- Email: [email protected]
Development of a collisional radiative model of X-ray lasers
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 to investigate the mechanism of lasing in 4d}4p transition of Ni-like ions at short wavelength by the transient pumping scheme. The model is designed for calculations of the ion abundance and soft X-ray gain in the short pulse laser-irradiated plasmas. We develop a compact collisional radiative model which combines the detailed level structure of Ni-like ion using atomic data calculated by HULLAC, with averaged levels over a wide range of charge states using the screened hydrogenic model. The ion abundance and soft X-ray gain are calculated by postprocessing the temperature and density of the laser-produced plasma obtained by the hydrodynamics code. It is found that a large abundance of Ni-like ion can be maintained in the plasma produced from an exploding foil target showing its usefulness as a gain medium of transient collisional X-ray lasers. For improvement of the model, sensitivity of the gain and averaged charge to the level structure included 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 with an optimized density and temperature pro"le, high gain of more than 30/cm and saturated ampli"cation have been obtained with small pump energies [1]. Since in short pulse laserproduced plasmas, the laser absorption, heat conduction, hydrodynamics expansion, atomic kinetics and radiative transfer are coupled, the computational model of X-ray lasers should solve these processes together. Di$culty in the modeling also arises from the complex atomic structure of
* Corresponding author. Tel.: #81-720-31-0709; fax: #81-7210-31-0596. E-mail address: [email protected] (A. Sasaki) 0022-4073/00/$ - 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
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Ni-like system. Furthermore, to develop a reliable model, not only theoretical investigations but comparison with experiments such as X-ray spectroscopic measurements are required. Nevertheless, a theoretical model of Ni-like transient collisional X-ray lasers will be very useful for optimizing the water-window (20}40 As ) laser and for substantial improvement of e$ciency to levels much greater than the present level (10~6). To obtain gain, "rstly a solid mid- to high-z target is irradiated by a prepulse laser. Secondly, the preformed plasma is irradiated by an intense short pulse laser and heated up where the Ni-like ions are excited by the electron impact to the upper laser level 3d94d (3, 3) J"0 to produce population 22 inversion to 3d94p (5, 3) J"1 level. Due to the very strong collisional excitation rate, quasi22 steady-state gain can be obtained. However, in the case when plasma is heated immediately to the temperature much higher than the value where steady-state gain is obtained, large transient population inversion is produced during the ionization phase before the lower level is populated. To avoid e!ects of opacity from the ground state and refraction of the X-ray laser beam due to density gradient, the optimum ion density of the laser medium is considered to be less than 1020/cm3. In subcritical density plasmas, the ionization time through the electron collisional ionization becomes '10 ps. Thus, a preformed plasma with the large abundance of Ni-like ion should be prepared at the time of arrival of the main pulse to obtain the transient gain. We develop an atomic kinetics code applicable to Ni-like collisional X-ray lasers. To calculate soft X-ray gain, the atomic model should include more than 100 "ne structure levels of Ni-like ions. On the other hand, since the model should calculate the ion abundance in solid targets as well as in high-temperature plasmas, it should also include many charge states. Ions with open M- and Nshells consist of a large number of levels. We have used the HULLAC code to calculate the detailed energy levels and rate coe$cients [5]. However, to calculate the output energy, spatial pro"le, and coherence of the X-ray laser, the atomic kinetics code should be coupled with the multidimensional plasma hydrocode and X-ray propagation code, where the number of levels in the model should be limited due to the limitation of computational time. We designed a hybrid model which combines the 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 abundance and the soft X-ray gain in a short pulse laser irradiated thin foil target, to "nd an e$cient way of producing Ni-like ions. Further, we have calculated the spatial pro"le of temperature and density and their temporal evolution using HYADES, one-dimensional Lagrangian hydrodynamics code [7], and then calculated the atomic kinetics in a post-process.
