by R4/2 values near 3.0. However, not all shape transition regions are phase transitional and certainly they need not resemble X (5). We will see this later in the context of Landau theory and the IBA. Thus, P ∼ 5 provides a ‘‘gene pool’’ of nuclei which might exhibit characteristics of X (5). One then needs to inspect the data more thoroughly to vet each candidate. In the A ∼ 150–190 region, the P ∼ 5 criterion suggests a sequence of possible candidates in such nuclei as 162 Yb, 166 Hf, and 178 Os, among others. All these have been studied extensively [74–77]. Some observables, usually including R4/2 , or the yrast B(E2) values (e.g., in 178 Os) agree quite well, often spectacularly so, with X (5). However, in each of these nuclei, other observables deviate from X (5). We illustrate the comparison for 178 Os in Fig. 18. We indicated earlier that a first order phase transition should be accompanied by a change in slope of the two-nucleon separation energies. We saw such an effect near the N = 90 nuclei in Fig. 1(bottom). Looking again at Fig. 1(bottom) we see that there is no similar slope change visible at all near 178 Os. This is consistent with our discussion that, while some observables in 178 Os seem to reflect the properties of X (5), others do not and that this nucleus is likely not representative of

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Fig. 18. Similar to Fig. 17, for 178 Os. From Ref. [64], based on results of Ref. [77].

an X (5)-like nucleus. We will return, however, to this point below after considering Landau theory and a symmetry triangle that captures the evolution of structure. This discussion highlights an important point – with two complementary facets – concerning how one can use various observables to pinpoint structure. On the one hand, it is important to realize that a single observable seldom determines the structure. We will return to this below. On the other hand, a typical nuclear level scheme often has dozens of observables (energy levels, transition rates, etc.) among the low lying levels. However, not all of these are independent and it is critical to determine those that independently narrow down the structural possibilities. A trivial case illustrates this. Given an R4/2 value, the rest of the yrast levels (below any backbend) are more or less determined quite well. For example, the standard general anharmonic vibrator (AHV) expression for yrast energies (shown in Eq. (12)), generally reproduces very well yrast energies stretching from the harmonic vibrator to the rigid rotor; that is, once R4/2 is given, the higher yrast energies, to first order, often do not tell us much more about the shape and collective properties of the nucleus. Although they can reveal subtle details and, indeed, additional relevant degrees of freedom or interactions at work, they generally do not change our understanding of whether the nucleus is close to magic, vibrational with an equilibrium spherical shape, transitional, axial or axially asymmetric, or well deformed. In some cases, of course, that structure changes with spin, and E-GOS plots are a nice way of spotting such changes. To summarize these ideas: To determine the basic structure of a nucleus, it is key to identify the essential observables that independently help to pinpoint its structure. Once this is done, then other observables, or, better, deviations of any observables from those predicted by that structure can help identify more subtle aspects. These ideas may seem trivial but they are often forgotten or ignored, and the literature is rife with examples in which structure has been mis-identified by a focus either on a single observable, or an inappropriate set of observables, or by blind χ 2 -fitting procedures that treat all observables equally and use only experimental uncertainties, thus weighting the fits by accidents of experimental accuracies far beyond any reasonable expectation for the observable-dependent quality of the model predictions. We started this discussion of a search for additional candidates for X (5)-like nuclei by using a phenomenological approach, the P-factor, based on the competition of pairing and p–n interactions, and using the value P ∼ 5 as a criterion for possible X (5) nuclei, but we stressed that this criterion only gives a set of possible X (5) nuclei and we also commented that, in general, more than one observable is needed to pin down the structure. We now pursue the discussion of optimal signatures of X (5). In an important paper, Clark et al. [78] surveyed a large number of nuclei to search for X (5) behavior using six + + criteria: yrast energies and B(E2) values, the ratio E (0+ 2 )/E (21 ), 02 -sequence (yrare) energies and B(E2) values, and intersequence B(E2) values. [Ref. [79] carried out a similar search for candidates for E (5) nuclei—see below.] This work was carried out before the simple explanations discussed above were developed for the overprediction of 0+ 2 -sequence energies (the sloped wall) and of the extreme sensitivity of the scale of intersequence B(E2) values to structure in the critical region. Therefore, in retrospect one might look at relative rather than absolute yrare energies and inter-sequence B(E2) values. Nevertheless, this survey identified, besides the N = 90 nuclei, 126 Ba and 130 Ca as possible X (5) nuclei. 130 Ce has subsequently been studied [80] with β -decay and showed some similarities with X (5) but also some significant deviations. In addition, Brenner also carried out a search [81] for X (5) candidates, in the A ∼ 80 and 100 regions. He focused first on the + 76,78,80 R4/2 and E (0+ Zr would be useful, as well as 2 )/E (21 ) criteria and then inspected further data, suggesting that study of 104 104 of Mo. A subsequent study [82] of Mo showed reasonable consistency with X (5) but later measurements [83] of yrast

