Symmetries in Nuclear Structure

The following sections are included:

• PHOTO

• CONTENTS

• PREFACE

The following sections are included:

• Symmetry

• Symmetry in Physics

• Acknowledgments

Eigenvalues of Hamiltonians with dynamical symmetries are given by algebraic expressions of certain quantum numbers. Degeneracies occur frequently and states must be specified by additional quantum numbers. Discussion of the latter shows that such degeneracies may occur also for Hamiltonians without dynamical symmetry. These degeneracies can be removed in certain cases and some examples are presented.

We introduce the notion of a partial dynamical symmetry for which a prescribed symmetry is neither exact nor completely broken. We survey the different types of partial dynamical symmetries and present empirical examples in each category.

VCS theory is perhaps the simplest and most effective way known for computing the matrix elements of a Lie algebra. It is a mathematical tool that noone who is serious about using algebraic methods in physics should be without. It encorporates the mathematical theories of induced representations and geometric quantization in a physically intuitive manner that makes it easy to construct the explicit representations of a desired Lie algebra in a chosen basis in a systematic manner. Its practical utility has been confirmed in numerous applications.

Symmetries, articulated mathematically through group theory, play a central role in nuclear physics. The Interacting Boson Model (IBM), introduced to our field by Franco Iachello, who's 60th birthday we honor through this symposium, is a clear and clever example of the use of group theory to model symmetries of atomic nuclei. While all present may not be IBM disciples, it is clear we are all at least IBM apostles and here to honor Franco and acknowledge his significant contributions to the field of nuclear physics and beyond. I will take the opportunity of this very special occasion to give a brief update on a complementary theory, the pseudo-SU(3) model, that builds on pseudo-spin symmetry in heavy nuclei, a theory that is applicable to deformed rare earth and actinide species.

We discuss the early evidence that led to introducing particle-hole excitations across closed shells as a means to understand low-lying shape coexisting configurations. We point out the essential features of proton-neutron forces in order to understand the salient features of these configurations and the very specific mass dependence of the excitation energy. In the central part, we discuss how to reconcile particle-hole excitations within the framework of the Interacting Boson Model (IBM) and specifically discuss the underlying symmetry structures. Applications in order to describe recently observed shape coexisting excitations in the neutron-deficient Pb nuclei are presented.

Recent developments in the study of shape-invariant Hamiltonians are briefly summarized. Relations between certain exactly solvable problems in many-body physics and shape-invariance are explored. Connection between Gaudin algebras and supersymmetric quantum mechanics is pointed out.

Following a brief reminder of how the pairing model can be solved exactly, we describe how this can be used to address two interesting issues in nuclear structure physics. One concerns the mechanism for realizing superconductivity in finite nuclei and the other concerns the role of the nucleon Pauli principle in producing sd dominance in interacting boson models of nuclei.

The nuclear shell model allows several analytical solutions which broadly can be divided in two classes: pairing models and rotational models. In this contribution a review is given of nuclear pairing models with an emphasis on their symmetry character. The most general SO(8) model which accommodates neutrons and protons as well as T = 0 and T = 1 pairing, is solvable in three limits: only T = 0 pairing, only T = 1 pairing and equal strengths in the two channels. In these limits, the superconducting ground-state solution of even-even N = Z nuclei exhibits a quartet structure. This fermionic model is used as a starting point for a mapping onto an interacting boson model which includes T = 0 as well as T = 1 bosons.

The crucial role played by symmetries in suggesting and defining phenomena that can be expected near the neutron and proton drip lines is demonstrated. Possible experimental signatures are discussed and new results for Coulomb energy differences in the A=53 mirror nuclei are presented.

The relevance of precision measurements on nuclear ground states is discussed in the frame of current experimental developments. Cooling and trapping techniques of low-energy radioactive ion beams are shortly presented with emphasis on high-precision measurements on the ground state properties of exotic nuclei. The impact of the new generation Penning traps on mass measurements of short-lived nuclei is discussed. Examples of recent precision measurements of masses of very neutron-rich isotopes of zirconium and rhodium are given. Possible impacts of nuclear structure calculations such as the Interacting Boson Model on understanding the fine structure of the mass surface are discussed.

