Z. Phys. D 40, 375–380 (1997) ZEITSCHRIFTFUR PHYSIK Dc© Springer-Verlag 1997

Giant resonances in atoms, atomic clusters, fullerenes,condensed media, and nucleiI.G. Kaplan

Instituto de Fısica, UNAM, Apartado Postal 20-364, 01000 Mexico, D.F. Mexico

Received 5 July 1996 / Final version: 15 September 1996

Abstract. The properties of giant resonances in atoms,atomic clusters, condensed media, and nuclei are discussed.We present a comparative analysis of similarities and dif-ferences between collective excitations in these physical ob-jects. The main conclusion: the existence of the giant reso-nance phenomenon does not depend on the nature of the par-ticles and the interparticle forces. The necessary conditionsare: the particles must be moving in a confining potentialand the number of them should not be small.

PACS: 73.20.Mf; 24.30.cz

1 Introduction

It is well established that in anN−particle system excited bya charged particle or an electromagnetic field, the excitationenergy can be absorbed by a single particle of the systemor by a part of the system, containing many particles. Thelast option is the topic of this study. I combine in one pa-per such different physical objects as nuclei, atoms, atomicclusters, and condensed media, because in all of them it ispossible to excite collective states involving many particles.Such collective excited states were first discovered in nu-clei [1] and were called giant dipole resonances [2, 3]. Theirexistence was anticipated in the two-fluid (protons and neu-trons) hydrodynamical model by Migdal [4] and Goldhaberand Teller [5] before their experimental observation.

In condensed media many different types of collectiveexcitations can exist. The low lying excitations are usuallydescribed in terms of quasi-particles [6]. To us, in connec-tion with this study, the most interesting are plasmons, orthe collective excitations of the free electron gas in metals.They were predicted by Bohm and Pines, [7] and were firstdetected experimentally in metals and very soon in dielectricmedia, see references in [8, 9]. In 1960 Fano [10] studied themodel for collective excitations in non-metallic media andnoted the formal analogy with giant resonances in nuclei.

In the 60th’s broad collective peaks located above theionization threshold were revealed in noble gas atoms [11,12] and lanthanide elements [13, 14]. Wendin [15] was the

first who emphasized the similarity of these atomic collectiveexcitations with the giant dipole resonances in nuclei.

In the last 80th’s, the collective resonance behavior wasalso revealed in sodium clusters irradiated with visible laserlight [16, 17]. Later on, the resonance peaks were detectedin some other metal clusters: AgN [18], LiN [19] and evenin fullerenes [20, 21]. It is remarkable that most of the the-oretical investigations of resonances in atomic clusters weremade by nuclear physicists due to the striking similarity ofsome aspects of this phenomenon in nuclei and atomic clus-ters, see Sect. 3 and [22–24].

So, now it is well established that the phenomenon ofgiant resonances is not specific to nuclear systems but canalso be observed in different physical objects. The energy ofgiant resonance peaks varies from∼ 2 eV in metal clusters to∼20 MeV in nuclei (on a range of 7 orders of magnitude!).In the following sections, we will discuss the differencesand common features of giant resonances in various physicalobjects.

A large number of reviews have been devoted to col-lective excited states in nuclei [25–27], atoms [28–30], con-densed media [8, 31–33], see also [34]. In several excellentreviews on collective excitations in clusters [30, 35–38] theirinterconnection with collective excitations in other physicalobjects is also discussed.

In this paper, despite of the shortage of space, I will tryto give a comparative analysis of similarities and differencesbetween collective excitations in all four types of physicalobjects mentioned in the title. A preliminary discussion waspublished in [39].

