Calibrated simulations of $Z$ opacity experiments that reproduce the experimentally measured plasma conditions

Recently, frequency-resolved iron opacity measurements at electron temperatures of 170–200 eV and electron densities of $(0.7–4.0)\times {10}^{22}{\mathrm{cm}}^{-3}$ revealed a $30–400\%$ disagreement with the calculated opacities [J. E. Bailey et al., Nature (London) 517, 56 (2015)]. The discrepancies have a high impact on astrophysics, atomic physics, and high-energy density physics, and it is important to verify our understanding of the experimental platform with simulations. Reliable simulations are challenging because the temporal and spatial evolution of the source radiation and of the sample plasma are both complex and incompletely diagnosed. In this article, we describe simulations that reproduce the measured temperature and density in recent iron opacity experiments performed at the Sandia National Laboratories $Z$ facility. The time-dependent spectral irradiance at the sample is estimated using the measured time- and space-dependent source radiation distribution, in situ source-to-sample distance measurements, and a three-dimensional (3D) view-factor code. The inferred spectral irradiance is used to drive 1D sample radiation hydrodynamics simulations. The images recorded by slit-imaged space-resolved spectrometers are modeled by solving radiation transport of the source radiation through the sample. We find that the same drive radiation time history successfully reproduces the measured plasma conditions for eight different opacity experiments. These results provide a quantitative physical explanation for the observed dependence of both temperature and density on the sample configuration. Simulated spectral images for the experiments without the FeMg sample show quantitative agreement with the measured spectral images. The agreement in spectral profile, spatial profile, and brightness provides further confidence in our understanding of the backlight-radiation time history and image formation. These simulations bridge the static-uniform picture of the data interpretation and the dynamic-gradient reality of the experiments, and they will allow us to quantitatively assess the impact of effects neglected in the data interpretation.

Figure 1

(a) The experimental configuration employs a half-moon target and spectrometers fielded at $0^{\circ }$ and $\pm 9^{\circ }$ with respect to the $z$ axis. This experimental geometry dictates the backlight radiation projected through different parts of the target and allows measurements of the FeMg-plus-tamper-attenuated and tamper-only-attenuated spectra in a single experiment. (b) Cross sections of the various half-moon target configurations, their labels, and the typical conditions. The tamper mass above the sample controls the sample conditions. Images are not to scale.

Figure 2

Gated pinhole images of the pre-refurbished ZPDH radiation measured for an experiment without the aperture. The images are azimuthally smoothed. The gate time, $t$, is indicated in ns with respect to the backlight radiation peak, and the frame numbers are shown in the upper right corner of each image.

Figure 3

Measured FeMg-attenuated, $I_{\nu }^{+9^{\circ }}$, and -unattenuated, $I_{\nu }^{-9^{\circ }}$, spectra from a single experiment. Mg lines are analyzed to characterize the sample $T_e$ and $n_e$, and FeMg transmission at those conditions can be inferred by $T_{\nu }^{\text{FeMg}}\approx I_{\nu }^{+9^{\circ }}/I_{\nu }^{-9^{\circ }}$.

Figure 4

The blue and red curves are the radiation power time history measured on the prerefurbished $Z$ facility with x-ray diodes and the backlight-radiance time history simulated in Sec. 5a, respectively. Dashed lines indicate the timing of each pinhole image. The blue and red dashed lines correspond to the frames closest to the axial-power peak and to the backlight-radiance peak, respectively.

Figure 5

visrad defines emitting components (i.e., ZPDH) and the sample location in a 3D space. visrad solves the 3D view factors between the sample and each emitting component and computes the heating radiation time history on the sample.

Figure 6

(a) Comparison of a simulated heating spectral irradiance at $t=0$ ns with a scaled Planckian distribution. (b) Simulated heating spectral irradiance at $t=-2,-1$, 0, +1, and +2 ns for the Thick CH sample (i.e., $h=1.4$ mm). (c) Heating radiance time histories computed at $h=1.4$, 1.5, and 2.2 mm. Gray solid and dashed lines are the scaled backlight radiance and the scaled axial power, respectively.

