Cluster sodium

Figure Cl. 1.2. (a) Mass spectmm of sodium clusters (Na ), N= 4-75. The inset corresponds to A = 75-100. Note tire more abundant clusters at A = 8, 20, 40, 58, and 92. (b) Calculated relative electronic stability, A(A + 1) - A(A0 versus N using tire spherical electron shell model. The closed shell orbitals are labelled, which correspond to tire more abundant clusters observed in tire mass spectmm. Knight W D, Clemenger K, de Heer W A, Saunders W A, Chou M Y and Cohen ML 1984 Phys. Rev. Lett. 52 2141, figure 1.

Knight W D, Clemenger K, de Heer W A, Saunders W A, Chou M Y and Cohen M L 1984 Electron shell structure and abundances of sodium clusters Phys. Rev. Lett. 52 2141... 

The other problem in the AIM approach is the presence of non-nuclear attractors in certain metallic systems, such as lithium and sodium clusters. While these are of interest by themselves, they spoil the picture of electrons associated with nuclei forming atoms within molecules. 

There are similarities and differences between the Hirsch rule and those developed by physicists, originally to rationalize the magic numbers observed experimentally for simple metal clusters. The reader is referred to the excellent review by de Heer for more information [71]. The magic numbers for sodium clusters, 8, 20,... 

Figure 1. Photoabsorption cross section for the dipole plasmon in axially deformed sodium clusters, normalized to the number of valence electrons N - The parameters of quadrupole and hexadecapole deformations are given in boxes. The experimental data [39] (triangles) are compared with SRPA results given as bars for RPA states and as the strength function (49) smoothed by the Lorentz weight with A = 0.25 eV. Contribntions to the strength function from p =0 and 1 dipole modes (the latter has twice larger strength) are exhibited by dashed curves. The bars are given in eVA. ...

The simple hard-wall boundary condition, eqn (5.1), does yield the correct ordering of the energy levels as shown in Fig. 5.1. The only exceptions are the 3s and lh states, which have their sequence reversed compared to that of the self-consistent jellium predictions. Experimentally, the most frequently occurring sodium clusters are indeed Na8 and Na20 as expected from their special stability in Fig. 5.3. 

Fig. 5.3 The total energy per atom of sodium clusters versus the number of atoms in the cluster, evaluated within the self-consistent jellium model. (From Cohen (1987).)...

Using the spherical jellium model explain the expected special stability of sodium clusters containing the magic number of atoms 2, 8, 18, 20,... 

Leaching out contaminants Glass may leach trace elements (sodium clustering MS analysis) plastic can produce plasticizers, which may confound MS analysis... 

The interaction between ions and valence electrons is described in a standard manner by pseudopotentials. Sodium clusters allow to use local pseudopotentials. We employ here a soft shape in terms of error functions... 

Microporous superbasic catalysts based on zeolites suitable for alkene isomerization were developed by Martens et al. [28-30]. They have high activity in the 1-butene isomerization and side-chain alkenylation of xylene [31], and are prepared by impregnating a dehydrated zeolite (NaY) with NaN3 in alcoholic solution. Subsequent decomposition of the azide inside the zeolite pores produces metallic sodium particles Nax°and cationic sodium clusters Na43+. The high catalytic activity was attributed to the ionic sodium clusters as they could be detected using ESR spectroscopy. 

There are exceptions from this trend. A noted example is the polarizability of sodium clusters significantly underestimated by semi-local functionals.73... 

Recent experiments by Citrin and coworkers (41) have clarified the role of the support in photoemission from small metal clusters. They condensed several monolayers of krypton onto either platinum or sodium metal substrates. By varying the thickness of the krypton from one to ten monolayers, the surface could be converted from metal to semimetal to insulator. The krypton peak position provides a direct measure of the sample vacuum level (32). The krypton layers are thin, less than 10 monolayers, so that the vacuum level is determined by the metal substrate. Onto the krypton layers, sodium clusters were deposited at varying coverages. Shifts in the Kr 4s and Na 2p binding energies were recorded relative to the Fermi level of the grounded substrate. 

The results obtained by Citrin and coworkers are shown in Figure 3. For sodium clusters on a metal support (wigl ML Kr/Pt, filled circles), the Kr 4s binding energy decreases with cluster coverage. This shows that the Fermi levels of the sodium and platinum equilibrate. As the sodium is added, the work function decreases from the value for platinum to the value for a sodium film. Conversely, the Na 2p peak position does not shift with cluster coverage. The rapid electron transfer between the sodium and platinum prevents any accumulation of charge on the cluster in the photoemission final state (41). 