Extensions of nuclear supersymmetry are discussed, together with a proposal for new, more stringent and precise tests that probe the susy classification and specific two-particle correlations among supersymmetric partners. The combination of these theoretical and experimental studies may play a unifying role in nuclear phenomena.

New experimental tests of nuclear supersymmetry are suggested. They involve the measurement of one- and two-nucleon transfer reactions between nuclei that belong to the same supermultiplet. These reactions provide a direct test of the ‘fermionic’ sector, i.e. of the operators that change a boson into a fermion or vice versa. We present some theoretical predictions for the supersymmetric quartet of nuclei: ¹⁹⁴Pt, ¹⁹⁵Pt, ¹⁹⁵Au and ¹⁹⁶Au.

Supersymmetry as applied to identical bands is discussed. A review of the work of the Koeln-Dubna group on this topic is given and examples in ^171,172Yb , ^173,174Hf and ^195,194pt are discussed. The role of pseudo-spin in the supersymmetry is investigated. A recent precision lifetime measurement for identical bands in ^171,172Yb is discussed.

Nuclear shell structure has been predicted to evolve as the N/Z ratio becomes much larger than characteristic of stable nuclei. To probe this expected evolution will require transfer reactions using beams of neutron-rich nuclei. The first measurements of a (d,p) reaction with a ⁸²Ge beam are reported. The prospects for future studies of single-particle transfer reactions on unstable neutron-rich species, and implications for future searches for boson-fermion symmetries and supersymmetries, are discussed.

The α-cluster bands of the ²⁰Ne, ¹⁹F and ¹⁸F nuclei are analysed in terms of a U(4|12) superalgebra. These systems are interpreted as members of the 0, 1 and 2 fermion sectors of the supersymmetry scheme, and the correlations between their spectroscopic properties are discussed according to this assumption.

A cluster model is applied to the description of the properties of the ground state alternating parity bands in actinides and some Ba, Ce, Nd and Sm isotopes and to the description of the superdeformed band in ⁶⁰Zn. The model is based on the assumption that cluster-type shapes are produced by a collective motion of the nuclear system in the mass asymmetry coordinate. The results of calculations of the parity splitting, electric multipole transition moments of the alternating parity bands and of the spectra, transitional quadrupole moment of the superdeformed band and transitions between superdeformed and ground bands in ⁶⁰Zn agree with the experimental data.

Phase transitions in nuclei are predicted in the mean-field approximation (e.g., Landau theory), but in the finite nucleus their singularities are smoothed out by the quantal and thermal fluctuations. We discuss thermal signatures of the shape and pairing transitions that are observed despite the large fluctuations.

The exact solution of the boson pairing hamiltonian given by Richardson in the sixties is used to study the phenomena of level crossings and quantum phase transitions in the integrable regions of the sd and sdg interacting boson models.

The recent proposal and discovery of examples of critical point symmetries [in particular X(5) in ¹⁵²Sm] has spurred the search for other empirical manifestations of these symmetries and given new understanding of how deformation arises in nuclei. This talk will review the characteristic signatures for X(5) and the resulting evidence for X(5) symmetry in nuclei near N = 90. New experiments on nuclei in this region will be mentioned. The different behavior of nuclei as a function of neutron number (using Sm and Ba as examples) will be discussed in terms of the phase structure of the nuclear structure symmetry triangle.

The nuclear phase/shape transition in the framework of the Interacting Boson Model ¹ is analyzed and predictions of observables along trajectories crossing the transition region are presented. The model calculations are compared with the evolution of basic observables and a mapping of the symmetry triangle in terms of a new proposed set of polar coordinates is realized for the N=82-104 region.

We consider properties of critical points in the interacting boson model, corresponding to flat-bottomed potentials as encountered in a second-order phase transition between spherical and deformed γ-unstable nuclei. We show that intrinsic states with an effective β-deformation reproduce the dynamics of the underlying non-rigid shapes. The effective deformation can be determined from the the global minimum of the energy surface after projection onto the appropriate symmetry. States of fixed N and good O(5) symmetry projected from these intrinsic states provide good analytic estimates to the exact eigenstates, energies and quadrupole transition rates at the critical point.

