Matter and Radiation at Extremes, Volume. 9, Issue 4, 047801(2024)

Measurement of 2p-3d absorption in a hot molybdenum plasma

Gang Xiong1,2, Bo Qing2, Zhiyu Zhang2, Longfei Jing2, Yang Zhao2, Minxi Wei2, Yimeng Yang2, Lifei Hou2, Chengwu Huang2, Tuo Zhu2, Tianming Song2, Min Lv2, Yan Zhao2, Yuxue Zhang2, Guohong Yang2, Zeqing Wu3, Jun Yan3, Yaming Zou1, Jiyan Zhang2, and Jiamin Yang2
Author Affiliations
  • 1Shanghai EBIT Laboratory, and Key Laboratory of Nuclear Physics and Ion-Beam Application (MOE), Institute of Modern Physics, Fudan University, Shanghai 200433, China
  • 2Laser Fusion Research Center, China Academy of Engineering Physics, Mianyang 621900, China
  • 3Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
  • show less

    We present measurements of the 2p-3d transition opacity of a hot molybdenum–scandium sample with nearly half-vacant molybdenum M-shell configurations. A plastic-tamped molybdenum–scandium foil sample is radiatively heated to high temperature in a compact D-shaped gold Hohlraum driven by ∼30 kJ laser energy at the SG-100 kJ laser facility. X rays transmitted through the molybdenum and scandium plasmas are diffracted by crystals and finally recorded by image plates. The electron temperatures in the sample in particular spatial and temporal zones are determined by the K-shell absorption of the scandium plasma. A combination of the IRAD3D view factor code and the MULTI hydrodynamic code is used to simulate the spatial distribution and temporal behavior of the sample temperature and density. The inferred temperature in the molybdenum plasma reaches a average of 138 ± 11 eV. A detailed configuration-accounting calculation of the n = 2–3 transition absorption of the molybdenum plasma is compared with experimental measurements and quite good agreement is found. The present measurements provide an opportunity to test opacity models for complicated M-shell configurations.

    I. INTRODUCTION

    Radiative opacity of hot dense plasmas is of great interest in astrophysics, inertial confinement fusion, and other areas of high-energy-density physics.1–4 To test the models used in opacity calculations, it is necessary to conduct radiative opacity experiments under specific plasma conditions, since these complicated opacity models inevitably involve various approximations. In recent decades, theoretical models have been validated and greatly improved with the development of more sophisticated experimental methods for measuring opacity.5–7

    There has been a particular focus on L-shell absorption structures corresponding to 2p-3d transitions in a number of experiments with medium-Z plasmas. The experimentally measured L-shell absorption spectrum of an open-M-shell germanium plasma was compared with the result of a detailed configuration-accounting (DCA) calculation involving an approximate treatment of term width.8 Subsequently, several theoretical calculations9–14 and experimental measurements15–24 of L-shell absorption have been reported on elements such as iron, nickel, copper, germanium, bromine, and niobium, with typical sample temperatures below 80 eV. Recently, opacity measurements have been performed at higher plasma temperatures above 150 eV at more powerful facilities, such as the Z-pinch facility25–28 and the National Ignition Facility (NIF).29,30 However, these experiments at higher temperatures have all concerned iron25–27,29,30 and its neighboring elements.28 Opacity measurements with higher-Z element are of great importance, owing to the complexities introduced into the associated opacity models by the need to treat, for example, bound-electron wave functions, orbital relaxation, and electron–electron interaction. The spin–orbit separation of 2p1/2-3d3/2 and 2p3/2-3d5/2 transition arrays is manifested in experimental spectra by a dependence on the atomic number and electron temperature,9,16 i.e., the spin–orbit separation tends to disappear as the electron temperature rises, while it becomes more significant with increasing atomic number. More importantly, a large number of L-shell transitions emerge in hot plasmas when half of the electrons in the M shell are stripped. The half-vacancy M-shell configurations produce the maximum possible number of atomic states, and the higher temperature further increases the populations of highly excited states. It is becoming more challenging for theoretical calculations to reproduce experimental absorption spectra in such complicated cases. However, there is a lack of available L-shell 2p-3d absorption data for relatively high-Z and high-temperature plasmas with half-vacancy M-shell configurations, owing to the great difficulties involved in obtaining accurate opacity measurements at high temperatures above the hundred-eV level at current laser facilities. The recently reported iron transmission measurements at a temperature of ∼150 eV driven by a total laser energy of 225 kJ at the NIF29,30 were achieved only after a great deal of effort had been expended in designing the spectrometer,31 backlighter,32 Hohlraum,33 and overall measurement procedure.34,35 The few reported opacity experiments with the higher-Z element niobium23,24 were conducted on almost full M-shell configurations, while the higher-temperature iron experiments25–30 that have been performed were on almost empty M-shell configurations.

