Transient ionization potential depression in nonthermal dense plasmas at high x-ray intensity
Rui Jin, Malik Muhammad Abdullah, Zoltan Jurek, Robin Santra, Sang-Kil Son
aa r X i v : . [ phy s i c s . p l a s m - ph ] J a n Transient ionization potential depression in nonthermal dense plasmas at highx-ray intensity
Rui Jin,
1, 2, ∗ Malik Muhammad Abdullah, Zoltan Jurek,
1, 4
Robin Santra,
1, 4, 5 and Sang-Kil Son
1, 4, † Center for Free-Electron Laser Science, DESY,Notkestrasse 85, 22607 Hamburg, Germany Department of Physics and Astronomy, Shanghai Jiao Tong University, 200240 Shanghai, China Deutsches Elektronen-Synchrotron DESY,Notkestrasse 85, 22607 Hamburg, Germany The Hamburg Centre for Ultrafast Imaging,Luruper Chaussee 149, 22761 Hamburg, Germany Department of Physics, Universit¨at Hamburg,Jungiusstrasse 9, 20355 Hamburg, Germany (Dated: January 26, 2021) bstract The advent of x-ray free-electron lasers (XFELs), which provide intense ultrashort x-ray pulses, has brought a newway of creating and analyzing hot and warm dense plasmas in the laboratory. Because of the ultrashort pulse duration,the XFEL-produced plasma will be out of equilibrium at the beginning and even the electronic subsystem may notreach thermal equilibrium while interacting with a femtosecond time-scale pulse. In the dense plasma, the ionizationpotential depression (IPD) induced by the plasma environment plays a crucial role for understanding and modelingmicroscopic dynamical processes. However, all theoretical approaches for IPD have been based on local thermalequilibrium (LTE) and it has been controversial to use LTE IPD models for the nonthermal situation. In this work,we propose a non-LTE (NLTE) approach to calculate the IPD e ff ect by combining a quantum-mechanical electronic-structure calculation and a classical molecular dynamics simulation. This hybrid approach enables us to investigatethe time evolution of ionization potentials and IPDs during and after the interaction with XFEL pulses, without thelimitation of the LTE assumption. In our NLTE approach, the transient IPD values are presented as distributionsevolving with time, which cannot be captured by conventional LTE-based models. The time-integrated ionizationpotential values are in good agreement with benchmark experimental data on solid-density aluminum plasma and othertheoretical predictions based on LTE. The present work is promising to provide critical insights into nonequilibriumdynamics of dense plasma formation and thermalization induced by XFEL pulses. I. INTRODUCTION
High energy density matter exists extensively in the Universe, from the hot dense plasmas such as thosein supernovae and stellar interiors [1, 2] to warm dense plasmas like those in planetary interiors [3–6].Understanding matter at extreme conditions is of crucial importance for not only astrophysical observation,but also emerging experiments in the laboratory such as inertial confinement fusion (ICF) [7, 8] and x-rayfree-electron laser (XFEL) [9, 10] facilities. The typical energy density is above 10 J / m [11]. In thelaboratory, high energy density is achieved by imposing energy sources such as high intensity lasers or highpressure onto fluids or solids within ultrashort time. In particular, highly brilliant, spatially coherent XFELpulses can uniformly heat bulk matter and transform it to a warm dense plasma on a femtosecond time scale,which makes XFELs an ideal tool for creating and probing solid-density plasmas [12].When a solid-density material is exposed to an intense x-ray pulse generated by XFELs, many electronsare ionized via x-ray photoionization and the system evolves through microscopic dynamical processes ∗ [email protected] † [email protected] ff ect of the plasma electrons and the dense environment, the atomic energy levelsare shifted and the ionization potential is lowered in comparison with that for the isolated atom case. Thisphenomenon is called ionization potential depression (IPD) and it plays a crucial role for understandingand modeling atomic processes in dense plasmas. To describe this IPD e ff ect, two distinct models havebeen available, Ecker–Kr¨oll (EK) [23] and Stewart–Pyatt (SP) [24], both of which are based on thermalequilibrium. Recent experiments using an XFEL [12, 25] and a high-power optical laser [26, 27] havetriggered a controversial debate on the validity of these models, which has been followed by extensivestudies on the IPD measurements [28–33] and the theoretical treatments of the IPD e ff ect [34–43], including ab initio electronic structure calculations [44–49]. It is worthwhile to note that all of the employed methodsare based on the LTE condition for the electronic subsystem. In other words, all above-mentioned methodsincorporate the assumption of thermal equilibrium at a given electron temperature. Therefore, the IPDcalculated may not be suitable for describing femtosecond dynamics of solid-density plasma formation byXFELs, where an equilibrium