Alkali jellium model

With respect to the thermodynamic stability of metal clusters, there is a plethora of results which support the spherical Jellium model for the alkalis as well as for other metals, like copper. This appears to be the case for cluster reactivity, at least for etching reactions, where electronic structure dominates reactivity and minor anomalies are attributable to geometric influence. These cases, however, illustrate a situation where significant addition or diminution of valence electron density occurs via loss or gain of metal atoms. A small molecule, like carbon monoxide,... 

Fig. 4.4. Surface energy predicted by the jellium model. A test case of the accuracy of the jellium model, conducted by Lang and Kohn (1970). Only for four alkali metals, Na, K, Rb, and Cs, are the predictions of surface energy by the jellium model fair. For most metals, with r, < 2.5 bohr, the surface energy predicted by the jellium model is negative, contradicting seriously with experimental facts. On the other hand, the calculated values of surface energies with crystal lattices agree much better with the values measured experimentally. (After Lang and Kohn, 1970).

The jellium model of the free-electron gas can account for the increased abundance of alkali metal clusters of a certain size which are observed in mass spectroscopy experiments. This occurrence of so-called magic numbers is related directly to the electronic shell structure of the atomic clusters. Rather than solving the Schrodinger equation self-consistently for jellium clusters, we first consider the two simpler problems of a free-electron gas that is confined either within a sphere of radius, R, or within a cubic box of edge length, L (cf. problem 28 of Sutton (1993)). This corresponds to imposing hard-wall boundary conditions on the electrons, namely... 

Various refinements of the above model have been proposed for example, using alternative spherical potentials or allowing for nonspherical perturbations,and these can improve the agreement of the model with the abundance peaks observed in different experimental spectra. For small alkali metal clusters, the results are essentially equivalent to those obtained by TSH theory, for the simple reason that both approaches start from an assumption of zeroth-order spherical symmetry. This connection has been emphasized in two reviews,and also holds to some extent when considerations of symmetry breaking are applied. This aspect is discussed further below. The same shell structure is also observed in simple Hiickel calculations for alkali metals, again basically due to the symmetry of the systems considered. However, the developments of TSH theory, below, and the assumptions made in the jellium model itself, should make it clear that the latter approach is only likely to be successful for alkali and perhaps alkali earth metals. For example, recent results for aluminium clusters have led to the suggestion that symmetry-breaking effects are more important in these systems. ... 

Particularly exciting in cluster science has been how the simple jellium model has been able to in first order explain many quantities for clusters of alkali and noble metals as the appearance of magic numbers. Examples were given in Fig. 3. We have in connection with our studies of the reactivity of... 

Quantum-mechanical models have been developed for the electrons in sp metals, the best known of these being the jellium model [4]. Sp metals are systems such as the alkali and alkaline earth metals in which d electrons are not involved as valence electrons. For these systems, reasonable estimates of p can be made these are summarized in table 8.3 for the alkali metals. It is interesting that this... 

When such clusters are formed in beams, essentially by many slow collisions of alkali atoms within a nozzle, it is found experimentally (by analysing the mass distribution using a time of flight spectrometer) that there are discontinuities in the intensity distribution of the peaks from which abundances can be computed for each cluster size. These discontinuities correspond to enhanced stability of metallic clusters around specific sizes (8, 20, 40, 58, etc). They are the same for all the different alkali metals, and are therefore referred to as magic numbers see fig. 12.5. They also turn out to be the same (at least, for the first few magic numbers) as those observed in nuclear physics. This similarity has led to an explanation based on the shell model and to the suggestion that the jellium model can be used to account for the properties of metallic clusters [683]. 

Fig. 12.7. Radial potential and charge density for an alkali cluster of 20 atoms, as obtained from the jellium model. Note the FYiedel oscillations in the density of electronic charge. The ion density is assumed to be a top hat function (after W. Ekardt [685]).