    In the present work, we focus on 2p-3d absorption measurements of radiatively heated hot molybdenum (Mo, Z = 42) plasmas at a temperature of 138 ± 11 eV, where about half of the electrons in the M shell are stripped. In comparison with the M-shell opacity of an Mo plasma measured at the SG-II laser facility at a temperature of 67 ± 4 eV, where the M-shell of the Mo plasma is just open, with one or two vacancies,36 the present design increases the temperature and creates half-vacant M-shell configurations, which provides an opportunity to validate the more challenging theoretical model of these extremely complicated configurations. A compact D-shaped Hohlraum was specially designed for the present measurement to radiatively heat the sample to a temperature above 130 eV with a total of ∼30 kJ laser driving energy. This was the first time that we have conducted an opacity experiment on hot plasma at the SG-100 kJ laser facility,37 and we will report in detail the experimental design, the radiative driving, the experimental diagnostics, the theoretical simulations of the sample condition, and finally the Mo 2p-3d absorption spectrum. The electron temperature was experimentally diagnosed by the K-shell absorption spectrum of a tracer doped in the sample. Instead of the usual magnesium and aluminum elements used at temperatures below 100 eV,8,17,36 scandium (Sc) was used for the first time in the present measurements for its higher sensitivity to electron temperatures above 130 eV. The X-ray radiation emitted from the D-shaped Hohlraum was characterized in detail by a multi-angle radiation flux-detecting system. The one-dimensional hydrodynamic code MULTI developed by Ramis et al.,38,39 in combination with the three-dimensional view-factor code IRAD3D,40 was used to estimate the temporal behavior and the spatial distributions of the temperature and density in the Mo/Sc sample. The experimentally measured absorption spectrum from n = 2–3 transitions of Mo plasma are presented and compared with the results of calculations based on the DCA model.41,42

    II. EXPERIMENTAL CONDITIONS

    A. Experimental design

    A sketch of the experimental arrangement is shown in Fig. 1. The D-shaped gold Hohlraum used here was designed to convert the high-power laser energy into a high-temperature X-ray radiation field that radiatively heated a sample placed in the Hohlraum in a uniform manner while ensuring that the sample was protected from direct impact by scattered laser light and M-band emission from the laser spots, and thus stayed in local thermodynamic equilibrium (LTE).5,17,34,35 The D-shaped design also reduced the influence of plasma filling, which would have disturbed the X-ray transmission measurements of the sample. The Hohlraum was 2.7 mm in maximum diameter and 3.6 mm in length, which was about three times larger in volume than those used in the SG-II experiment.43–45 The aim of this increase in size was to create a relatively clean Hohlraum environment under the stronger laser drive, since the accumulation of Au plasmas from the Hohlraum wall around the sample could have disturbed the absorption measurements. In fact, the increase in size was quite small in comparison with the increase in laser energy. The 32 laser beams with duration of 1 ns and laser spot size of 500 μm were injected into the Hohlraum through two laser entrance holes (LEHs) to produce X-rays to radiatively heat the sample. The laser beams were arranged in two cones on either side of the Hohlraum. Each beam delivered a nominal laser power of 0.85 TW, which means that a total energy of 27 kJ was injected into the Hohlraum. Although a 3 ns laser duration could have delivered more energy and thus further increased the electron temperature, the 1 ns duration was chosen to avoid Au plasma filling in the later period. As the opacity experiments at the SG-II laser facility used ∼2 kJ laser energy to drive a Hohlraum that had about one-third of the present Hohlraum volume V,44 the laser power densities PL/V in the present measurements were about five times higher than those used at SG-II. A rough estimate from the expression Tr4PL/V indicates that a 1.5-fold rise in radiation temperature could be expected in this experiment, which means a ∼130 eV sample temperature in the present measurements.

    Experimental setup for opacity measurements: (a) arrangement of Hohlraum, sample, backlighter, and diagnostics; (b) geometry of crystal spectrometers; (c) the view angles of FXRDs simulated by IRAD3D.

    Figure 1.Experimental setup for opacity measurements: (a) arrangement of Hohlraum, sample, backlighter, and diagnostics; (b) geometry of crystal spectrometers; (c) the view angles of FXRDs simulated by IRAD3D.

    The method of point-projection spectroscopy was used in the transmission experiments. The Mo/Sc sample was tamped by CH plastic of 1 μm thickness on both sides and placed at the center of the D-shaped Hohlraum. In all shots, the samples were fabricated by plasma sputtering as Mo/Sc mixtures with a nominal area density of 0.09 mg/cm2 Mo and 0.15 mg/cm2 Sc. The initial solid densities of Mo and Sc were 10.28 and 2.99 g/cm3, respectively, corresponding to atom number densities of 6.5 × 1022 and 4.0 × 1022 cm−3 for Mo and Sc elements, respectively. The other six laser beams with pulse duration of 500 ps and nominal laser spot sizes of 200 μm were focused on a planar Au target to create a bright X-ray point backlighter to be used in the absorption spectrum measurements. The backlighter was arranged in the side of the Hohlraum rather than the LEH direction to prevent the diagnostics from having a direct view of the Au plasma in the Hohlraum. The onset time of the backlighter beams was set at 1.7 ns after the heating beams shot. The sample condition was expected to undergo only minor changes during the backlighter probe time duration, thus ensuring that the transmission spectrum of the sample was measured while the latter was in a relatively stable state of temperature and density.