electron temperature often cannot be defined [50]. In particular, the waythat NLTE dynamics simulations are fed by LTE IPD input such as EK and SP might be questionablefor calculation of XFEL-created dense plasmas. Furthermore, nonthermal femtosecond phase transitionsinduced by an intense hard-x-ray pulse have been reported [51, 52], thus necessitating an IPD treatment thatis unrestricted, for both electrons and ions, by any thermal equilibrium condition.In this work, we propose an NLTE approach to calculate the IPD e ff ect, based on the real-space chargedistribution obtained directly from real-space molecular-dynamics (MD) simulation trajectories. In order toincorporate the NLTE plasma status, we employ a Monte Carlo-molecular dynamics (MC-MD) simulation3ool, XMDYN [53, 54]. The IPD values are calculated by subtracting the ionization potential for an isolatedatom and that for an atom embedded in the plasma with the same electron configuration. For both cases,the ionization potential is calculated quantum-mechanically, by using an atomic toolkit, XATOM [54, 55].To calculate the atomic electronic structure including the plasma e ff ect, we have developed a dedicatedtool, XPOT, interfacing between XMDYN and XATOM. This paper is organized as follows. In Sec. II, weprovide the theoretical description of how to construct the classical environmental micro field from MDsimulations and how to evaluate IPD values. In Sec. III, we present an ensemble of calculated IPD valuesduring an NLTE simulation of Al solid-density plasma created by an intense x-ray pulse. This result iscompared with experimental measurement and other theoretical results. We conclude with a short summaryand an outlook in Sec. IV. II. METHODOLOGYA. XMDYN: Monte Carlo-molecular dynamics scheme
We briefly review the theoretical background of XMDYN [53, 54], which is used in this study to simulatethe creation of a nonequilibrium dense plasma. XMDYN is a versatile code based on the classical moleculardynamics (MD) and Monte Carlo (MC) approaches. In this method, ionized electrons and atomic ionsare treated as classical particles roaming in 3D real space, and their dynamics are described by the MDtechnique. The dynamics of the electronic configuration of each atomic ion are computed in a stochasticframework using an MC algorithm. The electronic structure (orbitals and orbital energies) of each atomicion is solved quantum-mechanically on the fly (i.e., when needed in a given numerical time step) by usingthe atomic toolkit XATOM [54, 55], based on the Hartree–Fock–Slater (HFS) method. Cross sectionsand rates of photoionization, Auger decay, and fluorescence are calculated based on the orbitals and orbitalenergies within XATOM. Further, the treatment of collisional ionization and recombination in XMDYN alsorelies on computed orbital energies. The complex electron thermalization and energy transfer processesare realized in the real-time MD evolution by the elementary classical many-body collisions (includingelectron–electron scattering and electron–ion scattering) as well as collisional ionization and recombination.Note that a trajectory tracing algorithm [54] is employed to treat the recombination process. Essentially, aclassical electron is recombined with an atomic ion if it is found to orbit around the same trapping ion for n rec full periods, where the cycle-count threshold for recombination n rec is a small integer.This hybrid quantum–classical treatment is capable to treat some quantum e ff ects, but admittedly notall, and there are no significant limitations in its ability to capture many-body interactions, which is hard to4chieve within, for example, time-dependent density-functional theory (TDDFT) [56]. For the solid-densityAl experiment considered here, the created plasma is not deep in the regime of quantum degeneracy, oncethe system has been heated. Note that the standard functionals used in DFT and TDDFT do not treatexchange e ff ects, which are connected to quantum degeneracy, in a rigorous manner. At the moment, onlya mixed quantum-classical framework, as adopted within XMDYN, is available to tackle nonequilibriumdynamics in the regime of fairly strong coupling.This XMDYN–XATOM approach has been applied to explain many XFEL experiments, for example,explosion and fragmentation dynamics of C [53, 57] and nanoplasma formation and disintegrationdynamics of rare gas clusters [58–60]. It has been extended to simulate a bulk system within the supercellapproach in combination with periodic boundary conditions [61–63], and it has been an indispensable toolfor the start-to-end simulation of single-particle imaging at the European XFEL [64, 65]. However, beforethe present work, there was the caveat that the quantum treatment of the atomic electronic structure did nottake into account the plasma environment. The most obvious shortcoming of this was the lack of ionizationpotential depression emerging in the ionization dynamics of dense plasmas. In the following, we describehow we have addressed this challenge. B. Electronic structure calculation