Figure 7

(a) $T_e$ and (b) $n_e$ axial profiles simulated for the Thick CH case at $t=-2,-1$, 0, $+1$, and $+2$ ns with respect to the backlight-radiation peak. The dashed-rectangle range is blown up in the upper right corner to better show the FeMg sample region indicated by black lines.

Figure 8

Simulated $T_e$ and $n_e$ temporal profiles at the center of the sample for the Thick CH (green), CH+Be (blue), and Thin CH (red). The different tamper configurations produce different sample hydrodynamics.

Figure 9

Spectra extracted from the simulated half-moon data for $\pm 9^{\circ }$ spectrometers for (a) Thin CH, (b) CH+Be, and (c) Thick CH tamper configurations. Spectra extracted at the bright spots on the image data (Fig. 13) result in FeMg-attenuated (solid) and unattenuated (dashed) spectra. Strong line features at $\lambda <9.5\AA$ are Mg $K$-shell lines such as He-$\alpha ,\beta ,\gamma$ and Ly-$\alpha ,\beta$ lines, while lines at $\lambda >9.5\AA$ are absorption features due to Fe $L$-shell lines.

Figure 10

helios hydrodynamic simulations are performed for existing data with various target configurations. There are several Fe thicknesses for each tamper configuration. (a) $T_e$ and (b) $n_e$ values inferred from the simulated spectra agree well with the conditions inferred from the measured spectra.

Figure 11

Simulated spectral image for the Thin CH tamper-only case at (a) $t=-2\mathrm{ns}$ and (b) $t=1\mathrm{ns}$, and (c) time-integrated. Part (d) is the measured spectral image averaged over five datasets. The backlighter spectral images simulated (c) and measured (d) at $0^{\circ }$ show very good agreement.

Figure 12

Comparison in (a) spectral and (b) spatial profiles extracted from the simulated (red) and measured (black) image data. The spectral profiles are extracted at the brightest spot over 0.3 mm. The spatial profiles are extracted at $9\AA$ over $0.1\AA$. Gray dashed curves indicate experiment-to-experiment absolute variations, which includes variations in backlit brightness, uncalibrated crystal reflectivity, x-ray film sensitivity, and slit width over five data sets in two years. Black error bars show the standard deviations in the relative shapes. The reduced ${\chi }^2$ are computed based on the relative shapes.

Figure 13

(a) Schematics to illustrate the lateral shift in the apparent backlight bright spot on the $\pm 9^{\circ }$ image data. Images are not to scale. Parts (b) and (c) illustrate the simulated Thick CH half-moon data at $-9^{\circ }$ and $+9^{\circ }$, respectively. Green solid and dashed lines indicate the location of the backlight spatial intensity maximum (i.e., black filled circles in the schematics). $x=0$ is defined to correspond to the FeMg boundary. The images in (b) and (c) are restricted to the 7–$9.5\AA$ range for clarity.

Figure 14

Spectra taken at $x^{\prime \prime }=0$ (i.e., ZPDH stagnation point) from a simulated time-resolved spectral image for Thin CH tamper-only data at $t=-2,-1$, 0, 1, and 2 ns.

Figure 15

(a) Radiation contributed from a point on an emission surface depends on the view factor $\mathrm{\Omega }\left(x,y\right)cos\theta \left(x,y\right)$. (b) Schematic illustrating self-emission and backlighter radiation contributing to the infinitesimally small surface, $s$, on the detector.

Figure 16

(a) Schematic illustrating how each spatial point on the film observes radiation through the aperture and slits. Each point in space observes radiation averaged over $0.01\times 0.1{\text{mm}}^2$ centered at $x=x^{\prime \prime }$. (b) Measured source-to-sample distance (1.4–2.2 mm) is much smaller than sample-to-detector distance (4180 mm). Schematics are not to scale.