    The opacity is usually deduced from transmission measurements by comparing the absorption and backlighter spectra recorded by a crystal spectrometer. The geometry of the spectrometers used in the present experiment is shown in Fig. 1(b). TAM (2d = 8.78 Å) and a RbAP (2d = 26.12 Å) flat crystals were employed to measure the Sc K-shell and Mo L-shell spectra, respectively. The two spectrometers could not be used simultaneously in the same shot, owing to the limited detectable angle when the backlighter passed through the sample, and therefore the Mo and Sc spectra were measured in different shots with similar laser drives. Each of the Mo and Sc spectra were finally recorded by an image plate. A lead aperture and a lead baffle were placed in the front of the spectrometers to prevent the high-energy X ray from passing directly to the image plate. A piece of Be filter of 100 μm thickness was also mounted in front of the spectrometers and another similar piece in front of the image plate. Since an imaging plate is a time-integral detector, the time resolution of an absorption spectrum depends mainly on the duration of the backlighter beam. The 500 ps duration adopted here was intended to achieve a higher signal-to-noise ratio in the measurements of the absorption spectra. It should be noted that the actual time resolution has to be determined by the duration of the X rays generated by the laser beam. Generally speaking, a shorter backlighter pulse means a better time resolution of the measured transmission spectrum. To further improve the time resolution, the pulse width of the backlighter laser beams must be shortened in the future if the intensity of the backlighter is to be sufficient for absorption measurements. It should be possible to improve the intensity of the backlighter by optimizing the backlighter materials and using more backlighter laser beams in the future experiments.

    The geometries of the Sc and Mo spectrometers were designed to meet specific demands with regard in particular to the spectral resolution, the photon energy range, and the probe area of the sample. As shown in Fig. 1(b), we denote by D1 the distance between the backlighter and the sample, and by D2 and D3 the distances from the sample to the crystal and image plate, respectively. The distance D1 of the Sc spectrometer was shorter than that of the Mo spectrometer, but the distances D2 and D3 were longer, which resulted in the Sc spectrometer having a better spectral resolution. The photon energies of interest for the K-shell absorption spectrum of the Sc sample are those where the intensities of different spectral transitions are sensitive to the electron temperature, and they lie in the narrow range 4140–4320 eV. The Sc spectrometer was therefore designed with higher resolution to separate the transitions from Sc ions with different ionizations. By contrast, the photon energy range of interest for the Mo L-shell 2p-3d absorption spectra is much broader. Therefore, the distances D2 and D3 of the Mo spectrometer were shortened to obtain a wider photon energy range with the limited length of the crystal. The photon energy range of the Mo spectrometer was designed to cover 2200–2800 eV.

    The ratio D3/D1 represents the magnification of the probe zone of the sample through which the backlighter passes. A larger magnification means a smaller probe zone, owing to the limitation of the crystal width. The sample zones probed by the Sc and Mo spectrometers are sketched at the top left of Fig. 1(b). The shaded areas and the blank are correspond to the CH/Mo/Sc/CH portion and the pure CH portion, respectively. The blue dashed rectangles indicate the probe area through which the backlighter passes. The probe areas are 420 × 76 μm2 and 760 × 410 μm2 in the Sc and Mo absorption measurements, respectively. The X axis corresponds to the direction of the sample width and the Y axis to the X-ray spectral dispersion. The Sc spectrometer was designed with a larger magnification to ensure that only the center of the sample could be detected. The smaller detection zone also meant a smaller temperature gradient, which helped to improve the accuracy of the diagnosed sample temperature. However, it would have been quite difficult to measure the absorption and backlight spectra simultaneously in one shot with the relatively small sample size and relatively large backlighter, and therefore the backlight spectrum was measured in a separate shot. In contrast, the magnification of the Mo spectrometer was decreased to record the absorption and backlight spectra of the Mo sample simultaneously in the same shot, thereby avoiding the measurement uncertainties arising from shot-to-shot variations in sample conditions.

    Three flat-response X-ray detectors (FXRDs)46 were used to obtain the X-ray fluxes emerging from the LEHs of the Hohlraum at 16°, 20°, and 64° with respect to its axis. Each FXRD was composed of a specially designed compound filter and gold cathode X-ray diode to provide a nearly flat response in the photon energy range from 0.1 to 5.0 keV. The locations of the FXRDs are illustrated in Fig. 1(a). The exact observation angles of the FXRDs as provided by the IRAD3D view-factor code are shown in Fig. 1(c). The responses of the filter and gold cathode were absolutely calibrated at the Beijing Synchrotron Radiation Facility (BSRF) before the experiment. The time resolution and intensity measurement uncertainty of the FXRDs were 100 ps and 10%, respectively.

    B. Radiation field characteristics

    Knowledge of the characteristics of the radiation field used to heat the sample radiatively is very important in assessing the sample temperature and density status. The radiation temperature can be deduced from the X-ray flux measured by the FXRDs and then be used as the input in a subsequent hydrodynamic simulation of the heating of the sample.

    In total, three shots were launched for the transmission measurements: two for the Sc plasma and one for Mo. Shots N20087 and N20100 were intended to obtain the Sc absorption and backlighter spectra, respectively. The absorption and backlighter spectra of the Mo plasma were measured in a single shot, N20085. It was intended that the experimental conditions in all three shots were to be the same. However, the actual total laser energies injected into the Hohlraum were 27.3 and 29.5 kJ in shots N20087 and N20085, respectively. The total laser energy for the Mo spectrum measurement (shot N20085) was slightly higher than that in the Sc case (shot N20087) by about 8%. From the relationship Tr4PL between radiation temperature Tr and laser power PL, the temperature of the Mo plasma (shot N20085) was estimated to be higher than that of the Sc plasma (shot N20087) by about 2%, i.e., about 3 eV for the given experimental conditions. This difference in radiation temperature was confirmed by the FXRD measurements.

