NUCLEAR STRUCTURE AT THE LIMITS*

W. Nazarewicz

Department of Physics and Astronomy, University of TennesseeKnoxville, Tennessee 37996 U.S.A.

Physics Division, Oak Ridge National LaboratoryOak Ridge, Tennessee 3783 1 U.S.A.

to be uublished in

Proceedings ofEleventh Physics Summer School on Frontiers in Nuclear Physics:

From Quark-Gluon Plasma to SupernovaCanberra, AustraliaJanuary 12-23, 1998

*Research supported by the U.S. Department of Energy under ContractDE-AC05-960R22464 managed by Lockheed Martin Energy Research Corp.

‘7he submitted manuscript has been authored by a contractor of the U.S. Government under contract No. DE-AC05-%OR22464. 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 NAZAREWICZDepartment of Physics and Astronomy, University of Tennessee

Knoxville, Tennessee 37996, U.S.A.Physics Division, Oak Ridge National Laboratory

Oak Ridge, Tennessee 37831, U.S.A.E-mail: witek-nazarewictOutk.edu

One of the frontiers of today’s nuclear science is the “journey to the limits” ofatomic charge and nuclear mass, of neutron-to-proton ratio, and of angular mo-mentum. 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 informationabout the fundamental properties of the nucleonic many-body system, the natureof the nuclear interaction, and nucleonic correlations at various energy-distancescales. In this series of lectures, current developments in nuclear structure atthe limits are discussed from a theoretical perspective, mainly concentrating onmedium-mass and heavy nuclei.

1 Introduction

The atomic nucleus is a fascinating many-body system bound by strong interac-tion. The building blocks of a nucleus - protons and neutrons - are themselvescomposite aggregations of quarks and gluons governed by quantum chromo-dynamics (QCD) - the fundamental theory of strong interaction. However, inthe 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 prob-lem is overwhelming. Nuclei are exceedingly difficult to describe; they containtoo 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 nuclearmodes is close to the single-nucleonic time scale. This means that many con-cepts and methods applied successfully to other many-body systems, such assolids and molecules, cannot be blindly adopted to nuclei. In these lectures, Iintend to review - rather briefly - the enormous progress that has happenedin nuclear structure during recent years. After discussing fundamental nuclearmodes (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, wh?ch is nu-clear structure at the limits of isospin, angular momentum, mass, and charge(Sec. 4). This will be followed by a discussion of interdisciplinary aspects oftheoretical nuclear structure research (Sec. 5). A short summary is containedin Sec. 6. For a general overview of nuclear science, the reader is encouraged

1

to study the recent report l.

2 Nuclear Modes and Their Time Scales

Considering that the atomic nucleus is, in a good approximation, a cluster ofstrongly 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 equilibriumposition of slowly moving ions.

This point is nicely illustrated in Fig. 1, which shows the measured spec-trum of a diatomic molecule Ns. Each electronic excitation represents a band-head upon which vibrational and rotational states are built. Vibrational statesare indicated by a vibrational quantum number v (Y=O representing the zero-phonon state, v=l is a one-phonon state, and so on). Rotational levels are notplotted - there are far too many to be displayed. The time scale of molecu-lar modes is governed by the hierarchy of excitation energies. The electronicexcitations are of the order of lo5 cm-’ (which in “nuclear” units correspondsto an eV), vibrational frequencies are ~10~ cm-l, and rotational energies are-10-l cm-‘. This means that the single-particle electronic motion is fromtwo to six orders of magnitude faster than molecular collective modes. Con-sequently, the adiabatic assumption based on the separation of all moleculardegrees of freedom into fast and slow ones is justified due to different timescales.

In atomic nuclei, the total A-body wave function cannot, in general, be ex-pressed in terms of slow and fast components. This is because (i) the collectivenuclear coordinates are auxiliary variables which depend, in a complex way,on fast nucleonic degrees of freedom and (ii) the nuclear residual interactionsare not small.

How good is the time separation between single-particle and collectivenuclear motion? The typical single-particle period (i.e., the average time ittakes a neutron or a proton to go across the nucleus), T&=4R/VF [where Ris the nuclear radius and VF is the Fermi velocity (-O.Zlc)], is approximately3.10-22 sec. (The unit 1O-22 set is sometimes referred to as babysecond. Thesingle-particle time scale is of the order of several babysec.) The typical periodof nuclear rotation (Tr,tz10-21 set) is only -30 times greater than Ts.p., andfor nuclear vibrations the period of oscillations is only slightly greater than thesingle-particle period. It is truly amazing that these relatively small differencesin time scales seem to be sufficient to create rotating or vibrating potentials,common for all nucleons!

2

Excitation spectrum of N, molecule

I,, c-- -----N;xlq-------” ------- -

1o- II

h ‘I’”” excited ‘Xi and ‘l7, state

I

”B=111 -

z .:=-; . II- 4--,I- Yz . II- ,- ‘-

14 - ;; q I-. fl- I-. lo-,- I-�- ,-. 1-

,os- :=0-0%�

o-�- t1- Cl �L

. z-:=

Im - b’n.

I:= I-II -,‘ -II- 1-II -

ii- ,-

:=

‘- I-G-I-

L- ,-I-I-1- o-

Diabatic potential energysurfaces for excited electronicconfigurations of N,

Figure 1: Top: vibrational spectrum of Nz molecule. Bottom: collective potentials, corre-sponding to different electronic configurations, as functions of the internuclear radius r (fromRef. ‘).

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

3

“Molecular” vibrational-rotational spectrumH

232Th G g’b’ 7“I- -7-k

- !I.I -

L b-yy

-l&w

k4

beta gamma vibration

vibration (2 phonons)

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 associatedwith 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 fairlyregular. However, after closer inspection, many deviations from the perfectrotational and vibrational pattern can be seen. For example, the moment ofinertia of 232Th is by no means constant but shows local variations as a func-tion of angular momentum. Such deviations indicate that the nuclear motionis not completely collective, i.e., that the collective modes result from coherentsuperpositions of single-particle nucleonic excitations.

Another extreme case is shown in Fig. 3, which displays the excitationspectrum of .14*Gd This spectrum looks very irregular; it is characteristicof many-particle many-hole excitations in an almost spherical nucleus. Thereare many states of different angular momentum and parity connected via rela-tively weak electromagnetic transitions. In spite of the fact that the excitation

4

-I

Figure 3: Excitation spectrum of 148Gd It is a beautiful example of noncollecive nucleonic.motion (from Ref. 4).

pattern looks “chaotic”, most of the noncollective states of 148Gd can be beau-tifully described in terms of well-defined quantum numbers of the nuclear shell

5

model 5. The collective spectrum of 232Th discussed above was not perfectlycollective. Likewise, in the “mess” of noncollective levels in 148Gd one canfind some elements of collectivity. For instance, the energies of the four low-est states O+, 2+, 4+, 6+, when plotted as functions of angular momentum,exhibit a characteristic parabolic pattern. As was realized around forty yearsav6, such a sequence of states indicates the presence of pairing - a collectivephenomenon.

The next example illustrates the coexistence effect. As will be discussedbelow in Sec. 5.1, shape coexistence is a particular case of the large ampli-tude collective motion where several nuclear configurations, characterized bydifferent intrinsic properties, compete ‘. In the spectrum of 152Dy, shown inFig. 4, one can recognize the noncollective structure resembling that in 14*Gd,a collective rotational band (interpreted in terms of a deformed triaxial config-uration) and super-collective superdeformed bands corresponding to very largeshape-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 collectiv-ity - high-frequency nuclear vibrations. Figure 5 displays the cross section forthe scattering of the relativistic “*Pb on a xenon nucleus. The first maximumin the cross section, which appears at -14MeV, can be associated with theone-phonon giant dipole resonance (i.e., very fast vibration of protons againstneutrons). It has been suggested lo that the local maximum at an energy of~28MeV is a two-phonon giant dipole vibration. In the data shown in Fig. 5one can also find isoscalar and isovector giant quadrupole resonances. The gi-ant nuclear excitations are always strongly fragmented; their widths are usuallymore than 5 MeV. It is worth noting that the time scale corresponding to gianttwo-phonon vibrations is shorter than 1 babysec! The unusual harmonicity ofthis mode is not understood well.

One of the outstanding challenges in nuclear structure is to understand themechanism governing the nature of nuclear collective excitations. By studyingnuclear rotations and vibrations, one is probing the details of the nuclear forcein a strongly interacting medium.

