Effects of The Nanoplasma Electrons on Coulomb Explosion of Xenon

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J. Phys. Chem. C 2010, 114, 20636–20647

Effects of The Nanoplasma Electrons on Coulomb Explosion of Xenon Clusters† Andreas Heidenreich‡,§ and Joshua Jortner*,⊥ Kimika Fakultatea, Euskal Herriko Unibertsitatea and Donostia International Physics Center (DIPC), P.K. 1072, 20080 Donostia, Eukadi, Spain, IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Spain, and School of Chemistry, Tel AViV UniVersity, Ramat AViV, Tel AViV 69978, Israel ReceiVed: June 9, 2010; ReVised Manuscript ReceiVed: August 16, 2010

The traditional description of Coulomb explosion (CE) of multicharged clusters driven by ultraintense, nearinfrared, femtosecond laser fields (peak intensities of 1015-4 × 1016) W cm-2 at pulse lengths of 25-230 fs) under the initial conditions of complete and vertical outer ionization is transcended in the presence of a nanoplasma within the cluster. Unique features of the energetics of CE of many-electron elemental Xen (n ) 55-6099) clusters manifest the interrelations between electron dynamics of outer ionization and nuclear CE dynamics. Molecular dynamics simulations of high-energy electrons and ions focused on the cluster size and laser parameter domains, where the contribution of electron impact inner ionization is important, the population of the persistent nanoplasma is considerable, and the effects of the nanoplasma on CE nuclear dynamics and energetics are significant. Relations were established for the cluster size and the laser parameter dependence of CE energetic attributes, that is, the average and maximal kinetic energies and the energy distribution, which mark the breakdown of the initial conditions of the complete cluster vertical ionization and of the simple electrostatic models for this situation. Incomplete vertical outer ionization, which retains a persistent nanoplasma, is exhibited for short pulses and low intensities. The CE energetics is then described in terms of the lychee model, where the persistent nanoplasma is initially confined within a neutral sphere in the center of the cluster while ions from an exterior shell undergo CE. This model accounts for the cluster size dependence of the average and maximal CE energies, for their initial site dependence, and for the narrow kinetic energy distribution at low energies. Complete or incomplete nonvertical ionization, when the time scales for electron outer ionization dynamics and nuclear CE dynamics are inseparable, is exhibited for long pulses. For complete nonvertical ionization, we relied on an empirical analysis of the cluster size scaling of the average and maximal CE energies, which can be reasonably well described in terms of the simple electrostatic model for complete cluster vertical outer ionization. For incomplete nonvertical outer ionization, CE kinetic energy distribution for long pulses is broad with a pronounced structure, which is due to a distribution of {Xeq+} ions with different charges. The gross features of the initial site dependence of CE energies and the broad distribution of the kinetic energies for incomplete nonvertical outer ionization can be rationalized in terms of the lychee model, which is strictly applicable for incomplete vertical ionization. I. Introduction The ubiquity of fragmentation of multiply charged, large, finite systems driven by Coulomb (or pseudo-Coulomb) forces1-12 encompasses nuclei,2 molecules and molecular clusters,3,4 metallic and elemental clusters,5-7 nanostructures,8 proteins,9 droplets,10,11 and optical molasses.12 These multiply charged (or effectively charged) systems span a broad size domain of 12 orders of magnitude, from the femtometer structures of nuclei to the nanometer structures of large molecules and clusters, to tens of nanometers structures and proteins, to micrometer structures of irradiated ultracold gases, and to millimeter structures of droplets. A central question regarding the Coulomb instability of multicharged, large, finite systems pertains to the nature of the fragmentation channels and the conditions for their realization. These are determined by the Rayleigh fissibility parameter10 X ) E(Coulomb)/2E(surface), where E(Coulomb) is the Coulomb energy and E(surface) is the surface energy. †

Part of the “Mark A. Ratner Festschrift”. * To whom correspondence should be addressed. ‡ Donostia International Physics Center (DIPC). § IKERBASQUE, Basque Foundation for Science. ⊥ Tel Aviv University.

Computational1 and experimental7 evidence established two distinct fragmentation channels; spatially anisotropic fission into a small number of large, highly charged clusters is realized for X < 1, while spatially isotropic Coulomb explosion (CE) into a large number of charged ions occurs for X > 1. CE of multicharged systems is prevalent in chemistry, physics, and biophysics.1-12 Developments in the experimental and theoretical exploration of CE13-15 pertain to the consequences of electron and nuclear dynamics of clusters and nanostructures driven by ultraintense near-infrared laser fields (peak intensities IM ) 1015-1021 W cm-2). The response of clusters to ultraintense laser fields manifests sequential parallel processes of electron dynamics involving inner ionization, nanoplasma formation, and outer ionization.13-15 The response of the nanoplasma electrons and their outer ionization by the laser field are accompanied and followed by CE of extremely charged ions.13-33 Interesting applications of CE pertain to nuclear reactions driven by CE of homonuclear clusters of deuterium,16-19 of deuterium-containing heteroclusters, for example, (D2O)n, (CD4)n, and (DI)n,20-28 and of (DT)n heteroclusters.8,19 These involve dd(D++D+) and dt(D++T+) nuclear fusion within or outside of the macroscopic plasma filament produced by CE of

