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Recently, from the study of the absolute [Formula: see text] values for the ([Formula: see text]–[Formula: see text]) [Formula: see text]2 transitions, the different roles of the triaxiality parameter [Formula: see text] and [Formula: see text] parts were pointed out. Here, the use of the triaxial rotor model expressions for the [Formula: see text]2)s in producing the bell-shaped curve of [Formula: see text] is illustrated. The variation of certain ([Formula: see text]–[Formula: see text]) [Formula: see text] ratios versus [Formula: see text] for the states [Formula: see text] and [Formula: see text] are illustrated, reflecting the two regions of [Formula: see text]. The inter relation of the [Formula: see text] and [Formula: see text] variables is illustrated for the Os isotopes.

The anharmonicity in the [Formula: see text][Formula: see text]Cd, and [Formula: see text][Formula: see text]Te chain of isotopes, which lie close to magic number [Formula: see text], is analyzed from the study of the correlation of the energy levels in the ground band, and some anomalies in the various entities are pointed out. The split of the degeneracy of the phonon triplet and quintuplet is illustrated. Formation of multiple coexisting phonon multiplets, associated with higher lying excited [Formula: see text] and [Formula: see text] states, and intruder states, including 2p4h excitation, with varying deformations, in [Formula: see text]Cd is discussed. The interacting boson model (IBM) is applied for [Formula: see text] [Formula: see text]Cd. The results of a previous study of [Formula: see text] isotones in the dynamic pairing plus quadrupole model are cited for Cd and Te. The problem of collectivity as reflected in the quadrupole moments and [Formula: see text] values is analyzed. The enigma of deformation features in spherical [Formula: see text]Te, reflected in the transition probability [Formula: see text] is resolved. Charge dependence of [Formula: see text]2 transitions for vibrational nuclei, in producing enhanced [Formula: see text] strengths in Te versus Cd is illustrated.

The problem of mismatching of the level energies, in the ground band and the [Formula: see text]-band of triaxially deformed atomic nuclei, as predicted in the rigid triaxial rotor (RTR) approximation of Davydov and Filippov (DF) model, with experiment, is well known. Here, we illustrate the solutions suggested in the literature, and the deviations observed in the converted energy values, from the experiment. We analyze the source of problem of this mismatch with experiment. This enables a physical picture of the DF (or RTR) model spectra. Our analysis will help in understanding the merits and the limitation of the RTR model in this respect.

Recently, we illustrated the use of the $$\gamma $$ –g absolute B(E2) values in the Davydov–Filippov model (DFM), instead of the $$\gamma \! \!-\!\!g\, B(E2)$$ ratios, for analysing the spectral features of the $$(\gamma < 20^{\circ })$$ deformed nuclei. Here, we illustrate the application of the absolute $$\gamma \! \!-\!\!g \, B(E2)$$ values to the triaxial $$(\gamma > 20^{\circ })$$ nuclei to seek the new role of the $$\gamma $$ variable in these nuclei, and to derive the underlying physics. The decrease of the absolute $$\gamma -g\, B(E2)$$ value for $$\gamma > 20^{\circ }$$ , instead of an increase observed for $$\gamma < 20^{\circ }$$ deformed nuclei is explained. The increase of $$B(E2, 2_{\mathrm {\gamma }}- 0_{\mathrm {1}}^{\mathrm {+}})$$ in spite of the negative band mixing parameter, for deformed nuclei, reveals a new view of the band mixing angle, in contrast to its role for the triaxial nuclei. The three moment of inertia in DF model, for $${}^{\mathrm {158}}$$ Dy and $${}^{\mathrm {120}}$$ Xe display their degree of triaxiality.

