The first realizations of quanttun algebraic symmetries in nuclear and molecular spectra are presented. Rotational spectra of even-even nuclei are described by the quantum algebra SUq(2). The two parameter formula given by the algebra is equivalent to an expan- sion in terms of powers of j(j + 1), similar to the expansion given by the Variable Moment of Inertia (VMI) model. The moment of inertia parameter in the two models, as well as the small parameter of the expansion, are found to have very similar numerical values. The same formalism is found to give very good results for superdeformed nuclear bands, which are closer to the classical SU(2) limit, as well as for rotational bands of diatomic molecules, in which a partial summation of the Dunham expansion for rotation-vibration spectra is achieved. Vibrational spectra of diatomic molecules can be described by the q-deformed anhannonic oscillator, having the symmetry Uq(2)>Oq(2). An alternative de- scription is obtained in terms of the quantum algebra SUq(1,1). In both cases the energy formula obtained is equivalent to an expansion in terms of powers of (v+½) , where ν is the vibrational quantum number, while in the classical ST(1,1) case only the first two powers appear. In all cases the improved description of the empirical data is obtained with q being a phase (and not a real number). Further applications of quantum algebraic symmetries in nuclei and molecules are discussed.

The Δ I = 1 staggering (odd-even staggering) in octupole bands of light actinides. is found to exhibit a "beat" behaviour as a function of the angular momentum J, forcing us to revise the traditional belief that this staggering decreases gradually to, zero and then remains at this zero value. Various algebraic models (spf-Interacting Boson Model, spdf-IBM, Vector Boson Model, Nuclear Vibron Model) predict in their su(3) limits constant staggering for this case, being thus unable to describe the "beat" behaviour. An explanation of the "beat" behaviour is given in terms of two Dunham expansions (expansions in terms of powers of I ( I + 1) ) with slightly different sets of coefficients for the ground state band and the negative parity band, the difference in the values of the coefficients being attributed to Coriolis couplings to other negative parity bands.

Normed division rings are reviewed in the more general framework of composition algebras that include the split (indefinite metric) case. The Jordan–von Neumann–Wigner classification of finite euclidean Jordan algebras is outlined with special attention to the 27 dimensional exceptional Jordan algebra \({\mathcal {J}}\). The automorphism group \(F_4\) of \({\mathcal {J}}\) and its maximal Borel-de Siebenthal subgroups \(\frac{SU(3)\times SU(3)}{{{\mathbb {Z}}_3}}\) and Spin(9) are studied in detail and applied to the classification of fundamental fermions and gauge bosons. Their intersection in \(F_4\) is demonstrated to coincide with the gauge group \(S(U(2)\times U(3))\) of the Standard Model of particle physics.

We show that the potential in the radial equation in the model of coherent quadrupole-octupole motion (CQOM) in nuclei generates a sequence of superpotentials and subsequent series of effective potentials which satisfy the shape-invariance condition and correspond to a SUSY-QM hierarchy of Hamiltonians. On this basis we suggest that the original CQOM level scheme possesses a generic supersymmetric structure of the spectrum inherent for the coherent quadrupole-octupole mode. We outline the mechanism in which the real quadrupole-octupole spectra in even-even and odd-even nuclei deviate from the genuine symmetry. By using it we illustrate the possibilities to identify the signs of supersymmetry in the alternating-parity spectra of even-even nuclei and the quasi-parity-doublet levels of odd-mass nuclei described within the CQOM model approach.

Normed division rings are reviewed in the more general framework of composition algebras that include the split (indefinite metric) case. The Jordan von Neumann - Wigner classification of finite dimensional Jordan algebras is outlined with special attention to the 27 dimensional exceptional Jordan algebra J. The automorphism group F-4 of J and its maximal Borel-de Siebenthal subgroups SU(3) x SU(3)/Z(3) and Spin (9) are studied in some detail with an eye to possible applications to the fundamental fermions in the Standard Model of particle physics.

