Photodynamics of Nanoclusters 1588830640

Citation preview

Encyclopedia of Nanoscience and Nanotechnology

www.aspbs.com/enn

Photodynamics of Nanoclusters M. Belkacem, M. A. Bouchene Université Paul Sabatier, Toulouse, France

P.-G. Reinhard Universität Erlangen, Erlangen, Germany

E. Suraud Université Paul Sabatier, Toulouse, France

CONTENTS 1. Introduction 2. Optical Response in Nanoclusters 3. Nanoclusters in Intense Laser Beams 4. Toward Quantitative Models 5. Conclusions Glossary References

1. INTRODUCTION Cluster physics is by its very nature an interdisciplinary field with connections to well established domains such as chemistry, atomic physics, molecular physics, and solid state physics [1–4]. These various branches of science have found in clusters sort of a melting pot in which several concepts and techniques have merged. For a long time, light has been preferred tool of investigation of cluster properties. Incoherent light constitutes here the simplest, natural means of approach and allows one to address the linear response domain. This is the realm of the so-called optical response, with clusters responding in the visible part of the electromagnetic spectrum, in a way similar to the response of atoms in the sky, subject to solar radiations. This linear domain provides clues on both static and dynamic properties of clusters. In particular it is an especially powerful tool of investigation of structure properties, as we shall discuss. For a few years, though, in relation to the rapid progresses in laser technology, it has also become possible to investigate truly dynamic properties of clusters subject to lasers of various intensities. Beyond the linear domain accessible to low fluence lasers one can now routinely address problems in ISBN: 1-58883-064-0/$35.00 Copyright © 2004 by American Scientific Publishers All rights of reproduction in any form reserved.

the truly nonlinear domain with intense laser beams. In all these situations, both in the linear and nonlinear domains, it turns out that the basic electronic response of the cluster, the aforementioned optical response, plays a key role, both in metallic systems and in nonmetals. It is thus a key issue in this field, whatever dynamical regime one wants to address. We shall thus spend some time discussing its properties in some detail, before considering more “dynamical situations.” Large clusters at the nanometer scale contain hundreds or thousands of particles and modelling them thus constitutes a formidable task which requires simplifying assumptions. Small clusters have been attacked with a variety of detailed methods inherited mostly from chemistry and molecular physics (from the experimental as well as the theoretical points of view), and very large clusters can be studied with methods imported from solid state physics. Intermediate size clusters, typically in the nanometer range, constitute specific systems which are yet little explored in detail. They require dedicated approaches in between chemistry and bulk techniques. The building of simple and realistic models hence constitutes a key issue in the progress of the field, both to guide experiments and to understand the produced results. It is obvious that a detailed description, at the level of all electronic and ionic degrees of freedom (even when restricting the approach to the least bound electrons), is far beyond the possibilities of any practical calculation. One thus has to employ simple pictures, based on microscopic considerations, but dealing with more or less macroscopic or semimacroscopic quantities. Simplifications are even more crucial when one aims at describing cluster dynamics, which is a central issue for understanding photodynamics of nanoclusters. Most experiments on clusters indeed deal with light or electromagnetic

Encyclopedia of Nanoscience and Nanotechnology Edited by H. S. Nalwa Volume 8: Pages (575–591)

Photodynamics of Nanoclusters

576

we shall particularly focus on the most important and most studied case of metals.

2.1. Metal Nanoclusters 2.1.1. The Shining Colors of the Optical Response Clusters have been used for practical purposes for centuries. The tailoring of fine dispersed pieces of material (clusters!) inside bulk has been turned into an art by craftworkers, from the Romans to modern artists. At least from a practical point of view, how to tune the size of dispersed particles in a glass to produce various colors has been known for a long time. The understanding of colors of such small metal particles goes back to Mie [15]. He addressed the question of the response of small particles to light, and how this response might depend on the size of the considered particle. And Mie realized immediately the potential applications of such studies for our understanding of the properties of matter. Let us briefly quote him: “Because gold atoms surely differ in their optical properties from small gold spheres [it would] probably be very interesting to study the absorption of solutions with the smallest sub-microscopical particles; thus, in a way, one could investigate by optical means how gold particles are composed of atoms” [16]. Indeed the optical response of metal clusters to light is strongly dependent on their size. This is illustrated in Figure 1 where the peak response (to light) of clusters of various sizes is plotted precisely as a function of their size. As can be clearly seen, the position of the peak response is monotonously linked to cluster’s size, which explains the reflections of various colors exhibited by glasses with metallic inclusions. In fact, what we are seeing here is just the response of a metallic sphere to an external electric field. It should thus be easy to understand the basic mechanism responsible for this effect. And we go back here to Mie’s picture. Let us briefly see how such an approach works.

3.0

peak maximum (eV)

probes. The rapid development of laser technology in the last few decades has allowed the attainment of large electric fields (comparable to the ones binding clusters together) and on short time scales, typically of the order of electronic time scales [5, 6]. One is thus in a position of studying the response of clusters to short and strong electromagnetic signals which place the system far away from its ground state. In such dynamical situations, there is an urgent need for guidelines and the invention of simple tractable models is thus a key issue. It is the aim of this chapter to discuss the response of clusters of various types to irradiations by light (in particular with various laser lights). A major goal will thus be to describe the many types of measurements performed in the various dynamical regimes attained as a function of the intensity of the excitation process. And we shall emphasize the key role played by the collective response of clusters to light, the so-called optical response. We will also discuss examples of simple models dedicated to the study of large clusters irradiated by lasers of various intensities. We shall see that “back of the envelope models,” in particular the Mie model of plasmon response, provide a key ingredient for the understanding of the coupling between light and nanoclusters. The latter model particularly allows one to understand basic physical mechanisms at work. And yet a more detailed explanation, and at least a semiquantitative approach, requires more elaborate models, which can be built in a rigorous way on the basis of well defined microscopic approaches. The chapter is organized as follows. We first discuss the historical case of the coupling of metal clusters with light (optical response). This corresponds to a regime of small perturbation of the electron cloud. Elementary models (Mie plasmon) allow one to understand the mechanism underlying the cluster’s response, but not at a fully quantitative level. Similar plasmon behaviors are also observed in other types of clusters than in metal clusters, although with slightly different patterns, and we shall also discuss these situations. We next consider the case of more violent laser irradiations. We discuss in particular the very recent case of clusters subject to intense lasers and show that Mie plasmon again provides a key to understand the cluster behavior. However, once again details of the response go beyond simple arguments. The last part of the chapter is devoted to a discussion of how to make progress in the understanding of the clusters’s response by means of microscopic methods. We illustrate the achievements of such simplified approaches and we sketch the next steps of development to reach an even more coherent description of these various phenomena.

2.7 Agn (Ag0.75Au0.25)n (Ag0.5Au0.5)n (Ag0.25Au0.75)n

2.4

2. OPTICAL RESPONSE IN NANOCLUSTERS Because in metal clusters valence electrons move quasi freely, they respond particularly easily to an electromagnetic probe. This constitutes the so-called plasmon (or optical response) in metal clusters. Similar behavior can be observed in other not strictly metallic objects such as carbon systems. We shall discuss here these various aspects although

Aun

0.00

0.04

0.08

I/R (a.u)

Figure 1. Optical response of silver and gold clusters embedded in a rare gas matrix. The peak value of the response is plotted as a function of inverse cluster radius. Reprinted with permission from [14], M. Gaudry et al., Phys. Rev. B 64, 085407 (2001). © 2001, American Physical Society.

Photodynamics of Nanoclusters

577

2.1.2. A Simple Model of the Optical Response in Metals Massive metal clusters, such as the ones included in glass matrices, constitute typical examples of metallic spheres. The valence electrons form there an almost uniform electron gas not too tightly bound to the ionic background. It is thus no big surprise that a moderate external electric field as delivered by natural visible light may easily affect such clusters and possibly provide a fingerprint of some cluster properties. This is the essence of Mie theory describing the response of metallic spheres to light. Let us consider an extremely simple model of a spherical metal cluster XN in which the ionic background is taken as an homogeneous positively charged sphere of radius R (density ). Electrons are modelled by a negatively charged sphere with the same density . We thus assume that the cluster is a neutral system and that both electrons and ions occupy the same volume (which is an intuitive, although slightly excessive, approximation). In addition, we make the assumption that both electrons and ions will oscillate as rigid spheres against each other. This hypothesis may be realistic for ions. Electrons are probably not behaving that rigidly. But it fortunately turns out that the electronic response is primarily collective as long as the system is not too perturbed. Thus, the rigid sphere approximation makes sense in the regime of small excitations. If this system is put in a small uniform external electric field Eext electrons will slightly separate from ions, each one going in opposite directions. Still, because ions are much heavier than electrons, they basically remain fixed and one can consider that only electrons are displaced with respect to ions. The displacement builds up a strong Coulomb field which provides a restoring force on the displaced electron cloud counterweighting the effect of the external field Eext . In the limit of small values a of the electron–ion separation the net force acting on the electron cloud (along electric field Eext ) becomes proportional to a, F = −N

size N . The density  differs from one metal to the next. But metal clusters of a given material do all have the same average density. This is an effect of electron density saturation, much similar to the behavior observed for example in nuclei [17]. This saturation can be seen in terms of cluster radii R which follow a law of the form R ∼ rs N 1/3 , where rs is the Wigner–Seitz radius of the material. Any size dependence is obviously absent from the simple expression in Eq. (1) while experimental results do depend on cluster size. The Mie model thus provides an explanation for the basic mechanism of optical response in metal clusters but does not give access to the crucial size effects observed in this optical response. A more detailed approach is thus necessary, as we shall discuss.

