Interaction of Intense Vuv Radiation with Large Xenon Clusters

arXiv:physics/0510187v3 [physics.atom-ph] 19 Apr 2006 oitddnmc aebe are u o radXe and Ar for out Lezius carried by clusters been as- have the dynamics of sociated studies Experimental undergo nanoplasmas fragmentation. hot (keV) complete The hot extremely [5]. observed of [4]. was production emission electrons the x-ray clusters, strong xenon and In states accompanied effi- charge ionic nanoplasmas extremely high into energy by turned pulse ar- are 10 laser or They krypton of the ciently. of the intensities absorb consisting on pulse atoms clusters At durations gon gas pulse noble [3]. higher, and fs or eV 100 1 of pho- about order with of near-infrared, the energies in ton pulses laser using out ried in- This with interaction unique. radiation. their are electromagnetic by Clusters tense example, for to. highlighted, converge nor is they constituents bulk do their the of they properties of However, the mimic matter. simply between condensed not are link and which natural atoms clusters, a simple sense, represent this In nanomaterials, typical molecules. millions many or contain atoms others con- to monomers; of clusters few Some begin a only 2]. phase of [1, sist gas clusters or the systems, in bound form molecules and atoms density, int raeeteeyht es lsa,wihmay which plasmas, dense hot, extremely These create interac- bulk. a to laser–cluster in by tion the materials allow off combination with in carried occurs properties as not to bath, region, heat constrained that interaction large largely ensures the is clusters within cluster the lone stay the of by of size absorbed that finite energy over the interaction while atoms, atom–laser the enhances expanding the from ejected be to found are clusters. MeV 1 to up h aoiyo orsodn xeiet eecar- were experiments corresponding of majority The high sufficiently and temperatures low sufficiently At h ihdniyo tm ntecutrgreatly cluster the in atoms of density high The ihLaarmann with opr hresaedsrbto,cag tt jcine Wabnitz ejection of state experiment charge Hamburg distribution, the state charge rec and compare o ionization influence collisional the under photoionization, cluster heating, the of section evolution the cross l describe We photoionization non-pert for uses and which theories bremsstrahlung interaction inverse by laser–cluster the suggested of than intera r model laser–cluster stronger interaction the much the atom–laser probe interactions escaping the to experiments from enhances allowed energy greatly have prevents cluster cluster the an the within experimental extensive atoms attracted of has plasmas dense hot, ASnmes 32.80.-t,36.40.Gk,52.50.Jm,52.20.Fs numbers: PACS 1 eateto hsc n IA nvriyo ooao Bou Colorado, of University JILA, and Physics of Department h neato faoi lseswt hr,itnepuls intense short, with clusters atomic of interaction The .INTRODUCTION I. neato fitnevvrdainwt ag eo clust xenon large with radiation vuv intense of Interaction tal. et 6:in ihkntceege of energies kinetic with ions [6]: ahr .Walters, B. Zachary tal. et 2 ron ainlLbrtr,Agne lios649 USA 60439, Illinois Argonne, Laboratory, National Argonne Py.Rv Lett. Rev. [Phys. 16 tal. et Dtd ue5 2018) 5, June (Dated: 1 W/cm oi Santra, Robin 95 642(2005)]. 063402 , [Nature 2 ntr ev ssucsfrhg-nryprilso pho- or particles high-energy for sources tons. as serve turn in in ihteproeo dniyn h eeatheat- relevant Ditmire the identifying mechanisms. of purpose ing the with tions h olm edo egbrn os hsinterplay This ions. be- to neighboring lead interplay of and can laser the field the of on Coulomb field the concentrated Other electric quasistatic 11] strong observed. the states tween 10, charge [9, ionic high authors the to tribute nlsi lcrnincliin fte( the of collisions as electron–ion ions. to by Inelastic referred scattered is being ab- process are can This they but ion- whenever quasifree are photons the cluster sorb precisely, the inside More electrons ized dominates. heating collisional usswt ekitniyo lot10 almost of laser intensity of peak impact a destructive with The pulses energies. photon higher data. experimental considered simulations available the not between and was provided heating was comparison collisional of no 11] and since 10, importance assess, [9, relative to Refs. difficult in The somewhat is ionization 13]. enhanced [12, molecules atomic eepromd[7 nwihVndrWascutr of clusters Waals der Van experiments which [16], in Ref. [17] in performed documented were generated FEL, radiation TTF the the of displayedby properties already outstanding Motivated has the physics: by source the interesting laser exploring x-ray is for vuv capability ultrabright TTF its new an the The for of technology laser.) objectives the major of the development of Ger- (One Hamburg, in (TTF) many. Facility Test TESLA the of part ety n20,tefis aigi reeeto laser free-electron a re- lasing—in very first until the (FEL)—at accessible 2000, been In not cently. have energies McPherson photon by demonstrated al. was et nm 248 of wavelength 420 miain n xaso ftecutr We cluster. the of expansion and ombination, eea ruspromdetnienmrclsimula- numerical extensive performed groups Several itei nw bu ae–lse neatosa vor uv at interactions laser–cluster about known is Little 2 8 20) n jce lcrnspectra electron ejected and (2002)] 482 , o emnSila tmcpotentials. atomic Herman-Skillman for s raieRmti ehiust calculate to techniques R-matrix urbative n hi .Greene H. Chris and ege,adttleeg bobdwith absorbed energy total and nergies, to tvr ihpoo nris with energies, photon high very at ction 1] oee,itnelsrfilsa vnhigher even at fields laser intense However, [14]. h rcse fivrebremsstrahlung inverse of processes the f hoeia neet h ihdensity high The interest. theoretical d go.Rcn ehooia advances technological Recent egion. wrpoo nris epeeta present We energies. photon ower so ae ih ofr extremely form to light laser of es neato,wietefiiesz of size finite the while interaction, λ nacdionization enhanced dr ooao83904,USA 80309-0440, Colorado lder, 0 mwsrpre 1] h E is FEL The [15]. reported was nm 109 = 1 tal. et nes bremsstrahlung inverse rtdsoee ndi- in discovered first , 7 one u that out pointed [7] ers e, 19 2 e W/cm yecon- type ) 2 ta at [8]. 2 xenon atoms were exposed to 12.7-eV vuv photons, an II. PHOTOIONIZATION energy range which had been previously unexplored. Each pulse in the experiment lasted about 100 fs. A novel feature of the Hamburg experiment is that The highest intensity in the experiment was about 7 13 2 × the 12.7-eV photons are sufficient to overcome the 12.1- 10 W/cm . Under these conditions, isolated Xe atoms eV ionization potential of neutral xenon. Therefore, al- were found to produce only singly charged ions (see though the oscillating electric field is too weak for the Refs. [18] and [19] for recent developments). In large xenon atoms to undergo field ionization, as occurs in the clusters, however, each atom was found to absorb up to infrared domain, there is still an efficient optical process 400 eV, corresponding to 30 photons, and charge states for creating Xe+. of up to 8 plus were detected. Friedrich [24] gives the cross section for the transition These results were very surprising. The dominant pro- from the bound state φ to the continuum state φ as cesses in most models of infrared laser–cluster interac- | ii | f i tions are field ionization of atoms by the strong electric field of the laser, and heating of the cluster through in- σ (E)=4π2αω ~π ~r 2 , (1) elastic dephasing collisions by electrons oscillating in the fi | · fi| laser electric field. Both of these processes are strongly inhibited by the high frequency of the vuv photons. A where α is the fine-structure constant, ω is the photon relevant quantity in this context is the ponderomotive energy, ~π is the polarization vector for the radiation, and potential [20] of an electron oscillating in a laser field. ~rfi = φf ~r φi is the dipole matrix element coupling the h | | i At a laser intensity of 7 1013 W/cm2, the ponderomo- initial and final states of the electron. The wave function tive potential is only 62× meV, which is smaller than the of the photoelectron in Eq. (1) is energy-normalized. E ionization potential of atomic xenon (12.1 eV [21]) by stands for the kinetic energy of the photoelectron. three orders of magnitude. For linearly polarized light, chosen without loss of gen- In a previous Letter, two of us [22] proposed that the erality to be polarized in thez ˆ direction, the matrix ele- additional heating was due to the effects of atomic struc- ment that must be found is ture on the inverse bremsstrahlung rates. However, that initial study calculated the inverse bremsstrahlung rates ∗ φ z φ = I (l ,l ) dΩY (Ω)cos(θ)Y (Ω) . using perturbation theory for both the electron–photon h f | | ii R i f lf mf limi interaction and the electron–ion interaction. The method Z (2) implemented for this paper instead treats the electron– Here, ion interaction nonperturbatively using variational R- matrix methods, while retaining first-order perturbation ∞ theory for the electron–photon interaction. The result- IR(li,lf )= drUf (r)rUi(r) , (3) 0 ing photoionization and inverse bremsstrahlung cross sec- Z tions calculated are expected to be more realistic, even where U(r) = rR(r) denotes the rescaled radial wave though our description remains at the independent elec- function. Equations (1) and (2) refer to specific angular tron level. momentum quantum numbers l and m for the initial and In this paper, we present a model of the laser–cluster final states. At a photon energy of 12.7 eV, only the interaction that takes atomic structure and the effects 5p electrons of xenon can respond to the radiation field. of plasma screening into account more fully than pre- Hence, we can focus on a subshell with fixed l (here, vious approaches to the subject. Considering the lim- i li = 1). Let q stand for the number of electrons in this itations of the model and the poor experimental char- subshell. Then, within the independent particle model, acterization of the FEL radiation [23], we achieve good after averaging over the initial and summing over the agreement with the Hamburg results. We track photoion- final one-electron states, the total atomic photoionization ization, collisional ionization and recombination, inverse cross section is given by bremsstrahlung heating, evaporation of electrons from the cluster, and expansion of the cluster due to hydrody- 4 2 αω namic pressure of hot electrons and Coulomb repulsion σPI = q π (4) through the duration of the laser pulse and, later, as the 3 2li +1 2 2 cluster undergoes a Coulomb explosion. liIR(li,li 1)+(li + 1)IR(li,li + 1) . × − Atomic units are used throughout, unless otherwise noted. The identities [25, 26]