A systematic comparison between experimental spectra of neighboring odd-mass and even-even nuclei with octupole deformation has been done. As a result a difference in the angular momentum dependence of the parity shift in alternating-parity bands of even-even nuclei and of the parity-doublet splitting in odd-A nuclei was found. In even-even nuclei the parity shift is initially large and always decreases with the angular momentum. In odd-mass nuclei the parity-doublet splitting does not exhibit such uniform behavior, but shows several possible dependencies, namely continuous decrease, increase and decrease with subsequent increase. Also, it was found that the angular momentum dependence of the levels with given parity in the spectrum of an odd-mass nucleus is similar to the respective behavior of the levels with opposite parity in the spectra of the neighboring even-even nuclei. This finding gives an idea for possible explanation of the various dependencies of the parity-doublet splitting in odd-A nuclei.

We study the specific features of alternating-parity spectra in the region of Ba and Ce nuclei in comparison to the structure of analogous excitations in actinide nuclei. The spectra in Ba-Ce nuclei are characterized by larger parityshift at low angular momenta, compared to Ra and Th nuclei, which sharply decreases at the higher levels forming a single band with irregular-staggering structure. In addition, our analysis shows that the structure of the even-parity level-sequences strongly deviates from the pure rotation behaviour, whereas the odd-parity sequences exhibit rather well pronounced rotation structure. Given the above peculiarities of the considered spectra we examine the possibility to describe them within the framework of the Quadrupole-Octupole Rotation Model (QORM) earlier developed in Sofia group. The preliminary results obtained for the alternating-parity spectra of Ba and Ce support the applicability of the model in this region.

The model of coherent quadrupole and octupole motion (CQOM) is combined with the reflection-asymmetric deformed shell model (DSM) in a way allowing fully microscopic description of the Coriolis decoupling and K-mixing effects in the quasi parity-doublet spectra of odd-mass nuclei. In this approach the even-even core is considered within the CQOM model, while the odd nucleon is described within DSM with pairing interaction. The Corio- lis decoupling/mixing factors are calculated through a parity-projection of the single-particle wave function. Expressions for the Coriolis mixed quasi parity- doublet levels are obtained in the second order of perturbation theory, while the K-mixed core plus particle wave function is obtained in the first order. Expres- sions for the B(E1), B(E2) and B(E3)reduced probabilities for transitions within and between different quasi-doublets are obtained by using the total K-mixed wave function. The model scheme is elaborated in a form capable of describing the yrast and non-yrast quasi parity-doublet spectra in odd-mass nuclei.

A model of strongly coupled quadrupole and octupole vibrations and rotations is applied to describe non-yrast alternating-parity sequences in even-even nuclei and split parity-doublet spectra in odd-mass nuclei. In even-even nuclei the yrast alternating-parity sequence includes the ground-state band and the lowest negative-parity levels with odd angular momenta, while the non-yrast sequences include excited beta-bands and higher negative-parity levels. In odd-mass nuclei the yrast levels are described as low-energy rotation-vibration modes coupled to the ground single-particle (s.p.) state, while the non-yrast parity-doublets are obtained as higher-energy rotation-vibration modes coupled to excited s.p. configurations. We show that the extended model scheme describes the yrast and non-yrast quadrupole-octupole spectra in both even-even and odd-A nuclei. The involvement of the reflection-asymmetric deformed shell model to explain the single-particle motion and the Coriolis interaction in odd nuclei is discussed.

The model of coherent quadrupole and octupole motion (CQOM) is applie d to describe non-yrast split parity-doublet spectra in odd-mass nuclei. T h yrast levels are described as low-energy rotation-vibration modes coupled to the g round single-particle state, while the non-yrast parity-doublet structures are o btained as higher-energy rotation-vibration modes. It is shown that the extende d model scheme describes both the yrast and non-yrast quasi-parity doublet spectra and the related B(E1) and B(E2) transition rates in different regions of heav y odd-A nuclei. The involvement of the reflection-asymmetric deformed shell mo del to describe the single-particle motion and the Coriolis interaction on a deeper lev el is discussed.

