Nuclear Structure at the Limits* W

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NUCLEAR STRUCTURE AT THE LIMITS* W. Nazarewicz

Department of Physics and Astronomy, University of Tennessee Knoxville, Tennessee 37996 U.S.A. Physics Division, Oak Ridge National Laboratory Oak Ridge, Tennessee 3783 1 U.S.A.

to be published in

Proceedings of Eleventh Physics Summer School on Frontiers in Nuclear Physics: From Quark- Gluon Plasma to Supernova Canberra, Australia January 12-23, 1998

*Research supported by the U.S. Department of Energy under Contract DE-AC05-96OR22464 managed by Lockheed Martin Energy Research Corp.

"The submitted manuscript has been authored by a contractor of the U.S. Government under contract No. DE-AC05-96OR22464. Accordingly, the U.S. Government retains a nonexclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U.S. Government purposes." NUCLEAR STRUCTURE AT THE LIMITS

WITOLD NAZAREWICZ Department of Physics and Astronomy, University of Tennessee KnaxviUe, Tennessee 37996, U.&A. Physics Division, Oak Ridge National Laboratory Oak Ridge, Tennessee 37831, USA. E-mail: [email protected]

One of the frontiers of today's nuclear science is the "journey to the limits" of atomic charge and nuclear mass, of neutron-to-proton ratio, and of angular momentum. The tour to the limits is not only a quest for new, exciting phenomena, but the new data are expected, as well, to bring qualitatively new information about the fundamental properties of the nucleonic many-body system, the nature of the nuclear interaction, and nucleonic correlations at various energy-distance scales. In this series of lectures, current developments in nuclear structure at the limits are discussed from a theoretical perspective, mainly concentrating on medium-mass and heavy nuclei.

1 Introduction The atomic nucleus is a fascinating many-body system bound by strong interaction. The building blocks of a nucleus - protons and neutrons - are themselves composite aggregations of quarks and gluons governed by quantum chromo- dynamics (QCD) - the fundamental theory of strong interaction. However, in the description of low-energy nuclear properties, the picture of A "effective" nucleons interacting through "effective" forces is usually more than adequate. But even in this approximation, the dimension of the nuclear many-body problem is overwhelming. Nuclei are exceedingly difficult to describe; they contain too many nucleons to allow for an exact treatment and far too few to disre- gard finite-size effects. Also, the time scale characteristic of collective nuclear modes is close to the single-nucleonic time scale. This means that many concepts and methods applied successfully to other many-body systems, such as solids and molecules, cannot be blindly adopted to nuclei. In these lectures, I intend to review - rather briefly - the enormous progress that has happened in nuclear structure during recent years. After discussing fundamental nuclear modes (Sec. 2) and progress in the theory of the nuclear many-body problem (Sec. 3), I shall concentrate on the main subject of my lectures, which is nuclear structure at the limits of isospin, angular momentum, mass, and charge (Sec. 4). This will be followed by a discussion of interdisciplinary aspects of theoretical nuclear structure research (Sec. 5). A short summary is contained in Sec. 6. For a general overview of nuclear science, the reader is encouraged

1 to study the recent report 1.

2 Nuclear Modes and Their Time Scales Considering that the atomic nucleus is, in a good approximation, a cluster of strongly interacting nucleons (or pairs of nucleons) moving solo in all directions, the very existence of nuclear collective motions such as rotations or vibrations, with all particles moving in unison, is rather astonishing. In molecular physics, for example, the fast electronic motion is strongly coupled to the equilibrium position of slowly moving ions. This point is nicely illustrated in Fig. 1, which shows the measured spectrum of a diatomic molecule N2. Each electronic excitation represents a band- head upon which vibrational and rotational states are built. Vibrational states are indicated by a vibrational quantum number v (v=0 representing the zero- phonon state, v=\ is a one-phonon state, and so on). Rotational levels are not plotted - there are far too many to be displayed. The time scale of molecular modes is governed by the hierarchy of excitation energies. The electronic excitations are of the order of 105 cm"1 (which in "nuclear" units corresponds to an eV), vibrational frequencies are ~103 cm"1, and rotational energies are ~10~1 cm-'. This means that the single-particle electronic motion is from two to six orders of magnitude faster than molecular collective modes. Con- sequently, the adiabatic assumption based on the separation of all molecular degrees of freedom into fast and slow ones is justified due to different time scales. In atomic nuclei, the total A-body wave function cannot, in general, be expressed in terms of slow and fast components. This is because (i) the collective nuclear coordinates are auxiliary variables which depend, in a complex way, on fast nucleonic degrees of freedom and (ii) the nuclear residual interactions are not small. How good is the time separation between single-particle and collective nuclear motion? The typical single-particle period (i.e., the average time it takes a neutron or a proton to go across the nucleus), TB.P.=4R/VF [where R is the nuclear radius and vp is the Fermi velocity (~0.29c)], is approximately 3-10~22 sec. (The unit 10~22 sec is sometimes referred to as babysecond. The single-particle time scale is of the order of several babysec.) The typical period 21 of nuclear rotation (Trot^lO" sec) is only -30 times greater than Ts.p., and for nuclear vibrations the period of oscillations is only slightly greater than the single-particle period. It is truly amazing that these relatively small differences in time scales seem to be sufficient to create rotating or vibrating potentials, common for all nucleons! Excitation spectrum of N2 molecule

in 1- "-T 1 h % excited'S* and 'IIU state B = » —

j* .. li .-

N N

Diabatic potential energy surfaces for excited electronic

configurations of N2

Figure 1: Top: vibrational spectrum of N2 molecule. Bottom: collective potentials, corresponding to different electronic configurations, as functions of the internuclear radius r (from Ref. 2).

How "good" are the actual nuclear rotations and vibrations? Let us con- sider some representative experimental examples. A very nice case of a nuclear rotator-vibrator is the nucleus 232Th (Fig. 2). Its spectrum shows some similar-

3 "Molecular" vibrational-rotational spectrum

232Th

beta gamma vibration \ \ gammai (2phonons) \ iA beta octupole vibration vibratfan (2phonons) KMT

Figure 2: Excitation spectrum of 232Th. Low-lying states have characteristic rotational- vibrational structure. (Courtesy of D. Radford; data were taken from Ref. 3.) ity with the molecular pattern; several low-lying excitations can be associated with one-phonon states, and some of its levels are believed to have a two- phonon nature. At first sight, the collective structures seen in 232Th are fairly regular. However, after closer inspection, many deviations from the perfect rotational and vibrational pattern can be seen. For example, the moment of inertia of 232Th is by no means constant but shows local variations as a function of angular momentum. Such deviations indicate that the nuclear motion is not completely collective, i.e., that the collective modes result from coherent superpositions of single-particle nucleonic excitations. Another extreme case is shown in Fig. 3, which displays the excitation spectrum <148Gd. This spectrum looks very irregular; it is characteristic of many-particle many-hole excitations in an almost spherical nucleus. There are many states of different angular momentum and parity connected via relatively weak electromagnetic transitions. In spite of the fact that the excitation Noncollective excitations

148Gd

Figure 3: Excitation spectrum of i48Gd.It is a beautiful example of noncollecive nucleonic motion (from Ref. 4). pattern looks "chaotic", most of the noncollective states of 148Gd can be beautifully described in terms of well-defined quantum numbers of the nuclear shell

5 model 5. The collective spectrum of 232Th discussed above was not perfectly collective. Likewise, in the "mess" of noncollective levels in 148Gd one can find some elements of collectivity. For instance, the energies of the four lowest states 0+, 2+, 4+, 6+, when plotted as functions of angular momentum, exhibit a characteristic parabolic pattern. As was realized around forty years ago6, such a sequence of states indicates the presence of pairing - a collective phenomenon. The next example illustrates the coexistence effect. As will be discussed below in Sec. 5.1, shape coexistence is a particular case of the large amplitude collective motion where several nuclear configurations, characterized by different intrinsic properties, compete 9. In the spectrum of 152Dy, shown in Fig. 4, one can recognize the noncollective structure resembling that in 148Gd, a collective rotational band (interpreted in terms of a deformed triaxial configuration) and super-collective superdeformed bands corresponding to very large shape-elongations which are structurally isolated from the rest of the spectrum. The microscopic origin of such an isolation will be discussed later in Sec. 5.1. The last experimental example shows yet another case of nuclear collectivity - high-frequency nuclear vibrations. Figure 5 displays the cross section for the scattering of the relativistic 208Pb on a xenon nucleus. The first maximum in the cross section, which appears at ~14MeV, can be associated with the one-phonon giant dipole resonance (i.e., very fast vibration of protons against neutrons). It has been suggested 10 that the local maximum at an energy of ~28MeV is a two-phonon giant dipole vibration. In the data shown in Fig. 5 one can also find isoscalar and isovector giant quadrupole resonances. The giant nuclear excitations are always strongly fragmented; their widths are usually more than 5 MeV. It is worth noting that the time scale corresponding to giant two-phonon vibrations is shorter than 1 baby sec! The unusual harmonicity of this mode is not understood well. One of the outstanding challenges in nuclear structure is to understand the mechanism governing the nature of nuclear collective excitations. By studying nuclear rotations and vibrations, one is probing the details of the nuclear force in a strongly interacting medium.

3 The Nuclear Many-Body Problem 3.1 The Nuclear Force The common theme for the field of nuclear structure is the problem of force: the one acting between two colliding nucleons, the one which produces exotic topologies in light nuclei, and the one giving rise to the collective motion in heavy nuclei. The main challenges in understanding the nuclear force are shown Coexistence of collective and noncollective motion triaxial superdeformed band •» » bands I nonoollective states

Figure 4: Excitation spectrum of 152Dy. It is a spectacular example of the coexistence of collective and noncollecive nucleonic modes. (Courtesy of D. Radford; Experimental data were taken from Refs. 7'8.) in Fig. 6 in the context of the hadronic and nucleonic many-body problem.

