1. INTRODUCTION
Observations of astrophysical phenomena have been a continual source of X-ray spectra for many years. Through spectroscopic analysis, these spectra provide clues to understanding the processes that create the various plasmas seen throughout the universe. The spectra point out gaps in our understanding, such as important lines and atomic data details, and targeted laboratory experiments can help fill in those gaps (Beiersdorfer, Reference Beiersdorfer2003). Iron is an important element in solar (Rajmal et al., Reference Rajmal, Pradhan, Joshi, Shah, Trivedi, Kayasth, Shah and Deshpande2006; Phillips, Reference Phillips2012) and other star systems (Eze, Reference Eze2014; Ponti et al., Reference Ponti, Bianchi, Munoz-Darias, De Marco, Dwelly, Fender, Nandra, Rea, Mori, Haggard, Heinke, Degenaar, Aramaki, Clavel, Goldwurm, Hailey, Israel, Morris, Rushton and Terrier2015) and for that reason X-ray Fe spectra have been studied in laboratory experiments using many different plasma sources, including: Electron beam ion traps (Decaux et al., Reference Decaux, Beiersdorfer, Osterheld, Chen and Kahn1995, Reference Decaux, Beiersdorfer, Kahn and Jacobs1997, Reference Drake, Swartz, Beiersdorfer, Brown and Kahn1999), tokamaks (Beiersdorfer et al., Reference Beiersdorfer, Phillips, Jacobs, Hill, Bitter, von Goeler and Kahn1993, Reference Beiersdorfer, von Goeler, Bitter and Thorn2001), and X-pinches (Safronova et al., Reference Safronova, Kantsyrev, Esaulov, Ouart, Safronova, Shrestha and Williamson2009).
Iron is the heaviest element that can be created by the fusion process that occurs in stars as the energy created by the fusion becomes greater than the necessary energy to fuse after iron. This can only occur in the largest stars, however, when a star goes supernova it can create iron quickly along with heavier elements due to the high density of energy and neutrons. Additionally due to being the heaviest element with a positive energy of formation, heavier elements are likely to decay to iron over time. The presence of iron in an astrophysical spectrum can provide information on the history of the object.
Short pulse lasers have been a useful tool in studying X-ray radiation (for example, Magunov et al., Reference Magunov, Faenov, Skobelev, Pikuz, Dobosz, Schmidt, Perdrix, Meynadier, Gobert, Normand, Stenz, Bagnoud, Blasco, Roche, Salin and Sharkov2003 and Abdallah et al., Reference Abdallah, Batani, Desai, Lucchini, Faenov, Pikuz and Magunov2007). There have been many improvements in the understanding of Fe X-ray spectra, however, there is more to learn about the different processes involved in X-ray formation with respect to the roles of fast electron beams and X-ray pumping for spectra in laboratory and astrophysics for both low and high density plasmas. It has been found that the plasma created by irradiation from a 500 fs pulse laser on solid targets include a hotter region along the surface with the solid material remaining cold (Chen et al., Reference Chen, Patel, Chung, Kemp, Le Pape, Maddox, Wilks, Stephens and Beg2009). The radial direction of plasma from irradiation of a 330 fs pulse laser was also explored with spectroscopy (Zastrau et al., Reference Zastrau, Audebert, Bernshtam, Brambrink, Kämpfer, Kroupp, Loetzsch, Maron, Ralchenko, Reinholz, Röpke, Sengebusch, Stambulchik, Uschmann, Weingarten and Förster2010). One of the important points is to understand the possible role of X-ray photo pumping. In recent investigations with ultra-relativistic laser-produced plasma it was found