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EPJ manuscript No. (will be inserted by the editor) Dissipation and energy balance in electronic dynamics of Na clusters M. Vincendon1,2, E. Suraud1,2, and P.-G. Reinhard3,a 1 Institut fu¨r Theoretische Physik, Universität Erlangen, D-91058 Erlangen, Germany 7 2 Université de Toulouse; UPS; Laboratoire de Physique Théorique (IRSAMC); F-31062 Toulouse, France 1 3 CNRS; LPT (IRSAMC); F-31062 Toulouse, France 0 2 First draft: 31. 12. 2016 n a J Abstract We investigate the impact of dissipation on the energy balance in the electron dynamics of metalclustersexcitedbystrongelectro-magneticpulses.ThedynamicsisdescribedtheoreticallybyTime- 1 Dependent Density-Functional Theory (TDDFT) at the level of Local Density Approximation (LDA) 3 augmented by a self interaction correction term and a quantum collision term in Relaxation-Time Ap- ] proximation (RTA). We evaluate the separate contributions to the total excitation energy, namely energy l exported by electron emission, potential energy due to changing charge state, intrinsic kinetic and poten- l a tial energy, and collective flow energy. The balance of these energies is studied as function of the laser h parameters (frequency, intensity, pulse length) and as function of system size and charge. We also look - at collisions with a highly charged ion and here at the dependence on the impact parameter (close versus s e distantcollisions).Dissipationturnsouttobesmallwheredirectelectronemissionprevailsnamelyforlaser m frequenciesaboveanyionizationthresholdandforslowelectronextractionindistantcollisions.Dissipation is large for fast collisions and at low laser frequencies, particularly at resonances. . t a m PACS. 0 5.60.Cg,31.15.ee,31.70.Hq,33.80.Wz,34.10.+x,36.40.Cg - d n 1 Introduction affordable approach to dissipation, the Relaxation-Time o Approximation (RTA), which had been implemented re- c Time-Dependent Density-Functional Theory (TDDFT) is cently for simulations of finite electronic systems [24]. [ thestartingpointandtheleadingtooltosimulatethedy- namicsofmany-fermionsystems,inelectronicsystems[1, Althoughhighlydesirable,theoreticalinvestigationsof 1 dissipation in finite fermion systems have been hampered v 2,3]aswellasinnuclei[4,5,6].TheLocalDensityApprox- so far by the enormous computational demands for a mi- 4 imation (LDA) provides a robust and efficient mean-field croscopic description of two-body collisions in the quan- 2 descriptionofdynamicswhichallowstocoverahugerange tum regime. The way from the full many-body hierarchy 0 of phenomena from the linear regime of small-amplitude 9 oscillations (also known as random-phase approximation) down to a mean-field description augmented by dynami- 0 cal correlations has been thoroughly developed since long [7]tosystemspossiblyhighlyexcitedbystronglaserpulses . in classical non-equilibrium thermodynamics [25], leading 1 [8,9]orheftycollisions[10,11].However,moredetailedob- eventually to the much celebrated Boltzmann equation 0 servations and/or long-time evolution is often sensitive to 7 allsortsofmany-bodycorrelationsbeyondthemean-field to account for dynamical correlations in classical systems 1 [26]. A manageable scheme for a fully quantum mechan- approach[12].Aparticularlyimportantclassaredynami- : ical description in finite systems is still a matter of ac- v calcorrelationsfromtwo-fermioncollisions.Theyadddis- tualresearch.OneimportantquantumfeatureisthePauli i sipation to the mean-field motion which has important X principle. It can be accounted for by extending the Boltz- consequencesinagreatvarietyofdynamicalscenariosand r systems,e.g.,forcollisionalbroadeningofexcitationspec- manncollisiontermtotheBoltzmann-Uehling-Uhlenbeck a (BUU) form [27]. This semi-classical BUU approach (also tra[13],fornecessarythermalizationstepsinnuclearreac- knownasVlasov-Uehling-Uhlenbeck(VUU)equation)pro- tions [14,15,16], for thermalization in highly excited elec- videsanacceptablepictureatsufficientlylargeexcitations tronic systems [17,18]. Dissipation (and thermalization) wherequantumshelleffectscanbeignored.Ithasbeenex- is a particularly important and much discussed process tensivelyusedinnuclearphysics[28,29]andalsoemployed in the dynamics of small metal clusters, see e.g. [19,20, for the description of metal clusters in a high excitation 18,21,22,23].Inthepresentpaper,weaddressdissipation domain [30,31]. Although very successful, BUU/VUU is andenergytransportinsmallmetalclusterstakingupan valid only for sufficiently high excitation energies. And a e-mail: [email protected] even in the high-excitation domain, de-excitation by ion- 2 M. Vincendon et al.: Dissipation and energy balance in cluster dynamics ization can quickly evacuate large amounts of excitation coupling to the ions is mediated by soft local pseudopo- energy thus cooling the system down into a regime where tentials [45]. The electronic exchange-correlation energy quantumeffectscountagaindominantly.Inanycase,there functional is taken from Perdew and Wang [46]. isanurgentneedforaquantumdescriptionaugmentedby The Kohn-Sham potential is handled in the Cylindri- relaxation effects. cally Averaged Pseudo-potential Scheme (CAPS) [47,48], Suchdissipativequantumapproachesarestillwellman- which has proven to be an efficient and reliable approxi- ageableinbulksystemsandhavebeenextensivelystudied mation for metal clusters close to axial symmetry. Wave- intheframeworkofFermiliquidtheory[32].Itwasfound functionsandfieldsarethusrepresentedona2Dcylindri- that global features of dissipation can often be character- cal grid in coordinate space [49]. For the typical example ized by one dominant, exponential relaxation mode. This of the Na cluster, the numerical box extends up to 104 40 motivated the Relaxation Time Approximation (RTA) as a in radial direction and 208 a along the z-axis, while 0 0 was introduced in [33] and later on applied to a wide va- the grid spacing is 0.8 