2.1.3. Optical Response as a Function of Frequency Our discussion has shown that the optical response of a cluster is characterized by one typical frequency for which the Mie model provides a good guess. Thus the response of the cluster will strongly depend on the color of the light used. In other words, the optical response is a resonant phenomenon. Depending on the excitation frequency, the response will be more or less strong, as in any resonant situation. This resonant behavior is illustrated in Figure 2 in the case of massive

e2 a = NeEext 3 0

where N is the number of electrons. Once the external perturbation is switched off, there remains only the attracting (restoring) force between electronic and ionic clouds, proportional to the actual separation between ions and electrons. Electrons will thus undergo harmonic oscillations around ions, with a frequency

2Mie =

Ne2 e2 = 3 0 me 4 0 me R3

(1)

where me is the electron mass. This oscillation frequency

Mie is known as the Mie frequency. When a cluster is irradiated by photons, its response will be particularly strong for light frequencies around Mie . In simple alkaline metals the frequency Mie takes values in the visible part of the electromagnetic spectrum, hence the term “optical response” to characterize this phenomenon. The Mie model, as just outlined, does explain the gross features of the response of clusters to light, but it misses the crucial size dependence. As can be seen from Eq. (1),

Mie only depends on the density  and not on the cluster

Figure 2. Examples of optical response in massive Li and K clusters. Photoabsorption cross-sections are plotted as a function of photon energy. Reprinted with permission from [18], C. Bréchignac et al., Phys. Rev. Lett. 70, 2036 (1993). © 1993, American Physical Society.

Photodynamics of Nanoclusters

578

2.1.4. Size Dependence Optical spectra do depend on cluster size. The average resonance frequency follows a trend Mie N  = Mie  + cN −1/3 typically down to N ≈ 100 and levels off for smaller clusters. The slope c is generally small with different signs depending on the material. Negative c prevail for alkalines, positive c are found for most noble metals, and Hg is an example with zero slope [20], as can be seen in Figure 3. It is due to surface effects, in fact a combination of them. In a metal cluster electrons, mostly because of their quantal nature, do occupy a volume slightly larger than ionic cores. One usually calls this effect the spill out of the electron cloud. This spill out reduces the plasmon frequency [21] and small clusters have, relatively, more surface than volume which leads to a negative slope. The spill out effect dominates for example in alkaline clusters, hence the negative slope for these materials. But in noble metals, for example, the plasmon frequency is strongly influenced by the collective dipole oscillations of the d core electrons which redshifts the resonance. This effect is inactive at the surface which means that larger surfaces yield less redshift, which favors a positive slope. The core electron effect dominates in noble metals, hence the positive slope. The two effects, core electrons and spill out, seem just to compensate for Hg.

2.1.5. Temperature Effects As can be seen from Figure 4 the thermal effect plays a crucial role for the observed linewidth in optical absorption. The spectrum at low temperature shows three narrow

T = 537 K

Na7+

2

0

optical absorption

Li and K clusters. Both spectra exhibit a marked resonant behavior around a well defined peak. As a consequence one may expect a high sensitivity of the response to the excitation frequency; the stronger the response, the closer the excitation frequency to peak frequency. Practically the optical response is explored by means of a low intensity laser irradiating the metal cluster. By scanning the laser frequencies one can explore the full frequency spectrum and attain an optical response spectrum as presented in Figure 2.

T = 349 K 2

0 4 T = 35 K

2

0 2.0

3.0

2.5

3.5

ω [eV]

Figure 4. Photoabsorption strength for Na+7 at three different temperatures as indicated.

and well separated peaks reflecting the deformation of the considered cluster (see insert). We see here the collective splitting of the strongly oblate Na+ 7 and the lower peak is furthermore Landau fragmented [26] by one close 1ph state [27]. The picture changes systematically for larger temperatures. The peaks grow broader and the lower double peak merges into one. It is the temperature which causes the large width. Thermal effects at the side of the electrons cannot induce such a width because 400 K is a small energy at an electronic scale. It is the thermal ionic motion which is responsible for that line broadening. The ionic excitation energies reach down to below 100 K for Na+ 7 [28]. A thermal excitation of 400 K is large on that scale. Many ionic eigenmodes are excited to rather large amplitudes. The cluster thus undergoes substantial thermal shape fluctuations. But deformation has a strong influence on the dipole spectrum; the more elongated the cluster in one direction, the lower the eigenfrequency along this direction. Thus, each member of the thermal ensemble, with its different deformation, contributes a different spectrum to the total optical response. These spectra all add up incoherently to rather broad peaks. This thermal line broadening mechanism has been discussed first in [29, 30] on the grounds of the jellium model with quadrupole shapes. The crucial impact of octupole deformations was pointed out in [31] and confirmed in the fully microscopic analysis of [32].

2.2. “Optical” Response in Covalent Clusters Figure 3. Mie plasmon frequencies as a function of system size for a variety of metals. Reprinted with permission from [19], C. Bréchignac and Ph. Cahuzac, Com. Atom. Mol. Phys. 31, 215 (1995). © 1995, IOP Publishing.

Carbon clusters are halfway between metallic and covalent binding. The optical absorption spectra thus show pronounced resonance peaks together with a swamp of noncollective peaks. This is not so surprising. An interesting aspect of C is that its particular binding properties allow stable

Photodynamics of Nanoclusters

579

chains over a broad range of sizes. The case of C chains is thus presented in Figure 5. In a given chain the strength is distributed over a broad range of frequencies. But there sticks out a pronounced peak at low frequency. It is sort of a plasmon peak due to the metallic behavior of the p electrons which can travel freely along the whole chain, in the spirit of the Mie picture. The s electrons are more localized but take part by their large dipole polarizability (in a way similar to the d electrons in noble metals). The linear geometry of the chain altogether leads to a strong size dependence of the plasmon frequency. The picture is here that a chain of electrons is oscillating relative to the chain of ions. This produces basically a dipole field whose energy is ∝d 2 /L where L is the length of the chain and d is the displacement of the cloud. The spring constant is thus ∝1/L and so is the plasmon frequency. Figure 5 shows the trend of the low-lying dominant peak with size N of the chain. Theoretical and experimental results are in agreement and the trends line up nicely with the predicted trend ∝N −1/2 for the longitudinal mode.

3. NANOCLUSTERS IN INTENSE LASER BEAMS 3.1. From Low to High Laser Intensities

be smoothed, while the Mie plasmon mechanism remains a valuable tool for understanding the qualitative behaviors of the systems. This evolution with laser intensity is not surprising. At low laser intensity the coupling to light mostly leads to electron oscillations with only occasional electron emission. In turn, for sufficiently high laser intensity the dominant primary mechanism of excitation is electron emission. But electron emission is soon slowed down by the net charge acquired by the cluster which tends to retain the electrons, otherwise ready to leave. This, at least, lasts as long as the system has not disappeared, for example by Coulomb explosion, when the net charge becomes too high. During this transient regime with quasi free electrons stripped from their parent atoms, but still bound to the cluster by its net charge, electrons more or less behave the same, whatever the type of cluster they do belong too. For example, the difference between metal and rare gas clusters (both of which under experimental investigation) thus essentially lies in the initial amount of free electrons, which turns out not to be a decisive factor for high laser intensities. In that sense, the differences between the various types of clusters tend to disappear for these large intensities. For intermediate intensities, the situation is of course a mixture, with traces of specific behaviors (to a given cluster type) and generic pattern.

3.1.1. Similarity in Diversity

3.1.2. Intense Laser Domain

As we have seen, the interaction of low fluence lasers with clusters leads to different patterns from one cluster to another depending on the type of the cluster, metallic, covalent, rare gas, etc. Still, we have also seen that in many cases the electronic response exhibits a qualitatively similar, more or less collective, behavior, as exemplified by the Mie model in metal systems. With increasing laser intensity the differences between the various types of clusters tend to

The investigation of the (nonlinear) dynamical properties of atomic clusters has lived up with the advent of table-top devices that produce subpicosecond laser pulses with intensities easily exceeding 1017 W/cm2 [39–45]. Due to the collective electron dynamics discussed, there may be a strong electromagnetic coupling to the Mie frequency that leads to nearly 100% absorption rates [46]. The clusters being isolated objects, no surrounding environment will absorb their excitation energy, in contrast to solids. The emission of high energy (keV) electrons [43], highly charged and very energetic ions [42] and fragments [44], as well as X-ray production [39, 47] are then the spectacular manifestations of this laser–cluster interaction.

8

C chains

3.1.3. Basic Mechanisms at Work ω [eV]

6

4

exp. Cl 2

TDLDA YB TDLDA BRS

0

0

0.1

0.2

0.3

0.4

0.5

0.6

N–1/2

Figure 5. Systematics of longitudinal plasmon resonance frequencies for N = 2–32. Results from various sources are compiled, experiments from [33, 34], CI calculations from [35, 36], TDLDA YB are TDLDA calculations from [37], and TDLDA BRS from [38].

Several models have been developed to explain the experimental features observed in the regime of “high” laser intensities [89–94]. All these models emphasize the major role played by the electrons inside the cluster. The basic mechanisms at work in intense laser–cluster interactions can be summarized as follows. The laser very quickly strips a sizable number of electrons from their parent atoms. These electrons form a reservoir of quasi free electrons, which can strongly couple to the laser when the electron density matches a critical value characteristic of the dipole eigenfrequency of the system (Mie resonance). The global response of the cluster is characterized by a heating of the electron cloud and electronic emission, in addition to the possibly enhanced dipole oscillations. The net charge and the high excitation energy acquired by the cluster lead to its final explosion. It should be noted that this violent scenario is, to a large extent, independent of the nature of the irradiated clusters. The general scenario we have just outlined

580 does not precisely specify the relative importance and the time scales of the various mechanisms at work. As a consequence, although the global scheme is generally accepted, the debate on the relative importance of the competing processes (various ionization mechanisms inside the cluster, role of the plasma resonance, of electron temperature, of net charge,    ) remains largely open and various models are still investigated to explain experimental trends.

