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Electron emission to analyze cluster dynamics

We present a brief review of typical theoretical methods used to analyze cluster dynamics on the basis of electronic emission. After putting the various possible theories in perspective, we focus on a few examples of application, concerning in particular photoelectron spectroscopy and photoelectron angular distributions. We finally conclude by considering the case of high laser frequencies as delivered by free electron lasers and investigate the impact of such frequencies on cluster ionization.


Introduction
The physics of metal clusters has attracted much attention in the past decades. Early works were mostly devoted to structure properties [1,2,3]. Meanwhile, dynamical studies become increasingly important [4]. Observables from the electronic dynamics of clusters turn out to provide useful information on the underlying cluster properties. That holds in all dynamical regimes ranging from the linear to the strongly non-linear one.
Optical absorption spectra are the typical examples in the linear domain providing useful insights into structure properties of clusters. They have been successfully exploited in the early decades of cluster physics, for overviews see e.g. [1,5,6,7,8]. But optical response also plays an important role in many dynamical scenarios, as for example in the case of pump-probe dynamics [9,10,11,12].
As another key example let us mention Photo-Electron Spectra (PES), i.e. the distribution of kinetic energies of the emitted electrons for which exist several interesting experimental results, see e.g. [13,14,15]. These allow to gain an almost direct insight into the electronic shell structure of the cluster. They have also been explored in truly dynamical situations both experimentally [16] and theoretically [17,18,19]. Further complementing information can be obtained from the angular distributions of the emitted electrons (Photoelectron Angular Distribution, PAD). There exist already a few very promising measurements [20,21,22]. Angular distributions, for example, allow to distinguish between direct and thermal electron emission [20,23]. Direct emission prevails for short laser pulses or low intensities and one can expect that angular distributions from direct emission carries information on the spatial structure of the initial electron states [24,21] Figure 1. Sketch of various theories developed to analyze cluster dynamics. The axes correspond to the deposited excitation energy per electron (horizontal axis) and the cluster size (vertical axis). The higher the excitation energy the more dynamical the situation so that the horizontal axis represents so to say a dynamical arrow. The various theories are represented by overlapping grey squares. The boundaries are of course only indicative and mostly qualitative. The typical excitation energies can be loosely related to typical laser intensities in the optical range, as indicated by the intensity scales on top, which are, however, also strongly dependent on the response of the particular system, i.e. resonant or non-resonant.
onset of the relativistic regime, where retardation effects within the coupling begin to severely influence the dynamics.

Example of electronic emission
As discussed in the introduction, electronic emission constitutes a key tool of investigation of cluster properties. We take here two typical examples for illustration.