a . To solve the (time-dependent) 0 riety of homogeneous systems [34,35]. The quantum case Kohn-Sham equations for the single particle (s.p.) wave- for finite systems is much more involved. A full descrip- functions, we use time-splitting for time propagation [50] tion of detailed correlations has been carried through in andacceleratedgradientiterationsforthestationarysolu- schematic model systems [36] and in the time-dependent tion [51]. The Coulomb field is computed with successive configuration-interaction (TD-CI) method [37], both be- over-relaxation [49]. We use absorbing boundary condi- ing nevertheless limited to simple systems. A stochastic tions [43,52], which gently absorb all outgoing electron treatmentofthequantumcollisiontermpromisesatractableflow reaching the boundaries of the grid. The difference approach [38]. It has meanwhile been successfully tested between the initial number of electrons and the actual in one-dimensional model systems [39,40] and will be de- number of electrons left in the simulation box is thus a velopedfurther.Recently,RTAhasbeenbeensuccessfully measure for ionization in terms of N , the number of esc implementedasdissipativeextensionofTDLDAforfinite escaped electrons. systems and applied to the realistic test case of Na clus- Theexternallaserfieldisdescribedasaclassicalelectro- ters [24]. This now provides an affordable and efficient magnetic wave in the long wavelengths limit. This aug- approach to dissipation in finite fermion systems. ments the Kohn-Sham Hamiltonian by a time-dependent The present paper uses RTA to study systematically external dipole field the dynamics of Na cluster during and after laser excita- tionindependenceonthekeylaserparameters,frequency, Uext(r,t)=e2r·ezE0 sin(ωlast)f(t) , (1) intensity, and pulse length. At the side of observables, we (cid:18) t (cid:19) f(t)=sin2 π θ(t)θ(T −t) . (2) concentratehereontheenergybalance.Tothisendwein- T pulse pulse troducethevariouscontributionstotheexcitationenergy, namelyintrinsickineticandpotentialenergy,chargingen- The laser features therein are: the (linear) polarization ergy, and energy loss by electron emission. The paper is e along the symmetry axis, the peak field strength E z 0 organized as follows. In section 2, we summarize the nu- relatedtolaserintensityasI ∝E2,thephotonfrequency 0 0 merical handling of TDLDA and the RTA scheme. In sec- ω ,andthetotalpulselengthT .Thelatterisrelated las pulse tion, 3 we introduce in detail the key observables used tothefullwidthathalfmaximum(ofintensity)asFWHM in this study, the various contributions to the energy. In = T /3. pulse section4,wepresenttheresults,especiallytheenergybal- This basic building block, mean-field propagation of ance as function of the various laser parameters. Further thes.p.wavefunctionsφ (t)accordingtoTDLDA,canbe α technical details are provided in appendices. summarized formally as |φ (t)(cid:105)=Uˆ(t,t(cid:48))|φ (t(cid:48))(cid:105), (3a) α α 2 Formal framework (cid:32) (cid:90) t(cid:48) (cid:33) Uˆ(t,t(cid:48))=Têxp −i hˆ(t(cid:48)(cid:48))dt(cid:48)(cid:48) , (3b) 2.1 Implementation of TDDFT t pˆ2 hˆ(t)= +U [ρ(r,t)], (3c) Basisofthedescriptionismean-fielddynamicswithTime- 2m KS Dependent Density Functional Theory (TDDFT). Actu- ally, we employ it at the level of the Time-Dependent where Uˆ(t,t(cid:48)) is the unitary one-body time-evolution op- Local-Density Approximation (TDLDA) treated in the erator with Tˆ therein being the time-ordering operator, hˆ realtimedomain[1,2].ItisaugmentedbyaSelf-Interaction istheKohn-Shammean-fieldoperator,andU the(den- KS Correction (SIC) approximated by average-density SIC sity dependent) actual Kohn-Sham potential [53]. (ADSIC) [41] in order have correct ionization potentials [42], which is crucial to simulate electron emission prop- erly. The time-dependent Kohn-Sham equations for mean 2.2 Brief review on RTA field and single-electron wave functions are solved with standard techniques [43,44]. The numerical implementa- Mere TDLDA is formulated in terms of a set of occupied tion of TDLDA is done in standard manner [43,44]. The single-particle (s.p.) wavefunctions {|φ (t)(cid:105),α = 1...N} α M. Vincendon et al.: Dissipation and energy balance in cluster dynamics 3 propagating according to eq. (3). So far, TDLDA deals states are given occupations weights W(eq) according to α with pure Slater states. Dissipation leads inevitably to thermal equilibrium. The temperature T is tuned to re- mixed states. These can be described compactly by the produce the desired total energy E. For details of this one-body density operator, which reads, in natural or- cumbersome procedure see [24]. bitals representation, OncethisDCMFstepisundercontrol,theRTAscheme is straightforward. The collision term in Eq. (5a) is eval- Ω (cid:88) uated at time intervals ∆t, typically 0.25 fs and for high ρˆ= |φ (cid:105)W (cid:104)φ | (4) α α α laser frequencies somewhat shorter. In between, the s.p. α=1 wavefunctions in the one-body density are propagated by where Ω is the size of the configuration space, which is mean-field evolution Eq. (3b). Once one time span ∆t is significantly larger than the actual electron number N. completed, we stay at time t+∆t and dispose of a mean- The weights W represent the occupation probability for field propagated, preliminary one-body density ρãnd we α s.p.state|φ (cid:105).Thepuremean-fieldpropagationleavesthe evaluate the collision term. First, the actual (cid:37), j, and α occupation weights W unchanged and propagates only E are computed. These are used to determine the local- α thes.p.states,suchthatρˆ(t)=(cid:80)Ω |φ (t)(cid:105)W (cid:104)φ (t)|= instantaneous equilibrium state ρêq. This is used to step Uˆ(t,0)ρˆ(0)Uˆ−1(t,0) with Uˆ accordαi=n1g toαEq. (3αb).