7 High-frequency nuclear vibrations (giantresonances)

Giant Dipole Resonance

5 10 15 20 2S 30 35 40 50 Eneigy (MeV)

Figure 5: One-phonon and two-phonon giant resonances in 208Pb. (Experimental data were taken from Ref. 10.)

The low-energy interaction between nucleons has a complicated spin-isospin dependence dictated by the hadron's substructure. One of the main challenges of nuclear science, indicated by the first bridge in Fig. 6, is the derivation of a nucleon-nucleon (NN) interaction from the underlying quark-gluon dynamics of QCD. Experimentally, the NN force can be studied by means of NN scattering experiments. Examples of phenomenological parameterizations based on the NN scattering data are the Bonn12'13, Nijmegen, Reid14 and Argonne Nuclear Forces and The Nudear Many-Body Roblem

light nudei

medium-mass and heavy nuclei

Derivation of the effective NN force Derivation of the in nuclear medium bare NN force fromQCO

Figure 6: From hadrons to heavy nuclei: main challenges in understanding the nuclear force. (FromRef. n.)

15 potentials. While the long-range part of these free (but still effective!) NN forces is well described by one-pion exchange, their short-range behavior is purely phenomenological. Here the quark-gluon degrees of freedom must be considered explicitly. (It is believed that the short-range part can be well accounted for by a simple gluon exchange potential and the Pauli principle 16,17 \ While the very light nuclei can nowadays be described as A-body clusters bound by a free NN force (including higher-order interactions, such as a three- body force), the conceptual framework of larger nuclei is still that of the nuclear shell model. Here, the basic assumption is that the nucleons are moving almost independently in a mean potential obtained by averaging out the interactions between a single nucleon and all remaining A-l protons and neutrons. This picture is only a first approximation; it is modified by the presence of the residual interaction between the nucleons. The "effective" NN interaction in the heavy nucleus, used to determine the mean potential, differs considerably from the free NN force. In principle, it should be obtained by means of the complicated Bruckner renormalization procedure which corrects the free NN interaction for the effects due to the nuclear medium. This challenging task is represented by the second bridge in Fig. 6. Many features of the effective interaction such as short range; strong dependence on spins, isospins, and relative momenta of interacting nucleons; and the reduction of mass in the nuclear interior have been extracted from experimental data. Figure 6 shows the intellectual connection between the hadronic many- body problem (quark-gluon description of a nucleon) and the nucleonic many- body problem (nucleus as a system of Z protons and N neutrons). The free NN force can be viewed as a residual interaction of the underlying quark- gluon dynamics of QCD, similar to the intermolecular forces that stem from QED. Similarly, the effective NN force in heavy nuclei can be derived from the effective free NN interaction. It probably would be very naive to think of the behavior of a heavy nucleus directly in terms of the underlying quark-gluon dynamics, but, clearly, the understanding of the bridges in Fig. 6 will make this goal qualitatively possible.

3.2 Theoretical Strategies How to tackle the problem of A strongly interacting nucleons? The general theoretical strategy is illustrated in Fig. 7. The starting point is, of course, the exact solution of the A-body Schrodinger equation (or relativistic field equations) with the bare NN force. Today, such ab initio calculations can be performed for very light nuclei, but still it is a very difficult task. Firstly, the corresponding dimensions are huge. Secondly, the bare NN force is very complicated (e.g., it contains tensor terms and, often, non-local terms). Con- sequently, especially for heavier nuclei, one is forced to work with effective NN interactions (second bridge in Fig. 6). This can be achieved either microscopically by means of the Bruckner theory18-19'20 by fitting some specific force to experimental data, or by extracting interaction matrix elements from the data. Having determined the effective force, the two most commonly used strategies are (i) that of the nuclear shell model and (ii) the mean-field strategy. In the nuclear shell model, the effective two-body Hamiltonian is diagonalized in the limited configuration space. Here, the practical limitation is the size of the Hilbert space considered. Often, the realistic shell-model Hamiltonian can be further approximated by replacing it with a Hamiltonian which can be 21 22 23 24 25 diagonalized exactly using group theoretical techniques > > ' > . The wave function of the nuclear shell model is an eigenstate of fundamental symmetry

10 The Nuclear Many-Body Problem

bare NN force calculations O ab initio

renormalisation

effective interaction laboratory intrinsic system system description description shell model mean field + correlations

self-consistency limited Hartree-Fock configuration Hartree-Fock- space Bogoiyubov symmetry 8 a inclusion of dictated "35 approaches correlations 3' RPA GCM deformed TDHF shell model projections

Figure 7: The nuclear many body problem - various theoretical strategies. operators; it has good parity, angular momentum and isospin. From this point of view, it is a laboratory system wave function. Another strategy is that of the mean-field theory. Here, the main assumption is that the many-body wave function can be - to zero order - approximated by that of independently moving quasiparticles. Together with the variational principle applied to the effective density-dependent Hamiltonian, this leads to

11 the Hartree-Fock (HF) or Hartree-Fock-Bogolyubov (HFB) equations 20. In spite of the very simple form, the independent-quasiparticle wave function is a highly correlated state. In the mean-field theory, self-consistency is automati- cally guaranteed by the equations of motion. That is, the same wave function which generates the mean field is an eigenstate of the mean-field Hamiltonian. Since the mean-field Hamiltonian often breaks symmetries present in the laboratory system, the HFB state is an intrinsic wave function, which - in general - is not an eigenstate of angular momentum, parity, and particle number operators. By restoring the intrinsically broken symmetries, one takes into account correlations going beyond the mean-field description. This can be done by various theoretical techniques such as the Random Phase Approximation (RPA), the Generator Coordinate Method (GCM), or various projection methods. The two main theoretical routes shown in Fig. 7 are obviously very schematic. In practice, one often uses approaches which take advantage of both strategies. Examples are the modern shell-model calculations employing the mean- field basis obtained by solving the HFB equations with either the shell-model Hamiltonian or some auxiliary model Hamiltonian. We have learned a great deal about modes of nuclear excitations using phenomenological models, often based on ingenious intuition and symmetry considerations. These models, approaches, and approximations have been extremely successful in interpreting nuclear states and classifying nuclear states and decays. Based on this experience, and thanks to developments in theoretical modeling and computer technology, we are now on the edge of the microscopic description. By taking advantage of modern many-body algorithms, one can now shorten the cycle theory Oexperiment-f*theory. Most many-body methods of theoretical nuclear structure were introduced a long time ago (in the fifties and sixties). However, thanks to incredible progress in hardware and numerical techniques, it is only now that we can use these methods in their full glory. Some spectacular examples of modern-day many-body calculations are shown in Fig. 8 and discussed in the following section.

3.3 Recent Theoretical Developments The best NN force parameterizations not only describe the two-body on-shell properties but have been used in few-body and many-body calculations. Prob- ably the most advanced few-body calculations today are the Green's function Monte Carlo calculations with the Argonne-Urbana interaction for nuclei with A57 (Ref. 26). The variational Monte Carlo calculations with a free NN force have been carried out for relatively heavy systems such as 0 . Other ab initio methods which are currently undergoing a renaissance include the

12 coupled-cluster (or "exp(S)") method 28.29.30 (see Refs. 31'32 for recent applications) and the antisymmetrized molecular dynamics 33>34.

bare NPforce ineffective force •^experiment Theoretical experiment ^- (dfective force developments in the nuclear many body problem

GT strength in fp nuclei 2 KB3 interaction Monte Carlo shell model

mean field 44 46 48 SO 52 54 56 58 60 62 64 66 3:10'

• * Sy.l Neutron a HFB*SkM« ~ separation energies in 7\ , I > spherical nuclei Skyrme forces 1 a HFB*SU" " SkP i SkM Hartree-Fock- 6 Electromagnetic Form Factor in Li. . i Bogolyubov. Argonne vv,+Urbana IX force too Variational Monte Carlo Neutron Number

Figure 8: Examples of advanced nuclear structure many-body calculations. Left: Variational Monte Carlo calculations with the Argonne-Urbana bare interaction for the electromagnetic form factor of 6Li 35. Right: Shell Model Monte Carlo description of the Gamow-Teller strength in the fp nuclei36. Bottom: Hartree-Fock-Bogolyubov calculations for two-neutron separation energies in heavy spherical nuclei 37. Thanks to new-generation computers and novel numerical techniques, significant progress in the nuclear many-body problem has been achieved.

There has been substantial progress in the area of the traditional shell model. In 1971, Whitehead and Watt 38 succeeded in performing shell-model calculations for 24Mg in the full sd shell (with the dimension of the model space approaching 30,000). Today, this can be done in a few seconds on a modern workstation. Traditional shell-model techniques make it possible to approach the collective nuclei from the pi shell; the calculations involve model spaces with dimensions of several millions. However, progress in this area is going to be very slow due to exploding dimensions when increasing the number of va-

13 lence nucleons39'40 (see Fig. 9). Hence, traditional shell-model calculations for heavy nuclei appear to be out of reach in the near future, although the conven- tional shell-model calculations employing realistic NN interactions41'42'43'44'45 are becoming more and more efficient in handling large configuration spaces. The state-of-the-art shell-model studies of the A=47 and 49 fp nuclei in the full, Ofuj space 39 set the new standard in this area, although future progress is strongly limited by present-day computer resources. (Actually, it took two generations of hardware and software development to extend shell- model calculations from A=44 to A—4846!)