∗ −m1 l1 1 l2 l1 1 l2 dΩYl m (Ω)cos(θ)Yl2m2 (Ω) = (2l1 + 1)(2l2 + 1)( 1) , (5) 1 1 − m1 0 m2 0 0 0 − Z p 3

′ ′ ′ l1 l2 l3 l1 l2 l3 δ(l3,l3)δ(m3,m3) ′ = , (6) m1 m2 m3 m1 m2 m3 2l +1 m ,m 3 X1 2 and III. INVERSE BREMSSTRAHLUNG HEATING

2 l1 + 1 if l2 = l1 +1 l1 1 l2 A second effect of having a high density of free (2l1 + 1)(2l2 + 1) = l if l = l 1 0 0 0  1 2 1 − electrons in the cluster plasma is that these electrons 0 otherwise can themselves undergo both stimulated and inverse (7) bremsstrahlung, creating a second mechanism through have been exploited in the derivation of Eq. (4). which laser energy can be deposited into the cluster. l l l Stimulated (inverse) bremsstrahlung refers to photon 1 2 3 represents the Wigner 3-j symbol, related m1 m2 m3 emission into (absorption from) the laser mode by an to the Clebsch-Gordan coefficient by electron colliding with an ion in the plasma. We treat the collisions of an electron with the clus- − − l l l ( 1)l1 l2 m ter ions as independent events in both time and space. 1 2 = − (8) m1 m2 m √2l +1 This allows us to focus on a single collision of an electron . with a single ion embedded in the plasma. The cross sec- × 1 1 2 2| 1 2 − tion per unit energy for a free-free transition from initial state φ ′ ′ ′ to final state φ can be shown, using The radial integrals IR(li,lf ) [Eq. (3)] were calculated | E ,l ,m i | E,l,mi in the acceleration representation, using wave functions Fermi’s golden rule, to equal generated in a variational eigenchannel R-matrix calcula- 4π2α ∂V 2 tion [27], using a B-spline basis set to describe electrons σE,l,m←E′,l′,m′ = φE,l,m φE′,l′,m′ . (9) ω3 h | ∂z | i that experience a Hartree-Slater atomic potential. This ′ potential, which was calculated employing the program In the case of photon emission (absorption), E = E ω by Herman and Skillman [28], is spherically symmetric. (E = E′ + ω). Equation (9) describes the interaction − of Its eigenstates may therefore be chosen as eigenstates of linearly polarized radiation in the acceleration represen- orbital angular momentum. tation. V is the plasma-screened atomic potential expe- As a consequence of efficient photoionization, the elec- rienced by the scattered electron. trons and ions inside the cluster form a dense, nanoscale- As with photoionization cross sections, radial wave size plasma already at an early stage of the laser pulse. functions were calculated using a nonperturbative eigen- This plasma has the effect of screening the atomic poten- channel R-matrix approach. Matrix elements between tial from both bound and continuum electrons. This low- the energy-normalized wave functions were then calcu- ers the ionization potential and changes both the initial- lated in the acceleration gauge, where the 1/r2 long-range and final-state electron wave functions. Because of this, dependence of ∂V/∂z ensures that the radial integral will cross sections for photoionization become larger as the converge, although the continuum electron wave func- screening length in the plasma becomes shorter. With tions are not spatially normalizable. sufficient screening, it becomes possible for ions to un- Although microscopic reversibility ensures that ab- dergo additional photoionization. sorption and emission cross sections coincide, stimulated To account for this process, the screened radial ma- free-free transitions act as a powerful heating process be- trix elements were calculated using the same R-matrix cause lower energy states are more highly populated than methods as for isolated Xe atoms. However, before the higher energy states in a thermal distribution. Heating initial- and final-state wave functions were calculated, rates can then be calculated for any given electron dis- the Herman-Skillman potential was multiplied by a De- tribution. In this study, we assume that after each pho- bye screening factor exp ( r/λ ). (The electron Debye ton absorption or emission event the electron gas reequi- − D length is defined as λD = T/(4πne) [29]. The electron librates rapidly as a consequence of frequent electron– temperature T in this expression is measured in units of electron collisions. Thus, the electron probability distri- p energy. ne is the electron density.) Both the matrix ele- bution ρ(E) may be written at all times during the laser ments and the corresponding photoionization potentials pulse as a Maxwell-Boltzmann distribution: were then spline-interpolated in the process of calculat- E ing the photoionization cross section at a given screening −E/T ρ(E)=2 3 e . (10) length. For most of this paper, this screening length was rπT restricted to be no less than the Wigner-Seitz radius of The cross section defined in Eq. (9) describes a free- xenon at liquid density, 4.64 bohr. A discussion of shorter free transition between orbital angular momentum eigen- screening lengths is given in a later section. states. We therefore introduce ρ(E,l,m), which is the 4 probability per unit energy to find an electron in the a rate equation for the electron temperature, expressed state φE,l,m . Clearly, ρ(E) = l,m ρ(E,l,m). If the in terms of the cross sections for stimulated and inverse wave function| i is normalized within a large sphere of ra- bremsstrahlung [Eq. (9)]. When writing down the equa- dius R (not to be confused with theP cluster radius), then tion for the time evolution of ρ(E,l,m), we must take the largest l that contributes to this sum at a given ki- into consideration that ρ(E,l,m) refers to (spherical) box netic energy E is lmax = R√2E [30]. Since, in thermal normalization, while the cross section per unit energy in equilibrium, ρ(E,l,m) can depend only on E, we see that Eq. (9) is based on energy-normalized wave functions. For the sake of consistency, it is necessary to change the initial state in the free-free radiative transition in ρ(E) ρ(E,l,m)= (11) Eq. (9) from energy normalization to box normalization. 2R2E This has the effect of multiplying the cross section by π√2E/R. in the limit of large R (lmax 1). We are interested in radiation-induced≫ heating, i.e. in the change of the electron temperature due to photon absorption and emission. To this end, we will derive from