10.1021/jp105291u  2010 American Chemical Society Published on Web 09/08/2010

Nanoplasma Electrons and Coulomb Explosion of Xe Clusters an assembly of clusters18-31 or by CE within a single cluster.8,34 CE of ensembles of completely ionized large methane, ammonia, and water clusters (nanodroplets)35,36 or within a single methane nanodroplet8 will drive nucleosynthesis of protons with nuclei of carbon, nitrogen, or oxygen, which are of interest in nuclear astrophysics.37 Of considerable interest is CE of many-electron elemental Xen clusters, which is the subject matter of this paper. Experimental studies of Xen (n ) 200-(5 × 104)) clusters driven by ultraintense, femtosecond, near-infrared laser pulses reveal high ionization levels and high CE energies (10 keV-1 MeV).38-42 The general features of extreme ionization, ultrafast ion dynamics, and ultrahigh energetics were manifested in theoretical and computational studies13-15,30-33,43-45 of CE of Xen clusters. Simulation data were reported for Xen (n ) 13-6144) clusters driven by ultraintense lasers (IM ) (4 × 1014)-1021 W cm-2, with pulse lengths of τ ) 10-250 fs).13-15,30-33,44,45 Theoretical work on extreme CE of Xen clusters13-15,30-33,43 utilized electrostatic models to account for the dependence of the CE dynamics and energetics on the cluster size (expressed in terms of the initial cluster radius R0), on the inner ionization level (expressed in terms of the average charge qav per atom), and on the laser intensity (which determines the outer ionization level). These electrostatic models13-15,30-33,43 are applicable for cluster vertical ionization (CVI) under conditions of complete outer ionization, with the formation of an ionic cluster at the initial nuclear configuration of the neutral cluster. The CE velocities then scale as qav, being independent of R0; the CE average energies Eav and maximal energies EM 2 2 scale as Eav, EM ∝ qav R0, while the energy distribution P(E) -1 scales as P(E) ∝ EM (E/EM)1/2 for E e EM.14,15 To make contact with experimental data, one has to account for the cluster size distribution,36,39,40,43 and for the spatial intensity profile of the focused laser beam.43,46,47 A related study by Fennel, Meiwes-Broer, and their collaborators47 for the ion charge distribution in CE accounted for the laser intensity distribution using the z-scanning method (with a small slice along the laser focus offset axis being recorded).46,47 Rost et al.43 applied an approach based on the incorporation of both cluster size and intensity averaging for the calculation of the energy distributions of ions in CE of Xen and of other molecular clusters but using the complete CVI approximation13-15 for the single cluster. The experimental interrogation of P(E) in CE of Xen clusters was reported for an average size of 〈n〉 ) 2500 with IM ) 2 × 1016 W cm-2 and τ ) 150 fs38 and for 〈n〉 ) 9000 with IM ) 1.5 × 1016 W cm-2 and τ ) 170 - 230 fs.39 Under these conditions, the applicability of the complete CVI conditions used by Rost et al.43 has to be examined. Computational MD methods and more advanced electrostatic models are required to address the energetics of CE under these conditions. Complete CVI is inapplicable under two conditions, (1) incomplete outer ionization with a persistent nanoplasma being retained within the cluster and (2) nonvertical outer ionization with CE occurring concurrently with outer ionization. These deviations from complete CVI will be manifested for CE of large Xen clusters (n e 6099) driven by “moderately long” femtosecond pulses (τ ) 50-230 fs) in the intensity range of IM ) 1015-(4 × 1016) W cm-2. We report on MD simulations for electron and ion dynamics in many-electron Xe clusters for the experimentally relevant cluster size and laser parameters, where the effects of the persistent nanoplasma on CE dynamics and energetics are significant. We shall transcend the traditional description of CE, providing new information on CE energetics and dynamics under the conditions of incomplete and/or

J. Phys. Chem. C, Vol. 114, No. 48, 2010 20637 nonvertical cluster outer ionization, when the time scales for electron outer ionization dynamics and nuclear CE dynamics are inseparable. II. Methodology and Computational Details Our MD simulation scheme for high-energy electron dynamics and ion dynamics in a cluster interacting with an electric and a magnetic field of an ultraintense Gaussian laser pulse was previously described.30,32,48 In brief, the laser electric field Fl(t) was taken as

Fl(t) ) Fl0(t) cos(2πνt)

(1)

with the Gaussian envelope function

(

Fl0(t) ) FM exp -4 ln(2)

t2 τ2

)

(2)

a frequency of ν ) 0.35 fs-1 (photon energy, 1.44 eV), and a temporal length τ (the temporal fwhm of the intensity profile is τ/2). The electric field maximum FM is related to the peak (i.e., cycle-averaged) intensity IM (at t ) 0) by

FM )

( ) 2IM cε0

1/2

(3)

with the velocity of light c and the dielectric constant ε0 of the vacuum. The laser magnetic field is

(

Bl(t) ) BM exp -4 ln(2)

BM )

)

t2 cos(2πνt) τ2

( ) 2µ0IM c

(4)

1/2

(5)

with the permeability µ0 of the vacuum. The electrons are treated nonquantum mechanically but relativistically with an electronic time step of 10-3 fs and a nuclear time step of 0.02 fs. At every electronic time, the electric field, consisting of the external laser field and an inner field from all charged particles, is evaluated at every ion. Inner ionization occurs when the electric field reaches the critical strength for classical barrier suppression ionization (BSI), neglecting tunneling. Likewise, at every electronic time step, electron impact ionization (EII) is checked for, using experimental atomic electron impact cross sections. Unlike the BSI channel, the maximum EII level considered in our current simulation code is Xe11+, that is, the highest ionization level for which experimental EII cross sections are available.49 In our work, the cross sections for EII were adopted from the fit of the experimental data for a single atom/ion32,49 and did not include inner field effects on EII, which were considered by Fennel et al.45 Electrons formed by the BSI and the EII channels (inner ionization) produce a (transient or permanent) nanoplasma. Electron-ion recombination within the nanoplasma was neglected, in accord with the analyses of Heidenreich, Last and Jortner,30,32 Petrov et al.,44 and Fennel et al.46 An electron is removed permanently from the nanoplasma (outer ionization) when its distance to the center of mass exceeds six radii R(t) of the expanding cluster.

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Figure 1. The time evolution of the average inner ionization level qav per ion (- - -), of the number np of nanoplasma electrons per atom ( · · · ), and of the relative cluster radius R/R0, that is, the instantaneous radius relative to the initial cluster radius R0 (s) for the Xe6099 cluster and laser peak intensities IM ) 1015 (panels a-c) and 2 × 1016 W cm-2 (panels d-f), for three pulse lengths of τ ) 25, 100, and 230 fs. In each panel, the oscillating electric field of the laser is given in arbitrary units.