3 The Nuclear Many-Body Problem

3.1 The Nuclear Foi-ce

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 exotictopologies in light nuclei, and the one giving rise to the collective motion inheavy nuclei. The main challenges in understanding the nuclear force are shown

6

Coexistence of collectiveand noncollective motiontriaxial Iband % $b

whd”+

superdeformedbands

,m

; noncollective -+4TP-t+

152Dy

Figure 4: Excitation spectrum of 15*Dy. It is a specta&ar example of the coekistence ofcollective and noncollecive nucleonic modes. (Courtesy of D. Radford; Experimental datawere taken from Refs. 7,8.)

in Fig. 6 in the context of the hadronic and nucleonic many-body problem.

7

High-frequency nuclear vibrations(giant resonanfzes)

ElEl

El El

s *d- I I I I I I1-phonon

I I

G - - state2-phonon

a 1

u owl I I PI”“---“ecJ5 1 0 15 20 25 r) 35 40 45 50

Energy (MeV)

Figure 5: One-phonon and two-phonon giant resonances in *08Pb. (Experimental data weretaken from Ref. lo.)

The low-energy interaction between nucleons has a complicated spin-isospindependence dictated by the hadron’s substructure. One of the main challengesof nuclear science, indicated by the first bridge in Fig. 6, is the derivation of anucleon-nucleon (NN) interaction from the underlying quark-gluon dynamicsof &CD. Experimentally, the NN force can be studied by means of NN scat-tering experiments. Examples of phenomenological parameterizations basedon the NN scattering data are the Bonn12t13, Nijmegen, Reid14 and Argonne

8

Nuclear Forcesand

The Nuclear Many-Body Problem

light nuclei

Figure 6: From hadrons to heavy nuclei: main challenges in understanding the nuclear force.(From Ref. l1 .)

l5 potentials. While the long-range part of these free (but still effective!) NNforces is well described by one-pion exchange, their short-range behavior ispurely phenomenological. Here the quark-gluon degrees of freedom must beconsidered explicitly. (It is believed that the short-range part can be wellaccounted for by a simple gluon exchange potential and the Pauli principle16,17

-1

While the very light nuclei can nowadays be described as A-body clustersbound 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 nuclearshell model. Here, the basic assumption is that the nucleons are moving almostindependently in a mean potential obtained by averaging out the interactionsbetween a single nucleon and all remaining A-l protons and neutrons. Thispicture is only a first approximation; it is modified by the presence of theresidual interaction between the nucleons. The “effective” NN interaction in

9

the heavy nucleus, used to determine the mean potential, differs considerablyfrom the free NN force. In principle, it should be obtained by means of thecomplicated Bruckner renormalization procedure which corrects the free NNinteraction for the effects due to the nuclear medium. This challenging taskis represented by the second bridge in Fig. 6. Many features of the effectiveinteraction such as short range; strong dependence on spins, isospins, andrelative momenta of interacting nucleons; and the reduction of mass in thenuclear 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 2 protons and N neutrons). The freeNN force can be viewed as a residual interaction of the underlying quark-gluon dynamics of &CD, similar to the intermolecular forces that stem fromQED. Similarly, the effective NN force in heavy nuclei can be derived from theeffective free NN interaction. It probably would be very naive to think of thebehavior of a heavy nucleus directly in terms of the underlying quark-gluondynamics, but, clearly, the understanding of the bridges in Fig. 6 will makethis goal qualitatively possible.

3.2 Theoretical Strategies

How to tackle the problem of A strongly interacting nucleons? The generaltheoretical strategy is illustrated in Fig. 7. The starting point is, of course,the exact solution of the A-body Schrodinger equation (or relativistic fieldequations) with the bare NN force. Today, such ab initio calculations canbe 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 verycomplicated (e.g., it contains tensor terms and, often, non-local terms). Con-sequently, especially for heavier nuclei, one is forced to work with effective NNinteractions (second bridge in Fig. 6). This can be achieved either microscop-ically by means of the Bruckner theory 18~1g~20 by fitting some specific force toexperimental data, or by extracting interaction matrix elements from the data.Having determined the effective force, the two most commonly used strategiesare (i) that of the nuclear shell model and (ii) the mean-field strategy.

In the nuclear shell model, the effective two-body Hamiltonian is diagonal-ized in the limited configuration space. Here, the practical limitation is the sizeof the Hilbert space considered. Often, the realistic shell-model Hamiltoniancan be further approximated by replacing it with a Hamiltonian which can bediagonalized exactly using group theoretical techniques 21122Y23*24t25. The wavefunction of the nuclear shell model is an eigenstate of fundamental symmetry

10

The Nuclear Many-Body Problem

shell model mean field+ correlations

\ J

/ 7 /self-consistency

l limited\

Hartree-Fockconfiguration + - Hartree-Fock-space

l symmetry0, Bogoiyubov8

/

dictated a

approaches

;iss2%

rdeformedshell model

.

Figure 7: The nuclear many body problem - various theoretical strategies.

operators; it has good parity, angular momentum and isospin. From this pointof view, it is a laboratory system wave function.

Another strategy is that of the mean-field theory. Here, the main assump-tion is that the many-body wave function can be - to zero order - approximatedby that of independently moving quasiparticles. Together with the variationalprinciple applied to the effective density-dependent Hamiltonian, this leads to

11

the Hartree-Fock (HF) or Hartree-Fock-Bogolyubov (HFB) equations 20. Inspite of the very simple form, the independent-quasiparticle wave function is ahighly correlated state. In the mean-field theory, self-consistency is automati-cally guaranteed by the equations of motion. That is, the same wave functionwhich generates the mean field is an eigenstate of the mean-field Hamiltonian.Since the mean-field Hamiltonian often breaks symmetries present in the labo-ratory system, the HFB state is an inttinsic w’ave function, which - in general- is not an eigenstate of angular momentum, parity, and particle number oper-ators. By restoring the intrinsically broken symmetries, one takes into accountcorrelations going beyond the mean-field description. This can be done by var-ious 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 schemat-ic. In practice, one often uses approaches which take advantage of both strate-gies. Examples are the modern shell-model calculations employing the mean-field basis obtained by solving the HFB equations with either the shell-modelHamiltonian or some auxiliary model Hamiltonian.

We have learned a great deal about modes of nuclear excitations usingphenomenological models, often based on ingenious intuition and symmetryconsiderations. These models, approaches, and approximations have been ex-tremely successful in interpreting nuclear states and classifying nuclear statesand decays. Based on this experience, and thanks to developments in theo-retical modeling and computer technology, we are now on the edge of the mi-croscopic description. By taking advantage of modern many-body algorithms,one can now shorten the cycle theory*experiment*theory. Most many-bodymethods of theoretical nuclear structure were introduced a long time ago (inthe fifties and sixties). However, thanks to incredible progress in hardware andnumerical techniques, it is only now that we can use these methods in theirfull glory. Some spectacular examples of modern-day many-body calculationsare 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-shellproperties but have been used in few-body and many-body calculations. Prob-ably the most advanced few-body calculations today are the Green’s functionMonte Carlo calculations with the Argonne-Urbana interaction for nuclei withA57 (Ref. 26). T he variational Monte Carlo calculations with a free NNforce have been carried out for relatively heavy systems such as 160 27. Otherab initio methods which are currently undergoing a renaissance include the

12

coupled-cluster (or “exp(S)“) method 28~2g~30 (see Refs. 31,32 for recent appli-cations) and the antisymmetrized molecular dynamics 33*34.

bare NN force *effective force *experimentLiiffEgl vexpenment I, effective force

Electromagnetic Form Factor in 6Li.Ar@~nne v,,+IJrbana IX force

I;=iEj”

Variatio& Monte Carlo__ .-- ._” --”

Neutmn N u m b e r

Neutronseparationenergies inspherical nucleiSkyme forces

SkF i SkM’,Hattree-Fock-

Bo~ol~~bw.

Figure 8: Examples of advanced nuclear structure many-body calculations. Left: VariationalMonte Carlo calculations with the Argonne-Urbana bare interaction for the electromagneticform factor of 6Li 35. Right: Shell Model Monte Carlo description of the Gamow-Tellerstrength in the fp nuclei 36. Bottom: Hartree-Fock-Bogolyubov calculations for two-neutronseparation energies in heavy spherical nuclei 37. Thanks to new-generation computers andnovel numerical techniques, significant progress in the nuclear many-body problem has beenachieved.

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

13

lence nucleons3g340 (see Fig. 9). Hence, traditional shell-model calculations forheavy nuclei appear to be out of reach in the near future, although the conven-tional shell-model calculations employing realistic NIV interactions41~42*43~44~45are 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 nucleiin the full, OtW space 3g set the new standard in this area, although futureprogress is strongly limited by present-day computer resources. (Actually, ittook two generations of hardware and software development to extend shell-model calculations from A=44 to A=4846!)

