Giant resonances in clusters

Atomic giant resonances in clusters were observed by Brechignac et al [696] for clusters of Sb. Since the spectrum of the corresponding free atom was not available, the universal curve deduced from the uncertainty principle (see section 5.16) was used for analysis. The resulting point (see fig. 12.13) shows that the observed resonance fits closely onto the line for free atoms. 

Fig. 12.16. Examples of giant resonances in clusters containing magic and nearly-magic numbers of delocalised electrons, showing how the giant resonances split into structures indicative of the symmetry of the system (after C. Brechignac and J.-P. Connerade [714]).

Relative contributions of T-odd densities to a given mode should obviously depend on the character of this mode. Electric multipole excitations (plasmons in atomic clusters, E giant resonances in atomic nuclei) are mainly provided by T-even densities (see e.g. [19]). Instead, T-odd densities and currents might be important for magnetic modes and maybe some exotic (toroidal,. ..) electric modes. 

Vp(fO is peaked at the surface. Many collective oscillations manifest themselves as predominantly surface modes. As a result, already one separable term generating by (74) usually delivers a quite good description of collective excitations like plasmons in atomic clusters and giant resonances in atomic nuclei. The detailed distributions depends on a subtle interplay of surface and volume vibrations. This can be resolved by taking into account the nuclear interior. For this aim, the radial parts with larger powers and spherical Bessel functions can be used, much similar as in the local RPA [24]. This results in the shift of the maxima of the operators (If), (12) and (65) to the interior. Exploring different conceivable combinations, one may found a most efficient set of the initial operators. 

The sets (75)-(77) are optimal for description of electric collective modes EX plasmons in clusters and giant resonances in nuclei). For description of magnetic modes, the initial operator should resemble the T-odd magnetic external field. So, in this case we should start from the initial operators... 

The particular SRPA versions for electronic Kohn-Sham and nuclear Skyrme functional were considered and examples of the calculations for the dipole plasmon in atomic clusters and giant resonances in atomic nuclei were presented. SRPA was compared with alternative methods, in particular with EOM-CC. It would be interesting to combine advantages of SRPA and couled-cluster approach in one powerful method. 

The organisation of the present chapter is as follows first, we describe different kinds of clusters which can be formed and the shapes which are commonly found. Next, we give an example of how clusters can be used to bridge the gap from the atom to the solid and, finally, we discuss the subject of giant dipole resonances in clusters. 

Fig. 12.13. Universal curve for the giant resonances in free atoms with the point for the quasiatomic resonance in clusters of Sb included (adapted from J.-P. Connerade [225]).

Fig. 12.14. The quasiatomic giant resonances in Sb clusters of different sizes. Note the marked dependence of the amplitude of the giant resonance on the number of atoms in the cluster (after C. Brechignac et al. [696]).

The fact that they tend to be fairly symmetrical (at least when they occur below the ionisation threshold) is related to their time characteristics from the lifetime widths and resonance energies, one can deduce that the giant resonances in metallic clusters are many-body oscillations undergoing several periods. Giant resonances in metallic clusters can truly be considered as plasmons, and relate quite clearly to surface plasmons in solids. 

Clusters also demonstrate the ubiquity and generality of the basic principles of physics the stability of metal clusters is governed by a shell closure closely related to that of nuclear physics. Indeed, the collective, giant dipole resonances in clusters and in nuclei obey the same laws over changes of fourteen decades in scale size. 

TIggesbaumker J, Koller L, Lutz H O and Melwes-Broer K H 1992 Giant resonances In silver-cluster photofragmentation Chem. Phys. Lett. 190 42... 

The scope of this contribution is the comparison of metals clusters and nuclei. These systems have much in common as their structure and dynamics are dominated by the behavior of fermion liquids, the protons and neutrons in nuclei and the dense electron cloud in clusters. This gives rise to shell effects and a corresponding deformation pattern as well as pronounced resonance excitations related to zero sound in homogeneous matter, the giant resonances in nuclei and the surface plasmon in clusters. The structural aspects have already been much discussed in the past and are well documented in several review articles, see e.g. [1, 2]. A prominent feature here was the appearance of supershells which are only accessible in metal clusters with their unlimited pool of system sizes [3] and which have a particularly transparent explanation in the framework of semiclassical... 