Hence, in the presence of Na atomic scatterers (within ∞ ∂T 2 ∂ρ(E,l,m) the normalization volume) and a laser beam of intensity = dEE (12) I, the rate of change of ρ(E,l,m) is given by ∂t 3 0 ∂t Z Xl,m

∂ρ(E,l,m) I √2π ′ ′ ′ ′ = Na σE,l,m←E−ω,l′,m′ √E ωρ(E ω,l ,m )+ σE,l,m←E+ω,l′,m′ √E + ωρ(E + ω,l ,m ) ∂t ω R ′ ′ − − lX,m n σ ′ ′← √Eρ(E,l,m) σ − ′ ′← √Eρ(E,l,m) . (13) − E+ω,l ,m E,l,m − E ω,l ,m E,l,m o

The first row in the curly brackets in Eq. (13) describes where the population of φE,l,m via photoabsorption (photoe- ∞ mission) from states| withi energy E ω (E + ω); the ′ dV JR(l,l )= drUE+ω,l(r) UE,l′ (r) . (16) second row describes the depopulation− of φ due dr E,l,m Z0 to photoabsorption and photoemission from| thisi state. Equation (13) implies a nondegenerate electron gas. The rate of change of the electron temperature due to An electron state with energy E, then, communicates inverse bremsstrahlung is then with states of energy E ω, which are on average more − 3/2 densely populated than itself. Since the absorption- and ∂T 2 2π − = n I 1 e ω/T emission cross sections are equal, this tends to populate ∂t 9 a T − the state of energy E and depopulate the states of energy ∞ h i −E/T E ω, resulting in a net heating process. The state will dEe σE+ω←E . (17) − × also communicate with states of energy E + ω, which are Z0 less densely populated than itself, thereby again tending The parameter na stands for the number of atoms per to populate the higher-energy states while depopulating unit volume. In general, the ions in the plasma are the lower-energy state. not all in the same charge state. Denoting the frac- Combining Eqs. (10), (11), (12), and (13), we are led i+ (i) tion of Xe by f , σE+ω←E in Eq. (17) is replaced in a natural way to the following definition of the inverse (i) (i) (i) bremsstrahlung cross section (per unit energy): with i f σE+ω←E , where σE+ω←E is the inverse bremsstrahlung cross section in the field of Xei+. P σE+ω←E = σE+ω,l,m←E,l′,m′ . (14) Figures 1 and 2 illustrate the dramatic effects of the l,m l′,m′ ionic potential on the inverse bremsstrahlung cross sec- X X tion. In Fig. 1, as the scattering electron collides with Using Eqs. (5), (6), (7), and (9), this can be written as the ion at higher and higher initial energies, it probes regions of the ionic potential at which the ion nucleus is 4 2 α screened increasingly poorly by inner-shell electrons. As σE+ω←E = π (15) 3 ω3 a result, the inverse bremsstrahlung cross section rises lJ 2 (l,l 1)+(l + 1)J 2 (l,l + 1) , to many hundreds of times that of the naked Coulomb × R − R l potential. X 5

12000 Coulomb Potential Herman-Skillman Potential 10000 10000

8000 1000

6000

100 4000

10 2000

1 0 0 50 100 0 50 100 Inverse Bremsstrahlung Cross Section (Mb/H) Initial Electron Energy (Hartree) Inverse Bremsstrahlung Cross Section (Mb/H) Initial Electron Energy (Hartree)

FIG. 1: Inverse bremsstrahlung cross sections [Eqs. 14 and FIG. 2: The inverse bremsstrahlung cross section as a func- 15] for an electron with incident energy E to absorb a 12.7- tion of energy is shown for an electron in the field of a eV photon are given for an electron in the field of a purely Debye-screened Xe Herman-Skillman potential, with the De- Coulombic 1+ potential and for an electron in the field bye screening length λD ranging from 1 a.u. to 20 a.u. As of a Xe Herman-Skillman atomic potential. The effects of λD grows, the cross section approaches the limit of no plasma atomic structure on inverse bremsstrahlung rates are quite screening, shown in this graph by the dotted line. As the pronounced. Debye length of the cluster plasma shrinks, the charged ion is shielded more effectively from the scattering electron, and the inverse bremsstrahlung cross section is decreased.

Adding plasma screening to this picture has the effect of supplementing the screening effects of inner-shell electrons with the screening effects of plasma electrons. As a Section II. The ionization and recombination coefficients S and R for the reaction Xei++e− Xe(i+1)++2e− are result, the scattering electron feels the effects of the ionic i i → nucleus more strongly than in the pure Coulomb case, calculated later in this section. but less strongly than in the case of the unscreened ionic These rate equations, along with equations for the en- potential. This is seen in a steady decrease of the inverse ergy in the free electron gas and the radius of the clus- bremsstrahlung cross section as the screening range de- ter are integrated numerically through the duration of creases. the pulse. As a general rule, this set of equations will be quite stiff. We performed this integration using the Rosenbrock method [31]. IV. COLLISIONAL IONIZATION AND There are two requirements for a satisfactory treat- RECOMBINATION ment of collisional ionization and recombination in the cluster. First, both processes must occur at appropriate Although photoionization and inverse bremsstrahlung rates, and second, the rates for ionization and recom- are the only processes by which the cluster can absorb bination must be consistent with one another, in that photons from the laser beam, they are not by themselves they drive the cluster towards chemical equilibrium at all sufficient to explain the cluster’s evolution. As the pulse times. The second requirement is particularly significant progresses, large numbers of free electrons fill the cluster. because the usual treatment of collisional ionization (also used in this study) uses the semiempirical Lotz formula These electrons can liberate other electrons via collisional th ionization if they have sufficient energy, or they can un- [32] for ionization from the j subshell, dergo three-body recombination with an ion. j ∞ −x − a q 1 e Including the effects of ionization and recombination, Sj = 6.7 10 7 j i dx (19) the rate equation for the number per unit volume n of i × T 3/2 P /T x i j ZPj /T charge species i is given by ∞ b exp(c ) e−y cm3 j j dy , ∂ni I i−1 i −Pj/T + cj P /T +c y s = (σ ni−1 σ ni) Z j j ! ∂t ω PI − P I +Si−1ni−1ne Sinine (18) to find ionization coefficients, where aj ,bj,cj are semiem- 2 − 2 j +Ri+1ni+1ne Rinine , pirical constants, qi the number of equivalent electrons − i+ th Xe contains in the j subshell, Pj the ionization po- where ne is the number of electrons per unit volume. tential, and T the temperature. For charge states of i The photoionization cross sections σPI were calculated in 0,..., 5+, we choose semiempirical constants correspond- 6

