Dynamics of photo-activated Coulomb complexes
DDynamics of photo-activated Coulomb complexes
Christian Gnodtke , Ulf Saalmann , , Jan-Michael Rost , Max Planck Institute for the Physics of Complex SystemsN¨othnitzer Straße 38, 01187 Dresden, Germany Max Planck Advanced Study Group at the CFELLuruper Chaussee 149, 22761 Hamburg, GermanyE-mail: [email protected]
Abstract.
Intense light with frequencies above typical atomic or molecular ionizationpotentials as provided by free-electron lasers couples many photons into extendedtargets such as clusters and biomolecules. This implies, in contrast to traditionalmulti-photon ionization, multiple single-photon absorption. Thereby, many electronsare removed from their bound states and either released or trapped if the target chargehas become sufficiently large. We develop a simple model for this photo activation tostudy electron migration and interaction. It satisfies scaling relations which help torelate quite different scenarios. To understand this type of multi-electron dynamics onvery short time scales is vital for assessing the radiation damage inflicted by that typeof radiation and to pave the way for coherent diffraction imaging of single molecules.PACS numbers: 34.10.+x,42.50.Hz,33.80.Wz,41.60.Cr a r X i v : . [ phy s i c s . a t m - c l u s ] S e p ynamics of photo-activated Coulomb complexes
1. Introduction
The difference between multi-photon ionization at near-infrared frequencies ( ∼ >
50 eV) lies in the fact thatmulti-photon single -electron ionization is drastically suppressed in the latter case due tothe small dipole matrix elements for continuum-continuum transitions which typicallywould be required for an electron to absorb more than one photon. As a consequence,multi-photon ionization by XUV or X-ray radiation, as available from novel free-electronlaser sources [1–3], means single -photon ionization of many atoms in an extended targetsuch as a cluster or bio-molecule [4–6]. We call this process photo activation. As otherabsorption mechanisms, such as inverse bremsstrahlung, are less important for high laserfrequencies, the system is “driven” by this activation process only. In the following wewill develop a fairly general and simple model, which we call Coulomb complex (CC). Itfocuses on the multi-electron dynamics treating the ions created as a homogeneous andstatic background charge. The model shares similarities with the shell model for nucleior metal clusters [7] with the difference that the electrons are through photo activationfar above the ground-state which allows us to follow their dynamics classically.We will first discuss the initial multi-electron ground state in Sect. 2. The groundstate is the prerequisite for photo activation which we specify in Sect. 3 including the set-up of a universal time-dependent photo activation rate. In Sect. 4 we discuss a powerfulscaling property of the photo-activated CC which allows one to relate quite differentsituations in terms of photo activation time (length of the XUV pulse) T , excess energy ε ∗ of the electrons after photo absorption and size of the target designated by a radius R , to each other. In Sect. 5 we elaborate on a generic example and discuss relations ofthe present model to recently measured electron spectra of Xenon clusters [5] exposedto 90 eV photon energy pulses at FLASH [1]. In Appendix A we give an analyticalexpression for the electron spectrum for the case of sequential ionization.