Figure 9: Dimension of the shell-model configuration space for even-even N=Z (2012<40)nuclei. It is assumed that the shell model calculations are performed in the full pf shell inthe A4 scheme (here M=O) (courtesy of T. Otsuka).

One actively pursued alternative is to truncate the configuration space byapplying the projected self-consistent quasi-particle basis 47~48. Another fam-ily of novel shell-model techniques is based on the Monte Carlo method 4g150.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 thedimensions of the model space reach 102’.

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 withmany valence particles, the concept of valence nucleons is less useful, and thevalence and inner-shell nucleons have to be treated on an equal footing. Manyproperties of these heavy systems are well described by means of self-consistenttheories with the density-dependent effective NN interaction. Here, the staticmean-field description, based on the HFB method, or relativistic mean-field(RMF) theory, provides a useful starting point. Thanks to developments incomputational techniques, the approaches employing microscopic effective in-teractions are now widely used and - in terms of their predictive power - favor-ably compare with results of more phenomenological macroscopic-microscopicmodels. Examples of recent large-scale self-consistent mean-field calculationsinclude HF 54, HFB 55, and RMF 56 studies of ground-state nuclear properties.Figure 10 displays two-neutron separation energies for the Sn isotopes calcu-lated in several state-of-the-art models based on the self-consistent mean-fieldtheory: HFB-DlS (based on the finite-range Gogny interaction DlS 57), HFB-SkP and HFB-SLy4 (based on zero-range Skyrme parametrizations SkP 58 andSLyC 5Q), LEDF (local energy-density functional model with parametrizationFaNDFO 60), and RHB-NL3 (relativistic Hartree-Bogolyubov model with NL3parametrization61). All these models nicely describe the existing experimentaldata; some interesting deviations are seen when approaching the proton dripline.

Among other recent developments in theoretical nuclear structure, partic-ularly important is the elegant explanation of the pseudo-spin symmetry of thenuclear single-particle spectra proposed thirty years ago 62,63. Surprisingly, asdemonstrated by Ginocchio 64, the roots of the pseudo-spin can be traced backto the symmetry of the Dirac equation (see also Refs. 65166). Consequently, therelativistic approach explains, at the same time, the depth of the average po-tential (around 50 MeV), the magnitude of the spin-orbit term, and the smallpseudo-spin-orbit splitting. In this context, it should be noted that the tra-ditional picture of nuclear shells and magic gaps proposed fifty years ago67@should not be taken for granted. As discussed in Sec. 4.1 below, modificationsof 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 suchfrontiers is physics at subfemtometer distances where the internal quark-gluon

15

1 0 , (I I I fit II I L 11 11

5 0 60 70 8o N

Figure 10: Two-neutron separation energies, S2,, , for the Sn isotopes calculated in five self-consistent models: HFB-Dl (courtesy of J. DechargB), HFB-SkP and HFB-SLy4 (courtesyof J. Dobaczewski), LEDF (courtesy of S. Fayans), and RHB-NL3 (courtesy of G. Lalazissis).The experimental data are indicated by stars.

structures of nucleons overlap. For heavier nuclei, the frontiers are defined bythe extremes of (i) the N/Z ratio (Sec. 4.1), (ii) atomic charge and nuclearmass (Sec. 4.2), and (iii) angular momentum (Sec. 4.3). The tour to the limitsis not only a quest for new and unexpected phenomena (sometimes dubbedas a “fishing expedition”), but the new data are expected as well to bringqualitatively new information about the effective NN interaction and henceabout the fundamental properties of the nucleonic many-body system. Byexploring exotic nuclei, one can magnify certain terms of the Hamiltonianwhich are small in “normal” nuclei and thus difficult to test. The hope is thatafter probing these important interactions at the limits, we can later improvethe description of normal nuclei.

16

4.1 Limits of Extreme N/Z Ratios

One of the main frontiers of nuclear structure today is the physics of radioac-tive nuclear beams (RNB). One of the indications of the potential of this fieldis the large international interest 6Q~70~71~72. At present there are only a fewlaboratories with radioactive ion beam capabilities. However, the prospects

for new experiments and the success of the current programs have led to anumber 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) nu-clear astrophysics (the origin of the Universe), and (iii) physics of fundamentalsymmetries (tests of the Standard Model).

The nuclear landscape (the territory of the RNB physics) is shown inFig. 11. Black squares indicate stable nuclei; there are less than 300 stablenuclei, or those long-lived, with half-lives comparable to or longer than theage of Earth. Some of the unstable nuclei can be found on Earth, some areman-made, and several thousand nuclei are the yet-unexplored exotic speciesbelonging to nuclear “terra incognita”. Moving away from stable nuclei byadding either protons or neutrons, one finally reaches the particle drip lineswhere the nuclear binding ends. The nuclei beyond the drip lines are unboundto nucleon emission; that is, for those systems the strong interaction is unableto cluster A nucleons as one nucleus. (An exciting question is whether therecan possibly exist islands of stability beyond the neutron drip line. One ofsuch islands, indicated in Fig. 11, is a neutron star which exists thanks togravitation. So far, calculations for light neutron drops have not producedpermanent binding 73~74 .)

The uncharted regions of the (N, 2) plane contain information that cananswer many questions of fundamental importance for science: What are thelimits of nuclear existence? What are the properties of nuclei with an extremeN/Z ratio? What is the effective nucleon-nucleon interaction in the nucleushaving a very large neutron excess? There are also related questions in thefield of nuclear astrophysics. Since radioactive nuclei are produced in many as-trophysical sites, knowledge of their properties is crucial for our understandingof the underlying processes 75.

Experiments with beams of unstable nuclei will make it possible to lookclosely into many aspects of the nuclear many-body problem. Theoretically,exotic nuclei represent a formidable challenge for the nuclear many-body the-ories 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

pairing1 K-!Q

stable or long-lived terra incognita

known nuclei

\ ’superallowed

50 --

beta decays 5 0 -+ -.r

neutron stars

28\ -w . . . . . . . . . . -b

halos _ nm+rnnc

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

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

Theoretical Aspects of Physics at Extreme N/Z Ratios

From a theoretical point of view, spectroscopy of exotic nuclei offers a uniquetest of those components of effective interactions that depend on the isospin

76degrees of freedom . Effective interactions used to describe heavy nuclei areusually approximated by means of density-dependent forces with parameters

18

that are usually fitted to stable nuclei and to selected properties of the infinitenuclear matter. Hence, it is by no means obvious that the isotopic trends farfrom stability, predicted by commonly used interactions (such as Skyrme orGogny forces), are correct. In the models aiming at such an extrapolation, theimportant questions asked are: What is the N/Z behavior of the two-bodycentral force and the one-body spin-orbit force 77t78*7g? Does the spin-orbitsplitting strongly vary with N/Z 73? What is the form of the pairing interaction.in weakly bound nucleiso>*‘? 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 themedium effects (renormalization) and of the core polarization in the nuclearexterior (halo or skin region) where the nucleonic density is smalls2?

Exotic nuclei are wonderful laboratories to study superfluid correlations.Here, the main questions pertaining to the problem of pairing force are: Whatis the microscopic origin of the pairing interaction83y84? What is the role of fi-nite range and the importance of density dependence80y85? How can propertiesof the pairing force be tested? These questions are of considerable importancenot only for nuclear physics but also for nuclear astrophysics and cosmology86~87,88~8g. For instance, a better understanding of the density dependence ofthe nuclear pairing interaction is important for theories of superfluidity in neu-tron stars. It is impossible at present to deduce the magnitude of the pairinggaps in neutron stars with sufficient accuracy. Indeed, calculations of ISa pair-ing gaps in pure neutron matter, or symmetric nuclear matter based on freeNN 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 ratherlarge values of the pairing gap at saturation!

Nuclear life at extreme N/Z ratios is different from that around the stabil-ity line. The unique structural factor is the weak binding; hence the closenessto the particle continuum. For weakly bound nuclei, the Fermi energy lies veryclose 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 explicitcoupling between bound states and continuum, and the presence of low-lyingscattering states invite strong interplay and cross-fertilization between nuclearstructure and reaction theory.