The electronic structure of an isolated atom is obtained by solving the e ff ective single-electronSchr¨odinger equation (atomic units are used unless specified otherwise), " − ∇ + V iso A ( r ) ψ ( r ) = εψ ( r ) . (1)Here, V iso A ( r ) is the mean field potential for an isolated atom, V iso A ( r ) = − Zr + Z d r ′ ρ ( r ′ ) | r − r ′ | + V x (cid:2) ρ ( r ) (cid:3) , (2)where Z is the nuclear charge, V x (cid:2) ρ ( r ) (cid:3) = − (3 / (cid:2) (3 /π ) ρ ( r ) (cid:3) / is the Slater exchange potential [66],and ρ ( r ) = P N elec , A p ψ † p ( r ) ψ p ( r ) is the electron density. Here, N elec , A is the number of electrons assignedto atom A and p indicates the one-particle state index, i.e., p = ( n , l , m l , m s ), where n , l , m l , and m s arethe principal quantum number, the orbital angular momentum quantum number, the associated projectionquantum number, and the spin quantum number, respectively. In addition, the Latter tail correction [67]is employed to obtain the proper long-range potential. Assuming that the electron density is spherically5ymmetric, V iso A ( r ) is also spherically symmetric and Eq. (1) can be reduced to a radial Schr¨odinger equation.This equation is solved using a numerical grid technique with a su ffi ciently large radius [44].An atom embedded in a plasma environment experiences an additional potential V env A ( r ) from theenvironment. The electrons assigned to each atom are treated quantum mechanically via direct Coulombinteraction and exchange interaction with electron density, whereas the electrons and atomic ions in theenvironment are considered as classical particles. For each atom, we employ a microcanonical ensembleat every realization at every time step, implying a fixed electron configuration. Then, the potential for anatomic electron in a plasma calculation, V pla A , is given by V pla A ( r ) = V atom A ( r ) + V env A ( r ) for r < r c , V for r ≥ r c . (3)Here we introduce a flat potential tail V for r ≥ r c , using the mu ffi n-tin approximation [68]. The energylevels below V are considered as bound states, whereas above V are continuum states. Then, the excitationof an electron from a bound state to the continuum threshold located at V defines the inner ionization [44],and we use the associated excitation energy for defining the ionization potential. The determination of V and r c will be explained later. V atom A ( r ) has the same form as shown in Eq. (2), except ρ ( r ) is determined self-consistently in the presence of the additional environmental potential. Note that no Fermi-Dirac occupationfactor is involved in the density calculation, because we use a fixed electron configuration and the conceptof temperature is not defined in the NLTE condition. The Latter correction is not applied because of themu ffi n-tin approximation.In a given MC-MD simulation step, the environmental potential for atom A is simply given by the sumof the static electric potentials of all charged particles in a supercell, excluding atom A itself. In the MDsimulations, the electric potential is approximated with a soft-core potential [54] to avoid the Coulombicsingularity. Thus, V env A ( r ) is evaluated as V env A ( r ) = − X i , A q i p | r − r i | + a , (4)where q i is the current charge state of the particle: q i = − q i = Z i − N elec , i for atomicions. Here, a is the soft-core potential radius, which is chosen according to the numerical criteria suggestedin Ref. [54]. We examined convergence in XMDYN varying a as well as the time step ∆ t and the cycle-count threshold for recombination n rec , and found that a = ∆ t = n rec = a = . ffi ciently, we sample the MC-MD simulationresults using a time step of 0.25 fs, which is larger than the ∆ t used for propagation in XMDYN, becausethe environmental potential, especially after applying the averaging schemes to be introduced below, variessmoothly on the femtosecond time scale. We use the same time step for free electrons and ions for simplicity.In order to perform an atomic calculation, we need to assume a spherically-symmetric potential. Weimpose it for the environmental potential by performing spherical averaging of V env A ( r ), V env A ( r ) = R d Ω A V env A ( r ) R d Ω A = − π X i , A Z d Ω A q i p | r − r i | + a = − X i , A q i p ( r + r i ) + a − p ( r − r i ) + a rr i , (5)where Ω A is the solid angle around atom A . Note that the minimal image convention [62] is appliedto evaluate the potential for crystalline structures. The minimal image convention basically chooses atranslational equivalent image of the original supercell, so that the atom of interest sits exactly at the center.Next, we introduce another averaging scheme. The evaluation of Eq. (5) is relatively simple, but V env A ( r )has to be calculated and stored for every single A at every MC-MD step. In our calculation, one supercelltypically