development of a collisional radiative model of x-ray lasers

*Corresponding author. Tel.: #81-720-31-0709; fax: #81-7210-31-0596.E-mail address: [email protected] (A. Sasaki)

Journal of Quantitative Spectroscopy &Radiative Transfer 65 (2000) 501}509

Development of a collisional radiative model of X-ray lasers

Akira Sasaki*, Takayuki Utsumi, Kengo Moribayashi, Toshiki Tajima,Hiroshi Takuma

Advanced Photon Research Center, Kansai Research Establishment, Japan Atomic Energy Research Institute,25-1 Miiminami-cho, Neyagawa-shi, Osaka 572-0019, Japan

Abstract

A theoretical model of plasma hydrodynamics and atomic kinetics of X-ray lasers is developed toinvestigate the mechanism of lasing in 4d}4p transition of Ni-like ions at short wavelength by the transientpumping scheme. The model is designed for calculations of the ion abundance and soft X-ray gain in theshort pulse laser-irradiated plasmas. We develop a compact collisional radiative model which combines thedetailed level structure of Ni-like ion using atomic data calculated by HULLAC, with averaged levels overa wide range of charge states using the screened hydrogenic model. The ion abundance and soft X-ray gainare calculated by postprocessing the temperature and density of the laser-produced plasma obtained by thehydrodynamics code. It is found that a large abundance of Ni-like ion can be maintained in the plasmaproduced from an exploding foil target showing its usefulness as a gain medium of transient collisional X-raylasers. For improvement of the model, sensitivity of the gain and averaged charge to the level structureincluded in the model are discussed. ( 2000 Elsevier Science Ltd. All rights reserved.

1. Introduction

Recently, signi"cant progress in collisional X-ray lasers has been reported as in Refs. [1}4].Using a combination of pre- and main-laser pulses to irradiate the target to produce a plasma withan optimized density and temperature pro"le, high gain of more than 30/cm and saturatedampli"cation have been obtained with small pump energies [1]. Since in short pulse laser-produced plasmas, the laser absorption, heat conduction, hydrodynamics expansion, atomickinetics and radiative transfer are coupled, the computational model of X-ray lasers should solvethese processes together. Di$culty in the modeling also arises from the complex atomic structure of

0022-4073/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved.PII: S 0 0 2 2 - 4 0 7 3 ( 9 9 ) 0 0 0 9 2 - 8

Ni-like system. Furthermore, to develop a reliable model, not only theoretical investigations butcomparison with experiments such as X-ray spectroscopic measurements are required. Neverthe-less, a theoretical model of Ni-like transient collisional X-ray lasers will be very useful foroptimizing the water-window (20}40 As ) laser and for substantial improvement of e$ciency to levelsmuch greater than the present level (10~6).

To obtain gain, "rstly a solid mid- to high-z target is irradiated by a prepulse laser. Secondly, thepreformed plasma is irradiated by an intense short pulse laser and heated up where the Ni-like ionsare excited by the electron impact to the upper laser level 3d94d (3

2, 32) J"0 to produce population

inversion to 3d94p (52, 32) J"1 level. Due to the very strong collisional excitation rate, quasi-

steady-state gain can be obtained. However, in the case when plasma is heated immediately to thetemperature much higher than the value where steady-state gain is obtained, large transientpopulation inversion is produced during the ionization phase before the lower level is populated.To avoid e!ects of opacity from the ground state and refraction of the X-ray laser beam due todensity gradient, the optimum ion density of the laser medium is considered to be less than1020/cm3. In subcritical density plasmas, the ionization time through the electron collisionalionization becomes '10 ps. Thus, a preformed plasma with the large abundance of Ni-like ionshould be prepared at the time of arrival of the main pulse to obtain the transient gain.