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Fig. 19. Low-lying levels of E (5) (from Ref. [29]) compared with the data for 134 Ba. Based on Ref. [35].

Fig. 20. Comparison of E (5/4) with the low-lying levels and transitions in 135 Ba. Based on Ref. [85].

B(E2) values found discrepancies between X (5) and the data. Of course, the most fertile ground for seeking new examples of X (5) probably lies in the regions of the dark contours far from stability in Fig. 8. To this point we have only discussed the first order CPS X (5). However, nuclei spanning the transition region from a vibrator to a γ -soft rotor go through a second order phase transition for which Iachello developed [29] the first CPS, E (5). In E (5), V (γ ) = constant and so this description preserves all the selection rules (and relative intra-O(5) representation B(E2) values) of the O(5) symmetry. The low-lying levels are shown in Fig. 19. Since the branching ratios within the phononlike sequence of degenerate multiplets are just those of O(5), we do not show them. The key observables are the yrast energies and B(E2) values and the properties (energies and decay transitions) of the excited 0+ states. The level scheme of 134 Ba [35] is shown for comparison. The agreement, as far as it goes, is reasonable, but more data are clearly needed. Other possible E (5) candidates were discussed in Ref. [79]. E (5) is of interest for another reason: it spawned the first example of a bose-fermi CPS, E (5/4), which describes the coupling of a j = 3/2 (e.g., d3/2 ) particle to an E (5) case [84]. 135 Ba is the initial test case and E (5/4) and the 135 Ba level scheme [85] are shown in Fig. 20. In comparing the two, one first has to note that, near the 2d3/2 orbit, is the 3s1/2 single particle state and this level (and collective levels built on it) needs to be identified in the experimental level scheme. Comparing E (5/4) and 135 Ba one indeed finds areas of agreement (especially for B(E2) values) but also serious discrepancies (especially in the breaking of energy degeneracies). Clearly more data, and probably a refined CPS incorporating the j = 1/2

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orbit is needed. A more useful bose-fermi CPS model would correspond to odd-A nuclei near X (5) but, theoretically, this is difficult due to the need to incorporate several rather high-j single particle orbits (e.g., 1h9/2 , 2f7/2 or, alternately, 1i13/2 ). 3. Structural evolution Of course, the vast majority of nuclei are not at either critical point. Indeed, even in phase transitional regions there is no assurance that any given nucleus will resemble X (5) simply because nuclei exist only for integer numbers of nucleons and therefore their structure changes discretely with N and Z . It may easily happen that the exact critical point is jumped over. The A ∼ 100 region near 100 Zr exemplifies this where, for example, R4/2 (98 Zr) = 2.16, R4/2 (100 Zr) = 2.66, and R4/2 (102 Zr) = 3.15 and no nucleus in fact has R4/2 ∼ 2.9. More generally, most nuclei fall in regions of intermediate structure. Here the paradigms of structure, such as the symmetric rotor, the harmonic vibrator, the gamma soft rotor, E (5) and X (5), and others studied more recently [48–62] provide essential benchmarks that frame a given structure. The key task, if we are to understand how structure evolves in nuclei and what determines those variations, is to develop techniques to pinpoint a given structure. In this section, we will present a brief and highly simplified discussion of the possible range of structures in the context of Landau theory, we will show that the IBA-1 model nicely spans this set of structures, and then we will present a simple yet powerful technique, that of orthogonal crossing contours, that will allow us to closely identify structure in a very large suite of nuclei where at least minimal data on yrast energies and a couple of intrinsic excitations are known. 3.1. Landau theory Landau theory is an elegant approach to understanding phase transitional behavior in a wide variety of physical systems. Here we give a very simplified treatment that contains the essential physics but ignores some subtleties that arise in the γ degree of freedom (or, more generally, when there are two degrees of freedom or more). For a more rigorous treatment, the reader is referred both to Landau’s original paper [86] and to [87,88]. For the pedagogical treatment that is closely followed here, see Ref. [89]. Consider a physical system whose energy can be written as a functional of two variables. We assume that the energy surface can be written as an expansion of an energy functional in terms of the system coordinates. In the original theory, the variables were pressure and temperature. Here we will use β and γ and use the standard form of the dependency of the energy functional on γ , keeping terms up to quartic in β , the quadrupole deformation. Thus we write this energy functional (Φ , using Landau’s notation)