2. The atomic model Fig. 1 illustrates the level structure of Ni-like Xe. For few levels including 3d94d (3, 3)J"0, the 22 electron collisional excitation rate is exceptionally large [8], so that population inversion occurs to 3d94p (5, 3)J"1 level. In Ni-like system, the quantum e$ciency de"ned by the fraction of energy of 22 the laser transition to that of the collisional excitation is higher than Ne-like system. On the other hand, as the ratio of the rate of radiative decay between the upper and lower level is small, the population of the lower level can be increased by the e!ect of opacity which would cause reduction of gain [9]. Since 3d94s are metastable levels, repumping to 3d94p levels also can cause reduction of gain. As the energy required to excite one electron from 3d shell to 4l con"guration is 500}800 eV,
A. Sasaki et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 65 (2000) 501}509
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Fig. 1. Schematic level structure of Ni-like Xe.
whereas the ionization energy is 1495 eV, several double excited con"gurations 3d94l4l@ and inner shell excited con"gurations 3s4l and 3p54l are below the ionization limit. These levels may have signi"cant population. Those levels above the ionization limit are subject to dielectronic recombination and excitation autoionization. Therefore, both sets of levels near the ionization limit may have 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 states and one electron excited state with a principal quantum number of the electron up to 10 of Pd-like to Ar-like ions as superlevels. For lower excited states, l dependence of energy is considered using Perrot's model [10]. With an appropriate choice of screening constant, the screened hydrogenic model provides energy levels with reasonable accuracy. As shown in Fig. 2, the di!erence between the 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. Radiative and collisional rate coe$cients are calculated using empirical formulas [6] with hydrogenic oscillator strengths and level energies, except for transitions between nl levels, where approximate oscillator strengths are taken from those for Cu-like and K-like ions. Next, we have extended the model to include detailed level structure. However, there is no way to determine how important each level is nor how many levels should be included in the model. The appropriate set of levels should be determined by an iterative procedure using di!erent sets until convergence is reached. We have developed a #exible computer program with which we can perform calculations with a di!erent atomic model by making a small change in the input data. The #ow chart of the computer program is shown in Fig. 3. The program reads the list of con"gurations and creates an atomic model automatically following directives that indicate the method. In
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Fig. 2. Ionization energy of multiple charged Xe from (v) the present model compared with (s) experimental value [11].
Fig. 3. The #ow chart of the atomic kinetics code which combines simple and detailed atomic model.
general, the most important con"gurations relevant to the X-ray gain are treated level by level. For other Ni-like con"gurations such as 3d95l, 3d96l, 3d84l4l@, 3snl, 3p5nl, and lower excited con"gurations such as from Ge-like to Co-like ion, detailed atomic data are "rst calculated in detail and then averaged for each con"guration. For Rydberg states of Ni-like ions and con"gurations belonging to lower and higher charge state, in-line atomic data package are used to calculate energy levels and rate coe$cients. The program reads the output "les of HULLAC and extracts rate coe$cients for the corresponding levels to construct the rate matrix. HULLAC is a general purpose atomic data code suite which gives energy levels and almost all of rate coe$cients of multiple-charged ions required in the collisional radiative model, such as rates of spontaneous emission, autoionization, and cross sections of collisional excitations, collisional ionizations and photo ionizations. It calculates angular momentum coupling coe$cients based on jj coupling scheme using a graphical method [12]. The energy levels are calculated using the parametric potential method with con"guration interaction [13]. Collisional rates are calculated using distorted wave method with factorization and interporation [14]. Although the number of levels can be more than hundreds for a single con"guration, if an appropriate list of con"gurations is given, one can calculate large number of energy levels and rate coe$cients within a few hours with a fast workstation. Rate coe$cients of inverse processes such as three body recombination, collisional deexcitation and radiative recombination can be obtained from detailed balance. Rate coe$cients of dielectronic recombination and other multiple processes can be obtained by postprocessing the rates of autoionization and radiative decay.