The validity of the extended Grodzins product rule (GPR) for higher spins in the ground bands of even [Formula: see text], even [Formula: see text] nuclei of the light mass region of Cd to Zn chain of isotopes is studied. The plots of [Formula: see text]2, [Formula: see text]) versus [1/[Formula: see text]] ([Formula: see text], 6[Formula: see text] provide test of the linear relationship of the two entities. There seems to be a good correlation of the two entities at these higher spin states for most of the nuclei studied here. The deviations from linearity in specific cases are found to be useful for studying the variations in the nuclear structures involved. From these linearity plots, the structural change at higher spin in some nuclei involving subshell gaps, as reflected in the anomalous rise or saturation, leads to further insight into the under lying microscopic structure.

The Kr isotopes lying in between the lighter isotopes of (Zn, Ge and Se) and the heavier isotopes of (Sr and Zr) in the [Formula: see text]–80 region exhibit very interesting spectral features. The spectra of [Formula: see text]Kr isotopes display a striking contrast from those of Zn, Ge and Se isotopes. The role of spherical and oblate and prolate deformed subshell gaps at specific [Formula: see text] and [Formula: see text] and the resulting re-inforcement are strikingly evident in these contrasting features, with variation in [Formula: see text] or [Formula: see text]. The evolution of the spectral features in Kr isotopes with [Formula: see text] as reflected in the quadrupole deformation, [Formula: see text]-band structures, [Formula: see text]0 decay, [Formula: see text]2) values, [Formula: see text]-softness of the nuclear core and odd–even staggering in [Formula: see text]-bands is studied to explore the role of the under lying nuclear interactions. The correlations with odd–[Formula: see text] isotopes are explored. The shape co-existence displayed in some Kr isotopes is studied. The large deformation observed in the ground bands of [Formula: see text]Kr, as exhibited in the [Formula: see text]2) values, is especially interesting.

The well known, one-parameter Davydov–Filippov model (DFM) or its variants have been used often to study the inter band [Formula: see text]-g B(E2) ratios for the triaxially deformed nuclei. Here, its application for the axially deformed nuclei with [Formula: see text] less than [Formula: see text] is illustrated, which has not received attention so far. Instead of the usual practice of studying the gamma-ground B(E2) ratios, we focus on the absolute B(E2) values in DFM. It offers a wholly new role of the [Formula: see text]-variable in explaining the spectral features of the axially deformed nuclei. Further, it illustrates the intimate relationship of the [Formula: see text] variable and the quadrupole deformation variable [Formula: see text]. Additionally, a unified view of the various differing patterns of the variation of the [Formula: see text]-g B(E2) ratios versus [Formula: see text] is presented.

A host of alternative energy formulae for the ground bands of even Z even N nuclei are available in the literature. The usual approach is to compare the relative numerical accuracy of the predictions of the level energies by these formulae, for varying deformations of the nuclear core and for high spins. The soft rotor formula and variable moment of inertia model, the ab and pq formulae, the rotation vibration interaction and power index formulae are illustrated. Here, a new perspective is presented, with emphasis on the limitation of the region of their physical validity and on deriving useful meaning of their parameters.

The well-known Grodzins product [Formula: see text]2, 0[Formula: see text]2[Formula: see text] constancy rule was generalized to the Grodzins linearity relation in our recent work in the form of [Formula: see text] versus [Formula: see text] linear plots. In this form, besides testing the linear relation of [Formula: see text]) and kinetic moment of inertia, it also served as a tool for the study of the spectral details of the nuclei, and for the study of the variation of nuclear structure with [Formula: see text] and [Formula: see text] We now study its extension to the intra band [Formula: see text]2 transition from the higher spins of the ground bands of even [Formula: see text] even [Formula: see text] nuclei of the mid-mass region of Xe–Pt chain of isotopes. We use the plots of [Formula: see text]2, [Formula: see text] versus [Formula: see text] [Formula: see text] and 6[Formula: see text] to test their linear relationship. There seems to be a good correlation of the two entities even at the higher spins for all the nuclei studied here. The deviations from linearity in specific cases can be used for studying the nuclear structures involved.