Φ = Φ0 + Aβ 2 + Bβ 3 cos 3γ + C β 4

(15)

where Φ0 is the spherical solution, and A, B, and C are functionals of the control variables (N and Z ). Note that the use of this functional assumes that we are dealing with a nucleus with quadrupole deformation and a single minimum in the collective potential. It is therefore applicable to the types of nuclei we have been discussing but not, for example, to isotopic chains such as the light Pb or Hg nuclei. (It cannot encompass the small double minima envisaged in Fig. 4(left) for a first order QPT. However, the barriers between such minima are considered to be quite small.) To identify possible equilibrium solutions, we consider the minima of this equation in β and γ . The minima of Φ as a function of γ occurs for γ < 0◦ or 60◦ (for opposite signs of Φ0 ). We consider the prolate case γ = 0◦ . First, clearly one minimum occurs if A is positive and β = 0. This corresponds to spherical nuclei. For β non-zero, the first derivative of Eq. (15) must be zero and the second derivative positive. This gives

β(2A + 3Bβ cos 3γ + 4C β 2 ) = 0

(16)

2A + 6Bβ cos 3γ + 12C β > 0.

(17)

2

The solutions for β 6= 0 are obtained if we set B = 0 (this is a line in the A–C plane) in Eq. (16). Then, Eq. (16) gives 2A + 4C β 2 = 0 or

r β=±

−A 2C

.

(18)

This requires C and A to have opposite signs. Since C must be positive for bound solutions, A must be negative in the deformed phase. Of course, this makes sense: to have a minimum at finite β requires Φ to go negative for small β . This is assured if the β 2 term in Eq. (15) has a negative coefficient. The finite β solutions correspond to prolate (β > 0) and oblate (β < 0) equilibrium shapes, separated by a curve corresponding to B = 0. Thus there are, in total, three phases. The spherical-deformed phase transition is generated by a change in sign of A, while the prolate-oblate one corresponds to changing the sign of B. These solutions are illustrated in Fig. 21(left). The curves in the phase space defined by A = 0 and B = 0 are lines of first order phase transition. They meet at an isolated point of second order phase transition (A = B = 0), labeled t, which is a triple point.

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Fig. 21. Left: Simplified sketch relevant to the discussion of Landau theory [86]. Right: The sketch at left incorporated in a structural triangle for collective nuclear structure [87–89].

Fig. 22. The symmetry triangle of the IBA with the addition of the geometric critical point descriptions X (5) and E (5). See discussion of ζ and χ in the text.