    The typical radiation flux measured by the FXRDs in shot N20087 is shown in Fig. 2. The flux increases rapidly during 0–1 ns, which corresponds to the laser heating time. When the laser is turned off, the radiation flux decreases slowly and approaches zero at 3 ns. A extra peak can be observed around 1.7 ns, as shown in the shaded area in Fig. 2(a). This extra peak was contributed by the backlighter, since the FXRDs had a large field of view and recorded the X-ray flux from the Hohlraum and backlighter simultaneously. The differences in behavior of the extra peaks at the three different angles are due to the different observation directions of the FXRDs. The IRAD3D code used to provide the observation angle of the FXRDs, as shown in Fig. 1(c) is a three-dimensional (3D) view-factor code that incorporates the laser arrangement, the view angle of the diagnostics, and the angular radiation temperature simulation. The small circles on the sketches of the Hohlraum in Fig. 1(c) indicate the directions from which the backlighter came. As can be seen, the FXRDs at 16°, 20°, and 64° viewed respectively the side, the front, and near the back of the backlighter plane, which is consistent with the intensity of the extra peaks. The contribution of the Hohlraum to the radiation flux during the backlighter time can be simply extracted by a spline interpolation method. The measured fluxes coming respectively from the Hohlraum and backlighter are shown in Fig. 2(b). The duration of the X-ray backlighter, especially when viewed from the front (i.e., at 20°), provides reference information about the time resolution of the absorption spectrum measurement. It should be noted that the full width at half maximum (FWHM) of the radiation flux pulse of about 650 ps viewed at 20° was longer than the 500 ps laser pulse width, since the soft X rays could still be sustained for hundreds of picoseconds after the end of the laser pulse. However, the FWHM of the gold M-band X-ray emission, which corresponds to the photon energy of the backlighter in the Sc and Mo absorption measurements, was closer to the laser pulse width, since the high-temperature plasmas that emitted the M-band X rays tended to cool down immediately after the end of the laser pulse. We plan to use time-resolved recorders such as X-ray streak or framing cameras in our future measurements to replace the time-integrated image plate incorporated in the crystal spectrometer. X-ray streak and framing cameras can provide better time resolution than the current combination of ashort backlighter laser pulse and an image plate.

    Time evolution of radiation flux recorded by the three FXRDs at different view angles in shot N20087: (a) raw radiation flux; (b) contributions of Hohlraum and backlighter.

    Figure 2.Time evolution of radiation flux recorded by the three FXRDs at different view angles in shot N20087: (a) raw radiation flux; (b) contributions of Hohlraum and backlighter.

    The radiation temperature in the Hohlraum can be deduced from the radiation flux measured by the FXRD via the following equation46Tr=πFσAcosθ1/4,where σ is the Stefan–Boltzmann constant, A the area of the LEH of the Hohlraum, and θ the observation angle of the FXRD. The radiation temperatures at different angles measured by the FXRD from the laser entrance hole are shown in Fig. 3. The solid lines represent the radiation temperatures of shot N20087 for the Sc absorption measurement, while the dashed lines represent those of shot N20085 for the Mo case. The radiation temperatures exhibit an anisotropic distribution, with the peak values being shown in Fig. 3(b). The radiation temperature observed at 64° is higher than that at 16° and 20°, owing to direct viewing of the laser spot points, as simulated by IRAD3D in Fig. 1(c). Agreement is found between the two shots with regard to the peak radiation temperature, within measurement uncertainty. However, as mentioned above, the actual total laser energy of shot N20085 was higher than that of shot N20087. Consequently, the peak radiation temperature of shot N20085 was higher that of shot N20087 by about 1 eV at 16° and about 6 eV at 20° and 64°, as shown in Fig. 3. The difference between 16° and 20° may be attributed to the energy differences among the laser beams between the two shots. Combining the total laser energy and the radiation temperatures measured by the FXRD, the temperature in shot N20085 is finally estimated to be higher than that of N20087 by 3 eV.

    (a) Time evolution and (b) peak values of radiation temperature reconstructed from three FXRDs at different view angles.

    Figure 3.(a) Time evolution and (b) peak values of radiation temperature reconstructed from three FXRDs at different view angles.

    III. SAMPLE STATUS

    A. Experimental determination of sample temperature

    The electron temperature of the foil sample was diagnosed using the K-shell transmission spectrum of the Sc plasma. The raw images of the measured transmission spectra are shown in Fig. 4(a): the upper one is the backlighter spectrum (shot N20100) and the lower is the absorption spectrum (shot N20087). Several obvious absorption features are obvious around 4250 eV in the absorption image when it is compared with the backlighter one. The backlighter and absorption spectra are shown in Fig. 4(b), where the background counts have been subtracted. It should be noted that a slight count fluctuation is still observed between the two shots, even though efforts were made to keep the experimental conditions the same. Finally, the backlighter spectrum (shot N20100) was normalized to the absorption one (shot N20087) by the spectral counts in the photon energies from 4120 to 4160 eV, where no obvious bound–bound absorption arises and the transmissions of the Sc/Mo plasmas are about 0.9, as predicted by a detailed term-accounting (DTA) calculation45 [the top-right inset in Fig. 4(b)]. The transmission was calculated at a temperature of 131 eV and density of 0.006 g/cm3.

    (a) Raw images and (b) cross sections of backlighter (shot N20100) and absorption (shot N20087) spectra in the Sc transmission measurements.

    Figure 4.(a) Raw images and (b) cross sections of backlighter (shot N20100) and absorption (shot N20087) spectra in the Sc transmission measurements.