A further extension of a model of coherent quadrupole–octupole vibrations and rotations and its application to alternating-parity spectra in heavy even–even nuclei is presented. Within the model the yrast alternating-parity sequence includes the ground state band and the lowest negative parity levels with odd angular momenta, while the non-yrast sequences include excited β-bands and higher negative-parity levels. Electric transition operators reflecting the complex shape characteristics associated with the quadrupole–octupole vibration modes are introduced. By using them B(E1), B(E2) and B(E3) reduced transition probabilities within and between the different energy sequences are calculated. It is shown that the model successfully reproduces yrast and non-yrast alternating-parity bands together with the attendant B(E1)–B(E3) transition rates in the nuclei 154 Sm , 156 Gd and 100 Mo .

The model of coherent quadrupole and octupole motion is applied to describe non-yrast split parity-doublet spectra in odd-mass nuclei. The yrast levels are described as low-energy rotation-vibration modes coupled to the ground single-particle (s.p.) state, while the non-yrast parity-doublet structures are obtained as higher-energy rotation-vibration modes coupled to excited s.p. states. It is shown that the extended model scheme describes both the yrast and non-yrast quasi parity-doublet spectra and the related B(E1) and B(E2) transition rates in different regions of heavy odd-A nuclei. The involvement of the reflection-asymmetric deformed shell model to describe the single-particle motion and the Coriolis interaction on a deeper level is discussed.

Effects of the parity-mixed single-particle (s.p.) state on the collective properties of odd nuclei with reflection-asymmetric degrees of freedom are studied. The Coriolis strength and the average s.p. parity in the nuclei 219 Ra , 225 Ra , 225 Th and 241 Cm are examined in dependence on axial quadrupole and octupole deformation parameters β2 and β3 within a reflection-asymmetric deformed shell model. The obtained behaviour of the Coriolis decoupling factor in the (β2, β3)-plane is compared with values fitted in a collective quadrupole-octupole model, which allows one to determine physically reasonable deformation regions for the considered nuclei. The study provides a relation between deformation parameters, Coriolis interaction strength and the structure of collective spectra in odd-mass nuclei with quadrupole-octupole deformations.

A model assuming coherent quadrupole-octupole vibrations and rotations is applied to describe non-yrast energy sequences with alternating parity in even-even rare-earth and actinide nuclei. Within the model scheme the yrast alternating parity band is composed by the members of the ground-state band and the lowest negative-parity levels with odd angular momenta. The non-yrast alternating parity sequences unite the levels of the βbands with respective higher negative parity levels. It is shown that the model successfully reproduces the yrast and the first non-yrast alternating parity bands in the rare-earth nuclei Nd, Sm, Sm and Dy, and the actinides U. In the nuclei Gd and U the available second non-yrast sequences are also described.

The Coriolis decoupling strength in nuclear single-particle (s.p.) states with mixed parity is examined within a reflection-asymmetric deformed shell model. A strong coupling scheme between the parity mixed s.p. state and a coherent quadrupole–octupole vibration mode in the core is considered for odd-mass nuclei. The Coriolis decoupling factor is obtained in a projected form corresponding to good total parity of the system. The average parity of the s.p. state and the decoupling factor are evaluated in several nuclei as functions of the quadrupole and octupole deformation parameters β2 and β3. It is found that the average s.p. parity obtains various dominant values in the (β2, β3)-plane. It is shown that by comparing the behaviour of the decoupling factor in the (β2, β3)-plane to values obtained in a collective quadrupole–octupole model and by requiring consistency between the calculated dominant parity and the experimentally established parity in the ground state one can determine physically reasonable deformation regions for the considered odd-A nuclei.