According to the Gibbs phase rule for a single component system (here, we assume that nuclei are symmetric in proton and neutron degrees of freedom, an assumption that is well justified in medium mass and heavy nuclei at low excitation √ energies), there are the only three solutions, and hence these solutions (β = 0, ± ) represent the entire phase diagram. It is convenient to depict the phase structure of Landau theory in the context of a triangle. We show this on the right in Fig. 21 and its topological connection to Fig. 21(left) should be obvious. The lower left represents spherical nuclei (β = 0). Then there is a line of first order phase transition stretching from a point on the bottom leg to a point on the left leg. This is the spherical-deformed phase transitional line. Extending from this line to the center of the upper right leg of the triangle is another first order line corresponding to the phase transition between prolate and oblate shapes. (We note that this degree of freedom introduces an issue we will not delve into, namely, that, in a γ -flat potential, one can go smoothly from a prolate to an oblate shape through a variation in γ . In such a scenario, this transition is a structural transition but not a phase transition. In the present context, whether or not we are dealing with a rigorous phase transition or a sharp change in the sign of the deformation is not important.) The point where these two lines of first order phase transitions meet (labeled t on the left) is a singular point of second order phase transition, that is, a nuclear triple point. This discussion provides a complete description of the structural possibilities for quadrupole deformed collective nuclei as long as they are proton-neutron symmetric. This ignores nuclei with rigid triaxial shapes and, of course, higher order shape components. It is well known (in particular from analyses [90] of energy staggering in the γ bands) that, in nearly all nuclei with axial asymmetry, that asymmetry arises from dynamical excursions in γ due to zero point motion in a γ -soft potential rather than from a deep minima in γ . Therefore, it is a reasonable assumption to invoke the triangle in Fig. 21 as spanning the range of quadrupole collective structures applicable to collective nuclei at low energy and low spin. Oblate nuclei and nuclei in a prolate–oblate transition region have recently been discussed [87,89] in the context of Fig. 21. However, by far the vast majority of deformed nuclei are prolate. Henceforth in this overview, we will limit the discussion to such cases. Thus, the symmetry triangle of Fig. 21(right) reduces to that of Fig. 22 which will form the basis of the remainder of our discussion. This triangle will be considered in the framework of a particular model, the Interacting Boson Approximation (or IBA) model [28], to which we now turn. 3.2. Evolutionary trajectories For the most part we have been discussing regions of critical point behavior or other paradigms of structure such as the vibrator, symmetric rotor, and γ -soft rotor. Of course, most nuclei are not represented by such idealizations but have

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some sort of intermediate structure. How can one deal with such cases? Clearly, one needs a collective model (we are only discussing so-called collective nuclei here: one can use R4/2 >∼ 2.0 as a practical definition of this class) that is general, that contains a variety of structures embodied in its Hamiltonian, and which is easy to deal with. Preferably, it should have as few parameters as possible, and the parameters should hopefully have a simple physical interpretation. While, as a consequence of the interest in the CPS, a number of solutions to the Bohr Hamiltonian have recently been intensely studied (and some, such as the Davidson potential, β 2n potentials and the CBS, were discussed above), the IBA model is, by far, the most general and parameter-efficient collective model ever proposed. This model has several incarnations—here we discuss only the simplest, the so-called IBA-1, which assumes proton-neutron symmetric structure so that the Hamiltonian does not depend on separate proton and neutron degrees of freedom. Extensions to this, such as the IBA-2, contain all the solutions that the IBA-1 does, plus others, but nearly always have a considerably larger number of parameters. Here we deal only with the IBA-1 which we call simply the IBA. This model straddles the divide between collective and microscopic models. Phrased alternately, it is a collective model with a microscopic underpinning. The model has been reviewed innumerable times (see, for example, early Refs. [91,92]) and need not be re-hashed here except for the following brief comments. The model represents a drastic truncation of the shell model to only those valence space configurations in which two nucleons couple to spin zero (s bosons) or 2 (d bosons). The number of bosons in a given nucleus, NB , is thus a fixed number and the predictions depend on this boson number, lending a true microscopic aspect to the model. The Hamiltonian consists of boson energy terms and boson-boson interactions. The s–d boson structure also leads to a group theoretical framework in terms of the group U (6), and its subgroups. Analysis of the generators, representations, and Casimir operators of these groups leads to three dynamical symmetries, denoted U (5), SU (3) and O(6) corresponding closely to the three geometrical symmetries of the vibrator, rotor, and γ -soft rotor. It is nearly universal practice to discuss the IBA in terms of a symmetry triangle given either as on the right in Fig. 21 (relevant to a description including both prolate and oblate shapes) or, more commonly, as in Fig. 22 (relevant for prolate shapes only). We consider the latter here. The vertices correspond to the three symmetries. Intermediate structures correspond to internal points in the triangle. In this figure we also show the CPS X (5) and E (5). This requires an important caveat. The Bohr Hamiltonian and the IBA are not equivalent. This has been much discussed. The models are different: the boson number dependence in the IBA is absent in the geometric model, and potentials such as those of the CPS X (5) and E (5) are not contained within the IBA Hamiltonian, nor are they dynamical symmetries. Therefore, formally, it is incorrect to place dynamical symmetries and geometrical solutions on the same diagram. Nevertheless, with this caveat, it remains pedagogically useful and highly intuitive to do so. With such warnings, we therefore include the CPS in Fig. 22. In Fig. 22, we have connected the E (5) and X (5) CPS with a first order phase transition line in accordance with both the Landau analysis above and with an analysis of the phase structure of the IBA model [93,94] using the coherent state formalism. This line of first order phase transition terminates in an isolated point of second order phase transition at E (5). Viewed in light of Fig. 21, E (5) is simultaneously a point of second order phase transition and the meeting of two lines of first order phase transition, that is, a nuclear triple point. We now turn to a general description of structural evolution within the triangle in the context of the IBA. The full IBA Hamiltonian for a given total boson number NB (that is, a given nucleus) has 6 parameters. However, by far, the majority of calculations are carried out with a simplified Hamiltonian [96] with just three parameters, namely,