2 Atoms

As we mentioned in the Introduction, the broad peaks lo-cated above the ionization threshold were detected in no-ble gas atoms and lanthanide elements [11–14]. Compari-son with Hartree-Fock calculations clearly showed that thesingle-particle spectrum was partly, or in some cases, com-pletely wiped out and the oscillator strength was pushed tohigher energies. Absorption of synchrotron radiation by thelanthanide elements revealed strong, broad and asymmet-ric maxima in the energy region 10–20 eV above the 4d

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absorption edge [13, 14]. These features of the absorptionspectra of lanthanide elements and Ba were confirmed in thesubsequent photoabsorption, photoemission, and electron-spectroscopy experiments. In the theoretical study of thephotoionization cross section of the 5p6 shell in Xe, Wendin[40] came to the conclusion, that the process of photoion-ization of the 5p6 shells is entirely a many-particle process,but it is not characterized by any dielectric instability, givingrise to plasmon-type oscillations. But for thed10 shell in XeWendin [41] found that the real dynamical particle-hole pairinteraction has poles due to zeros in the effective dielectricfunction. Since the damping is strong, there are no free col-lective oscillations. Nevertheless there is a strong evidencefor a broad collective resonance involving the coherent mo-tion of all electrons in the 4d10 shell.

The point is that in heavy atoms the screening effectsand the centrifugal barrier for electrons with large angularmoment ( ≥ 2) crucially change the simple Coulomb pic-ture which takes place in light atoms. Still in 1928, Fermi[42] suggested that the centrifugal term can explain the or-der of filling the electronic shells in the Periodical Table ofElements. The balance between repulsive centrifugal termand the screened Coulomb potential near the core leads toa deep internal well, while at large distances an outer long-range well is created. So for heavy atoms the effective radialpotential has two wells [30].

It was stressed by Connerade [43, 29, 30] that it is theshort-range inner well that gives rise to giant resonancesabove an ionization threshold. The probability of resonanceis enhanced if the wave functions of initial and final statesstrongly overlap inside the inner well (this situation takesplace in excitations of the d-shell in lanthanides). The con-finement of the excited electrons during the ionization pro-cess due to the centrifugal barrier leads to resonance phe-nomenon. In the outer well, the long-range Coulomb poten-tial is unable to reflect efficiently an escaping electron andthe ionization occurs easily. This physical picture helps usto understand why the giant resonance phenomenon takesplace in 4d10Xe shell but is not observed in the 5p6Xe shell.

The mechanism of broadening of the resonance peaksin rare-earth elements was explained by Fano’s group [44]soon after their detection. It was shown that the exchangeinteraction between thef electrons and thed vacancy is re-sponsible for the≥ 20 eV shift in the high energy region.Due to the centrifugal effects associated withd → f tran-sition, the transitions to the 5f and high discrete levels aresuppressed by the barrier. ForZ ≥ 57, the 4f orbit has asmall radius which fits inside the inner well. This favoursthe observation of intense 4d→ 4f transitions. Most of theoscillator strength is concentrated in transitions to the higherlevels of the 4d94fn+1 configuration decaying by autoioniza-tion channel [45] mainly to the configuration 4d94fnεf .

The state with a hole in an inner shell (this is just the caseof 4d94fn+15s25p2 configuration) beside autoionization hasan additional decay channel: the Auger process [46]. Thischannel is discussed in [47], where it is named as an addi-tional autoionization channel. According to [47], about 20%of decay probability be due to this two-electron channel.

3 Metal clusters and fullerenes

In 1984 Knight et al. [48] revealed that the stability ofNaN clusters is characterized by “magic numbers”, later onthe same observation was made in Agn clusters [49]. Magicnumbers were independently predicted also by Eckart [50] injellium model calculations. It is remarkable that the valuesof the magic numbersNm=2,8,20,34,40,58,92.... are exactlythe same as the nuclear magic numbers. In the last case,these numbers correspond to the shell closing. Thus, themetal clusters have a shell structure as nuclei and atoms.From this it also follows that valence electrons in clusters aremoving in a finite-size mean potential. For sufficiently largemetals clusters (N > 1000) it was predicted [51, 52] andthen confirmed experimentally [53] that besides the shells, asupershell structure exists. This phenomenon is in principleimpossible in atoms and nuclei where the total number ofparticles cannot be more than several hundred.

There can be two types of resonances in clusters: 1.Atomic resonances of the electrons in the inner-shells (agood example is the large resonance peak at 88 eV assignedto the 4d → εf shape resonance in antimony clusters SbN

[30]). In the previous section we described this type of reso-nance. 2. Collective resonances in the valence electron sys-tem, belonging to the whole cluster. This type of resonanceswill be discussed below.