In this paper, we focus on the ion kinetic energies as the experimental observable. Since the temporal length of the trajectories is limited, the electrostatic potential energy is not completely converted into kinetic energy. To correct the final ion kinetic energies Ti ) miV2i /2, we assume a spherical charge distribution of the exploding cluster, so that the kinetic energy correction for ion i at distance ri from the center of the cluster can be taken as the sum of the pairwise Coulomb interactions, where only ions j and electrons k closer to the cluster center (rj < ri, rk < ri) contribute

Ti(corr) ≈ Ti +

∑

j(rj 0. For a fixed laser pulse length, the threshold size increases with increasing IM. (ii) Cluster size dependence of np/qav. For fixed laser parameters (IM and τ), the population of the nanoplasma electrons (above the threshold size) increases with increasing initial cluster size. This size dependence of np is attributed to the reduction of the outer ionization level of large clusters. (iii) Intensity dependence of np. For fixed values of n and τ, the nanoplasma electron population above the threshold cluster size decreases with increasing IM, marking the enhancement of outer ionization at higher laser intensities. (iv) Pulse length dependence of np. At a fixed cluster size and IM, the nanoplasma electron population above the threshold cluster size markedly decreases with increasing pulse length, marking the enhancement of outer ionization at long τ. For the long pulses (τ ) 230 fs) and higher intensities (IM > 2 × 1016 W cm-2), outer ionization is nearly complete, marking the upper limit of the laser parameters’ domain for the prevalence of the persistent nanoplasma. (v) Complete outer ionization. This is accomplished for all laser parameters below the size threshold (when outer ionization is complete and vertical; see section III.a), for large clusters with large intensities and relatively short pulses (e.g., IM > 4 × 1016 W cm-2, τ ) 100 fs) when complete CVI prevails,

Heidenreich and Jortner

Figure 3. The cluster size, laser intensity, and pulse length dependence of the long-time relative nanoplasma population np/qav per ion. Values of np/qav ≈ 1-0.2 indicate a persistent nanoplasma, and np/qav ≈ 0 indicates a depleted one. The presentation and choice of cluster sizes, peak intensities, and pulse lengths are as those in Figure 2.

and for large clusters with long pulses and relatively high intensities (e.g., n ) 6099, τ ) 230 fs, and IM ) 2 × 1016 W cm-2) when complete but nonvertical outer ionization occurs. Under these conditions of complete outer ionization, EII is minor. (vi) Cluster size dependence above the size threshold for different pulse lengths. A qualitative difference is manifested between the nearly sharp increase of np/qav with increasing n for shorter pulses (τ ) 50 and 100 fs) and the gradual increase of np/qav with increasing n for long pulses (τ ) 230 fs). This distinction reflects on the difference between incomplete CVI for the shorter pulses and incomplete and nonvertical outer ionization for the long pulses. The population of the nanoplasma electrons is of considerable importance as input data for the elucidation of CE energetics. From the foregoing analysis (sections III.a, III.b, and III.d), we established the conditions for complete CVI when simple electrostatic models are applicable for the energetics,14,15 as well as for incomplete vertical ionization, where modified electrostatic models, that is, the lychee model,33,50 are applicable. We have also provided information on the ionization dynamics of outer ionization on the nanoplasma electron populations (sections III.a and III.d) and on nuclear dynamics (section III.b) for complete and nonvertical outer ionization, where simulation results for CE energetics data have to be utilized. IV. Energetics of Coulomb Explosion IV.a. Average and Maximal CE Energies. Our important feature of cluster dynamics at extremes pertains to the high

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Figure 4. The cluster size dependence of the long-time average ion kinetics energies Eav for laser peak intensities of (a) 1015, (b) 3 × 1015, and (c) 2 × 1016 W cm-2 and pulse lengths of 25, 50, 100, 150, and 230 fs.

Figure 5. The cluster size dependence of the long-time maximum ion energies EM for laser peak intensities of (a) 1015, (b) 3 × 1015, and (c) 2 × 1016 W cm-2 and pulse lengths of 25, 50, 100, 150, and 230 fs. The presentation is as that in Figure 4.

(3-120 keV) ion energies attained in CE of {Xeq+}n clusters (Figures 4 and 5). In Figures 4 and 5, we portray the cluster size dependence of two parameters that specify the CE energy distribution, P(E), which are the average ion kinetic energy Eav ) ∫EP(E) dE and the maximum ion energy EM. Following our previous classification for time-resolved outer ionization and nanoplasma dynamics (sections III.a and III.b), we shall invoke the two physical situations for the dependence of Eav and EM (Figures 4 and 5), case A for short pulses (τ ) 25 and 50 fs) and case B for long pulses (τ ) 150 and 230 fs). In the cluster size and laser parameter domains, case A corresponds to incomplete CVI, while case B corresponds to either incomplete or complete nonvertical outer ionization. In case A, for short pulses and low intensities (i.e., all intensities in Figure 4), the Eav data reveal an initial linear dependence on R02 and saturation of Eav for larger cluster sizes, that is

Eav )

γavR20

2 Eav ) γav(R(I) 0 )

R0 e

R(I) 0

R0 > R(I) 0

case A, EM reveals an initial scaling with R02 followed by linear scaling with R0 (Figure 5), that is

EM ) γMR20

R e R(I) 0

EM ) γMR(I) 0 R0

R > R(I) 0

(8a) (8b)

where γM is a numerical constant and the cutoff radius R0(I) for EM (eqs 8a and 8b) is identical to that for Eav (estimated from Figure 4 and eq 7a and from Figure 5 and eq 7b). In case B (complete nonvertical ionization, panels (c) of Figures 4 and 5), we observe that

Eav ) γ˜ avR20

(9)

EM ) γ˜ MR20

(10)

(7a)

(7b)

where γav is a numerical constant and R0(I) is the border radius for complete outer ionization, which depends on the laser intensity and on the pulse length.21,22,31,33,50 Concurrently, for

The numerical factors estimated from Figures 4 and 5 for IM ) 2 × 1016 W cm-2 at τ ) 230 fs are γ˜ av ) 3.3 × 10-2 keV Å-2 and γ˜ M ) 5.5 × 10-2 keV Å-2. Equations 9 and 10 are empirical, obtained from analyses of simulation data for complete and nonvertical ionization. It is very surprising to note that these γ˜ av and γ˜ M data are in reasonable agreement (within a numerical factor of 2) with the results of the simple electrostatic model for complete CVI, which predicts33 that Eav