How do well-established microscopic models of nuclear structure performwhen extrapolated to exotic nuclei? Of course, there are many uncertainties.As an example, Fig. 12 shows the predicted two-neutron separation energiesfor the tin isotopes using the same models as in Fig. 10. Clearly, the differences

19

90 100 110 120 NFigure 12: Similar to Fig. 10 except for very neutron-rich Sn isotopes.

between forces are greater in the region of “terra incognita” than in the regionwhere masses are known. As seen in Fig. 12, the position of the neutron dripline for the Sn isotopes slightly depends on the effective interaction used; itvaries between N=120 (HFB-DlS) and N=126 (RHB-NL3). Therefore, theuncertainty due to the largely unknown isospin dependence of the effective forcegives 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 whichof the forces discussed should be the preferred one since one is dealing withdramatic extrapolations far beyond the region known experimentally. However,a detailed analysis of the force dependence of results may give us valuableinformation on the relative importance of various force parameters.

Recent research relating to exotic nuclei has already demonstrated thepotential for exciting new nuclear physics. Some examples are shown in Fig. 11.They include the exotic structure of halo nuclei, the surprising fragility ofmagic numbers which hints at some of the marked changes in the uhderlying

20

foundations of nuclear structure, and the production and study of the long-searched doubly closed shell nuclei 78Ni and loo%. In the following, I shallbriefly 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 neutrondrip-line region is driven not only by the substantial uncertainties in theoret-ical predictions of its location, but also by the expectation that qualitativelynew features of nuclear structure will be discovered in this exotic territory.What makes neutron-rich nuclei so unusual? Firstly, they have very largesizes - as implied by their weak binding. Secondly, they are very diffused;their properties are greatly dominated by surface effects. Thirdly, they arevery superfluid; the close-lying particle continuum provides a giant reservoirfor scattered neutron Cooper pairs 58y80. So, roughly speaking, they are large,fuzzy superfluids.

Extreme cases are halo nuclei - loosely bound few-body systems with aboutthrice more neutrons than protons go They are symbols of RNB physics. The.halo region is a zone of weak binding in which quantum effects play a criticalrole in distributing nuclear density in regions not classically allowed. Much hasbeen learned about halo nuclei. Few-body calculations, especially those basedon the method of hyperspherical harmonics and the adiabatic hypersphericalmethod, have reached a high level of sophisticationg11g2. They give the basicunderstanding of many observed phenomena (momentum distributions, elec-tromagnetic strength distributions) but, often, neglect the important structureaspects. Some of the questions asked in the context of halo nuclei are: Whatare the main dynamical degrees of freedom which affect the core-halo coupling?What is the degree of clusterization of the cores.7 Is the clusterization mecha-nism enhanced close to the neutron drip line? If so, what are its manifestationsin heavier systems? What are the modifications of the effective interaction inthe halo region .827 Through halos, one hopes to learn more about the mi-croscopic mechanism of clusterization found in “normal” nuclei (see the Ikedadiagram shown in Fig. 13), such as in the hypothetical three-alpha state of12C.

In the heavier, neutron-rich nuclei, where the concept of mean field isbetter applicable, the separation into a “core” and “valence nucleon& seemsless justified. However, also in these nuclei the weak neutron binding impliesthe existence of the neutron skin (i.e., a dramatic excess of neutrons at largedistances). In the skin region of heavier, very neutron-rich nuclei, one mayfind the opportunity to study in the laboratory nearly pure neutron matter at

21

(7.16) (11.89) (21.21)

I

Mass number

Figure 13: Ikeda diagram for light nuclei Q3. The threshold energy for each decay mode (inMeV) is indicated. The halo nucleus “Li, viewed as a three-body borromean system, is alsoshown.

densities much less than the normal nuclear density. In addition, the existenceof a neutron skin should lead to new collective vibration modes in such nucleiin which, for example, the neutron skin may oscillate out of phase with awell-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 NN interaction is largely un-known, the structure of single-particle states, collective modes, and the behav-ior of global nuclear properties is very uncertain in nuclei with extreme N/Zratios (see Fig. 12 and related discussion).

Due to the systematic variation in the spatial distribution of nucleonic den-sities and the increased importance of the pairing field, the average nucleonic

22

g4potential is modified . This results in a new shell structure characterized bya more uniform distribution of normal-parity orbits and the unique-parity in-truder orbit which reverts towards its parent shell 76,g5,g4. According to othercalculations g6, a reduction of the spin-orbit splitting in neutron-rich nucleiis expected. The gradual change of shell structure with neutron number isbelieved to give rise to new sorts of collective phenomenag7yg8.

24

20

16

128

4

50

z2 24m

20

16

12

8

4

0SkP - N=80- N=82

z ;g i

0181, III,,

62 54 46 38

Proton Number

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

The quenching of shell effects manifests itself in the behavior of two-neutron separation energies Szn. This is illustrated in Fig. 14, which displaysthe two-neutron separation energies for the N=80, 82, 84, and 86 sphericaleven-even isotones calculated in the HFB model with the SkP and SLy4 effec-tive interactions. The large N=82 magic gap, clearly seen in the nuclei closeto the stability valley and to the proton drip line, gradually closes down whenapproaching the neutron drip line.

23

It is to be noted that the experimentally observed collapse of magic gapsseen in some neutron-rich light nuclei (the so-called islands of inversion, seeRefs. 1oo*101~102~103) might also be related to the predicted quenching of magic

lo4gaps . The effect of the weakening of known shell effects in drip-line nucleihas significant consequences for the description of the astrophysical r-process.Although. many of the astrophysically important neutron-rich nuclei, belong-ing to the terra incognita of Fig. 11, will not be experimentally accessible inthe foreseeble future, their part of the (2, N) chart has already been visitedby Mother Nature in cataclysmic events such as supernovae. Consequently,valuable information on nuclear structure in terra incognita can be offered byastrophysical data. In particular, the analysis of the r-process abundancesprovides an indirect experimental confirmation of the shell-quenching effect105,99

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

On the proton-rich side of the valley of stability, physics is different than innuclei with a large neutron excess. Because of the Coulomb barrier whichtends to localize the proton density in the nuclear interior, nuclei beyond theproton drip line are quasibound. However, in spite of the stabilizing effectof the Coulomb barrier, the effects associated with the weak binding are alsopresent in proton drip-line nuclei. They are not as dramatic as on the otherside of the stability valley, but nevertheless important 76.

A unique aspect of proton-rich nuclei with N=Z is that neutrons andprotons occupy the same shell-model orbitals. Consequently, due to the largespatial overlaps between neutron and proton single-particle wave functions,the proton-rich N=Z nuclei are expected to exhibit unique manifestationsof proton-neutron (pn) pairing 1o6Jo7 carried by isotropic (S=O, T=l) andanisotropic, doughnut-shaped lo8 (S=l, T=O) proton-neutron Cooper pairs.

At present, it is not clear what the specific experimental fingerprints ofthe pn pairing are, whether the pn correlations are strong enough to form astatic pair condensate, and what their main building blockslog are. So far, thestrongest evidence for enhanced pn correlations around the N=Z line comesfrom the measured binding energies. An additional binding (the so-calledWigner energy) found in these nuclei manifests itself as a spike in the isobaricmass parabola as a function of TZ=$(N-Z) lloJ1l. Recent calculations 112have revealed a rather complex mechanism responsible for the nuclear bindingaround the N=Z line. In particular, it has been found that the Wigner termcannot 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 playa role in beta decay 113,114, deuteron transfer reactions ii5, structure of highspins 107~116*117, and also in nuclear matter 88,118.

Proton-rich nuclei also offer a unique opportunity to study life beyondthe drip line. Although the protons in this region are not strictly bound tothe 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 enoughthat one can measure their spectroscopic properties llgJ20. Experimentally, anumber of proton emitters have now been discovered in mass regions A-110,150 and 170121. However, it is anticipated that new regions of proton-unstablenuclei will be explored in the near future using RNB’s.

Proton emission from deformed nucle

0% Z+U (N, 23

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

Proton radioactivity is an excellent example of the elementary 3D tunnel-ing. Experimental and theoretical investigations of proton emitters (or the-oretically predicted ground-state di-proton emitters) will open up .a wealthof exciting physics associated with the residual interaction coupling betweenbound states and extremely narrow resonances in the region of very low densityof single-particle levels. In general, proton emission half-lives depend mainly onthe 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 theparameters of the proton-core potential. This suggests that the lifetimes ofdeformed proton emitters will provide direct information on the angular mo-mentum content of the associated Nilsson state, and hence on the nuclear shape122y123,124. Figure 15 shows, schematically, a possible scenario of proton-gammacompetition in a deformed proton emitter. Of course, the energy window forsuch a process is expected to be fairly narrow. A slightly different phenomenon,gamma-delayed proton emission, has been recently observed in 58Cu 125 and56Ni 126 (see also discussion in Sec. 4.3). Here the strongly deformed excited in-truder bands gamma-decay to the lower-lying states, but their proton-unstableband 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 berapid. Especially for the neutron-rich heavy elements, it will be a real struggleto obtain basic spectroscopic information. To put things in perspective, it isinstructive to recall the discovery of polonium by Maria and Pierre Curie atthe end of June 1898. The longest-lived isotope of polonium is 2ogPo with ahalf-life of 102 y. The isotope 218Po had already been studied by Rutherfordin 1904 127; its half-life is 3.1 min 12’ Amazingly, it took ninety-four years to.add one more neutron to 218Po; only recently have the neutron-rich isotopes21gy220Po been produced at GSI by the international collaboration led by M.Pfiitzner 12s. The lightest polonium known, lgoPo, was found in 1988 at GSIi3’; its half-life, 2.4 ms 131, is long enough to enable detailed spectroscopicstudies.