contains a few hundred atoms. Each atomic case represents one realization of a stochastic process,and we can make an ensemble average with individual atomic realizations. One extreme is to average allindividual atomic potentials in a supercell (global averaging scheme), V envglobal ( r ) = N A all X A V env A ( r ) . (6)Alternatively, one can group the atomic potentials according to individual charge states and average them,assuming that the short-range shape of the environmental potential is dominated by the ionic charge state(charge-selective averaging scheme), V env q ( r ) = N q q A = q X A V env A ( r ) , (7)where q A = Z A − N elec , A and N q is the number of atoms corresponding to q A = q in a given time step. Thischarge-selective averaging scheme preserves locality of the environmental potential. A comparison amongthe di ff erent averaging schemes with respect to calculated ionization potentials will be made at the end of7ec. II C.The global potential experienced by a quantum electron is obtained by connecting all atomic potentials V pla A such that the resulting potential is continuous in the interstitial regions via V . Moreover, we assumethat the atomic potentials are spherically symmetric. Thus, the connecting potential V must be the samefor all atoms. This approximation has been widely used in solid-state calculations and its validity hasbeen tested by precise band structure studies [69–72]. Here, we use a similar procedure for determining amu ffi n-tin potential as described in Ref. [46]. Strictly speaking, determination of V requires informationon all atomic potentials within a supercell. The total atomic potential for each atomic site, however, is to bedetermined self-consistently including V , which means that a self-consistent-field (SCF) calculation for thewhole supercell is necessary. To avoid such a complication, exclusively for determining V we approximatethe total atomic potential as V approx A ( r ) = − ( q A + / r + V env A ( r ), where ( q A +
1) accounts for the Lattercorrection [67]. We match this potential with the B th neighboring atom at a distance r AB . The touchingpotential of A and B is given by V AB = V approx A ( r T ) = V approx B ( r AB − r T ), where r T is the touching sphere radiuswith respect to atom A . The lowest value of this touching potential V AB for all atom combinations is chosenas the global mu ffi n-tin potential V ( = min { V AB } ). -60-40-20 0 0 1 2 3 4 5 6 7 8V r c Al P o t en t i a l ( a . u . ) Radius (a.u.)V atom (r)V env (r)V approx (r)V pla (r)V iso (r) -8-4 0 4 8 0 1 2 3V r c FIG. 1. Potentials for Al + in the charge-selective averaging scheme. V pla is the total potential used for the electronic-structure calculation on an atom in a plasma environment. It is the sum of the atomic potential V atom and theenvironmental potential V env . The approximation potential V approx is very close to V pla in the vicinity of r = r c .The mu ffi n-tin energy tail V is fixed during the SCF iterations. V env A ( r ) is constructed for all individual atomic sites and V is determinedwith the above procedure, before entering individual atomic SCF calculations. The final expression for theplasma potential (with the charge-selective averaging scheme) is V pla A ( r ) = V atom A ( r ) + V env q = q A ( r ) for r < r c , V for r ≥ r c . (8)Here r c is determined by matching the potential V atom A ( r ) + V env q = q A ( r ) with the fixed mu ffi n-tin energy tail V in each SCF iteration. Figure 1 shows all potentials involved in the plasma calculation of Al + (electronconfiguration: 1 s s p ): V pla A ( r ), V atom A ( r ), V env q = q A ( r ), and V approx A ( r ). The approximation potential V approx A approaches V atom A ( r ) + V env q = q A ( r ) as r approaches r c , indicating the approximation is valid. For comparison, V iso A ( r ) in the isolated-atom calculation is also shown with the dashed line. The subscripts are dropped inFig. 1 as no confusion will occur for a single atomic species. As shown in Fig. 1, the curve of V pla ( r ) showsa kink at r c because of the way the plasma potential is constructed. In the numerical grid employed here,the maximum radius is much larger than r c (typically r max =
50 a.u.) and smoothing around this kink positionprovides almost no change in calculated orbital energies and orbitals.Note that V pla A ( r ) is only used for the bound electrons in atoms and atomic ions to account for the plasmaenvironment e ff ect. The classical particles (free electrons and atomic ions) move on the potential surfacegiven by the sum of Coulomb interactions with all other classical particles in the system. C. Ionization potential depression