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3. Result and discussion Firstly, we have carried out plasma hydrodynamics calculations to "nd the target geometry with which the X-ray gain can be obtained e$ciently. We have mainly investigated temporal evolution of the density and temperature of the plasma from a thin foil target irradiated by two short laser pulses. It is found from particle-in-cell simulation [15] that when a short pulse laser with a duration of 1 ps irradiates a thin foil target with a thickness shorter than the scale length of the temperature gradient, 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 skin e!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 an advantage as a medium of the X-ray lasers. In the absence of loss of the energy due to heat conduction into solid material, the absorbed energy is expected to cause strong heating of the plasma and ionization of the target material to produce multiple-charged ions. By short pulse irradiation, the plasma temperature becomes very high ('1 keV), while the density is still around the solid density, so that the fast collisional ionization take place to ionize the target element to Ni-like. After the "rst laser pulse, the plasma from foil target will expand rapidly, causing the recombination rate to slow down signi"cantly, so that the population of Ni-like ion can be maintained for a long time ('100 ps) until the plasma density becomes low enough for the opacity of the lower laser level to become less signi"cant. Exciting the plasma by the second short pulse laser, 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, from a HYADES calculation. Solid Xe(z"54) target with a thickness of 1000 As is chosen as a typical mid-z element. The laser intensity absorbed in the target in 20 fs is assumed to be 1018 W/cm2. It is found that initially the plasma temperature increases up to more than 1 keV, then drops very rapidly within 1 ps to 100 eV. The fast cooling is due to radiation. High z plasmas can emit
Fig. 4. The temporal evolution of (a) density and (b) electron(solid line) and ion(dashed line) temperature of a short pulse laser-irradiated thin foil.
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radiation at a rate that can increase strongly with temperature, e.g., as T4. Therefore, no matter how high the initial temperature is, it drops below 100 eV where the radiation loss becomes less signi"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 and density at the center of the plasma from the thin foil as shown in Fig. 4. This calculation used the screened hydrogenic levels from Pd-like to Ar-like ion. It is found that the plasma is in the ionizing phase until !100 fs, then turns to recombination phase as plasma temperature decreases. It is found that even in a solid density plasma, the ionization time to produce Ni-like ion is not less than a few 100 fs. Due to the large di!erence of the ionization potential between M- and N-shell ions, the recombination 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 of calculations changing the intensity and heating duration. The temperature and density of the plasma is shown in Fig. 6, and resultant temporal evolution of ion abundance is shown in Fig. 7. In these 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 to radiative cooling. Fast recombination takes place while plasma density is high, so that the abundance of Ni-like ion becomes small even for higher laser intensity where initial temperature is higher. As the heating duration is increased to 0.5 ps, the temperature of the plasma is kept high enough until the expansion causes decrease of the density which leads to the larger abundance of Ni-like ion at 100 ps. For longer pulse duration beyond 1 ps, the peak temperature becomes too low for ionization to Ni-like ion. The abundance of Ni-like ion has a maximum of around 0.1 for pulse duration of 0.5 to 1 ps.
Fig. 5. The temporal evolution of the ion abundance corresponds to the time history of temperature and density shown in Fig. 4. Thick line corresponds to abundance of Ni-like ion.
Fig. 6. The temporal evolution of temperature and density at 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.
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Fig. 7. Ion abundance of Xe plasma for di!erent heating durations (indicated by P) keeping the total absorbed energy constant. (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 the averaged charge and soft X-ray gain on the level structure is shown in Fig. 8, for n "1019/cm3 and i 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 structure only for 3d94l, to a complex model which includes con"guration averaged levels with additional 3d95l and 3d96l single electron excited con"gurations 3d94l4l@ double excited con"gurations, and 3s4l, 3p54l inner shell excited con"gurations. Although the ratio of populations between two adjacent levels converges to the value determined by Saha}Boltzmann relation in the limit of high density, as the number of levels in the model is increased the averaged charge is increased and the soft X-ray gain is decreased. When we include more multiple excited con"gurations 3d84l4l@ and inner shell excited states 3l4l@, the total population of Ni-like ion is decreased without signi"cant
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Fig. 8. Comparison of results calculated from di!erent atomic model. (a) (s) averaged charge and (v) soft X-ray gain of 3d94d (3, 3)J"0P3d94p (5, 3)J"1 transition. (b) calculated populations of Xe; (s) Co-like ground state 3d9, (v) Ni-like 22 22 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 additional ionization channels through multiple and inner shell excited con"gurations. Calculated gain is almost proportional to populations of ground and lower excited con"gurations, 3d10 and 3d94l. If we included the inner shell excited levels not by using the con"guration-averaged autoionization rates 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, deexcitation and ionization are small compared to that of radiative decay. Those e!ects should be examined at the 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 in the ion abundance and soft X-ray gain. For atomic data itself, for few but very important levels such as 3d94d, the atomic structure calculation is di$cult because of strong electron correlation and 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.
