ION-SIZE EFFECTS IN THE GROWTH SEQUENCES OF METAL-ION-DOPED NOBLE GAS CLUSTERS

We have studied the stability and the structure of doped noble gas cluster ions of the type 
M


INTRODUCTION
After intensive experimental and theoretical investigations in the last ten years, it is well established that the geometrical structure of the atomic van der Waals clusters can be inferred from hard sphere packing models. This is a direct consequence of the electron *Also at: Fachbereich Physik, Universit/it Rostock, Universit/itsplatz 3,18051 Rostock, Germany Also at: Department of Physics, University of Crete, 710 03 Heraklion, Greece localization on the individual atomic orbitals. As a result, atoms can be treated as hard spheres, thereby allowing the application of geometrical models for determining the cluster structure. The appearance of magic numbers patterns, observed in massspectroscopic studies on Xen clusters by Echt et al. [1] and on Arn by Harris et al. [2] could be explained by the adaptation of icosahedral sphere packing. These results are also confirmed by electron diffraction studies from Farges et al. [3] on Arn clusters. The atoms in these systems arrange themselves in shells around a central atom.
When the total number of atoms is n 13,55,147,... shell closing occurs, giving rise to very stable clusters, which have icosahedral geometry (Mackay-icohedra [4]). Stable clusters also are observed for n--19, 23, 26, 29, 32,... where in this case subshells are filled and the cluster geometry is characterized as polyicosahedral [5]. The completion of shells and subshells takes place at a certain sequence (growth sequence), such that every additional atom occupies the position which offers the maximum number of neighbors (contacts). In this report we present our experimental results for several metaldoped noble gas cluster ions. While for some of them the familiar icosahedral magic numbers are observed, others show magic numbers which cannot be explained with icosahedral packing. We present a simple sphere packing model which is based on capped square antiprism structure and is able to explain the observed non-icosahedral magic numbers.

EXPERIMENTAL
Details of the employed experimental setup have been reported previously [9,10]. In brief, clusters ions of the type MXn are produced using a combination of laser ablation of the metallic target and supersonic expansion of the noble gas into vacuum. The ablation laser is a Nd:YAG laser (A 1064 nm, pulse duration 10 ns, repetition rate 12.5 Hz) with an output energy of 10-100 mJ/pulse. The noble gas is expanded from a pulsed nozle (diameter 0.8 nm, backing pressure 2-8 bar, room temperature) and is mixed with the metal ions contained in the plasma plume at 0.5-5 mm away from the nozzle orifice. The cluster ions produced from this mixing are then analyzed with a Timeof-Flight (TOF) device, which can be used either as a linear or as a reflectron spectrometer. The pulsed nozzle, the ablation laser firing and the triggering of the TOF are synchronized through coupled delay generators, which are sequentially optimized for maximum signal. The TOF-signal is recorded with a digital storage scope and can be accumulated for an arbitrary number of laser shots. In situations when the ion intensity is low the mass-spectra are recorded with the linear TOF setup.