The ionization potential depression (IPD) for an atom is defined by the di ff erence between the ionizationpotential of the atom in isolation and the ionization potential of the atom in a plasma environment. For the j th orbital in the electronic configuration I , the IPD is calculated as ∆ E I , j = IP iso I , j − IP pla I , j , (9)where the ionization potentials (IPs) for an isolated atom and for an atom in a plasma are defined byIP iso I , j = − ε iso I , j , (10)IP pla I , j = V − ε pla I , j , (11)9hich are obtained from the isolated-atom calculation using Eq. (2) and from the plasma calculation usingEq. (8), respectively. For the plasma case, the continuum states start at the mu ffi n-tin flat potential V , sothe ionization potential is defined by the di ff erence between V and the orbital energy calculated with theplasma environment.In the following, we apply this procedure to XFEL-irradiated Al solid [12, 25, 73] and investigate IPand IPD values for given Al atomic charges. It is known that, for Al solid at room temperature, 3 s and3 p electrons are not bound to the atom, but are delocalized and form conduction bands. In our previousaverage-atom-based study [44], states were considered bound if their energy was below the mu ffi n-tin flatpotential. For example, for Al solid at T = M -shell states (3 s and 3 p orbitals) are not bound tothe atom, whereas at T =
80 eV they become bound in the average-atom calculation. However, the recentall-electron quantum-mechanical crystalline calculation of Al plasmas [46] indicates that the states above2 p are not fully localized even at high temperature ( T ≤
100 eV), leading to finite valence energy bands.Following Ref. [46], for the assignment of the atomic charge we count only electrons in 1 s , 2 s , and 2 p orbitals. Thus, for the electronic configuration I A = ( n A s , n A s , n A p ), regardless of M -shell occupation number,the atomic charge is given by Q A = Z A − s , s , p X j n Aj , (12)where n Aj is the occupation number in the j th orbital of atom A . This assignment is consistent with thespectroscopic notation used in Refs. [12, 73], where the occupation numbers of only the K - and L -shellsare counted and the energy resolution in the experiment was insu ffi cient to resolve the M -shell occupationnumber.Note that Q A in Eq. (12) used to assign an atomic charge for the IP and IPD calculation is di ff erentfrom q A in Eq. (7) used for charge-selective averaging. This originates from a subtlety of how 3 s and 3 p electrons are treated. In our quantum-mechanical calculations, the Al electronic configuration contains 1 s ,2 s , 2 p , 3 s , and 3 p for both isolated-atom and plasma cases. More precisely, V atom A in Eq. (8) contains theelectron density including 3 s and 3 p electrons. In this way, 3 s and 3 p states are explicitly treated as ifthey are bound to the atom. However, this is an ad hoc treatment because they may not be bound statesas discussed above. In the current implementation, 3 s and 3 p states are included in our atomic quantum-mechanical treatment, but they are excluded when defining atomic charge. Thus, quantum e ff ects in freeelectrons are partially incorporated, but at the same time they could be overestimated because 3 s and 3 p electrons cannot be delocalized in the current approach, until they are ionized and become classical particles.This limitation could be overcome by introducing a quantum charge transfer treatment among individual10toms [59]. During the interaction with an intense x-ray pulse, many electrons are ionized, turning 3 s and3 p electrons into classical particles within XMDYN, so that the di ff erence between Q A and q A vanisheswhen no electron remains in the M shells. Alternatively, electrons in 3 s and 3 p states could be treated asclassical particles from the beginning and only the K and L shells could be treated quantum mechanically.In this extreme case, quantum e ff ects in free electrons would be completely neglected. To overcome thisdrawback, one could apply the electron force field [74, 75] to free electrons.We perform a calculation of IP and IPD at every single MC-MD step, which provides the time evolutionof the IP and IPD values. We will examine not only time-resolved IP but also time-integrated IP values tobe compared with time-integrated measurements. In Fig. 2, time-integrated IP values for Al + based on theaveraging schemes of Eqs. (6) and (7) are compared with that obtained with no averaging. Computationaldetails regarding the time-integrated IP and its distribution will be discussed later in Sec. III. As shown inFig. 2 the time-integrated IP distributions (manually convolved with a Gaussian of 4.7 eV full width at halfmaximum, FWHM) for Al + ions in the global and charge-selective averaging schemes look similar. The no-averaging scheme leads to a wider and blue-shifted distribution, indicating that environmental fluctuationsmatter. However, the mean values of the IP distributions calculated with the three di ff erent schemes arevery similar to each other within 1 eV. The no-averaging scheme is computationally less attractive than the m ean A bundan c e ( a r b . un i t s ) Ionization potential (eV)global averagingcharge-selective averagingno averaging