that photo-pumping was affecting the spectra of medium z targets such as Al (Colgan et al., Reference Colgan, Abdallah, Faenov, Pikuz, Wagenaars, Booth, Culfa, Dance, Evans, Gray, Kaempfer, Lancaster, McKenna, Rossall, Skobelev, Schulze, Uschmann, Zhidkov and Woolsey2013; Pikuz et al., Reference Pikuz, Faenov, Colgan, Dance, Abdallah, Wagenaars, Booth, Culfa, Evans, Gray, Kaempfer, Lancaster, McKenna, Rossall, Skobelev, Schulze, Uschmann, Zhidkov and Woolsey2013; Hansen et al., Reference Hansen, Colgan, Faenov, Abdallah, Pikuz, Skobelev, Wagenaars, Booth, Culfa, Dance, Tallents, Evans, Gray, Kaempfer, Lancaster, McKenna, Rossall, Schulze, Uschmann, Zhidkov and Woolsey2014; Faenov et al., Reference Faenov, Colgan, Hansen, Zhidkov, Pikuz, Nishiuchi, Pikuz, Skobelev, Abdallah, Sakaki, Sagisaka, Pirozhkov, Ogura, Fukuda, Kanasaki, Hasegawa, Nishikino, Kando, Watanabe, Kawachi, Masuda, Hosokai, Kodama and Kondo2015a ) and Si (Colgan et al., Reference Colgan, Faenov, Pikuz, Tubman, Butler, Abdallah, Dance, Pikuz, Skobelev, Alkhimova, Booth, Green, Gregory, Andreev, Lӧtz, Uschmann, Zhidkov, Kodama, McKenna and Woolsey2016). It was shown in (Faenov et al., Reference Faenov, Colgan, Hansen, Zhidkov, Pikuz, Nishiuchi, Pikuz, Skobelev, Abdallah, Sakaki, Sagisaka, Pirozhkov, Ogura, Fukuda, Kanasaki, Hasegawa, Nishikino, Kando, Watanabe, Kawachi, Masuda, Hosokai, Kodama and Kondo2015a ) that femtosecond laser pulses with intensity reaching 1021 W/cm2 at the surface of some microns Al foils could generate X-ray source with intensity of 1017 W/cm2 and such intensity is already enough for photo-pumping generation of hollow KK and KL ions. Understanding KK and KL ions are of value for diagnosing non-ideal plasma and warm dense matter (Colgan et al., Reference Colgan, Abdallah, Faenov, Pikuz, Skobelev, Fortov, Fukuda, Akahane, Aoyama, Inoue and Yamakawa2008; Faenov et al., Reference Faenov, Skobelev, Pikuz, Pikuz, Kodama and Fortov2015b ). Here spectra of higher z material (Fe) collected from the same experimental conditions as seen in the Al experiments are analyzed to see the roles of fast electron beams and X-ray pumping.
2. EXPERIMENT
The experiments are performed using the J-KAREN (Kiriyama et al., Reference Kiriyama, Mori, Nakai, Shimomura, Sasao, Tanoue, Kanazawa, Wakai, Sasao, Okada, Daito, Suzuki, Kondo, Kondo, Sugiyama, Bolton, Yokoyama, Daido, Kawanishi, Kimura and Tajima2010; Nishiuchi et al., Reference Nishiuchi, Sakaki, Esirkepov, Nishio, Pikuz, Faenov, Skobelev, Orlandi, Sako, Pirozhkov, Matsukawa, Sagisaka, Ogura, Kanasaki, Kiriyama, Fukuda, Koura, Kando, Yamauchi, Watanabe, Bulanov, Kondo, Imai and Nagamiya2015), an optical parametric chirped-pulse amplification (OPCPA) Ti:Sapphire hybrid laser system at Kansai Photon Science Institute of Japan Atomic Energy Agency. Laser pulses have the energy on the target up to ~7 J, duration of 35 fs (full width at half maximum, FWHM, with respect to intensity), wavelength of 0.8 µm and a typical contrast of 1010 (with respect to power; at several picoseconds before the pulse peak). The contrast is improved with a saturable absorber inserted between the high-energy CPA oscillator and stretcher. The ultrafast Pockels cell with the extinction ratio of 200 is applied at 500 ps before the main pulse to secure the contrast on the nanosecond time scale. Using F/2.1 off-axis parabolic mirror, p-polarized laser pulses are focused onto a target at an incidence angle of 84°, producing a spot with the FWHM diameter of ~4.5 µm (FWHM) and the peak laser intensity of 1021 W/cm2.