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4. Conclusions An atomic kinetics model of collisional X-ray lasers using Ni-like ions is developed. The model is applied to calculations of the ion abundance in a plasma produced by irradiating a thin foil target by a short laser pulse. An optimum heating pulse duration to produce a plasma with a large abundance of Ni-like ion is obtained. Irradiating the plasma by another intense laser pulse, a large transient gain will be obtained e$ciently. For more accurate calculation of the ion abundance and gain, improvement of the model is suggested. Optimization of the X-ray laser are being studied.
References [1] Dunn J, Osterheld AL, Shepherd R, White WE, Shlyaptsev VN, Stewart RE. Phys Rev Lett 1998;80:2825. [2] Nickles PV, Shlyaptsev VN, Kalachnikov M, SchnuK rer M, Willi I, Sandner W. Phys Rev Lett 1997;78:2748. [3] Zhang J, MacPhee AG, Nilsen J, Lin J, Barbee Jr. TW, Danson C, Key MH, Lewis CLS, Neely D, O'Rourke RMN, Pert GJ, Smith R, Tallents GJ, Wark JS, Wolfrum E. Phys Rev Lett 1997;78:3856. [4] Daido H, Kato Y, Murai K, Ninomiya S, Kodama R, Yuan G, Oshikane Y, Takagi M, Takabe H, Koike F. Phys Rev Lett 1995;75:1074. [5] Klapisch M, Bar-Shalom A. JQSRT 1997;58:687. [6] Lokke WA, Grasberger WH. 1977 Lawrence Livermore Laboratory Report No. UCRL-52276 (unpublished). [7] Larsen JT, Lane SM. JQSRT 1994;51:179. [8] Daido H, Ninomiya S, Imani T, Okaichi Y, Takagi M, Kodama R, Takabe H, Kato Y, Koike F, Nilsen J, Muirai K. J Modern Phys 1997;11:945. [9] Goldstein WH, Oreg J, Zigler A, Bar-Shalom A, Klapisch M. Phys Rev 1988;A38:1797. [10] Perrot F. Phys Scripta 1989;39:332. [11] Cowan RD. Theory of atomic structure and spectra, Berkeley, University of California Press, 1981. [12] Grant IP. Comput Phys Commun 1973;5:263. [13] Klapisch M, Schwob JL, Fraenkel BS, Oreg J. J Opt Soc Am 1977;61:148. [14] Bar-Shalom A, Klapisch M, Oreg J. Phys Rev 1988;A38:1773. [15] Zhidkov A, Sasaki A. Jaeri-Research Report 98-068. [16] Li Y, Nilsen J, Dunn J, Osterheld AL. Phys Rev 1998;A58:2668.