Photodynamics of Nanoclusters

hν = 1.75 keV

hν = 2.25 keV

3.2. Rare Gas Clusters 3.2.1. High Energy Phenomena Experimental studies of large rare gas clusters with high intensity lasers started in the early 1990s. Multiphoton induced X-ray emissions have been observed by McPherson et al. in Kr clusters as early as 1993 [39]. The emission of radiation in the keV energy range was clear evidence of the appearance of new phenomena specific to clusters. This was later confirmed by many other groups; see Figure 6a [47, 50]. Sustained experimental efforts were then devoted to laser–cluster interaction in the high intensity regime, mainly in the Lawrence Livermore National Laboratory, the Blacket Laboratory (Imperial College), and in the CEA-Saclay. Most of these studies dealt with rare gas clusters because of the possibility to produce such large clusters in supersonic beams. The most striking experimental results are the observation of emitted ions with charges larger than those expected for isolated atoms exposed to the same laser intensity, the appearance of extremely energetic ions, and the production of electrons with kinetic energies in the keV range. We show in Figure 6b typical results obtained for xenon clusters. Here, a cluster of 2500 xenon atoms was irradiated by a laser with a peak intensity of 2 × 1016 W/cm2 [48]. Although most of the emitted ions have kinetic energies of a few keV, the mean kinetic energy is 45 ± 5 keV. Another remarkable aspect of this energy distribution is the presence of ions with energies up to 1 MeV. Figure 6c shows the energy distribution of electrons in conditions close to the previous ones (2100 xenon clusters irradiated by a laser with 15 × 1016 W/cm2 maximum intensity) [49]. Two bands appear. The first one is centered around 0.8 keV and is expected to be due to warm electrons emitted before the plasmon resonance occurs and the second one around 2.4 keV corresponds to hot electrons released by the cluster during the occurrence of plasmon resonance when important heating of electrons has been realized through collisions.

3.2.2. Dependence on Cluster Size and Laser Intensity Systematic studies of the response of rare gas clusters to intense laser fields have been performed by different groups. In Figure 7, we show the results of a systematic study of average electron and ion energies as a function of the cluster-jet backing pressure, which basically fixes cluster sizes in the case of rare gas clusters irradiated by long-pulse laser fields obtained by the CEA-Saclay group [51, 52]. The results from three different clusters, argon, krypton and xenon, are shown. We note that the mean size

Figure 6. Experimental results obtained after irradiation of clusters by intense laser beams. (a) X-ray generation from irradiation of large krypton clusters (∼5 × 105 atoms) with 130-fs, 790-nm laser with peak intensity 4 × 1017 W/cm2 . Reprinted with permission from [47], S. Dobosz et al., Phys. Rev. A 56, R2526 (1997). © 1997, American Physical Society. (b) Ion energy spectrum from 2500-atom xenon clusters irradiated by a peak intensity of 2 × 1016 W/cm2 . Reprinted with permission from [48], T. Ditmire et al., Phys. Rev. A 57, 369 (1998). © 1998, American Physical Society. (c) Electron energy distribution from 2100-atom xenon clusters irradiated by a laser with peak intensity 15 × 1016 W/cm2 . Reprinted with permission from [49], Y. L. Shao et al., Phys. Rev. Lett. 77, 3343 (1996). © 1996, American Physical Society.

of clusters grows approximately quadratically with the stagnant pressure and goes from 3 × 104 to 5 × 106 atoms in the entire pressure range. It appears that the electron-to-ion energy ratio depends on the constituents of the clusters and the highest ion energies were observed for krypton clusters in combination with relatively low electron energies. The huge difference between the electron and ion energies also

Photodynamics of Nanoclusters

581 Rare gas clusters

Rare gas clusters 200 30 ps, 5×1013 W/cm2, 1064 nm

400

Xe Kr

Krypton Xenon

350

Max ion energy (keV)

Electron Temperature [eV]

Argon

300

150

100

50

250 0

Electrons

102

200

103

104

105

Atoms per cluster 0

5

10

15

20

25

30

Max ion energy (keV)

Stagnant Pressure [bar]

30 ps, 5×1013 W/cm2, 1064 nm

18

Argon Krypton Xenon

Ion Temperature [keV]

16

100

10

14

Xe Kr

12

1016

1015

Intensity (W cm–2)

10

Ions 8 0

5

10

15

20

25

30

Stagnant Pressure [bar]

Figure 7. Average kinetic energies of electrons (upper panel) and ions (lower panel) for various backing pressures of the cluster gas. These results were obtained by irradiating rare gas clusters with a 30-ps, 1064-nm laser pulse with peak intensity 5 × 1013 W/cm2 . Reprinted with permission from [51], M. Lezius and M. Schmidt, in “Molecules and Clusters in Intense Laser Fields” (J. Posthumus, Ed.), p. 142. Cambridge Univ. Press, Cambridge, UK, 2001. © 2001, Cambridge University Press.

observed in the case of subpicosecond intense laser pulses (see Fig. 6) resists a simple interpretation but seems to lead, in this case, to the formation of a strongly heated, nonthermalized plasma. A similar systematic investigation has been conducted by the Blackett Laboratory group at Imperial College [53–55]. Figure 8 shows the maximum ion energies obtained for krypton and xenon clusters irradiated by high-intensity laser pulses as a function of cluster size (upper panel) and as a function of laser intensity (lower panel). The maximum ion energies as a function of cluster size (upper panel) were obtained with a 170-fs laser pulse with a peak intensity of 13 × 1016 W cm−2 . The maximum ion energy of Xe clusters rises from 8 keV at a cluster size of about 1000 atoms to a

Figure 8. Maximum ion energies from xenon (filled circles) and krypton (open circles) clusters measured as a function of cluster size (upper panel) with a 170-fs, 780-nm laser pulse with peak intensity 13 × 1016 W/cm2 , and as a function of laser intensity (lower panel) with a 230-fs, 780-nm laser pulse and for 5300-atom Xe and 6200-atom Kr clusters. Reprinted with permission from [53], E. Springate el al., Phys. Rev. A 61, 063201 (2000). © 2000, American Physical Society.

peak of 150 keV for clusters of ∼5000 atoms before falling to 100 keV as the cluster size is increased beyond 10,000 atoms. Kr clusters show a similar trend as Xe clusters, but the ion energies are lower. The experimental setup limits did not allow in this case an increase of the cluster size to the point where an optimum cluster size could be observed. The scaling of ion energies with laser intensity (lower panel) has been obtained with Xe clusters containing ∼5300 atoms (corresponding to a 43 Å radius) and Kr clusters with ∼6300 atoms (41 Å). Again, a similar behavior is observed for both rare gas species, with about 20% lower energies for Kr.

3.2.3. X-Ray Production Another systematic study concerning X-ray generation has been conducted by the same CEA-Saclay group for rare gas clusters submitted to ultrashort and intense lasers [47]. Figure 9 shows the obtained mean photon number per pulse as a function of the laser intensity (upper panel) and as a

Photodynamics of Nanoclusters

582

3/2

Kr . These the figure. Accordingly, the X-ray yield scales as N results suggest that the onset of X-ray production occurs at a pressure offset of Pc ≈ 3 bar, corresponding to a mean

Kr close to 104 atoms. This value would then cluster size N indicate the minimum cluster size for significant X-ray production in the present experimental conditions.

3.3. Metal Clusters

Figure 9. X-ray photon yield measured as a function of laser peak intensity (upper panel) for Kr clusters of estimated mean size 7 × 105 , and as a function of backing pressure and the corresponding mean cluster size on the upper scale (lower panel) with a laser peak intensity of 4 × 1017 W/cm2 . Reprinted with permission from [47], S. Dobosz et al., Phys. Rev. A 56, R2526 (1997). © 1997, American Physical Society. The solid line in the upper panel and the dashed line in the lower one represent power law fits to the experimental results (see text).