The targets are 2 µm stainless steel Fe foils placed at the laser focus position with a 5 µm accuracy along the laser-propagation axis, well within the Rayleigh length of 20 µm. For high-resolution spectroscopy measurements (Faenov et al., Reference Faenov, Colgan, Hansen, Zhidkov, Pikuz, Nishiuchi, Pikuz, Skobelev, Abdallah, Sakaki, Sagisaka, Pirozhkov, Ogura, Fukuda, Kanasaki, Hasegawa, Nishikino, Kando, Watanabe, Kawachi, Masuda, Hosokai, Kodama and Kondo2015a ) a Focusing Spectrometer with Spatial Resolution (FSSR) (Faenov et al., Reference Faenov, Pikuz, Erko, Bryunetkin, Dyakin, Ivanenkov, Mingaleev, Pikuz, Romanova and Shelkovenko1994; Blasco et al., Reference Blasco, Stenz, Salin, Faenov, Magunov, Pikuz and Skobelev2001) was implemented. The instrument was equipped with a spherically bent quartz (31–40) crystal with a lattice spacing 2d ~ 2.36 Å and a radius of curvature of R = 150 mm. The crystal was aligned to record K-shell emission spectra of multi-charged (Heα line of Fe XXV and Lyα line of Fe XXVI) and neutral (i.e. K α, K β lines) Fe ions in 1.7–1.97 Å wavelength range. The FSSR spectral resolving power was approximately 3000. The spectrometer observed the laser-irradiated front surface of the target at an angle of ~40° to the target surface normal and the target-to-crystal distance of 2045 mm. Background fogging and crystal fluorescence due to intense fast electrons were limited using two pairs of 0.5 T neodymium-iron-boron permanent magnets that formed a slit ~15 mm wide in front of the crystal. Spectra were recorded on X-ray Andor DX-420 back-illuminated charge-coupled device (CCD) with 26 µm pixel size. The X-ray CCD was protected against exposure to visible light using two layers of 1 µm thick polypropylene coated from both sides with 0.2 µm Al. Additionally, polypropylene filter of 6 µm thickness was used to reduce the background noise. Spectra were measured in single laser shots.
The spectra from the J-KAREN experiments are compared with Stainless Steel (69% Fe, 19% Cr, 9% Ni) X-pinch experiments performed on the Zebra Generator at the Nevada Terawatt Facility at the University of Nevada, Reno. The Zebra generator is a pulsed power device capable of storing up to 150 kJ of energy. That energy is released to provide ~1 MA of current with a 100–110 ns rise time (see, for example, Kantsyrev et al., Reference Kantsyrev, Rudakov, Safronova, Fedin, Ivanov, Velikovich, Esaulov, Chuvatin, Williamson, Ouart, Nalajala, Osborne, Shrestha, Yilmaz, Pokala, Laca and Cowan2006). The X-pinches were composed of four wires crossed in an X pattern with a 31.7° angle between the wires on the anode and cathode sides. The wires were 40 µm thick providing a 415 µg/cm linear mass through a 20 mm anode to cathode gap. X-ray spectra were recorded using a time integrated spatially resolved spectrometer using a LiF crystal (2d = 4.028 Å). The spectrometer was capable of recording over a spectral range of 1.5–2.3 Å and used a 8 µm Kapton Polyimide, 0.3 µm Al, and 2 µm Polypropylene filter.
3. THEORETICAL MODELING
The K-shell Fe experimental spectrum from plasmas irradiated by ultra-intense laser beams with femtosecond scale pulse duration is expected to be primarily dominated by three distinct regions. The first is the hottest region. Despite the high 1010 laser contrast in our experiments, at laser intensities ~1021 W/cm2 preplasma is still created. This preplasma will be energized prior to the main pulse, which would then arrive and the already excited material would further excite to the hottest region of the experiment. This would be most strongly composed of H-like, He-like, and Li-like Fe. It would be expected to have an electron temperature (T e) between 3000 and–4000 eV with optimal absorption of the main pulse when the electron density (N e) is around 2 × 1021 cm−3, which for our case corresponded laser critical intensity.
The second region would be characterized by the portions of solid material that remain after the prepulse that are hit by the main pulse. This would create cooler plasma than the first region due to the higher density of the material. For neutral Fe, the solid density is 8.25 × 1022 atoms/cm−3. This region would be expected to be dominated by ionizations between Li-like and Ne-like Fe with an average ion charge of +22–24, which would suggest N e for this region should be around 2 × 1024 cm−3.
The third region would be the material that does not directly interact with either the prepulse or main pulse. This region would be heated by the radiation transfer and hot electrons from the previous two regions. This would create relatively cold temperature plasma (10–50 eV) that is likely to be dominated by ionizations near neutral Fe. This plasma would be close to solid density with an average ion charge around +1–3. This suggests N e should be around 8 × 1022–2 × 1023 cm−3.