choose constants corresponding to the 5s sublevel. Be- 15 a) cause this formula is only a semiempirical approximation, Total Energy Absorbed it is important to use recombination coefficients which are Electron Kinetic Energy Energy Absorbed due to Photoionization consistent with the ionization coefficients to prevent the model from settling into an incorrect equilibrium charge 10 distribution. The ratio between ionization and recombination coefficients can be obtained using the concept of equilibrium constants. In a plasma at equilibrium, the rate of collisions ionizing Xei+ to form Xe(i+1)+ + e− must be equal 5 to the rate at which Xe(i+1)+ + e− recombines to form Xei+. The recombination coefficients found in this man- Energy (Hartree) per Atom ner can then be applied to modeling the cluster plasma, which is not, in general, in a state of chemical equilib- 0 rium. -200 -100 0 100 200 Time (fs) Relative to Pulse Maximum These two rates are given respectively by

Sinine = rate of ionizing collisions (20)

1 and b) 0+ 1+ 2 Ri+1ni+1ne = rate of recombining collisions . (21) 0.8 2+ 3+ Hence, the ratio S/R is given by

0.6 eq eq Si ni+1ne = eq . (22) Ri+1 ni 0.4 eq eq eq The fraction (ni+1ne )/ni is known as the equilibrium constant for the reaction, and can be calculated thermo- Fraction of Population 0.2 dynamically. In any reaction A B + C at equilibrium, the chemical potentials for the→ forwards and backwards reactions 0 -200 -100 0 100 200 must be balanced µA = µB + µC. The chemical poten- Time (fs) Relative to Pulse Maximum tial of each species is given by a partial derivative of the Helmholtz free energy, ∂F FIG. 3: Evolution of a 1500 atom cluster exposed to a 100 fs, µ = . (23) 13 2 i 7 × 10 W/cm pulse, employing only photoionization and ∂Ni T,V inverse bremsstrahlung heating. a) Energy absorbed vs. time. 2+ 3+ b) Ionic population vs. time. Xe and Xe are produced The Helmholtz free energy is given by F = T ln Ztot, − eﬃciently via photoionization. where Ztot is the partition function for the system as a whole. Factoring the total partition function into the product ing to the 5p sublevel. For charge states of 6+ and of individual particle partition functions (which implies 7+, which have no 5p electrons in the ground state, we independent particles),

NA NB NC zA(V,T ) zB(V,T ) zC(V,T ) Ztot(NA,NB,NC ,V,T )= ZA(NA,V,T )ZB(NB,V,T )ZC(NC ,V,T )= , (24) NA! NB! NC !

and assuming Ni >> 1 yields Imposing balanced chemical potentials yields

∂ ln(Zi(V,T )) zi NBNC zB(V,T )zC(V,T ) µi = T = T ln( ) . (25) = (26) − ∂Ni − Ni NA zA(V,T ) 7 or equivalently the Zel’dovich and Raizer correction, but is larger for longer screening lengths and smaller at shorter screening zB (V,T ) zC (V,T ) nBnC V V lengths. Keq(T )= = z (V,T ) , (27) nA A One advantage to the equilibrium constant approach V is that it conceptually separates information about thermodynamic balance from the rate at which the system where the equilibrium constant Keq is a function of temperature only. seeks that balance. As a result, any formula for ioniza- i+ tion or recombination coefficients could be substituted for If the ionization potential of Xe is given by Pi, then the partition functions are given by the Lotz formula, with the accuracies of the overall rate and of the equilibrium constant used the only criteria for ∞ −E/T validity of the formula. zi = dEe ρi(E) , Including the effects of collisional ionization and re- 0 Z ∞ combination has a pronounced effect on the evolution −Pi/T −E/T zi+1 = e dEe ρi+1(E) , (28) of the cluster. In Fig. 3, the evolution of the cluster 0 ∞ Z is calculated employing only photoionization and inverse −E/T bremsstrahlung. In contrast, Fig. 6 shows the evolution ze = dEe ρe(E) . of the same cluster employing photoionization, inverse Z0 bremsstrahlung, collisional ionization and recombination, Through most of the lifetime of the pulse, tight plasma and evaporation of energetic electrons from the cluster. screening destroys the Rydberg states and most of the Allowing ionization and recombination has the effect of internal degrees of freedom of the various ions, leaving producing charge states up to Xe8+ in substantial quanti- the density of states ρ(E) dominated by the center of ties, and of nearly doubling the energy per atom absorbed mass term and by a combinatorial term by the cluster. m D(i)= (29) n V. CLUSTER DYNAMICS DURING THE LASER PULSE accounting for the number of ways n electrons can be distributed in m orbitals. For charge states up to 6+, we use m = 6,n = 6 i. For 7+ and 8+, we use m = As the cluster absorbs energy from the laser field, some 2,n = 8 i. If we exploit− this by setting ρ (E)/D(i) = of the electrons become so energetic that they are no − i longer bound to the cluster. In addition, the cluster ex- ρi+1(E)/D(i + 1), the common integral in zi and zi+1 falls out of the equilibrium constant, yielding pands and cools due to hydrostatic forces from the hot electrons and Coulomb repulsion as escaping electrons 3 leave a charge imbalance behind. These in turn affect i+ (i+1)+ − 2 (Xe →Xe +e ) −Pi/T T D(i + 1) Keq = e 3 (30) the microscopic processes inside the cluster, since all such √2π 2 D(i) processes depend on the concentrations of charge species within the cluster. Collisional ionization and recombina- Equation (30) can now be combined with Eqs. (20) and tion are also sensitively dependent on the temperature of (22) to yield recombination rate coefficients which have the electron gas relative to electron binding energy. appropriate magnitude and which, in combination with For the evolution of the cluster during the period of the the ionization coefficients, drive the system toward the laser pulse, we employed a simple model [35] of the clus- correct equilibrium distribution at all times. ter expansion which tracks only the radius of the cluster, A gas of charged particles has different thermodynamic the evaporation of electrons away from the cluster, and properties from an ideal gas due to Coulomb interactions the loss of heat from the electron gas accompanying both between the constituent particles. Zel’dovich and Raizer processes. We did not consider the possibility of either [33] calculate the adjustment to K due to a Debye- eq gross movement of electrons or spatial inhomogeneity of Hückel potential. The equilibrium constant including charge species within the cluster, processes which a re- Coulomb effects can be written cent theoretical study [36] has suggested may account 3 i+ (i+1)+ − 2 for the formation of highly charged ions detected at the (Xe →Xe +e ) −(Pi+∆Pi)/T T D(i + 1) Keq = e 3 , Hamburg experiment. √ 2 2π D(i) The equation for the radius of the cluster is given by (31) 2 where the change in ionization potential due to Coulomb ∂ r Pe + PCoul 1 effects is ∆Pi = (Qi + 1)/λD, the Coulomb potential 2 =3 , (32) − ∂t nXemXe r between the ion core and an electron held at distance λD. In our approach, by explicitly calculating bound state where Pe = neTe is the electron pressure and PCoul = energies for Debye-screened Hartree-Slater potentials, Q2/(8πr4) is the Coulomb pressure resulting from the we calculate this adjustment to the ionization poten- charge built up as electrons evaporate away from the clus- tial directly. Our adjustment behaves similarly to ter. 8