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 to investigate the mechanism of lasing in 4d}4p transition of Ni-like ions at short wavelength by the transient pumping scheme. The model is designed for calculations of the ion abundance and soft X-ray gain in the short pulse laser-irradiated plasmas. We develop a compact collisional radiative model which combines the detailed level structure of Ni-like ion using atomic data calculated by HULLAC, with averaged levels over a wide range of charge states using the screened hydrogenic model. The ion abundance and soft X-ray gain are calculated by postprocessing the temperature and density of the laser-produced plasma obtained by the hydrodynamics code. It is found that a large abundance of Ni-like ion can be maintained in the plasma produced from an exploding foil target showing its usefulness as a gain medium of transient collisional X-ray lasers. For improvement of the model, sensitivity of the gain and averaged charge to the level structure included 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 with an optimized density and temperature pro"le, high gain of more than 30/cm and saturated ampli"cation have been obtained with small pump energies [1]. Since in short pulse laserproduced plasmas, the laser absorption, heat conduction, hydrodynamics expansion, atomic kinetics and radiative transfer are coupled, the computational model of X-ray lasers should solve these processes together. Di$culty in the modeling also arises from the complex atomic structure of
* Corresponding author. Tel.: #81-720-31-0709; fax: #81-7210-31-0596. E-mail address: [email protected] (A. Sasaki) 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
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Ni-like system. Furthermore, to develop a reliable model, not only theoretical investigations but comparison with experiments such as X-ray spectroscopic measurements are required. Nevertheless, a theoretical model of Ni-like transient collisional X-ray lasers will be very useful for optimizing the water-window (20}40 As ) laser and for substantial improvement of e$ciency to levels much greater than the present level (10~6). To obtain gain, "rstly a solid mid- to high-z target is irradiated by a prepulse laser. Secondly, the preformed plasma is irradiated by an intense short pulse laser and heated up where the Ni-like ions are excited by the electron impact to the upper laser level 3d94d (3, 3) J"0 to produce population 22 inversion to 3d94p (5, 3) J"1 level. Due to the very strong collisional excitation rate, quasi22 steady-state gain can be obtained. However, in the case when plasma is heated immediately to the temperature much higher than the value where steady-state gain is obtained, large transient population inversion is produced during the ionization phase before the lower level is populated. To avoid e!ects of opacity from the ground state and refraction of the X-ray laser beam due to density gradient, the optimum ion density of the laser medium is considered to be less than 1020/cm3. In subcritical density plasmas, the ionization time through the electron collisional ionization becomes '10 ps. Thus, a preformed plasma with the large abundance of Ni-like ion should be prepared at the time of arrival of the main pulse to obtain the transient gain. We develop an atomic kinetics code applicable to Ni-like collisional X-ray lasers. To calculate soft X-ray gain, the atomic model should include more than 100 "ne structure levels of Ni-like ions. On the other hand, since the model should calculate the ion abundance in solid targets as well as in high-temperature plasmas, it should also include many charge states. Ions with open M- and Nshells consist of a large number of levels. We have used the HULLAC code to calculate the detailed energy levels and rate coe$cients [5]. However, to calculate the output energy, spatial pro"le, and coherence of the X-ray laser, the atomic kinetics code should be coupled with the multidimensional plasma hydrocode and X-ray propagation code, where the number of levels in the model should be limited due to the limitation of computational time. We designed a hybrid model which combines the 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 abundance and the soft X-ray gain in a short pulse laser irradiated thin foil target, to "nd an e$cient way of producing Ni-like ions. Further, we have calculated the spatial pro"le of temperature and density and their temporal evolution using HYADES, one-dimensional Lagrangian hydrodynamics code [7], and then calculated the atomic kinetics in a post-process.
2. The atomic model Fig. 1 illustrates the level structure of Ni-like Xe. For few levels including 3d94d (3, 3)J"0, the 22 electron collisional excitation rate is exceptionally large [8], so that population inversion occurs to 3d94p (5, 3)J"1 level. In Ni-like system, the quantum e$ciency de"ned by the fraction of energy of 22 the laser transition to that of the collisional excitation is higher than Ne-like system. On the other hand, as the ratio of the rate of radiative decay between the upper and lower level is small, the population of the lower level can be increased by the e!ect of opacity which would cause reduction of gain [9]. Since 3d94s are metastable levels, repumping to 3d94p levels also can cause reduction of gain. As the energy required to excite one electron from 3d shell to 4l con"guration is 500}800 eV,
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Fig. 1. Schematic level structure of Ni-like Xe.