Theoretical spectra were obtained using two models: One for ionic transitions and one for characteristic transitions. The ionic model was previously developed to study stainless steel Z-pinch wire arrays (Safronova et al., Reference Safronova, Kantsyrev, Esaulov, Ouart, Yilmaz, Williamson, Shrestha, Osborne, Greenly, Chandler, McBride, Chalenski, Hammer, Kusse and LePell2008, Reference Safronova, Kantsyrev, Esaulov, Ouart, Safronova, Shrestha and Williamson2009; Ouart et al., Reference Ouart, Safronova, Faenov, Pikuz, Gasilov, Calegari, Nisoli, De Silvestri, Stagira, Poletto and Villoresi2011). The model is based on atomic data calculated by the Flexible Atomic Code (Gu, Reference Gu2008). Ground states for ionization levels from bare to neutral were used, while excited states for H-like to Al-like were detailed with the largest focus being on O-like to Al-like states. For K-shell transitions, singly excited states up to n = 6 were used for H-like ions, n = 5 for He- and Li-like ions, and up to n = 4 for Be- to F-like. Doubly excited states were included for He- and Li-like ions up to n = 3 and for Be-like up to n = 2. The model is run assuming nonlocal thermodynamic equilibrium conditions and synthetic spectra are created by broadening line emissions using Voigt line profiles. The characteristic model includes ground states for all ionization levels with excited states focusing on the K- to Fe-like ionization levels. Singly excited states up to n = 3 were included for the K- to Fe-like ionization levels.
Figure 1 shows the spectrum recorded from the J-KAREN experiment (Nishiuchi et al., Reference Nishiuchi, Sakaki, Esirkepov, Nishio, Pikuz, Faenov, Skobelev, Orlandi, Sako, Pirozhkov, Matsukawa, Sagisaka, Ogura, Kanasaki, Kiriyama, Fukuda, Koura, Kando, Yamauchi, Watanabe, Bulanov, Kondo, Imai and Nagamiya2015) with modeling. Three models are included that correspond with the expected predicted regions. Figure 1a covers the first expected region. The plasma parameters used to model this region were T e = 3500 eV, N e = 3 × 1021 cm−3, and 0.1% hot electrons. T e was determined by matching the relative ratios of the Heα and Lyα (1.78 Å) lines. N e was determined by matching the relative ratios of the Heα line with the intercombination (I.C.) line (1.86 Å). The hot electron concentration was most noticeable in the intensities of the H-like, He-like, and Li-like transitions. Increasing the hot electron concentration further emphasized the He-like transitions while the reducing the H-like and Li-like transitions.
Theoretical modeling matched to experimental data from laser experiments performed at JKAREN facility (Colgan et al., Reference Colgan, Faenov, Pikuz, Tubman, Butler, Abdallah, Dance, Pikuz, Skobelev, Alkhimova, Booth, Green, Gregory, Andreev, Lӧtz, Uschmann, Zhidkov, Kodama, McKenna and Woolsey2016). All models include 0.1% hot electrons described with a Gaussian distribution at 10 keV with a 50 eV half width at half max (HWHM). (a) 3500 eV and 8 × 1022 cm−3 modeling matches Heα and Lyα transitions. (b) 700 eV and 1.8 × 1024 cm−3 modeling matches B-like and C-like structures between 1.88 and 1.9 Å. (c) Modeling for cold Kα and Kβ transitions suggest 10 eV electron temperatures with a 8 × 1022 cm−3 electron density.
The modeling in Figure 1b covers the second expected region. The modeling suggest plasma parameters of T e = 700 eV, N e = 1.8 × 1024 cm−3, and 0.1% hot electrons. This modeling primarily covers Li-like to C-like transitions. Changes to T e shift the ionization balance of the plasma. The T e for this region was determined by adjusting the ionization balance such that Be-like transitions (~1.87 Å) are the most prominent. Increasing N e broadens the ionization distribution while decreasing N e narrows the distribution. N e was determined by matching the relative intensities of the Be-like, B-like and C-like structures. The presence of hot electrons effected the higher ionizations most significantly. Therefore the hot electron concentration was chosen by looking at the He-like, Li-like, and Be-like contributions to the spectrum.