This model of the laser–cluster dynamics also distin- electrons with energy E = Eesc + Ekin escaped from the guishes between inner and outer ionization. Inner ioniza- cluster using Eq. (33). Integrated through the timescale tion, which takes place due to photoionization and col- of a pulse until the evaporation has stopped, this yields lisional ionization, is the process by which electrons be- an ejected electron spectrum for a single cluster exposed come liberated from their parent ion and join the cluster to the pulse. Since the clusters are located randomly plasma, where they can undergo inverse bremsstrahlung with respect to the center of the laser pulse, we fur- heating or collisional ionization/recombination. Outer ther performed a spatial integration over the radial di- ionization is the process by which electrons with sufficient mension of the pulse, assuming a Gaussian laser profile 2 2 energy escape the cluster and cease to have interactions I(r) e−r /σ from 0 to 3 σ. The length of the interac- with it. tion∝ region in the Hamburg experiment was comparable The rate of evaporation from a Maxwell distribution to the Rayleigh range for the laser; accordingly, we as- of electrons can be calculated knowing the size of the sumed a constant laser intensity along the direction of cluster, the mean free path of electrons in the cluster, propagation. After performing the spatial integration, and the temperature of the electron plasma. The rate at we found that on average .22 electrons per xenon atom which electrons escape from the cluster is then given by evaporated from the cluster in this way. The spectrum ∞ of ejection energies for these electrons shown in Fig. 4, π λ W = dv e (12r2 λ2)vf(v) (33) although not exponential, is nevertheless quite similar to fs 4 r − e Zvesc the electron spectrum found in Ref. [37]. where vesc = 2(Q + 1)/r is the velocity required for an electron to escape from a cluster of charge Q, The largest discrepancy between our calculated spec- p trum and the spectrum from [37] occurs at low ejection 2 −3/2 2 − v energy. In addition, our model of the cluster expansion f(v)=4πn (2πT ) v e 2T e predicts that the majority of electrons will comprise electron plasmas which remain bound to the cluster ions and is the Maxwell distribution, and λe is the mean free path in the cluster plasma, given by become quite cold during the process of expansion. These electrons would reach the detector at low energies and T 2 after long delay times, further boosting the spectrum at λe = et al. 4πne(Z + 1) ln Λ low energies. However, Laarmann note that for Ekin < 2.5 eV, coinciding with the region of largest dis- for a plasma with average ion charge Z. The Coulomb crepancy, the spectrum cannot be evaluated due to large logarithm, ln Λ, is set equal to 1 in our calculation of the levels of noise in the background spectra. mean free path. λe is constrained to be no greater than 2r, the diameter of the sphere. Since electrons faster than about 1 eV are ejected from As electrons evaporate from the cluster, the remaining the cluster during the pulse rather than during the slower cluster becomes ever more highly charged, and a corre- process of cluster expansion, the ejected electron spec- spondingly lower fraction of the Maxwell distribution has trum has the potential to serve as a window into the enough energy to escape the cluster, thereby choking off nature of the laser–cluster interaction. Accordingly, we the evaporation rate. give the spectra for 1500 atom clusters exposed to a 100 It is likely that nearly all high-energy electrons de- fs, 7 1013 W/cm2 pulse, and for 2500 atom clusters tected in the experiment escape during this original pe- exposed× to a 50 fs, 2.5 1013 W/cm2 pulse in Fig. 5. riod of evaporation. As the cluster expands, the temper- × ature of the electron plasma falls very quickly as electron thermal energy is converted into ion kinetic energy, while After spatial averaging, we find that 1500 atom clus- the energy required to escape the cluster falls only as 1/r. ters exposed to a 100 fs, 7 1013 W/cm2 pulse eject A recent experiment [37] has for the first time mea- 0.22 electrons per atom during× this early evaporation pe- sured the energy spectrum for electrons emitted from riod using the Wigner-Seitz cutoff model for the screening rare gas clusters exposed to intense VUV light. They give length (see section VI for a discussion of plasma screen- ejection spectra for 70 atom xenon clusters exposed to a ing). Using the Attard model, 0.07 electrons per atom 4.4 1012 W/cm2 pulse of VUV light at the same pho- are evaporated during this period. For 2500 atom clus- ton× energy as the original Hamburg experiment, finding ters exposed to a 50 fs, 2.5 1013 W/cm2 pulse, the cor- an electron distribution which decreases approximately responding numbers are 0.13× electrons per atom for the exponentially according to I = I exp( E /E ), with Wigner-Seitz cutoff model and 0.02 electrons per atom 0 − kin 0 E0 =8.9 eV. for the Attard model. In contrast to this, the Hamburg We calculated a spectrum of ejected electrons by step- experiment measured an average charge per ion of 2.98. ping through a laser pulse using small timesteps. For Hence, the electrons which comprise these ejected elec- each timestep, we calculated the electron density, mean tron spectra correspond to only a few percent of all free free path, cluster radius, and plasma temperature. Us- electrons at the time when the expanding clusters reach ing these parameters, we calculated the rate at which the detector. 9

1 1 Laarmann et al Wigner-Seitz screening a) Wigner-Seitz Screening Attard screening Attard Screening 0.75 0.8

0.6 0.5

0.4 0.25 Ejected Electrons (arb. units) 0.2

0 Ejected Electrons (arbitrary units) 0 10 20 30 40 0 Ejected Electron Energy (eV) 0 20 40 60 80 100 Kinetic Energy (eV)

FIG. 4: Ejected electron spectrum. Comparison between data from [37] and spatially-averaged spectra calculated using 70 atom clusters exposed to a 4.4 × 1012 W/cm2, 100 fs pulse for 1 b) two diﬀerent models of plasma screening. The Wigner Seitz Wigner-Seitz Screening cutoﬀ model uses the ordinary Debye length as the screen- Attard Screening ing radius, but the screening radius is not allowed to fall be- 0.8 low xenon’s Wigner-Seitz radius at liquid density, 4.64 bohr. The Attard model of screening calculates the screening ra- 0.6 dius according to equation (34), discussed in Section VI. The spectrum calculated using xenon’s Wigner-Seitz radius as a minimum screening distance displays a strong similarity to 0.4 the experimental curve. The intensity of the experimental spectra is arbitrary; magnitudes were chosen by setting each curve equal at the beginning of the exponential tail in the 0.2 experiment. Ejected Electrons (arbitrary units)

0 0 20 40 60 80 100 VI. NONIDEAL PLASMA SCREENING Kinetic Energy (eV)