The nuclei in the [Formula: see text] [Formula: see text] and [Formula: see text] regions, lying on both sides of the [Formula: see text]-stability line, continue to be of interest for their complex nuclear structures. The Grodzins product rule (GPR) viz. [Formula: see text], for the ground bands of even-[Formula: see text] even-[Formula: see text] nuclei provides a useful approach to study these structures. The utility of our method, displaying the linear relation of [Formula: see text] to [Formula: see text], is illustrated for the [Formula: see text] Zn to [Formula: see text] Cd series of isotopes. The spread of the data on the linear plots enables a quick view of the shape phase transitions. The role of the shells and the subshells, at spherical and deformed shell gaps for neutrons and protons, with their mutual re-inforcement and the shape phase transition are vividly visible on our plots. The development of collectivity in this region is also linked to the effective number of valence nucleons above the magic number of [Formula: see text], and 28 rather than [Formula: see text], for Mo to Cd isotopes for a microscopic calculation.

The decay pattern of the low lying levels in the shape transitional nucleus 150 Sm is analyzed in terms of the quasi-vibrational and quasi-rotational collective model. The Dynamic Pairing plus quadrupole Model in the microscopic theory of the collective model is employed to predict extensive structure characteristics. The potential energy surface (PES), the spectroscopic factors and the E2, E0 transition rates from the DPPQ model are illustrated. Comparison is made with the predictions in the Interacting Boson Model (IBM-1). A correspondence is demonstrated of the effect of the control parameter in the two models on the calculated nuclear structure. The study of the wave functions of the two spin I = 2 excited states in IBM is carried out. The alternative view of an anharmonic vibrator and a soft rotor is discussed in terms of the E2 transition rates and other structural characteristics.

The power law expression [Formula: see text] offers a single-term formula with just two parameters for expressing the level energies in the spectra of even-[Formula: see text] even-[Formula: see text] nuclei. Its application to ground band spectra for a wide range of nuclei has been demonstrated in our earlier works. Here, we extend its application to the rotational bands built on an excited state of [Formula: see text] [Formula: see text]-vibration band and [Formula: see text] beta band. A novel assumption of a virtual level with spin zero for [Formula: see text]-bands is made and its validity and use is illustrated. Here, the constancy of the parameters “[Formula: see text]” and “[Formula: see text]” with spin, offers a more realistic view of the dependence of the nuclear core deformation on spin, in the excited bands. Also, it enables a spinwise view, not available in the other energy fit expressions.

In the collective spectra of atomic nuclei, the level energy [Formula: see text] varies with atomic number [Formula: see text] and neutron number [Formula: see text]. Also the [Formula: see text]2 decay-reduced transition probability [Formula: see text] is related to the energy [Formula: see text]. The product [Formula: see text] is constant according to Grodzins product rule, independent of the vibration or rotational status of the nucleus. The product rule is often used for determining [Formula: see text] from the known [Formula: see text]. However, the variation of the product with various parameters is also suggested in the literature. Hence, a detailed global study of this rule for [Formula: see text] region is warranted. We use a novel method of displaying the linear relation of [Formula: see text] with [Formula: see text] for the isotopes of each element (Xe–Pt), instead of their variation with [Formula: see text] or [Formula: see text]. Through our work, we firmly establish the global validity of the Grodzins relation of [Formula: see text], being proportional to the moment of inertia, except for the deviation in specific cases. Our [Formula: see text] versus [Formula: see text] plots provide a transparent view of the variation of the low-energy nuclear structure. This gives a new perspective of their nuclear structure. Also the various theoretical interpretations of [Formula: see text]s and the energy [Formula: see text] are reviewed.

The structure of theK π = 2+ gamma vibrational bands and the quasi-gamma bands of even-Z-even-N nuclei is investigated on a global scale, vis-a-vis the variation of band head, the moment of inertia of the band and the odd-even spin staggering. The variation withN andZ and with spinJ of the odd-even spin energy staggering index is studied and a unified view of the same is presented.