N=Z pf nuclei (Ohco space)

1010

Figure 9: Dimension of the shell-model configuration space for even-even N=Z (20<2<40) nuclei. It is assumed that the shell model calculations are performed in the full pf shell in the M scheme (here M=0) (courtesy of T. Otsuka).

One actively pursued alternative is to truncate the configuration space by applying the projected self-consistent quasi-particle basis ' . Another fam- ily of novel shell-model techniques is based on the Monte Carlo method 49'50. Applications of the Monte Carlo shell model have been remarkably success-

14 ful in describing many structural properties of heavy nuclei 51>52-53 where the dimensions of the model space reach 1020. For nuclei close to the magic ones, the low-energy properties depend pri- marily on the behavior of a few valence nucleons. However, for nuclei with many valence particles, the concept of valence nucleons is less useful, and the valence and inner-shell nucleons have to be treated on an equal footing. Many properties of these heavy systems are well described by means of self-consistent theories with the density-dependent effective NN interaction. Here, the static mean-field description, based on the HFB method, or relativistic mean-field (RMF) theory, provides a useful starting point. Thanks to developments in computational techniques, the approaches employing microscopic effective interactions are now widely used and - in terms of their predictive power — favor- ably compare with results of more phenomenological macroscopic-microscopic models. Examples of recent large-scale self-consistent mean-field calculations include HF 54, HFB55 , and RMF5 6 studies of ground-state nuclear properties. Figure 10 displays two-neutron separation energies for the Sn isotopes calculated in several state-of-the-art models based on the self-consistent mean-field theory: HFB-D1S (based on the finite-range Gogny interaction D1S 57), HFB- SkP and HFB-SLy4 (based on zero-range Skyrme parametrizations SkP 58 and SLy4 59), LEDF (local energy-density functional model with parametrization FaNDF0 60), and RHB-NL3 (relativistic Hartree-Bogolyubov model with NL3 parametrization61). All these models nicely describe the existing experimental data; some interesting deviations are seen when approaching the proton drip line. Among other recent developments in theoretical nuclear structure, particularly important is the elegant explanation of the pseudo-spin symmetry of the nuclear single-particle spectra proposed thirty years ago 62'63. Surprisingly, as demonstrated by Ginocchio , the roots of the pseudo-spin can be traced back to the symmetry of the Dirac equation (see also Refs. 65>66). Consequently, the relativistic approach explains, at the same time, the depth of the average potential (around 50 MeV), the magnitude of the spin-orbit term, and the small pseudo-spin-orbit splitting. In this context, it should be noted that the traditional picture of nuclear shells and magic gaps proposed fifty years ago ' should not be taken for granted. As discussed in Sec. 4.1 below, modifications of shell structure are expected in the limit of a large N/Z ratio.

4 Nuclear Structure at the Limits

What are the frontiers of nuclear structure today? For light nuclei, one of such frontiers is physics at subfemtometer distances where the internal quark-gluon

15 Figure 10: Two-neutron separation energies, S2n , for the Sn isotopes calculated in five self- consistent models: HFB-D1 (courtesy of J Decharge), HFB-SkP and HFB-SLy4 (courtesy of J Dobaczewski), LEDF (courtesy of S Fayans), and RHB-NL3 (courtesy of G. Laleizissis). The experimental data are indicated by stars.

structures of nucleons overlap. For heavier nuclei, the frontiers are defined by the extremes of (i) the N/Z ratio (Sec. 4.1), (ii) atomic charge and nuclear mass (Sec. 4.2), and (iii) angular momentum (Sec. 4.3). The tour to the limits is not only a quest for new and unexpected phenomena (sometimes dubbed as a "fishing expedition"), but the new data are expected as well to bring qualitatively new information about the effective NN interaction and hence about the fundamental properties of the nucleonic many-body system. By exploring exotic nuclei, one can magnify certain terms of the Hamiltonian which are small in "normal" nuclei and thus difficult to test. The hope is that after probing these important interactions at the limits, we can later improve the description of normal nuclei.

16 4-1 Limits of Extreme NIZ Ratios

One of the main frontiers of nuclear structure today is the physics of radioactive nuclear beams (RNB). One of the indications of the potential of this field is the large international interest 69>70>71>72. At present there are only a few laboratories with radioactive ion beam capabilities. However, the prospects for new experiments and the success of the current programs have led to a number of RNB facilities under development and a number of further pro- posals, including the construction of the next-generation facilities in Europe, U.S., and Japan. There are Several key themes behind the RNB program 71. They include: (i) nuclear structure (the nature of nucleonic matter), (ii) nuclear astrophysics (the origin of the Universe), and (iii) physics of fundamental symmetries (tests of the Standard Model). The nuclear landscape (the territory of the RNB physics) is shown in Fig. 11. Black squares indicate stable nuclei; there are less than 300 stable nuclei, or those long-lived, with half-lives comparable to or longer than the age of Earth. Some of the unstable nuclei can be found on Earth, some are man-made, and several thousand nuclei are the yet-unexplored exotic species belonging to nuclear "terra incognita". Moving away from stable nuclei by adding either protons or neutrons, one finally reaches the particle drip lines where the nuclear binding ends. The nuclei beyond the drip lines are unbound to nucleon emission; that is, for those systems the strong interaction is unable to cluster A. nucleons as one nucleus. (An exciting question is whether there can possibly exist islands of stability beyond the neutron drip line. One of such islands, indicated in Fig. 11, is a neutron star which exists thanks to gravitation. So far, calculations for light neutron drops have not produced permanent binding 73>74 .) The uncharted regions of the (N, Z) plane contain information that can answer many questions of fundamental importance for science: What are the limits of nuclear existence? What are the properties of nuclei with an extreme N/Z ratio? What is the effective nucleon-nucleon interaction in the nucleus having a very large neutron excess? There are also related questions in the field of nuclear astrophysics. Since radioactive nuclei are produced in many astrophysical sites, knowledge of their properties is crucial for our understanding of the underlying processes 75. Experiments with beams of unstable nuclei will make it possible to look closely into many aspects of the nuclear many-body problem. Theoretically, exotic nuclei represent a formidable challenge for the nuclear many-body theories and their power to predict nuclear properties in nuclear terra incognita. What makes this subject both exciting and difficult is: (i) the weak binding

17 stable or long-lived tena incognita pn known nuclei

2 8

Figure 11: The nuclear landscape: territory of radioactive ion beam physics. Some of the important physics themes are indicated schematically. They include weakly bound nuclear halos, neutron skins and their excitation modes, modifications of shell structure at large N/Z ratios, proton emitters, proton-neutron superconductivity, nuclear tests of the Standard Model, and nucleosynthesis.

and corresponding closeness of the particle continuum, implying a large dif- fuseness of the nuclear surface and extreme spatial dimensions characterizing the outermost nucleons, and (ii) access to the exotic combinations of proton and neutron numbers which offer prospects for completely new structural phenomena.

Theoretical Aspects of Physics at Extreme N/Z Ratios

From a theoretical point of view, spectroscopy of exotic nuclei offers a unique test of those components of effective interactions that depend on the isospin degrees of freedom76. Effective interactions used to describe heavy nuclei are usually approximated by means of density-dependent forces with parameters

18 that are usually fitted to stable nuclei and to selected properties of the infinite nuclear matter. Hence, it is by no means obvious that the isotopic trends far from stability, predicted by commonly used interactions (such as Skyrme or Gogny forces), are correct. In the models aiming at such an extrapolation, the important questions asked are: What is the N/Z behavior of the two-body central force and the one-body spin-orbit force 77>78>79? Does the spin-orbit splitting strongly vary with N/Z73? What is the form of the pairing interaction in weakly bound nuclei80'81? What is the importance of the effective mass (i.e., the non-locality of the force) for isotopic trends 37? What is the role of the medium effects (renormalization) and of the core polarization in the nuclear exterior (halo or skin region) where the nucleonic density is small82? Exotic nuclei are wonderful laboratories to study superfluid correlations. Here, the main questions pertaining to the problem of pairing force are: What is the microscopic origin of the pairing interaction83'84? What is the role of finite range and the importance of density dependence80'85? How can properties of the pairing force be tested? These questions are of considerable importance not only for nuclear physics but also for nuclear astrophysics and cosmology 86,87,88,89_ por instance; a better understanding of the density dependence of the nuclear pairing interaction is important for theories of superfluidity in neutron stars. It is impossible at present to deduce the magnitude of the pairing gaps in neutron stars with sufficient accuracy. Indeed, calculations of 1So pairing gaps in pure neutron matter, or symmetric nuclear matter based on free NN interactions, suggest a strong dependence on the force used; in general, the singlet-S pairing is very small at the saturation point. On the other hand, nuclear matter calculations with effective finite-range interactions yield rather large values of the pairing gap at saturation! Nuclear life at extreme N/Z ratios is different from that around the stability line. The unique structural factor is the weak binding; hence the closeness to the particle continuum. For weakly bound nuclei, the Fermi energy lies very close to zero, and the decay channels must be taken into account explicitly. As a result, many cherished approaches of nuclear theory must be modified. (For an extensive discussion of the theoretical perspectives far from stability, see the recent review76.) But there is also a splendid opportunity: the explicit coupling between bound states and continuum, and the presence of low-lying scattering states invite strong interplay and cross-fertilization between nuclear structure and reaction theory. How do well-established microscopic models of nuclear structure perform when extrapolated to exotic nuclei? Of course, there are many uncertainties. As an example, Fig. 12 shows the predicted two-neutron separation energies for the tin isotopes using the same models as in Fig. 10. Clearly, the differences

19 90 100 110 120

figure 12: Similar to Pig. 10 except for very neutron-rich Sn isotopes. between forces are greater in the region of "terra incognita" than in the region where masses are known. As seen in Fig. 12, the position of the neutron drip line for the Sn isotopes slightly depends on the effective interaction used; it varies between AT=120 (HFB-DIS) and N=126 (RHB-NL3). Therefore, the uncertainty due to the largely unknown isospin dependence of the effective force gives an appreciable theoretical "error bar" for the position of the drip line. Unfortunately, the results presented in Fig. 12 do not tell us much about which of the forces discussed should be the preferred one since one is dealing with dramatic extrapolations far beyond the region known experimentally. However, a detailed analysis of the force dependence of results may give us valuable information on the relative importance of various force parameters. Recent research relating to exotic nuclei has already demonstrated the potential for exciting new nuclear physics. Some examples are shown in Fig. 11. They include the exotic structure of halo nuclei, the surprising fragility of magic numbers which hints at some of the marked changes in the underlying

20 foundations of nuclear structure, and the production and study of the long- searched doubly closed shell nuclei 78Ni and 100Sn. In the following, I shall briefly comment on some of the themes related to the RNB physics.

