Geometrical Structure of Metal Clusters

Extended Abstract

The geometrical structure of clusters as the size decreases is strongly influenced by the increasing importance of the surface energy. However for very small clusters electronic effects, such as the lowering of energy by Jahn-Teller distortions, have to be taken into account. In the first part we thus give some experimental and theoretical results for large clusters (diameter > 20 Å) where a wealth of experimental information has been obtained by electron microscopy and electron diffraction. In the second part we describe theoretical results obtained for very small alkali metal clusters and compare them with the available experimental data.

It is well known that for some metals, e.g. platinum and aluminium, most of the aggregates are single crystals which retain the bulk structure, while for other metals, e.g. gold and silver, the surface induces changes in the structure which lead to the multiply twinned particles (MTP) described by Ino [1, 2]. The present day electron microscopes allow to study in detail the MTP’s and their defects using high resolution lattice imaging, dark field and microdiffraction techniques. In a comparative electron diffraction study [3, 4] the lattice parameter changes and the integrated width of the (220) diffraction line have been observed as a function of size for gold and platinum particles of diameters comprised between 30 Å and 200Å. For both gold and platinum the lattice parameter a varies linearly as a function of the inverse diameter D, in agreement with a simple drop model [5]. The average surface stress <y>, deduced from the drop model and the variation of a, is approximately equal to the surface tension σ of the bulk metal in the case of platinum. It is larger than σ for gold particles, which is in agreement with the elastic deformation of the tetrahedra forming the MTP [6]. The width △ of the integrated diffraction peak follows the usual “Scherrer straight line” △=1.3/D for platinum clusters. For gold clusters the different behavior reflects their multiply twinned structure.

Theoretical studies of the geometrical structure and electronic properties of particles having more than 100 atoms have only been carried using semiempirical methods. The change of structure when the size decreases from a cubooctahedral geometry to an icosahedral geometry has been studied for Ni and Pt clusters [7, 8]. A transition from a bcc to a fee structure has been found for transition metal clusters using a tight-binding approximation [9]. Recently the cohesive energy of 3 d transition metals has been calculated [10, 11] for different bcc and fee clusters, using empirical N-body potentials proposed by Finnis and Sinclair [12]. No transition from bcc to fcc was found down to the smallest sizes. Using the same effective potential and starting from a dodecahedra, the forces were calculated on each atom and the relaxed geometry was studied for different size clusters (15–1,695 atoms).

With the cluster sources now available aggregates of essentially all elements, comprising between 3 and 100 atoms, can be formed in a beam. However the experimental information available on their structural geometry is very small and the comparison between the growing number of ab-initio calculations [13] and the experimental data is difficult. In the case of alkali aggregates a complete study of the Born-Oppenheimer surface of Li₃, K₃, Na₃ has been reported [14] using an approach based on the density functional theory. For Na, K and Li the equilibrium geometries as well as the dynamic behavior (passage from one equilibrium configuration to another through pseudorotation) are in agreement with the electron spin resonance measurements [15, 16] of alkali trimers in a rare gas matrix. The vibrational frequencies of the normal modes of sodium trimers in its ground state (obtuse isosceles triangle) have been obtained in analyzing the hot bands observed during a systematic study of the excitation spectrum of Na₃ [17, 18]. The measured values (49, 87 and 139 cm⁻¹) are in agreement with the theoretical results (58, 94, 142 cm⁻¹) deduced from an analytic representation [19] of the potential energy surfaces.

The structural and electronic properties of the neutral and ionized sodium clusters with n ≤ 8 and n = 13 have been investigated in a selfconsistent pseudopotential local spin-density calculation [20]. The equilibrium geometries have been obtained in starting from randomly generated cluster geometries and letting them relax under the action of the forces on the atoms [21]. The clusters with up to five atoms have closepacked planar equilibrium geometries, the six-atom cluster is quasiplanar with a planar isomer lying only 0.04 eV higher in energy, real three dimensional structures only occur when the number of atoms is greater than or equal to seven. Similar geometrical structures have been obtained for lithium [13]. The results are in good agreement with the measured ionization potentials [22–24]. For a seven atoms cluster, which is predicted to be a pentagonal bipyramid, the spin densities deduced from the hyperfine structure of electron spin resonance data [25] are in excellent agreement with the theoretical prediction. A simple model which takes into account the delocalized nature of the valence electrons and the Jahn-Teller effect explains the main features of the equilibrium geometries.