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2 2 ) Γavqav R0 (with Γav ) 6.5 × 10-4 keV Å-2 e-2) and that EM 2 2 ) ΓMqavR0 (with ΓM ) 10.3 × 10-4 keV Å-2 e-2). For IM ) 2 × 1016 W cm-2, we take qav ) 10.2 for τ ) 230 fs (Figure 2); therefore, for the electrostatic model, we estimate the proportionality constants in eqs 9 and 10 to be γ˜ av(complete CVI) ) 6.8 × 10-2 keV Å-2 and γ˜ M(complete CVI) ) 10.7 keV Å-2. Furthermore, the vertical CVI model predicts that γ˜ M(complete CVI)/γav(complete CVI) ) 1.67,33 while our simulation results for complete nonvertical ionization give γ˜ av/γ˜ M ) 1.58 for τ ) 150 fs and 1.67 for τ ) 230 fs. From this analysis of the size scaling of the energy parameters Eav and EM, three conclusions emerge. First, for the case of incomplete vertical outer ionization, the size dependence of the energy parameters is given by eqs 7a and 7b. A description of the CE energetics for a cluster with a persistent nanoplasma will be given in section IV.b. Second, for complete nonvertical outer ionization, the size scaling of the energy parameters is given by eqs 9 and 10, with the proportionality constants γ˜ av and γ˜ M being reasonably well described (within a numerical factor of 2) by the electrostatic model for complete CVI. This analysis establishes relations between CE energetics and electron outer ionization dynamics. IV.b. Lychee Model for CE. Under the conditions of incomplete vertical ionization, CE occurs from the ionic cluster, which still contains persistent nanoplasma electrons. We shall adopt the lychee model33,50 to treat the persistent nanoplasma in Xen clusters. The persistent nanoplasma is assumed to form an initially neutral inner sphere with a radius Rp. The ions form an electron-free spatial exterior shell in the range of Rp e r e R0. This exterior ion shell undergoes CE, while ions from the interior range of 0 e r e Rp of the neutral nanoplasma are characterized by a negligible kinetic energy. From a previous analysis of the lychee model,33,50 the nanoplasma radius for Xen clusters is

Rp ) R0(np /qav)1/3

(11)

The width of the ionic exterior shell s ) R0 - Rp is

s ) R0[1 - (np/qav)1/3]

(12)

For the relevant size and laser parameters (noi/qav) , 1; therefore, eq 12 gives

s ) (R0 /3)(noi /qav)

(13)

The border radius for outer ionization R0(I) (refs 33 and 50) is given by

R(I) 0 ) 3s

(14)

The data of Figure 3 allow for the determination of Rp, s, and R0(I). A simple electrostatic treatment of the lychee model provides the following expressions for the average energy Eav (R0 . s) and the maximal energy EM (R0 . s) at saturation33,50 2 (I) 2 j Eav(R0 . R(I) 0 ) ) (2π/3)BFXeqav(R0 )

and

(15a)

2 (I) j EM(R0 . R(I) 0 ) ) (4π/3)BFXeqavR0 R0

(15b)

j ) 14.4 eV Å and FXe ) 0.017 Å-3 is the initial number where B density of the Xe cluster. Equations 15a and 15b are isomorphous with eqs 7b and 8b, respectively. The lychee model accounts for the gross features of the size dependence of the CE energetics. A direct demonstration for the validity of this model will be reported in section IV.d from the simulations for the initial site dependence of the CE energetics. IV.c. Kinetic Energy Distributions. In Figure 6, we present the simulation results for the kinetic energy distribution P(E) of Xeq+ ions from CE of Xe6099. The energy distribution is expressed as P(E) ) n(E)/(nδE), where n(E) is the number of ions in the energy interval δE around E. The simulation data for P(E) correspond both to incomplete vertical outer ionization (panels (a), (b), (c), (f), and (g) in Figure 6) or incomplete/ complete and nonvertical outer ionization (panels (d), (e), (i), and (j) in Figure 6). The energy distributions reveal a marked difference from the complete CVI result P(E) ∝ E1/2; E e EM. The energy distribution curves (Figure 6) reveal distinct features in the following two physical situations, for short pulses (corresponding to case A in sections III.a, III.b, and IV.a) and for long pulses (corresponding to case B in sections III.a, III.b, and IV.a). In case A, P(E) reveals a sharp decrease with increasing kinetic energies in the narrow energy range of 0 e E < EM, while in case B, a large spread of P(E) over the broad energy range from 0 to EM is exhibited. As demonstrated by our MD simulations (section III.a), cases A and B correspond to large and small populations of nanoplasma electrons, respectively. Another interesting feature, which is exhibited only for long pulses and being most pronounced at the highest intensity (Figure 6), is the structure in P(E), which shows at least four peaks. This structure is attributed to the contribution to CE of {Xeq+} ions with different charges q. We now proceed to the analysis of simulation results for energetic and dynamic CE data, which will determine the features of P(E) on the basis of the distribution of ions. IV.d. Initial Site Dependence of CE Energies. Additional information on the energetics of CE was inferred from the final kinetic energy E(r) of the {Xeq+} ions initially located at distance r from the cluster center. In Figure 7, we present the simulation results for E(r)/EM versus r/R0 of Xe6099 clusters driven by a laser with IM ) 2 × 1016 W cm-2 over a broad range of τ values (which correspond both to classes A and B defined in section IV.c). The maximal energies EM are marked on the panels. The initial site CE energies E(r)/EM (Figure 7) j p)/EM = 0 for r e R j p. This j p, where E(R reveal an onset at r ≡ R j p is interpreted shape of the simulated E(r)/EM curves for r g R in terms of an electrostatic model where the incomplete outer j p. The ionized clusters contain a neutral sphere within a radius R existence of these onsets provides a compelling argument for the applicability of the lychee model, where the persistent, neutral nanoplasma is confined within a neutral sphere of radius Rp in the center of the cluster. We shall identify the onsets Rp in the initial site-dependent CE energy curves (Figure 7) with the initial radii Rp of the lychee model (eq 11), that is, Rp ≡ Rp. The CE simulation data for Rp (Table 1) reveal a decrease of Rp with increasing pulse length. In Table 1, we present the estimates of Rp/R0 obtained from the qav and np/qav data (Figures 2 and 3), together with the Rp/R0 obtained from the onsets of the initial site CE energies (Figure 7), which were taken for E(Rp/R0)/EM ) 5%. The agreement between these two sets of

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Figure 6. Distribution functions P(E) for the ion kinetic energies of the Xe6099 cluster at long time. (a-e) IM ) 1015 W cm-2 and pulse lengths of τ ) 25, 50, 100, 150, and 230 fs, respectively; (f-j) IM ) 2 × 1016 W cm-2 with the pulse lengths marked in panels a-e. The average and maximum ion energies, Eav and EM, are marked in each panel.