Physics of Large Neutron Excess

The intense current interest in the experimental exploration of the neutron drip-line region is driven not only by the substantial uncertainties in theoretical predictions of its location, but also by the expectation that qualitatively new features of nuclear structure will be discovered in this exotic territory. What makes neutron-rich nuclei so unusual? Firstly, they have very large sizes — as implied by their weak binding. Secondly, they are very diffused; their properties are greatly dominated by surface effects. Thirdly, they are very superfluid; the close-lying particle continuum provides a giant reservoir for scattered neutron Cooper pairs58'80. So, roughly speaking, they are large, fuzzy superfluids. Extreme cases are halo nuclei - loosely bound few-body systems with about thrice more neutrons than protons90. They are symbols of RNB physics. The halo region is a zone of weak binding in which quantum effects play a critical role in distributing nuclear density in regions not classically allowed. Much has been learned about halo nuclei. Few-body calculations, especially those based on the method of hyperspherical harmonics and the adiabatic hyperspherical method, have reached a high level of sophistication91'92. They give the basic understanding of many observed phenomena (momentum distributions, electromagnetic strength distributions) but, often, neglect the important structure aspects. Some of the questions asked in the context of halo nuclei are: What are the main dynamical degrees of freedom which affect the core-halo coupling? What is the degree of clusterization of the cores? Is the clusterization mechanism enhanced close to the neutron drip line? If so, what are its manifestations in heavier systems? What are the modifications of the effective interaction in the halo regioi?2? Through halos, one hopes to learn more about the microscopic mechanism of clusterization found in "normal" nuclei (see the Ikeda diagram shown in Fig. 13), such as in the hypothetical three-alpha state of 12c. In the heavier, neutron-rich nuclei, where the concept of mean field is better applicable, the separation into a "core" and "valence nucleons" seems less justified. However, also in these nuclei the weak neutron binding implies the existence of the neutron skin (i.e., a dramatic excess of neutrons at large distances). In the skin region of heavier, very neutron-rich nuclei, one may find the opportunity to study in the laboratory nearly pure neutron matter at

21 (7.27) (14.44) (19.17) (28.48) s) (7.16) (11.89) (21.21) Qw**

(4.73) o 1 (13.93)

(9.32) "Li Mg

Mass number

Figure 13: Ikeda diagram for light nuclei 93. The threshold energy for each decay mode (in MeV) is indicated. The halo nucleus 11Li, viewed as a three-body borromean system, is also shown. densities much less than the normal nuclear density. In addition, the existence of a neutron skin should lead to new collective vibration modes in such nuclei in which, for example, the neutron skin may oscillate out of phase with a well-bound proton-neutron core.

Nuclear Shell Structure far from Stability A significant new theme concerns shell structure near the particle drip lines. Since the isospin dependence of the effective AW interaction is largely unknown, the structure of single-particle states, collective modes, and the behavior of global nuclear properties is very uncertain in nuclei with extreme N/Z ratios (see Fig. 12 and related discussion). Due to the systematic variation in the spatial distribution of nucleonic densities and the increased importance of the pairing field, the average nucleonic

22 potential is modified94. This results in a new shell structure characterized by a more uniform distribution of normal-parity orbits and the unique-parity intruder orbit which reverts towards its parent shell 76>95>94. According to other calculations 96, a reduction of the spin-orbit splitting in neutron-rich nuclei is expected. The gradual change of shell structure with neutron number is believed to give rise to new sorts of collective phenomena97'98.

N=80 N=82 N=84 N=86

54 46 38 Proton Number

Figure 14: Two-neutron separation energies for the JV=80, 82, 84, and 86 spherical even- even isotones calculated in the HFB+SkP and HFB+SLy4"5 models as functions of the proton number. The arrows indicate the proximity of neutron and proton drip lines for small and large proton numbers, respectively (from Ref. 76).

The quenching of shell effects manifests itself in the behavior of two- neutron separation energies S2n- This is illustrated in Fig. 14, which displays the two-neutron separation energies for the N=80, 82, 84, and 86 spherical even-even isotones calculated in the HFB model with the SkP and SLy4 effective interactions. The large N=82 magic gap, clearly seen in the nuclei close to the stability valley and to the proton drip line, gradually closes down when approaching the neutron drip line.

23 It is to be noted that the experimentally observed collapse of magic gaps seen in some neutron-rich light nuclei (the so-called islands of inversion, see Refs. 100,101,102,103^ ^g^ alSo be related to the predicted quenching of magic gaps 104 The effect of the weakening of known shell effects in drip-line nuclei has significant consequences for the description of the astrophysical r-process. Although, many of the astrophysically important neutron-rich nuclei, belonging to the terra incognita of Fig. 11, will not be experimentally accessible in the foreseeble future, their part of the (Z, N) chart has already been visited by Mother Nature in cataclysmic events such as supernovae. Consequently, valuable information on nuclear structure in terra incognita can be offered by astrophysical data. In particular, the analysis of the r-process abundances provides an indirect experimental confirmation of the shell-quenching effect 105,99

Physics of Proton-Rich Nuclei: Nuclear Life at and Beyond the Drip Line

On the proton-rich side of the valley of stability, physics is different than in nuclei with a large neutron excess. Because of the Coulomb barrier which tends to localize the proton density in the nuclear interior, nuclei beyond the proton drip line are quasibound. However, in spite of the stabilizing effect of the Coulomb barrier, the effects associated with the weak binding are also present in proton drip-line nuclei. They are not as dramatic as on the other side of the stability valley, but nevertheless important 76. A unique aspect of proton-rich nuclei with N=Z is that neutrons and protons occupy the same shell-model orbitals. Consequently, due to the large spatial overlaps between neutron and proton single-particle wave functions, the proton-rich N=Z nuclei are expected to exhibit unique manifestations of proton-neutron (pn) pairing 106'107 carried by isotropic (5=0, T=l) and anisotropic, doughnut-shaped108 (5=1, T—0) proton-neutron Cooper pairs. At present, it is not clear what the specific experimental fingerprints of the pn pairing are, whether the pn correlations are strong enough to form a static pair condensate, and what their main building blocks are. So far, the strongest evidence for enhanced pn correlations around the N=Z line comes from the measured binding energies. An additional binding (the so-called Wigner energy) found in these nuclei manifests itself as a spike in the isobaric mass parabola as a function of T^^iN-Z) ' . Recent calculations have revealed a rather complex mechanism responsible for the nuclear binding around the N=Z line. In particular, it has been found that the Wigner term cannot be solely explained in terms of correlations between the proton-neutron

24 J=l, T=O (deuteron-like) pairs. The pn correlations are also expected to play a role in beta decay113'114, deuteron transfer reactions 115, structure of high spins ^'U6-117, and also in nuclear matter 88>118. Proton-rich nuclei also offer a unique opportunity to study life beyond the drip line. Although the protons in this region are not strictly bound to the nuclear core, nonetheless their escape is impeded by the Coulomb force. Quantum barrier tunneling allows these nuclei to decay by proton emission, but with lifetimes ranging from microseconds to a few seconds — long enough that one can measure their spectroscopic properties U9'120. Experimentally, a number of proton emitters have now been discovered in mass regions A-110, 150, and 170121. However, it is anticipated that new regions of proton-unstable nuclei will be explored in the near future using RNB's.

IProton emission from deformed nudei

(N, Z+l) (N,Z)

Figure 15: Competition between gamma and proton decays in a deformed proton emitter.

Proton radioactivity is an excellent example of the elementary 3D tunneling. Experimental and theoretical investigations of proton emitters (or theoretically predicted ground-state di-proton emitters) will open up -a wealth of exciting physics associated with the residual interaction coupling between bound states and extremely narrow resonances in the region of very low density of single-particle levels. In general, proton emission half-lives depend mainly on the proton separation energy and orbital angular momentum and also rather

25 weakly on the details of the intrinsic structure of proton emitters, e.g., on the parameters of the proton-core potential. This suggests that the lifetimes of deformed proton emitters will provide direct information on the angular momentum content of the associated Nilsson state, and hence on the nuclear shape 122,123,124 piguj-g 15 shows, schematically, a possible scenario of proton-gamma competition in a deformed proton emitter. Of course, the energy window for such a process is expected to be fairly narrow. A slightly different phenomenon, gamma-delayed proton emission, has been recently observed in 58Cu 125 and 56Ni 126 (see also discussion in Sec. 4.3). Here the strongly deformed excited intruder bands gamma-decay to the lower-lying states, but their proton-unstable band heads directly feed into the excited state in the daughter system.