    The Sc transmission spectrum obtained from the absorption and backlighter spectra is shown in Fig. 5 by the black solid line. As can be seen, the absorption is mainly contributed by C-, B-, Be-, and Li-like ions. It should be noted that the photon energy in the transmission spectrum was first determined using a ray-tracing method,47 taking account of the geometry of the spectrometer and Bragg’s law of diffraction, and then further calibrated using a reference line method,48 since the actual position and posture of the spectrometer may have deviated slightly from their design values. The 7f5/2-3d and 6d5/2-3p transitions of nickel-like gold ions measured by May et al.49 at an electron beam ion trap facility (EBIT) were taken as the reference lines in the calibration. The DTA model under the LTE approximation was used to calculate the transmission spectrum. The best fit of the measured transmission spectrum with the calculation was found at a temperature Te = 131 eV and density ρ = 0.006 g/cm3, as shown in Fig. 5 by the red solid line, where the spectral linewidths due to instrument broadening have been included. The absorption contribution from the Mo plasma, which is almost constant in this photon energy range, has also been included in the spectrum. Overall, the experimental spectrum is reproduced well by the theoretical calculation at a temperature Te = 131 eV. The overall uncertainty in the temperature is estimated to be ±7% (i.e., Te = 131 ± 9 eV). Systematic assessment of the temperature uncertainty is complicated owing to the existence of a variety of sources of uncertainty.29,50 The main sources that contribute to the total uncertainty are the contributions from experimental errors in the transmission spectrum, nonuniformity of sample conditions, density uncertainty from the hydrodynamic simulation, and model uncertainty from the atomic data. The measurement errors include contributions from static uncertainty, shot-to-shot variations, the normalization procedure, the background count, and the spectral structures of the backlighter. The density uncertainty of the sample may also influence the inferred Te, since the transmission spectrum predicted by the model is sensitive to the density, although quite weakly. The LTE assumption is reasonable, because the sample was heated radiatively by a near-equilibrium thermal X-ray source from the D-shaped Hohlraum. The hydrodynamic simulation presented in Sec. III B also indicates that the radiation temperature Tr and sample electron temperature Te are almost the same at the probe time. Generally speaking, the impact of possible deviations from LTE is small compared with the other uncertainties mentioned above, as assessed by the pioneering work of Perry et al.5

    Measured Sc transmission spectrum obtained by comparing the absorption of the Mo/Sc mixture sample in shot N20087 and of the backlighter in shot N20010, and the spectra calculated with a DTA model at electron temperatures of 131, 122, and 140 eV and a density of 0.006 g/cm3.

    Figure 5.Measured Sc transmission spectrum obtained by comparing the absorption of the Mo/Sc mixture sample in shot N20087 and of the backlighter in shot N20010, and the spectra calculated with a DTA model at electron temperatures of 131, 122, and 140 eV and a density of 0.006 g/cm3.

    To assess the temperature sensitivity of the calculations, the theoretical spectra at temperatures Te = 122 eV and Te = 140 eV are also shown in Fig. 5 by dashed and dash-dotted lines, respectively. The density in the calculation was again set to ρ = 0.006 g/cm3. The comparison demonstrates that the ionization distribution and the line shape of the K-shell spectrum of Sc plasma are very sensitive to the temperature. When the temperature increases or decreases by ±7%, deviations to a certain degree can be observed at the absorption structures of C-like and Be-like ions.

    B. Simulations of sample temperature and density

    The spatial distribution of the radiation temperature at the sample was estimated by the IRAD3D code. The laser condition of shot N20087 and the peak radiation temperature at 1.05 ns were used in the simulation. The simulation result was then extrapolated to the center of the probe time, 1.7 ns, under the simple assumption that the radiation drive at the sample had the same temporal behavior as that measured from the LEH of the Hohlraum. The temperature of the Mo probe zone extrapolated to 1.7 ns is shown in Fig. 6. The dashed square in the center of the sample indicates the Sc probe zone. As can be seen, the radiation temperature has a symmetric spatial distribution in the sample, which is due to the symmetries of the radiation field. The simulation also reveals a small temperature gradient, especially in the X direction, which is due to the different distances from the initial radiation source. The simulated average temperature in the Sc probe zone is 131 eV, which is exactly the same as that diagnosed from the experimental Sc transmission spectrum. Owing to the larger probe zone, the simulated average temperature is 134 eV in the Mo probe zone, which is 3 eV higher than that in the Sc probe zone. As a consequence of the limited probe zone, the temperature of the Sc plasma exhibits an extraordinarily small gradient, which is favorable for temperature determination by the Sc transmission spectrum.

    Radiation temperature felt by the sample at 1.7 ns simulated by IRAD3D.

    Figure 6.Radiation temperature felt by the sample at 1.7 ns simulated by IRAD3D.

    The time behaviors of the temperature and density of the sample were simulated by the radiative hydrodynamic code MULTI. The determination of the sample condition was actually an iterative process involving the IRAD3D and MULTI simulations and the Sc K-shell transmission spectrum. The peak radiation temperature predicted by IRAD3D was used as the initial radiative drive source in the MULTI simulation. The density given by the MULTI simulation was then adopted in the calculation of the Sc K-shell transmission spectrum. The electron temperature deduced from the Sc spectrum was then fed back to the MULTI simulation again to revise the peak radiation temperature felt by the sample. The time behavior of the radiation temperature in shot N20087 observed at 16° was finally scaled to a peak value of 160 eV and used as the driving source in the MULTI code simulation. The flux limiter in the simulation was set to be 0.03.