H =  nd − κ Q · Q = c (1 − ζ )nd −

ζ 4NB

 Q ·Q

(19)

where nd is the number operator (dĎ d)0 , giving the number of d bosons,  is the d boson energy, and the boson quadrupole operator Q is defined as Q = sĎ d˜ + dĎ s + χ[dĎ d˜ ]2

and

ζ =

4NB 4NB + /κ

.

(20)

The second form of Eq. (19) avoids the infinities inherent in the use of the ratio of /κ as ζ varies from 0 to 1 (see below). The factor c in Eq. (19) is only a scale factor of little interest, given by c =  − 4NB κ , and ζ and χ are therefore the two parameters that determine the structure. Note that since we are interested in excitation energies there is an arbitrary energy zero which we have fixed by setting the s boson to zero energy (hence there is no ns term in H). Thus, in practice we have a two-parameter model of Ising [98] type. Let us consider this Hamiltonian and the effects of its two parameters. Clearly, if ζ = 0 we have only the first term. Hence the eigenvalues are simply  times the number of d bosons. The ground state clearly will have nd = 0 and spin 0+ , the first excited state will have nd = 1 and spin 2+ . The nd = 2 configuration will be a triplet of states formed by angular momentum coupling two d bosons to J = 0+ , 2+ , and 4+ . Clearly this equally spaced spectrum is that of the geometric vibrator. √ On the other hand, if ζ = 1, the first term vanishes and one has only a boson quadrupole interaction. If χ = − 7/2 = −1.32, the resulting spectrum is that of SU (3), a special limit of a symmetric rotor with degenerate β and γ bands. If χ = 0, one has the γ -soft O(6) limit. These three limiting cases, their relevant parameters, and mini-level schemes are indicated in Fig. 22. The figure also shows the meaning of ζ and χ based on the above discussion. ζ represents the distance to a given point in the triangle from the U (5) vertex: it gives a measure of the spherical-deformed character at that point. χ gives the positioning of this radius

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√

Fig. 23. Left: Behavior of E6/0 for several IBA calculations (for large NB ) corresponding to the bottom leg of the symmetry triangle (χ = − 7/2), the U (5) to O(6) leg (χ = 0), and to an intermediate case. Note that E6/0 crosses unity very rapidly in the transitional region. Right: Data on E6/0 in the A ∼ 150 region of first order phase transitional behavior reflecting the predicted peaking behavior on the left. From Ref. [95].