2. The Coulomb complex before photo activation
We assume a system with N electrons which will be eventually photo-activated.They are bound by positive ions which are assumed to be fixed in space during therelevant electron dynamics. Moreover, we approximate the ions within the CC as ahomogeneously charged sphere, with total charge Q and radius R , producing a radialpotential for the electrons of the form V ( r ) = (cid:40) Q R ( r /R −
3) for r ≤ R − Qr for r > R. (1)The potential is within the CC ( r < R ) harmonic with the frequency Ω = ( Q/R ) / andfor r > R of pure Coulomb nature. The full Hamiltonian of the interacting electronicsystem is given by H = N (cid:88) j =1 (cid:32) p j V ( r j ) (cid:33) + 12 N (cid:88) j,k =1 (cid:48) r jk , (2) ynamics of photo-activated Coulomb complexes r jk = | r j − r k | and the prime excluding the term j = k . The total ground stateenergy E gs may be approximated by considering a homogeneous electron distribution.Then we can replace the sum over all electrons by an average potential energy v c andan average electron-electron interaction energy v ee , i. e., E gs = N (cid:88) j =1 V ( r j ) + 12 N (cid:88) j,k =1 (cid:48) r jk ≡ N v c + N v ee . (3)This assumption should be valid for sufficiently many electrons N , but applies also forquite small N as we will see below. Within this approximation the average energy fromthe background potential reads v c = 3 R (cid:90) R d r r V ( r ) = − QR . (4)The interaction energy v ee cannot be calculated this way, since we have to excludeexplicitly the self-interaction term. Instead we estimate the average energy of an electron ε = v c + v ee in the ground state. This energy can be approximated under the assumptionthat the electron density is identical to the ion density apart from a “hole” with volume(4 π/ R /N . The corresponding potential energy of such a hole is most easily calculatedfor a sphere (with a radius R/N / ) and gives ε = − N / R . (5)This single-particle binding energy depends exclusively on the density of the CC. Finallywe can calculate the average electron-electron repulsion energy v ee = ε − v c = − N / R + 65 QR , (6)which leads to the explicit approximation E gs = N v c + N v ee = − N R − N / R , (7)for the total energy of Eq. (3) of a neutral CC with N = Q . Note, that due to the non-linear dependence of the average electron-electron interaction upon the electron density, E gs does not scale quadratically with N . Table 1 presents the analytically estimatedenergies ε and E gs from Eqs. (5) and (7), respectively, in comparison to the numericallyobtained values for 9 different values of N which only for the smallest value N = 7 differby more than 3 %.The numerical energies were obtained by propagating N electrons with theHamiltonian (2) while reducing the particle velocities by a factor of 0 .
75 every 2.4 fsfor N ≤
123 and every 0.24 fs for
N >
123 to arrive at a minimal energy configurationof the system with corresponding optimized positions { ¯ r j } for the electrons. The bindingenergies of the N electrons are given by ε i = V (¯ r i ) + N (cid:88) k =1 (cid:48) r ik . (8)Figure 1 shows these binding energies for various CCs with the same radius R butdifferent electron numbers N . Interestingly, not only the average values ε agree quite ynamics of photo-activated Coulomb complexes Figure 1.
Individual electron energies ε i according to Eq. (8) as a function of theelectron’s distance r i = | r i | from the origin of the Coulomb complex for R = 26 a (Bohr radius) and various electron numbers N . The corresponding approximatedaverage energies ε from Eq. (5) are shown with lines. well with the analytical estimates (shown by lines in Fig. 1), there is also relatively smallvariation in the individual numerical electron energies (shown by symbols in Fig. 1). Thisapplies down to such small electron numbers as N = 7. In the figure one also clearlyrecognizes the formation of electron shells as known from so called Coulomb crystals [8].
3. Photo activation of the Coulomb complex
In the previous section we have characterized the electrons of the CC in energy andprepared them in space to absorb photons. Due to the small dipole matrix elements inthe continuum it is much more likely that N ph photons of XUV energy from an intensepulse are absorbed by N ph different bound electrons compared to the situation of multi-photon absorption by a few electrons. Hence, we have the N ph photons absorbed each byone bound electron. The single-photon absorption rate is proportional to the intensity N − ε [eV] (num.) 2.9 4.1 4.6 5.9 7.7 9.8 12.0 15.3 19.4(anl.) − E gs /N [eV] (num.) 6.1 14.4 19.7 39.4 81.9 163.5 296.7 605.1 1226.7(anl.) Table 1.