    The final results provided by the MULTI simulation are shown in Figs. 7(a) and 7(b) for Sc and Mo, respectively. The red solid and black dash-dotted lines are the sample temperature and density, respectively. The radiation temperature used in the simulation is also shown by the dash lines. The temperatures of the Mo and Sc plasmas are almost the same after 1.3 ns, with the temperature of the Mo being slightly higher than that of the Sc by about 1 eV. The shaded areas in Fig. 7 indicate the ranges of possible temperatures and densities arising from the spatial distribution of the radiation temperature felt by the sample. Taking all the differences together, the average temperature of Mo in shot N20085 is found to be around 138 eV, which is higher than that of Sc in shot N20087 by 7 eV. The overall uncertainty is estimated to be about 8%, which includes the uncertainties in the electron temperature, the radiation temperature difference between shots N20087 and N20085, and the electron temperature difference between Sc and Mo simulated by the MULTI code. The densities of the Sc and Mo plasmas at 1.7 ns are 0.006 and 0.009 g/cm3, respectively, corresponding to an electron density of ne ∼ 1.4 × 1021 cm−3. The ion densities of Sc and Mo at 1.7 ns are 8.8 × 1019 and 5.6 × 1019 cm−3, respectively.

    Time evolution of electron temperature and density in (a) Sc and (b) Mo plasmas predicted by a hydrodynamic simulation by MULTI, along with the radiation temperature (dash-dotted line) used in the simulation.

    Figure 7.Time evolution of electron temperature and density in (a) Sc and (b) Mo plasmas predicted by a hydrodynamic simulation by MULTI, along with the radiation temperature (dash-dotted line) used in the simulation.

    IV. Mo L-SHELL TRANSMISSION

    The raw image of the Mo transmission measurement (shot N20085) is shown in Fig. 8(a). The lower half is the gold backlight spectrum and the upper half is the Mo absorption spectrum. The counts taken from the sides of the backlight and absorption spectral images are taken as the background. Two absorption features can be clearly observed in the raw image. The gold backlighter and Mo absorption spectra deduced from the raw image are shown in Fig. 8(b), where background counts have been subtracted. The photon energy of the spectrum was calibrated with the n = 4–3 transitions of the copper-like and nickel-like gold ions. The transition energies of the lines were again taken from the measurement at EBIT.49 The n = 2–3 transmission spectrum obtained by comparing the absorption and backlighter spectra is shown in Fig. 8(c). The point-backlighter method and the simultaneous measurement of absorption and backlight spectra in a single shot effectively avoid the likelihood of spatial nonuniformity in the backlighter and absorption regions, which improves the accuracy of the transmission spectrum. A sufficient number of counts were accumulated in both the absorption and backlighter spectra, thereby providing data with high signal-to-noise ratio and helping to reduce statistical uncertainty.

    Experimental Mo transmission measurement in shot N20085: (a) raw image; (b) cross sections of backlighter and absorption spectra; (c) cross section of transmission spectrum.

    Figure 8.Experimental Mo transmission measurement in shot N20085: (a) raw image; (b) cross sections of backlighter and absorption spectra; (c) cross section of transmission spectrum.

    In Fig. 8(c), two strong absorption structures at photon energies around 2450 and 2550 eV can clearly be seen in the experimental transmission spectrum, and these can be attributed to the 2p3/2-3d5/2 and 2p1/2-3d3/2 transition arrays, respectively. A relatively weak absorption structure is also observed around 2650 eV, which has been attributed to the 2s-3d transition array. The 2p-3d spin–orbit splitting is still visible at temperatures above 130 eV. However, when compared with the reported neighboring niobium plasma at a temperature of about 45 eV where the 2p-3d spin–orbit splitting is well separated,23,24 the 2p3/2-3d and 2p1/2-3d transition arrays in the present work exhibit significantly broader profiles, and the edges of the two structures nearly overlap. It is thought that the large number of transitions, the wide ionization fraction, the temperature gradient, and the instrument resolution all contribute to this broadening.

    Figure 9 shows the ionization fraction in the Mo plasma at temperatures of 132, 138, and 144 eV. The main ions are Mo XXIII to Mo XXVII, corresponding to the ground configurations of [Ne]3s23p4 to [Ne]3s23p63d2. Compared with the nearly full M-shell configurations of niobium absorption spectra at ∼45 eV,23,24 a large number of transitions arise, stay close together in photon energy, and merge into unresolved transition arrays in the present configurations.

    Charge state fractions in the Mo plasma at Te = 132, 138, and 144 eV and ρ = 0.009 g/cm3.

    Figure 9.Charge state fractions in the Mo plasma at Te = 132, 138, and 144 eV and ρ = 0.009 g/cm3.

    The contributions of each ion in the 2p-3d region at a temperature of 138 eV were calculated and are shown together with the experimental spectrum in Fig. 10. All of the transition arrays were calculated using the DCA model, with the term structures treated using an unresolved transition array (UTA) method.41,42 A fully relativistic treatment and quantum defect theory were adopted in the model to provide a more efficient calculation of the X-ray absorption spectrum with it huge number of open M-shell configurations, even with high principal quantum numbers. The maximum value of the principal quantum number n was set to n = 8 with a specific orbital angular momentum quantum number l ≤ 5. The linewidth contributions from the natural lifetime, electron impact, Doppler broadening, and term broadening were taken into account. An artificial optical depth was used in the calculated transmission spectra in Fig. 10, just to make the comparison more illustrative.