The relation of the recently proposed E(5) critical point symmetry with the interacting boson model is investigated. It is established that the large N limit of IBM at the critical point in the transition from U(5) to O(6) dynamical symmetries is given by a β⁴ potential rather than by an infinite square well as in E(5).

The solutions of the E(5) and X(5) Hamiltonians for finite well depth are described, and the effects of finite depth on observables are discussed.

The main aim of this contribution is to discuss the analytical solution of the collective Bohr equation with a Coulomb-like and a Kratzer-like γ–unstable potentials in quadrupole deformation space. Eigenvalues and eigenfunctions are given in closed form and transition rates are calculated for the two cases. The corresponding SO(2,1) × SO(5) algebraic structure is discussed. A few remarks concerning the possibility to study an approximate solution in the γ–stable case with these potential are treated briefly.

We present a general analysis of phase-shape transitions in the framework of the Interacting Boson Model (IBM) using the catastrophe theory. The application to the the rare earth region shows that ¹⁴⁸Nd and ¹⁵⁰Sm are critical nuclei.

Lifetimes of excited states were measured in the stable nucleus ¹⁵⁰Nd and the neutron-rich unstable nuclei ¹⁴⁴Ba and ^104,106Mo isotopes by means of the recoil distance method. In ¹⁵⁰Nd excited states were populated via Coulomb excitation while the excited states in the neutron-rich Mo and Ba isotopes were populated in the spontaneous fission of ²⁵²Cf. For ¹⁵⁰Nd and ¹⁰⁴Mo energies and B(E2) values are compared to the predictions for the X(5) critical point symmetry discovered by Iachello. While ¹⁵⁰Nd is found to be a good example for the empirical realization of the X(5) symmetry, the B(E2) values in ¹⁰⁴Mo clearly follow a rotational behavior. In ¹⁴⁴Ba the dipole moment of the octupole deformed band was measured for the first time directly and was found to be in agreement with theoretical predictions while the quadrupole deformation of this band seems to be significantly higher than that of the ground state band.

Within the Interacting Boson Model a very simple hamiltonian can be constructed by combining a vibrational one body term with a quadrupole-quadrupole interaction. This hamiltonian which essentially depends on two control parameters, has four dynamical symmetries, three first order phase transitions and an isolated second order phase transition. The shapes can be related to Landau theory of phase transitions. Extensions of the simple hamiltonian to more complicated situations are proposed and studied.

Excited, non-yrast states in ¹⁶²Yb were populated through β decay and studied through off-beam γ-ray spectroscopy. New coincidence data provided evidence for the elimination of a previously reported low-lying 0⁺ state. The revised level scheme of ¹⁶²Yb is compared to the predictions of the X(5) critical point model.

Reliable and precise lifetimes of excited states in ¹⁵⁴Gd and ¹⁵⁶Dy were measured using the recoil distance Doppler shift (RDDS) technique. Excited states of ¹⁵⁴Gd were populated via Coulomb excitation with a ³²S beam at 110 MeV delivered by the FN tandem accelerator at the University of Cologne. For ¹⁵⁶Dy a coincidence plunger experiment was performed at the Laboratori Nazionali di Legnaro with the GASP spectrometer and the Cologne coincidence plunger apparatus using the reaction ¹²⁴Sn(³⁶S,4n)¹⁵⁶Dy at a beam energy of 155 MeV. The measured transition probabilities in ¹⁵⁶Dy and ¹⁵⁴Gd as well as the corresponding energy spectra are compared with the predictions of the recently proposed X(5) model and in the case of ¹⁵⁶Dy also with an IBA fit. In addition, criteria for finding new X(5) regions are given and the Os nuclei around A=180 were found to be very good X(5) candidates.