FIG. 2. Time-integrated ionization potential distributions for Al + in solid-density Al plasma using di ff erent averagingstrategies. C ha r ge pulse shape − Q 0 0.2 0.4 0.6 0.8 0 20 40 60 80 100 120 140 160(b) P opu l a t i on Time (fs)+3+4+5 +6+7+8 +9
FIG. 3. XMDYN simulation for a solid aluminum target irradiated by a short laser pulse with a photon energy of1850 eV and a fluence of 1.0 × ph / µ m . (a) Average charge state ( ¯ Q ) and Gaussian pulse profile. (b) Timeevolution of charge populations. other two schemes, because snapshots of classically treated electron density without averaging often causedi ffi culties in SCF convergence and every atomic ion in a supercell has to be treated separately. On the otherhand, the global averaging scheme completely ignores di ff erent charge environments within the supercell.Given these reasons, we decide to use the charge-selective averaging scheme for further calculations. III. RESULTS AND DISCUSSION
In order to test our NLTE IP and IPD calculation procedure, we apply it to the simulation of solid densityaluminum plasma and compare with a recent experiment [12, 25]. In this experiment, a solid aluminumtarget was irradiated with intense XFEL pulses with a pulse duration of 80 fs (FWHM) and a peak intensityof 1.1 × W / cm , corresponding to a peak fluence of 1.0 × ph / µ m , providing time-integrated K-shellIP values for di ff erent charge states.In our simulation of the solid density aluminum target ( n i = / cm = − ), we employ XMDYNwith the supercell approach [61–63]. The number of atoms in the supercell should be su ffi cient to guaranteethat stochastic x-ray interactions are properly described. We choose 500 atoms in the supercell with alattice constant of 20.23 Å, containing 5 × × e =38 eV E l e c t r on nu m be r Kinetic energy (keV) 30 fsfitted 30 fs 0 5 10 15 20 25 30 0 0.5 1 1.5 2(b) T e =125 eVKinetic energy (keV) 80 fsfitted 80 fs 0 5 10 15 20 25 30 0 0.5 1 1.5 2(c) T e =191 eVKinetic energy (keV) 160 fsfitted 160 fs FIG. 4. Maxwell–Boltzmann distribution fitting of electron kinetic energies for (a) t =
30 fs, (b) t =
80 fs, and (c) t =
160 fs. from the same crystalline geometry where the atoms are all initially at rest. The photon energy is fixed at1850 eV. The fluence is fixed at 1.0 × ph / µ m and we assume that atoms experience the same fluencethroughout the supercell. When an XFEL pulse is focused onto a target, the x-ray fluence value has aspatially nonuniform distribution in the focal spot, demanding volume integration for calculating physicalobservables [76]. This spatial fluence distribution is not considered for simplicity. The XFEL pulse shapeis chosen as a Gaussian function with 80 fs FWHM and the peak is centered at 80 fs, which is plotted asthe green dashed line in Fig. 3(a). The simulation is performed up to 160 fs. The time evolution of theaverage charge, ¯ Q ( t ) = P Q Qp Q ( t ), and the individual charge populations, p Q ( t ), from XMDYN simulationare shown in Figs. 3(a) and (b), respectively. As we define the atomic charge via K - and L -shell occupations,the initial charge state is + d = / √ n e , where n e = ¯ Qn i ) is much larger than the thermal de Brogliewavelength [77] ( λ e = h / √ π m e kT e , where m e is the electron mass and k is the Boltzmann constant),the electron plasma can be treated as a classical plasma and the kinetic-energy distribution follows theMaxwell–Boltzmann distribution upon thermalization. If d is smaller than λ e , quantum degeneracy e ff ects13ecome important. In our plasma condition at the end of the pulse ( ¯ Q ∼ + T e ∼
200 eV),we have d = .
33 Å and λ e = .
49 Å, i.e., d > λ e but they are comparable to each other. Figure 4 showskinetic-energy distributions of plasma electrons (a) at the early stage [ t =
30 fs], (b) at the peak of the pulse[ t =
80 fs], and (c) at the end of the pulse [ t =
160 fs]. One can see that they follow the Maxwell–Boltzmanndistribution, indicating that the plasma electrons may be thermalized. This could be explained by the factthat a relatively long pulse (80 fs) is applied, which provides enough time for thermalization at least for freeelectrons, during the pulse. The discrepancies between the simulated electron spectrum and the Maxwell–Boltzmann distribution could be attributed to strong correlation between ions and electrons expected fromthe dense plasma. At the same time, quantum electrons bound to individual atoms are far from equilibriumduring the pulse, as clearly shown in the charge-state population dynamics in Fig. 3. Therefore, it maynot be straightforward to justify thermalization of classical and quantum electrons and to define a unifiedelectron temperature T e . Only e ff ective temperatures could be defined in the NLTE regime [13].The IPs in the NLTE plasma environment, namely IP pla , can be readily obtained by applying our approachto the real-time simulation results. The IP pla values for a specific atomic ion species form a distributiondue to di ff