As shown in Fig. 7, when plasma screening of the Xe ions becomes strong enough to allow photoionization of FIG. 5: Ejected electron spectra, calculated for the two sets Xe+ into Xe2+, large numbers of extremely low-energy of parameters and the two models of screening. The Wigner electrons are added to the plasma. As a result, the ratio Seitz cutoff model uses the ordinary Debye length as the of electron kinetic energy to electrostatic potential energy screening radius, but the screening radius is not allowed to falls dramatically, the Debye length of the plasma falls fall below xenon’s Wigner-Seitz radius at liquid density, 4.64 bohr. The Attard model of screening calculates the screen- abruptly below the Wigner-Seitz radius of xenon, and ing radius according to equation (34), discussed in section VI. the plasma enters a regime of strong correlation. In this a) Nature parameters: 1500 atom clusters exposed to a 100 regime, a number of the assumptions of Debye-Hückel fs, 7 × 1013 W/cm2 pulse. b) Thesis parameters: 2500 atom screening model break down, and the Debye length loses clusters exposed to a 50 fs, 2.5 × 1013 W/cm2 pulse. Since its meaning as a screening distance [43]. If the plasma electrons faster than about 1 eV are ejected from the cluster cools sufficiently, screening lengths can become complex, during the pulse the ejection spectra could serve as a window and result in oscillatory electron–ion correlation func- into the dynamics of the laser-cluster interaction. tions [44, 45]. Another possibly important effect of the strongly coupled plasma was identified in a recent study [46], which sity as a minimum value below which the screening was has identified electron dynamics in a strongly coupled not allowed to fall. Clearly, with the precise nature plasma as having a very large impact upon rates of many- of screening unknown in the strongly correlated regime, body recombination and hence upon energy absorption our method of calculating atomic properties based on a by the cluster as the re-combined ions undergo multiple Debye-screened atomic potential acquires a correspond- episodes of photoionization. ing uncertainty. In an attempt to estimate this uncer- Most calculations performed in this paper were per- tainty, we have described the evolution of the cluster us- formed using xenon’s Wigner-Seitz radius at liquid den- ing different models for the screening length in a highly 10

20 30 Wigner-Seitz Cutoff Total Energy Absorbed Attard Screening Electron Kinetic Energy 25 15 a)

10 15

10 5 Screening Length (bohr)

Energy (Hartree) per Atom 5

0 0 0 -200 -100 0 100 200 Time (fs) Relative to Pulse Maximum Time (fs) Relative to Pulse Maximum

FIG. 7: The interaction of plasma screening with atomic potentials is unknown as the screening length becomes very 1 0+ short. Here the screening length vs time is given for two b) 1+ 0+ simulations of a 1500 atom cluster exposed to a 100 fs, 1+ 7 × 1013 W/cm2 pulse, using two models for screening. In 0.8 2+ 3+ the ﬁrst model, the screening length is not allowed to fall be- 4+ 2+ 5+ low xenon’s Wigner-Seitz radius at liquid density. The second 3+ 6+ 7+ model for screening uses a formula given by Attard. 0.6 5+ 4+ 8+

0.4 Debye-H¨uckel length λD =1/κD according to 6+

Fraction of Population 8+ κD 0.2 κ = . (34) 7+ 1 (κ d)2/2 + (κ d)3/6 − D D p 0 -200 -100 0 100 200 This effect becomes important in the domain where λD Time (fs) Relative to Pulse Maximum d. ≤ Qualitatively, the effect of considering screening lengths in this model which are shorter than the Wigner- FIG. 6: The effects of collisional ionization and recombina- Seitz radius is twofold. First, the tighter screening tion are to allow the formation of charge states beyond Xe3+ slightly decreases inverse bremsstrahlung heating. Sec- Pictured is the time evolution of a single 1500 atom cluster ondly, it allows photoionization of Xe3+ and higher exposed to a 100 fs, 7 × 1013 W/cm2 pulse. These states charge states. Directly substituting the Attard screen- enhance the rate of inverse bremsstrahlung heating. As the ing length for the Debye length with Wigner-Seitz cutoff plasma expands and cools, the chemical equilibrium shifts to- therefore gives some insight as to how sensitive our re- ward lower charge states on a timescale much longer than the sults are to different models of the ionic potential under laser pulse, until decreasing plasma density causes recombi- very strong screening. As can be seen in Fig. 8 the At- nation and ionization rates to go to zero. a) Energy absorbed vs. time b) Ionic population vs. time during laser pulse tard screening model has a relatively small impact on our prediction for the energy absorbed by the cluster. More prominent is the formation of higher charge states, which is abetted by the reduced ionization potentials resulting correlated plasma. from the tighter screening in the Attard model. Figure 9 Our simplest approximation applied xenon’s Wigner- shows the plasma coupling parameter, a measure for the Seitz radius at liquid density as a minimum value below nonideality of a plasma, for the two models, demonstrat- which the screening was not allowed to fall. A second ing that the Attard screening model gives rise to a more model, proposed by Attard [45], deals with ions having strongly coupled plasma than the pure Debye model. In a nonzero radius. Strictly speaking, the Debye-Hückel addition, the two models give different populations for model for plasma screening is invalid except in the limit the various charge states at the end of the pulse; how- of ions which have zero size. Attard has shown that in ever, the combined effects of the cluster expansion and the case where ions have a nonzero hard-sphere radius spatial averaging over the beam profile act to destroy d, the screening length λ =1/κ differs from the classical much of this information. 11

30 80 a) a) Total Energy Absorbed Electron-Electron Coupling 25 Electron Kinetic Energy Electron-Ion Coupling

15 40

Coupling Parameter 20

Energy (Hartree) per Atom 5

0 0 -200 -100 0 100 200 -200 -100 0 100 200 Time (fs) Relative to Pulse Maximum Time (fs) Relative to Pulse Maximum

1 600 b) 0+ 1+ b) 0+ Electron-Electron Coupling 1+ 500 Electron-Ion Coupling 0.8 2+ 3+ 4+ 5+ 2+ 6+ 400 0.6 3+ 7+ 5+ 8+ 4+ 300

0.4 6+ 200 8+ Coupling Parameter Fraction of Population 0.2 7+ 100

0 0 -200 -100 0 100 200 -200 -100 0 100 200 Time (fs) Relative to Pulse Maximum Time (fs) Relative to Pulse Maximum

FIG. 8: Near the center of the pulse, the evolution of the FIG. 9: Plasma coupling parameter vs. time for the two cluster using Attard screening is very similar to the evolution models of plasma screening. The coupling parameters are 1 Z 3/2 using Wigner-Seitz screening, shown in Figure 6. For a 1500 deﬁned by Γee = aT and Γei = Γee where the average × 13 2 3 1 3 atom cluster exposed to a 7 10 W/cm , 100 fs pulse: a) distance between electrons a is given by a = ( ) / and 4πne Energy absorbed vs time for the Attard screening model, b) Z is the average charge of the ions. The plasma becomes Charge species population vs time for the Attard screening very strongly coupled early in the pulse, but the strength of model. the coupling decreases as the plasma absorbs more energy in the course of the cluster heating. a) Coupling parameters vs. time using Wigner-Seitz cutoﬀ. b) Coupling parameters vs. VII. HYDROGENIC MODEL OF INVERSE time for the Attard screening model. BREMSSTRAHLUNG

Most previous approaches to the problem of laser- using Coulomb and Herman-Skillman potentials for ions cluster interactions have considered the ionic poten- of charge 5. tial seen by the electron as a pure Coulomb poten- As can be seen in figure 6b, when the laser reaches tial. This is not an unreasonable approximation: as the maximum intensity, most of the cluster has been ionized charge of the ion increases, the difference between inverse to such high charge states. Thus, models of the inverse bremsstrahlung cross sections calculated using Herman bremsstrahlung process which use Coulombic potentials Skillman potentials and cross sections calculated using should be able to see comparable levels of heating to those Coulomb potentials is much smaller than in the case of using cross sections derived using Herman-Skillman po- the bare ion. This can be seen in figure 10, which con- tentials. trasts inverse bremsstrahlung cross sections calculated To investigate this proposition, we simulated the laser- 12