How Far is Far?

There is very little doubt that progress in the RNB research is not going to be rapid. Especially for the neutron-rich heavy elements, it will be a real struggle to obtain basic spectroscopic information. To put things in perspective, it is instructive to recall the discovery of polonium by Maria and Pierre Curie at the end of June 1898. The longest-lived isotope of polonium is 209Po with a half-life of 102 y. The isotope 218Po had already been studied by Rutherford in 1904 127; its half-life is 3.1 min , Amazingly, it took ninety-four years to add one more neutron to Po; only recently have the neutron-rich isotopes 1 Po been produced at GSI by the international collaboration led by M. Pfiitzner 129. The lightest polonium known, 190Po, was found in 1988 at GSI 130; its half-life, 2.4 ms 131, is long enough to enable detailed spectroscopic studies. The doubly magic N—Z=50 nucleus 100Sn is a paradigm of RNB physics on the proton-rich side. Although it was found experimentally three years ago 134 132,133^ ^ t00]i more than two years to roughly determine its mass , and it will still take quite a few years to find its first excited state. On an optimistic note, many nuclei very close to the N=Z=50 corner have already been approached spectroscopically in recent years 135, so the prospects for more spectroscopic news on 100Sn are good. It will be much harder to approach another doubly magic nucleus, the neutron-rich 78Ni. It was produced in 1995 136, but our knowledge about this system and its neighbors is very scarce. Indeed, the heaviest nickel isotope known spectroscopically is Ni which is as far as eight neutrons away! An experimental excursion into uncharted territories of the chart of the nuclides exploring new combinations of Z and N will offer many excellent opportunities for nuclear structure research. What is most exciting, however,

26 is that there are many unique features of exotic nuclei that give prospects for entirely new phenomena likely to be different from anything we have observed to date. We are only at the beginning of a most exciting journey.

4.2 Limits of Mass and Charge One of the fundamental and persistent questions of nuclear science concerns the maximum charge and weight that the nucleus may attain. Amazingly, even after thirty-or-so years of our quest for superheavy elements, the borders of the upper-right end of the nuclear chart are unknown. Theoretically, the mere existence of the heaviest elements with Z>104 is entirely due to quantal shell effects. Indeed, for these nuclei the classical nuclear droplet, governed by surface tension and Coulomb repulsion, fissions immediately due to the huge electric charge. At the end of the sixties, a number of theoretical predictions were made that pointed towards the existence of long-lived superheavy (SHE) nuclei 138'139>14O>141_ However, it is only during recent years that significant progress in the production of the heaviest nuclei has been achieved 142>143. No- tably, three new elements, Z=110, 111, and 112, were synthesized by means of both cold and hot fusion reactions 144'145>146.147. These heaviest isotopes decay predominantly by groups of a particles (a chains) as expected theoretically 148,149_ (The experimental a-chain that identifies the element 277112 is shown in Fig. 16.) All the heaviest elements found during recent years are believed to be well deformed. Indeed, the measured a-decay energies, along with complementary syntheses of new neutron-rich isotopes of elements Z=10f> and Z=108, have furnished confirmation of the special stability of the deformed shell at iV=162 predicted by theory 150'151. Still heavier and more neutron-rich elements are expected to be spherical and even more strongly stabilized by shell effects. Beautiful experimental confirmation of large quadrupole deformations in this mass region comes from gamma-ray spectroscopy. In recent experimental work, Reiter et al. 154 succeeded in identifying the ground-state band of No (the heaviest nucleus studied in gamma-ray spectroscopy so far!). Figure 17 (top) shows the experimental gamma spectrum corresponding to the ground- state band of No. The quadrupole deformation of No can be inferred from the energy of the deduced 2+ state. The resulting value, /32«0.27, is in nice agreement with theoretical predictions (see Fig. 17, bottom,, and also Refs 152,155,156,157 and references quoted therein). As a matter of fact, the nucleus 254No has probably the largest ground-state quadrupole deformation in the whole actinide-transfermium region. Figure 16 displays the shell energy calculated in the HF+Sly7 model of

27 Synthesis of New Elements

170 176 182 Neutron Number

Figure 16: Contour map of shell energy for the superheavy elements obtained with the self- consistent HF+SLy7 model of Ref. 152. The experimental chain of a-particle decays that identifies the element with A=277 and Z—\Y1 147 is indicated together with the currently observed 2=114 a-chains corresponding to isotopes with A=289 and 287 153. Still heavier superheavy nuclei will probably be reached with the new-generation RNB facility (NISOL).

Ref. 152. Here the shell energy is defined as the difference between the self- consistent ground-state energy and the spherical macroscopic energy of Ref. 156. The calculation clearly shows the presence of strong shell stability around the "doubly magic" nucleus 310126. This result markedly differs from the most macroscopic-microscopic approaches where the first island of shell stability in the superheavy region is concentrated around Z=114. The HF prediction of Fig. 16 is consistent with results of other self-consistent studies based on 158 157 non-relativistic and relativistic mean-field theories ' j which predict the spherical neutron closure at 7V=184 and the proton gap at Z—120 or 126. In this context, the recent result by the Dubna/Livermore collaboration, who reported the synthesis of two isotopes of the Z=114 element in the 48Ca+244Pu and 48Ca+242Pu "hot fusion" reactions 153, is both exciting and important. The evidence is based on o-decay chains which are claimed to be candidates

28 208 a=3.4(ib

_ z.,o2x.ray. GAMMASPHERE and FMA 20 *""127J 254K • i4u s» Nogsb transitions 8 10

0 100 200 300 400 500 600

\ quadrupole deformation .0.28 HF+\BCS+SLy4+5 0.24 10.20 10.16 8 0.12 0.08 0.04 0.00 136 142 148 154 160 166 172 178 184 190 N Figure 17: Top: Gamma spectrum corresponding to the ground-state band of 254No obtained in the recent GammaSphere-FMA work at ATLAS 154. Bottom: Contour map of the ground- state quadrupole deformation 02 obtained in the Skyrme-HF calculations of Ref. 152.

for the decay of 289114 and 287114 (see Fig. 16). The measured values of a energies are consistent with predictions of the HF theory 159

29 4-3 Limits of Angular Momentum

Let me begin by discussing nuclear rotations in the context of the variety of rotational motions in the Universe. Figure 18 displays, in a log-log scale, the characteristic rotational frequency as a function of the characteristic size of the rotating body 160.

rotations in the universe molecules nuclei: mechanical hadrons

io30

10]

1020- galaxy clusters

10,20 1010 10° 10"10 10'20 typical size (cm)

Figure 18: Typical rotational frequency u (in sec"1) versus characteristic dimension of a rotating object ]C°. Interestingly, the data spanning around forty orders of magnitude in L and UJ and different interaction ranges (from gravitational to strongest) cluster around the "universal" line given by a simple power law.

A few general comments regarding the details of Fig. 18 are in order. The largest bodies shown in Fig. 18 are galaxy clusters, with a typical period of rotation of 1017 sec. The pulsars, with their dimensions of several km, are, considering their huge masses, amazingly fast; they are wonderful laboratories of high-spin nuclear superfluidity161. One of the dizziest mechanical man-made objects is an ultra-centrifuge used for isotope separation. The Oak Ridge gas centrifuge could make ~104 rotations/sec162, but there exist even faster ones.

30 Molecules offer many beautiful examples of a very rigid quantum-mechanical rotation. Discrete molecular states with angular momenta of the order of 7=80 A are measured routinely today (see, e.g., Ref. 163 for rotational bands in 120Sn16O). The smallest objects shown in Fig. 18 are hadrons and nuclei. Some excited states of hadrons can be interpreted as rotational bands. The moment of inertia of the nucleon is Jr«20-25 MeV/7t2 164; the corresponding rotational frequencies 24 are of the order of HUJ=0.5 GeV (T~10~ sec). In this company, atomic nuclei, with their typical dimensions of several fm and rotational frequencies ranging from 1020 to 1021 Hz, are champions of rotational motion. What makes nuclear rotation so special is the overwhelming presence of the Pauli principle, quantal shell effects, and superconductivity. The phenomenon of nuclear rotation has a long history. As early as 1937, N. Bohr and Kalckar 165, estimated for the first time the energies of lowest rotational excitations and introduced the notion of the nuclear moment of inertia. More than sixty years later, nuclear high-spin physics is a mature field. We have learned a lot about the basic mechanisms governing the behavior of fast nuclear rotation, especially the interplay between collective and non-collective degrees of freedom, competition between rotation, deformation, nuclear superconductivity, and many global and detailed aspects of high-spin gamma-ray spectroscopy. However, a lot of shocking surprises are still being encountered. The new-generation experimental tools, such as multidetector arrays EUROBALL and GAMMASPHERE, combined with the new-generation particle detectors and mass/charge separators, enable us to study discrete nuclear states up to the fission limit as well as a high-spin quasi-continuum, and explore new limits of excitation energy, angular momentum, and energy resolution. In- deed, these very precise tools made it possible to probe very small effects on a keV scale. New high-quality data tell us more about the structure of intrinsic wave functions of high-spin states. Advances in high-resolution gamma-ray detector systems are also responsible for a revolution in our study of low-spin nuclear behavior. Here, new insights have been gained on the nature of collective nuclear vibrations. In particular, long-searched vibrational multiphonon states have been found experimentally. The existence of these very harmonic modes of nuclei, both at high 10 and low frequencies 166, challenges our understanding of the Pauli principle. Figure 19 displays, schematically, the territory of high-spin science. The preferred pathways in the de-excitation process relate to favorable arrange- ments of protons and neutrons and can often be associated with specific nuclear deformations. Some spectacular examples of intrinsic shapes are illustrated in

31 Fig. 19: superdeformed and hyperdeformed shapes corresponding to huge elongations of nuclear density, magnetic deformations represented by shears bands, or static pairing deformations. Superdeformed states have now been found in all mass regions: in very light nuclei (cluster configurations), in the Ni-Zn region, and in the actinides.