Comparison of numerical (num.) and analytical (anl.), cf. Eqs. (5) and (7),binding ( ε ) and ground state ( E gs ) energies of a Coulomb complex with N electronsrelaxed in a background potential of Eq. (1) with Q = N and radius R = 26 a . ynamics of photo-activated Coulomb complexes I ( t ) of the pulse, dn ( t ) dt = N at σ ω I ( t ) ω (9)with n ( t ) the number of electrons that have absorbed a photon up to time t , while N at is the number of atoms and σ ω the photo-absorption cross section at frequency ω .Assuming for convenience (in fact one can use any pulse shape) a Gaussian pulse withfull width at half maximum T we get n ( t ) = N at σ ω I ω (cid:90) t −∞ dt (cid:48) exp( − t (cid:48) /T ) )= N ph (cid:20) (cid:18) √ ln 2 tT (cid:19)(cid:21) , (10)with N ph = N at σ ω I T ( π/ / /ω the total number of photons absorbed from thepulse. The expression for N ph requires that each atom is singly ionized by the pulsewith the same cross section σ ω . This is of course an idealization, since realisticallythe cross section changes even if each atom is only singly ionized through ionizationinto a continuum which already contains electrons. Moreover, not the highest occupiedmolecular orbital may get predominantly ionized leading to more than one electron perphoton through auto-ionization, and finally, simply more than one photon could beabsorbed by each atom leading to multiple ionization. Nevertheless, Eq. (10) provides areasonably general yet realistic form for the number of absorbed photons.Within our model we account for the photo absorption by the activation of electrons,which means that only after a certain time do the electrons take part in the dynamicswith an initial momentum specified below. This implies that each electron j is heldfixed at its original position until its individual time t j of activation. Since the photoabsorption is a statistical process we treat the activation process statistically. Thereforewe first calculate N ph random numbers { x j } between 0 and 1. The activation time t j ofelectron j is then given by solving the equation n ( t ) = x j N ph which is implicit in t . Inorder to keep the computational expense as low as possible the number of electrons N of the CC is set equal to the number of activated electrons: N = N ph . For simplicity,we consider a CC which is neutral in the beginning, so that finally Q = N = N ph .We determine the initial momenta ¯ p j of the photo-activated electrons under thecondition that activation of a single electron should lead asymptotically (if this electronis removed from the CC) to the atomic excess energy ε ∗ . This implies for the initialkinetic energy for each activated electron in the complex¯ p j ε ∗ − V (¯ r j ) − N (cid:88) k =1 (cid:48) r jk . (11)For large N we can estimate the initial average kinetic energy per electron ε kin byaveraging Eq. (11) over all electrons and obtain with Eq. (5) ε kin = 1 N N (cid:88) j =1 ¯ p j = ε ∗ − ( v c + v ee ) = ε ∗ + 32 N / R . (12) ynamics of photo-activated Coulomb complexes p j / ε kin . Note that ω appears here only indirectly throughdetermining ε ∗ = ω − E ip with the true atomic ionization potential E ip .Formally we can express the activation by means of an activation function ‡ A τ ( X, Y, t ) = Θ( τ − t ) X + Θ( t − τ ) Y (13)in order to rewrite the original Hamiltonian (2) as H A = N (cid:88) j =1 (cid:32) A t j (¯ p j , p j , t ) V ( A t j (¯ r j , r j , t )) (cid:33) + 12 N (cid:88) j,k =1 (cid:48) (cid:12)(cid:12)(cid:12) A t j (¯ r j , r j , t ) − A t k (¯ r k , r k , t ) (cid:12)(cid:12)(cid:12) . (14)For this Hamiltonian the position of electron j in phase space is kept constant at (¯ r j , ¯ p j )until its activation at time t = t j . One should note that, however, the single-electronenergy ε j may change already for times t < t j due to the interaction with previouslyactivated electrons. This is quite important since in highly charged systems the effectsof the surrounding charges (binding, screening, etc.) easily exceed atomic properties byfar and may change on a femtosecond time scale. The calculation of all interactions(including those with the not yet activated electrons) is therefore crucial.Note that, although the Hamiltonian H A is time-dependent through the activationfunctions A , the corresponding total energy E ( t ) is conserved if one starts thepropagation at the initial point in phase space, i. e., r j ( t j ) = ¯ r j and p j ( t j ) = ¯ p j for j = 1 , . . . , N . It is lim δt → [ E ( t j + δt ) − E ( t j − δt )] = 0 for any j .