The following sections are included:

• Introduction
  • Phase transitions and new symmetries in nuclei
  • Phase transitions in the octupole degree of freedom

• Octupole excitations in nuclei with stable quadrupole deformation
  • Previous investigations of the Octupole + Quadrupole case
  • A simple model for the critical point in the axial octupole mode
  • Comparison with the experimental data
  • The A = 224 puzzle

• A general theoretical frame for simultaneous axial quadrupole and octupole excitations
  • The different steps of a consistent calculation
  • A new parameterization
  • The classical expression of the kinetic energy
  • The case of permanent quadrupole deformation

• References

During recent years the nuclear decay modes of discrete prompt proton and alpha-particle emission from deformed or superdeformed high-spin states have been discovered in nuclei in the vicinity of doubly-magic ⁵⁶Ni. The particle decays may be viewed as self-regulated two-dimensional quantum tunnelling processes. Due to the decay the remaining nuclear mean-field potential is rearranged dramatically. Quantum-mechanical tunnelling is a wide-spread phenomenon in the natural sciences. Therefore, a full understanding of this process may be of importance far beyond nuclear physics. Significant experimental progress has been made lately. It illustrates that prompt particle decays are “natural” decays in proton rich nuclei in the mass 60 region. New experiments are being planned, involving new instrumentation for charged particle detection.

We have measured the quadrupole moments of the 11⁻ isomers in ^194,196Pb and have proven that the particle-hole intruder excitation across the Z = 82 shell strongly polarizes the core: the quadrupole moments of these states are nearly one order of magnitude larger compared to the neutron hole states and a factor of two larger than the normal π(1h_9/21i_13/2)₁₁₋ excitation in the Po nuclei. We have also measured the quadrupole moment of the magnetic-rotational band head in ¹⁹³Pb, |Q_s| = 2.84(26) eb. The results are compared to different theoretical calculations in an attempt to conclude on the deformation of the “shears” states.

The analysis of the new experimental data on neutron-rich palladium isotopes, performed in the framework of the IBA-2 model, supports our previous findings on the importance of mixed-symmetry components for a correct description of the collective positive-parity bands observed in the mass region A ≃ 80 and ≃ 110.

By means of the (p,t) reaction the excitation spectra of 0⁺ states in ¹⁵⁸Gd [¹], ²²⁸Th, ²³⁰Th, and ²³²U have been studied using the Q3D magnetic spectrograph facility at the Munich tandem accelerator. The 0⁺ transfer angular distributions have very large cross sections at very small reaction angles, a feature that allows to identify these states in otherwise very complicated and dense spectra. We resolved for each of these nuclei typically 12 excited states with safe 0⁺ assignments. The studied excitation energy range is up to 3.1, 2.5, 2.5, and 2.1 MeV, resp. As for ¹⁵⁸Gd [²], we compare the data with spdf/-IBA calculations. The parameters are chosen to reproduce the low lying spectra of these axially symmetric, statically deformed nuclei, especially the bands of negative parity. For the energy ranges considered, the IBA predicts five excited 0⁺ states of pure sd (quadrupolar) bosonic structure for all these nuclei, but three, six, seven, and four excited 0⁺ states resp., which have two bosons in the pf boson space. They are related to – or represent – octupole two phonon excitations. The collective model descriptions provide nearly quantitatively the number of the observed excited 0⁺ states in these actinide nuclei.

We show the results of calculations of the energy levels and electromagnetic properties of the positive-parity states as well as the beta-decay rates in the odd-mass Xe and Cs isotopes in the proton-neutron interacting boson-fermion model (IBFM2). The properties are well described by the model calculation.

We review the applications of the Interacting boson-fermion plus broken-pairs model in the description of the structure of high angular momentum states in deformed, transitional and spherical nuclei.

The shell model in N ≈ 50 nuclei with few valence nucleons predicts high-spin states, whose decays are severely inhibited, due to selection rules of the single-particle transitions and/or the need to recouple both types of nucleons. Such seniority isomers play a dominant role in the interpretation of shell model structures, as will be discussed on the basis of lifetime measurements in ⁹³Tc and ⁹⁵Ru.