— ,\chaos, rotational , continuum, \ complete i —7 / spectroscopy

Angular Momentum

Figure 19: Territory of nuclear high spins. Some of the important physics bullets are indicated schematically (from Refs. n'167).

A great challenge for nuclear theory is the existence of very regular rotational "shears" bands (with rather constant moments of inertia) in nuclei which are spherical. These and other exciting high-spin phenomena have been recently reviewed in a beautiful booklet167. Discussed below are some of the bullets shown in Fig. 19.

32 Superdeformation: Spectroscopy at a keV Scale

The observation of SD states constitutes an important confirmation of the shell structure of the nucleus. Quantum-mechanically, the remarkable stability of SD states can be attributed to strong shell effects that are present in the average nuclear potential at very elongated shapes168'169'170. For the oscillator potential, this happens when the frequency ratio is 2:1. (For more realistic average potentials, strong shell effects appear even at lower deformations.) The structure of single-particle states around the Fermi level in SD nuclei is significantly different from the pattern at normal deformations. Indeed, the SD shells consist of states originating from spherical shells having different principal quantum numbers, and hence having very different spatial character. This unusual situation produces new effects in nuclear structure. Superdeformed states have been discovered in several mass regions. These are fission isomers in the actinides; high-spin bands around 152Dy (Ref. m), 192Hg (Ref.172), 83Sr (Ref. 173), and 62Zn (Ref. 174); and "molecular" (cluster) configurations in light nuclei175. Intrinsic configurations of SD states are well characterized by the intruder orbitals carrying large principal oscillator numbers 176>177. Because of their large intrinsic angular momenta and quadrupole moments, these orbitals strongly respond to the Coriolis interaction and to the deformed average field. The increased precision of experimental tools of gamma-ray spectroscopy has allowed us to look much more closely at very weak effects which are, ener- getically, at an eV or a keV scale. Among the most startling new phenomena is that of identical bands, i.e., the observation of sequences of ten or more identical gamma rays associated with rotational bands in different nuclei 178. Another recent discovery in SD nuclei is the observation of very small (tens of eV!), but systematic, shifts in the energy levels of certain bands (AJ=2 staggering)179-180. A satisfactory explanation of both phenomena is still lacking178 (see also recent Refs. 181>182). It is rather clear that the puzzle of identical bands is ultimately related to the questions normally addressed in the context of large- amplitude collective motion such as: What are the "strong" quantum numbers that stabilize collective rotation? Indeed, recent studies show that nuclear systems at very large deformations and high angular momenta are the best examples of an almost-undisturbed single-particle motion (i.e., extreme one- body picture 183.184). A spectacular example of today's superdeformed spectroscopy is the inves- tigation of Ref.185 where as many as thirteen superdeformed bands have been found in 149Gd and all have been classified in terms of simple particle-hole con-

33 figurations with respect to the "doubly magic" superdeformed core of 152Dy. Another important achievement has been the identification of single-particle excitations in the rotating SD well through their electric and magnetic properties. This has been accomplished by measurements of intraband magnetic transitions 186>187 and relative electric quadrupole moments 188489,i90

Magnetic Rotation In nearly spherical nuclei, sequences of gamma rays were observed reminis- cent of collective rotational bands, but consisting of unusually strong magnetic dipole transitions 191. The basic mechanism leading to the unusually large magnetic collectivity is explained theoretically 192 in terms of a gradual alignment of valence protons and neutrons along the axis of the total spin (shears mechanism). Since quadrupole deformations of these bands are extremely small, the rotational-like pattern observed cannot be attributed to the rotation of the electric charge distribution. It is believed that the collective properties of shears bands are linked to magnetic-type static intrinsic deformations. However, the exact nature of the intrinsic magnetic fields responsible for this magnetic rotation is not fully understood at present.

Complete Spectroscopy and Qua&continuum: from Chaos to Order Gamma-ray spectroscopy with the new generation of detector systems offers a unique possibility to probe quantum chaos, roughly defined as a regime where the quantum numbers that may be used to characterize low-lying (cold) states of a many-body system are gone. Today, the important issues under study in the so-called "complete spectroscopy" experiments are: At what energy does chaos set in? What are the unique fingerprints of the transition from regular to chaotic motion? The signatures for the onset of chaos can be observed not only in the distribution of energy levels but also in the properties of electromagnetic transition intensities. Progress in gamma-ray spectroscopy has resulted in the discovery of discrete lines linking superdeformed bands to low-deformation states 193>194. This has allowed the determination of the excitation energies and spins of superdeformed states, and also the mixing between the superdeformed states and the nearly spherical configurations in the first well. Although the general prin- ciples seem to be under control, the satisfactory microscopic understanding of the complicated multidimensional tunneling, resulting in a decay out of a superdeformed band, is still lacking. The dynamics of a metastable state is a part of a multidisciplinary field of the large-amplitude collective motion (which includes such phenomena as fusion, fission, and coexistence, see Sec. 5). This

34 particular area of the nuclear many-body problem is expected to greatly ben- efit from the improved computational resources. One of the most important tasks for experiment and theory is to identify the islands of stability at high excitation energies, and to understand the underlying quantum numbers, that is, to find order in chaos.

Gamma-Delayed Proton Emission As mentioned earlier, a new exciting avenue is the competition between gamma- radiation and the emission of prompt protons. Here, spectacular examples are proton-emitting intruder bands in 58Cu 125 and 56Ni 126. In the doubly magic nucleus 56Ni, where two intruder bands have been observed, the lower rotational band can be explained by large-scale shell-model calculations in the pf shell. The results of cranked mean-field calculations indicate that this band is built upon a 4p-4h excitation (see Fig. 20, 4°4° band). The second band, however, is expected to involve particles in the I39/2 orbit, which is supported by its nearly identical behavior to the band in 58Cu. However, the best scenario for this configuration involves only one proton promoted to the I59/2 orbit (4°41 band in Fig. 20), while a neutron and a proton occupy this orbit in 58Cu! Radioactive medium-mass nuclei such as 56Ni are fantastic territories where various theoretical approaches can be confronted: state-of-the-art shell model, mean-field models, and cluster models. The spherical structures in 56Ni are well described by the large-scale shell model, the collective 4°41 band can be understood in terms of the self-consistent mean-field theory, and the 4°4° band can be described by both methods.

Theoretical Developments Unexpected experimental data from large arrays have generated a great deal of theoretical interest and activity. As a result, many new insights have been gained regarding the behavior of atomic nuclei at high spins. The new-generation high-spin data tell us a lot about effective NN interaction, and about pairing force in particular. For instance, self-consistent HF studies of moments of inertia of SD bands indicate the importance of higher- order pairing interactions such as the rotation-induced Kn—\+ pairing forces 197 198 199 195,196 ancj me density-dependent pairing forces ' ' . Interestingly, the question of the density dependence of pairing interaction is also of great interest for physics of nuclear radii, deep hole states, and properties of drip-line nuclei80 (see Sec. 4.1). That is, this important problem has also surfaced in a different corner of nuclear structure.

35 20

12 -

8 So - o band 1 ° band 2 4 - + SM

0 - EEHF

0 4 8 12 16 20 Angular Momentum

Figure 20: Excitation energy versus angular momentum for experimental and calculated intruder bands in 56Ni. Lines indicate the 4°4°, 4°41, and 4141 bands calculated in the cranked HF+SLy4 model, while the KBF shell-model calculations (SM) are indicated by crosses (from Ref. 126).

In the microscopic theory of a rotating nucleus: the average nucleonic field is obtained self-consistently from the nucleonic density. Consequently, in the presence of a large angular momentum, the intrinsic density is strongly polar- ized, i.e., the nucleus shows phenomena and behaviors characteristic of condensed matter in the magnetic field: ferromagnetism, Meissner effect, Joseph- son effect. The nuclear magnetism is caused by the time-odd components in the average potential 200,201,202 -phe understanding of the structure of these terms will rely mostly on the expected increase of information on high-spin properties and on magnetic properties. Here, a good example is recent calculations based on a RMF theory; as a consequence of broken time-reversal invariance, the spatial contributions of the vector meson fields have a strong influence on the moments of inertia 203.

36 5 Nucleus as a Finite Many-Fermion System The atomic nucleus is a complex, finite many-fermion system of particles interacting via a complicated effective force which is strongly affected by the medium. As such, it shows many similarities to other many-body systems involving many degrees of freedom, such as molecules, clusters, grains, mesoscopic rings, quantum dots, and others. There are many topics that are common to all these aggregations: existence of shell structure and collective modes (e.g., vibrations in nuclei, molecules, and clusters; superconductivity in nuclei and grains), various manifestations of the large-amplitude collective motion (such as multidimensional tunneling) and nonlinear phenomena (many-fermion systems are wonderful laboratories to study chaos), and the presence of dynamical symmetries. Historically, many concepts and tools of nuclear structure theory were brought to nuclear physics from other fields. Today, thanks to the wide arse- nal of methods, many ideas from nuclear physics have been applied to studies of other complex systems. There are many splendid examples of such interdisciplinary research: applications of the nuclear mean-field theory and its extensions to studies of static and dynamical properties of metal clusters, treatment of finite-size effects in the description of superconductivity of ultrasmall grains, use of symmetry-dictated approaches to describe collective excitations of complex molecules, applications of the nuclear random matrix theory to various phenomena in mesoscopic systems (nuclei and quantum dots can be used as complementary laboratories to study transport properties and the onset of chaos in Fermi systems), and the description of short- and long-range correlations in many-fermion systems (nuclear theorists have pioneered the study of quantum helium liquid droplets by exact quantum Monte Carlo methods; the nuclear shell model has been applied to the treatment of electron correlations in solids). The count can go on, of course. Given below are several examples of nuclear physics research which goes well beyond the frames of the nuclear many-body problem.