Supersonic expansions have been used to form small metal aggregates, (2 < n < 4). Emphasis is placed on the analysis of bound-free transitions in these small metal clusters. Discussion focuses on the characterization of variously produced sodium supersonic expansions and the analysis of laser Induced atomic fluorescence resulting from the photodissociation of triatomic sodium clusters. We will consider (1) the nature of observed "fluctuation" bands corresponding to bound-free transitions involving a repulsive excited state which dissociates to yield (Na-D line) sodium atoms and ground state,, ... 

The transition from the atom to the cluster to the bulk metal can best be understood in the alkali metals. For example, the ionization potential (IP) (and also the electron affinity (EA)) of sodium clusters Na must approach the metallic sodium work function in the limit N - . We previously displayed this (1) by showing these values from the beautiful experiments by Schumacher et al. (36, 37) (also described in this volume 38)) plotted versus N". The electron affinity values also shown are from (39), (40) and (34) for N = 1,2 and 3, respectively. A better plot still is versus the radius R of the N-mer, equivalent to a plot versus as shown in Figure 1. The slopes of the lines labelled "metal sphere" are slightly uncertain those shown are 4/3 times the slope of Wood ( j ) and assume a simple cubic lattice relation of R and N. It is clear that reasonably accurate interpolation between the bulk work function and the IP and EA values for small clusters is now possible. There are, of course, important quantum and statistical effects for small N, e.g. the trimer has an anomalously low IP and high EA, which can be readily understood in terms of molecular orbital theory (, ). The positive trimer ions may in fact be "ionization sinks" in alkali vapor discharges a possible explanation for the "violet bands" seen in sodium vapor (20) is the radiative recombination of Na. Csj may be the hypothetical negative ion corresponding to EA == 1.2 eV... 

Before leaving the jellium model, it seems appropriate to mention some results for sodium clusters of up to 22000 atoms. The abundances observed can be explained in terms of the extra stability associated with both completed jellium shells (for less than about 3000 atoms) and with completed icosahedral or cuboctahedral geometric shells for larger sizes. The results are especially interesting because of the observation of supershells that occur because of an interference effect, which has been explained using semiclassical arguments. ... 

Figure 10.1 Illustration of electronic magic numbers abundance of sodium clusters as a function of size measured by mass spectromet7 in a molecular beam of sodium seeded in Ar. Relatively high stability is found for...

Sodium Clusters. - Since the pioneering work of Knight et al.61 more than 20 years ago on mass abundance spectra of sodium clusters, these have been the prototype for metallic clusters. Knight et al. measured the relative abundance of sodium clusters with up to somewhat more than 70 atoms. Instead of finding a smooth curve as a function of N, certain sharp peaks were observed, indicating that for certain values of N, particularly large amounts of this cluster were found, i.e., that clusters with these sizes were particularly stable. 

Figure 15 Melting temperature and latent heat for sodium clusters of different sizes. Open and closed symbols mark results for the singly charged and the neutral clusters, respectively, and the squares and the circles represent results from calculations using the tight-binding and the embedded-atom model, respectively. Finally, experimental results are shown with the triangles. Reproduced with permission of American Institute of Physics from 52...

L.R.M. Martens, PJ. Grobert, and P.A. Jacobs, Preparation and Catalytic Properties of Ionic Sodium Clusters in Zeolites. Nature London), 1985, 315, 568-570. 

However, the most striking features of Figure 5.10 are the pronounced peaks at AT = 2, 8, 18, 20, 34,40 and 58. At the magic numbers the hardness shows a local maximum. Increased stability is accompanied by increased hardness. Identical results have been calculated for sodium clusters.Similar, but not so spectacular, resuls may be calculated for small silicon clusters. 

Accurate ab initio calculations on lithium and sodium clusters have been performed by Fantucci et al [20, 21,22, 23], These studies have yielded detailed information about the equilibrium geometries, stabilities, and ionisation potentials of both neutral and cationic clusters. In this section the moments of inertia are used as a chemical tool to interpret the gross shape of the cluster. The deviation from sphericality is defined by the parameter L, derived from the moments of inertia by... 

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