whereas the ionization energy is 1495 eV, several double excited con"gurations 3d94l4l@ and inner shell excited con"gurations 3s4l and 3p54l are below the ionization limit. These levels may have signi"cant population. Those levels above the ionization limit are subject to dielectronic recombination and excitation autoionization. Therefore, both sets of levels near the ionization limit may have 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 states and one electron excited state with a principal quantum number of the electron up to 10 of Pd-like to Ar-like ions as superlevels. For lower excited states, l dependence of energy is considered using Perrot's model [10]. With an appropriate choice of screening constant, the screened hydrogenic model provides energy levels with reasonable accuracy. As shown in Fig. 2, the di!erence between the 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. Radiative and collisional rate coe$cients are calculated using empirical formulas [6] with hydrogenic oscillator strengths and level energies, except for transitions between nl levels, where approximate oscillator strengths are taken from those for Cu-like and K-like ions. Next, we have extended the model to include detailed level structure. However, there is no way to determine how important each level is nor how many levels should be included in the model. The appropriate set of levels should be determined by an iterative procedure using di!erent sets until convergence is reached. We have developed a #exible computer program with which we can perform calculations with a di!erent atomic model by making a small change in the input data. The #ow chart of the computer program is shown in Fig. 3. The program reads the list of con"gurations and creates an atomic model automatically following directives that indicate the method. In
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Fig. 2. Ionization energy of multiple charged Xe from (v) the present model compared with (s) experimental value [11].
Fig. 3. The #ow chart of the atomic kinetics code which combines simple and detailed atomic model.
general, the most important con"gurations relevant to the X-ray gain are treated level by level. For other Ni-like con"gurations such as 3d95l, 3d96l, 3d84l4l@, 3snl, 3p5nl, and lower excited con"gurations such as from Ge-like to Co-like ion, detailed atomic data are "rst calculated in detail and then averaged for each con"guration. For Rydberg states of Ni-like ions and con"gurations belonging to lower and higher charge state, in-line atomic data package are used to calculate energy levels and rate coe$cients. The program reads the output "les of HULLAC and extracts rate coe$cients for the corresponding levels to construct the rate matrix. HULLAC is a general purpose atomic data code suite which gives energy levels and almost all of rate coe$cients of multiple-charged ions required in the collisional radiative model, such as rates of spontaneous emission, autoionization, and cross sections of collisional excitations, collisional ionizations and photo ionizations. It calculates angular momentum coupling coe$cients based on jj coupling scheme using a graphical method [12]. The energy levels are calculated using the parametric potential method with con"guration interaction [13]. Collisional rates are calculated using distorted wave method with factorization and interporation [14]. Although the number of levels can be more than hundreds for a single con"guration, if an appropriate list of con"gurations is given, one can calculate large number of energy levels and rate coe$cients within a few hours with a fast workstation. Rate coe$cients of inverse processes such as three body recombination, collisional deexcitation and radiative recombination can be obtained from detailed balance. Rate coe$cients of dielectronic recombination and other multiple processes can be obtained by postprocessing the rates of autoionization and radiative decay.
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3. Result and discussion Firstly, we have carried out plasma hydrodynamics calculations to "nd the target geometry with which the X-ray gain can be obtained e$ciently. We have mainly investigated temporal evolution of the density and temperature of the plasma from a thin foil target irradiated by two short laser pulses. It is found from particle-in-cell simulation [15] that when a short pulse laser with a duration of 1 ps irradiates a thin foil target with a thickness shorter than the scale length of the temperature gradient, 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 skin e!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 an advantage as a medium of the X-ray lasers. In the absence of loss of the energy due to heat conduction into solid material, the absorbed energy is expected to cause strong heating of the plasma and ionization of the target material to produce multiple-charged ions. By short pulse irradiation, the plasma temperature becomes very high ('1 keV), while the density is still around the solid density, so that the fast collisional ionization take place to ionize the target element to Ni-like. After the "rst laser pulse, the plasma from foil target will expand rapidly, causing the recombination rate to slow down signi"cantly, so that the population of Ni-like ion can be maintained for a long time ('100 ps) until the plasma density becomes low enough for the opacity of the lower laser level to become less signi"cant. Exciting the plasma by the second short pulse laser, 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, from a HYADES calculation. Solid Xe(z"54) target with a thickness of 1000 As is chosen as a typical mid-z element. The laser intensity absorbed in the target in 20 fs is assumed to be 1018 W/cm2. It is found that initially the plasma temperature increases up to more than 1 keV, then drops very rapidly within 1 ps to 100 eV. The fast cooling is due to radiation. High z plasmas can emit
Fig. 4. The temporal evolution of (a) density and (b) electron(solid line) and ion(dashed line) temperature of a short pulse laser-irradiated thin foil.