TABLE 1: Data for Rp/R0 Obtained from the Lychee Model and from the Threshold of the Initial Site Dependence of CE Energetics E(r)/EMa Rp/R0 τ (fs)

np

qav

Lychee model

25 50 100 150 230

6.93 6.15 2.32 0.367 0.219

9.38 10.8 11.1 10.8 10.2

0.90 0.93 0.59 0.32 0.28

b

onset of E(r)/EMc 0.88 0.81 0.43 0.21 0.17

a Cluster size, n ) 6099; laser peak intensity, IM ) 2 × 1016 W cm-2. b Equation 11 and data from Figures 2 and 3. c Data for onsets were taken for E(r)/EM ) 0.05 at r ) Rp.

Figure 7. Initial site specificity for high-energy ion production. The dependence of the normalized ion energies E/EM at long times on the corresponding normalized initial distances r/R0 from the cluster center is presented. The data in the four panels correspond to a Xe6099 cluster, for a peak intensity of IM ) 2 × 1016 W cm-2 and for four different pulse lengths, τ ) 25, 50, 100, and 230 fs. Average and maximum ion energies are marked in each panel. For short pulses, 25 and 50 fs (panels a and b), only ions at distances of r/R0 J 0.9 and 0.8, respectively, attain notable energies. For τ ) 230 fs (panel d), the onset of appreciable final energies has reached the cluster interior (r/R0 ≈ 0.1). Thereby, pronounced branching of ion populations takes place (panels c and d). The branching for τ ) 230 fs was attributed to populations of ion charges 8-12 (panel d), which are marked on the curves.

data is good for the shorter pulses (τ ) 25 and 50 fs), where incomplete CVI prevails and the lychee model is expected to be strictly valid. For the longer pulses (τ ) 150 and 230 fs), incomplete nonvertical ionization occurs; therefore, the deviations from the lychee model are larger, that is, ∼50%. The overall agreement between the two sets of Rp/R0 data (Table 1) provides strong support for the validity of the lychee model.

Another interesting feature of the simulation results for E(r)/ EM is the distribution of {Xeq+} ionic charges involved in CE. Simple counting of the ions produced by CE of Xe6099 driven by a laser with IM ) 2 × 1016 W cm-2 and τ ) 230 fs shows a broad initial state distribution of the ions, with q ) 12 in the midrange of r/R0 ) 0.5-0.65, q ) 11 in the broad range of r/R0 ) 0.1-1.0, q ) 10 in the range of r/R0 ) 0.6-1.0, q ) 9 in the range of r/R0 ) 0.7-1.0, and q ) 8 in the surface region in the range of r/R0 ) 0.88-1.0. For each of the ions, a separate E(r)/E curve will be exhibited. Such splitting of the E(r)/EM as r/R0 curves is indeed exhibited for the long pulses (τ ) 100 and 230 fs) in Figure 7, where each branch corresponds to a different q value, which is marked on panel (d) of Figure 7. The spatial range of each branch, which manifests surface ions for q ) 7, midrange ions for q ) 12, and a broad ion distribution for q ) 11, is in accord with the ion counting results presented above. We also obtained additional information on the dynamics of CE from the time dependence of the ion kinetic energies T (see section II). In Figure 8, we present the initial site dependence of T/TM at different times during the pulse, where TM is the maximal kinetic energy at each time, which is marked on each panel. These data were obtained for Xe6099 driven by a laser with IM ) 2 × 1016 W cm-2 and τ ) 230 fs. The longtime T/TM curves (panel (f) in Figure 8) converge to the E(r)/ EM curve (panel (d) in Figure 7). The structure with different branches due to different ion charges in CE sets in at t ) -48

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[ ( )]

(CVI) EM ) EM 1-

Rp R0

3

(17)

Equations 16 and 17 result in

E(r) ) EM

[( ) ( ) ( )] [ ( )] r R0

Rp R0 Rp 1R0 2

-

2

Rp r

Rp e r e R0

3

(18)

The kinetic energy distribution in CE is

P(E) )

(dn(r)/dr) (dE(r)/dr)

(19)

where dn(r)/dr ) 4πFXer2 (at Rp e r e R) and E(r) is given by eq 18. This analysis results in

( )[ ( )] [ ( ) ( )( )] 3

P(E) )

Figure 8. Time evolution of the site-specific ion energy production for the case presented in Figure 7d, that is, the Xe6099 cluster irradiated by a laser pulse of IM ) 2 × 1016 W cm-2 and τ ) 230 fs. Snapshots of the normalized ion kinetic energies T/TM at various times during the laser pulse are plotted versus the normalized initial ion distance r/R0 from the cluster center. The times t indicated in the panels refer to the peak of the laser envelope function at t ) 0. The instantaneous average and maximum ion kinetic energies are given in the panels. The branching of ion populations begins to emerge notably around t ) -48 fs (panel d). For this time domain, a large part of the nanoplasma electrons is depleted by outer ionization, compare Figure 1f.

fs (panel (d) of Figure 8). At this time, the nanoplasma electrons are markedly depleted (panel (f) in Figure 1), facilitating CE. IV.e. Lychee Model Revisited. We now proceed to utilize the lychee model for calculations of the initial site-dependent CE energies (section IV.d) and of the kinetic energy distributions (section IV.c). We consider again incomplete CVI with sharp boundaries between the internal sphere of radius Rp containing the neutral nanoplasma and the ionic outer shell and the surface of the ionized clusters. The initial site CE energy, as obtained from the electrostatic model, is