The doubly magic N=Z=50 nucleus looSn is a paradigm of RNB physicson the proton-rich side. Although it was found experimentally three years ago132y133, it took more than two years to roughly determine its mass134, and it willstill 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 approachedspectroscopically in recent years 135, so the prospects for more spectroscopicnews on looSn are good. It will be much harder to approach another doublymagic nucleus, the neutron-rich 78Ni. It was produced in 1995 136, but ourknowledge about this system and its neighbors is very scarce. Indeed, theheaviest nickel isotope known spectroscopically is 70Ni 137 which is as far aseight neutrons away!

An experimental excursion into uncharted territories of the chart of thenuclides exploring new combinations of 2 and N will offer many excellentopportunities for nuclear structure research. What is most exciting, however,

26

is that there are many unique features of exotic nuclei that give prospects forentirely new phenomena likely to be different from anything we have observedto 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 concernsthe maximum charge and weight that the nucleus may attain. Amazingly,even after thirty-or-so years of our quest for superheavy elements, the bordersof the upper-right end of the nuclear chart are unknown. Theoretically, themere existence of the heaviest elements with 2>104 is entirely due to quanta1shell effects. Indeed, for these nuclei the classical nuclear droplet, governed bysurface tension and Coulomb repulsion, fissions immediately due to the hugeelectric charge. At the end of the sixties, a number of theoretical predictionswere made that pointed towards the existence of long-lived superheavy (SHE)nuclei 138,13s~140~141. However, it is only during recent years that significantprogress in the production of the heaviest nuclei has been achieved 142y143. No-tably, three new elements, Z=llO, 111, and 112, were synthesized by means ofboth cold and hot fusion reactions 144,145,1461147. These heaviest isotopes decaypredominantly by groups of cy particles (0 chains) as expected theoretically148,149. (Th e experimental a-chain that identifies the element 277112 is shownin Fig. 16.)

All the heaviest elements found during recent years are believed to be welldeformed. Indeed, the measured a-decay energies, along with complementarysyntheses of new neutron-rich isotopes of elements 2=106 and 2=108, havefurnished confirmation of the special stability of the deformed shell at N=162predicted by theory 1501151. Still heavier and more neutron-rich elements areexpected to be spherical and even more strongly stabilized by shell effects.

Beautiful experimental confirmation of large quadrupole deformations inthis mass region comes from gamma-ray spectroscopy. In recent experimentalwork, Reiter et al. 154 succeeded in identifying the ground-state band of 254N~(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 254N~. The quadrupole deformation of 254N~ can be inferredfrom the energy of the deduced 2+ state. The resulting value, ,&x0.27, isin nice agreement with theoretical predictions (see Fig. 17, bottom,. and alsoRefs 152*155~156,157 and references quoted therein). As a matter of fact, thenucleus 254 No has probably the largest ground-state quadrupole deformationin the whole actinide-transfermium region.

Figure 16 displays the shell energy calculated in the HFfSly7 model of

27

Synthesis of New Elements

126-

108

170 176 182 188

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 thatidentifies the element with A=277 and Z=ll2 147 ’1s indicated together with the currentlyobserved 2~114 a-chains corresponding to isotopes with A=289 and 287 153. Still heaviersuperheavy 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 mostmacroscopic-microscopic approaches where the first island of shell stabilityin the superheavy region is concentrated around 2=114. The HF predictionof Fig. 16 is consistent with results of other self-consistent studies based onnon-relativistic and relativistic mean-field theories 158,157, which predict thespherical neutron closure at N=184 and the proton gap at 2=120 or 126.In this context, the recent result by the Dubna/Livermore collaboration, whoreported the synthesis of two isotopes of the 2=114 element in the 48Ca+244Puand 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

“8Pb(%a,2n)254No E,,=215MeV o=3.4p.b

F t,wX-nw GAMMASPHERE and FMA20

gt 15

z10

254No gsb transitions

5

0 100 200 300 400 500 600

q WV)\ quadrupole deformation

P2

130 } HF+\BCS+SLv4+60.28

N 112 t

106 t

1001 m0.08

1

1

0.04^ ^^

N

Figure 17: Top: Gamma spectrum corresponding to the ground-state band of 254N~ obtainedin the recent GammaSphere-FMA work at ATLAS 154. Bottom: Contour map of the ground-state quadrupole deformation p2 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 ofrotational motions in the Universe. Figure 18 displays, in a log-log scale, thecharacteristic rotational frequency as a function of the characteristic size ofthe rotating body 160.

1030-

i 1(-p-<g lOlO- sidereal rotation

a2 100 -Id

lo-lo- daneb

1020 1o1O 100 10-10 10-20

typical size (cm)

Figure 18: Typical rotational frequency w (in set-l) versus characteristic dimension of a160rotating object L . Interestingly, the data spanning around forty orders of magnitude in

L and w 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. Thelargest bodies shown in Fig. 18 are galaxy clusters, with a typical period ofrotation of 1017 sec. The pulsars, with their dimensions of several km, are,considering their huge masses, amazingly fast; they are wonderful laboratoriesof high-spin nuclear superfluidity 16r . One of the dizziest mechanical man-madeobjects is an ultra-centrifuge used for isotope separation. The Oak Ridge gascentrifuge could make -lo4 rotations/set ‘s2 but there exist even faster ones.,

30

Molecules offer many beautiful examples of a very rigid quantum-mechanicalrotation. Discrete molecular states with angular momenta of the order ofI=80 A are measured routinely today (see, e.g., Ref. 163 for rotational bands in120Sn160).

The smallest objects shown in Fig. 18 are hadrons and nuclei. Some excitedstates of hadrons can be interpreted as rotational bands. The moment of inertiaof the nucleon is Jz20-25 MeV/fi2 164; the corresponding rotational frequenciesare of the order of fw=O.5 GeV (T-lo-24 set). In this company, atomic nuclei,with their typical dimensions of several fm and rotational frequencies rangingfrom 102’ to 1021 Hz, are champions of rotational motion. What makes nuclearrotation so special is the overwhelming presence of the Pauli principle, quanta1shell 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 lowestrotational excitations and introduced the notion of the nuclear moment ofinertia. More than sixty years later, nuclear high-spin physics is a maturefield. We have learned a lot about the basic mechanisms governing the be-havior of fast nuclear rotation, especially the interplay between collective andnon-collective degrees of freedom, competition between rotation, deformation,nuclear superconductivity, and many global and detailed aspects of high-spingamma-ray spectroscopy. However, a lot of shocking surprises are still beingencountered. The new-generation experimental tools, such as multidetectorarrays EUROBALL and GAMMASPHERE, combined with the new-generation par-ticle detectors and mass/charge separators, enable us to study discrete nuclearstates up to the fission limit as well as a high-spin quasi-continuum, and explorenew limits of excitation energy, angular momentum, and energy resolution. In-deed, these very precise tools made it possible to probe very small effects on akeV scale. New high-quality data tell us more about the structure of intrinsicwave functions of high-spin states.

Advances in high-resolution gamma-ray detector systems are also respon-sible for a revolution in our study of low-spin nuclear behavior. Here, newinsights have been gained on the nature of collective nuclear vibrations. Inparticular, long-searched vibrational multiphonon states have been found ex-perimentally. The existence of these very harmonic modes of nuclei, bothat high lo and low frequencies no, challenges our understanding of the Pauliprinciple.

Figure 19 displays, schematically, the territory of high-spin science. Thepreferred pathways in the de-excitation process relate to favorable arrange-ments of protons and neutrons and can often be associated with specific nucleardeformations. Some spectacular examples of intrinsic shapes are illustrated in

31

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

,-/. 5Gig - ‘94’H

\ manetic

/Angular Momentum

Figure 19: Territory of nuclear high spins. Some of the important physics bullets are indi-cated schematically (from Refs. 11,167).