The question of the integrity of the closed doubly magic cores at Z, N = 20,28 is examined from the perspective of g factors of states in ⁴⁴Ca and other nuclei in the f_7/2 shell. The g factors were measured by the transient field technique and Coulomb excitation in inverse kinematics. A model which describes the states as a mixture of approximately equal parts of spherical four valence neutron configurations and core-excited, deformed, configurations agrees well with the measured value .

Recent results are summarized on low-lying electric and magnetic dipole excitations in heavy nuclei studied systematically in Nuclear Resonance Fluorescence (NRF) experiments. The systematics of the M1 Scissors Mode in deformed even-even and odd-mass nuclei are shown. New results are reported on strong E1 excitations in spherical nuclei near the N=82 and Z=50 shell closures. The corresponding J^π = 1⁻ states are interpreted as two-phonon excitations due to the coupling of quadrupole and octupole vibrations (2⁺ ⊗ 3⁻). A comprehensive systematics of the lowest E1 excitations in the entire mass region 130 ≤ A ≤ 200 is discussed in view of various excitation modes. Recent experimental developments and future new applications of low-energy, photon-induced reaction studies are discussed.

Improved experimental techniques allow to measure the dipole strength distribution in nuclei up to the particle threshold with very high sensitivity and precision. A concentration of E1 strength exhausting up to 1 % of the isovector energy weighted sum rule is found in the 1 ħω region. Results on N=82 isotones are presented and different interpretations of the strength are discussed.

Recent γ-ray spectroscopy of off-yrast low-spin states of ⁹⁴Mo yielded evidence for one-phonon and two-phonon states with mixed proton-neutron symmetry, a phenomenon anticipated by Franco Iachello and collaborators in the framework of the interacting boson model (IBM-2). The mixed-symmetry assignments are based on the measurement of absolute M1 matrix elements, ≈ 1 μ_N in size. Corresponding structures were lateron found in other soft nuclei. We give a brief overview over the recent investigations of mixed-symmetry multiphonon structures and we discuss for the first time the observation of F-vector E1 transitions involving mixed-symmetry states.

We study nuclear responses to the spin-isospin dependent Gamow-Teller operator. We focus on proton rich nuclei of mass around 70-80. We perform QRPA calculations on a selfconsistent deformed single particle basis. Pairing correlations in T= 1 channel are included in the BCS approach, while J=1 pairing force is included as a residual particle-particle force in the QRPA calculation, along with the particle-hole force. The sensitivity of the results to pairing and deformation is discussed. Oblate and prolate shape isomers are found in several isotopes. We argue that present data on half-lives and beta-decay spectra can be used to conclude on the strength of pairing and on the amount of shape coexistence in those nuclei.

In celebrating Iachello's 60th birthday we underline many seminal contributions for the study of the degrees of freddom relevant for the structure of nuclei and other hadrons. A dipole degree of freedom, well described by the spectrum generating algebra U(4) and the Vibron Model, is a most natural concept in molecular physics. It has been suggested by Iachello with much debate, to be most important for understanding the low lying structure of nuclei and other hadrons. After its first observation in ¹⁸O it was also shown to be relevant for the structure of heavy nuclei (e.g. ²¹⁸Ra). Much like the Ar-benzene molecule, it is shown that molecular configurations are important near threshold as exhibited by states with a large halo and strong electric dipole transitions. The cluster-molecular Sum Rule derived by Alhassid, Gai and Bertsch (AGB) is shown to be a very useful model independent tool for examining such dipole molecular structure near thereshold. Accordingly, the dipole strength observed in the halo nuclei such as ⁶He, ¹¹Li, ¹¹Be,¹⁷O, as well as the N=82 isotones is concentrated around threshold and it exhausts a large fraction (close to 100%) of the AGB sum rule, but a small fraction (a few percent) of the TRK sum rule. This is suggested as an evidence for a new soft dipole Vibron like oscillations in nuclei.

We are planning to measure the spin entanglement of two protons (p-p) in the spin-singlet state [¹S₀]. Results will be compared with predictions by the quantum mechanics as well as by the Bell's inequality. The [¹S₀] state will be produced by the (d, ²He) reaction and their spin-correlation will be measured by the newly constructed polarimeter EPOL.