α tothenewone-bodydensityρ(t+∆t)=ρ˜+(∆t/τrelax)(cid:0)ρˆ− (cid:1) ρˆ [(cid:37),j,E] . In a final clean-up, this new state ρ(t+∆t) is Dynamicalcorrelationsgeneratetime-evolutionchanges eq mapped into natural orbitals representation Eq. (4), thus also for the occupation weights. The RTA describes this delivering the new s.p. wavefunctions ϕ (t+∆t) and oc- in terms of the density-matrix equation [24] α cupation weights W (t+∆t) from which on the next step α ∂ ρˆ+i(cid:2)hˆ[(cid:37)],ρˆ(cid:3)= 1 (ρˆ−ρˆ [(cid:37),j,E]) , (5a) is performed. For more details see again [24]. t τ eq relax wherehˆ[(cid:37)]istheKohn-ShamHamiltonianEq.(3c)inLDA 3 Energies as key observables (with ADSIC) depending on the actual local density distribution (cid:37)(r,t) = (cid:80) W |φ (r,t)|2. The right-hand-side In our previous paper on RTA, we have concentrated on α α α stands for the collision term in RTA. It describes relax- thermalizationprocesses,inparticularonrelaxationtimes ation towards the local-instantaneous equilibrium state [24].Here,wearegoingtoemployRTAtotheenergybal- ρˆ [(cid:37),j,E] for given local density (cid:37), current distribution j ance in metal clusters excited by strong laser fields. The eq andtotalenergyE.Therelaxationtimeτ isestimated keyobservablesarethevariouscontributionstotheenergy relax in semi-classical Fermi liquid theory. For the metal clus- whichwewillintroduceinthissection.Theexpressionsas- ters serving as test examples in the following, it becomes sume tacitly a numerical representation of wavefunctions and fields on a spatial grid in a finite box with absorbing (cid:126) σ E∗ =0.40 ee intr , (5b) boundaries.Particularlytheboundariesrequiresomecare τrelax rs2 N as we will see. The basic question we aim to analyze here is how the where E∗ is the intrinsic (thermal) energy of the sys- intr energy absorbed by the laser is ”used” by the cluster and tem,N theactualnumberofelectrons,σ thein-medium ee redistributedintovariouswellidentifiedcomponents.The electron-electron cross section, and r = (3/(4π(cid:37)))2/3 is s key starting quantity will thus be the energy absorbed the Wigner-Seitz radius of the electron cloud [24]. It em- by the laser which we denote by E . The basic energy ploys an average density (cid:37) because τ is a global pa- abs relax branching channelsof theclusterconsistinelectron emis- rameter.Thisapproximationislegitimateformetallicsys- sion and intrinsic heating [24] and we thus have to ana- tems where the electron density is rather homogeneous lyze both these components separately. Electron emission remaining generally close to the average. Note that the corresponds to charge loss associated with energy loss be- in-medium cross section σ also depends on this average ee cause the emitted electrons carry some energy outwards. density through the density dependence screening effects. We denote this energy by E . But electron emission The actual σ is taken from the careful evaluation of [54, ch,loss ee also affects the cluster itself, net cluster charge leading to 55] computing electron screening for homogeneous elec- an associated change in potential energy E . The re- tron matter in Thomas-Fermi approximation. This yields ch,pot maining energy dekivered by the laser is shared between σ = 6.5 a2 for the case of Na clusters for r ≈ 3.7 a . ee 0 s 0 collective motion of electron leading to collective kinetic These are the values which are used throughout this pa- energy E and ”intrinsic” excitation energy of the elec- per. coll tron cloud itself consisting out of a kinetic E and Themostdemandingtaskistodeterminetheinstanta- intr,kin a potential E component. All terms, of course, sum neousequilibriumdensity-operatorρˆ [(cid:37),j,E]intheRTA intr,pot eq up to E : equation Eq. (5a). It is the thermal mean-field state of abs minimumenergyundertheconstraintsofgivenlocalden- E =E +E +E +E +E , (6) abs ch,loss ch,pot intr,kin intr,pot coll sity (cid:37)(r), local current j(r), and total energy E. For the wavefunctions we use the density constrained mean-field Letusnowspecifythesevariouscontributionsinmorede- (DCMF) techniques as developed in [56], extended to ac- tail.Thisimpliesthatwealsodetailsmallcomponentsre- count also for the constraint on current j(r). The s.p. latedtothetreatmentofabsorbingboundariesconditions 4 M. Vincendon et al.: Dissipation and energy balance in cluster dynamics andwhichhavetobeproperlyaccountedforintheenergy 6. E = intrinsic potential energy : intr,pot balance.Moreover,weintroduceasauxiliaryquantitythe actual total energy E(t) of the system which is a crucial E =E (ρ,j=0,T=0)−E (Q) . (13) intr,pot DCMF g.s. input for the RTA step. The various energy components This is the “potential” energy stored in the constraint are thus computed as follows: on given ρ & j at T =0. 1. E = Energy absorbed from the laser field: abs 7. E = collective flow energy: coll (cid:90) t (cid:90) E = dt(cid:48) d3rE (t(cid:48))·j(r,t(cid:48))−E(mask) (7) (cid:90) j2(r) abs 0 abs E = d3r (14) 0 coll 2mρ(r) where E(mask) is a correction for the particle loss at This is the kinetic energy which is contained in the abs average momentum distribution j(r). It is to be noted the absorbing bounds (for details see appendix A). that E = E (ρ,j,T=0)−E (ρ,j=0,T= coll DCMF DCMF 0).ThisshowsthatE ispartoftheintrinsicenergy. 2. E(t) = total energy: coll For the balance plots below, we consider also the rela- E(t)=ET∗DLDA(t)+Epot,bc , (8) tivecontributionsEch,loss/Eabs,Ech,pot/Eabs,Eintr,kin/Eabs, E /E ,andE /E addinguptoone.Moreover, (cid:90) t (cid:90) intr,pot abs coll abs Epot,bc = dt(cid:48) d3r(1−M2)UKSρ(r,t(cid:48)). (9) we use the completeness Eq. (6) to deduce Eintr,pot from 0 the other energies. This saves another costly DCMF evaluation for E ((cid:37),j = 0,T = 0) in the definition Eq. TherebyE∗ (t)=E (t)−E istheenergy DCMF TDLDA TDLDA g.s. (13). E (t)computedwiththegivenLDA+ADSICfunc- TDLDA Finally, we mention that the evaluation of the intrin- tional taken relative to the static ground state energy sic kinetic energy Eq. (12) had been used in the past of- E . The E is a correction for the small amount g.s. pot,bc ten with a semi-classical estimate [43], for details see ap- ofbindingenergycarriedintheabsorbedelectrons,an pendix B. This is much simpler to evaluate, but not pre- artifact which arises due to finite numerical boxes. Al- cise enough for the present purposes. Moreover, we need together, E(t) accounts for the energy left within the the expensive DCMF state anyway and so get the correct simulationboxasresultofenergyabsorptionfromthe quantum mechanical value Eq. (12) for free. laser and energy loss through ionization. 