Feature

Clustering Together

At what point does a solid acquire its characteristic properties?

Blair Hall

Subdivide a solid into smaller and smaller pieces and, eventually, only a handful of atoms will remain. Will the properties of this microscopic speck differ from the original solid? Such is the conundrum of cluster physics.

Bulk solids are usually characterised by physical properties that are independent of quantity -- conductivity, density and structure are all intrinsic properties of a solid; they do not depend on how much material there actually is. Yet the word "structure" suggests the underlying arrangement of the atoms that that a material has and, clearly, the properties of these individual atoms are very different from those of the bulk solid.

"Clusters" or, perhaps more aptly, "nano-sized particles" are the names given to this class of objects that bridges the gap between atoms and solids. The study of these small particles, containing anything from two to several thousand atoms, is concerned with the physics of matter in the transition regime that must occur as atoms aggregate and grow to form a bulk solid.

How many atoms must a copper cluster have before it is capable of conducting electricity? How many atoms must a gold cluster contain in order to have that distinctive yellow colour of the bulk solid? In answering such questions, cluster physics forms a link between two more established branches of science -- condensed matter physics and atomic physics.

Interest in clusters is not new. In the latter part of the 19th century, Michael Faraday investigated the nature of "finely divided metals". Cluster physics, however, has had slow beginnings due to very severe technical problems that made experimental work and theoretical work too difficult to undertake.

It was not until 1976 that the first international conference on the subject was held. Since then, activity in cluster research has snowballed, partly because of the enormous technological possibilities that it will open up (better catalysts, new chemistry, new materials, miniature electronic devices, microscopic lasers) and partly because of the fascinating new science that is being uncovered.

The difficulty in performing experiments on clusters lies in creating a suitable sample for study. There are three experimental criteria to be met:

• a large number of clusters are needed
• they must be of unique size
• they must be free from interaction with each other or another body

Large numbers are required because clusters are so small. Each contributes only a very weak signal in any measurement and it is necessary to have a large number of identical clusters, whose contributions add together, to give a measurable signal. Freedom from interaction is needed to ensure that what is measured is a property of the cluster alone.

These requirements are not easy to satisfy -- freedom from interaction usually means a low density of clusters. If a sample is "mono-disperse" (consisting of one type of cluster) then cluster numbers are small; if there are a lot of clusters, then, unfortunately, there will also be a range of cluster sizes present.

Cluster Production

The inert-gas-aggregation technique is one cluster production method, whose underlying principle is familiar to anyone who has watched water being heated on a stove and seen a cloud of droplets swirling in the air above.

The technique requires a source of vapour (usually provided by melting the material in a crucible) and a controlled atmosphere containing a chemically inert buffer gas. The role of the gas is to cool the hot vapour as it leaves the crucible. The exchange of heat with the buffer gas causes the vapour to supersaturate, providing favourable conditions for nucleation and growth of tiny particles -- clusters.

The clusters grow, according to the laws of thermodynamics and according to the availability of atoms and other clusters to which they can join. This is a statistical process and is guaranteed to produce a range of sizes of particles. The average size can be influenced by the temperature of the liquid material and by the pressure of the inert gas atmosphere, but the range of cluster sizes is largely beyond the experimenter's control. Inert-gas-aggregation has been applied to materials that can be readily melted under experimental conditions. There are, however, a large number of substances that cannot be used because their melting points are too high.

Another more recent, and now widely used, method is the laser-vaporisation technique, which can be applied to a much wider range of materials. Cluster production is only part of the work needed to produce a sample for measurement. Both these production techniques will produce a range of sizes which must be filtered to produce a mono-disperse sample. Size selection is achieved by combining a cluster source with a mass spectrometer, which can be tuned to allow only a particular size of cluster to pass through.

Magic Numbers

A cluster source and a mass spectrometer already form the bare bones of a cluster physics experiment, and many researchers have investigated the size-related stability of clusters in this way. By comparing the intensity, or number, of different cluster sizes in a mass spectrum of a cluster source, indirect information can be obtained about the relative stability of clusters. A strong peak in the mass spectrum is indicative of a stable configuration; a smaller peak indicates a cluster structure which is likely to either absorb or eject an atom during the growth process and thus change its size.