(CVI) E(r) ) EM

[( ) ( ) ( )] r R0

2

-

Rp R0

2

Rp r

Rp e r e R0

(16)

(CVI) 2 2 j FXeqav ) (4π/3)B R0 is the maximal energy under where EM complete CVI conditions (section IV.a). The maximal CE energy EM ) E(R0) is

r R0

2

1-

Rp R0

Rp r EM 2 + R0 R0

3

3

R0 r

2

(20)

In panel (a) of Figure 9, we present the results of the model calculations, based on eq 18, of E(r)/EM versus the initial ion position r/R0 within the cluster, which was outer-ionized under incomplete CVI conditions. The contribution of the persistent nanoplasma was subsumed into the parameter Rp/R0. In the limit of a transient nanoplasma, that is, Rp/R0 ) 0, the parabolic (CVI) (r/R0)2, holds. relation for complete CVI, that is, E(r) ) EM For finite values of Rp/R0, a threshold in each E(r)/EM curve is exhibited at r ) Rp. These features of the CE energetics are in accord with the simulation results (Figure 7). In panel (b) of Figure 9, we present the kinetic energy distribution functions P(E) of ion energies obtained by CE under conditions of incomplete CVI, as presented for Rp values in the range of Rp/R0 ) 0-0.95. The model calculations were carried out for qav ) 10 and R0 ) 43.7 Å. For Rp/R0 ) 0, the P(E) curve is given by the complete CVI result P(E) ) (3/2E(CVI) M )(E/ (CVI) 1/2 ) . For low values of Rp/R0 ) 0.1-0.5, the P(E) curves EM become finite at E ) 0 and span a broad energy range. For large values of Rp/R0 ) 0.8-0.95, P(E) is described by a step function with a large value of P(0) at E ) 0. With increasing Rp/R0, the value of P(0) increases, and P(E) spans a narrower energy domain (panel (b) of Figure 9). From the dependence of P(0) on the nanoplasma parameter Rp/R0 (inset to panel (b) of Figure 9), we observe a “transition” at (Rp/R0)c = 0.8, marking the onset of the persistent nanoplasma size for the attainment of a large low-energy and narrow-energy contribution to P(E). These results of the model calculations based on the lychee model are in accord with the gross features of the full simulation results for P(E) (Figure 7). In order to account for the details of the simulation results, the simple and useful lychee model used herein has to be extended in several directions. First, a more realistic description is required for the interface between the inner sphere of the persistent neutral nanoplasma and the outer ionic shell. The step function used herein for the description of this interface has to be replaced by a continuous function. Second, the contribution of different ion charges to

Nanoplasma Electrons and Coulomb Explosion of Xe Clusters

Figure 9. Results of the Lychee model. (a) The normalized ion energies E/EM, eq 18, versus the normalized initial ion distances r/R0 for various degrees of nanoplasma depletion, ranging from a completely depleted nanoplasma, Rp/R0 ) 0, to a persistent nanoplasma with very little depletion, Rp/R0 ) 0.9. The dependence of E/EM on r/R0 for a fixed value of Rp/R0 is valid for any cluster size. (b) The ion energy probability density function P(E), calculated by eq 20 for qav ) 10 and R0 ) 43.7 Å, corresponding to Xe6099. The inset shows the ion energy probability density at E ) 0 as a function of Rp/R0, manifesting the transition to a narrow energy probability density function for Rp/R0 > 0.8.

the CE energetics has to be incorporated. This effect will lead to the “smearing” of the E/EM curves and to a splitting of the P(E) curves. Third, and most challenging, in order to make contact between model calculations and physical reality, one has to advance a more sophisticated description of CE under conditions of incomplete and nonvertical outer ionization when the separation between electron ionization dynamics and CE nuclear dynamics breaks down. V. Concluding Remarks The description of CE of multicharged molecular4 and elemental13-15,30-33 clusters is quite simple and straightforward for small clusters and for superintense laser fields (e.g., IM > 1017 W cm-2) when outer ionization of the nanostructures is complete and vertical. These conditions of complete CVI allow for the separation of the time scales between electron dynamics of outer ionization and nuclear CE dynamics. Simple electrostatic models are then applicable for the description of the nuclear dynamics and energetics of CE.14,15 The complete CVI model will be transcended when one of the following three scenarios is realized: (i) incomplete vertical ionization, when the time scales for incomplete outer ionization and nuclear CE are separable; (ii) complete nonvertical ionization, when the time scales for complete outer ionization and nuclear CE are inseparable; or (iii) incomplete nonvertical ionization. Laserdriven outer ionization dynamics of clusters qualitatively differs from electron dynamics, which involves changes in electronic states of atoms and molecules, with the nuclear motion being frozen.51-56 Such ultrafast processes with the temporal resolution of electron motion were recently interrogated by attosecond electron tunneling,55 inner shell electron dynamics,54 and nonsequential double ionization (e,2e) from recollision processes.53,56 For outer ionization electron dynamics, the nuclear motion is not necessarily frozen as outer ionization of large clusters driven by long (in the range of hundreds of femtoseconds) laser pulses occurs on the same time scale as CE, precluding the separation of time scales between electron and nuclear dynamics. We shall now allude to relations between outer ionization dynamics and CE nuclear dynamics. In sections III.a, III.b, and IV.c, we distinguished between two sets of laser parameters