A great challenge for nuclear theory is the existence of very regular ro-tational “shears” bands (with rather constant moments of inertia) in nucleiwhich are spherical. These and other exciting high-spin phenomena have beenrecently reviewed in a beautiful booklet 167. Discussed below are some of thebullets shown in Fig. 19.

32

Superdeformation: Spectroscopy at a keV Scale

The observation of SD states constitutes an important confirmation of theshell structure of the nucleus. Quantum-mechanically, the remarkable stabilityof SD states can be attributed to strong shell effects that are present in theaverage nuclear potential at very elongated shapes 16s~169Y170. For the oscillatorpotential, this happens when the frequency ratio is 2:l. (For more realisticaverage potentials, strong shell effects appear even at lower deformations.)The structure of single-particle states around the Fermi level in SD nuclei issignificantly different from the pattern at normal deformations. Indeed, theSD shells consist of states originating from spherical shells having differentprincipal 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. Theseare fission isomers in the actinides; high-spin bands around 152Dy (Ref. 171),lg2Hg (Ref.172), 83Sr (Ref. 173), and 62Zn (Ref. 174); and “molecular” (cluster)configurations in light nuclei 175. Intrinsic configurations of SD states are wellcharacterized by the intruder orbitals carrying large principal oscillator num-bers 176,177. Because of their large intrinsic angular momenta and quadrupolemoments, these orbitals strongly respond to the Coriolis interaction and to thedeformed 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 phenomenais that of identical bands, i.e., the observation of sequences of ten or moreidentical gamma rays associated with rotational bands in different nuclei 17*.Another recent discovery in SD nuclei is the observation of very small (tensof eV!), but systematic, shifts in the energy levels of certain bands (Al=2staggering) 17g*180.

A satisfactory explanation of both phenomena is still lacking178 (see alsorecent Refs. 181y182). It is rather clear that the puzzle of identical bands isultimately related to the questions normally addressed in the context of large-amplitude collective motion such as: What are the “strong” quantum numbersthat stabilize collective rotation? Indeed, recent studies show that nuclearsystems at very large deformations and high angular momenta are the bestexamples of an almost-undisturbed single-particle motion (i.e., extreme one-body picture 183y184).

A spectacular example of today’s superdeformed spectroscopy is the inves-tigation of Ref. is5 where as many as thirteen superdeformed bands have beenfound 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-particleexcitations in the rotating SD well through their electric and magnetic prop-erties. This has been accomplished by measurements of intraband magnetictransitions 186~187 and relative electric quadrupole moments 188J89~1go.

Magnetic Rotation

In nearly spherical nuclei, sequences of gamma rays were observed reminis-cent of collective rotational bands, but consisting of unusually strong mag-netic dipole transitions M. The basic mechanism leading to the unusuallylarge magnetic collectivity is explained theoretically lg2 in terms of a gradualalignment of valence protons and neutrons along the axis of the total spin(shears mechanism). Since quadrupole deformations of these bands are ex-tremely small, the rotational-like pattern observed cannot be attributed to therotation of the electric charge distribution. It is believed that the collectiveproperties of shears bands are linked to magnetic-type static intrinsic defor-mations. However, the exact nature of the intrinsic magnetic fields responsiblefor 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 aunique possibility to probe quantum chaos, roughly defined as a regime wherethe quantum numbers that may be used to characterize low-lying (cold) statesof a many-body system are gone. Today, the important issues under study inthe so-called “complete spectroscopy” experiments are: At what energy doeschaos set in? What are the unique fingerprints of the transition from regular tochaotic motion? The signatures for the onset of chaos can be observed not onlyin the distribution of energy levels but also in the properties of electromagnetictransition intensities.

Progress in gamma-ray spectroscopy has resulted in the discovery of dis-crete lines linking superdeformed bands to low-deformation states 1g3*194. Thishas allowed the determination of the excitation energies and spins of superde-formed states, and also the mixing between the superdeformed states and thenearly spherical configurations in the first well. Although the general prin-ciples seem to be under control, the satisfactory microscopic understandingof the complicated multidimensional tunneling, resulting in a decay out of asuperdeformed band, is still lacking. The dynamics of a metastable state is apart of a multidisciplinary field of the large-amplitude collective motion (whichincludes 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 importanttasks for experiment and theory is to identify the islands of stability at highexcitation energies, and to understand the underlying quantum numbers, thatis, 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 areproton-emitting intruder bands in 58Cu 125 and 56Ni 126. In the doubly magicnucleus 56Ni, where two intruder bands have been observed, the lower rota-tional band can be explained by large-scale shell-model calculations in the pfshell. The results of cranked mean-field calculations indicate that this bandis built upon a 4p-4h excitation (see Fig. 20, 4O4O band). The second band,however, is expected to involve particles in the 1g9/s orbit, which is supportedby its nearly identical behavior to the band in 58Cu. However, the best sce-nario for this configuration involves only one proton promoted to the 1gsj2orbit (4O4l band in Fig. 20), while a neutron and a proton occupy this orbitin .58C~l Radioactive medium-mass nuclei such as 56Ni are fantastic territorieswhere various theoretical approaches can be confronted: state-of-the-art shellmodel, mean-field models, and cluster models. The spherical structures in 56Niare well described by the large-scale shell model, the collective 4’4l band canbe 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 dealof theoretical interest and activity. As a result, many new insights have beengained regarding the behavior of atomic nuclei at high spins.

The new-generation high-spin data tell us a lot about effective NN inter-action, and about pairing force in particular. For instance, self-consistent HFstudies of moments of inertia of SD bands indicate the importance of higher-order pairing interactions such as the rotation-induced K”=l+ pairing forceslg5,1g6 and the density-dependent pairing forces 197~1g8*1g9. Interestingly, thequestion of the density dependence of pairing interaction is also of great in-terest for physics of nuclear radii, deep hole states, and properties of drip-linenucleigo (see Sec. 4.1). That is, this important problem has also surfaced in adifferent corner of nuclear structure.

35

16

Figure 20: Excitation energy versus angular momentum for experimental and calculatedintruder bands in 56Ni. Lines indicate the 4O4’, 4O4l, and 4l4l bands calculated in thecranked HF+SLy4 model, while the KBF shell-model calculations (SM) are indicated bycrosses (from Ref. 126).

In the microscopic theory of a rotating nucleus: I-he average nucleonic fieldis obtained self-consistently from the nucleonic density. Consequently, in thepresence of a large angular momentum, the intrinsic density is strongly polar-ized, i.e., the nucleus shows phenomena and behaviors characteristic of con-densed matter in the magnetic field: ferromagnetism, Meissner effect, Joseph-son effect. The nuclear magnetism is caused by the time-odd components inthe average potential 2oo~201~202. The understanding of the structure of theseterms will rely mostly on the expected increase of information on high-spinproperties and on magnetic properties. Here, a good example is recent cal-culations based on a RMF theory; as a consequence of broken time-reversalinvariance, the spatial contributions of the vector meson fields have a stronginfluence on the moments of inertia ‘03.

36

5 Nucleus as a Finite Many-Fermion System

The atomic nucleus is a complex, finite many-fermion system of particles in-teracting via a complicated effective force which is strongly affected by themedium. As such, it shows many similarities to other many-body systemsinvolving many degrees of freedom, such as molecules, clusters, grains, meso-scopic rings, quantum dots, and others. There are many topics that are com-mon to all these aggregations: existence of shell structure and collective modes(e.g., vibrations in nuclei, molecules, and clusters; superconductivity in nucleiand grains), various manifestations of the large-amplitude collective motion(such as multidimensional tunneling) and nonlinear phenomena (many-fermionsystems are wonderful laboratories to study chaos), and the presence of dy-namical symmetries.

Historically, many concepts and tools of nuclear structure theory werebrought to nuclear physics from other fields. Today, thanks to the wide arse-nal of methods, many ideas from nuclear physics have been applied to studiesof other complex systems. There are many splendid examples of such in-terdisciplinary research: applications of the nuclear mean-field theory and itsextensions to studies of static and dynamical properties of metal clusters, treat-ment of finite-size effects in the description of superconductivity of ultrasmallgrains, use of symmetry-dictated approaches to describe collective excitationsof complex molecules, applications of the nuclear random matrix theory to var-ious phenomena in mesoscopic systems (nuclei and quantum dots can be usedas complementary laboratories to study transport properties and the onset ofchaos in Fermi systems), and the description of short- and long-range correla-tions in many-fermion systems (nuclear theorists have pioneered the study ofquantum helium liquid droplets by exact quantum Monte Carlo methods; thenuclear shell model has been applied to the treatment of electron correlationsin solids). The count can go on, of course. Given below are several examplesof nuclear physics research which goes well beyond the frames of the nuclearmany-body problem.