Beyond the shape phase transition from the spherical vibrator to the deformed rotor regime at [Formula: see text], the interplay of [Formula: see text]- and [Formula: see text]-degrees of freedom becomes important, which affects the relative positions of the [Formula: see text]- and [Formula: see text]-bands. In the microscopic approach of the dynamic pairing plus quadrupole model, a correlation of the strength of the quadrupole force and the formation of the [Formula: see text]- and [Formula: see text]-bands in [Formula: see text]Dy is described. The role of the potential energy surface is illustrated. The [Formula: see text] transition rates in the lower three [Formula: see text]-bands and the multi-phonon bands with [Formula: see text] and [Formula: see text] are well reproduced. The absolute [Formula: see text] [Formula: see text] serves as a good measure of the quadrupole strength. The role of the single particle Nilsson orbits is also described.

The anharmonity in shape transitional nuclei, observed earlier, is studied and an alternative form is derived. The dichotomy of a constant anharmonicity along with a changing nuclear structure is resolved. The evolution of the collective nuclear structure from the spherical vibrator to the deformed rotor is studied through the variation of energy ratio R10/2(E10/E2) with R4/2, for Ba – Dy and for Dy – Hf (N12/2. The role of the Z = 64 subshell and the N = 88–90 shape phase transition are illustrated in the Mallmann plot. The relative merits of the empirical formulae: rotation–vibration linearity model, the soft rotor formula and the power index formula are compared.

A linear relation for the level energy ratios in the ground band of even–even nuclei, based on the rotation–vibration expression, is derived and its application on a universal scale, including the O(6), E(5) and X(5) symmetries, is illustrated. A microscopic view of this relation and of the collective model is given. Also, an approximate relationship with single term power index formula E = aIb is demonstrated.

The level energies of ground band of even Z, even N nuclei may be reproduced well with good accuracy by using the power index formula E = aIb. In an earlier study of the dependence of the kinetic moment of inertia (MoI) J(1) on spin I, a possible correlation of the MoI J(1) with power index "b" was suggested. Here we illustrate that the slope of the kinetic MoI versus spin I corresponds to the magnitude of the index "b" for several isotopes in the A = 100–150 region. The validity of the formula is illustrated for light nuclei in A = 100 region and its use for studying shape phase transition at N = 60.

Dynamic deformation theory based on Kumar and Baranger's pairing-plus-quadrupole model (1968) is employed to calculate the collective states of 150Sm. A delicate balance of the pairing and quadrupole forces, leading to an interchange of the K-identification of the second and the third 2+ states at a critical value, is exhibited. An improved solution of the collective Hamiltonian, compared Kumar's calculation (1974), removes most of the discrepancies between the calculated and experimental branching ratios. Comparison with the experimental X(EO/E2) values is also given.

Using the deformed quasiparticle wavefunctions obtained in the pairing-plus-quadrupole model at the equilibrium deformation ( beta 0, gamma 0) of the N=88 isotones Nd, Sm and Gd, the spectroscopic factors Sj+=(2j+1)uj2 have been calculated for the single-proton states in the N=4, 5 shells. Comparison is made with the data from the proton stripping reactions (3He,d) and ( alpha ,t) giving the product nuclei 142Pm, 151Eu and 153Tb. The effect of a small deformation of the even-even core is also investigated.

Dynamic deformation theory based on the pairing plus-quadrupole model is employed for a detailed study of the N=88 transitional nucleus 152Gd. Comparisons with experiment and with other models are presented for the potential minimum and related properties, B(E2) values, rho (E0), X(EO/E2) and delta (E2/M1) values for transitions from seven 2+ states and a number of other states below 2 MeV. Brief comparisons with two N=88 isotones (148Nd and 150Sm) and with five Z=64 isotopes (150,154,156,158,160Gd) are also given.

Dynamic deformation theory based on a pairing plus quadrupole model (DPPQ model) is extended to 25 levels belonging to various multi phonon vibrational bands in 154Gd. Although the calculated excited band energies are too high by factors of 1�4-1�7, there is generally reasonable agreement with the experimental energies, B(E2) values, magnetic moments, E2-Ml mixing ratios and EO moments. Many new values are predicted. A brief comparison with some results derived from a new version of dynamic deformation theory, the so-called DNSB model, is also given.