We develop an atomic kinetics code applicable to Ni-like collisional X-ray lasers. To calculatesoft X-ray gain, the atomic model should include more than 100 "ne structure levels of Ni-like ions.On the other hand, since the model should calculate the ion abundance in solid targets as well as inhigh-temperature plasmas, it should also include many charge states. Ions with open M- and N-shells consist of a large number of levels. We have used the HULLAC code to calculate the detailedenergy levels and rate coe$cients [5]. However, to calculate the output energy, spatial pro"le, andcoherence of the X-ray laser, the atomic kinetics code should be coupled with the multidimensionalplasma hydrocode and X-ray propagation code, where the number of levels in the model should belimited due to the limitation of computational time. We designed a hybrid model which combinesthe detailed model for the most important Ni-like levels with simpler screened hydrogenic model[6] for the rest of the con"gurations and charge states. We have investigated the ion abundanceand the soft X-ray gain in a short pulse laser irradiated thin foil target, to "nd an e$cient way ofproducing Ni-like ions. Further, we have calculated the spatial pro"le of temperature and densityand their temporal evolution using HYADES, one-dimensional Lagrangian hydrodynamics code[7], and then calculated the atomic kinetics in a post-process.

2. The atomic model

Fig. 1 illustrates the level structure of Ni-like Xe. For few levels including 3d94d (32, 32)J"0, the

electron collisional excitation rate is exceptionally large [8], so that population inversion occurs to3d94p (5

2, 32)J"1 level. In Ni-like system, the quantum e$ciency de"ned by the fraction of energy of

the laser transition to that of the collisional excitation is higher than Ne-like system. On the otherhand, as the ratio of the rate of radiative decay between the upper and lower level is small, thepopulation of the lower level can be increased by the e!ect of opacity which would cause reductionof gain [9]. Since 3d94s are metastable levels, repumping to 3d94p levels also can cause reduction ofgain. As the energy required to excite one electron from 3d shell to 4l con"guration is 500}800 eV,

502 A. Sasaki et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 65 (2000) 501}509

Fig. 1. Schematic level structure of Ni-like Xe.

whereas the ionization energy is 1495 eV, several double excited con"gurations 3d94l4l@ and innershell excited con"gurations 3s4l and 3p54l are below the ionization limit. These levels may havesigni"cant population. Those levels above the ionization limit are subject to dielectronic recombi-nation and excitation autoionization. Therefore, both sets of levels near the ionization limit mayhave signi"cant e!ects in the ion abundance.

We have started the modeling of Ni-like ion from a simple model. We included the ground statesand one electron excited state with a principal quantum number of the electron up to 10 of Pd-liketo Ar-like ions as superlevels. For lower excited states, l dependence of energy is considered usingPerrot's model [10]. With an appropriate choice of screening constant, the screened hydrogenicmodel provides energy levels with reasonable accuracy. As shown in Fig. 2, the di!erence betweenthe calculated ionization energy of multiple-charged Xe ion and experiment is less than 15% [11].To take into account the e!ect of large statistical weight of each level from partially "lled subshells,the total statistical weight of the level which belongs to each nl superlevel are calculated. Radiativeand collisional rate coe$cients are calculated using empirical formulas [6] with hydrogenicoscillator strengths and level energies, except for transitions between nl levels, where approximateoscillator strengths are taken from those for Cu-like and K-like ions.

Next, we have extended the model to include detailed level structure. However, there is no way todetermine how important each level is nor how many levels should be included in the model. Theappropriate set of levels should be determined by an iterative procedure using di!erent sets untilconvergence is reached. We have developed a #exible computer program with which we canperform calculations with a di!erent atomic model by making a small change in the input data. The#ow chart of the computer program is shown in Fig. 3. The program reads the list of con"gurationsand creates an atomic model automatically following directives that indicate the method. In

A. Sasaki et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 65 (2000) 501}509 503

Fig. 2. Ionization energy of multiple charged Xefrom (v) the present model compared with (s)experimental value [11].

Fig. 3. The #ow chart of the atomic kinetics code which com-bines simple and detailed atomic model.

general, the most important con"gurations relevant to the X-ray gain are treated level by level. Forother Ni-like con"gurations such as 3d95l, 3d96l, 3d84l4l@, 3snl, 3p5nl, and lower excited con"g-urations such as from Ge-like to Co-like ion, detailed atomic data are "rst calculated in detail andthen averaged for each con"guration. For Rydberg states of Ni-like ions and con"gurationsbelonging to lower and higher charge state, in-line atomic data package are used to calculateenergy levels and rate coe$cients.