The collective resonance peaks were first observed insodium clusters [16, 17] in the last 80th’s, but the theory oflight absorption by a small metal sphere has been developedsince the beginning of our century by Mie [54]. Many of hisresults are valid in quantum theory. The quantum mechan-ical study of this problem was made by Eckardt [55], whocalculated the photoabsorption spectrum of sodium clustersin the time-dependent local-density approximation. It wasfound that the spectrum should consist of narrow lines cor-responding to single-particle transitions and of the collectivemode. The collective mode is strongly size-dependent.

The physical picture of the resonance in metal clusterscan be described as the collective dipole oscillations of thevalence-electron cloud against the positive background ofthe cores (cf. proton-neutron oscillations in nuclei). As inthe case of nuclei, the cluster shape depends on the numberof fermions. Clusters with closed shells are expected to bespherical and have one resonance peak. This simple picturewas observed first in neutral sodium clusters [16], then inionized potassium [56], sodium [57] and silver [18] clusters,see Fig. 1.

As was shown by the Haberland group [58], the one-peakbehavior of Na+9 is a high temperature result, at low tempera-ture the resonance peak is splitting on two peaks. Accordingto Clemenger [59], the equilibrium shape of metal clustersis in many cases ellipsoidal rather than spherical, in anal-ogy with the Nilsson model for nuclei [3]. Two resonancepeak were observerd also for neutral closed-shell clusterssuch as Na20 [60]. This splitting was interpreted in [60] as“a delicate balance between shell structure and residual in-teraction”. For splitting in Na+9 it was recently shown [61]that it is caused by ionic structure effects.

For alkali-metal clusters with increasing size, a blue shiftof the resonance frequency is observed [62, 63]. In contrastto alkali-metal clusters, silver clusters exhibit a red shift in

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Fig. 1. Giant resonances forclosed-shell(N = 8 and 20 valence electrons)andopen-shell(N = 10) ionized sodium [57] and silver [18] clusters

the resonance frequency [18, 64]. This shift and some otherspecific properties of silver clusters can be attributed to thepresence of the d-shell in the Ag atom [30]. It is also anindication that the simple jellium model fails to explain thefine features of the giant resonance in clusters.

The mechanisms of damping of giant resonances inatomic clusters are different in small and large clusters. Thedominant mechanism of the broadening of the resonanceline in small clusters is the coupling of surface plasmonto quadrupole surface vibrations. For small sodium clusters(8 ≤ N ≤ 20) it was demonstrated in [22] by taking intoaccount the quadrupole surface fluctuations of the clustershape. As was shown by Bertsch and Tomanek [23], thethermal fluctuations in the shape of the small clusters couldbe an essential source of the resonance line broadening. Asimilar mechanism has been suggested in nuclear physics todescribe the thermal broadening of nuclear collective exci-tations [65].

For large clusters (R > 10 nm, orN > 105 atoms),studied in [17], all experimental results indicate that there isno coupling with phonons. The decay channel is completelyelectronic. The plasmon energy is converted into a localizedsingle electron excitation. So the photodesorption of atomsin sodium clusters, withN > 105 atoms, is due to the decayof the plasmon collective energy into a repulsive state of oneof the atom-atom bonds.

After the discovery of the icosahedrical hollow “bucky-ball” moleculeC60 [66], its electronic properties were inten-sively studied by means of various spectroscopic techniques.In high resolution electron-energy-loss spectra of thin filmsof C60 [20], two collective peaks were revealed:

(i) the intensive peak at 6.3eV, attributed to the molecularπ-electron system, localized inside the ringsC6;

Fig. 2. Energy-loss spectra for 80keV electrons for two polymers [67]

(ii) the intensive broad peak at 28 eV attributed to aplasmon excitation of all 240(σ+π) valence electrons of thewholeC60.

The collective resonance absorption in freeC60 andC70molecules irradiated in vacuum with ultraviolet light wasstudied in [21]. The authors [21] measured the photo-ionsignal as a function of the wavelength of the light whichionizes a beam of neutral fullerene molecules and founda strong broad resonance peak located at about 20 eV withwidth about 11.5 eV. The peak position is in good agreementwith the theoretical prediction [24].

