Odd-even oscillation

Besides these many cluster studies, it is currently not knovm at what approximate cluster size the metallic state is reached, or when the transition occurs to solid-statelike properties. As an example. Figure 4.17 shows the dependence of the ionization potential and electron affinity on the cluster size for the Group 11 metals. We see a typical odd-even oscillation for the open/closed shell cases. Note that the work-function for Au is still 2 eV below the ionization potential of AU24. Another interesting fact is that the Au ionization potentials are about 2 eV higher than the corresponding CUn and Ag values up to the bulk, which has been shown to be a relativistic effect [334]. A similar situation is found for the Group 11 cluster electron affinities [334]. 

Fig. 6 compares the nuclearity effect on the redox potentials [19,31,63] of hydrated Ag+ clusters E°(Ag /Ag )aq together with the effect on ionization potentials IPg (Ag ) of bare silver clusters in the gas phase [67,68]. The asymptotic value of the redox potential is reached at the nuclearity around n = 500 (diameter == 2 nm), which thus represents, for the system, the transition between the mesoscopic and the macroscopic phase of the bulk metal. The density of values available so far is not sufficient to prove the existence of odd-even oscillations as for IPg. However, it is obvious from this figure that the variation of E° and IPg do exhibit opposite trends vs. n, for the solution (Table 5) and the gas phase, respectively. The difference between ionization potentials of bare and solvated clusters decreases with increasing n as which corresponds fairly well to the solvation free energy of the cation deduced from the Born solvation model [45] (for the single atom, the difference of 5 eV represents the solvation energy of the silver cation) [31]. 

Some calculations have been made also to derive the microelectrode potential E°(M, M .,/M ) for silver and copper from the data in the gas phase (nuclearity-dependent M-M bond energy and IPg(M )). The potential E°(M, M .,/M ) presents odd-even oscillations with n, (more stable for n even) as for IPg, but again the general trends are opposite, and an increase is found in solution due to the solvation energy. 

The strong fluctuation of IP or of the mass abundance is an electronic-structure effect, reflecting the global shape of the cluster, but not necessarily its detailed ionic structure. This is demonstrated in Fig. 6, where the ionization potentials of sodium clusters obtained by the spheroidal jellium model [32] are compared with their experimental values [46]. The odd-even oscillation of IP for low N is reproduced well. The amplitude of these oscillations is exaggerated, but this is corrected by using the spin-dependent LSDA, instead of the simple LDA [47]. The same occurs for the staggering of d2 N) [48]. 

From the odd-even effects observed in the ionization potential and electron affinity of small clusters (see Sect. 2.4) we can also expect odd-even oscillations in the reactivity. For instance, the ionization potential I(Np) of a small alkali cluster with an even number of atoms (Np) is larger than I(Np -i- 1) and I(Np — 1). At the same time the electron affinity A(Np) is smaller than A(Np -I- 1) and A(Np — 1). Consequently I(Np) —A(Np) will be larger than I(Np-I- 1) — A(Np -I- 1) or I(Np — 1) — A(Np — 1). In summary, the spin pairing effect induces odd-even oscillations in the reactivity. 

The effect of the functional group (X) joining the mesogenic unit (Ms) and the spacer in a main-chain polymer such as -(Ms-X-(CH2) -X) c- was first pointed out by Roviello and Sirigu [1982], who found that the odd-even oscillation of the latent entropy A ni with n became substantially smaller when the carbonate group was used for X in place of ether or ester groups in polymer LCs. The odd-even characteristics of the phase transition behaviors 7ni = have been studied extensively for... 

As an illustration of the physical content of Eq. (4) (which also serves as a motivating example for the SCM), we show in Figure 4.1 the size-evolutionary pattern of the ionization potentials (IPs) of Na/v clusters, which exhibits odd-even oscillations in the observed... 

We also include for comparison results obtained by KS-LDA calculations [77] for deformed Na v clusters restricted to spheroidal (axial) symmetry (Figure 4.4, top panel). As expected, except for very small clusters (N < 9), these results do not exhibit odd-even oscillations. In addition, significant discrepancies between the calculated and experimental results are evident, particularly pertaining to the amplitude of oscillations at shell and subshell closures. 

Whereas a negative value for A E(n) indicates that the process 2 M - M i + is energetically feasible, a positive value signals that the process M M , + M is less favorable than the reaction M +, M + M. When one plots the quantity A E(/i) against the cluster size n, the odd-even oscillations in the cluster stability become quite obvious (Fig. 2 4). For alkali metal clusters, the sharp maxima found at n = 2, 4, 6 and 8 indicate that clusters having odd numbers of atoms undergo fragmentation more easily than clusters with even nuclearity. [25, 91]... 

The IP s of several low nuclearity clusters have been computed by various methods with different degrees of success. As shown by the following examples, the trend in the IP s as a function of the cluster size strongly depends on the metal considered. For the alkali metal clusters Li and Na , odd-even oscillations have been observed, [112] whereby the higher IP s correspond to those clusters having closed shells and higher stability (n even). In general, however, the IP... 

Bredow, T., Giordano, L., Cinquini, R, and Pacchioni, G. 2004. Electronic properties of rutile TiOj ultrathin films Odd-even oscillations with the number of layers. Phys. Rev. B 70 035419. 

The results discussed above are consistent with the principle of Maximum Hardness, proposed by Pearson [23], which states that molecular systans at equilibrium present the highest values of hardness. Therefore, it is reasonable that the clusters with the highest hardness are those with closed electronic shells. Also consistent with this principle is the odd-even oscillation in the reactivity. DPT calculations also display an odd-even oscillation of the hardness of sodium and copper clusters [24,25]. 

It was found that the EAM-binding energy per atom increases rapidly and more monotonically for the smaller cluster size than the DFTB results and that leads to faster stabilization of the structures. A larger number of particularly stable clusters was identified. The odd-even oscillation of the DFTB stability function implies that even-numbered clusters are more stable than the odd-numbered ones. In contrast, the EAM calculations identify magic clusters with both even and odd number of atoms, and a correlation between the point group and the stability of the cluster was evident, that is, the stable clusters possess high symmetry. 

There is a clear odd-even effect the smectic layer thickness d increases with increasing n, but with an odd-even oscillation in which d for even n is relatively larger than that for odd n. 

The second-order difference in total energy (A ) can be considered as a measure of relative stability of clusters. Accordingly, a high value of A indicates a higher stability of the size as compared to its two left and right neighbors. From Fig. 6b, a consistent odd-even oscillation is easily found for the curves of A . The closed-shell systems with an odd number of B-atoms for cation BlJ" (with even number of B-atoms for neutral B ) reveal local maximum peaks. Interestingly, the A plot of cationic clusters B shows the enhanced peaks at n = 5 and 13, while the closed-shell systems Bg" " and Big correspond to local minimum peaks. These observations are consistent with experimental results of mass spectroscopy previously... 

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