Fig. 1. Schematic view of dynamical mechanisms involved in the processes described in this review: collisions with rapid highly charged ions and irradiation with intense femtosecond laser pulses. The various associated times scales are indicated with typical values (see text for details). The picture is split into four panels corresponding to various stages of the cluster's response: the first panel corresponds to immediate (electronic) response; the second one, still in the electronic domain, describes the electronic relaxation following the excitation phase; the third panel describes the long time electronic relaxation and the gradual coupling to ionic degrees of freedom, while the fourth panel shows the coulomb explosion due to high ionization.

Fig. 2. Illustration of the various time scales in sodium cluster dynamics. Times have been drawn as a function of the temperature T to accommodate the two processes which depend strongly on temperature. The processes depending little on temperature are shown as constant lines.

Fig. 3. The external potential (dashed lines) and the total (Kohn–Sham+external) potential (full lines) in Na₈ with soft jellium background for two different excitation processes, laser on resonance (left part) and collision with an Ar⁸⁺ ion passing by at impact parameter b=25a₀ with velocity v_I=v_F (right part). Three typical times are shown in each case: the ground state before external fields become active (uppermost panels), an early stage for the laser case and the time of closest approach for the collision (middle panels), and a late stage where plasmon response of the system has shaken up the mean field (lower panels). The dashed lines in the uppermost panels indicate the energies of the occupied single electrons states. The note “^*3” in the middle right panel indicates that this potential is to be rescaled by factor 3. The rescaling was needed to fit the enormous Coulomb well into the frame.

Fig. 4. Full 3D TDLDA calculation of the optical response of Na₉⁺ with explicit ionic background (144 configuration, as shown in the inset), compared to experimental data (diamonds) measured at 35 K. Figure from [173], data from courtesy of H. Haberland.

Fig. 5. Spectral strength distributions for the dipole signal (upper panels) and the spin-dipole signal (middle panels) after initialization with a spin-dipole shift. The left part shows results from the triaxial ground state of Na₁₂ and the right part from the polarized isomer, both optimized with detailed ionic structure. In the lowest panel the energies of the corresponding 1ph transitions between levels with opposite spins are plotted. The assignment of line types is: x-mode=full line, y-mode=dotted, and z-mode=dashed. From [174].

Fig. 6. Upper panels: contour plots of the electronic density in the x–z plane for the planar ground state (left) and the free-like isomer (right) of Na₈ adsorbed on NaCl(100). The projection of cluster ions are indicated by squares and the first monolayer of the NaCl(100) substrate is labeled by circles (Cl⁻) and stars (Na⁺). Lower panels: spectral dipole strength distributions for modes in x-, y-, and z-direction. From [182].

Fig. 7. Dipole strength for Li₁₄₇⁺. The experimental points are from [189]. The TDLDA has been computed for two different assumptions about the geometry of the ionic background as indicated, taken from [17].

Fig. 8. Dipole strength for the organic molecule C₆H₆. The experimental curve is from [191] and the TDLDA results, computed with explicit ionic configuration. From [163].

Fig. 9. Dipole strength distributions for a sequence of clusters with N_el=92 and different net charges as indicated. The ionic background was described in the soft jellium model.

Fig. 10. Key spectral features deduced from dipole strength distributions . Upper part: plasmon peak positions in units of ω_Mie=3.45 eV versus N^−1/3 for the experimental peaks and the peaks evaluated from separable RPA for charged clusters (soft jellium model). The dotted lines show the slope fitted to the finite surface calculations. Lower part: full-width at half-maximum (FWHM) versus N^−1/3 for the two cases as above, evaluated at the flanks of the strength distribution. The dotted lines show the asymptotic linear trends as fitted through the results for large samples in case of charged clusters (soft jellium model). From [196].

Fig. 11. Trend of key spectral features with Wigner–Seitz radius r_s for a fictitious cluster with N=20 and N_ion=21 using the soft jellium model for the ionic background [107]. The heavy line shows the average position of the Mie plasmon resonance, the dash–dotted line the diabatic ionization energy (i.e. the “continuum” threshold), the horizontal bars indicate 1ph states, and the faint dotted lines serve to mark the bands of 1ph states. The chemical symbols at the bottom indicate several simple metals associated with the corresponding r_s.

Fig. 12. Trends of the Mie plasmon resonance in x-, y-, and z-direction with deformation computed in a triaxial soft jellium model [199]. The first block shows the evolution with increasing deformation for an axially symmetric prolate system. The second block shows the trends when the triaxiality angle γ is cranked through 0–60° at fixed overall deformation β=0.3. And the third block shows again an axially symmetric, but now oblate, system for a deformation which decreases from 0.3 back to 0.

Fig. 13. Snapshots (in the x–z plane with a parallel perspective along y, all lengths measured in a₀) of a proton-Na₉⁺ collision. The initial kinetic energy of the proton is E_p=10 keV and its impact parameter is b=15a₀. White spheres denote the centroids of the Gaussians on which the Wigner distribution is projected. The actual distribution is smoothed by the Gaussian form factor of the numerical test particles. The proton is represented by a larger sphere. The instant of closest approach corresponds to t=5 fs. After data from [21].