5.1 Large-Amplitude Collective Motion

The question of whether the motion of a finite many-body quanta1 system canbe divided into a slow collective motion and a fast single-particle motion isolder than nuclear physics itself. As early as 1927, Born and Oppenheimer*04 successfully described molecular motion by assuming that fast electronswere strongly coupled to the equilibrium position of the slow and much heav-ier ions. However, it was a decade later when Jahn and Teller discovered *05that 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 astable symmetry-breaking deformation provided the coupling between degen-erate electronic excitations and collective motion is strong. This phenomenonis usually referred to as the spontaneous symmetry-breaking or the Jahn-Tellereffect *06**07

It was quickly recognized that the spontaneous symmetry-breaking wasnot exclusively related to molecular physics. It is a universal phenomenonknown in all areas of physics: field theory (Higgs mechanism), physics of su-perconductors (Meissner effect), atomic, and nuclear physics. The commondenominator is the existence of an intrinsic deformed system determined bythe instantaneous position of slow degrees of freedom. The importance ofsymmetry-breaking in nuclear physics was noticed early by Hill and Wheeler208 and by Bohr 2og. The unifying theme is the concept of (self-consistent)mean field and its intrinsic system determined by the instantaneous positionof slow degrees of freedom. The static mean-field equilibria are characterizedby self-consistent (dynamical) symmetries or “effective” quantum numbers, re-sponsible for the near-equilibrium (local) motion of the system. It should beemphasized that the choice of proper nuclear collective coordinates is a long-standing problen?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, thereare here no heavy particles to provide the necessary rigidity of the structure.However, nuclear matter appears to have some of the properties of coherentmatter which makes it capable of types of motion for which the effective massis 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 presenceof some symmetry. Such a degeneracy leads to a reduced stability, and evenan infinitely small perturbation produces a violent transition to a state that isdeformed. Or, in simple words, if a nucleus can remove degeneracy by meansof a symmetry breaking, it will always go for it!

Hence the symmetries are important. They provide necessary initial con-ditions for the existence of collective modes and moments which allow us tounderstand complex systems simply. The atomic nucleus is a very rich andbeautiful laboratory of these. The very existence of nuclear rotational statestells us that in many situations the nucleus prefers to choose a deformedshape over the (more symmetric!) spherical shape. Spectacular examplesof symmetry-breaking are static reflection-asymmetric pear-like deformationsfound in certain’nuclei. These systems exhibit features familiar from molec-ular physics: alternating parity bands, intrinsic dipole moments, and paritydoublets (see Fig. 21). The exciting aspect of intrinsic symmetry-breaking is

38

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

r

L

238 U =‘Ra

Figure 21: For most deformed nuclei, like 23sU, a description as an axially and reflectionsymmetric spheroid is adequate to describe the band’s spectroscopy. Because such a shapeis symmetric with respect to the space inversion operation, all members of the rotationalband will have the same parity. However, it has been found that some nuclei might havea shape that is asymmetric under reflection, such as a pear shape. The rotational band of220Ra consists of levels of both parities, hence the “molecular” name parity doublet. Thelevels having the same parity are connected by E2 photons while the states of opposite parityare connected by El photons. The enhanced El transitions are characteristic of an intrinsicshape having an electric dipole moment, i.e., the shift between the center of charge and thecenter-of-mass of the nucleus.

39

The ultimate test of the theories of large-amplitude collective motion isprovided by experimental data. Of particular interest are measurements of ex-cited states built upon coexisting configurations ‘. This kind of experimentalinformation tells us about mixing between stable HF minima and the depen-dence of this mixing on excitation energy, i.e., it addresses explicitly the issueof diabaticity of the nuclear collective motion. A spectacular example of coex-istence between excited states built upon different mean fields is provided bythe excitation spectrum of 184Pt Figure 22 displays the experimental situa-.tion, characteristic of neutron-deficient Pt isotopes. The ground state K”=O+rotational band has a large moment of inertia. Built upon this ground state is aK”=2+ “gamma” band. These two structures coexist with the excited K”=O+and Kn=2+ bands with considerably lower moments of inertia. Mean-field cal-culations z3 predict two minima to coexist in the potential energy surface ofiB4Pt. The prolate minimum, 0.2<&<0.25, can be associated with the groundstate in the Pt isotopes, while the less-deformed oblate minimum, -0.15-&<-0.1, forms an excited intruder O+ state.

Of course, the comparison between experiment and theory, although sug-gestive, cannot be too quantitative, as the static theory does not take intoaccount many important effects such as mixing between states with differentdeformations. (Experimentally, the interaction between prolate and oblatestates in Pt isotopes is around 200-400 keV 214 .) In Fig. 22 the two minimaare separated by a very small barrier (the static path connecting the minimais shown by a thin dashed line) which suggests a very strong shape-mixingeffect - hence an adiabatic situation. However, there are significant differencesbetween the underlying intrinsic configurations. It was noticed long ago 215,that prolate, oblate and spherical configurations in the Pt-Pb region can bewell classified in terms of particle-like proton intruder hgiz and fri2 orbitalscrossing the Z=82 spherical gap.

How to understand the separation between different minima correspondingto different intrinsic configurations? One possible way is suggested by thetime-dependent Hartree-Fock (TDHF) theory. The TDHF wave function canbe represented by a time-dependent Slater determinant 2o

xfij(t)ataj - h.c.i j

where (@e) is the stationary HF state and fij is a time-dependent Thoulessmatrix. It is interesting to note that the TDHF (or TDHFB) equations can be

40

00 0 . 0 5 0 . 1 0 0.15 0.20 0.25 0.30

Figure 22: Shape coexistence in rs4Pt. Top panel shows the two coexisting O+ bands in184Pt 216. The arrows indicate AI=0 transitions between the bands. As can be seen fromthe energy spacings, the lower band is the more strongly deformed. It can be associated withthe prolate-deformed minimum in the calculated potential energy surface (bottom panel) 213.The excited band corresponds to the ablate-deformed secondary shallow minimum.

written in a canonical form

where the complete set of canonical variables {Cij, C&} is obtained from the

41

matrix fij through the variable transformation 217

%= ( ‘i$f},, C,j..= {f+si$}ij. (3)

In Eq. (2) H=H(C, C*)=(*(C, C*)I&l@(C, C*)> represents the classical en-ergy surface of the system in the TDHF coordinate space {CQ, CG}, Equation(2) demonstrates that the TDHF manifold has the same symplectic structureas the classical phase space 218 . As seen in Fig. 23, the nuclear motion is repre-sented by a trajectory in the phase space. The minimum points in the TDHFmanifold are stable points (solutions to the static HF(B) problem). The motionaround the minima is regular; it corresponds to nuclear collective modes, rota-tions and vibrations. These are well classified by means of quantum numbersassociated with the corresponding stable point. The complicated trajectoriesassociated with complex nuclear states form the chaotic sea. Here, the twotrajectories with very similar initial conditions develop differently. Althoughthe barrier separating the two minima in Fig. 22 is small when plotting thepotential energy surface as a function of some selected collective coodinates(here 0 and r), the corresponding structures can be very well separated in thefull TDHF manifold 21g.

The variety of phenomena classified under the name of the large-amplitudecollective motion, such as coexistence, tunneling in complex systems, andchaos, can be found in other fields of physics and chemistry. The related ques-tions 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 nucleusare quite far apart and they very seldom interact with each other. This is dueto 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 awell-defined surface; it can be viewed as a Fermi liquid.

The very existence of magic nuclei is a beautiful consequence of indepen-dent particle motion. Nucleons having identical or similar energies are saidto belong to the same shell, and different shells are separated by energy gaps(see Fig. 24). The way the energy bunchings (called shell structure) occurdepends on the form and the shape of the average potential in which particlesare moving.

The way the energy bunchings, called shell structure, occur depends onthe form and the shape of the average potential in which particles are moving.

42

stable points (HF minima)

regular (collective) motionL

Figure 23: Schematic diagram representing the motion of the nucleus in the TDHF mani-fold. The HF minima are stable points. The collective rotations and vibrations take placearound the minima. The collective submanifold is submerged in the chaotic sea representingcomplicated nuclear modes.