The program reads the output "les of HULLAC and extracts rate coe$cients for the corre-sponding levels to construct the rate matrix. HULLAC is a general purpose atomic data code suitewhich gives energy levels and almost all of rate coe$cients of multiple-charged ions required in thecollisional radiative model, such as rates of spontaneous emission, autoionization, and crosssections of collisional excitations, collisional ionizations and photo ionizations. It calculatesangular momentum coupling coe$cients based on jj coupling scheme using a graphical method[12]. The energy levels are calculated using the parametric potential method with con"gurationinteraction [13]. Collisional rates are calculated using distorted wave method with factorizationand interporation [14]. Although the number of levels can be more than hundreds for a singlecon"guration, if an appropriate list of con"gurations is given, one can calculate large number ofenergy levels and rate coe$cients within a few hours with a fast workstation. Rate coe$cients ofinverse processes such as three body recombination, collisional deexcitation and radiative recombi-nation can be obtained from detailed balance. Rate coe$cients of dielectronic recombination andother multiple processes can be obtained by postprocessing the rates of autoionization andradiative decay.


Fig. 4. The temporal evolution of (a) density and (b) electron(solid line) and ion(dashed line) temperature of a short pulselaser-irradiated thin foil.

3. Result and discussion

Firstly, we have carried out plasma hydrodynamics calculations to "nd the target geometry withwhich the X-ray gain can be obtained e$ciently. We have mainly investigated temporal evolutionof the density and temperature of the plasma from a thin foil target irradiated by two short laserpulses. It is found from particle-in-cell simulation [15] that when a short pulse laser with a durationof 1 ps irradiates a thin foil target with a thickness shorter than the scale length of the temperaturegradient, 20}40% of the incident energy is absorbed uniformly. For shorter heating pulse((0.1 ps) main absorption mechanism for the solid target is vacuum heating and anomalous skine!ect. If the pulse duration is long enough ('0.5 ps) to produce the plasma with a density gradient'0.1j, the resonance absorption becomes signi"cant. We expected that foil targets have anadvantage as a medium of the X-ray lasers. In the absence of loss of the energy due to heatconduction into solid material, the absorbed energy is expected to cause strong heating of theplasma and ionization of the target material to produce multiple-charged ions. By short pulseirradiation, the plasma temperature becomes very high ('1 keV), while the density is still aroundthe solid density, so that the fast collisional ionization take place to ionize the target element toNi-like. After the "rst laser pulse, the plasma from foil target will expand rapidly, causing therecombination rate to slow down signi"cantly, so that the population of Ni-like ion can bemaintained for a long time ('100 ps) until the plasma density becomes low enough for the opacityof the lower laser level to become less signi"cant. Exciting the plasma by the second short pulselaser, one obtains large transient gain. Moreover, this plasma will have a less steep density gradient,so that X-ray laser light will propagate longer distances reaching saturation.

The typical spatial pro"le of temperature and density of a plasma is shown in Fig. 4, froma HYADES calculation. Solid Xe(z"54) target with a thickness of 1000 As is chosen as a typicalmid-z element. The laser intensity absorbed in the target in 20 fs is assumed to be 1018 W/cm2. It isfound that initially the plasma temperature increases up to more than 1 keV, then drops veryrapidly within 1 ps to 100 eV. The fast cooling is due to radiation. High z plasmas can emit


Fig. 5. The temporal evolution of the ion abundance cor-responds to the time history of temperature and densityshown in Fig. 4. Thick line corresponds to abundance ofNi-like ion.

Fig. 6. The temporal evolution of temperature and densityat the center of a short pulse irradiated thin solid Xe foil,for (v) absorbed intensity"1018 W/cm2, pulse duration"20 fs, (s) 2]1017 W/cm2, 100 fs, (j) 2]1016 W/cm2,1 ps, and (h) 2]1015 W/cm2, 10 ps.

radiation at a rate that can increase strongly with temperature, e.g., as T4. Therefore, no matterhow high the initial temperature is, it drops below 100 eV where the radiation loss becomes lesssigni"cant than that due to hydrodynamics expansion. The plasma expands up to 20 lm in 100 ps.