4. Scaling in the dynamics of Coulomb complexes
The CCs including their photo activation are fully determined by a number of externalparameters, namely (
Q, R, ε ∗ , T ). Before we discuss the dynamics of the CC with theinitial conditions of the electrons in phase space (¯ p j , ¯ r j ), we elaborate on a universalscaling of driven Coulomb explosion dynamics. It is based on the fact that all potentials(including the ionic background potential) originate ultimately from the homogeneousCoulomb forces. It will reveal the dynamics is the same for parameter sets emergingfrom an arbitrary reference set ( Q, R, ε ∗ , T ) by a scaling transformation which we willspecify.The system Hamiltonian Eq. (2) exhibits a global scaling with scaled variables ˜ x according to ( p , r ) = ( η / ˜ p , η − ˜ r ) ( E, t ) = ( η ˜ E, η − / ˜ t ) , (15)which applies to Coulomb systems. Note, that the external potential Eq. (1) belongs tothis class since it is given as the Coulomb potential of an extended charge distribution. ‡ The Heaviside step function is Θ( t ) = (cid:26) t <
01 for t > ynamics of photo-activated Coulomb complexes Q, R, ε ∗ , T ) → ( Q, η − R, ηε ∗ , η − / T ). E. g., with η = ( T /T ∗ ) / we can map( Q, R, ε ∗ , T ) → ( Q, η − R, ηε ∗ , T ∗ ) , (16)where T ∗ is any reasonable chosen reference pulse length. It should be emphasized, thatthe activation Hamiltonian H A in Eq. (14) has the same scaling properties, since the stepfunction Θ does not posess a time scale, i. e., Θ( t − τ ) = Θ (cid:16) η / (˜ t − ˜ τ ) (cid:17) = Θ(˜ t − ˜ τ ).We illustrate this scaling property in Fig. 2 where we have shown the convergedelectron spectra dP/dE of three CCs with parameters related by the scaling propertyEq. (16). In Fig. 2a each spectrum displays the same basic features, mainly a narrow peaknear E ≈ ηdP ( η − E ) /dE to reveal that theyare indeed identical within numerical accuracy. Figure 2.
Electron spectra dP/dE for reference parameters (
Q, R [a ] , ε ∗ [eV] , T [fs]) =(123 , , ,
10) (red, full), (123 , , , .
54) (green, dashed) and (123 , , , . η = 2 and η = 1 / dP/dE andpanel (b) the rescaled spectra ηdP ( η − E ) /dE that demonstrate the scaling invarianceof the underlying dynamics to within numerical accuracy.
5. Time-resolved electron spectrum
In order to understand the general feature of a double peaked final electron spectrumfor an activated CC, as shown in the previous section, it is necessary to appreciate thatCCs are open systems and as such are susceptible to electron loss. Indeed, it is theemission of energetic electrons that facilitates the relaxation of the remaining electronsfrom the highly excited activation state. This comes about as the remaining electronsin the CC lose the interaction energy with respect to emitted electrons, leading to adeeper effective binding potential. ynamics of photo-activated Coulomb complexes
Q, R [a ] , ε ∗ [eV] , T [fs]) = (1000 , , , ε ∗ . Figure 3.