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radiation at a rate that can increase strongly with temperature, e.g., as T4. Therefore, no matter how high the initial temperature is, it drops below 100 eV where the radiation loss becomes less signi"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 and density at the center of the plasma from the thin foil as shown in Fig. 4. This calculation used the screened hydrogenic levels from Pd-like to Ar-like ion. It is found that the plasma is in the ionizing phase until !100 fs, then turns to recombination phase as plasma temperature decreases. It is found that even in a solid density plasma, the ionization time to produce Ni-like ion is not less than a few 100 fs. Due to the large di!erence of the ionization potential between M- and N-shell ions, the recombination 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 of calculations changing the intensity and heating duration. The temperature and density of the plasma is shown in Fig. 6, and resultant temporal evolution of ion abundance is shown in Fig. 7. In these 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 to radiative cooling. Fast recombination takes place while plasma density is high, so that the abundance of Ni-like ion becomes small even for higher laser intensity where initial temperature is higher. As the heating duration is increased to 0.5 ps, the temperature of the plasma is kept high enough until the expansion causes decrease of the density which leads to the larger abundance of Ni-like ion at 100 ps. For longer pulse duration beyond 1 ps, the peak temperature becomes too low for ionization to Ni-like ion. The abundance of Ni-like ion has a maximum of around 0.1 for pulse duration of 0.5 to 1 ps.
Fig. 5. The temporal evolution of the ion abundance corresponds to the time history of temperature and density shown in Fig. 4. Thick line corresponds to abundance of Ni-like ion.
Fig. 6. The temporal evolution of temperature and density at 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.
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Fig. 7. Ion abundance of Xe plasma for di!erent heating durations (indicated by P) keeping the total absorbed energy constant. (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 the averaged charge and soft X-ray gain on the level structure is shown in Fig. 8, for n "1019/cm3 and i 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 structure only for 3d94l, to a complex model which includes con"guration averaged levels with additional 3d95l and 3d96l single electron excited con"gurations 3d94l4l@ double excited con"gurations, and 3s4l, 3p54l inner shell excited con"gurations. Although the ratio of populations between two adjacent levels converges to the value determined by Saha}Boltzmann relation in the limit of high density, as the number of levels in the model is increased the averaged charge is increased and the soft X-ray gain is decreased. When we include more multiple excited con"gurations 3d84l4l@ and inner shell excited states 3l4l@, the total population of Ni-like ion is decreased without signi"cant
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Fig. 8. Comparison of results calculated from di!erent atomic model. (a) (s) averaged charge and (v) soft X-ray gain of 3d94d (3, 3)J"0P3d94p (5, 3)J"1 transition. (b) calculated populations of Xe; (s) Co-like ground state 3d9, (v) Ni-like 22 22 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 additional ionization channels through multiple and inner shell excited con"gurations. Calculated gain is almost proportional to populations of ground and lower excited con"gurations, 3d10 and 3d94l. If we included the inner shell excited levels not by using the con"guration-averaged autoionization rates 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, deexcitation and ionization are small compared to that of radiative decay. Those e!ects should be examined at the 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 in the ion abundance and soft X-ray gain. For atomic data itself, for few but very important levels such as 3d94d, the atomic structure calculation is di$cult because of strong electron correlation and 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.
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4. Conclusions An atomic kinetics model of collisional X-ray lasers using Ni-like ions is developed. The model is applied to calculations of the ion abundance in a plasma produced by irradiating a thin foil target by a short laser pulse. An optimum heating pulse duration to produce a plasma with a large abundance of Ni-like ion is obtained. Irradiating the plasma by another intense laser pulse, a large transient gain will be obtained e$ciently. For more accurate calculation of the ion abundance and gain, improvement of the model is suggested. Optimization of the X-ray laser are being studied.
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