function of stagnant pressure (or cluster size, lower panel) for krypton clusters. The mean photon number as a function of laser intensity (upper panel of the figure), obtained for clusters of estimated mean size 7 × 105 atoms corresponding to a backing pressure P0 = 195 bar, has been fitted by

p ∝ IL3/2 (solid line). This scaling is typical a power law N for a saturated process in the focus of an intense laser field with a Gaussian intensity profile [47]. The dependence of the mean photon number on the stagnant pressure (lower panel), obtained with a laser intensity of 4 × 1017 W/cm2 ,

p = P0 − Pc 3 with Pc ≈ 3 bar. has also been fitted with N

Kr is also shown at the top of The mean cluster size scale N

The first experiment on metal clusters in the nonperturbative regime was performed by Gerber and co-workers on mercury clusters [56, 57]. The mercury atom has a 5d 10 6s 2 closed electronic shell configuration. Diatomic and small mercury clusters are predominantly van der Waals bound systems. However, the electronic structure strongly changes with increasing cluster size and finally converges toward the bulk electronic structure where the 6s and 6p bands overlap, giving mercury its metallic properties. Hgn (n ≤ 80) clusters were produced in a cold supersonic beam and were ionized by femtosecond laser pulse of 50–120 fs duration in the wavelength range from 255 to 800 nm. The originality of these experiments arises from the study of the cluster behavior in the transition region, going from the low field regime (intensity ∼ 1011 W/cm2 ) to the strong field one (intensity ∼ 1013 W/cm2 ). Figure 10 represents the time of flight mass spectra of the mercury clusters obtained at 800 nm. For a laser intensity of 1011 W/cm2 (Fig. 10a) a “normal” cluster size distribution is observed constituted by ionic fragments q+ Hgn in the range q = 1 2. At 1012 W/cm2 (Fig. 10b), the + Hg signal increases while the signal of other fragments decreases. Moreover, the ratio between singly and doubly charged clusters of a given size does not change, which indicates that the small fragments come from the fragmentation of larger cluster ions. Increasing the intensity to 2 × 1012 W/cm2 (Fig. 10c), highly charged mercury atoms (up to Hg5+ ) appear. Note that the direct multiphoton ionization of Hg atom requires an intensity of about one order of magnitude higher. Finally, at the highest applied intensity of 1013 W/cm2 , the signal due to molecules and clusters disappears and only atomic ions, singly and multiply charged, are observed. In this case the whole cluster breaks into pieces by Coulomb explosion. These results clearly show that the energy absorption is strongly enhanced at relatively small laser intensity and the interaction process is beyond the classical multiphoton ionization. Other investigations on metal clusters have been performed with the high intensity regime. Exhaustive studies were performed at the CEA-Saclay by Lebeault et al. [58] for lead clusters exposed to an intense laser field of 1015 W/cm2 intensity, whereas platinum clusters (and lead clusters as well) excited by strong femtosecond laser pulses with intensities up to 1016 W/cm2 were studied by MeiwesBroer and co-workers [44, 59–61]. At such high intensities, it turns out that the metallic character of the clusters does not play an important role and the results can be interpreted in terms of plasmon resonance. In the Meiwes-Broer experiments, the cluster size distribution is estimated to be below about 100 atoms. The interaction with the laser pulse makes no clusters survive the excitation. The net result is the production of highly charged ions. Ion charges as high as Pt20+ were observed at 2 × 1015 W/cm2 whereas only quadruple

Photodynamics of Nanoclusters

583

Figure 11. Abundances of platinum ions Ptz+ (z = 1     11) for different laser pulse widths ranging from 140 fs to 1 ps at a fixed pulse energy of 5 mJ. From left to right, the pulse intensity drops by a factor 7. Reprinted with permission from [44], L. Köller et al., Phys. Rev. Lett. 82, 3783 (1999). © 1999, American Physical Society.

model (see Section 4.5). One can then conclude that the fragments with charges below 10+ are driven by Coulomb forces, whereas the higher charged fragments are accelerated by hydrodynamics forces. Figure 10. Time-of-flight mass spectra obtained by irradiating Hgn clusters (n ≤ 80) with femtosecond (50–100 fs) 800-nm laser pulses with peak intensities (a) 1011 W/cm2 , (b) 1012 W/cm2 , and (c) 2 × 1012 W/cm2 . Reprinted with permission from [56], B. Lang et al., Z. Phys. D 40, 1 (1997). © 1997, Springer-Verlag.

ionization is obtained if isolated atoms were irradiated by the same laser intensity. The production of multiply charged platinum and lead cluster ions shows a significant increase of efficiency when increasing the laser pulse length. Figure 11 shows the abundance of platinum ions as a function of the pulse width for a given pulse energy of 5 mJ. The important feature is that the optimum pulse width of 600 fs that gives the largest abundances corresponds to neither the shortest pulse width for which the highest intensity is obtained nor the longest time duration for which the interaction time is longest. In the Saclay experiments, small size lead clusters with a few hundred of atoms are irradiated with laser pulses with duration between 60 fs and 2.5 ps and a laser intensity of about 1015 W/cm2 . The kinetic energies and ionic charge of fragments were studied as a function of the laser intensity and pulse duration. Highly charged Pbn+ ions up to n = 26 have been detected with kinetic energies up to 15 keV. Moreover, the measure of the kinetic energy of ionic fragments versus the charge state exhibits a quadratic dependence for charge states lower than 10 and a linear dependence for higher charge states. These results were interpreted in the framework of the so-called nanoplasma

3.4. Other Materials Experiments with molecular (or mixed species) clusters irradiated by high-intensity ultrafast laser pulses have also been conducted by several groups [63–65]. Mixed species clusters offer the advantage to allow the use of elements which may hardly bind together to form “pure” clusters. For example, the formation of large hydrogen clusters in gas jets (or its isotopes for fusion experiments; see next section) requires very high backing pressures and cryogenic cooling. Hydrogen iodide (HI) gas, investigated by Tisch et al. [63], or deuterated methane (CD4 ), investigated by Grillon et al. [65], forms large clusters very readily at low pressures even at room temperature. Figure 12 shows the energy spectra of iodine ions (upper panel) and protons (lower panel) obtained by irradiating HI clusters of an estimated mean size of 60,000 molecules by 260-fs, 780-nm laser pulses at a peak intensity of 2 × 1016 W cm−2 . The maximum kinetic energy of iodine ions is ∼92 keV, while the mean value is about 8.6 keV. The mean and maximum kinetic energies of protons are approximately 270 eV and 2.5 keV, respectively. A monotonous increase of both iodine and proton energies was also observed with increasing the size of HI clusters [63]. The observation of energetic protons and iodine ions indicates that molecular clusters can indeed be used to produce energetic ions from a range of much broader elements than could be expected otherwise. This was also the case for the experiment by Grillon et al. [65] using deuterated methane (CD4 ) to produce energetic deuterium ions for fusion experiments.

Photodynamics of Nanoclusters

584 Hl clusters

Figure 12. Measured energy spectra of iodine ions (upper panel) and protons (lower panel) from explosions of 60,000-molecule HI clusters irradiated by 260-fs, 780-nm pulses at a peak intensity of 2 × 1016 W/cm2 . Reprinted with permission from [63], J. W. G. Tisch et al., Phys. Rev. A 60, 3076 (1999). © 1999, American Physical Society.

Photoexcitation of other types of nanoclusters (covalent, molecular) has also been performed by several groups. Bescós et al. have studied the photoionization and photofragmentation of large neutral silicon clusters Sin with sizes up to n ≈ 6000 irradiated by both nanosecond pulses with intensities 104 –105 W cm−2 and femtosecond pulses with intensities ranging from 108 to 5 × 1011 W cm−2 [66]. Dobosz et al. have studied X-ray generation from carbon dioxide (CO2 ) clusters with sizes of 80–100 nm irradiated by a 60-fs laser pulse with peak intensity of 1018 W cm−2 [64]. Schumacher et al. have investigated the photoionization of small covalent carbon clusters submitted to strong femtosecond laser pulses with up to 1016 W cm−2 [59]. Their results are shown in Figure 13 compared to the results obtained with platinum and lead clusters submitted to the same laser pulse. Photoionization and photofragmentation experiments of carbon fullerenes C60 have also been performed using subpicosend intense laser pulses by Campbell and co-workers [68–70]. Photoelectron as well as Cn+ 60 spectra have been determined for different pulse durations and intensities. As an example, the Cn+ 60 ion spectra are shown in Figure 14 for fullerenes submitted to a 35-fs, 795-nm laser pulse with a peak intensity of 1014 W/cm2 . The most striking feature of these mass spectra is the appearance of multiply 2+ charged species. Metastable fragments C+ 60−2n , C60−2n , and 3+ C60−2n are also clearly observed indicating a strong coupling of electronic excitation energy to vibrational degrees of freedom leading to fragmentation.

Figure 13. Mass spectra of ionized carbon, platinum, and lead clusters and atomic ions after irradiation of neutral clusters by femtosecond laser pulses of 3 × 1015 W/cm2 at 800 nm. In contrast to carbon where only a small fraction of the ion intensity originates from the higher charged ions, the metal cluster spectra show that the main intensity emerges from the abundance of atomic Ptz+ and Pbz+ . Reprinted with permission from [60], T. Döppner et al., Int. J. Mass Spectr. 192, 387 (1999). © 1999, Elsevier Science.

3.5. Fusion Large opportunities are opened by the studies of laser– cluster interactions. This can be illustrated by the experimental realization of nuclear fusion in deuterium by Ditmire et al. at LLNL [67]. In this experiment one irradiates a large cluster of deuterium D2 by a laser with a peak intensity of 1016 W/cm2 . The explosion of clusters results in their fragmentation into D+ ions with kinetic energies of a few keV. This is sufficient to realize nuclear fusion in the collisions D+ + D+ → He3 + n that take place in the cluster beam. The detection of neutrons with 2.45 MeV kinetic energy clearly confirms the phenomenon. The experiment was later repeated by Grillon et al. using, instead of D2 clusters, deuterated methane (CD4 ) clusters leading to much higher D+ energies of the order of 45 keV compared to the 2.5 keV obtained by Ditmire et al. [65] (see Fig. 15). In this experiment, the number of neutrons generated per shot at the maximum intensity of several 1017 W cm−2 was estimated as 7000 (over the whole solid angle). The pulse width of neutrons produced by intense laser irradiation of deuterium clusters was measured by Zweiback et al. and found to be under 100 picoseconds in duration near the source [71]. Such a short time-scale neutron pulse may represent a table-top source of neutrons that enables a new class of ultrafast neutron pump–probe experiments.

Photodynamics of Nanoclusters

585 D - D fusion

C60 ionization and fragmentation 1000

Laser focal diameter d

(a)

Fast deuterons

uls

nse

rp ase

e in

l

Inte

Hot plasma filament Fusion burn time ~ d/vion

104

(b)

6 4

Neutron yield

2

103 6 4 2

102 6 4

2

(c)

3

4

5

6

7

Figure 14. Ionization and fragmentation patterns of C60 irradiated by a 35-fs, 795-nm laser pulse with a peak intensity of 1014 W/cm2 : (a) the whole spectrum; (b) the spectrum of singly ionized fragments; (c) the spectrum of doubly ionized fragments. Reprinted with permission from [68], M. Tchaplyguine et al., J. Chem. Phys. 112, 2781 (2000). © 2000, American Institute of Physics.