The effect of parity mixing in the single particle (s.p.) states of odd-mass nuclei with quadrupole-octupole deformations is examined through a reflection-asymmetric deformed shell model. A strong coupling scheme between the parity mixed s.p. state and a coherent quadrupole-octupole vibration mode in the core is considered. The Coriolis decoupling factor is obtained in a projected form corresponding to the good total parity of the system. The average parity of the s.p. state and the decoupling factor are evaluated in several nuclei as functions of the quadrupole and octupole deformation parameters β2 and β3. It is found that the average s.p. parity obtains various dominant (+ or −) values in the (β2,β3)-plane, while the s.p. wave function is strongly fragmented into components with different parities. It is shown that by comparing the behaviour of the decoupling factor in the (β2,β3)-plane to values obtained in a collective quadrupole-octupole model one can determine physically reasonable regions for the deformation parameters.

Based on a recent application of the collective model of coherent quadrupole–octupole oscillations and rotations in odd-mass nuclei we develop an algorithm for a microscopic calculation of the Coriolis interaction strength. It is realized by using the reflection asymmetric deformed shell model. The single-particle (s.p.) wavefunction is obtained in the basis of the axially deformed harmonic oscillator (ADHO). The Coriolis interaction strength is calculated after transforming the ADHO decomposition coefficients of the wavefunction into coefficients in the basis of the spherical harmonic oscillator (SHO). The transformation brackets relating the ADHO and SHO basis functions are numerically integrated. Calculations were implemented for several nuclei in which the parity doublet spectra are known or supposed to be built on a single-particle orbital with Ω = 1/2. The results show the applicability of this approach to study the effects of Coriolis interaction in nuclei with quadrupole and octupole deformations as well as to examine their s.p. and shape characteristics.

We apply the collective axial quadrupole—octupole Hamiltonian to describe the rotation—vibration motion of odd nuclei with Coriolis coupling between the even-even core and the unpaired nucleon.We consider that the core oscillates coherently with respect to the quadrupole and octupole axialdeformation variables. The coupling between the core and the unpaired nucleon provides a split paritydoublet structure of the spectrum. The formalism successfully reproduces the parity-doublet splitting in a wide range of odd-A nuclei. It provides model estimations for the third angular-momentum projection K on the intrinsic symmetry axis and the related intrinsic nuclear structure.

A systematics of the atomic nuclei in the frame of the nucleon number $A = Z + N$ and the proton-neutron difference $F = Z - N$ is considered. The classification scheme is provided by means of the non-compact algebra $sp(4,R)$. In this scheme the nuclei are ordered into isobaric multiplets, for which $A={fix}$, as well as in F-multiplets, for which $F={fix}$. The dependence of the mass excess $Δ$, the first exited states $E_{2^+}$ and the ratio $R_2=E_{4^+}/E_{2^+}$ on the nucleon number $A$ is empirically investigated within the $F$-multiplets. Appropriate filters are used to study the properties of the mass excess. Many structural effects are observed. The mirror symmetry is clearly indicated for the energy levels of the nuclei with the same value of $A$ and opposite $F$-values.

A collective model describing coherent quadrupole-octupole oscillations and rotations with a Coriolis coupling between the even-even core and the unpaired nucleon is applied to odd nuclei. The particle-core coupling provides a parity-doublet structure of the spectrum, whereas the quadrupole-octupole motion leads to a splitting of the doublet energy levels. The formalism successfully reproduces the split parity-doublet spectra and the attendant B(E1) and B(E2) transition probabilities in a wide range of odd-A nuclei. It provides estimations for the influence of the Coriolis interaction on the collective motion and subsequently for the value of angular momentum projection K on which the spectrum is built. The analysis of the energy splitting and B(E1) transition probabilities between opposite parity counterparts suggests degenerate doublet structures at high angular momenta. The study provides information about the evolution of quadrupole-octupole collectivity in odd-mass nuclei.