Time evolution of an activated Coulomb complex with the followingparameters (
Q, R [a ] , ε ∗ [eV] , T [fs]) = (10 , , , t trap (red) and through thermal emissionfrom a trapped plasma after t trap (blue) and activation rate of electrons (green). (b)Time-resolved electron spectrum of Coulomb complex including as yet not activatedelectrons, (d) final electron spectrum at t = 100 fs, separated by activation time ofelectrons before t trap (direct, red) and after t trap (thermal, blue) as well as totalspectrum (green). Gray shaded area marks the analytical value for a sequentialspectrum (see Appendix). (c) Mean energy of the electrons in time from a Maxwell-Boltzmann fit to the kinetic energy spectrum of electrons (see (Fig. 4a)) within a sphereof radius 2 R (purple) and expressed through the energy scale ˜ E (see Fig. 4b) from theexponential fit to the spectrum of the thermally emitted electrons (blue). ynamics of photo-activated Coulomb complexes Directly emitted electrons
In the early stage of the activation, with a rate shown in green in Fig. 3a, activatedelectrons have a positive energy and they may leave the CC, a process which we call direct electron emission. With each emitted electron the total charge of the CC increasesthereby deepening the trapping potential for the subsequently activated electrons, whichis visible as the decrease in energy of the main peak in the time-resolved electronspectrum.The direct emission process continues until the main peak of mostly still dormantelectrons falls below the threshold of ε = 0 and newly activated electrons find themselvesbound to the CC, which they can no longer leave. We refer to this as trapping of electrons[9–11] by the now positively charged CC. In terms of the direct emission rate, which isshown in red in Fig. 3a, this corresponds to a sharp drop-off. We call the time when theemission rate reaches zero the trapping time t trap which here occurs near −
10 fs. Theprocess of direct emission of electrons is identical to the previously described multi-stepionization [12]. It results in a plateau-shaped spectrum shown in red in Fig. 3d. Thesequential process is amenable to an analytic description giving an exact expressionfor the height of the plateau of purely geometrical origin as Λ R , with Λ ≈ .
84 (cf.Appendix A for details). The number of directly emitted electrons may therefore bereadily approximated as N direct ≈ Λ ε ∗ R which here amounts to less than 5% of the totalnumber of electrons. Plasma formation and equilibration
With the onset of trapping at t = t trap the activated electrons within the CC undergoenergy-exchanging collisions that lead to relaxation. This has two consequences: onthe one hand the spectral peak undergoes a broadening as a thermalized plasma isformed. On the other hand particularly fast electrons in the plasma may leave thecomplex contributing to a second emission peak in Fig. 3a and resulting in a continueddecrease of the mean energy of the plasma. Since emitted electrons carry away energythe mean electron energy in the complex is reduced. This opens a gap in the spectrumthat individual electrons need to overcome in order to leave the cluster. However, theplasma temperature eventually does not suffice to produce fast enough electrons andthe emission decreases. From this time on the spectrum has essentially assumed its finalshape. The peak at negative energy describes the thermalized plasma and the peak atpositive energies contains the electrons emitted early during the activation. The gapbetween these two peaks must necessarily span a multiple of the plasma temperature toprevent further electron emission.The kinetic energy distribution of the plasma electrons follows closely a Maxwell-Boltzmann distribution, as can be seen in Fig. 4a. Since the photo activation continues atthe time for which the distributions are determined, we consider only electrons activatedbefore t − t Ω , where t Ω = 2 π ( R /Q ) / is the oscillation period within the Jelliumpotential which provides an intrinsic time-scale of the electron dynamics. Obviously, ynamics of photo-activated Coulomb complexes Figure 4. (a) Kinetic energy spectra of plasma electrons dP ( ε kin ) /dε kin and fittedMaxwell-Boltzmann distributions ∼ (cid:112) E/πk B T exp( − E/k B T ) at different times inthe evolution of the Coulomb complex. (b) Spectrum dP thermal /dE of thermallyemitted electrons at various times and exponential fit ∼ exp( − E/ ˜ E ). this time is sufficient for the equilibration of newly activated electrons.The plasma temperature extracted from the Maxwell-Boltzmann fit exhibits asharp rise beginning at time t trap towards a maximum close to the time of maximumactivation rate (purple, Fig. 3c). The rise reflects the increasing depth of the positivebackground potential allowing to trap faster electrons in a hotter plasma. Eventually,the temperature decreases due to evaporative cooling effects and levels off asymptoticallyslightly below 25 eV. Note, that this temperature remains consistently below thetemperature inferred from the initial kinetic energy of the electrons, Eq. (12), whichis due to a broadened spatial distribution of the plasma with respect to the groundstate. Emission from the plasma
As can be seen in Fig. 4b and as has recently been similarly described elsewhere [13, 14],the spectrum dP thermal /dE of the electrons emitted by the thermalized plasma forms anexponential distribution characterized by the energy scale ˜ E , dP thermal /dE = N thermal ˜ E − exp( − E/ ˜ E ) , (17)where N thermal is the number of electrons thermally emitted from the plasma. Althoughoriginating from the tail of the equilibrated plasma inside the cluster the “temperature”˜ E of the thermally emitted electrons (blue, Fig. 3c) is considerably lower than k B T ofthe trapped plasma (purple, Fig. 3c). One should therefore not take this experimentallydirectly accessible energy ˜ E as the temperature of the trapped electron plasma.The exponential distribution Eq. (17) has been measured recently [5] in anexperiment, where ionization of clusters with 90 eV photons from intense pulses supplied ynamics of photo-activated Coulomb complexes ε ∗ < ∼
0, as measured recentlyat SCSS [2] for neon clusters [15]. Whereas direct emission is not possible, the energyof the excited electrons is redistributed through collisions with some electrons beingemitted.