RESULTS AND DISCUSSION
In Figures and 2 we present the TOF-spectra of the In+Xn, and AI+Xn (X=Ar, Kr, Xe) respectively and in Figure 3 the spectra of Na+Xn (X Ar, Kr). These spectra are averaged over 2000 laser shots. The pronounced intensity irregularities (magic numbers) in the spectra are labeled with the total number N of atoms in the corresponding clusters. We check the reproducibility of the magic numbers for each system, by recording a series of averaged spectra obtained under different source conditions (backing pressure, laser fluence, etc.). The magic numbers indicated in Figures 1-3 correspond to peaks whose intensity differs from that of the neighboring ones by at least 10%, and simultaneously show the same behavior in all recorded spectra for a given system. (27), 30. As N increases this sequence changes into the icosahedral one.
The magic numbers N 9, 11, 14, 17, 21 have been previously observed from Saito et al. [11] in the Al+Xen clusters formed by ion sputtering of an A1 target with Xe ions, the origin of these magic numbers however could not be explained.
indicated in Figure 4. These polyhedra can be formed by joining two identical regular polygons, each of them being made from k n/2--2, 3, 4, 5,...atoms, and one of which being twisted by an angle of 7r/k with respect to the other one. The first four polyhedra formed in this way are the tetrahedron (n 4), the octahedron (n 6), the square antiprism (n 8), and the pentagonal antiprism (n 10).
From the above polyhedra, only the tetrahedron and the octahedron are closed shell structures, while for k=4 and k 5, two additional atoms are needed in order to acquire the closed shell structures, the capped square antiprism (n=10) and the icosahedron (n=12) respectively. It should be also mentioned, that only the tetrahedron, the octahedron and the icosahedron are regular polyhedra. We expect that the polyhedra of Figure  (1) where, k n/2 represents the numbers of atoms per ring. This formula gives the maximum radii ratio for each symmetry in order to keep the exterior atoms in contact with each other. Thus we obtain the maximum R *-values of Table I for k 2 to 5. That is, for 0.902 >_ R* > 0.645 icosahedral packing is preferred, for 0.645 _> R > 0.414 CSA-packing, etc.
The case of icosahedral packing is well known and the corresponding magic numbers sequence has been deduced by Harris et al. [2] and Regularly the FP is preferred when the second shell is relatively empty, whereas the EP is favorable when the shell is nearly complete. As a result, a crossover region of cluster sizes exist, where no particular stable clusters are expected. This is a consequence of the balance between the surface density of atoms and the number of nearestneighbor contacts [5], or equivalently is due to the balance between the surface energy and the interior energy of the cluster [13]. Since this simple model seems to reproduce very well the observed magic number sequences for icosahedral "aufbau", it will also be worthwhile to attempt to deduce similar growth sequences for the other polyhedra of Figure 4. In the cluster systems M+Xn studied here, we successively go from lighter to heavier noble gases X. This is equivalent to decreasing the radius of the central ion in comparison with that of the noble gas. Since the first packing appearing (e.g. in In+Xn) is the icosahedral one, then according to the results of the model presented above, the square antiprism can be the basic structure for the new magic numbers sequence appearing in the case of e.g. In+Xen.
A cluster of the kind MXn which has the same symmetry as the square antiprism includes N 9 atoms. This structure contains, apart from the triangular faces, two quadratic faces, which are able to accept capping atoms in order to result in a more closed structure. This capped square antiprism (CSA) structure with 11 atoms (see Fig. 5a), has then 16 triangular faces, 10 vertexes and 24 edges.
Generally, the total number of atoms Nm in a cluster with m filled shells and having the symmetries shown in Figure 4, is given by: where V is number of vertices and/w the number of triangular faces. For all the polyhedra N 1, while N2 13 for the icosahedron and for the CSA-structure, we get N-11,45,119,249,... Similar formulas for a variety of polyhedra are given by Martin et al. [14].
In the remainder of the present paper, we concentrate only on the case of the CSA and we work in a similar manner as Harris et al. did for the icosahedral clusters. In order to represent in two dimensions what happens in three dimensions, we use the so called Sehlegel diagram [15] shown in Figure 5b for the eleven-atom CSA-core. Such a diagram shows schematically all the elements (faces, edges and vertexes) of a three dimensional polyhedron in two dimensions. The numbers at the triangular faces of the Figure 5b represent the successively added atoms, whereas the small subscripts accompanying, represent the number of the neighboring atoms which are present before the atoms are added i.e. the number of newly formed bonds. In Figure 6 we display these numbers as function of the corresponding cluster size. Several maxima occur at N 17,21,24,27,30 and 32-35.
These numbers coincide with the observed ones in In+Xe, for N _< 21, in AI+Xe, for N _< 30, in Na+Ar, for N <_ 21 and in Na+Kr, for N<30. Assuming that the contribution of every new added atom to the total binding energy of the cluster, is proportional to the number of nearest neighbor bonds, these maxima should then correspond to the binding energy differences of the clusters. This quantity reflects the magic numbers in the cluster stability [2]. According to the above points, the magic number sequence N=9,11,17, 21, 24, 27, 30 may be due to CSA-packing in these clusters. The corresponding cluster geometries are plotted in Figure 7.
It is interesting to compare the experimental results with the predictions of the model presented above, using the radius of the involved elements obtained from the literature. Generally the effective ionic radius depends on the coordination number of the metal ion [16], but due to electronic configuration of In+,Al+,and Na + we expect that R{n > RI > Ra. Butterfield and Carlson [17], using Hartree-Fock-Slater calculations, could obtain the ionic radius for many closed N=ll N=I7 N=21 N--24 N=27 N=30 FIGURE 7 The geometrical structure of the magic numbers clusters whose growth is based on the capped square antiprism symmetry. The gray-shaded atoms represent the atoms needed to form 6-atom pentagonal caps to result in partial shell closures.
shell ions. Although their absolute values cannot be taken as very accurate, since these have been obtained in an uniform manner, they allow comparisons between different ions in the Periodic  Figure 8 (solid symbols) for the systems considered here. The solid lines in Figure 8 are the limiting radii ratios from Table I for each symmetry. We observe that, in almost all cases the radii ratio renders very well the observed symmetry. Only in the case of Al+Krn there is a disagreement, where R* seems to be lower than the observed icosahedral symmetry allows.
Our results indicate that, RI/RKr > 0.645. Therefore we increase slightly (6%) the ionic radius of A1 + to the value R- 1.29 (open circles in Fig. 8) in order to fit our experimental observations. This value can be considered as the lower limit of R. shell is not as strong as it is for the atoms which are in direct contact with the central ion (polarization energy). Thus as more and more noble gas atoms are added on the third shell to build larger clusters, the noble gas-noble gas interaction becomes more important, resulting in an increase of the cluster surface energy, which forces the noble gas atoms to be placed in a more compact fashion. This corresponds effectively to a decrease of the Rx-value, so that the ratio RM/Rx increases. These changes of RM and Rx radii as a function of cluster size, can be seen in the numerical results from molecular dynamics calculation in the K+Arn clusters in Re/'. 19. Thus, the cluster at some critical cluster size, rearranges from CSA-symmetry to icosahedral one, which is more favorable for larger R*-values (see Tab. I). This critical cluster size for structural changes seems to depend also on the radius ratio R* (e.g. compare In+Xe to AI+Xe,). The examination of this topic, based on molecular dynamics simulations of the cluster energetics and structure, will be presented in a forthcoming paper [20].

CONCLUSIONS
In mass spectrometric studies concerning the stability and structure of metal ion-doped noble gas clusters, we could observe the icosahedral packing as well as a new cluster packing, which is consistent with a capped square antiprism geometry. A simple hard sphere packing model, which assumes that the metal ion resides in the center of the cluster, can adequately predict the experimental observations and the cluster structure, as a function of radius ratio of metal ion to noble gas atom. Furthermore, we observed structural transitions from the CSAgeometry to icosahedral one as the cluster size increases. To our knowledge, the present study of metal ion-doped noble gas clusters provides the first example of such a change of geometrical structure in small atomic cluster systems.