3. Ech,loss = energy loss by electron emission: 4 Results and discussion E =E −E(t) (10) In the previous RTA paper [24], we had briefly looked at ch,loss abs dissipation effects as function of laser frequency for con- It represents the kinetic energy carried away by the stant intensity and found that dissipation is strong if the emitted electrons. laser is in resonance with a system mode and weak oth- erwise. This is a trivial result in view of Eq. (5b): The 4. Ech,pot(Q) = charging energy: relaxation rate increases with excitation energy and excitation energy is large at resonance. In order to eliminate E (Q)=E (Q)−E −E (11) ch,pot g.s. g.s.,initial pot,bc this trivial trend, we consider here variation of laser parameters for fixed absorbed energy E tuning the inten- abs whereE (Q)isthegroundstateenergy(i.e.temper- g.s. sity such that the wanted value for E is maintained. abs atureT =0)foragivenchargestateQandE = g.s.,initial We calibrate the laser parameters this way for the case of E(t=0)theinitialgroundstateenergy.Forcompensa- pure TDLDA and use the same parameters then also for tionofdefinition(8),itisaugmentedbythecorrection RTA.TheresultingE isinmostsituationsthesame.A abs for lost potential energy. The E (Q) accounts for ch,pot difference in E between RTA and TDLDA, if it occurs, abs theexcitationenergyinvestedforchargingthecluster. is then already a message. One of the interesting topics related to energy bal- 5. E = intrinsic kinetic energy : intr,kin ance is the question of appearance size, the limit of fis- sion/fragmentation stability of a metal cluster for a given Eintr,kin =ETDLDA(t)−EDCMF((cid:37),j,T=0) (12) charge state [57,19]. It is the lower the more gentle one canarrangeionization.Thesystematicsofenergybalance where E (t) is the actual LDA+ADSIC energy TDLDA willtellushowtoionizemostgentlyor,inreverse,toheat and E ((cid:37),j,T =0) the DCMF energy at T = 0 DCMF most efficiently. (= ground state for fixed (cid:37) and j). The computation issimplifiedbyexploitingthefactthat(cid:37)andjremain frozen in DCMF and thus also the Kohn-Sham poten- 4.1 Typical time evolution of energies tial. This allows to take the difference of the sums of s.p. kinetic energies between the two configurations. The lower panel of figure 1 shows the time evolution of the five contributions Eq. (6) to the energy stacked in a M. Vincendon et al.: Dissipation and energy balance in cluster dynamics 5 RTA Na40, CAPS, ω=2.7 eV, E0=0.017 eV/a0, T=100 fs TDLDA alreadyatmeanfieldlevel.Asaconsequence,collectiveki- netic energy becomes negligible soon after the laser pulse Nesc 0 .18 is extinguished. We will ignore it in the following analysis ionization 000...246 eevnaelrIugtayitseErdematarlfakrtaoebmlsettathgheeastlaoRsfTetrAh,eaacllltlouhwsotsuegtrhodayebnxsaaomcrtbilcyms.tuhcehsmaomree abs 0 pulse is used in both cases. This is a particular feature 0.4 0.3 of resonant excitation related to Rabi oscillations [60]. dipole [a]0 -- 0000 ....02112 Tthheeeleexctterronnacllofiuedld.Tqhuiisckdliypoilnedeuxcceistadtiiopno,leonocsecislulaffiticoinenstolyf -0.3 large, leads to stimulated emission and so reduces exci- -0.4 tation. This can be seen from oscillations of E where 1102 EEiinnttrr,,pkoint EchargeE,locossll phases of energy absorption are interrupted byapbhsases of V] Echarge,pot Eabs energylossbacktothelaserfield.NowinRTA,dissipation energies [e 468 ssetirmveuslaatseadceommipsseitoinngadned-esxociptaavtieosntchheawnnaeyltwohmichorreedsuticmes- 2 ulated absorption. This mechanism is less important off 0 resonance where we observe generally less differences be- 0 50 100 150 200 250 0 50 100 150 200 250 time [fs] time [fs] tween RTA and TDLDA as we will see in the upper panel Figure1. Timeevolutionofionization(upperpanels),dipole of figure 2. moment (middle panels), and energies (lower panels) for the case of Na in CAPS excited by a laser with frequency ω = 40 2.7 eV, total pulse length T = 100 fs, and intensity I = 4.2 Trends with laser frequency pulse 1.31010 W/cm2. Left panels show results from RTA and right panelsfromTDLDA.Thelowerpanelsshowthetotalabsorbed The main intention of the study is to figure out trends energy (black line) ad the four different contributions stacked with laser parameters. To this end, we simulate each case one above the other. for a time of 300 fs and collect the results at this final time. This is a safe procedure for the majority of non- resonant cases. It is incomplete for resonant excitation, balance manner. Each colored band represents the contri- at least with TDLDA. In the latter case we have to keep butionindicatedinthekeytotherightsideofthepanels. in mind that the contribution of emission is somewhat Upper and middle panels show as complementing infor- underestimatedandthatofintrinsicenergyoverestimated. mation dipole moment and ionization. The case ω = 2.7 The major trends remain, nonetheless, the same. eVshowninFigure1correspondstoaresonantexcitation Figure2showstheenergycontributionsandotherob- at the Mie plasmon frequency. We see this from the time servables as function of laser frequency ω. The laser in- evolutionofionizationNesc ≡Qanddipole.TheTDLDA tensityistunedforeachfrequencysuchthattheabsorbed result(rightpanels)showsongoingdipoleoscillationsand, energy is about the same, namely E ≈ 8.2 eV, for abs connected with that, ionization carries on long after the TDLDA. The same field strength is then used also for laser pulse has