6. Summary
We have discussed a simple and fairly general scenario for the electron dynamics in densematter exposed to short and intense laser pulses of high intensity. It applies to variousfrequencies ω of the light up to X-rays. The lower limit for ω is set by the conditionthat single-photon ionization of many atoms dominates which we call photo activation.We have provided a simple formalization of the photo activation in time which relatesthe successive photo-ionization events to the time-dependent shape of the laser pulseand the total number N of ionized electrons generated. The process leads inevitablyto a Coulomb complex, a many particle system of ions and electrons whose Coulombinteraction dominates all other forces. From this follows a simple scaling property ofCoulomb systems which relates seemingly disparate experimental szenarios with respectto the size of the system (characterized by the radius R of atomic positions), the timescale of photo activation (determined by the pulse width T ) and the electron energy N ( ω − E ip ) made available through the activation. The latter depends through N onthe total energy contained in the laser pulse.Furthermore, we have discussed the time-evolution of the electrons during and afterphoto activation including the formation of a plasma and its equilibration in detail.Photo activation in Coulomb complexes is not limited to the examples of intense XUVand X-ray pulses illuminating clusters of almost solid density as discussed here. Thevery same phenomenon occurs, e.g., in the activation of ultracold plasmas [16] fromatom clouds held in magneto-optical traps at appropriately adopted time and energyscales. Acknowledgments
The authors thank Ionut¸ Georgescu for helpful discussions. US and JMR acknowledgesupport by the KITP at UC Santa Barbara during the program
X-ray Frontiers . ynamics of photo-activated Coulomb complexes Appendix A. Electron spectrum for sequential ionization
We derive an analytical expression for the electron spectrum in the case of high excessenergy ε ∗ (cid:29) Q/R and long pulses T → ∞ . In these limits all electrons leave the CCdirectly upon activation, without the possibility of exchanging energy with other activeelectrons, i. e., the emission occurs sequentially. Furthermore we assume a homogeneouselectron distribution within r < R of the non-activated electrons, which gives rise to theprobability distribution in the radial coordinate for the electron to be activated next as dPdr = 3 r R . (A.1)Released electrons leave a charged CC behind. This remaining charge, denoted by ˜ Q , iswithin our approximation also homogeneously distributed. It modifies the asymptotic(or measured) energy E of the next to be activated electron. Therefore the energydepends on the radial coordinate r , E ˜ Q ( r ) = ε ∗ −
32 ˜ QR + ˜ Q R r , (A.2)with the modification given by the potential in Eq. (1). The dependence on theinstantaneous charge ˜ Q is emphasized by the subscript. This expression allows one (a)(b) Figure A1. (a) Probability distribution in energy dP ˜ Q /dE from Eq. (A.3) of nextactivated electron for CC with parameters ( Q, R [a ] , ε ∗ [au]) = (500 , ,
40) and charge˜ Q = 0 , , . . . ,
500 on the CC. (b) Analytical electron spectrum dP/dE from Eq. (A.4)from integration of probability distributions. ynamics of photo-activated Coulomb complexes dP ˜ Q dE = dPdr (cid:30) dE ˜ Q dr = 3˜ Q r = 3˜ Q (cid:32)(cid:32) E − ε ∗ + 32 ˜ QR (cid:33) R ˜ Q (cid:33) / , (A.3)where the lower line uses the inverse function of Eq. (A.2). As the radial coordinate isrestricted to 0 ≤ r ≤ R , so is the energy in Eq. (A.3), namely − (3 /
