Introduction

The behavior of the nucleus at high angular momenta is strongly affected by the single-particle (s.p.) structure, i.e., shell effects. Properties of the s.p. orbits around the Fermi level determine the deformability of the nucleus, the amount of angular momentum available in the lowest-energy configurations, the moment of inertia, and the Coriolis coupling. Consequently, nucleonic shells can be seen and probed through the measured properties of rapidly rotating nuclei.

The independent particle model is a first approximation to the nuclear motion. Here, the nucleons are assumed to move independently of each other in an average field generated by other nucleons. Each nucleon occupies a s.p. energy level, and levels with similar energies are bunched together into shells. The wave function of a given many-body configuration uniquely characterized by s.p. occupations is an antisymmetrized product of one-particle orbitals (the Slater determinant). In the next step, the residual interaction between particles needs to be considered. This is the essence of the configuration interaction method or the interacting shell model. For heavier nuclei, where the number of s.p. orbits becomes large, a customary approximation is to divide the configuration space into the (inert) core states and the (active) valence orbits and to perform configuration mixing in the valence subspace.

The basic idea behind the additivity principle for one-body operators is rooted in the independent particle model. The principle states that the average value of a one-body operator $\hat{O}$ in a given many-body configuration $k$, $O(k)$, relative to the average value in the core configuration $O^{\mbox{\rm\scriptsize {core}}}$, is equal to the sum of effective contributions of particle and hole states by which the $k$-th configuration differs from that of the core. Such a property is trivially valid in the independent particle model. However, the presence of residual interactions and resulting configuration mixing could, in principle, spoil the simple picture. In particular, in the interacting shell model, the polarization effects due to additions of particles or holes are significant and they give rise to strong modifications of the mean field. So the essence of the additivity principle lies in the fact that these polarizations are, to a large extent, independent of one another and thus can by treated additively.

The additivity principle for strongly deformed nuclear systems was emerging gradually in the 1990s. First, it was found in Ref. [1] that effective (relative) angular momentum alignments are additive to a good precision in the superdeformed (SD) bands around $^{147}$Gd. However, the analysis was only restricted to a few bands. Later, the statistical analysis of Ref. [2] in the $A\sim 150$ and 190 mass regions clearly demonstrated that the so-called phenomenon of band twinning (or identical bands) is more likely to occur in SD than in normal-deformed bands. It was shown that a necessary condition for the occurrence of identical bands is the presence of the same number of high-$N$ intruder orbitals (see also Ref. [3]). In addition, it was concluded that the configuration-mixing interactions such as pairing and the coupling to the low-lying collective vibrational degrees of freedom act destructively on identical bands by smearing out the individuality of each s.p. orbital. Such individuality is an important ingredient for the additivity principle: it is expected that this principle works only in the systems with weak residual interaction, in particular, pairing [2,4].

The principle of additivity at superdeformation was explicitly and thoroughly formulated for the case of the $Q_{20}$ quadrupole moments in the non-relativistic study of quadrupole moments of SD bands in the $A\sim 150$ mass region in Ref. [5] within the cranked Hartree-Fock (CHF) approach based on Skyrme forces. It was shown that the charge quadrupole moments calculated with respect to the doubly magic SD core of $^{152}$Dy can be expressed very precisely in terms of effective contributions from the individual hole and particle orbitals, independently of the intrinsic configuration and of the combination of proton and neutron numbers.

Following this work, it was shown that the principle of additivity of quadrupole moments works also in the framework of the microscopic+macroscopic method (in particular, the configuration-dependent cranked Nilsson+Strutinsky approach) [6,7]. However, contrary to self-consistent approaches, the effective s.p. quadrupole moments of the microscopic+macroscopic method are not uniquely defined due to the lack of self-consistency between the microscopic and macroscopic contributions.

The study of additivity of quadrupole moments and effective alignments was also performed in the framework of the cranked relativistic mean field (CRMF) approach, but it was restricted to a few configurations in the vicinity of the doubly magic SD core of $^{152}$Dy [8]. It was suggested in this work that the additivity principle when applied to the angular momentum operator (i.e., effective alignments) does not work as well as for the quadrupole moment. In addition, the effective alignments of high-$N$ intruder orbitals seem to be less additive than the effective alignments of non-intruder orbitals. The latter can be attributed to a pronounced polarization of the nucleus by high-$N$ intruder orbitals at high spin.