    Contributions of Mo ions to the total transmission spectrum. An artificial optical depth was used in the calculation to enable a illustrative comparison with the experimental spectrum.

    Figure 10.Contributions of Mo ions to the total transmission spectrum. An artificial optical depth was used in the calculation to enable a illustrative comparison with the experimental spectrum.

    As shown in Fig. 10, each of the experimental structures contains the contributions of the Mo XXII to Mo XXVIII ions. The absorption structures gradually shift to higher energy region with the increase of the ionization state. The 2p-3d transition arrays of the neighboring ions are still strongly overlapped in this region, since the spectral lines are wider than their separations. Besides, in both the 2p3/2-3d5/2 and 2p1/2-3d3/2 transitions, the absorption structure is contributed by two groups, especially in Mo XXIII and Mo XXIV, where splitting of these groups can be observed. The two groups are transitions from ground and excited configurations, respectively. To illustrate the contribution of the excited configurations clearly, the calculated transmission spectra of Mo XXIII and Mo XXV ions are shown together in Fig. 11 with another calculation at a maximum principal quantum number n = 3. As shown in Fig. 11, the first group labeled with 2p3/2-3d5/2 and 2p1/2-3d3/2 indicates that the absorption is contributed mainly by the transitions from the base states, for example, [Ne]3l10-[He]2s22p73l11 for Mo XXIII ion, and the other group labeled with 2p3/2-3d5/2(n ≥ 4) and 2p1/2-3d1/2(n ≥ 4) is contributed mainly by the transitions from excited states, for example [Ne]3l9nl′(n ≥ 4)-[He]2s22p73l10nl′ (n ≥ 4) for Mo XXIII ion. As can be seen, the transitions from the excited configurations lead to significant broadening of the spectrum. It is also found that the energy of the transitions 2p3/2-3d5/2 (n ≥ 4) of Mo XXIII around 2417 eV is close to the energy of the transitions 2p3/2-3d5/2 of the neighboring Mo XXIV ion. This feature of the transition energy can also be found in other neighboring ions and for the 2p1/2-3d3/2 absorption structures. Thus, the transitions from the excited configurations further increase the spectral overlap between different ions. The contribution from excited configurations decreases for more highly ionized ions, for example, the Mo XXV ion shown in Fig. 11. This is due to the fact that for more highly ionized ions, a higher temperature is needed to excite electrons to higher states. The overlaps and the contribution from excited configurations eventually lead to a wider absorption structure, which makes it hard to clearly separate the contributions of specific ions from the measurement. It should be possible to experimentally identify the transmission lines among the strong absorption structures, such as the Mo XXIV and Mo XXVI ions at the temperature considered here, if the resolution of the spectrometer could be further improved by for example, increasing the detection distance [D2 or D3 in Fig. 1(b)] or using a bent crystal, and we plan to investigate this in our future work.

    Transmissions of Mo ions calculated by the DCA model with different configurations.

    Figure 11.Transmissions of Mo ions calculated by the DCA model with different configurations.

    The overall absorption calculated by the DCA model is shown in Fig. 12 with solid lines and compared with the experimental measurements. Taking into account the temperature gradient of the sample, the Mo L-shell transmission spectra were calculated at electron temperatures of 132, 138, and 144 eV with a density of 0.009 g/cm3 and a nominal area density of 0.09 mg/cm2. The transmission calculated at the average temperature of 138 eV is shown by the thicker solid line. The calculated transmission spectra were convolved with an extra instrument width to adapt to the experimental spectrum. The contribution from the Sc plasma, which is almost constant in this photon energy range, was also included in the calculation. As can be seen from Fig. 12, the transmissions are similar in shape at the three electron temperatures, but the absorption structure shifts to higher energy with increasing temperature. The reason for this is that the ions are excited to higher configurations or ionized to higher ionization states as the temperature increases, which results in the shift in the photon energy.

    Comparison between Mo L-shell transmission spectra obtained from experimental measurements and from DCA model calculations at different electron temperatures.

    Figure 12.Comparison between Mo L-shell transmission spectra obtained from experimental measurements and from DCA model calculations at different electron temperatures.