Bertsch et al. [24] studied the electromagnetic responsefunction of theC60 molecule (or cluster) using the linearresponse theory of different carbon cluster structures. Ac-cording to their results, the main effect of the Coulombelectron-electron interaction is to collect many one-electrontransitions into a single collective excitation, some kind ofa Mie-type plasmon. The giant plasmon resonance peaks ataround 22 eV and has total integrated oscillator strength of71. The authors [24] confirmed these results by the jelliummodel calculation of a conducting spherical shell with radiusR ' 3.5A and 240 conduction electrons.

4 Condensed medium

The probability of energy losses by a charge particle in acondensed medium is described by the energy-loss function

Im

[ −1ε(ω)

]=

ε2(ω)ε2

1(ω) + ε21(ω)

, (1)

where ε1 and ε2 are the real and imaginary parts of thedielectric permitivity, respectively; we do not write the de-pendence ofε on the momentumq transferred to the media.Equation (1) has two kinds of maxima determining the peaksin the energy-loss spectra: the maxima ofε2(ω) which corre-spond to one-electron transitions in atoms or molecules andthe zeros of the denominator corresponding to the excitationsof collective states.

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Table 1. Comparative properties of giant resonances in different physical systemsa

Property Nuclei Atoms Clusters Condensed media1. Number of N < 300 N < 115 N has not the N →∞

particles,N upper limit2. Mean density

of particles unchanged increases unchanged unchanged(with increasingN )

3. Location approxim. positive charge approxim. approxim.of charge uniform at center uniform uniform

4. Shell structure yes yes yes no5. Main type strong screened screened screened

of interaction interaction Coulomb Coulomb Coulomb6. Effective short-range two-well finite-range finite-range

potential deep well potential potential potential7. Surface effects important absent important non-important8. Shape yes no yes no

deformation9. Dependence of red shift, no blue shift no

Er upon size ∼ A−1/3 (alkali-metals),red shift (silver)

10. Γ/Er 0.3 - 0.4 0.1 - 0.5 0.1 - 0.5 0.1 - 0.811.Σfi/Nv 0.6 - 0.8 0.7 -0.95 0.2 - 0.312. Mechanism a) coupling with localization small clusters: localization

of giant quadrupole sur - in 1p-1h coupling with surface in 1p-1hresonance face vibrations, excitation: vibrations excitation:damping b) direct escape a) autoionization,large clusters: localization a) interband

of neutrons b) the Auger in single-electron transition,(∼ 15%) effect excitations, atomic b) autoionization

fragmentationa Notation:Er is the energy in the maximum of resonance peak,Γ is its width,fi is the oscillator strength, andNv

is the number of valence electrons (nucleons)

In the Fig. 2 the energy-loss spectra [67] for two poly-mers are presented. Narrow peaks in the energy region 6-8 eV correspond to one-electron monomer transitions, thebroad peaks in the region∼ 21 eV correspond to the col-lective resonance absorption.

Recently, Fano [33] reviewed his old study [10] andstressed the common features in collective excitations incondensed matter, superconductivity and nuclear physics Hederived necessary conditions for appearance of the resonancecollective states in a condensed medium, the discussion ofit see in [32, 9].

The plasmon-like peaks are characterized by large widthand consequently, by short lifetimes. In atomic-molecularmedia the damping of plasmon states is due to the inter-action of plasmon oscillations with electrons, phonons, andimpurities. The electron-plasmon interaction is a long-rangeone. As a result of plasmon decay its energy is transferredto a single electron, which is knocked out from an atom ormolecule [68]. In crystal structures the most probable pro-cess, in which the plasmon decays, is the creation of oneelectron-hole pair by an interband transition [69, 31].

The separate discussion of the peculiarities of the giantresonance in nuclei we will publish elsewhere. Althoughpartly, it is included in the following comparative discussion.

5 Comparative discussion

In Table 1, we present some characteristics of the four phys-ical objects under consideration and properties of the giantresonances in them. In all these objects, the giant resonance

is characterized by strong broad peaks corresponding to acollective excitation of many particles. The resonance peakexhausts a large part of the dipole oscillator strengths (up to80% in nuclei and up to 95% in some clusters). Althoughthe energy of the resonance peak,Er, varies from 2 eV forthe sodium clusters to 15-20 MeV for nuclei, the ratioΓ/Eris located between 0.1-1.0 for all objects.