terminated. However, the RTA ionization RTA and the emerging E may then be different. This abs (upper left panel) turns gently to a constant Nesc. This isindeedseenintheleftmiddlepanelwherejustnearthe is achieved by the dissipation in RTA which damps the Mie plasmon resonance (≈ 2.7 eV) RTA absorbs much dipole signal. This highly resonant case reveals a marked more energy, as was discussed already in connection with qualitativedifferencebetweenTDLDAandRTA.Wethus figure 1. see that long-time TDLDA simulations have to be taken TheupperleftpanelofFigure2showsthefieldstrength with care because they overestimate the long-lasting re- E .TheMieplasmonresonanceisvisibleasmarkeddipat 0 verberations of the dipole. ω = 2.7 eV because resonance means that more response The difference in ionization also shows up as a dif- is achieved with less impact. The steady growth of E 0 ference in the energy loss by ionization (green and blue for larger frequencies complies with the Keldysh formula areas) such that eventually TDLDA produces relatively where the effective field strength shrinks ∝ω−2 [61]. less intrinsic excitation energy in than RTA. ThemiddlerightpanelshowsionizationN .Atlower esc Thelowerpanelsoffigure1alsoshowthecollectiveki- frequencies, RTA suppresses emission significantly. Obvi- neticenergyEq.(14).Itplaysaroleintheinitialstagesof ously, more of the absorbed energy is turned to intrinsic excitation. The reason is that the dipole field of the laser excitation (thermalization). Quite different is the behav- couples to the collective dipole operator thus depositing ior at high frequencies above ionization potential (IP) in itsenergyfirstincollectivedipoleflow.However,thelarge whichcaseTDLDAandRTAdeliveralmostthesameion- spectralfragmentationofthedipolemode(Landaudamp- ization. ing)[58,59]spreadsthecollectiveenergyveryquicklyover Thelowerpanelsdisentangletheabsorbedenergyinto thedipolespectrum.Thelargefragmentationwidthofthe its four relevant contributions (6). Again, we see that actual test case Na produces a relaxation time below 1 TDLDA and RTA differ most at the side of lower ener- 40 fsforthisLandaudampingandthisrelaxationtakesplace gies, particularly near the Mie plasmon resonance. There 6 M. Vincendon et al.: Dissipation and energy balance in cluster dynamics ω=2.7eV, RTA TDLDA 1 RTA,TDLDA ω=0.8eV, RTA TDLDA ω=6.1eV, RTA TDLDA E [eV/a]00 01.21 TDRLDTAA TDRLDTAA 11..23 RTATDLDAE/Eabsabs 11112 .....1224682 ω=0ωω.8== 26e..V71 eeRVVy N/E [1/eV]escabs 0000 0....0111.18246 E [eV]abs 11 8901 000011.....67891 ionization Nesc 0 .1R8T 5A0 100Tpulse [fs] 150 0.06 50 100TTpDulLseD [fAs] 150 7 0.5 ω=2.7 eV energies/Eabs 0000 ....12468RT 1A 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9TD 10LD 1A1EEcchhaEEargriingnetet,rrl,,o,ppksooinstt Fenergies/Eabsig 0000u ....02468r 5e0 3. L 10o0Twpulsee r[fs]p 1a50nel: En 50ergy b 1a00lTapunlsec [fes] a15s0 functioEEcnchhaEEargriingneotet,rrl,,o,ppfksooinsttpulse 0 length T for Na excited by a laser with three frequency 1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11 pulse 40 ω [Ry] ω [eV] ω = 2.7 eV and intensity tuned to E(TDLDA) ≈ 8.2 eV. Left abs Figure 2. Various observables from RTA (full lines) and upper panel: The ratio E(RTA)/E(TDLDA) for three different abs abs TDLDA (dashed lines) evaluated at final time of the simu- frequenciesasindicated,resonantω=2.7Ryandoff-resonant lations at 300 fs. Pulse length was Tpulse =100 fs throughout. ω=0.8,6.1eV.Rightupperpanel:RatioNesc/Eabsofemitted Intensityhasbeentunedsuchthat√Eabs ≈8.2eVforTDLDA. electrons per absorbed energy for the three frequencies as in Upper left: field strength E0 (∝ I). Middle left: total ab- the left panel and separately for RTA (full lines) as well as sorbedenergyEabs.Middleright:ionizationNesc.Lower:Bal- TDLDA (dashed lines). anceofrelativeenergies(energycontributionsdividedbytotal absorbed energy E ). Left panel for RTA and right one for abs TDLDA. output is achieved near the point from which on all elec- tronscanberemovedbyonephotonwhichis6.1eVinthe presentcase.Mostheatingisobtainedbelow,particularly is practically no difference from ω ≈ 6.1 eV on. This near resonance or for very low frequencies. ω = 6.1 eV is a very prominent point. It is just the frequency from which on all occupied valence electrons of Na can be emitted by a one-photon process. The IP at 40 4.3 Trends with pulse length T 3.5 eV Ry sets the frequency where the HOMO can be pulse removed by one photon. The region 3.5-6.1 eV covers the transition from the onset of one-photon processes for the Figure 3 shows the effect of laser pulse length Tpulse. The least bound state to an “all inclusive” one-photon ioniza- lowerpanelsshowtheenergybalance.asfunctionofTpulse tion. And we see, indeed, how TDLDA and RTA results for the resonant case ω = 2.7 eV. The trends with Tpulse come stepwise closer to each other in this region. are extremely weak, even for the most sensitive case of resonant excitation. They are equally weak for other fre- The lower panels of figure 2 shows the results in form quencies. Thus these are not shown. of energy balance where the filled areas visualize a given contribution, as indicated. Blue and green areas together One interesting aspect pops up, again, concerning the showtheamountofenergyspentforionizationwhilepur- amountofabsorbedenergy.Thisisillustratedintheupper ple and yellow together illustrate the part of the intrin- panel of figure 3 showing the ratio from RTA to TDLDA, sic energy. The balance plot makes the trends of intrinsic E(RTA)/E(TDLDA), for three frequencies standing for the abs abs energy immediately visible. Its fraction is largest around three typical regions, very low frequency (0.8 eV), res- resonance and smallest above the point of “all one pho- onance (2.7 eV), and above threshold for direct ioniza- ton”ionizationnearω =6.1eV.ThistrendholdsforRTA tion of all shells (6.1 eV). The energy ratio increases dra- as well as for TDLDA. What differs are