The ratio between the anomalous magnetic moments of proton and neutron has recently been suggested to be connected to the ratio of proton momentum fractions carried by valence quarks. This relation has been obtained within a parametrization of the Generalized Parton Distributions (GPD) ¹, but it is completely independent of such a parametrization.

It will be shown that using different CQMs this relation holds within a few percent accuracy. This agreement is based on what all the CQMs have in common: the effective degrees of freedom of the three constituent quarks and the underlying SU(6) symmetry. On the other hand, the experimental value of the ratio is not reproduced by CQMs. This means that the SU(6)-breaking mechanism contained in the phenomenological partonic distributions does not correspond to the SU(6) breaking mechanism implemented in the CQMs we have analyzed ².

We will also show how this relation can be used in order to understand in which way to implement an SU(6)-breaking mechanism and to test models.

The hypercentral CQM, which is inspired by Lattice QCD calculations for quark-antiquark potentials, is presented, stressing its underlying symmetry. Its results for the spectrum, the helicity amplitudes and the elastic form factors are briefly reported. In the latter case the model has allowed to show, for the first time in the framework of a quark model, that relativistic effects are responsible for a deviation from the usually accepted dipole behaviour, in agreement with recent data taken at the Jefferson Lab.

In this contribution we present a specific application of a result obtained by Franco Iachello (in collaboration with R. Bijker and A. Leviatan), which concerns the inelastic electromagnetic form factors on the nucleons. In particular we show examples where symmetries inherent to the structure of the nucleon resonances can manifest in complicated processes of the strong interaction.

Using all the available empirical information, we analyse the spacing distributions of low-lying 2⁺ levels in even–even nuclei by comparing them with a theoretical distribution characterized by a single parameter (the chaoticity parameter f). We use the method of Bayesian inference. We show that the necessary unfolding procedure generally leads to an overestimate of f. We find that f varies strongly with the ratio R_4/2 of the excitation energies of the first 4⁺ and 2⁺ levels and assumes particularly small values in nuclei that have one of the dynamical symmetries of the Interacting Boson Model.

Nuclear dynamics is investigated in shape-transitional regions of the Interacting Boson Model - version 2 (IBM-2). Strong suppression of chaotic behavior is found even far from the dynamical symmetry limits, thus confirming the role played by partial dynamical symmetries and suggesting that also possible symmetries at the critical point may contribute to the persistence of regular motion in transitional nuclei.

A large part of the community considers the macroscopic superconducting wavefunction in the cuprates to be of near pure d–symmetry. The pertinent evidence has been obtained by experiments in which mainly surface phenomena have been used such as tunneling or the well known tricrystal or tetracrystal experiments¹. However recently, data probing the property in the bulk gave mounting evidence that inside the cuprate cuperconductor a substantial s–component is present, and therefore a changing symmetry from pure d at the surface to more s inside, at least, was proposed². The suggestion was made to reconcile the observations stemming from the surface and bulk. But such a behaviour would be at variance with the accepted classical symmetry properties in condensed matter^1,3. In this respect, Iachello, applying the Interacting Boson–Model, successful in nuclear theory, to the C_4ν symmetry of the cuprates, showed that indeed a crossover from a d–phase at the surface, over a d + s, to a pure s–phase could be present⁴. An attempt to estimate this crossover from known NMR experiments will be presented. It makes also plausible why the phase stiffness of the d– component is preserved over the a whole sample, i.e. in a Superconducting Quantum Interferometer Device (SQUID). Of interest is also the compatibility of the Interacting Boson Model with supersymmetry: in the hole doped cuprate superconductors there is since a number of years substantial evidence that there are two types of quasiparticles present, one of more bosonic and one of more fermionic character5. Due to the dynamic state in which one type transforms into the other the presence of a supersymmetry could be real.

Note from Publisher: This article contains the abstract and references only.