Figure 1: (color online) Single-particle energies for neutrons (left) and protons (right) in $^{128}$Ba as a function of the proton quadrupole moment calculated in the HF+SLy4 model. Solid and dashed lines mark positive and negative parity states, respectively. The orbitals are labeled by the asymptotic (Nilsson) quantum numbers $[{\cal N}n_z\Lambda ]\Omega$ of the dominant component of the s.p. wave function. The neutron intruder orbitals originating from the ${\cal N}$=6 shell are additionally labeled by the main oscillator quantum number and a subscript denoting the position of the orbital within the ${\cal N}$ shell.

For quadrupole moments, the additivity principle was experimentally confirmed in the $A$$\sim$140-150 mass region of superdeformation. It was shown that the quadrupole moments of the SD bands in $^{142}$Sm [9] and $^{146}$Gd [10] could be well explained in terms of the $^{152}$Dy SD core and effective s.p. quadrupole moments of valence (particle and hole) orbits. All of these studies, together with the previous results for moments of inertia [11,12] and effective alignments [13,8], strongly suggest that the SD bands in the $A$$\sim$140-150 mass region are excellent examples of an almost undisturbed s.p. motion. This is especially true at rotational frequencies above $\hbar\omega$=0.5 MeV [12,8] where pairing is expected to be of minor importance. (For other excellent examples of an almost undisturbed s.p. motion at high spins, see Refs. [14,15,16].)

In the mass $A$$\sim$135($Z$=58-62) light rare-earth region, large $Z$=58 and $N$=72 shell gaps (see Fig. 1 and Refs. [17,18]) lead to the existence of rotational structures with characteristics typical of highly deformed and SD bands. These bands were observed up to high and very high spins (see Refs. [19,20] and references quoted therein). For example, the yrast SD band in $^{132}$Ce extends to $\sim$68$\hbar$, which represents one of the highest spin states ever observed in atomic nuclei [19]. At such high spins, pairing is expected to play a minor role [11,18,21], which is a necessary condition for the additivity principle to hold. In this mass region, experimental studies of the additivity principle were performed in Refs. [21,20]. Differential lifetime measurements, free from common systematic errors, were performed for over 15 different nuclei (various isotopes of Ce, Pr, Nd, Pm, and Sm) at high spin within a single experiment [21,20].

There are several notable differences between the $A$$\sim$135 and $A$$\sim$140-150 regions of superdeformation. In particular, the rotational bands in the $A$$\sim$135 region are calculated to correspond to the local energy minima that are characterized by much larger $\gamma$-softness than those in the $A$$\sim$140-150 mass region [17,18]. Thus, one of the main goals of the present manuscript is to find the impact of the $\gamma$-softness on the additivity principle. The second goal is a detailed study of the additivity principle not only for quadrupole moments but also for angular momentum alignments. The present work is the first study where the additivity of relative alignments has been tested within the CHF and CRMF frameworks in a systematic way along with the additivity of quadrupole moments. Some results of this study have been reported in Refs. [21,20].

This paper is organized as follows. The principle of additivity, definitions of physical observables, the way of finding effective s.p. quantities, and details of theoretical calculations are discussed in Sec. 2. Analysis of the additivity principle for quadrupole moments and relative alignments, and the discussion of associated theoretical uncertainties are presented in Sec. 3. Finally, Sec. 4 contains the main conclusions of our work.

Figure 2: (color online) Neutron s.p. energies (routhians) in the self-consistent rotating potential (CHF+SLy4) as a function of rotational frequency for the core configuration (the lowest highly deformed band of $^{131}$Ce). Occupied (empty) states are denoted by thick (thin) lines. Dotted, solid, dot-dashed, and dashed lines indicate parity and signature quantum numbers $(\pi ,r)$=$(+,+i)$, $(+,-i)$, ($-,+i)$, and $(-,-i)$, respectively. The orbitals are also labeled by the asymptotic quantum numbers $[{\cal N}n_z\Lambda ]\Omega$ of the dominant harmonic oscillator component of the wave function. The neutron intruder orbitals originating from the ${\cal N}$=6 shell are additionally marked. At intermediate rotational frequencies, the lowest intruder level 6$_1$ becomes occupied and this leads to the presence of the large gap in the spectrum at $N$=73.

Figure 3: (color online) Similar to Fig. 2 except for proton s.p. states. The large proton $Z$=58 gap in the s.p. spectrum is present at all frequencies considered.