5.1 Large-Amplitude Collective Motion The question of whether the motion of a finite many-body quantal system can be divided into a slow collective motion and a fast single-particle motion is older than nuclear physics itself. As early as 1927, Born and Oppenheimer 204 successfuHy described molecular motion by assuming that fast electrons were strongly coupled to the equilibrium position of the slow and much heavier ions. However, it was a decade later when Jahn and Teller discovered 205 that this concept breaks down around the points of degeneracy between elec- 37 tronic states. They proved that a configuration of atoms or ions can develop a stable symmetry-breaking deformation provided the coupling between degen- erate electronic excitations and collective motion is strong. This phenomenon is usually referred to as the spontaneous symmetry-breaking or the Jahn-Teller effect206'207 It was quickly recognized that the spontaneous symmetry-breaking was not exclusively related to molecular physics. It is a universal phenomenon known in all areas of physics: field theory (Higgs mechanism), physics of su- perconductors (Meissner effect), atomic, and nuclear physics. The common denominator is the existence of an intrinsic deformed system determined by the instantaneous position of slow degrees of freedom. The importance of symmetry-breaking in nuclear physics was noticed early by Hill and Wheeler 208 and by Bohr 209. The unifying theme is the concept of (self-consistent) mean field and its intrinsic system determined by the instantaneous position of slow degrees of freedom. The static mean-field equilibria are characterized by self-consistent (dynamical) symmetries or "effective" quantum numbers, responsible for the near-equilibrium (local) motion of the system. It should be emphasized that the choice of proper nuclear collective coordinates is a long- standing problem?0'210 (see Sec. 2 for the discussion of nuclear time scales). Indeed, as stated by A. Bohr 211: "In contrast with the molecular case, there are here no heavy particles to provide the necessary rigidity of the structure. However, nuclear matter appears to have some of the properties of coherent matter which makes it capable of types of motion for which the effective mass is large as compared with the mass of a single nucleon." Microscopically, intrinsic deformations are always associated with single- particle states that are very close in energy, for instance, due to the presence of some symmetry. Such a degeneracy leads to a reduced stability, and even an infinitely small perturbation produces a violent transition to a state that is deformed. Or, in simple words, if a nucleus can remove degeneracy by means of a symmetry breaking, it will always go for it! Hence the symmetries are important. They provide necessary initial conditions for the existence of collective modes and moments which allow us to understand complex systems simply. The atomic nucleus is a very rich and beautiful laboratory of these. The very existence of nuclear rotational states tells us that in many situations the nucleus prefers to choose a deformed shape over the (more symmetric!) spherical shape. Spectacular examples of symmetry-breaking are static reflection-asymmetric pear-like deformations found in certain'nuclei. These systems exhibit features familiar from molecular physics: alternating parity bands, intrinsic dipole moments, and parity doublets (see Fig. 21). The exciting aspect of intrinsic symmetry-breaking is

38 that sometimes it enhances the "real" symmetry violation. For instance, the presence of parity doublets (i.e., close-lying states of opposite parity) in pear- like nuclei generates very favorable conditions for parity violation (see Ref. 212 and references quoted therein).

center of center of mass charge

> 4 01

01 U § 2

231 220Ra

Figure 21: For most deformed nuclei, like 238U, a description as an axially and reflection symmetric spheroid is adequate to describe the band's spectroscopy. Because such a shape is symmetric with respect to the space inversion operation, all members of the rotational band will have the same parity. However, it has been found that some nuclei might have a shape that is asymmetric under reflection, such as a pear shape. The rotational band of 22 Ra consists of levels of both parities, hence the "molecular" name parity doublet. The levels having the same parity are connected by E2 photons while the states of opposite parity are connected by El photons. The enhanced El transitions are characteristic of an intrinsic shape having an electric dipole moment, i.e., the shift between the center of charge and the center-of-mass of the nucleus.

39 The ultimate test of the theories of large-amplitude collective motion is provided by experimental data. Of particular interest are measurements of excited states built upon coexisting configurations 9. This kind of experimental information tells us about mixing between stable HF minima and the dependence of this mixing on excitation energy, i.e., it addresses explicitly the issue of diabaticity of the nuclear collective motion. A spectacular example of coexistence between excited states built upon different mean fields is provided by the excitation spectrum of 184Pt. Figure 22 displays the experimental situation, characteristic of neutron-deficient Pt isotopes. The ground state K*=0+ rotational band has a large moment of inertia. Built upon this ground state is a Kn=2+ "gamma" band. These two structures coexist with the excited K*=0+ and Kn=2+ bands with considerably lower moments of inertia. Mean-field calculations 213 predict two minima to coexist in the potential energy surface of 184Pt. The prolate minimum, 0.2Slater determinant 20

= exp < 2^ fij{t)a\a) - h.c. ) |$o), (1) U

where |$0) is the stationary HF state and fij is a time-dependent Thouless matrix. It is interesting to note that the TDHF (or TDHFB) equations can be 40 Figure 22: Shape coexistence in 184Pt. Top panel shows the two coexisting 0+ bands in 184 Pt 216. The arrows indicate A/=0 transitions between the bands. As can be seen from the energy spacings, the lower band is the more strongly deformed. It can be associated with the prolate-deformed minimum in the calculated potential energy surface (bottom panel) 213. The excited band corresponds to the oblate-deformed secondary shallow minimum. written in a canonical form

Q rr = {CihH)i and (2) where the complete set of canonical variables {Cij, C,* } is obtained from the 41 matrix fij through the variable transformation 217 ^L <3) In Eq. (2) H=H{C, C*)=(*(C, C*)|H|*(C, C")> represents the classical energy surface of the system in the TDHF coordinate space {Cy, C*j}. Equation (2) demonstrates that the TDHF manifold has the same symplectic structure as the classical phase space218. As seen in Fig. 23, the nuclear motion is represented by a trajectory in the phase space. The minimum points in the TDHF manifold are stable points (solutions to the static HF(B) problem). The motion around the minima is regular; it corresponds to nuclear collective modes, rotations and vibrations. These are well classified by means of quantum numbers associated with the corresponding stable point. The complicated trajectories associated with complex nuclear states form the chaotic sea. Here, the two trajectories with very similar initial conditions develop differently. Although the barrier separating the two minima in Fig. 22 is small when plotting the potential energy surface as a function of some selected collective coodinates (here /3 and 7), the corresponding structures can be very well separated in the full TDHF manifold219. The variety of phenomena classified under the name of the large-amplitude collective motion, such as coexistence, tunneling in complex systems, and chaos, can be found in other fields of physics and chemistry. The related questions are common threads to all areas of the many-body problem217'220'210.

5.2 From Atoms and Nuclei to Clusters: Shells and Collective Phenomena In spite of the fact that the nuclear force is "strong", the nucleons in a nucleus are quite far apart and they very seldom interact with each other. This is due to the Pauli principle which prevents the nucleons from approaching too close. Consequently, the nucleus is a system which is not very dense, but it has a well-defined surface; it can be viewed as a Fermi liquid. The very existence of magic nuclei is a beautiful consequence of independent particle motion. Nucleons having identical or similar energies are said to belong to the same shell, and different shells are separated by energy gaps (see Fig. 24). The way the energy bunchings (called shell structure) occur depends on the form and the shape of the average potential in which particles are moving. The way the energy bunchings, called shell structure, occur depends on the form and the shape of the average potential in which particles are moving. 42 stable points (HF minima)

chaotic •/ motion

regular (collective) motion

Figure 23: Schematic diagram representing the motion of the nucleus in the TDHF manifold. The HF minima are stable points. The collective rotations and vibrations take place around the minima. The collective submanifold is submerged in the chaotic sea representing complicated nuclear modes.