vector along the O(6) [χ = 0] to SU (3) [χ = −1.32] leg, that is, the angle relative to the bottom axis, and specifies the axial asymmetry. Thus ζ and χ give the position in the triangle corresponding to a given calculation in terms of a radius vector, ρ , from the U (5) vertex, and an angle Φ of this vector off the bottom axis given by [95]

√

3ζ

ρ= √

3 cos Φχ − sin Φχ

Φ=

π 3

+ Φχ

(21) (22)

√ χ = − 7/2 corresponds to the bottom leg of the triangle and to an axially symmetric nucleus. (Beyond the scope of this review is that there is zero point motion in γ which is boson number-dependent but this has only minor repercussions in the present discussion.) χ = 0 corresponds to the line connecting U (5) and O(6) and to a γ -independent potential. Note that such a geometric shape has the group structure O(5), which persists all along this leg. This is perhaps the simplest example of what is called a partial dynamical symmetry [99] in which all the states exhibit part of a dynamical symmetry. Note that E (5) also partakes of this character. Before discussing structural evolution generally throughout the triangle, it is worthwhile inspecting the results of calculations approximating the line of first order phase transition from X (5) to E (5) and to discuss a newly discovered [97] observable that has a unique value along most of that line and which can distinguish first and second order phase transitional behavior. Of course, the distinction between first and second order behavior will become muted along that line as the second order phase transition at E (5) is approached. Therefore, any distinguishing signature will lose sensitivity, given typical experimental errors, near E (5). Nevertheless, a recent study [97], in the context of the IBA in the large boson number limit, has shown that there is at least one very simple-to-observe quantity whose behavior uniquely identifies nuclei along this line and can also distinguish between first and second order cases. This observable was referred to earlier + E6/0 = E (6+ 1 )/E (02 )

(23)

E6/0 = 1.5 for a harmonic vibrator and descends to values well under unity for typical deformed nuclei. It takes on the value of ∼ unity at the phase transitional point as noted earlier in our initial discussion of the characteristics of X (5). The interesting new results are [97] that its behavior is very steep precisely across the phase transitional region and that this value of ∼1 is preserved along the phase transitional line. Moreover, E6/0 just prior to (on the vibrator side) the critical point shows a peak above the vibrator value for a first order phase transition but none for the second order case. This is illustrated in Fig. 23(left) for several IBA calculations. Experimentally, such a peaking is observed in the data for the N ∼ 90 region, as seen on the right. Interestingly, in a region where there is no first order phase transition, no peak is seen [97]. We noted earlier that 178 Os has some of the characteristics of X (5) such as yrast B(E2) values but that other observables differed from X (5). As one progresses from the bottom leg along the first order phase transitional line in Fig. 22 towards E (5), the principle structural change is a softening of the potential in γ (recall that E (5) and the entire U (5) to O(6) line are γ -independent), a muting of the difference in the coexisting values of β and of the width in the triangle of the coexistence region. The β values corresponding to the coexisting minima merge at the point of second order phase transition. The Os nuclei are well known to exhibit increasing γ -softness with increasing mass. Therefore it may well be that 178 Os has a structure close to that of a first order critical point nucleus, but one that lies off the bottom leg of the triangle. We will see that this is validated below by the results of a structural mapping of rare earth nuclei in the symmetry triangle. Such an interpretation also explains the lack of a kink in the S2n values in Fig. 1(bottom). We now turn to the general issue of structural behavior and to the task of locating a nucleus in the triangle. We discuss here a powerful technique, that of orthogonal crossing contours (OCC), whose basic idea goes back a long time [95,100–102]

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Fig. 24. Contours of R4/2 = 2.9 and E02 = +0.4 in the IBA symmetry triangle, calculated in the IBA with the Hamiltonian of Eq. (19) with NB = 10.

Fig. 25. Comparison of the IBA using the OCC technique with the level scheme of figure.