4. TOWARD QUANTITATIVE MODELS 4.1. General Strategy Our discussions have shown that while simple models allow one to understand basic mechanisms of interaction and qualitative behaviors of irradiated nanoclusters, they fail in providing quantitative access to experimental results. It is thus highly desirable to work out more elaborated approaches for such problems. The starting point of any quantum many body problem is, of course, the many body Schrödinger equation serving as basis for sequences of approximations. In nanometer scale systems far from equilibrium, a quantum description remains far beyond today’s computers, even at an approximate level (e.g., mean field approaches). Mean field approaches, however, provide crucial guidelines for deducing simplified models [7]. It has to be remembered

Most probable energy (keV)

Laser intensity (1017 W/cm2) 160

C+

120

80

D+ 40

0 1

2

3

4

5

6

Laser intensity (1017 W/cm2) Figure 15. Upper part: Experimental concept for an ultrafast neutron source. Reprinted with permission from [71], J. Zweiback et al., Phys. Rev. Lett. 85, 3640 (2000). © 2000, American Physical Society. Middle panel: Intensity dependence of the total neutron yield over the whole solid angle. Lower panel: Most probable energies of the measured distributions of C+ and D+ . Reprinted with permission from [65], G. Grillon et al., Phys. Rev. Lett. 89, 065005 (2002). © 2002, American Physical Society.

here that we want to consider situations in which electrons are deeply involved into the dynamics. Irradiating a cluster by a laser will primarily involve electrons, which will respond in various ways (collective oscillations, emission,    ) depending both on the laser characteristics and on the nature of the cluster (metallic, covalent, rare gas,    ). One thus needs to explicitly treat electronic dynamics and this eliminates the powerful approaches based on atomic molecular dynamics, which by construction cannot accommodate electronic excitations. The basic challenge is thus an efficient description of electronic dynamics. Of course, ions

Photodynamics of Nanoclusters

586 will also couple to the laser field. But the ionic response takes place on a much longer time scale (typically 100– 1000 fs) than the electronic one (typically 1–10 fs). A key constraint is hence to describe electronic dynamics both on the short and the long time scales and in a sufficiently simple approach to allow the study of sufficiently large systems. Along the line of simplifying the original quantum many body problem, the step from the quantum to the semiclassical world is certainly an efficient, though reliable, step. Indeed, semiclassical methods have been developed and used in cluster physics for several years now, in particular in the case of metal clusters [8–10]. The term semiclassical deserves some explanation. By semiclassical we mean approaches in which electrons are treated as classical particles but augmented with a simplified Pauli exclusion principle (namely, a given maximal occupation in elementary phase space cells). The step from quantum to semiclassical worlds allows one to replace electronic wavefunctions by a phase space occupation function f r p representing the ensemble of all electrons together. Still, basic quantum properties such as the Pauli principle are fulfilled by f , hence the label semiclassical. A great advantage of semiclassical approaches is that they allow the treatment of large clusters [10], but with a “maximum” of microscopic inputs. Furthermore, as is well known from basic quantum mechanics, the larger the involved energies, the less necessary the quantum details. Semiclassical models thus constitute an even more justified approximation in situations far from equilibrium. Finally, it should be noted that, from the more formal side, semiclassical methods open the door to the world of kinetic theory with its well established techniques for dealing with strongly interacting particles [11]. Indeed, semiclassical approximations to quantum mean-field calculations constitute a well established starting point for going beyond a mere mean-field description by including explicit electron–electron collisions, in the spirit of kinetic theory in electronic systems. This “beyond the mean-field” path has been attacked only very recently [12, 13] but provides a useful, and sometimes indispensable, tool for investigating the dynamics of clusters in the strongly out of equilibrium domain. We shall present some of these developments at the end of this chapter. Before reaching these latest developments we shall consider even more simplified approaches and show their interest and limitations for the problems we are interested in here.

4.2. Quantal Framework The simplest semiclassical approximation can be derived directly from the quantal many body problem. Even in simple small systems, solving the quantal many body problem represents a formidable task, still on the edge of modern computers. At the nanometer scale it is hardly conceivable to consider a detailed microscopic approach containing all electronic correlations. One has in any case to recur to simplified descriptions. The major difficulty lies here at the side of electrons which do behave quantally in many situations, while ions can often be treated as classical particles. A most efficient and robust, and still quantal, scheme is provided here by density functional theory [7], which

allows one to reformulate the quantum many body electronic problem in terms of an effective mean field Hamiltonian hˆ involving the local electronic density e r only. The many body Schrödinger equation can then be recast into a set of coupled, one-electron equations following the famous prescription of Kohn and Sham [72]. When time dependent problems are considered the former strategy can be extended and the time dependent Schrödinger equation for the many electron system replaced by a set of time dependent one-electron equations (time dependent Kohn–Sham equations). Of course a key ingredient of this approximation ˆ scheme lies in the choice of the one-electron Hamiltonian h. We shall not discuss here the many (interesting) researches on the topic, which actually go much beyond cluster physics. Rather we shall mention the simplest approximation, the so-called LDA (local density approximation) in which hˆ is expressed as a function of the density e r, namely hˆ = ˆ ˆ e r. This simple, although robust, approximahr = h tion provides a starting point for many efficient applications, both in cluster and outside cluster physics, also in the case of time dependent problems [73].

4.3. Vlasov Equation The semiclassical limit thus starts from the time dependent version of LDA, the so-called TDLDA (time depenˆ dent LDA). In terms of the one-body density operator , TDLDA takes a simple, compact form, well suited to apply semiclassical approximations [74]: 1 ˆ ˆ ! ˆ ˙ˆ = h i


(2)

It furthermore offers a form adapted for going “beyond” the mean field in the realm of kinetic equations. The semiclassical limit of Eq. (2) is the so-called Vlasov equation, well known in plasma physics [75]. But here it emerges as a semiclassical approximation of a truly quantal equation. Following the usual recipe, it can be formally obtained from Eq. (2) by replacing the density operator ˆ by a one-body phase space distribution f r p t and the commutator with Poisson brackets [74]. This then leads to #f r p t + $f  h% = 0 #t

(3)

where the mean field Hamiltonian h is now classical but still a functional (or a function, in the case of LDA) of the (semiclassical) electron density &r t. The latter is computed
by 3integrating f r p t over momentum space &r t = d p f r p t. It should, however, be noted that the semiclassical approximation leading from Eq. (2) to Eq. (3) has to be performed with due caution. The problem is here of a formal nature. Indeed a semiclassical approximation such as the one performed previously corresponds to the lowest order term of an expansion in terms of
. And the convergence of this
expansion may raise some difficulties, in particular when the mean field Hamiltonian is selfconsistent, namely itself depending on the actual density &r t. A thorough discussion of this problem can be found in [76–78] and we shall not further elaborate on it here. Another difficulty with Eq. (3) concerns the capability of

Photodynamics of Nanoclusters

4.4. Time-Dependent Thomas–Fermi Approximation The Vlasov approach provides a phase-space description of electron dynamics. This allows one to treat a wide range of electronic excitations in systems of various sizes and in very different dynamical contexts. In particular it can accommodate reasonably nonlinear situations, as encountered, for example, in the interactions between clusters and intense laser beams. In particular the full phase space description provided by the phase space distribution f r p t allows one to treat possibly large distortions of the local Fermi sphere, a feature typical of strongly out of equilibrium dynamics. But in the case of metal clusters two effects tend to suppress these local distortions. First, the dominating presence of the Mie plasmon mode, which carries little momentum space anisotropy, tends to soften effects linked to actual distortions of the Fermi sphere. In addition, electron–electron collisions (which go beyond the actual mean field picture; see Section 4.6) act on very short time scales of order a few fs, so that they tend to remove remaining momentum sphere anisotropies very quickly. To a very good approximation, the local momentum distribution of electrons thus deviates only a little from sphericity [82]. This provides a justification for going one step further down in terms of the semiclassical approximation: integration over momentum space then leads to the so-called time dependent Thomas–Fermi (TDTF) approximation. The TDTF approach is exactly a hydrodynamical reformulation of the original problem, assuming a Thomas–Fermi kinetic energy for the electrons. It is interesting to note here that the TDTF approximation can be formally derived either from the Vlasov equation by integration over momentum space, or directly from TDLDA by a proper choice of a model wavefunction [82]. In both cases one ends up with the same set of coupled equations for the local electron density &r and local velocity field ur = ' ( [82]. At this stage it should also be noted that TDTF represents a genuine density-functional method (see [83]), as it only involves the actual local density &r and its associated flow. In cluster dynamics, the full three-dimensional (3D) electronic TDTF approximation has been explored in [82, 84]. It was shown in these publications that TDTF provides a fairly good approach to TDLDA, even for moderate excitations, as is illustrated in Figure 16. Furthermore, the simplicity of the picture allows one to treat systems of large sizes, contrarily to more detailed approaches. It is thus an ideal tool of investigation of nanocluster dynamics. It was also used in simpler versions of the theory (restricted to 1D geometry) in direct relation to irradiation by intense lasers [85–87]. Furthermore, an explicit account of the ionic background by means of pseudo-potentials allows one to accede the local

+

N a 93 Ar b i t r a r y u n i ts

numerical simulations to preserve Fermi statistics on long times [79]. Indeed, as already mentioned, the term semiclassical precisely requires that the theory does preserve some quantal features. This holds in particular for the fermionic nature of the involved particles which is not guaranteed in a purely classical Vlasov equation. This question, however, is highly technical and we refer the reader to appropriate references on the topic [80, 81].