Photoelectron spectroscopy (PES)
Examples of PES are presented in Figure 2 which compares experimental results [35] to TDLDA calculations. Experiments were performed with photons in the UV range, ω photon = 6.42 eV, obtained from frequency tripling of a high-intensity laser pulse. That energy is above the ionization potential (IP) so that we are considering "simple" one-photon processes in this case. The TDLDA analysis of the emission pattern is here performed at specific "measuring points". Two choices are demonstrated in the figure : A doublet of points along the polarization axis (forward plus backward emission) and one point for outflow, orthogonal to the axis. Both  Figure 2. Examples of PES in the case of sodium cations. Experimental values are taken from [35]. The PES is analyzed along the laser polarization axis and perpendicular to it, which provides a first clue to angular distributions. From [19].
measuring points show different patterns due to a different selectivity on angular momentum l z as can be seen from the figure. The effect is visible in both cation clusters presented, namely Na + 41 and Na + 93 . This preliminary example of angular resolved PES taken at two different directions hints that much more information could be disentangled when measuring angular distributions of PES. This is illustrated below. Figure 3 shows examples of angular distributions for Na + 9 and for two photon frequencies, a very low one and another one somewhat above ionization threshold. Note that the system contains eight electrons which corresponds to electronic shell closure. The electronic distribution is thus, up to small perturbations from the detailed ionic background, close to spherical. The PAD have been furthermore disentangled with respect to the electron state from which they are emitted. This thus corresponds to an energy-resolved measurement of angular distributions (or a double differential cross-section, with respect to angle and energy), as is presently measured [21]. In addition, calculations have been performed for a thermal ensemble of ionic configurations. The faint lines in the figures show results from a few single members of the ensemble. They generally share the same pattern. The thermally averaged results with ionic structure are shown with heavy lines and filled dots as symbols. The low-frequency case shows overall pattern for each state almost as from a spherical system. Note, e.g., the pronounced zero at ϑ = 0 • and 180 • for the 1p state with l z = 1. Interestingly, the corresponding jellium results are very similar. In that case the ionic background potential has been modeled with exact spherical symmetry. The high-frequency case, on the other hand, looks much different. The angular distributions show marked structure which very much differ from jellium results (mind the logarithmic scales). These distributions are obviously strongly influenced by detailed ionic structure and, in turn, they carry information on structure. This raises two interesting questions : How to explicitly extract ionic effects from the PAD, and do higher frequencies enhance such an effect? The answer to the second might help to analyze the first one. We shall thus focus on the second issue. The analysis of the PAD obtained with even larger frequencies is in progress and we shall thus only focus on the simpler analysis of how levels are depleted in high frequency fields as a first clue to understand the impact of VUV-FEL laser on electronic emission.  Figure 3. Example of PAD in a small sodium cluster. Calculations with an explicit account of ionic structure at a temperature of 1000 K are compared to jellium ones. They match in the low laser frequency case but significantly differ in the high frequency case. From [24].

New dynamics with free electron lasers
The recent progress in the development of Free Electron Lasers allow the use of high frequency laser beams with possibly large intensities. The nature of the laser cluster coupling fundamentally changes when going from the IR regime towards excitation with VUV-, XUV-, or even X-ray pulses. This concerns ionization processes as well as the mechanisms of energy absorption. Still, we shall not discuss these details here and only illustrate the impact of large frequencies on the ionization of a small sodium cluster. The test case is Na + 13 cluster and we remain in a regime of small ionization to comply with the PAD case of the previous section. Such a study complements earlier results obtained in rare gases, both experimentally [36] and theoretically [32], and in which it was shown that moderate intensities already allow to attain large ionization states at the side of the atoms inside the cluster. Figure 4 shows the level depletion for various laser frequencies between 0.1 and 2.4 Ry as a function of total ionization. The larger the frequency, the larger the intensity required for a given net ionization, whence this choice of plotting. The depletion is simply computed as the number of electrons emitted from a given level. We consider two particular levels in Figure 4, namely the most deeply bound one (with 0.62 Ry binding) and the least bound one (0.44 Ry binding). The most striking feature of this figure is the difference in depletion observed between the various laser frequencies. For low frequencies, ionization dominantly comes from the least bound state, while the most bound state contributes very little (typically one order of magnitude less). The situation is the reverse in the case of high laser frequencies, in which ionization from  . Level depletion in a Na + 13 cluster irradiated by lasers of various frequencies between 0.1 and 2.4 Ry, as a function of net ionization of the cluster. Both the most and the least bound states are considered and compared. While low frequencies deplete preferentially high lying states, the situation is the reverse for high frequency cases for which low lying states are more depleted than high lying ones. the most bound state dominates. This case is not accidental. We checked a similar trend in a neutral Na 8 cluster and the case of small organic molecules exhibits similar patterns. The example indicates that low frequency pulses withdraw the electrons preferably from the surface while high frequency pulses allow to explore the whole volume of the cluster.

Conclusions and perspectives
We have presented a short theoretical review of the description of electron emission to analyze cluster dynamics, concentrating on a few examples in simple metal clusters. After a brief sketch of available theories, we focused on the most flexible and robust TDLDA which allows to make numerous exploratory studies. We have illustrated our point on the especially interesting cases of photoelectron spectra and photoelectron angular distribution for which several experimental results have been recently obtained. Finally we have roughly discussed the case of VUV-FEL frequencies in the moderate ionization regime and found interesting behaviors which will deserve more extensive investigations. Work along that direction is in progress.