The Coulomb potential acting on electrons in an atom is very different from the average potential in a nucleus. This explains the difference between magic

43 Pronounced Shell structure shell structure absent

she gap shell gap shell 0 closed trajectory trajectory does not (regular motion) close (chaos)

Figure 24: Left: shell structure of the single-particle potential. The energy bunchings (shells) are separated by large energy gaps. Right: the absence of shells in the spectrum. The classical closed trajectories are counterparts of quantum orbits that are bunched in shells. The absence of closed trajectories indicates the disappearance of the bunchiness in the spectrum; hence the presence of chaos. numbers seen in atoms and nuclei. The shell structure is a quantal thing. Magic gaps, positions, and sizes of shells are governed by the strict rules of quantum mechanics. But, amazingly enough, the origin of shell effects can be traced back to the the geometry of periodic orbits of the corresponding classical Newton equation168'221'222'169. That is, there exists a close link between the nuclear shell structure and the non-linear dynamics of the classical problem. Indeed, the single-particle level density g(e)=^2i S(e — €j) can be represented by means of the Gutzwiller trace

44 formula223 oo EX COS [* where 5 is the average level density, while the second term in Eq. (4) is the fluctuating part of g responsible for shell effects. The shell energy can then be expressed as a sum over periodic orbits L. (SL is the action integral associated with the orbit L and /z/, is the Maslov index.) Consequently, the shell structure of the many-body system (and hence the presence or absence of deformation) has its deep roots in the non-linear dynamics of the corresponding classical Hamiltonian and the geometry of classical orbits169. The microscopic analysis of the link between the phenomenon of spontaneous symmetry-breaking and the non-linear dynamics (and chaos) is one of the most vigorously pursued recent avenues in theoretical nuclear physics. Among many examples of such analyses are the explanation of supershell effects in superdeformed nuclei168'169 and the discussion of shell structure in nuclei with permanent octupole defor- mations224. The small clusters of metal atoms (typically made up of thousands of atoms or fewer) represent an intermediate form of matter between molecules and bulk systems. When the first experimental data on the cluster's structure were obtained225, it was immediately realized that the shell-model description could be applied to valence electrons in clusters 226. Since metal clusters can be made neutral, there is no limitation to their size, in contrast to nuclei. Magic numbers corresponding to filled shells up to -3000 electrons have been seen 227 The effect of the bunchiness in the energy spectrum can'be illuminated by looking at the quantal shell energy, i.e., the contribution to the total energy of the system resulting from the shell structure. The nuclear shell energy and the shell energy for small sodium clusters are shown in Fig 25. In both cases, the same nuclear theory technique of extracting the shell correction has been used. The sharp minima in the shell energy correspond to the magic gaps. Nuclei and clusters which are not magic have non-spherical shapes. In Fig. 25 the deformation effect is manifested through the reduction of the shell energy for particle numbers that lie between magic numbers. How can one see whether the clusters are spherical or not? The deformation of the clusters can be deduced by studying a collective vibrational mode, the so-called dipole surface plasmon, which corresponds to an almost-rigid oscillation of the electrons with respect to the positive ion background. This is a direct analog of the nuclear giant dipole resonance which is an oscillation of protons with respect to neutrons 23°, The occurrence of the deformation of the

45 Nuclei

60 100 140 Number of neutrons Na Clusters ; experiment

V V 58 92 138 LJ 1 theory spherical : clusters V7

¥ deformed ! 1* clusters I 50 100 150 200 Number of electrons

Figure 25: Top: experimental and calculated nuclear shell energy as a function of the neutron number228. The sharp minima at 20, 28, 50, 82, and 126 are due to the presence of nucleonic magic gaps. Bottom: experimental and calculated shell energy of sodium clusters 229 as a function of the electron number. Here, the magic gaps correspond to electron numbers 58, 92, 138, and 198. In both cases, the shell energy has been calculated by means of the same nuclear physics technique (macroscopic-microscopic method), assuming the individual single-particle motion (of nucieons or electrons) in an average potential. The reduction of the shell energy for particle numbers that lie between the magic numbers is due to deformation (the Jahn-Teller effect).

cluster implies that the dipole frequency will be split. This and many other

46 properties of collective plasmon states in clusters have been initially predicted by nuclear theorists, and their existence has been subsequently confirmed experimentally. Other applications of nuclear methods to clusters include the application of the theory of nuclear rotation to cluster magnetism 229 and the application of experimental and theoretical nuclear reaction techniques to cluster fragmentation31. The experience accumulated in nuclear physics plays an important role in understanding many of the properties of atomic clusters.

5.3 Nuclear Shell Model in Condensed Matter and Molecular Physics Interacting electrons moving on a thin surface under the influence of a strong magnetic field exhibit an unusual collective behavior known as the fractional quantum Hall effect. At special densities, the electron gas condenses into a remarkable state - an incompressible liquid - while the resistance of the system becomes very accurately quantized. These special densities (or fillings), measured in units of the elementary magnetic flux penetrating the surface, correspond to rational numbers. Although the fractional quantum Hall effect seems quite different from phenomena in nuclear many-body physics - the electron-electron interaction is long-range and repulsive rather than short- range and attractive - it was shown recently that these incompressible states have a shell structure similar to light nuclei 232. Nuclear and atomic physicists recognized very early that the behavior of complex many-body systems is often governed by symmetries. Some of these symmetries reflect the invariance of the system with respect to fundamental operations such as translations, rotations, inversions, exchange of particles, etc. Other symmetries can be attributed to the features of the effective interaction acting in the system 21,233,23,25 Although they are more difficult to visualize, these "interaction symmetries" (dynamical symmetries) can often dramatically simplify the description of otherwise very complicated systems. The natural language of symmetries is mathematical group theory. In this language, dynamical symmetries are associated with a special type of group, a Lie group. These methods have made it possible to describe the structure and dynamics of molecules in a much more accurate and detailed way than before 234. In particular, it has been possible to describe the rotational and vibrational motion of large molecules such as buckyballs (60C).

5.4 From Chaos to Order A topic of great, interest is the signatures of classical chaos in the associated quantum system, a sub-field known as quantum chaos. A nuclear physics theory, random matrix theory, developed in the 1950% and 60's to explain

47 the statistical properties of the compound nucleus in the regime of neutron resonances 235, is now used to describe the universality of quantum chaos. At the excitation energies of the neutron resonances (several MeV), the density of states (i.e., the number of states per energy unit) becomes very high, and a statistical description is appropriate. Today, the random matrix theory is the basic tool of the interdisciplinary field of quantum chaos, and the atomic nucleus is still a wonderful laboratory of chaotic phenomena. Other excellent examples of interplay between chaotic and ordered motion in nuclei are parity- violation effects amplified by the chaotic environment 236 and the appearance of very excited nuclear states (symmetry scars) well characterized by quantum numbers237. The study of collective behavior, of its regular and chaotic aspects, is the domain where the unity and universality of all finite many-body systems is beautifully manifested.

5.5 Mesoscopie Ptysics: From Compound Nuclei to Quantum Dots

Recent remarkable advances in materials science permit the fabrication of new systems with small dimensions, typically in the nanometer to microm- eter range. Mesoscopic physics (meso originates from the Greek word mesos, middle) describes an intermediate realm between the microscopic world of nuclei and atoms, and the macroscopic world of bulk matter. Quantum dots are an example of mesoscopic microstructures that have been under intensive in- vestigation in recent years; they are small enough that quantum and finite-size effects are significant, but large enough to be amenable to statistical analysis. For "open" dots, electrons can move classically into the dot. On the other hand, for "closed" dots, the movement of electrons at the dot-leads interfaces is classically forbidden, but allowed quantum-mechanically by a quantal tunneling. Tunneling is enhanced when the energy of an electron outside the dot matches one of the resonance energies of an electron inside the dot. The conductance of quantum dots displays rapid variations as a function of the energy of the electrons entering the dot. These fluctuations are aperiodic and have been explained by analogy to a similar phenomenon in nuclear reactions (known as Ericson fluctuations). The electron's motion inside the closed dot is generically chaotic. Recently, the same nuclear random matrix theory that was originally invoked to explain the fluctuation properties of neutron resonances has been used to develop a statistical theory of the conductance peaks in quantum dot2^8. That is, a quantum dot can be viewed as a nanometer- scale compound nucleus!

48 6 Summary

In years to come, we shall see substantial progress in our understanding of nuclear structure - a rich and many-faceted field. An important element in this task will be to extend the study of nuclei into new domains. There are many frontiers of today's nuclear structure. For very light nuclei, one such frontier is physics at subfemtometer distances where the internal quark-gluon structures of nucleons overlap. For heavier nuclei, the frontiers are defined by the extremes of the N/Z ratio, atomic charge and nuclear mass, and angular momentum. The journey to "the limits" is a quest for new and unexpected phenomena which await us in the uncharted territory. However, the new data are also expected to bring qualitatively new information about the effective NN interaction and hence about the fundamental properties of the nucleonic many-body system. By exploring exotic nuclei, one can magnify certain terms of the Hamiltonian which are small in "normal" nuclei, thus difficult to test. The hope is that after probing nuclear properties at the extremes, we can later improve the description of normal nuclei (at ground states, close to the valley of beta stability, etc.). There are many experimental and theoretical suggestions pointing to the fact that the structure of exotic nuclei is different from what has been found in normal systems. New RNB facilities, together with advanced multi-detector arrays and mass/charge separators, will be essential in probing nuclei in new domains 71. The field is extremely rich and has a truly multidisciplinary character. Experiments with radioactive beams will make it possible to look closely into many exciting aspects of the nuclear many-body problem. A broad international community is enthusiastically using existing RNB facilities and hoping and planning for future-generation tools. Advances in computer technology and theoretical modeling will make it possible to (i) better understand the bare NN interaction in terms of quarks and gluons, and effective interactions in complex nuclei in terms of bare forces, and to (ii) answer fundamental questions concerning nuclear dynamics. These questions on the microscopic mechanism behind the small- and large-amplitude collective motion, on the impact of the Pauli principle on nuclear collectivity, and on the origin of short-range correlations have interdisciplinary character. Particularly strong are overlaps between nuclear structure and condensed matter physics (many-body methods, superconductivity, cluster physics, physics of mesoscopic systems), atomic and molecular physics (treatment of correlations, physics of particle continuum, dynamical symmetries), nonlinear dynamics (chaotic phenomena, large-amplitude collective motion), astrophysics (nucleosynthesis, neutron stars, supernovae), fundamental symmetries physics,

49 and, of course, computational physics.

Acknowledgments

This research was supported by the U.S. Department of Energy under Contract Nos. DE-FG02-96ER40963 (University of Tennessee) and DE-AC05-96OR22464 with Lockheed Martin Energy Research Corp. (Oak Ridge National Labora- tory).

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