Theory of Cluster Dynamics

Why study cluster dynamics ?

Cluster dynamics represents a fast developing area of cluster physics. The field covers various phenomena with impact both on fundamental cluster research and on potential applications, for example in cluster engineering. We show in the following a few emblematic examples of the field.

Clusters are made of electrons and ions. Both are charged particles which can then be excited by an electromagnetic field. A favorite and fashionable tool of investigation of cluster dynamics is thus provided by lasers. The latter deliver to the system electromagnetic pulses whose characteristics can be tailored almost at will both in terms of deposited energy and time profile. Some experiments are also performed by means of collisions between clusters and highly charged projectiles, as delivered by heavy-ion sources and facilities.

In both cases (lasers, ions), electrons and ions strongly couple to the delivered electromagnetic field, but at different time scales. Indeed, electrons are light particles which thus react and evolve at short time scales, typically the fs (10^-15s). In turn, the much heavier ions (several thousand times heavier than electrons) evolve on a much longer time scale of order 100-1000 fs. Of course these time scales are not fully independent of each other, through the natural coupling between electrons and ions, and the actual relation between these two time scales somewhat depends on the deposited energy.
Let us illustrate the two coupled electron and ion dynamics on a few examples.

Metal clusters couple especially well to an electromagnetic perturbation because their electrons are only moderately bound to the ionic cores. They thus react strongly, for instance to a laser excitation. The response, called "optical response" (because the emitted light is to a large extent visible), is the fingerprint of this coupling.

The optical response is caused by the collective oscillations of the cluster electrons following an excitation by the electromagnetic pulse. The electron cloud, elastically bound to the ionic cores, oscillates around them, once displaced from its original position, and radiates visible light. This collective response provides a signature of the underlying structure of the irradiated cluster. The "color" of the irradiated cluster, for example, significantly depends on the size of the cluster. We thus have here an example where electron dynamics provides a direct means of investigation of structure properties. The case is illustrated on Figure 1 where the frequency (the color) of the optical response of mixed gold and silver clusters (embedded in an inert glass) is plotted as a function of cluster size. One can see that the cluster color significantly depends on size. It means that such golden inclusions in a glass (of course of various sizes) would deliver a variety of colors, as it was already well-known by ancient artcrafters (see in the cluster introduction page).

Fig.1: Optical response of mixed gold and silver clusters, embedded in inert glass, as a function of size (details).

The optical response is a rather simple process, involving mostly electrons (although ions may also interfere, for example when temperatures are involved). Another interesting case is provided by cluster fission where ionic motion then plays a key role. When sufficiently charged (for example after a laser irradiation and escape of several electrons) a metal cluster may become unstable with respect to fission, exactly as massive atomic nuclei. It then becomes preferable for the system to break into two smaller clusters, the fission fragments. In such processes electrons play a relatively passive role (once the system is properly charged) and tend to follow the ions during the fission process. Fission is furthermore characterized by a potential barrier over which the system has to pass in order to evolve from one piece to two. This is illustrated in Figure 2 where the fission barrier of a small metal cluster is shown, together with the different shapes taken by the system at different deformations (from the smallest: 1 piece, to the largest: 2 pieces). Figure 3 presents an example of fission dynamics for another cluster.

Fig.2: Potential energy of K₁₂⁺⁺ as a function of the extension of the cluster (details).

Fig.3: Movie of the fission of Na₁₄, induced by a laser irradiation (details).

A most interesting situation is attained when both electron and ion dynamics explicitely couple to produce elaborate dynamical scenarios. This is illustrated on the third example we want to present here. We consider the case of embedded silver clusters, the shape of which can be tailored, as one can see on Figure 2. Metal clusters possess a specific frequency at which they couple to light (the optical response frequency seen above). If one shines a cluster with a laser precisely tuned at that frequency, one will so much excite the cluster that it will emit several electrons. This is a typical resonant behavior as is well known in any oscillating system.

In this case , if several electrons are stripped during the exposure to the laser, the cluster may become highly charged and will consequently expand because of the net charge acquired, as in fission. But full ionic expansion is hindered here by the fact that the cluster is included in a matrix. The final result is a somewhat expanded cluster. This expansion can be further analyzed by irradiating again the cluster and recording its optical response (pump and probe experiment). As seen above the optical response provides a signature of the cluster size. A variation in the optical response thus indicates a structure modification. This is exactly what one can see on Figure 4. The peak is at the same time broadened and shifted to a higher wavelength. Moreover the optical response depends on the laser polarization, reflecting a non-spherical shape of the cluster. The laser and its preferred coupling to the cluster has thus allowed to tailor the system shape. This allows to envision potential applications in laser assisted tailoring of materials at the nanometer scale.

Fig.4: Optical response of silver clusters embedded in an inert glass (details).