the actual frac- matically with pulse length in the resonant case ω = 2.7 tions of intrinsic energy, generally being somewhat larger eV. Although the partition of energies changes very little, for RTA. But the fractions are not so dramatically differ- the total output becomes much larger with RTA for long ent as one may have expected from the plot of energies pulses.Thishappensbecausedissipationsteadilyremoves as such in figure 1. Division by E and the often larger energy from the coherent dipole oscillations thus keep- abs E in RTA reduces the effect for the fractions of energy. ing the door open for ongoing energy absorption while in abs Already at this point, we can give a first answer to TDLDA energy loss by stimulated emission limits energy the question of how to ionize most gently or to heat most take-up, see the discussion in section 4.1. For off-resonant efficiently. Least intrinsic energy relative to most electron cases,theratioE(RTA)/E(TDLDA)staysclosetooneascan abs abs M. Vincendon et al.: Dissipation and energy balance in cluster dynamics 7 be seen here for low frequency ω = 0.8 eV and for high large parts of the s.p. states out of the one-photon regime frequency 6.1 eV. back to the multi-photon regime. Differences between the frequencies shrink with increasing E because the frac- 0 tion of intrinsic energy decreases with E for the low and 0 4.4 Trends with field strength (laser intensity) medium frequencies thus approaching the high frequency case (related to direct emission). Convergence is better visible within the given span of E for RTA while it re- 0 quires even larger E for TDLDA. The effect is plausible 0 1.35 because large E means that we come into the field dom- (TDLDA)Eabs 11 11..12..2355 iannadtewdhreergeimdierecw0thfieerledfermeqiusseinocnietsakbeescoomveer [l6e2ss].important (RTA)E/abs 01 1..90 .1155 enerTghyeEu(pRpTeAr)p/Ean(eTlDoLfDfiAg)uares4fusnhcotwiosnthoef firaetldiosotfreanbgstohrbfeodr abs abs RTA 0.9 0.005 0.01 0.02 TDLDA the resonant frequency ωlas = 2.7 eV. This case, unlike the non-resonant frequencies, shows a peak at a certain 1 ω=2.7 eV field strength. This emerges as combination from several 0.8 energies/Eabs 000...246 EEcchhaEEargriingnetet,rrl,,o,ppksooinstt fedweniiaffettherugrryimensdcuresceephaeonsfsirinbotgemedfofi,TreteDlhd.uLAssDttlriAstemt.nlMegatldlohifirseewsilepdhnaiscetthrirogenyinsgbactneohcdnsov,RmetThretAseeradedboispseorslreibtfneteorldet- 0 0.005 0.01 0.02 0.005 0.01 0.02 ably to intrinsic energy in the resonant case opening sub- RTA TDLDA sequently the pathway to more absorption. This explains 1 ω=6.1 eV the increase from low E0 on upwards. For larger amounts energies/Eabs 0000....2468 EEcchhaEEargriingnetet,rrl,,o,ppksooinstt orqinefucsaeornbencasaoysn.irncbTegehdfitihseeulnedsexrpsrgtleyrade,inuntshcgitetnhhegsen.hrdeaesncorcneeaadnsdetiosrsfeistpphaoetnirosanetibaortofoaprdeefaunkrstfhtrheeer- 0 0.05 0.07 0.1 0.14 0.05 0.07 0.1 0.14 RTA TDLDA 1 ω=0.8 eV 4.5 Impact of cluster charge 0.8 energies/Eabs 000...246 EEcchhaEEargriingnetet,rrl,,o,ppksooinstt Fiveaodrriletathsyeerooefnxelcairsteeafrteiropenanrcwaemitsheytseetrxestm.enWNsievae4a0e,rxewpneloohrwaatvigeoonsinoogffattrhoesvtruaicrdhy- 0 0.1 0.14 0.20.1 0.14 0.2 the systems under consideration, studying clusters of the E0 [eV/a0] E0 [eV/a0] form Na+Q which have N = 40 electrons and varied Figure 4. Lower three panels: Energy balance as function of 40+Q el charges state Q. It would not make sense to unfold all fieldstrengthE forNa excitedbyalaserwiththreedifferent 0 40 the laser variations for each system anew. Thus we take frequenciesasindicatedandpulselengthT =100fs.Upper pulse as a means of comparison an instantaneous dipole boost. panel:RatioofabsorbedenergybetweenRTAandTDLDAfor ϕ → exp(−ip z)ϕ applied to all s.p. wavefunctions in the resonant case (ω =2.7 eV) as function of field strength. α 0 α las the same manner [43,44]. The boost momentum p regu- 0 latesitsstrengthassociatedwiththeinitialexcitationen- The three lower panels of figure 4 show the effect of ergy E = Np2/(2m) which can be compared with the abs 0 laser field strength E for three frequencies, low ω = 0.8 absorbed energy in the laser case. The boost excitation 0 eV, resonant ω = 2.7 eV, and high ω = 6.1 eV which is touches all modes of a system at once with some bias on ontheonsetoftheone-photonregimeforalloccupieds.p. resonant excitation and it has only one parameter which states.Forthelowfrequencyandtheresonantcase,intrin- simplifies global comparisons between different systems. sic energy shrinks with increasing E . The reason is that We will thus use boost excitation in this section for vari- 0 higherorderphotonprocessesbecomeincreasinglyimpor- ation of cluster charge and in the next section for cluster tantwhich,inturn,enhancesthecontributionfromdirect size. (multi-photon) emission leaving less energy to dissipate. There is another subtle problem when varying clus- Forresonantexcitation,wehavetheadditionaleffectthat ter charge: the ionic geometry changes with charge state. the Mie plasmon frequency is increasing with increasing This can become particularly pronounced for deformed E becauseionizationisstrongerandenhancesthecharge clusters. Thus we consider variation of charge for a magic 0 state of the cluster [58]. Thus the resonance frequency is electron number, actually N = 40. This forces all sys- el running away from the laser frequency which also reduces tems for any charge state to near spherical geometry. We dissipation. For the high-frequency case ω = 6.1 eV, the go one step further and exclude any geometry effect by intrinsic energy increases with E . This is, again, an ef- using a soft jellium density for the ionic background [63, 0 fect of ionization which drives the IP up and thus moves 43]. The result for charge balance after boost excitation 8 M. Vincendon et al.: Dissipation and energy balance in cluster dynamics Echarge,loss 1EchEERTAariingnTDLDAtetrr,,,ppkoointt