The dependence ofEr on the size of the physical sys-tems shows different trends, even in the case of clusters: ablue shift for alkali-metal clusters and a red shift for silverclusters. This is an indication of the importance for a gi-ant resonance in clusters of the peculiarities of the structureof the constituent atoms. In the case of heavy and mediumnuclei, the approximate analytical dependenceEr on the nu-clear mass A was found to be [3]

Er = 79A−1/3 MeV. (2)

The heavier is the nucleus the lower is the resonance peakenergy.

The clusters can be considered as an intermediate objectbetween atoms and solids. But from Table 1 it follows thatsuch properties as: 1) the constancy of particle density; 2)the uniform charge distribution; 3) the importance of surfaceeffects; 4) the possibility of shape deformation; are commonto nuclei and atomic clusters. This is the reason why in spiteof the different nature of the interaction forces, the theoret-ical methods developed in nuclear physics are applicable incluster physics. The mechanism of giant resonance dampingin small metal clusters is the same as in nuclei: the couplingwith quadrupole surface vibrations. While in an atom wehave a frozen positive charge in the core, in clusters (as in

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condensed media) the positive ions can undergo thermal vi-brations. The thermal fluctuations of the cluster shape leadsto the broadening of the resonance peak as in the case ofhot nuclei.

On the other hand, for large atomic clusters the influenceof the surface is less important and their electronic structureis rather close to that of solids. In consequence, the mecha-nism of giant resonance damping in large clusters is similarto that in condensed media. Thus, from the point of view ofthe mechanism of giant resonances damping, clusters are lo-cated between nuclei and solids (contrary to a more naturallocation between atoms and solids).

The crucial condition for the formation of the giant res-onance is the form of the mean field in which the excitedparticles move. The mean field must guarantee the confine-ment. The effective potential in such field is characterized bya finite radius (it decreases with distance faster than 1/r, notethat the Coulomb potential has an in finite radius). Anotherimportant condition on the effective potential is the steep-ness of the boundaries. This results in the effective reflectionof the excited particles which gives rise to the creation ofcoherent oscillations. The short-range potential in nuclei ful-fils this conditions. In atoms, it is the inner well which leadsto the confinement of excited electrons during the ionizationprocess. The shape of the well is not very important, it canbe flat (as a square well for metals). It is necessary onlythat the effective potential provides a confinement duringthe excitation process.

Another important point is the condition on the numberof particles which participate in the resonance. When can wesay that the system contains “many” particles? It seems thata rigorous answer to this question does not exist. It dependsupon the system under consideration. Therefore, let us turnto experimental data.

For nuclei, the smallest nucleus in which the giant reso-nance is found is6 Li [70], so for nucleiN ≥ 6. In atoms,the giant resonance revealed in atoms with filled d-shell,i.e. N ≥ 10. The giant resonance is very well establishedin such clusters as Na8 and K+

9. In these systems there are8 valence electrons, so for clustersN ≥ 8. Thus, the min-imum number of particles needed for the giant resonancephenomenon can be estimated between 6 and 10.

When the many-particle system is exposed to electro-magnetic radiation, the region of the instant excitation isabout the mean wave lengthλ0. Evidently, for simultaneousexcitation of several particle, the mean interparticle distancer0 must be smaller thanλ0.

In summary, we can conclude that the existence of thegiant resonance phenomenon does not depend on the natureof the particles and the interparticle forces. The convincingevidence for this is the excitation of giant resonances both innuclei and in different atomic systems. The sufficient condi-tions which the system must fulfil can be formulated in thefollowing way:

1. The particles are moving in an effective potential whichprovides the confinement of excited particles.

2. The number of particles must not be small,N ≥ (6−10).3. The mean wave length of the exciting fieldλ0 ≥ r0,

where ¯r0 is the mean distance between interacting parti-cles.

I am grateful to J.P. Connerade, F.A. Gareev, A. Mondragon, and Yu.F.Smirnov for helpful discussions and references.

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