104

Mo. I am grateful to E.A. McCutchan for providing this fit and the

but which has recently [103] been formulated into an easy to use, highly efficient approach where the physics is intuitively clear. The technique allows one to easily fit the data for collective nuclei with a 2-parameter calculation [95,104,105]. Consider Fig. 24 and R4/2 . R4/2 = 2.0 at the (harmonic) U (5) vertex and 3.33 for SU (3). So, presumably, somewhere along the U (5) to SU (3) leg, R4/2 passes through each value in between these limits, such as 2.9. Now consider the SU (3) [R4/2 = 3.33] to O(6) (R4/2 = 2.5) leg. Again, somewhere along that leg R4/2 = 2.9. Finally, consider a point, A, on the SU (3) to O(6) leg between the point where R4/2 = 2.9 and SU (3). At point A, 2.9 ≤ R4/2 ≤ 3.33. Hence, somewhere along the radius vector from U (5) (where R4/2 = 2.0) to point A on the SU (3) to O(6) leg, R4/2 must cross a value of 2.9. Continuing this argument for all possible points A we see that there must be a contour of constant R4/2 = 2.9 crossing the interior of the triangle from a point on the U (5) to SU (3) leg to a point on the SU (3) to O(6) leg. This contour is shown in Fig. 24 (the exact locus is boson number-dependent: This example is for NB = 10). Thus we see that R4/2 is an excellent signature of structure but it is not enough. All it specifies is a contour in the triangle. R4/2 is, quite literally, only half the story. We need a second observable to pinpoint a position in this 2-dimensional triangle. What observables can we use? This has been extensively discussed and that discussion need not be repeated here. Suffice it to say that most obvious observables tend to run parallel to R4/2 and therefore are of little help. However, there is a general class of observables that tends to run orthogonal to R4/2 . These consist of differences in the energies of pairs of intrinsic states. One of the best of these, which is frequently known experimentally, is E02 =

+ E (0+ 2 ) − E (22 )

E (2+ 1 )

(24)

where the denominator is a convenient normalization. In Fig. 24, we include the contour for E0/2 = +0.4 and see that it nicely cuts across the R4/2 = 2.9 contour. So, if we have some nucleus with R4/2 = 2.9 and E0/2 = +0.4 the crossing of these approximately orthogonal contours gives its location in the triangle. These values happen to apply to 104 Mo. Thus, given these observables, we pinpoint ζ and χ and can then predict many observables in 104 Mo. A comparison of the 104 Mo level scheme with the IBA predictions using such a technique is shown in Fig. 25. Fig. 26 shows a fuller set of R4/2 and E0/2 contours (again, for NB = 10) [103]. Note one critically important point. Structure does not vary linearly in the triangle. This is seen by the clustering of R4/2 contours between 2.5 and 3.1. Indeed, the rapid

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Fig. 26. Contours of R4/2 and E02 in the IBA for NB = 10. Based on Ref. [103].

Fig. 27. Comparison of IBA calculations using the OCC Technique with the data for 158 Dy. I am grateful to E.A. McCutchan for this figure.

Fig. 28. Comparison of the empirical level scheme of 180 Pt with an IBA calculation based on the OCC technique. Based on Ref. [104].

change seen there is the essence of the idea of a rapid phase transition. For higher boson numbers, the effect is more extreme. Thus, short distances near the middle of the triangle correspond to large structural changes while even rather large distances near the U (5) and SU (3) vertices yield only small changes. This is related to the idea of quasi-dynamical symmetries [106]. This and similar plots for other boson numbers are invaluable in placing nuclei within the triangle and in discerning the trajectories of structural evolution. Using a slight variant of such an approach, a large number of nuclei in the A ∼ 150–200 region have been mapped [95,104,105]. It is useful to illustrate how this works in practice. We consider a more or less random example, 158 Dy. From R4/2 and E0/2 we obtain the parameter values ζ and χ . Using these we get the level scheme shown in Fig. 27 (see Ref. [95]). Fig. 28 shows another example [104], for the neutron deficient nucleus 180 Pt. Clearly, in both cases, the agreement with the data is very good, especially considering the simplicity of the calculation. Fine tuning of the parameters can, of course, produce small improvements, as can adding further terms in the Hamiltonian, at the cost of additional parameters. Note that in the Pt cases there is no need to invoke intruder states. This has been discussed in Ref. [104].