erent electronic configurations and environmental fluctuations in di ff erent MC-MD simulation FIG. 5. The time dependent IP pla values from XMDYN simulation trajectories for a solid aluminum target irradiatedby an intense x-ray pulse with a photon energy of 1850 eV and a fluence of 1.0 × ph / µ m . The values are groupedinto individual charge states with di ff erent colors. pla distributioninto individual charge states with di ff erent colors. Note that only the atomic ions with closed K shells areconsidered, and IP values for single- or double- K -hole states are 100–200 eV higher than the closed K shells. As can be seen in Fig. 5, initially the IP pla values tend to increase with time. This is a consequenceof the fact that the M -shell electrons are ionized rapidly in the early stage, such that their contribution toscreening is reduced as they are being ionized. (The magnitude of this e ff ect may be overestimated becauseof the way the 3 s and 3 p electrons are included in our atomic quantum-mechanical treatment.) This processis in competition with increasing screening by plasma electrons, which eventually leads to a decreasing IPuntil the number of plasma electrons is equilibrated.In our approach, the time evolution of the IP distributions can be obtained, signaling the evolution ofenvironmental e ff ects and the energy structures in an NLTE system. As an example the IP distributions forAl + at selected times are shown as vertical colored shades in Fig. 6. The energy distribution is obtainedby manually broadening the discrete lines with a Gaussian of 4.7 eV FWHM. The mean IP values asa function of time are plotted on the bottom plane, when the charge population of Al + is higher than0.5%. The distribution corresponds mainly to the 1 s s p electronic configuration. Again, the widely
0 20 40 60 80 100 120 140 160 1520 1540 1560 1580 1600 1620 1640 1660 0 5 10 15 Al (b) ( c ) Mean20 fs40 fs60 fs80 fs100 fs120 fs140 fs160 fs T i m e ( f s ) IP (eV) A bundan c e ( a r b . un i t s ) ( a ) i n t e g r a t e d E - δ E - δ E + FIG. 6. The distribution of the Al + IP in an x-ray-driven solid-density Al plasma. The distribution functions for eighttimes are shown as an example, the time-integrated distribution is shown as the thick red curve on the top left, withthe corresponding mean value ( ¯ E ) and asymmetric deviation ( δ E − and δ E + ). M -shell electrons treated as quantum electrons. For instance,the configuration group represented by 1 s s p M k ( k ∈ [0 , K - and L -shellconfiguration 1 s s p under di ff erent environments.Figure 6 also contains the time-integrated IP distribution (thick red curve on the top left). We characterizethis asymmetric distribution though its mean value ¯ E and the asymmetric deviation δ E ± by identifying anenergy interval with a probability equal to 68%, i.e., P ( ¯ E < E ≤ ¯ E + δ E + ) =
34% and P ( ¯ E − δ E − < E ≤ ¯ E ) = ff erent electronic configurationstaking into account M -shell electrons [35]. In contrast, our approach achieves both aspects: the ensemble ofplasma environments via MC-MD simulations and detailed electronic configurations via atomic electronic-structure calculations.In Fig. 7(a), the mean value ¯ E and the asymmetric deviation δ E ± of the time-integrated IP distributions IV Al V Al VI Al VII Al VIII Al IX (a) E ne r g y ( e V ) Charge stateexperimental datatwo-step HFSunscreened IPpresent work (b) Abundance (arb. units)+8+7+6+5+4+3
FIG. 7. (a) Time-integrated K -shell IP values (mean and asymmetric deviation) for di ff erent charge states in a solid-density Al plasma generated by an x-ray pulse with a photon energy of 1850 eV, a fluence of 1 × ph / µ m , and apulse duration of 80 fs FWHM. The experimentally observed IP [12, 25], the theoretical calculation using the XATOMtwo-step model [44], corresponding IP values for unscreened atoms are plotted for comparison. (b) Time-integrated K -shell IP distribution. ff erent charge states are compared with experimental data [12, 25] and theoretical calculation withthe XATOM two-step model [44], which is based on the LTE condition. The isolated-atom IPs for thecorresponding electronic configurations are also plotted for comparison. Note that the accuracy of themodel is limited by the HFS description of the binding energy, with a typical relative uncertainty of 1%,so we apply a constant energy shift of + . ff erence between the mean K -shell IP value (1543.1 eV) of the Al + ion in the cold plasma environment (atthe beginning of the simulation) and the experimental K -shell IP value (1559.6 eV) [78]. The asymmetricdeviation is obtained from the time-integrated IP distribution shown in Fig. 7(b). The time-integrated IPvalues calculated within the NLTE framework we have developed match the experimental data. They arealso in good agreement with theoretical LTE-based results. It is currently unknown whether both LTE andNLTE calculations would be equally accurate in reproducing energetically more highly resolved XFEL datathat reveal the detailed shapes of IP distributions in solid-density plasmas.We use Eq. (9) to calculate transient IPD values from the di ff erence between IP pla and IP iso for the samebound-electronic configuration. The time-integrated IPD values from the time-resolved ones are calculatedin the same procedure as used for the IP values. Figure 8 shows the time-integrated IPD values for 1 s ,2 s , and 2 p orbitals, calculated from the IPD distribution for each case. The di ff erence in IPD between 2 s