    Overall, the calculation reproduces the main features of the absorption structures well. The 2p1/2-3d3/2 absorption shows good agreement between calculation and measurement, and the 2s-3p absorption is also predicted by the calculation. However, the measurement indicates weaker absorption of the 2p3/2-3d5/2 transition array. This difference is due to both the approximation involved in the DCA model and the uncertainties arising in the measurements. Accurate assessment of the uncertainty in the experimental transmission spectrum is still a challenging task.29,34,44,51 The difference cannot be attributed to temperature uncertainty, since no obvious change in the intensity ratio of 2p3/2-3d5/2 and 2p1/2-3d3/2 is observed among the Mo XXIII to Mo XXVII ions, as can be seen in Fig. 10, and the total transmissions of the 2p3/2-3d5/2 and 2p1/2-3d3/2 transitions at the three different temperatures all exhibit a similar trend of variation, as shown in Fig. 12. Apart from the statistical uncertainty in the measurements, there are two likely significant contributions to the uncertainty in the experimental transmission spectrum, namely, the self-emission and area density of the sample. The transmission T can be expressed asT=exp(ρlκ)=IbsI0b0,where κ and ρl are the opacity and area density of the sample, respectively, I, b, and s are the intensity of the absorption spectrum, the background count, and the self-emission of the sample, respectively, and the subscript 0 denotes the backlighter. The self-emission of an L-shell Mo plasma at electron temperatures above 130 eV may not be weak enough to be ignored in comparison with the backlighter, but it has not yet been considered in the transmission measurement. According to Eq. (2), a weaker backlighter enhances the influence of the self-emission on transmission spectrum. As shown in Fig. 8, the backlighter intensity is weaker at 2450 eV (2p3/2-3d5/2) than that at 2550 eV (2p1/2-3d3/2). As a result, the experimental transmission of the 2p3/2-3d5/2 transition will be reduced more compared with the 2p1/2-3d3/2 transition if the self-emission is removed. The self-emission can be ignored if the intensity of the backlighter is overwhelming that of the self-emission. In the future, it should be possible to further improve the spectral intensity in the photon energy ranges of both the 2p3/2-3d5/2 and 2p1/2-3d3/2 transitions by using a compound material (e.g., a tungsten–gold mixture) or a capsule implosion to provide a smooth X-ray continuum source.32 Uncertainties in the sample area density will arise during the sputtering process in the fabrication of the molybdenum–scandium mixture nanometer foil. Equation (2) shows that where the opacity κ is larger, the effect of the area density ρl on transmission is greater. Therefore, the 2p3/2-3d5/2 transition will be more susceptible to uncertainties in the sample area density. The overall uncertainty in the experimental transmission estimated from the main contributions of the self-emission, the area density and the statistic uncertainty, is sketched in Fig. 12. The experimental uncertainty ΔT varies from ±0.1 to ±0.15 when the transmission T changes from nearly unity to 0.4. Taking into account the uncertainties above, the calculation shows reasonable agreement with the measurement.

    V. CONCLUSION

    The experiments described here were devoted to demonstrating opacity measurements of a moderate-Z Mo plasma through radiative heating to higher electron temperature at the SG-100 kJ laser facility. A compact D-shaped Hohlraum was designed to increase the temperature of the Mo sample with limited laser driving energy. K-shell absorption of the element Sc, instead of the Mg and Al elements usually used, was used to experimentally determine the electron temperature in specific temporal and spatial regions. The spatial distribution and temporal behavior of sample conditions were simulated with a combination of the IRAD3D view factor code and the MULTI hydrodynamic code. The Mo sample was finally heated to an average temperature of 138 ± 11 eV by a total of 29.5 kJ laser energy. The L-shell transmission spectrum of the Mo plasma was experimentally measured through a point-projection spectroscopic method, with the backlighter at the side of the Hohlraum. Both 2p-3d and 2s-3p absorption structures were observed in the Mo transmission spectrum. The 2p3/2-3d5/2 and the 2p1/2-3d3/2 spin–orbit separation was clearly visible at the temperature examined. The DCA opacity model was compared with the transmission spectrum of the Mo plasmas, and it was found that the experimental spectrum was reproduced well by the theoretical model, subject to measurement uncertainties.

    Efforts toward improving the accuracy of measurements to be investigated in the future will include experimental determination of sample density and specific measurements of individual backgrounds and self-emission. It will be intriguing to measure the 2p3/2-3d5/2 and 2p1/2-3d3/2 structures with higher spectral resolution and thereby provide more detailed insight into half-vacant M-shell configurations. It is planned to improve resolution by for example, increasing the detection distance or using bent geometry crystals. To test the opacity models in a more extreme environment, it will be necessary to further increase the electron temperature, which could be done, for example, by reducing the size of the D-shaped Hohlraum, adopting a foam-baffled Hohlraum, and utilizing the total laser energy more efficiently. The intensity of the backlighter should also be enhanced simultaneously to suppress the self-emission from the sample at higher electron temperatures.

    ACKNOWLEDGMENTS

    Acknowledgment. This work was supported by the National Nature Science Foundation of China (Grant Nos. 12335015, 12375238, 12374261, 11734013, and 11704350), the Presidential Foundation of the China Academy of Engineering Physics (Grant No. YZJJLX2017010), the CAEP Foundation (Grant No. CX2019023), the Science Challenge Project (Grant Nos. TZ2018001 and TZ2018005) and the National Key R&D Program of China (Grant No. 2017YFA0403200). The authors would like to thank the laser operation and target fabrication staff for their hard work.

    [48] P.Fan, Z.Zhang. Dispersion curve and wavelength determination in flat-crystal X-ray spectrograph. Chin. J. Laser, 18, 88(1991).

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    Gang Xiong, Bo Qing, Zhiyu Zhang, Longfei Jing, Yang Zhao, Minxi Wei, Yimeng Yang, Lifei Hou, Chengwu Huang, Tuo Zhu, Tianming Song, Min Lv, Yan Zhao, Yuxue Zhang, Guohong Yang, Zeqing Wu, Jun Yan, Yaming Zou, Jiyan Zhang, Jiamin Yang. Measurement of 2p-3d absorption in a hot molybdenum plasma[J]. Matter and Radiation at Extremes, 2024, 9(4): 047801

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    Paper Information

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    Received: Aug. 24, 2023

    Accepted: Apr. 18, 2024

    Published Online: Aug. 13, 2024

    The Author Email:

    DOI:10.1063/5.0172662

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