Figure 1c compares the experimental spectrum with a model with plasma parameters of: T e = 10 eV, N e = 8 × 1022 cm−3, and 0.1% hot electrons with 10 keV temperature with 50 eV half width at half max. These plasma parameters describe the Kα and Kβ transitions near 1.93 and 1.75 Å, respectively. The T e was determined by aligning the location of the Kβ lines in the experimental and theoretical spectra. The Kβ transitions shift toward lower wavelengths with increasing T e quicker than the Kα transitions. The N e is determined by matching the relative intensities of the two peaks in the Kα region. The Kα peaks are also influenced by the concentration of hot electrons. The presence of hot electrons increases the relative intensity of peak at the longer wavelength. While the presence of hot electrons was a significant change to the modeling, varying the concentration of the hot electrons created very small changes to the modeling. This prevented a precise determination of the hot electron concentration in this region. A 0.1% concentration was used for the modeling as it was found to be a good match for the other regions in which the hot electron concentration is more noticeable.
Figure 2 compares the combined theoretical spectra from Figure 1 with the experimental spectra. The combined spectrum shows good comparison with the cold Kα and Kβ regions and with the Be-like, B-like, and C-like transitions. The most noticeable differences are surrounding the I.C. line. This is possibly due to some differences in wavelengths between the theoretical calculations and the experimental spectrum. The theoretical line for the Li-like peak just to the right of the I.C. line is closer to the I.C. line leading to more of a build up instead of a split peak. Also the theoretical line for the peak between the Heα and I.C. line is shifted close to the I.C. line. This leads to the peak getting absorbed into the I.C. line when trying to broaden the He-like transitions rather than rising to fill the region.
Combination of theoretical models shown in Figure 1 matched to the experimental spectrum. The modeling matches most structure between 1.85 and 1.95 Å.
Figure 3 is a comparison of the experimental spectrum with three models that include a very high concentration of hot electrons in a mild T e (300 eV) bulk plasma with the intent to describe the hotter plasma using one plasma region. The three concentrations are (a) 10%, (b) 25%, and (c) 40%, which were chosen to display the effects of hot electrons on the theoretical spectrum. The 40% is beyond any expectations of what would exist physically but it is useful for highlighting the effects of hot electrons. Figure 3a displays the 10% hot electron concentration and it covers the lower ionization states without any significant presence of the He- and Li-like ions. Figure 3b is the 25% hot electron concentration and it has a balance between the higher (He- and Li-like) and the lower (B- and C-like) ionization levels. This balance does not fit with the experimental spectrum as the higher ionizations are underrepresented and the lower ionizations are slightly overestimated. Figure 3c is the 40% concentration and it matches the higher ionizations closer while the lower ionizations are underestimated. This figure confirms that the hotter plasma cannot be a single region of mild T e with a high hot electron concentration.
Theoretical modeling of three different hot electron concentrations, (a) 10%, (b) 25%, and (c) 40%, with a bulk T e = 300 eV and N e = 1.8 × 1024 cm−3. The modeling shows the ionization balance does not fit the experimental spectrum which confirms the need for multiple regions shown in Figure 1.
Figure 4 displays a K-shell Fe experimental spectrum recorded during a stainless steel X-pinch experiment performed on the Zebra generator at 1 MA. Three theoretical spectra are compared with the experimental spectrum to point out the differences of the X-pinch spectrum and the laser-produced spectrum. The theoretical spectra were modeled at T e = 170 eV and N e = 1 × 1020 cm−3 with varying hot electron concentrations (1, 2.5, and 4%). The laser-produced spectrum was dominated by fewer regions that were relatively easy to distinguish, while the X-pinch spectrum is composed of significant contributions from all the ionizations between He-like and Ne-like Fe, which occurs as the plasma ionizes throughout the experiment.
Theoretical modeling showing the influence of different ionizations on the K-shell Fe experimental spectrum from a stainless steel X-pinch experiment performed on the Zebra generator at 1 MA. Modeling was done with T e = 170 eV and N e = 1 × 1020 cm−3 and three different hot electron concentrations (1, 2.5, and 4%). The experimental spectrum shows evidence of He-like to F-like ions.