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Fig. 29. Trajectories of structural evolution for the mass region A ∼ 150–200. I am grateful to E.A. McCutchan for this figure, updated from the work in Refs. [95,104,105].

From decades of experience with reproducing experimental level schemes with the IBA, it is highly recommended to follow a procedure such as this and to avoid at all costs blind least squares fitting procedures. There are several reasons for this. One is that the relevant errors in applying such procedures are not the experimental ones (especially for energies) but rather some expectation of the likely accuracy of the model—and such estimates are very difficult to quantify. Secondly, least squares fits to all the data, for example, those in Figs. 25, 27 and 28, multiply-count the same physics. For example, including all the yrast states largely repeats a fit to the same basic inertial structure. It is essential in evaluating a nucleus to choose observables that are structurally independent, as we have done in the OCC technique. In general, a couple of yrast energies, and either energies of intrinsic excitations or B(E2) values connecting them to each other or to the yrast levels are appropriate. Stated more concretely, fitting the IBA parameters using, say, the γ bandhead is far more physically meaningful than fitting one more yrast state. A systematic mapping of the rare earth region has been carried out in the last few years [95,104,105] and has led to a considerably revised understanding of the structural evolution of nuclei in the Z = 50–82 and N = 82–126 region. Not all nuclei have been re-mapped (for example, the nuclei with Z < 74 and N > 104) but an extensive mapping nonetheless exists. In Fig. 29 we show the trajectories of structural evolution resulting from this. These trajectories give an overview of how structure evolves in the best-studied major shell region in heavy nuclei. We see a rather complex behavior. Some nuclei (such as Sm-Gd) start near the bottom leg of the triangle and move from spherical to deformed, largely axially symmetric, shapes. Dy also starts near the bottom leg but rises into the interior for the neutronrich nuclei. Others, such as Yb and Hf, have more complicated loops through the triangle, while others such as Os and many of the neutron-deficient Pt isotopes seem stationary at fixed locales in the triangle. The fits for the Pt isotopes also showed that, while one needs to involve cross-Z = 82 proton intruder states to account for the Hg and Pb data, the Pt nuclei do not require a separate consideration of such configurations and can be successfully understood in the IBA in the context of the normal valence shells [104]. 4. Final comment The perspective that Fig. 29 provides on the evolution of structure in a broad mass region in terms of the collective symmetries of the nuclear systems and the spherical-deformed equilibrium phase diagram brings us back to where we started, namely the confluence of the femtoscopic and collective perspectives on nuclear structure come together. Collective models are able to correlate huge amounts of data, often with an extreme parameter efficiency, but they cannot indicate why these nuclei exhibit the structures that they do—understanding this latter question is the scope and purview of models based at the nucleonic level where the mean field and residual interactions determine and give rise to the coherent collective motions that nuclei exhibit. The issue at stake in the linking of the two mantras given at the beginning of this review— understanding the forces that determine structure in nuclei and understanding how those forces can give rise to the astonishing regularities and simple patterns that atomic nuclei exhibit. Linking these two approaches is one of the great challenges of modern nuclear structure and one that will become even more important as we delve more and more into nuclei further and further from stability where shell structure and the interactions themselves are likely to change and reveal new phenomena, new aspects of collective behavior and, possibly, new many-body symmetries. Acknowledgements I am grateful to many people, both for collaborations and discussions, without which this review, and much of the work discussed within, would not exist. It is impossible to mention all, but a few stand out and I would like to express my special gratitude to them: F. Iachello, D.D. Warner, N.V. Zamfir, E.A. McCutchan, M.A. Caprio, R.B. Cakirli, J. Jolie, P. von Brentano,

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D. Bonatsos, V. Werner, S. Pittel, W. Nazarewicz, K. Heyde, D.S. Brenner, P. Regan, N. Pietralla, Y. Oktem, and M. Fetea. Thanks are also due to P. Farnsworth for essential and invaluable help in preparing this paper. Work supported by the US DOE under Grant No. DE-FG02-91ER-40609. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72]

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