587

2

2.5

3

3.5

4

Energy (eV) Figure 16. Time Dependent Thomas Fermi calculation of the optical response of a medium size metal cluster compared to experimental data. Cluster is Na+93 ; experimental data have been obtained at temperature T = 105 K, which explains part of the spreading of the optical peak. Reprinted with permission from [82], A. Domps et al., Phys. Rev. Lett. 80, 5520 (1998). © 1998, American Physical Society.

behavior of collective currents during Mie plasmon oscillations. The TDTF picture, with its underlying hydrodynamical structure, thus provides an ideal tool of investigation of the nature (collective or not) of the optical response [84]. Coupling to ion dynamics is also possible and has already been investigated at the Born–Oppenheimer level in [88]. Extensions to nonadiabatic electron dynamics are also in reach, although they have not yet been investigated.

4.5. Rate Equations and High Intensity Regime Along the series of more and more simplified models, the next step beyond the hydrodynamical picture provided by TDTF is to consider rate equations for describing the dynamics. The relevant variables then reduce to a set of integral quantities (radius, ionization,    ), the evolution of which being followed in time. Such an oversimplified picture may, however, make sense when only gross properties of the system may be attained. This is typically the case of high intensity irradiations of clusters. At sufficiently high laser intensities, the details of the excitation mechanism tend to be washed out to the benefit of a global plasmalike behavior of the set of quasi free electrons. It is then sufficient, at least for a first step, to remain at the level of a gross description of the system, in terms of a few global variables, the primary goal being to understand the key mechanisms at work. This is the spirit of the so-called nanoplasma model developed by Ditmire et al. at Lawrence Livermore National Laboratory [95]. This model offers, at the level of global cluster characteristics, a complete scenario of the interaction process, taking into account ionization, heating, electronic emission, and cluster expansion simultaneously. In contrast to other models, it treats the polarization field radiated by the charges explicitly. In this model, the cluster is treated as a spherical plasma—a nanoplasma—where the plasmon resonance takes place. Whether the cluster can be treated as a plasma

Photodynamics of Nanoclusters

588 or not has given rise to a controversy, though. For example, classical dynamics simulations including the Coulomb field of the ions but neglecting the polarization field indicate that the electrons are quickly removed at the beginning of the interaction even from large clusters [96]. That makes the existence of a nanoplasma questionable. Until now there has not been any time-resolved measurement at the femtosecond scale to settle the question. The nanoplasma model consists essentially of a few rate equations expressing the time variation of the number Nj of ions of a given charge state j due to ionization    tun  dNj coll = Wjtun + Wjcoll Nj−1 − Wj+1 + Wj+1 Nj dt

(4)

with j running from 0 to the maximum charge state considered. Here, Wjtun and Wjcoll represent the rate of direct optical ionization and collisional ionization of ions with charge j, respectively. These rate equations are supplemented by an additional equation expressing charge conservation inside the cluster and two other equations expressing the cluster expansion and heating. For more details, we refer the reader to the original paper [95]. As an example, the computed temporal evolution of an exploding cluster of 5000 Xe atoms irradiated by a 100-fs, 780-nm laser pulse with a peak intensity of 1016 W/cm2 is illustrated in Figure 17 [97]. Time t = 0 fs is the time at which the laser reaches its maximum intensity. We have represented in the upper part of the figure the time variation

of the electron density inside the cluster normalized to the critical density nc and in the lower part the electric field inside the cluster (solid line) together with the external one (dashed line). At the rising edge of the pulse, quasi free electrons are produced inside the cluster from neutral atoms by tunnel ionization. The rapid increase of the number of quasi free electrons leads the system through a first modest resonance at t ≃ −135 fs when the density matches three times the critical density nc . The electric field is then shielded due to the high electron densities reached. The tunnel ionization rate falls off, but electrons are still created through thermal collisions. From times t ≃ −100 fs onward, the electron temperatures are high enough for some electrons to leave the cluster leading to a fall of the density inside the cluster until time t ≃ 10 fs when the second, more important resonance takes place. The inner field is then amplified to 1.5 times the external field and electronic temperature reaches 1.2 keV. The rate of electron emission increases consequently giving a highly positively charged cluster which will explode due to the huge Coulomb repulsion between ions. Even though the model provides a good description of the gross features of the interaction of intense lasers with clusters, it fails to give a good account of all the experimental results observed, especially those far from average values. For example, the model agrees with the average ion charges observed experimentally but fails in describing the far tails of ion charge distributions which are at the origin of X-ray generation.

102

4.6. Beyond Mean Field—The Vlasov–Uehling–Uhlenbeck Equation ne/nc

1

Xe5000 10–2

λ = 780 nm I = 1016 W/cm2 FWHM = 100 fs

–4

10

Eext Eint

Field [1011V/m]

3

2

1

0 –300

–200

–100

0

100

200

300

Time [fs]

Figure 17. Temporal history of an exploding 5000-atom Xe cluster irradiated by a 100-fs, 780-nm laser pulse with peak intensity 1016 W/cm2 . Upper part: electron density normalized to critical density nc ; lower part: amplitudes of internal (solid line) and external (dashed line) electric fields.

As is well known, the Vlasov equation represents the basic level of the hierarchy of many body dynamical equations [98]. The TDTF approach of course does not go beyond Vlasov in that respect, as it only represents an average of the Vlasov dynamics over momentum space. Neither do, of course, rate equations, which provide an even grosser picture of the dynamics. Furthermore, as already discussed, both TDTF as well as Vlasov remain at the pure mean field level and are thus a priori only suited to situations where dissipative effects are not too large, actually just as the quantal cousin TDLDA. But dynamical correlations, not included in mean field, do play a key role in the dynamics of highly dissipative systems. Remembering the underlying kinetic theory picture, it is no surprise that these correlations show up at the level of two-body effects. They furthermore usually take place at a faster time scale than the mean field motion. This is the justification of treating them by instantaneous two body collisions, which formally leads to the inclusion of a Markovian collision term to complement the mean field theory. This yields the well known Vlasov–Boltzmann equation for classical systems. In the case of dense fermion systems, such as metal clusters, atomic nuclei, and liquid helium, Fermi statistics plays a central role in many situations. This then leads to the so-called Vlasov–Uehling–Uhlenbeck (VUU) scheme, as proposed in the late 1920s and early 1930s [99, 100]. This VUU scheme has furthermore been intensely studied during the last two

Photodynamics of Nanoclusters

decades in nuclear physics [101–103]. The resulting VUU equation reads #f + $f  h% = IVUU f r p t! (5) #t
with IVUU = dp2 dp3 dp4 W 12 34f12in f34out − f12out f34in  where W 12 34 is the collision rate, proportional to the elementary cross section d,/d , which constitutes a key ingredient of the approach. In Eq. (5), “in” and “out” label the distribution of particles entering or exiting a two body collision (12 ↔ 34), such that fijin = fi fj , fklout = 1 − fk 1 − fl , with the short hand notation fi = f r pi  t. The weights are in a way similar to a standard Boltzmann collision term but augmented by the Pauli blocking factors in the exit channel fklout . The latter terms impose that no more than one particle, or two with opposite spins, can occupy an elementary phase space cell of volume 2
3 . This blocking factor plays a dramatic role for electronic systems [104], in particular at zero temperature where it fully blocks electron–electron collisions, as expected. In dynamical situations, when the system is hot or sufficiently excited, phase space opens up and two body collisions start to play a (possibly dominating) role. It is important to note here that the VUU scheme only represents a first exploratory step in the direction of accounting for dynamical correlations. Of course one should be cautious with the possible double counting of interactions between the effective interaction in the mean field term and the collision term, an effect which has not yet been truly explored in the case of clusters. However, results obtained in related fields of physics, in which a similar question has been addressed in full detail, show that with minimal precautions one can circumvent this problem. Relying on the experience gathered in nuclear dynamics [105] and in the physics of plasmas [106], or liquid helium [107], one can assume that there is no problem as long as collisions are treated in a Markovian approximation (i.e., as instantaneous) [105]. The rule is then to use together the LDA for the mean field and a screened Coulomb cross section for the collision integral. In the case of metal clusters an explicit reevaluation of the cross section [12] actually has lead to values in agreement with the bulk values of condensed matter calculations [104]. With such an established framework, one can now envision direct applications in realistic cases [13], even with coupling to ionic molecular dynamics [108].

5. CONCLUSIONS We have seen that the interaction of light with clusters may lead to various scenarios depending on the intensity of the lasers used for probing the clusters. In the optical response regime clusters are only slightly perturbed and respond in a correlatively gentle manner. Optical response nevertheless provides a bunch of interesting information on clusters, in particular because of its marked size dependence. The case of irradiations by intense lasers opens the door to a bunch of new challenging phenomena such as the production of highly energetic species (photons, electrons, ions). In this domain of physics the interaction process between the cluster and the laser is so strong that it leads to a complete destruction

589 of the cluster. At variance with the optical response regime it thus does not provide direct information on the clusters themselves, which constitute laboratories rather than true objects of studies. However, and this time in a similar way as in the low intensity regime, we have also seen that, again, the capability of the electron cloud to respond collectively to the external field remains a key issue. The dominant role of this plasmon is obvious for metallic systems, whatever the laser intensity. But is also indeed present in the case of rare gas clusters irradiated by sufficiently intense lasers. In this spirit irradiation of clusters by intense lasers leads us to plasma physics in the framework of the so-called nanoplasma model. The latter original model, similarly to the simple Mie model of the optical response, does provides interesting insights into the physical mechanisms at work. Both these simple models, however, only allow a qualitative description of the experiments. They miss many details, starting with the crucial finite size effects. The step from basic simple models toward more elaborate approaches for describing these various dynamical situations is not an easy one. Indeed fully quantal calculations are much too involved to be applied to such huge systems in strongly out of equilibrium situations. One is thus bound to find appropriate approximations, keeping as much as possible of the microscopic world. Semiclassical methods offer here an appealing possibility. They have already been a little explored in the context of metal cluster dynamics, but one is still far away from having fully exploited the many opportunities they offer, in particular for large systems out of equilibrium. More developments of these methods are thus to be expected in the field of the dynamics of nanometer scale clusters irradiated by lasers of various intensities.