40 60 80 100 120 140 3 4 5 6 7 8 I P D ( e V ) Charge state 1s 2s 2p
FIG. 8. Mean and asymmetric deviation of time-integrated IPD values of 1 s , 2 s and 2 p orbitals for di ff erent chargestates in a solid-density Al plasma generated by an x-ray pulse of 1850 eV, 1 × ph / µ m , and 80 fs FWHM. p orbitals is at most 0.2 eV, the IPD values for 1 s are at most 0.9 eV higher than those for 2 s . Thisbehavior is similar to the XATOM two-step model [44] and Debye-screened HFS model [79]. Note that,strictly speaking, the IPD values calculated here are not universal properties. They apply to a specific case:a plasma generated by an x-ray pulse with 1850 eV, 1.0 × ph / µ m , and 80 fs FWHM. However, weexpect that the time-integrated IPD values are similar for other x-ray parameter sets, because the IPD foreach charge state mainly depends on the plasma environment and time-integration could wash out some ofthe consequences of fluctuations.Using our NLTE calculations, we now investigate, in an internally consistent manner, to what degreestandard LTE-based models can describe time-resolved IPD values. To this end, the average charge state ¯ Q and the plasma electron density n e are obtained as a function of time from our XMDYN simulations. Usingthose inputs, we compute transient IPDs using the LTE-based modified Ecker–Kr¨oll (mEK) [23, 25] and theStewart–Pyatt (SP) [24, 25] models, ∆ E SP ( Q ) = Q + r SP , (13a) ∆ E mEK ( Q ) = C EK ( Q + r EK , (13b) FIG. 9. Transient IPD values calculated with our NLTE approach (scattered data points), in comparison with mEK(solid line) and SP (dashed line) for (a) Al + and (b) Al + . r SP = [3( Q + / (4 π n e )] / , r EK = [3 / { π ( n e + n i ) } ] / , n i is the ion density, and the coe ffi cient C EK istaken as 1 in the high-density regime. (These expressions are taken from Refs. [25, 29].) Figure 9 showstransient LTE IPD results obtained from mEK (solid line) and SP (dashed line) models, in comparison withour NLTE IPD values (scattered data points) for (a) Q = + Q = +
7. As can be seen in Fig. 9,not only are the standard LTE-based approaches incapable of capturing the fluctuations giving rise to IPDdistributions, the associated IPD values do not coincide with the mean value of the NLTE IPD distributionas a function of time.
IV. CONCLUSION
In this work, we propose an NLTE approach to calculate IPD in dense plasmas. It is a hybrid approachbased on a quantum-mechanical electronic-structure calculation of atomic ions embedded in a plasmaenvironment treated by combining Monte Carlo and classical molecular dynamics. To do so, we developa toolkit, XPOT, as an interface between the MC-MD simulation code XMDYN and the atomic structurecode XATOM. In the current framework, XPOT takes the plasma environment from XMDYN and gives thecalculated micro field to XATOM, in order to calculate atomic parameters a ff ected by IPD. However, themodified binding energy values and atomic data are not yet plugged back into the XMDYN simulation. Animplementation of such IPD feedback into the dynamics simulation is in progress.We apply this approach to describe IPD in a solid-density Al plasma generated by intense XFELpulses. Our NLTE approach allows us to track down the time evolution of transient IP and IPD values forindividual atomic ions in a supercell. In this way, we can examine time-resolved IP and IPD distributionsand obtain time-integrated quantities for each charge state. The mean values of time-integrated IPs arein good agreement with experimental IP data [12, 25] and theoretical calculations based on the LTEassumption [44, 46]. On the other hand, transient IPDs under nonequilibrium conditions show non-monotonic evolutions with time, which are not reproducible by standard LTE-based IPD models. Our NLTEapproach for IP and IPD calculation provides critical insight to understand ultrafast formation dynamicsand fluctuation properties of dense plasmas induced by XFEL pulses, particularly for the early time scale( ∼
100 fs), where electron thermalization is not fully guaranteed. We expect that our computational approachprovides detailed atomic data for NLTE kinetic simulation tools, avoiding usage of standard IPD modelsbased on LTE. It is worthwhile to note that the transient IP values reported here can be potentially measuredby using single-color [80] and two-color [81] pump-probe schemes at XFEL facilities.19
CKNOWLEDGMENTS
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