RTATDLDA RTATDLDA RTATDLDA ditional ionization 000...234 TDRLDTAA mpact par. b [a]0 1223350505 velocity [a/fs]0 1 1000 energies/Eabs000...468 adIP [eV] 0----.76541 i 1 R50TA 0.1 T1pulse [fs] 10 1 0.1 T T1DpuLlsDe [Afs] 10 0.2 -8 1 -9 0 eFleigcturroen 0s5.andRch 1eavrsgaue rslittaested 2 fQocrhaclr ug3setsersstaNtea+4Q0Q+.Q 0Twhehcichi 1oahrngehi csatavtse e2t QrNucetlu =3re4i0s energies/Eabs 0000....2468 EEcchhaEEargriingnetet,rrl,,o,ppksooinstt approximated by soft spherical jellium model with Wigner- 0 0.1 1 10 0.1 1 10 Seitzradiusr =3.65a andsurfaceparametersσ=1a [63, S 0 0 Tpulse [fs] Tpulse [fs] 43].Allclustersareexcitedinitiallybyaninstantaneousboost Figure 6. Lower panel: Energy balance as function of pulse with boost energy E = 2.7 eV. Left panel: Energy balance abs length T for Na excited through a bypassing ion. The for Na, plotting RTA and TDLDA side by side. Lower right: pulse 40 impact parameter b is tuned to provide E(TDLDA) ≈ 8.1 eV. Ionization potential (IP). Upper right: Ionization induced by abs Upper panel: Pulse length T and excitation strength can boost. pulse be translated to an impact parameter b (upper left) and ion velocity v (upper right). This is done here for an Ar ion with with initial energy of 2.7 eV is shown in figure 5. We see charge Q=8. againthetypicalpattern:aboutequalshareofintrinsicki- netic and intrinsic potential, about factor 2 more energy investedchargingtheclusterthanenergylostbyemission, the extend that magic systems gather more thermal en- and somewhat more intrinsic energy in RTA as compared ergy. This shell effect is going away for the higher excita- to TDLDA. The new feature here is that we see a strong tions. It is to be noted that the lower excitation strength trend of the intrinsic energy versus energy loss by emis- Eboost/Nel =0.027eVleadsinallfivesystemtoatemper- sion. Electron emission decreases with increasing charge aturearound1500KwhilethehigherexcitationEboost/Nel = state Q because the IP increases with Q which enhances 0.14eVisassociatedwithtemperatureabout3000K.This the cost of emission. In turn, less energy is exported by matcheswithobservationsfromshellstructureinNaclus- emissionandinvestedintochargingenergywhilemoreen- ters where the disappearance of shell effects is located at ergy is remaining in the clusters for dissipation into in- about 2000 K [64,65]. The lower excitation strength here trinsic excitation energy. The trend is clear, simple, and isbelowthiscriticalpointandthehigherexcitationabove. monotonous. It will apply equally well in other systems (with varying IP) and other observations. For example, laser frequency scans for different charge states will show 4.7 Excitation with by-passing ions the same pattern as function of frequency, but with an increasingoffsetofintrinsicenergywithincreasingcharge An alternative excitation mechanism is collision through state. aby-passingion.Wesimulatethatbyasingledipolepulse Eq. (1) with frequency ω = 0. The result is shown in the lower panel of figure 6. There are clearly two very differ- 4.6 Impact of cluster size entregimes.ForT ≤1fs,weencounterpracticallyan pulse instantaneous excitation by a Dirac δ pulse, practically a We have also compared RTA with TDLDA for clusters boost.Here,therelationsbetweenionizationandintrinsic of different size considering a series of closed-shell sys- energy are similar to laser excitation in the multi-photon temsNa+,Na+,Na+,aswellasopen-shellsystemsNa+, regime(frequenciesbelowIP),seefigure2.Muchdifferent 9 21 41 15 Na+. This sample allows to explore trends with system looks the regime of very slow ions (large T ). The in- 33 pulse size as well as the effect of shell closures. As for varia- trinsic excitation shrinks dramatically. Almost all energy tion of charge in the previous section, we avoid a tedious flows into ionization. The efficiency of ionization is here scan of frequencies and other laser parameters by using evenbetterthanfortheone-photonregime(highfrequen- simply a boost excitation. Two boost strengths are con- cies) in figure 2. Thus we can conclude that collision by sidered, E /N = 0.027 eV still in the linear regime veryslow,highlychargedionsisthesoftestwayofioniza- boost el and a higher E /N =0.14 eV. Note that these boost tion. boost el strength are scaled to system size. This should provide The field exerted by a highly charged ion passing by comparablethermodynamicconditions(e.g.temperatures). was simulated for simplicity by a single, zero frequency No clear trend with system size could be found. How- pulse. This can be translated into collision parameters. ever, at lower excitation energies, we see a shell effect to We have done that for an Ar ion with charge Q = 8 as M. Vincendon et al.: Dissipation and energy balance in cluster dynamics 9 example. The peak field strength E is related to the im- regime drives the energy balance to become more simi- 0 pact parameters b as E = 8Q/b2 and the passing time lar for the different frequencies (i.e. independent of fre- 0 is identified as the FWHM of field strength in the pulse quency). Field emission in the strong field regime comes which yields an estimate for the velocity as v =2b/T . along with producing less intrinsic energy. pulse Theresultofthisidentificationforfixedexcitationenergy The impact of system charge and system size was in- E = 8.1 eV is shown in the upper panel of figure 6. vestigatedforsimplicitywithaninstantaneousdipoleboost abs The sample of T produces a huge span of collisional excitation.Thechargestateofaclusterchangessystemat- pulse conditions. ically the relation between electron emission and intrinsic A word is in order about the “ideal case” of slow col- heating in an obvious manner: the higher the charge, the lisions. It may be not as ideal as it looks at first glance. harder it becomes to emit an electron and thus a larger Mind that the impact parameter b cannot be controlled fraction of the absorbed energy is kept in