GLOSSARY Density functional theory (DFT) A systematic way to develop a mean-field theory with purely local potentials. It proves the existence of a local representation for the nonlocal exchange as well as correlation terms and it set the framework for practical approximations (see LDA). Landau damping or fragmentation A mechanism, first introduced by Landau, which describes the damping of collective degrees of freedom due to their coupling to single-particle degrees of freedom. Local density approximation (LDA) The most widely used realization of DFT. It uses the exchange-correlation energy of the homogeneous electron and interprets it as a function of the now local density. By density variation, one thus derives a local single-particle potential depending on the local density. The kinetic energy remains approximated (in contrast to the Time dependent Thomas-Fermi approximation) and the single-particle wavefunctions remain the relevant degrees-of-freedom. Mean field theory The description of a many-particle system in terms of independent particles moving each in a common effective single-particle potential. This mean-field potential, in turn, depends on the particle density summed up from the single-particle densities. A typical example is the Hartree-Fock theory where the mean-field potential becomes non-local in the exchange term.

590 Mie plasmon, Mie model Collective description of the dominant dipole mode as small oscillations of the frozen electron cloud against the ionic background. The original formulation was done for a metal sphere using dielectric theory. Nanoplasma model A phenomenological model which describes the interaction of intense laser pulses with large atomic clusters in terms of macroscopic variables. The laser field ionizes individual atoms creating a plasma of quasi-free electrons and ions of the size of the nanocluster, the nanoplasma. Time-dependent LDA (TDLDA) The time-dependent generalization of the LDA employing a time-dependent variational principle instead of the stationary Ritz variational principle. Time dependent Thomas-Fermi approximation An approximation to TDLDA in which one also replaces the kinetic energy by a functional of the local density, called the Thomas-Fermi kinetic energy. The system is then described purely in terms of the local one-particle density. Vlasov equation The semi-classical limit of timedependent mean-field theory. It describes the propagation of the single-particle phase-space distribution under the influence of a mean-field potential. Vlasov-Uehling-Uhlenbeck equation An extension of the Vlasov equation taking into account collisions among particles (here electrons) in a Boltzmann-like collision term including Pauli exclusion principle.

ACKNOWLEDGMENTS We thank G. Bertsch, M. Brack, F. Calvayrac, A. Domps, C. Guet, G. Gerber, H. Haberland, C. Kohl, S. Kümmel, K. H. Meiwes-Broer, L. Schweikardt, S. Voll, and C. Ullrich for many inspiring and motivating discussions. We also thank exchange program PROCOPE number 99074 and Institut Universitaire de France for financial support during the realization of this work.

REFERENCES 1. Small particles and inorganic clusters, in “ISSPIC8” (H. H. Andersen, Ed.), Z. Phys. D 40, 1–578 (1997). 2. S. Sugano and H. Koizumi, “Microcluster Physics,” Springer Series in Materials Science, Vol. 20. Springer-Verlag, Berlin, 1998. 3. T. P. Martin, “Large Clusters of Atoms and Molecules,” NATO ASI Series in Applied Sciences. Plenum, New York, 1995. 4. H. Haberland, “Clusters of Atoms and Molecules 1—Theory, Experiment, and Clusters of Atoms,” Springer Series in Chemical Physics, Vol. 52. Springer-Verlag, Berlin, 1994; H. Haberland, “Clusters of Atoms and Molecules 2—Solvation and Chemistry of Free Clusters, and Embedded, Supported and Compressed Clusters,” Springer Series in Chemical Physics, Vol. 56. SpringerVerlag, Berlin, 1994. 5. B. Piraux, A. L’Huillier, and K. Rzazewski, “Super-intense LaserAtom Physics,” NATO ASI Series B, Vol. 316. Plenum, New York, 1993. 6. “Proceedings International Conference on Laser Physics,” Laser Phys. 11 (2001). 7. E. K. U. Gross, J. F. Dobson, and M. Petersilka, Top. Curr. Chem. 181, 81 (1996).

Photodynamics of Nanoclusters 8. M. Gross and C. Guet, Z. Phys. D 33, 289 (1995). 9. L. Féret, E. Suraud, F. Calvayrac, and P. G. Reinhard, J. Phys. B 29, 4477 (1996). 10. L. Plagne and C. Guet, Phys. Rev. A 59, 4461 (1999). 11. R. Balescu, “Equilibrium and Non-Equilibrium Statistical Mechanics.” Wiley, New York, 1975. 12. A. Domps, P.-G. Reinhard, and E. Suraud, Phys. Rev. Lett. 81, 5524 (1998). 13. A. Domps, P.-G. Reinhard, and E. Suraud, Ann. Phys. NY 280, 211 (2000). 14. M. Gaudry, J. Lerm, E. Cottancin, M. Pellarin, J.-L. Vialle, M. Broyer, B. Prvel, M. Treilleux, and P. Mlinon, Phys. Rev. B 64, 085407 (2001). 15. G. Mie, Ann. Phys. Leipzig 25, 377 (1908). 16. U. Kreibig and M. Vollmer, “Optical Properties of Metal Clusters,” Springer Series in Materials Science, Vol. 25. SpringerVerlag, Berlin, 1993. 17. P. Ring and P. Schuck, “The Nuclear Many-Body Problem.” Springer-Verlag, Berlin, 1980. 18. C. Bréchignac, Ph. Cahuzac, J. Leygnier, and A. Sarfati, Phys. Rev. Lett. 70, 2036 (1993). 19. C. Bréchignac and Ph. Cahuzac, Com. Atom. Mol. Phys. 31, 215 (1995). 20. U. Kreibig and M. Vollmer, “Optical Properties of Metal Clusters,” Springer Series in Materials Science, Vol. 25. SpringerVerlag, Berlin, 1993. 21. M. Brack, Rev. Mod. Phys. 65, 677 (1993). 22. P.-G. Reinhard, O. Genzken, and M. Brack, Ann. Phys. Leipzig 5, 1 (1996). 23. J. Babst and P.-G. Reinhard, Z. Phys. D 42, 209 (1997). 24. C. Yannouleas and R. Broglia, Ann. Phys. 217, 105 (1992). 25. C. Ellert, M. Schmidt, C. Schmitt, T. Reiners, and H. Haberland, Phys. Rev. Lett. 75, 1731 (1995). 26. E. M. Lifschitz and L. P. Pitaevskii, “Physical Kinetics.” Pergamon Press, Oxford, 1981. 27. P.-G. Reinhard and E. Suraud, in “Metal Clusters” (W. Ekardt, Ed.). Wiley, Chichester, UK, 1999. 28. P.-G. Reinhard and E. Suraud, Eur. Phys. J. D 21, 315 (2003). 29. G. F. Bertsch and D. Tomanek, Phys. Rev. B 40, 2749 (1989). 30. C. Yannouleas, J. M. Pacheco, and R. A. Broglia, Phys. Rev. B 41, 41 (1989). 31. B. Montag and P.-G. Reinhard, Phys. Rev. B 51, 14686 (1995). 32. M. Moseler, U. Landman, and C. Yannouleas, Phys. Rev. Lett. 87, 053401 (2001). 33. K. Chang and W. Graham, J. Chem. Phys. 77, 4300 (1982). 34. D. Forney, P. Freivogel, M. Grutter, and J. P. Maier, J. Chem. Phys. 104, 4954 (1996). 35. G. Pacchioni and J. Koutecky, J. Chem. Phys. 88, 1066 (1988). 36. M. Kolbuszewski, J. Chem. Phys. 102, 3679 (1995). 37. K. Yabana and G. F. Bertsch, Z. Phys. D 42, 219 (1997). 38. T. Berkus, P.-G. Reinhard, and E. Suraud, Int. J. Mol. Sci. 3, 69 (2002). 39. A. McPherson, T. S. Luk, B. D. Thompson, A. B. Borisov, O. B. Shiryaev, X. Chen, K. Boyer, and C. K. Rhodes, Phys. Rev. Lett. 72, 1810 (1994). 40. A. McPherson, K. Boyer, and C. K. Rhodes, J. Phys. B 27, L637 (1994). 41. A. McPherson, B. D. Thompson, A. B. Borisov, K. Boyer, and C. K. Rhodes, Nature 370, 631 (1994). 42. T. Ditmire, J. W. G. Tisch, E. Springate, M. B. Mason, N. Hay, R. A. Smith, J. Marangos, and M. H. R. Hutchinson, Nature 386, 54 (1997). 43. M. Lezius, S. Dobosz, D. Normand, and M. Schmidt, Phys. Rev. Lett. 80, 261 (1998). 44. L. Köller, M. Schumacher, J. Kohn, S. Teuber, J. Tiggesbaumker, and K. H. Meiwes-Broer, Phys. Rev. Lett. 82, 3783 (1999).