J. Phys. Chem. C, Vol. 114, No. 48, 2010 20645 which are of experimental interest.38-41 Scenario (i) for incomplete vertical outer ionization, which retains a persistent nanoplasma, is exhibited for short pulses and lower intensities. The CE energetics is then described by the lychee model (sections IV.b and IV.e), which accounts for the cluster size dependence of the average and maximal CE energies, for their initial site dependence, and for the narrow kinetic energy distribution P(E) at low energies. Scenario (ii) for complete nonvertical ionization is exhibited for long pulses. In this case, we relied on an empirical analysis of the cluster size scaling of the average and maximal CE energies, which can be reasonably well described in terms of the simple electrostatic model for complete cluster vertical outer ionization. For scenario (iii) of incomplete nonvertical outer ionization, the CE kinetic energy distribution for long pulses is broad with a pronounced structure, which is due to a distribution of {Xeq+} ions with different charges. The gross features of the initial site dependence of CE energies and the broad distribution of the kinetic energies for incomplete nonvertical outer ionization can be rationalized in terms of the lychee model, which is strictly applicable for incomplete vertical ionization. Further theoretical work on the energetics of CE, without the separation of time scales between electron dynamics of outer ionization and nuclear CE dynamics, will be of interest. A general result emerging from the present study is the quantification of the retardation of the energetics and dynamics of CE of elemental large clusters by the presence of the persistent nanoplasma. This situation is also expected to be realized for very large molecules, for molecular clusters, and for proteins driven by near-infrared lasers in the appropriate range of the laser parameters. In particular, the case of retardation of CE of proteins by the persistent nanoplasma will be of considerable interest in the context of the determination of the time-resolved structure of biomolecules by free electron lasers.9 The presence of a persistent nanoplasma is expected to retain structural information for longer times (on the fs time scale) than that under the conditions of complete outer ionization. These studies will require the exploration of the outer ionization dynamics of the nanoplasma electrons for proteins interacting with high-energy (tens of eV) photons,57 where the electron-ion nanoplasma is strongly coupled, being characterized by multielectron-ion, many-body collisions.58,59 We proceed to a brief confrontation between the computational-theoretical results and experiment38-40 for the effects of the nanoplasma electrons on the energetics of Xen clusters. At present, no single-cluster data are experimentally available, while the results reported herein describe the CE energetics for a fixed cluster size. To make contact with experimental reality, one has to perform a double averaging of the single-trajectory simulation data to account for the cluster size distribution39,40,43 and for the spatial intensity profile of the laser beam.43,46 The analysis of experimental data for elemental clusters is inherently limited by a broad cluster size distribution, which is accounted for by a log-normal distribution.39,40,43 The spatial laser intensity distribution, which originates from the focusing of the laser pulse by a lens or a parabolic mirror, was utilized.46,47 We have applied the double averaging of our computational results for P(E) over the cluster size39,40,43 and laser intensity47 distributions to obtain the information for CE energetics (i.e., EM, Eav, and kinetic energy distribution) for an assembly of clusters under realistic conditions of laser-cluster interactions. These results will be published separately. We note in passing that the double averaging procedure leads to considerable loss of information due to the smearing out of the size and intensity dependence of

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CE energetics and dynamics. This difficulty bears a close analogy to the inhomogeneous broadening of the spectra of molecules and biomolecules in the condensed phase, which was overcome by the advent of single-molecule60 and single-protein61 spectroscopy and dynamics. Future experimental progress in single-cluster CE dynamics will be of great interest. Before comparing our doubly averaged computational data with the experimental results,38-40 the reliability of both the theoretical and the experimental information has to be assessed. Previous theoretical work by Rost et al.43 applied the electrostatic model for complete CVI as input data for the single-cluster CE energetics. The present analysis reveals that under the relevant experimental conditions, complete CVI does not occur. Furthermore, additional modifications of the kinetic energy distribution curves, due to a distribution of charges, preclude the use of simple electrostatic models. A cursory examination of Figure 6 for the single-cluster distribution functions for the ion kinetic energy reveals the inapplicability of the complete CVI electrostatic model. Rather, single-cluster simulation data for P(E) have to be utilized, as applied in the present work. Regarding the assessment of the available experimental data,38-40 the early pioneering work of Ditmire et al.38 reported on exceedingly high CE energies, for example, for Xen clusters with a (average) size of 〈n〉 ) 2500 driven by a laser with IM ) 2 × 1016 W cm-2 and τ ) 150 fs, EM ) 600 MeV (estimated at P(E) ) 10-4 keV-1), and Eav ) 45 ( 5 keV. This value of EM (ref 38) is considerably higher than the value of EM ) 130 keV for 〈n〉 ) 2500 and similar laser parameters (IM ) 1.3 × 1016 W cm-2 and τ ) 170 fs) reported by Springate et al.39 Our simulation results with double averaging, which will be published separately, support the latter experimental data39 with the calculated value of EM ) 85 keV. Our doubly averaged simulation results account reasonably well (within 40%) for the experimental data of Springate et al.39 and of Mendham et al.40 for the cluster size dependence of CE energetics39 and its dependence on the laser intensity39 and pulse length,40 as well as for the anisotropy in the angular distribution of the kinetic energy of Xeq+ ions.39 Prospective extensions of the conceptual and experimental framework of CE in the presence of nanoplasma electrons should be considered. It will be interesting to overcome the conventional wisdom regarding the uniform CE of homonuclear clusters,13-15,23,24 which is realized for one-pulse driving of a spatially homogeneous initial structure (with the exception of the narrow surface profile). A transient inhomogeneous cluster structure can be produced within an expanding homonuclear multicharged cluster by two-pulse excitation with different intensities.34,63,64 According to the scheme advanced by Peano et al.62,63 and analyzed by Last et al.,34 the first weaker pulse (IM ) 1015-1016 W cm-2) drives “slow” CE in the presence of a persistent nanoplasma, while the expanding exterior ionic region produces a spatially inhomogeneous structure, which serves as a target for the second, intense pulse (IM ) 1018-1020 W cm-2). The second pulse will drive additional extreme inner ionization and complete outer ionization. The resulting ultrafast CE of a homonuclear nanostructure will reveal kinematic overrun effects,8,21,22,34 which will trigger high-energy collisions between Xeq+ ions within a single, exploding, large cluster. This two-pulse, different intensities excitation scheme34,62,63 is also of considerable interest for the control of the ionization level of Xen clusters by laser pulse shaping64 and by laser pulse length stretching.65 Acknowledgment. We are greatly indebted to Professor Isidore Last for many discussions and for extensive exchange of information. Computation time and technical and manpower