2) ˜
Q/R ≤ E − ε ∗ ≤− ˜ Q/R . The reduction of the energy E − ε ∗ is bounded by the potential energy at thecentre and the surface. The probability distribution of Eq. (A.3) is shown for the specificcase of ε ∗ = 40 au, R = 25 a und various instantaneous charges ˜ Q = 0 , , . . . ,
500 inFig. A1a. The larger the charge on the CC is, the wider the distribution in energybecomes and the further it moves to lower energy.As can be seen in Fig. A1 only certain values of ˜ Q contribute to the spectrum ata given energy E . These values are given by ˜ Q ( E ) ≤ ˜ Q ≤ ˜ Q ( E ) with ˜ Q ( E ) =(2 / R ( ε ∗ − E ) and ˜ Q ( E ) = min ( R ( ε ∗ − E ) , Q ). Remember that Q is the total chargereached after the activation and removal of all electrons. Thus one obtains by integration dPdE = (cid:90) ˜ Q ( E )˜ Q ( E ) d ˜ Q dP ˜ Q dE = (cid:40) Λ R for − Q/R ≤ E − ε ∗ ≤ ,λ Q,R ( E ) for − (3 / Q/R ≤ E − ε ∗ ≤ − Q/R. (A.4)Interestingly the electron spectrum for higher energies is independent of energy E with a system-independent constant given by Λ = 3 (cid:104) √ (cid:16) √ (cid:17) −√ − (cid:105) ≈ . E it falls off monotonically according to λ Q,R ( E ) =3 R (cid:104) √ (cid:16) √ − χ (cid:17) −√ χ ) − √ − χ (cid:105) with χ = ( ε ∗ − E ) R/Q .The entire analytical spectrum is shown in Fig. A1b for the specific parameters Q = 500, R = 25 a und ε ∗ = 40 au. In particular the plateau in the region from Figure A2.
Comparison of numerical (symbols) and analytical (lines and shaded)electron spectra for various parameters. Analytical spectra calculated according toEq. (A.4). A pulse length of T = 10 fs, sufficiently long for sequential ionization, wasused for all three spectra. ynamics of photo-activated Coulomb complexes ε ∗ − Q/R to ε ∗ is a characteristic feature of the sequential ionization as seen in anexperiment [12] and numerical simulations [17]. While the height of the spectrum issolely determined by the radius R , the width of the full spectrum and of the plateau-region are given by 3 Q/ R and Q/R respectively. Note that this implies that in thecase of ε ∗ < Q/ R trapping of electrons will occur and the full sequential spectrumis not realized. However, for those electrons emitted before the onset of trapping thesequential spectrum as derived above is valid.We compare the analytical spectrum of Eq. (A.4) with numerical results for threesets of parameters in Fig. A2, in each case with a pulse length of T = 10 fs which provessufficiently long for the CC to exhibit sequential ionization behaviour. Indeed, we find anoverall excellent agreement between the numerical and the analytical result, in particularas pertains to the appearance of an extended plateau region, well described in width,height and position by Eq. (A.4). Nevertheless, some minor discrepancies owing to thecontinuous approach employed in the analytical calculation can be observed. A narrowpeak at ε ∗ in the numerical spectrum arises from the first activated electron with exactlythe excess energy ε ∗ . Furthermore, in the energy domain − Q/ R ≤ E − ε ∗ ≤ − Q/R ,the shell-like structure of the initial configuration of the CC, as shown in Fig. 1, leadsto a step-like spectrum, as opposed to the smooth increase in the analytical case.
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