We apply in a systematic fashion the vibron model to obtain the complete description of vibrational spectra of molecular chains of finite size, such as n-alkanes (paraffin molecules). The one-dimensional model is extended to include infrared spectra of both CH stretching and bending modes. We describe the possible effect of anharmonic (Fermi) resonances in the spectra of fundamental and overtone energy regions. In the present framework, we show that in such molecular systems the algebraic treatment leads to a reliable, consistent set of parameters providing a fair description of the infrared spectrum without including Fermi resonances, even if they can have an appreciable role at higher resolution. It also seems that the parameters are applicable to extend the computation to longer and longer chains eventually describing a polymer chain (polyethylene).

Advantages of adopting algebraic approaches to vibration of polyatomic molecules are demonstrated. It is shown that the algebraic force-field expansion developed recently in our group can reproduce the experimental vibrational term values of H₂O and CO₂ in the wide energy range with a much smaller number of basis functions than the conventional force-field expansion.

An algebraic description of molecular non rigidity and the shape phase transition associated to the vibrational bending degree of freedom is presented. The algebraic approach, based on the dynamical symmetry breaking of an underlying U(3) Lie algebra, allows for an unified description of radically different situations, which range from the rigid linear to the rigid bent limit. The use of the intrinsic state formalism sheds light on the connection to other approaches to this problem. The study of the methinophosphide (HCP) ù A″ – X̃¹Σ⁺ emission spectrum, where the ù A″ electronic manifold shows quasi-rigid features and the X̃¹Σ⁺ state is rigidly-linear, is presented as an example. The underlying dynamical algebra is U(2) ⊗ U(3) ⊗ U(2), which embodies both stretching and bending degrees of freedom and permits the simultaneous quantitative description of term energies and Franck-Condon vibronic intensities.

A coupled U(2) algebraic theory has been employed to perform detailed analyses on absorption and emission spectra recorded for the C̃¹ A′ − X̃¹ A′ (π* ← π) electronic system of jet-cooled disulfur monoxide (S₂O) molecules. Vibronically-resolved features possessing up to 20 quanta of excitation in the ν₂ S–S stretching mode of the X̃ state (E_vib <14000cm⁻¹) and up to 8 quanta of excitation in the analogous ν′₂ vibration of the C̃ state (E_vib < 3500cm⁻¹) have been examined. Aside from providing an economical description for the inherently anharmonic and strongly coupled patterns of energy levels that distinguish highly-excited polyatomic species, the algebraic approach enables facile evaluation of multidimensional Franck-Condon factors required for the interpretation of spectral intensities. This ability to extract wavefunction information directly from spectroscopic data sets has revealed pronounced differences in the vibrational dynamics supported by the C̃¹A′ and X̃¹A′ manifolds, with the latter found to be substantially more “local” in character than the former.

As this meeting is to honour Franco on the occasion of his 60 birthday I thought that it might be fitting to report on some early reminiscences of Franco of the pre-IBA days. Franco first came to Groningen in 1972 for a seminar on the invitation of Alex Lande. Alex and Franco had known each other from the Niels Bohr Institute in Copenhagen, where they had collaborated. In 1972 both Alex and I had been freshly appointed at Groningen, Alex on the Faculty of the Theory Department, and I myself as the new director of the KVI. A position for a Senior Scientist in theory had been newly created at the KVI with the aim to establish a strong in-house theory group. Needless to say that everyone who met Franco was deeply impressed by him. We thus were extremely happy to be able to entice Franco to join the KVI as a Senior Scientist in 1974, after he had spent a few weeks in Groningen in 1973 as a visitor. So characteristic of Franco he immediately took a strong interest in the experimental program as evidenced by the following publications on the weak-coupling description of three-nucleon pickup in the (p, α) reaction [1] and the spreading width of deep-hole states [2]. Both topics appear to have maintained their actuality, looking at the many papers that have been published since on these and related topics. But this brief citation of the "other Franco" would not do justice to him without mentioning the diverse palette of Franco's work also listed in the KVI 1974 Annual Report, reflecting Franco's extremely broad and diversified scientific interests. [3-10]…

The following sections are included:

• Happy 60th to Franco

• List of participants

• Photographs