The spherical jellium model has been applied to alkali metal clusters by many authors (see Ref. [6]). Fig. (1) shows the self-consistent effective potential for a sodium cluster with twenty atoms. The degenerate levels are filled up to electron number = 20. In a spherical cluster with 21 electrons, the last electron will have to occupy the If level above (dashed line). This electron is less... 

Experiments on noble metal clusters (Cun, AgN, Aun) indicate the existence of shell-effects, similar to those observed in alkali clusters. These are reflected in the mass spectrum [10] and in the variations of the ionization potential with N. The shell-closing numbers are the same as for alkali metals, that is N = 2,S,20,40, etc. Cu, Ag and Au atoms have an electronic configuration of the type nd °(n + l)s so the DFT jellium model explains the magic numbers if we assume that the s electrons (one per atom) move within the self-consistent, spherically symmetric, effective jellium potential. 

In general, the presence of the impurity atom induces a strong perturbation of the electronic cloud of an alkali cluster. The different nature of the impurity can be accounted for by a simple extension of the jellium model. The foreign atom is assumed to be at the cluster centre, and both subsystems - impurity and host - are characterized by different ionic densities in a jellium-like description. The following positive-charge background is then assumed ... 

Linear response theory (TDLDA) applied to the jellium model follows the Mie result, but only in a qualitative way the dipole absorption cross sections of spherical alkali clusters usually exhibit a dominant peak, which exausts some 75-90% of the dipole sum rule and is red-shifted by 10-20% with respect to the Mie formula (see Fig. 7). The centroid of the strength distribution tends towards the Mie resonance in the limit of a macroscopic metal sphere. Its red-shift in finite clusters is a quantum mechanical finite-size effect, which is closely related to the spill-out of the electrons beyond the jellium edge. Some 10-25% of the... 

It was mentioned in Sect. 4 that electronic-shell effects appear in the mass abundance [10,43], ionization potentials [88], and electron affinities [89] of noble metal clusters that are very similar to those observed for alkalis. These can be readily interpreted within the spherical jellium model if we treat the noble metal atoms as monovalent, that is, each atom contributes its external s-el tron only. Even more, odd-even effects are also observed for small N in the properties mentioned above, and have been explained by Penzar and Ekardt [32] within the context of the spheroidally deformed jellium model. 

Plotted as a function of N, the calculated ionization potentials of alkali clusters display the expected drops at N = 8, 18, 20, 34, 40,... Nevertheless, the magnitude of the oscillations of I is overestimated in the jellium model. 

The deformation parameter rj describes how prolate or oblate the cluster is. This parameter was determined by minimizing the total energy calculated by adding the eigenvalues of the occupied electronic states. For alkali clusters with N less than 100, values up to q = 0.5 are estimated for open-shell clusters. The main first order effects of the ellipsoidal model are energy shifts that are proportional to q. The ellipsoidal model explains well the fine-structure features of the mass spectra [25], that is, those features which are beyond the realm of the spherical jellium model. 

Early interest in heteroatom clusters having alkali metals as the host was academic rather than dictated by precise observations. The main question regarded the extent to which the jellium-derived shell model retained its validity. However, this question was approached on the basis of oversimplified structural models in which the heteroatom (typically a closed-shell alkali-earth such as Mg) was located at the center of the cluster [235, 236]. In this hypothetical scheme, the perturbation of the electronic structure relative to that of the isoelectronic alkali cluster is somewhat trivial for instance, in the Na Mg system the presence of Mg would only alter the sequence of levels of the shell jellium model from Is, Ip, Is, 2s,. .. (appropriate to sodium clusters) to Is, Ip, 2s, Id,. .. (see also [236]). This would lead to the prediction that Na6Mg and NasMg are MNs. 

The glass-coloring experiments have been performed with gold, silver, nickel and other metals, which are much more difficult to handle theoretically than the alkalis. Among the latter, sodium is the best representative of the nearly free electron gas or jellium model which forms the basic assumption of some of the articles found here. Therefore this review is restricted to sodium clusters, and more specifically to their optical and thermal properties. 

---