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

43

Pronouncedshell structure

s h e l l

WP

shell

Shell structureabsent

WPshell

0closed 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 clas-sical closed trajectories are counterparts of quantum orbits that are bunched in shells. Theabsence 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 quanta1 thing. Magic gaps, positions, and sizes of

shells are governed by the strict rules of quantum mechanics. But, amazinglyenough, the origin of shell effects can be traced back to the the geometry ofperiodic orbits of the corresponding classical Newton equation1g83221y222J6g.That is, there exists a close link between the nuclear shell structure and thenon-linear dynamics of the classical problem. Indeed, the single-particle leveldensity g(c)=C( 6(~ - Q) can be represented by means of the Gutzwiller trace

44

g(e) = g(f) + 7 F au(f) co8 [k (S~(~)lfi - ;PL)] ,L k=l

(4)

where 3 is the average level density, while the second term in Eq. (4) is thefluctuating part of g responsible for shell effects. The shell energy can then beexpressed as a sum over periodic orbits L. (Sr, is the action integral associatedwith the orbit L and PL is the Maslov index.) Consequently, the shell structureof the many-body system (and hence the presence or absence of deformation)has its deep roots in the non-linear dynamics of the corresponding classicalHamiltonian and the geometry of classical orbitsnjg. The microscopic analysisof the link between the phenomenon of spontaneous symmetry-breaking andthe non-linear dynamics (and chaos) is one of the most vigorously pursuedrecent avenues in theoretical nuclear physics. Among many examples of suchanalyses are the explanation of supershell effects in superdeformed nuclei168J6gand the discussion of shell structure in nuclei with permanent octupole defor-mations224.

The small clusters of metal atoms (typically made up of thousands ofatoms or fewer) represent an intermediate form of matter between moleculesand bulk systems. When the first experimental data on the cluster’s structurewere obtained225, it was immediately realized that the shell-model descriptioncould be applied to valence electrons in clusters 226. Since metal clusters can bemade neutral, there is no limitation to their size, in contrast to nuclei. Magicnumbers corresponding to filled shells up to -3000 electrons have been seen227

The effect of the bunchiness in the energy spectrum can’be illuminated bylooking at the quanta1 shell energy, i.e., the contribution to the total energyof the system resulting from the shell structure. The nuclear shell energy andthe 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 beenused. 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. 25the deformation effect is manifested through the reduction of the shell energyfor particle numbers that lie between magic numbers.

How can one see whether the clusters are spherical or not? The deforma-tion of the clusters can be deduced by studying a collective vibrational mode,the so-called dipole surface plasmon, which corresponds to an almost-rigid os-cillation of the electrons with respect to the positive ion background. This isa direct analog of the nuclear giant dipole resonance which is an oscillation ofprotons with respect to neutrons 230 . The occurrence of the deformation of the

45

NucleiI I I I I I

10 experiment

60 100Number of neutrons

Na ClustersI I

1

Figure 25: Top: experimental and calculated nuclear shell energy as a function of the neutronnumberzzs. The sharp minima at 20, 28, 50, 82, and 126 are due to the presence of nucleonicmagic gaps. Bottom: experimental and calculated shell energy of sodium clusters 22g as afunction 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 thesame nuclear physics technique (macroscopic-microscopic method), assuming the individualsingle-particle motion (of nucleons or electrons) in an average potential. The reduction of theshell energy for particle numbers that lie between the magic numbers is due to deformation(the J&n-Teller eflect).

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

46

properties of collective plasmon states in clusters have been initially predictedby nuclear theorists, and their existence has been subsequently confirmed ex-perimentally. Other applications of nuclear methods to clusters include theapplication of the theory of nuclear rotation to cluster magnetism 22g and theapplication of experimental and theoretical nuclear reaction techniques to clus-

231ter fragmentation . The experience accumulated in nuclear physics plays animportant 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 strongmagnetic field exhibit an unusual collective behavior known as the fractionalquantum Hall effect. At special densities, the electron gas condenses into aremarkable state - an incompressible liquid - while the resistance of the sys-tem 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 ef-fect 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 stateshave a shell structure similar to light nuclei 232.

Nuclear and atomic physicists recognized very early that the behavior ofcomplex many-body systems is often governed by symmetries. Some of thesesymmetries reflect the invariance of the system with respect to fundamentaloperations such as translations, rotations, inversions, exchange of particles,etc. Other symmetries can be attributed to the features of the effective in-teraction acting in the system 21V233~23~25. Although they are more difficultto visualize, these “interaction symmetries” (dynamical symmetries) can oftendramatically simplify the description of otherwise very complicated systems.The natural language of symmetries is mathematical group theory. In thislanguage, dynamical symmetries are associated with a special type of group,a Lie group. These methods have made it possible to describe the structureand dynamics of molecules in a much more accurate and detailed way thanbefore 234. In particular, it has been possible to describe the rotational andvibrational 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 associatedquantum system, a sub-field known as quantum chaos. A nuclear physicstheory, random matn’x theory, developed in the 1950% and 60’s to explain

47

the statistical properties of the compound nucleus in the regime of neutronresonances 235, is now used to describe the universality of quantum chaos. Atthe excitation energies of the neutron resonances (several MeV), the densityof states (i.e., the number of states per energy unit) becomes very high, anda statistical description is appropriate. Today, the random matrix theory isthe basic tool of the interdisciplinary field of quantum chaos, and the atomicnucleus is still a wonderful laboratory of chaotic phenomena. Other excellentexamples of interplay between chaotic and ordered motion in nuclei are parity-violation effects amplified by the chaotic environment 236 and the appearanceof very excited nuclear states (symmetry scars) well characterized by quantumnumbers237. The study of collective behavior, of its regular and chaotic aspects,is the domain where the unity and universality of all finite many-body systemsis beautifully manifested.

5.5 Mesoscopic Physics: Jkom Compound Nuclei to Quantum Dots

Recent remarkable advances in materials science permit the fabrication ofnew 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 nu-clei and atoms, and the macroscopic world of bulk matter. Quantum dots arean example of mesoscopic microstructures that have been under intensive in-vestigation in recent years; they are small enough that quantum and finite-sizeeffects are significant, but large enough to be amenable to statistical analysis.For “open” dots, electrons can move classically into the dot. On the otherhand, for “closed” dots, the movement of electrons at the dot-leads interfacesis classically forbidden, but allowed quantum-mechanically by a quanta1 tun-neling. Tunneling is enhanced when the energy of an electron outside the dotmatches one of the resonance energies of an electron inside the dot.

The conductance of quantum dots displays rapid variations as a functionof the energy of the electrons entering the dot. These fluctuations are aperiodicand have been explained by analogy to a similar phenomenon in nuclear reac-tions (known as Ericson fluctuations). The electron’s motion inside the closeddot is generically chaotic. Recently, the same nuclear random matrix theorythat was originally invoked to explain the fluctuation properties of neutron res-onances has been used to develop a statistical theory of the conductance peaks

238in quantum dots . 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 ofnuclear structure - a rich and many-faceted field. An important element inthis task will be to extend the study of nuclei into new domains.

Thereare many frontiers of today’s nuclear structure. For very light nuclei,one such frontier is physics at subfemtometer distances where the internalquark-gluon structures of nucleons overlap. For heavier nuclei, the frontiersare 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 andunexpected phenomena which await us in the uncharted territory. However,the new data are also expected to bring qualitatively new information aboutthe effective NN interaction and hence about the fundamental properties ofthe nucleonic many-body system. By exploring exotic nuclei, one can magnifycertain terms of the Hamiltonian which are small in “normal” nuclei, thusdifficult to test. The hope is that after probing nuclear properties at theextremes, we can later improve the description of normal nuclei (at groundstates, close to the valley of beta stability, etc.).

There are many experimental and theoretical suggestions pointing to thefact that the structure of exotic nuclei is different from what has been foundin normal systems. New RNB facilities, together with advanced multi-detectorarrays and mass/charge separators, will be essential in probing nuclei in newdomains 71. The field is extremely rich and has a truly multidisciplinary char-acter. Experiments with radioactive beams will make it possible to look closelyinto many exciting aspects of the nuclear many-body problem. A broad inter-national community is enthusiastically using existing RNB facilities and hopingand planning for future-generation tools.

Advances in computer technology and theoretical modeling will make itpossible to (i) better understand the bare NN interaction in terms of quarksand gluons, and effective interactions in complex nuclei in terms of bare forces,and to (ii) answer fundamental questions concerning nuclear dynamics. Thesequestions on the microscopic mechanism behind the small- and large-amplitudecollective 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 mat-ter physics (many-body methods, superconductivity, cluster physics, physicsof mesoscopic systems), atomic and molecular physics (treatment of correla-tions, physics of particle continuum, dynamical symmetries), nonlinear dy-namics (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 ContractNos. DE-FG02-96ER40963 (University of Tennessee) and DE-AC05-960R22464with Lockheed Martin Energy Research Corp. (Oak Ridge National Labora-tory).

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