20 Coulomb Potential a) Herman-Skillman Potential Total Energy Absorbed Electron Kinetic Energy 10000 15

5 Energy (Hartree) per Atom

1000 0 5 10 Inverse Bremsstrahlung Cross Section (Mb/H) Initial Electron Energy (Hartree) 0 -200 -100 0 100 200 Time (fs) Relative to Pulse Maximum FIG. 10: Inverse bremsstrahlung cross sections [Eqs. 14 and 15] calculated for an electron in the field of a purely Coulombic 5+ potential and for an electron in the field of a Xe Herman- Skillman atomic potential of the same charge. In comparison 1 b) 0+ 6+ with figure 1, it can be seen that at higher charge states, the 0+ impact of atomic structure on inverse bremsstrahlung cross 1+ 0.8 2+ sections is decreased. 3+ 4+ 5+ 6+ 0.6 7+ 8+ cluster interaction for a 1500 atom cluster exposed to a 100 fs, 7 1013 W/cm2 using our model, but with a physi- 0.4 cal picture× chosen to emulate that of Siedschlag and Rost [36]. In the simulation, we used inverse bremsstrahlung Fraction of Population 1+ cross sections calculated with Debye-screened Coulomb 0.2 potentials. We used the same ionization potentials and 2+ 5+ 5+ photoionization cross sections as in our other simulations, 3+ 4+ 0 and used the unaltered Debye length as the screening -200 -100 0 100 200 length. Collisional ionization and recombination were not Time (fs) Relative to Pulse Maximum considered. The results of this simulation are presented in figure 11.

We found levels of energy absorption very comparable to 20 those in our own model but very different behavior of the c) ionic populations with time. Xe7+ and Xe8+, which make up almost half of the population of the cluster at the end of the pulse in our model, were present in negligible 15 quantities. Both differences between the two physical pictures are attributable to the effects of collisional ionization and re- 10 combination. Recombination slows the growth of high charge state populations by allowing some photoionized ions to recombine into a lower charge state, while colli- 5 sional ionization allows the population of charge states Screening Length (bohr) which cannot be created via sequential photoionization.

0 -200 -100 0 100 200 VIII. CONCLUSIONS Time (fs) Relative to Pulse Maximum

When a xenon cluster is irradiated by intense VUV light, there are four phases in its evolution. In the ﬁrst FIG. 11: Simulation of the laser-cluster interaction using phase, electrons are liberated from the xenon atoms and a physical model taken from [36]. In this model, inverse form a plasma. As the number of free electrons grows, bremsstrahlung cross sections are calculated using hydrogenic the screening length of the plasma shrinks. potentials and all high charge states are produced via sequential photoionization. Collisional ionization and recombination are not considered. a)Energy absorbed vs time. b)Charge state population vs time. c)Debye length vs time. 13

Once the screening length of the plasma reaches 10.6 neous model of the cluster expansion implicitly requires bohr, Xe1+ can undergo photoionization into Xe2+. This that all charge states in the same cluster have the same results in the addition of large numbers of low-energy average kinetic energy; this obviously conflicts with the electrons to the plasma, cooling it and decreasing the quadratic dependence of energy vs charge state detected screening length still further. The ratio of kinetic energy in the Hamburg experiment. Also, it is likely that high to potential energy falls dramatically, and the plasma charge states escape the cluster more quickly than low temporarily becomes strongly coupled. Ionization poten- charge states, spending less time in regions of high elec- tials for higher charge states fall with increased screening, tron density and having less opportunity to recombine. facilitating their creation. Thus, a more sophisticated model of the cluster expan- In the third phase, the plasma undergoes rapid in- sion is necessary in order to predict final charge state and verse bremsstrahlung heating. High charge states are ionic energy distributions with confidence for comparison formed through collisional ionization and recombination, with experiment. and the cluster becomes charged as energetic electrons At the center of a gaussian laser pulse using parameters evaporate away from its surface. The charge state dis- taken from the Nature paper and a Wigner-Seitz debye tribution shifts rapidly toward higher charges, with the length cutoff, each cluster absorbs on average 682 eV per average ionic charge reaching 5.5 at the pulse peak. This atom. At a distance of 3 sigma from the center of such a distribution changes only slowly on the timescale of the gaussian pulse, each cluster absorbs only .4 eV per atom. pulse. Spatial averaging over the gaussian pulse profile from 0 Finally, the cluster expands due to the pressure of the to 3 sigma gives an average of 195 eV per atom absorbed. electron gas and the cluster’s own charge. As the clus- Using parameters taken from Wabnitz’s thesis gives 219 ter expands, the electron plasma cools and becomes more eV per atom at the center, 0.2 eV per atom at 3 sigma, diffuse. Screening lengths increase, and charge state equi- and 65 eV per atom on spatial averaging. librium shifts toward lower charge states. Using a time of flight detector which could detect only Of these four phases, our current model describes the charged ions, Wabnitz et al. reported an average ion first and third phases well; the second more crudely. The energy of 400 eV, subsequently revised to 650 eV. dynamics of the expanding cluster are a challenging prob- Clearly, a spatial average such as we perform could be lem in their own right, and demand a treatment more altered by averaging over a different beam profile or by sophisticated than our simple homogeneous expansion changing the limits of the radial average and including model. more clusters which are exposed to only a tiny fraction For strongly coupled plasmas, it is unclear whether of the beam’s peak intensity. It is also clear that most of our treatment of plasma screening adequately describes the atoms in the clusters which are exposed to very small the potential seen by scattering or photoionizing elec- fractions of the peak intensity will never be ionized and trons. As the Debye length falls below the Wigner-Seitz thus would not register in a time-of-flight ion detector radius, the interaction of screening effects due to inner- such as was used in the Hamburg experiment. Thus, shell electrons and effects due to screening by continuum in the absence of better information about the beam’s plasma electrons should be considered. It is known that spatial and temporal profile and a more comprehensive the screening length diverges from the Debye length in model of the cluster expansion after the conclusion of the this limit, but the precise nature of the electron–ion po- laser pulse, it is impossible to make precise comparisons tential is unknown. between our model and the Hamburg results. There is some difficulty in comparing our results to the Nevertheless, our model of the laser-cluster interaction Hamburg experiment, due to experimental uncertainty explains some surprising features of the laser-cluster in- in laser intensity, temporal profile, spatial profile, and teraction in the VUV regime quite well. Primary among cluster size. Whereas in the Nature paper the Hamburg these is the surprising efficiency by which the clusters group described the laser pulse as 100 fs, 7 1013 W/cm2 absorb photons. Second, we explain the origin of the incident on 1500 atom xenon clusters, Wabnitz’s× thesis high charge states observed in the Hamburg experiment. [47] subsequently describes these pulses as 50 fs, 2.5 Third, we have with the same model calculated the early 1013 W/cm2 pulses incident on 2500 atom clusters. In× electron ejection spectrum measured in [37] and achieved addition, the temporal profile of the laser pulses is not great similarity to experiment, despite a cluster size and Gaussian, and varies in an unpredictable way from pulse pulse intensity which differ significantly from those of to pulse due to the nature of the SASE amplification the original Hamburg experiment. We have shown that process, which starts from shot noise. such spectra can depend strongly on the model of plasma Our model also has difficulty explaining the properties screening or the precise parameters of the experiment, of the clusters long after the laser-cluster interaction is and can therefore serve as a possible window into the na- over. As the clusters expand and cool, they continue to ture of the laser-cluster dynamics during the time period undergo collisional ionization and recombination. The of the pulse. distribution of charge states measured at the experimen- In conclusion, we have introduced a model of the laser– tal detectors bears no simple relationship to the distribu- cluster interaction in the VUV regime which takes into tion we calculate at the end of the pulse. Our homoge- account improved calculations of inverse bremsstrahlung 14 heating, photoionization, collisional ionization and re- about their experiments and Jan-Michael Rost for stim- combination. The effects of plasma screening on all of ulating discussions. This work was supported in part these processes are included, and an alternative model of by the Office of Basic Energy Sciences, Office of Science, very strong plasma screening has been considered. U.S. Department of Energy; R.S. under Contract No. W- 31-109-ENG-38.