the cluster and in a collision. We encounter always a mix of impact pa- converted to intrinsic energy. Effects of cluster size are rameters thus leaving clusters in very different excitation weak. Shell structure still plays a role for small excita- stages.Afairinvestigationhastoproducethewholeexci- tions and becomes unimportant for higher energies. tationcrosssection,integratedoverallimpactparameters. Wehavealsoinvestigatedexcitationbyahighlycharged Onlythenwecanjudgefinallywhetherslowcollisionsare ion passing by the cluster. There is a dramatic change of a good means for cold ionization. energybalancewithimpactparameter.Closecollisionsex- ertashortpulsewhichleadstosignificantintrinsicenergy (more than 50%) if dissipation is accounted for. Distant collisions soak off electrons very gently and achieve high 5 Conclusion ionization while depositing very little intrinsic energy. The trends of the energy balance with pulse profile Inthispaper,wehaveinvestigatedfromatheoreticalper- and pulse parameters are all plausible. It is interesting spective the effect of dissipation on the energy balance to check these effects for other systems (bonding types, in metal clusters under the influence of strong electro- geometries). Research in this direction is underway. magneticpulses.Particularattentionwaspaidtothebranch- ing between thermalization (intrinsic energy) and ionization (energy export by electron emission). Basis of the Acknowledgments description was time-dependent density functional theory at the level of the Time-Dependent Local-Density Ap- This work was supported by the CNRS and the Midi- proximation (TDLDA) augmented by an averaged self- Pyrénées region (doctoral allocation number 13050239), interaction correction. For a pertinent description of dis- and the Institut Universitaire de France. It was granted sipation, we include also dynamical correlations using the access to the HPC resources of IDRIS under the alloca- Relaxation-TimeApproximation(RTA).TestcasesareNa tion 2014–095115 made by GENCI (Grand Equipement clusters, mainly Na complemented by a few cases with National de Calcul Intensif), of CalMiP (Calcul en Midi- 40 different size and charge state. Pyrénées) under the allocation P1238, and to the Re- We have investigated laser excitation looking at the gionalesRechenzentrumErlangen(RRZE)oftheFriedrich- dependence of energy balance on the main laser parame- Alexander university Erlangen/Nu¨rnberg. ters,frequency,intensity(fieldstrength),andpulselength. Frequencyisfoundtobethemostcriticalparameter.Dis- sipation is much more important for resonant excitation A Boundary correction to laser energy thanfornon-resonantcases.Ittakesawayenergyfromthe coherent dipole oscillations induced from the laser field Startingpointforthecomputationoftheenergyabsorbed and converts it to intrinsic energy. This, in turn, reduces fromanexternallaserfieldisthedefinitionintermsofthe the energy loss by induced emission and so enhances sig- current j which reads nificantly the energy absorption from the laser field. The (cid:90) t (cid:90) effect continues steadily and thus grows huge the longer E(j)(t)= dt(cid:48) d3rE (t(cid:48))·j(r,t(cid:48)) (15) the laser pulse. Another crucial mark is set by ionization abs 0 0 threshold. For frequencies below, the fraction of intrinsic This is turned, by virtue of the continuity equation ∂ ρ= excitation is generally larger than for frequencies above. t ∇·j, into an expression in terms of ∂ ρ, namely: Direct emission (one-photon processes) is fast and leaves t dissipation no chance. Thus dissipative effects are negligi- (cid:90) t (cid:90) bleforhighfrequenciesandRTAbehavesalmostidentical Ea(ρb)s(t)= dt(cid:48) d3rE0(t(cid:48))·r∂tρ(r,t(cid:48)) (16) with TDLDA. The other two laser parameters, intensity 0 and pulse length shows much less dramatic trends in the This form is easier to evaluate because ρ is readily avail- energy balance. Noteworthy are here two effects. First, ablewhilejneedstobecomputedseparately.Theproblem the dissipative enhancement of energy absorption in the is that the continuity equation holds only for Hermitian resonant case increases linearly with pulse length. Sec- propagationofthes.p.wavefunctions.Tobemorespecific, ond, with increasing intensity (field strength), the transi- we have to write tionfromthefrequencydominatedtothefielddominated ∂ ρ =∇·j (17) t herm 10 M. Vincendon et al.: Dissipation and energy balance in cluster dynamics 0.4 with the collective energy from Eq. (14). The two defini- from DCMF semi classical tionsarecomparedinfigure7.Thesemi-classicalE(ETF) intr,kin is a robust order-of-magnitude estimate which works par- V] 0.3 ticularly well in the early phases of excitation. e [n The case is more involved than it appears in figure 7. ki Actually, the mismatch starts at t = 0. But we shift the E 0.2 c valueofE(ETF) tomatchatt=0,preciselybecauseitisa si intr,kin n semiclassicalestimate,thusnotfullyvanishinginground ri nt 0.1 state. The punishment is then a mismatch at large times. i This may have to be discussed. Na , ω =2.7 eV, I =1.4*1010 W/cm2 40 las las 0 0 50 100 150 200 250 300 References time [fs] 1. E.K.U. Gross, W. Kohn, Adv. Quant. Chem. 21, 255 Figure 7. Comparison of intrinsic kinetic energy Eintr,kin (1990) fromfullyquantum-mechanicalDCMFdefinition(12)withthe 2. E.K.U. Gross, J.F. Dobson, M. Petersilka, Top. Curr. semi-classicalestimateE(ETF) fortheexampleofNa excited Chem. 181, 81 (1996) intr,kin 40 by a laser pulse with ω =2.7 eV, I =1.4×1010 W/cm2, 3. M.A.L. Marques, N.T. Maitra, F.M.S. Nogueira, E.K.U. las las and T =96 fs. Gross, A. Rubio, Fundamentals of Time-Dependent Den- pulse sity Functional Theory (Lect. Notes in Phys. vol 837,Springer-Verlag, Berlin, 2012) where∂ ρ isthepartstemmingfromHermitianprop- 4. J.W. Negele, Rev. Mod. Phys. 54, 913 (1982) t herm agation ∂ ψ = (cid:104)hˆ,ψ (cid:105). Absorbing boundaries introduce 5. 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