Photodynamics of Nanoclusters 45. R. Schlipper, R. Kusche, B. von Issendorff, and H. Haberland, Phys. Rev. Lett. 80, 1194 (1998). 46. T. Ditmire, R. A. Smith, J. W. G. Tisch, and M. H. R. Hutchinson, Phys. Rev. Lett. 78, 3121 (1997). 47. S. Dobosz, M. Lezius, M. Schmidt, P. Meynadier, M. Perdrix, and D. Normand, Phys. Rev. A 56, R2526 (1997). 48. T. Ditmire, E. Springate, J. W. G. Tisch, Y. L. Shao, M. B. Mason, N. Hay, J. P. Marangos, and M. H. R. Hutchinson, Phys. Rev. A 57, 369 (1998). 49. Y. L. Shao, T. Ditmire, J. W. G. Tisch, E. Springate, J. P. Marangos, and M. H. R. Hutchinson, Phys. Rev. Lett. 77, 3343 (1996). 50. T. Ditmire, T. Donnelly, R. W. Falcone, and M. D. Peny, Phys. Rev. Lett. 75, 3122 (1995). 51. M. Lezius and M. Schmidt, in “Molecules and Clusters in Intense Laser Fields” (J. Posthumus, Ed.), p. 142. Cambridge Univ. Press, Cambridge, UK, 2001. 52. M. Lezius, S. Dobosz, D. Normand, and M. Schmidt, J. Phys. B 30, L251 (1997). 53. E. Springate, N. Hay, J. W. G. Tisch, M. B. Mason, T. Ditmire, M. H. R. Hutchinson, and J. P. Marangos, Phys. Rev. A 61, 063201 (2000). 54. K. J. Mendham, N. Hay, M. B. Mason, J. W. G. Tisch, and J. P. Marangos, Phys. Rev. A 64, 055201 (2001). 55. E. Springate, N. Hay, J. W. G. Tisch, M. B. Mason, T. Ditmire, J. P. Marangos, and M. H. R. Hutchinson, Phys. Rev. A 61, 044101 (2000). 56. B. Lang, A. Vierheilig, E. Wiedenmann, H. Buchenau, and G. Gerber, Z. Phys. D 40, 1 (1997). 57. B. Bescos, B. Lang, J. Weiner, V. Weiss, E. Wiedenmann, and G. Gerber, Eur. Phys. J. D 9, 399 (1999). 58. M. A. Lebeault, J. Viallon, J. Chevaleyre, C. Ellert, D. Normand, M. Schmidt, O. Sublemontier, C. Guet, and B. Huber, Eur. Phys. J. D 20, 233 (2002). 59. M. Schumacher, S. Teuber, L. Koller, J. Kohn, J. Tiggesbaumker, and K. H. Meiwes-Broer, Eur. Phys. J. D 9, 411 (1999). 60. T. Döppner, S. teuber, M. Schumacher, J. Tiggesbäumkerand, and K. H. Meiwes-Broer, Int. J. Mass Spectr. 192, 387 (1999). 61. T. Döppner, S. teuber, M. Schumacher, J. Tiggesbäumkerand, and K. H. Meiwes-Broer, Appl. Phys. B 71, 357 (2000). 62. S. Teuber, T. Doppner, T. Fennel, J. Tiggesbaumker, and K.-H. Meiwes-Broer, Eur. Phys. J. D 16, 59 (2001). 63. J. W. G. Tisch, N. Hay, E. Springate, E. T. Gumnbrell, M. H. R. Hutchinson, and J. P. Marangos, Phys. Rev. A 60, 3076 (1999). 64. S. Dobosz, M. Schmidt, M. Perdrix, P. Meynadier, O. Gobert, D. Normand, K. Ellert, and T. Blenski, J. Exp. Theo. Phys. 88, 1122 (1999). 65. G. Grillon, Ph. Balcou, J.-P. Chambaret, D. Hulin, J. Martino, S. Moustaizis, L. Notebaert, M. Pittman, Th. Pussieux, A. Rousse, J.-Ph. Rousseau, S. Sebban, O. Sublemontier, and M. Schmidt, Phys. Rev. Lett. 89, 065005 (2002). 66. B. Bescós, R. Hoch, H.-J. Schmidtke, and G. Gerber, Appl. Phys. B 71, 373 (2000). 67. T. Ditmire, J. Zweiback, V. P. Yanovsky, T. E. Cowan, G. Hays, and K. B. Wharton, Nature 398, 489 (1999). 68. M. Tchaplyguine, K. Hoffmann, O. Dühr, H. Hohmann, G. Korn, H. Rottke, M. Wittmann, I. V. Hertel, and E. E. B. Campbell, J. Chem. Phys. 112, 2781 (2000). 69. E. E. B. Campbell, K. Hansen, K. Hoffmann, G. Korn, M. Tchaplyguine, M. Wittmann, and I. V. Hertel, Phys. Rev. Lett. 84, 2128 (2000). 70. E. E. B. Campbell, K. Hoffmann, H. Rottke, and I. V. Hertel, J. Chem. Phys. 114, 1716 (2001).

591 71. J. Zweiback, T. E. Cowan, R. A. Smith, J. H. Hartley, R. Howell, C. A. Steinke, G. Hays, K. B. Wharton, J. K. Crane, and T. Ditmire, Phys. Rev. Lett. 85, 3640 (2000). 72. W. Kohn and L. J. Sham, Phys. Rev. 140, 1133 (1965). 73. F. Calvayrac, P.-G. Reinhard, E. Suraud, and C. Ullrich, Phys. Rep. 337, 493 (2000). 74. A. Domps, P. L’Eplattenier, P.-G. Reinhard, and E. Suraud, Ann. Phys. Leipzig 6, 455 (1997). 75. A. A. Vlasov, “Many-Particle Theory and Its Application to Plasma.” Gordon & Breach, New York, 1950. 76. K. Takahashi, Progr. Theor. Phys. Supp. 98, 109 (1989). 77. P. L’Eplattenier, E. Suraud, and P.-G. Reinhard, Ann. Phys. N.Y. 224, 426 (1995). 78. P. L’Eplattenier, P.-G. Reinhard, and E. Suraud, J. Phys. A 28, 787 (1995). 79. P.-G. Reinhard and E. Suraud, Ann. Phys. N.Y. 239, 193 (1995). 80. P.-G. Reinhard and E. Suraud, Z. Phys. A 355, 339 (1996). 81. A. Domps, P.-G. Reinhard, and E. Suraud, Ann. Phys. (N.Y.) 260, 171 (1997). 82. A. Domps, P.-G. Reinhard, and E. Suraud, Phys. Rev. Lett. 80, 5520 (1998). 83. R. M. Dreizler and E. K. U. Gross, “Density Functional Theory: An Approach to the Quantum Many-Body Problem.” SpringerVerlag, Berlin, 1990. 84. A. Domps, E. Suraud, and P.-G. Reinhard, Eur. Phys. J. D 2, 191 (1998). 85. M. Brewczyk, K. Rzazewski, and C. W. Clark, Phys. Rev. Lett. 78, 191 (1997). 86. M. Brewczyk, C. W. Clark, M. Lewenstein, and K. Rzazewski, Phys. Rev. Lett. 80, 1857 (1998). 87. M. Brewczyk and K. Rzazewski, Phys. Rev. A 60, 2285 (1999). 88. P. Blaise, S. A. Blundell, and C. Guet, Phys. Rev. B 55, 15856 (1996). 89. K. Boyer and C. K. Rhodes, J. Phys. B 27, L633 (1994). 90. C. Rose-Petruck, K. J. Schafer, K. R. Wilson, and C. P. J. Barty, Phys. Rev. A 55, 1182 (1997). 91. K. Ishikawa and T. Blenski, Phys. Rev. A 62, 063204 (2000). 92. M. Rusek, H. Lagadec, and T. Blenski, Phys. Rev. A 63, 013203 (2001). 93. I. Last and J. Jortner, Phys. Rev. A 62, 013201 (2000). 94. V. P. Krainov and A. S. Roshchupkin, J. Phys. B 34, L297 (2001). 95. T. Ditmire, T. Donnelly, A. M. Rubenchik, R. W. Falcone, and M. D. Perry, Phys. Rev. A 53, 3379 (1996). 96. I. Last and J. Jortner, J. Phys. Chem. A 102, 9655 (1998). 97. F. Megi, M. Belkacem, M. A. Bouchene, E. Suraud, and G. Zwicknagel, J. Phys. B 36, 273 (2003). 98. J.-P. Hansen and I. R. MacDonald, “Theory of Simple Liquids.” Academic Press, London, 1976. 99. E. A. Uehling and G. Uhlenbeck, Phys. Rev. 43, 552 (1933). 100. L. W. Nordheim, Proc. Roy. Soc. London Ser. A 119, 689 (1928). 101. G. F. Bertsch and S. Das Gupta, Phys. Rep. 160, 189 (1988). 102. A. Bonasera, F. Gulminelli, and J. Molitoris, Phys. Rep. 243, 1 (1994). 103. Y. Abe, S. Ayik, P.-G. Reinhard, and E. Suraud, Phys. Rep. 275, 49 (1996). 104. D. Pines and P. Nozieres, “The Theory of Quantum Liquids.” Benjamin, New York, 1966. 105. K. Gütter, K. Wagner, P.-G. Reinhard, and C. Toepffer, Ann. Phys. N.Y. 225, 339 (1993). 106. S. Ichimaru et al., Phys. Rep. 149, 91 (1987). 107. J. W. Serene and D. Rainer, Phys. Rep. 101, 221 (1983). 108. E. Giglio, P.-G. Reinhard, and E. Suraud, Ann. Phys. Leipzig 11, 291 (2002).