Secondly, the ion abundance, in Fig. 5, is calculated for the time history of the temperature anddensity at the center of the plasma from the thin foil as shown in Fig. 4. This calculation used thescreened hydrogenic levels from Pd-like to Ar-like ion. It is found that the plasma is in the ionizingphase until !100 fs, then turns to recombination phase as plasma temperature decreases. It isfound that even in a solid density plasma, the ionization time to produce Ni-like ion is not less thana few 100 fs. Due to the large di!erence of the ionization potential between M- and N-shell ions, therecombination rate of Ni-like ion in expanding plasma becomes slow leaving it abundant at 100 ps.To deteremine the condition for obtaining larger Ni-like abundance, we have carried out a series ofcalculations changing the intensity and heating duration. The temperature and density of theplasma is shown in Fig. 6, and resultant temporal evolution of ion abundance is shown in Fig. 7. Inthese calculations, the pulse duration is changed keeping total absorbed energy in the foil constant.It is found that for a short pulse duration ((100 fs), the plasma temperature drops too fast due toradiative cooling. Fast recombination takes place while plasma density is high, so that theabundance of Ni-like ion becomes small even for higher laser intensity where initial temperature ishigher. As the heating duration is increased to 0.5 ps, the temperature of the plasma is kept highenough until the expansion causes decrease of the density which leads to the larger abundance ofNi-like ion at 100 ps. For longer pulse duration beyond 1 ps, the peak temperature becomes toolow for ionization to Ni-like ion. The abundance of Ni-like ion has a maximum of around 0.1 forpulse duration of 0.5 to 1 ps.


Fig. 7. Ion abundance of Xe plasma for di!erent heating durations (indicated by P) keeping the total absorbed energyconstant. (a) intensity"2]1017 W/cm2, pulse duration"100 fs, (b) 2]1016 W/cm2, 1 ps, and (c) 2]1015 W/cm2,10 ps.

Thirdly, calculations using detailed atomic model were carried out. The dependence of theaveraged charge and soft X-ray gain on the level structure is shown in Fig. 8, for n

i"1019/cm3 and

¹%"200 eV. The results are compared with respect to atomic model from a simple model which

includes screened hydrogenic levels up to n"9 from Pd-like to Ar-like ion, with the "ne structureonly for 3d94l, to a complex model which includes con"guration averaged levels with additional3d95l and 3d96l single electron excited con"gurations 3d94l4l@ double excited con"gurations, and3s4l, 3p54l inner shell excited con"gurations. Although the ratio of populations between twoadjacent levels converges to the value determined by Saha}Boltzmann relation in the limit of highdensity, as the number of levels in the model is increased the averaged charge is increased and thesoft X-ray gain is decreased. When we include more multiple excited con"gurations 3d84l4l@ andinner shell excited states 3l4l@, the total population of Ni-like ion is decreased without signi"cant


Fig. 8. Comparison of results calculated from di!erent atomic model. (a) (s) averaged charge and (v) soft X-ray gain of3d94d (3

2, 32)J"0P3d94p (5

2, 32)J"1 transition. (b) calculated populations of Xe; (s) Co-like ground state 3d9, (v) Ni-like

ground state 3d10, (h) 3d94l, (v) 3d95l, (s) 3d96l, (h) 3s4l, (j) 3s4l, (e) 3d84l4l@.

change of the ratio of populations between any pair of con"gurations. It might be due to additionalionization channels through multiple and inner shell excited con"gurations. Calculated gain isalmost proportional to populations of ground and lower excited con"gurations, 3d10 and 3d94l. Ifwe included the inner shell excited levels not by using the con"guration-averaged autoionizationrates but by calculating dielectronic recombination and resonant excitation for each excited levels,population would be modi"ed. However, these rates are valid when collisional mixing, deexcitationand ionization are small compared to that of radiative decay. Those e!ects should be examined atthe density used in X-ray lasers.

Moreover, the e!ects of Rydberg levels and their depletion at high density need to be examined.The level structure of not only Ni-like ion but other close M- and N-shell ions will have e!ects inthe ion abundance and soft X-ray gain. For atomic data itself, for few but very important levelssuch as 3d94d, the atomic structure calculation is di$cult because of strong electron correlationand relativistic e!ects. More accurate calculation of energy levels and collisional radiative rates[16] and its e!ects on the X-ray gain may be of interest.


4. Conclusions

An atomic kinetics model of collisional X-ray lasers using Ni-like ions is developed. The model isapplied to calculations of the ion abundance in a plasma produced by irradiating a thin foil targetby a short laser pulse. An optimum heating pulse duration to produce a plasma with a largeabundance of Ni-like ion is obtained. Irradiating the plasma by another intense laser pulse, a largetransient gain will be obtained e$ciently. For more accurate calculation of the ion abundance andgain, improvement of the model is suggested. Optimization of the X-ray laser are being studied.

References

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development of a collisional radiative model of x-ray lasers

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