Experiments have shown that patterns of stability occur and can be associated with certain sequences of cluster sizes. These "magic number" sequences depend on the cluster material, but also show striking similarities between groups of materials. Alkali metals such as lithium, sodium and potassium, for example, form stable clusters of atoms which have a magic number sequence of 8, 20, 40, 58, ..., showing that there is something special about the energy associated with the formation of clusters with these numbers of atoms.

Initially, researchers tried to understand this sequence by the way that individual atoms might pack together, but this approach implied a different sequence. An explanation was found when it was realised that the magic cluster sizes depended, not on the number of atoms in the cluster, but on the number of free electrons. It turns out that each alkali atom shares one weakly-bound valence electron with the whole cluster. Supplying an extra electron to a cluster whose size is one smaller than "magic" will render that cluster just as stable.

It has now been shown that cluster stability depends on the solution to the quantum mechanical problem of a number of electrons confined within a spherical potential. Formulated in this way, there are striking similarities with the shell model of the atomic nucleus, and indeed the cluster theory has also adopted the name "shell model".

The shell model in clusters has now been extensively studied. It is of particular interest because, unlike protons and neutrons in an atomic nucleus, there is no limit on the number of atoms that a cluster can have. Investigations of clusters containing thousands of atoms have been carried out, and the shell model has been found to apply up to about 1,500 atoms. A very different sequence was observed for clusters of rare gases such as neon, argon and krypton, where the mass spectra show magic numbers with 7, 13, 55, 147... atoms. In this case, the sequence is related to the actual geometry of the cluster.

A good place to start in analysing the stability of this cluster sequence is to imagine adding atoms, one by one, to form a compact structure based on the bulk crystal solid that would eventually be produced. As solids, rare gases crystallise in a structure known as face-centred-cubic (fcc) and it turns out, with the exception of the number 7, that building a cluster in this way works well. The sequence 13, 55, 147 and so on is just right to construct a particular polyhedron, called a cubo-octahedron, using increasing layers of atoms in an fcc lattice. However, this is not the only interpretation possible.

Forbidden Symmetry

The attraction or repulsion between a pair of rare gas atoms depends only on how far apart they are. The atoms within a cluster behave like hard spheres that are attracted to each other. For a large number of atoms, the fcc structure corresponds to the closest possible packing of spheres, which is clearly what is to be expected of rare-gas clusters.

However, when the number of atoms is quite small, it can be shown that the most compact packing arrangements are ones with five-fold symmetries, called icosahedra and dodecahedra, which are "forbidden" in crystals. The sequence of stable structures based on these models is 7, 13, 55, 147, ....

Regular tiling of a two-dimensional surface with units that have five-fold axes of symmetry cannot be done. Nor in three dimensions is it possible to fill space in a uniform and regular pattern using polyhedra with five-fold symmetry. It is surprising, therefore to find such structures as precursors of bulk solids.

Confirmation that five-fold symmetry structures are adopted by rare gas clusters was obtained experimentally and corroborated by computer modelling of the way in which the atoms in a cluster move and organise themselves.

Cluster Watching

The availability of high resolution electron microscopes in the early 1980s provided new impetus for nano-sized particle studies. With a resolution of about 0.2 nanometres (0.0000000002 metres), these instruments can show images of the lattice planes within the nano-sized objects. Microscopy allows a single particle to be selected from the field of view, so that individual clusters can be studied.

Discoveries in cluster studies continue to provide surprises. When, in the mid-1980s, the highly stable carbon cluster C60 was found in routine laser vaporisation studies of carbon, it was entirely unexpected. This hollow, soccer ball-shaped particle, later named Buckminsterfullerene, after the architect famous for his geodesic dome constructions, has since generated enormous interest. At one point, C60 was considered as a candidate for a long-standing mystery regarding an unexplained band of absorption of radiation in interstellar space. Unfortunately, its measured behaviour did not fit well.

Several years later, a microscopist, trying to observe C60 at high resolution, stumbled on a new form of carbon cluster: a giant, layered spherical particle of incredible stability. Appropriately named a "carbon onion", measurements of the absorption spectrum of this cluster provides a striking resemblance to the mysterious interstellar absorption band. The origin of this effect may at last be resolved.

Without question cluster research will provide more surprises in the future. Working at the limits of technology in both experiments and in computer calculations of cluster dynamics, it is often an incremental improvement in one or other that advances the current state of our knowledge. Ideas for applications abound, from novel rocket propellants and new lubricants to superconductors and miniature lasers. The future of cluster physics is full of promise and adventure.

Dr Blair Hall is in Massey University's Department of Physics.