Figure 5 is the experimental spectrum for the X-pinch compared with two theoretical spectra that describe the maximum and minimum temperature regions represented in the experimental spectrum. The cold region is estimated with T e = 170 eV and N e = 1 × 1020 cm−3. This is primarily Ne-like transitions that fill in the largest peak in the spectrum. T e is estimated by matching the location of the Kα peak with the theoretical spectrum. N e is chosen by using the value from the hot region as the density cannot be properly estimated using one line. The hottest region is estimated with T e = 1700 eV and N e = 1 × 1020 cm−3. N e is estimated by matching the relative intensities of the Heα line with the I.C. line. T e is estimated with less accuracy as there is not a Lyα line presence with which to compare the Heα line.
Theoretical modeling for Stainless Steel X-pinch spectra estimating the plasma parameters of the highest T e (1700 eV, 1 × 1020 cm−3) and lowest T e (170 eV, 1 × 1020 cm−3) regions.
4. DISCUSSION
The laser-produced spectrum was describable with the three predicted plasma regions: (1) The preplasma region irradiated by both the prepulse and the main pulse, (2) the solid density plasma from center of focusing spot region irradiated by only the main pulse, and (3) the solid density plasma periphery region excited by the radiation and hot electrons emitted from the previous two regions. This creation of separate regions can occur due to the fast and ultra-intense interaction of the laser with the target. The fs pulse duration occurs so fast and its intensity so high that due to optical field ionization (Ammosov et al., Reference Ammosov, Delone and Krainov1986) the atoms are ionized faster than they can radiate, which prevents radiation from intermittent ions. The spectrum from the stainless steel X-pinch experiment was different because the intermediate ions were present. This is due to the much longer time scale of the experiment where the current typically rises for 30–60 ns prior to the first explosion of the X-pinch. The X-pinch spectrum had a strong Kα presence suggesting colder plasma surrounding the hotter core. This is similar to the Kα presence in the laser experiment; however the T e for each experiment is significantly different (~10 eV laser and ~170 eV X-pinch). The colder laser-produced plasma is heated by X-ray radiation and fast electrons from the hotter regions while the colder plasma in the X-pinch experiment is also energized by the kinetic energy acquired while being accelerated by the explosions and implosions. This is seen in the different ionization levels of the transitions that dominate the Kα peaks. The hottest regions for each experiment are also quite different. The value of the electron densities differ by an order of magnitude (3 × 1021 cm−3 laser, 1 × 1020 cm−3 X-pinch) and the maximum values of the electron temperatures are quite different (3500 eV laser, 1700 eV X-pinch). The T e difference is quickly noticeable by the presence of the Lyα in the laser-produced spectrum and not in the X-pinch spectrum.
Along with estimating the conditions of the laser-produced plasma, there was the hope to observe effects of X-ray pumping and hot electrons by analyzing KK and KL hollow ion emissions. The spectral regions where the KK and KL hollow ion transitions would be seen are at approximately 1.8–1.85 Å and 1.86–1.92 Å, respectively. Unfortunately the KK hollow ions spectral region has no evidence of spectral lines. The KL hollow ion spectral region includes multiple peaks of greater intensity that are also influenced by singly excited state transitions. The modeling of the laser-produced spectrum was improved with the inclusion of a small percentage of hot electrons for each of the three regions. The hot electrons were generated in the two hotter regions and dispersed to the coldest region to excite the near neutral atoms, which then produced the Kα emissions. The hot electrons also broadened the ionization distribution for the two hotter regions. Though the X-ray source intensity ~1017 W/cm2 (Faenov et al., Reference Faenov, Colgan, Hansen, Zhidkov, Pikuz, Nishiuchi, Pikuz, Skobelev, Abdallah, Sakaki, Sagisaka, Pirozhkov, Ogura, Fukuda, Kanasaki, Hasegawa, Nishikino, Kando, Watanabe, Kawachi, Masuda, Hosokai, Kodama and Kondo2015a ) was enough to create KK hollow ions in aluminum, it was insufficient to create KK hollow ions with iron. The KL spectral region does have a reasonable presence though it is difficult to distinguish between the KL transitions and the singly excited state transitions. By considering the KK and KL emission regions it was determined that there were no sufficient effects from X-ray pumping on the spectra of hollow ions of iron in laser-produced plasma experimental conditions. At the same time it was shown that influence of hot electrons for formation of K-shell spectra of iron is rather large.
ACKNOWLEDGEMENTS
This work was partly supported by Russian Foundation for Basics Research (grant # 14-22-02089) and by RAS Presidium Program for basic research #11. This work was also supported by NNSA under DOE Cooperative Agreement DE-NA0001987 and grant DE-NA0003047.