Acknowledgments

We would like to thank Thomas M¨oller, Hubertus Wabnitz and Tim Laarmann for valuable information

[1] R. F. Service, Science 271, 920 (1996). (2003). [2] S. Sugano and H. Koizumi, Microcluster Physics [23] We use the laser parameters published in Ref. [17]. Re- (Springer, Berlin, 1998). cent analysis suggests that the pulse duration in the [3] M. H. R. Hutchinson, T. Ditmire, E. Springate, J. W. G. Hamburg experiment was probably closer to 50 fs and Tisch, Y. L. Shao, M. B. Mason, N. Hay, and J. P. Maran- that the intensity was not higher than 3 × 1013 W/cm2. gos, Phil. Trans. R. Soc. Lond. A 356, 297 (1998). (Thomas Möller, private communication.). [4] T. Ditmire, T. Donnelly, R. W. Falcone, and M. D. Perry, [24] H. Friedrich, Theoretical Atomic Physics (Springer Ver- Phys. Rev. Lett. 75, 3122 (1995). lag, Berlin, 1991). [5] Y. L. Shao, T. Ditmire, J. W. G. Tisch, E. Springate, [25] M. Rotenberg, The 3-j and 6-j Symbols (Technology J. P. Marangos, and M. H. R. Hutchinson, Phys. Rev. Press, Cambridge, Massachusetts, 1959). Lett. 77, 3343 (1996). [26] A. R. Edmonds, Angular Momentum in Quantum Me- [6] M. Lezius, S. Dobosz, D. Normand, and M. Schmidt, chanics (Princeton University Press, Princeton, New Jer- Phys. Rev. Lett. 80, 261 (1998). sey, 1996). [7] T. Ditmire, T. Donnelly, A. M. Rubenchik, R. W. Fal- [27] M. Aymar, C. H. Greene, and E. Luc-Koenig, Reviews cone, and M. D. Perry, Phys. Rev. A 53, 3379 (1996). of Modern Physics 68, 1015 (1996). [8] J. F. Seely and E. G. Harris, Phys. Rev. A 7, 1064 (1973). [28] F. Herman and S. Skillman, Atomic Structure Calcula- [9] C. Rose-Petruck, K. J. Schafer, K. R. Wilson, and C. P. J. tions (Prentice-Hall, Englewood Cliffs, N.J., 1963). Barty, Phys. Rev. A 55, 1182 (1997). [29] W. L. Kruer, The Physics of Laser Plasma Interactions [10] I. Last and J. Jortner, Phys. Rev. A 60, 2215 (1999). (Westview Press, Boulder,Colorado, 2003). [11] C. Siedschlag and J.-M. Rost, Phys. Rev. Lett. 89, [30] In practice, matrix elements involving large values of l 173401 (2002). were found to converge very quickly to zero. Because of [12] T. Seideman, M. Y. Ivanov, and P. B. Corkum, Phys. this, the total transition rate from states of energy E to Rev. Lett. 75, 2819 (1995). states of energy E′, and hence the heating, is unaffected [13] T. Zuo and A. D. Bandrauk, Phys. Rev. A 52, R2511 by the existence of a cutoff. (1995). [31] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. [14] A. McPherson, B. D. Thompson, A. B. Borisov, K. Boyer, Flannery, Numerical Recipes in Fortran 77 (Cambridge and C. K. Rhodes, Nature 370, 631 (1994). University Press, Cambridge, England, 2001). [15] J. Andruszkow, B. Aune, V. Ayvazyan, N. Baboi, [32] W. Lotz, Zeitschrift fur Physik 216, 241 (1968). R. Bakker, V. Balakin, D. Barni, A. Bazhan, M. Bernard, [33] Y. B. Zel’dovich and Y. P. Raizer, Physics of Shock A. Bosotti, et al., Phys. Rev. Lett. 85, 3825 (2000). Waves and High-Temperature Hydrodynamic Phenomena [16] V. Ayvazyan, N. Baboi, I. Bohnet, R. Brinkmann, (Academic Press, New York and London, 1966). M. Castellano, P. Castro, L. Catani, S. Choroba, [34] Y. Hahn, Physics Letters A. 231, 82 (1997). A. Cianchi, M. Dohlus, et al., Phys. Rev. Lett. 88, 104802 [35] T. Ditmire, T. Donnely, A. M. Rubenchik, R. W. Falcone, (2002). and M. D. Perry, Phys. Rev. A 53, 3379 (1996). [17] H. Wabnitz, L. Bittner, A. R. B. de Castro, [36] C. Siedschlag and J.-M. Rost, Phys. Rev. Lett. 93, R. Döhrmann, P. Gürtler, T. Laarmann, W. Laasch, 043402 (2004). J. Schulz, A. Swiderski, K. von Haeften, et al., Nature [37] T. Laarman, M. Rusek, H. Wabnitz, J. Schultz, A. R. B. 420, 482 (2002). de Castro, P. Gürtler, W. Laasch, and T. Möller, Phys. [18] H. Wabnitz, A. R. B. de Castro, P. Gürtler, T. Laarmann, Rev. Lett. 95, 063402 (2005). W. Laasch, J. Schulz, and T. Möller, Phys. Rev. Lett. 94, [38] C. Sack and H. Schamel, Physics Reports 156, 311 023001 (2005). (1987). [19] R. Santra and C. H. Greene, Phys. Rev. A 70, 053401 [39] F. Robicheaux and J. D. Hanson, Phys. Plasmas 10, 2217 (2004). (2003). [20] N. H. Burnett and P. B. Corkum, Journal of the Optical [40] M. Rusek, H. Lagadec, and T. Blenski, Phys. Rev. A. 63, Society of America B. 6, 1195 (1989). 013203 (2000). [21] F. Brandi, I. Velchev, W. Hogervorst, and W. Ubachs, [41] T. Pattard, T. Pohl, and J. M. Rost, Nucl. Inst. and Phys. Rev. A 64, 032505 (2001). Meth. in Phys. Res. 233, 132 (2005). [22] R. Santra and C. H. Greene, Phys. Rev. Lett. 91, 233401 [42] T. Pohl, T. Pattard, and J. M. Rost, Phys. Rev. Lett. 15

92, 155003 (2004). [45] P. Attard, Phys. Rev. E 45, 3604 (1993). [43] V. E. Fortov and I. T. Iakubov, The Physics of Non-Ideal [46] C. Jungreuthmayer, L. Ramunno, J. Zanghellini, and Plasma (World Scientiﬁc, Singapore, 2000). T. Brabec, J. Phys. B. 38, 3029 (2005). [44] B. P. Lee and M. E. Fisher, Phys. Rev. Lett. 76, 2906 [47] H. Wabnitz, Ph.D. thesis, Universit¨at Hamburg (2003). (1996).