Heidenreich and Jortner support of the computation center IZO-SGI SDIker of the University of the Basque Country (UPV/EHU), provided by the Spanish Ministry of Science and Education (MICINN), by the Basque Government (GV/EJ) and by the European Social Fund (ESF), are gratefully acknowledged. This research was supported in part by the Saiotek Program (GV/EJ), by the Binational German-Israeli James Franck Program on lasermatter-interaction at Tel Aviv University and by the Deutsche Forschungsgemeinschaft (DFG) SFB 450 program on “Analysis and Control of Ultrafast Photoinduced Reactions” at the Humboldt University and at the Free University of Berlin. References and Notes (1) Last, I.; Levy, Y.; Jortner, J. Proc. Natl. Acad. Sci. U.S.A. 2002, 99, 9107. (2) Bohr, N.; Wheeler, J. A. Phys. ReV. 1939, 56, 426. (3) Purnell, J.; Snyder, S.; Wei, S.; Castleman, A. W., Jr. Chem. Phys. Lett. 1994, 229, 333. (4) Arion, T.; Flesch, R.; Schlathoelter, T.; Alvarado, F.; Hoekstra, R.; Morgenstern, R.; Ruehl, E. Int. J. Mass Spectrom. 2008, 277, 197. (5) Sattler, K.; Muhlbach, J.; Echt, O.; Pfau, P.; Recknagel, E. Phys. ReV. Lett. 1981, 47, 160. (6) Bre´chignac, C.; Cahuzac, Ph.; Carlier, F.; de Frutos, M. Phys. ReV. Lett. 1990, 64, 2893. (7) Hoener, M.; Bostedt, C.; Schorb, S.; Thomas, H.; Foucar, L.; Jagutzki, O.; Schmidt-Bo¨cking, H.; Do¨rner, R.; Mo¨ller, T. Phys. ReV. A 2008, 78, 021201(R). (8) Last, I.; Jortner, J. Phys. ReV. A 2008, 77, 033201. (9) Neutze, R.; Wouts, R.; van der Spoel, D.; Weckert, E.; Hajdu, J. Nature 2000, 406, 752. (10) Lord Rayleigh, L. Philos. Mag. 1884, 14, 184. (11) Duft, D.; Lebius, H.; Huber, B. A.; Guet, C.; Leisner, T. Phys. ReV. Lett. 2002, 89, 084503/1. (12) Pruvost, L.; Serre, I.; Duong, H. T.; Jortner, J. Phys. ReV. A 2000, 61, 053408/1. (13) Krainov, V. P.; Smirnov, M. B. Phys. Rep. 2002, 370, 237. (14) Saalmann, U.; Siedschlag, Ch.; Rost, J. M. J. Phys. B 2006, 39, R39. (15) Heidenreich, A.; Last, I.; Jortner, J. In Analysis and Control of Ultrafast Photoinduced Processes, Ku¨hn, O., Wo¨ste, L., Eds.: SpringerVerlag: Heidelberg, Germany, 2007; Vol. 87, p 575. (16) Zweiback, J.; Smith, R. A.; Cowan, T. E.; Hays, G.; Wharton, K. B.; Yanovsky, V. P.; Ditmire, T. Phys. ReV. Lett. 2000, 84, 2634. (17) Springate, E.; Hay, N.; Tisch, J. W. G.; Mason, M. B.; Ditmire, T.; Hutchinson, M. H. R.; Marangos, J. P. Phys. ReV. A 2000, 61, 063201. (18) Zweiback, J.; Cowan, T. E.; Smith, R. A.; Hartlay, J. H.; Howell, R.; Steinke, C. A.; Hays, G.; Wharton, K. B.; Krane, J. K.; Ditmire, T. Phys. ReV. Lett. 2000, 85, 3640. (19) Last, I.; Jortner, J. Phys. ReV. A 2001, 64, 063201. (20) Grillon, G.; Balcou, Ph.; Chambaret, J. P.; Hulin, D.; Martino, J.; Moustaizis, S.; Notebaert, L.; Pittman, M.; Pussieux, Th.; Rousse, A.; Rousseau, J.-Ph.; Sebban, S.; Sublemontier, O.; Schmidt, M. Phys. ReV. Lett. 2002, 89, 065005. (21) Last, I.; Jortner, J Phys. ReV. Lett. 2001, 87, 033401. (22) Last, I.; Jortner, J. J. Phys. Chem. A 2002, 106, 10877. (23) Last, I.; Jortner, J. J. Chem. Phys. 2004, 121, 3030. (24) Last, I.; Jortner, J. J. Chem. Phys. 2004, 121, 8329. (25) Madison, K. W.; Patel, P. K.; Price, D.; Edens, A.; Allen, M.; Cowan, T. E.; Zweiback, J.; Ditmire, T. Phys. Plasmas 2004, 11, 270. (26) Karsch, S.; Du¨sterer, S.; Schwoerer, H.; Ewald, F.; Habs, D.; Hegelich, M.; Pletzler, G.; Pukhov, A.; Witte, K.; Sauerbrey, R. Phys. ReV. Lett. 2003, 91, 015001. (27) Ter-Avetisyan, S.; Schnu¨rer, M.; Hilscher, D.; Jahnke, U.; Busch, S.; Nickles, P. V.; Sandner, W. Phys. Plasmas 2005, 12, 012702. (28) Heidenreich, A.; Jortner, J.; Last, I. Proc. Natl. Acad. Sci. U.S.A. 2006, 103, 10589. (29) Davis, J.; Petrov, G. M.; Velikovich, A. L. Phys. Plasmas 2006, 13, 064501. (30) Last, I.; Jortner, J. J. Chem. Phys. 2004, 120, 1336. (31) Last, I.; Jortner, J. J. Chem. Phys. 2004, 120, 1348. (32) Heidenreich, A.; Last, I.; Jortner, J. J. Chem. Phys. 2007, 127, 074305. (33) Heidenreich, A.; Last, I.; Jortner, J. Phys. Chem. Chem. Phys. 2009, 11, 111. (34) Last, I.; Peano, F.; Jortner, J.; Silva, L. O. Phys. Plasmas 2010, 17, 022702. (35) Last, I.; Jortner, J. Phys. ReV. Lett. 2006, 97, 173401. (36) Last, I.; Jortner, J. Phys. Plasmas 2007, 14, 123102. (37) Research Highlights Stars in the Lab. Nature 2006, 444, 126.

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