J. Tiggesbauker, L. Roller, H.O. Lutz, and K.H. Meiwes-Broer, Giant Resonances in Silver-Cluster Photofragmentation , Chem. Phys. Lett. 190, 42 (1992). 

Since the observation of giant resonances in the photofragmentation of free Ag clusters indicating collective electronic excitations small silver clusters have attracted considerable experimental and theoretical attention. Additional interest in these small metal clusters was aroused by the unexpected observation of a very sharp absorption line of Ags clusters inside helium droplets.Because of the large capture cross sections of helium droplets (Sec. 2.2) sufficient numbers of metal atoms such as silver can be picked up by passing the droplets through an oven heated to 800°C with a vapor pressure of only 10 mbar. 

Of course, since clusters are made up of atoms, and since atomic giant resonances are localised excitations which survive even in solids, they must clearly be present in clusters also. We thus have the possibility of observing two kinds of collective resonance, one localised on individual atoms, and the other delocalised on the whole cluster, within the same sample. 

The resonances exhibit interesting variations as a function of atomic number, the origin of which is unexplained. As can be seen in fig. 12.14, their intensity relative to the rest of the spectrum fluctuates strongly, with the giant resonance actually disappearing from view for n = 8, 12, 16, etc atoms in the cluster. This behaviour seems in some way to be related to the fact that Sb clusters tend to be made up from smaller groups of four atoms stuck together. 

The second, and more important kind is the giant dipole resonance intrinsic to the delocalised closed shell of a metallic cluster. Such resonances have received a great deal of attention [684]. They occur at energies typically around 2-3 eV for alkali atoms, and have all the features characteristic of collective resonances. In particular, they exhaust the oscillator strength sum rule, and dominate the spectrum locally. 

Experimental evidence supporting this view comes from measurements in which the cluster beam is cooled to low temperatures [709] it has been found that the giant resonance splits up into more than one component even for clusters containing magic numbers of atoms. 

Fig. 12.19. Trend of the giant resonance or surface plasmon frequencies towards the bulk limit as a function of the number of atoms in the cluster (after J.H. Parks and S.A. Donald [712]).

The first depletion spectra obtained for neutral sodium clusters N = 2-40 were characterized by structureless broad features containing one or two bands. The results were interpreted in terms of collective resonances of valence electrons (plasmons) for all clusters larger than tetramers [2, 52-55]. The analogies between findings for metallic clusters and observations of giant dipole resonances in nuclei have attracted a large attention. Therefore the methods employed in nuclear physics, such as different versions of RPA in connection with the jellium model, have also been applied for studying the optical properties of small clusters. Another aspect was the onset of conductivity in metal-insulator transitions. 

As hinted above, a comparable systematics of giant resonance splittings cannot be produced for nuclei because too many of them are so soft that the signal is smeared out, as is the case for the few triaxially soft clusters in the sample of Figure 7.4. One thus tries to retrieve the information on nuclear deformation from the amplitude of the low-energy quadrupole modes which is closely related to the transition strength, often called the B(E2) value [65]. This access has been used in [5] for the systematic comparison of cluster and nuclear deformation. 

The shell model for metal clusters, described above, has an important implication which will not have escaped the reader if electrons become delocalised from individual atoms and can roam freely over the whole cluster to form a closed shell, then this shell should be able to oscillate collectively, and should therefore exhibit giant dipole resonances analogous to those which were described in chapter 5 for free atoms. 

In a next step, we compare nuclear and cluster response in the generic case of Coulomb excitation, as modeled by an initial shift of the electrons (respectively neutrons) with respect to ions (respectively protons). We first consider the nuclear giant dipole resonance. The lower panel of Figure 7.9 shows the power spectrum of the dipole along the z axis (symmetry axis) of after Coulomb excitation for several amplitudes with average excitation energies as indicated. The small-amplitude case represents the nuclear excitation spectrum in the linear regime as it is known from nuclear RPA calculations. We... 

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