Fig. 14. Time evolution of basic observables for a collision of Ar⁸⁺ with a Na₄₀ cluster described in the soft jellium model. The left part shows a fast collision at ionic velocity v=v_F and the right part a slower case at v=0.1v_F. The impact parameter is b=26a₀ in both cases. The uppermost panel shows the dipole signal, the second panel from the top the intrinsic energy E_int, the third panel the number of escaped electrons (full line: escape from the analyzing box (volume ), dotted line: escape from the whole numerical box), and the lowest panel shows probabilities for a few selected ionization states as indicated.

Fig. 15. Cluster charge as a function of the impact parameter of Ar⁸⁺ hitting a Na₄₀ cluster with velocity v_I0.6v_F. Various calculations done with various formalisms and codes are compared: 3D TDLDA of [17], 2D TDLDA of [156] (present work) Vlasov of [72] (1) and Vlasov of [21 and 22] ((2), present work); assignment of symbols as indicated. In order to comply with the calculations of [17 and 72] the (outdated) steep jellium background has been used here to describe ionic background.

Fig. 16. Deposited excitation energy as a function of projectile charge in a Na₁₉₆ cluster (steep jellium) hit by ions of various charges, as indicated. The projectile velocity is 0.83v_F. The excitation energy is defined in Eq. (12) of [72]. Data points from courtesy of [72].

Fig. 17. Intrinsic excitation energy versus number of escaped electrons for a collision of Ar⁸⁺ with a Na₄₀ cluster described in the soft jellium model and for two ionic velocities, as indicated. Additionally, a result from laser excitation (see next Section 7) is shown, using a cos² pulse with FWHM of 50 fs and frequency ω_laser=2.7 eV.

Fig. 18. Trends with impact parameter for the asymptotic values of the basic observables for a collision of Ar⁸⁺ with a Na₄₀ cluster described in the soft jellium model. Two cases of ionic velocity are considered, v=v_F and 0.1v_F, as indicated. The uppermost panel shows the intrinsic excitation energy E_int, the middle panel the number of escaped electrons, and the lowest panel shows a few selected ionization states for the case v=v_F, as indicated.

Fig. 19. Ionization cross sections (in a₀²) for Na₄₀ hit by an Ar⁸⁺ ion at two velocities v_I=0.1v_F and v_F.

Fig. 20. Electron intrinsic energy E_int as a function of time during a proton-Na₉⁺ collision, as obtained from Vlasov (full line) or VUU (dashed line) computations. The Na₉⁺ was described within the soft jellium model. From [109].

Fig. 21. Time evolution of basic observables for laser excitation of a Na₄₁⁺ cluster with ionic background in CAPS. The laser parameters are indicated in the plots. The left part stems from a laser off resonance (ω=1.5 eV) and the right part from a laser slightly above resonance (ω=3.0 eV). The uppermost panel shows the intrinsic excitation energy E_int, the second panel the number of escaped electrons (full line: escape from the analyzing box, dotted line: escape from the whole numerical box), the third panel the amount of energy absorbed from the laser field, and the lowest panel the dipole signal.

Fig. 22. Time evolution of absorbed norm from the various single-electron states of Na₂₁⁺ with soft jellium background. The left part shows results for two different laser frequencies, one close to resonance (lower part) and one safely above ionization threshold (upper part). The right panel shows the stationary single-electron spectrum with the amount of depletion of each state. The single-particle states are indicated in each plot by their standard spectroscopic assignment.

Fig. 23. Number of emitted electrons N_esc (upper panel) versus laser frequency ω_laser for a laser irradiating a Na₉⁺ cluster (with soft jellium background) at various intensities, as indicated. The pulse profile was a Gaussian with FWHM of 25 fs. For completeness the power spectrum of Na₉⁺ (in the linear regime) is also indicated. From [206].

Fig. 24. Lower panel: schematic view of the various forces on the electrons as a function of laser intensity I. The force from the laser field (dotted) grows . The actual force on the valence electrons is amplified by electronic response and shown for two typical cases, on (full) and off (short dashed) the Mie resonance. The limits for binding in the ground state of Na₉⁺ (soft jellium) and for full ionization (“last valence”) are indicated as well as the binding force of the nearest core electrons (all dash-dotted). Upper panel: number of escaped electrons N_esc as a function of laser intensity I, for the two cases on and off resonance. Adapted from [207].

Fig. 25. Lower panel: trends of maximum field strength versus frequency for the clusters Na₉⁺ (dotted line) and Na₉₃⁺ (full line) irradiated with a laser beam of intensity I=10¹⁰ W/cm². Both clusters are described with the soft jellium model. For comparison we also provide the maximum field strength of the external laser field. Upper panel: the dipole power spectrum for the two clusters as indicated. From [207].