White Dwarf (WD) stars are the most common stellar remnant in the universe. WDs usually have a hydrogen or helium atmosphere, and helium WD (called DB) spectra can be used to solve outstanding problems in stellar and galactic evolution. DB origins, which are still a mystery, must be known to solve these problems. DB masses are crucial for discriminating between different proposed DB evolutionary hypotheses. Current DB mass determination methods deliver conflicting results. The spectroscopic mass determination method relies on line broadening models that have not been validated at DB atmosphere conditions. We performed helium benchmark experiments using the White Dwarf Photosphere Experiment (WDPE) platform at Sandia National Laboratories' Z-machine that aims to study He line broadening at DB conditions. Using hydrogen/helium mixture plasmas allows investigating the importance of He Stark and van der Waals broadening simultaneously. Accurate experimental data reduction methods are essential to test these line-broadening theories. In this paper, we present data calibration methods for these benchmark He line shape experiments. We give a detailed account of data processing, spectral power calibrations, and instrument broadening measurements. Uncertainties for each data calibration step are also derived. We demonstrate that our experiments meet all benchmark experiment accuracy requirements: WDPE wavelength uncertainties are <1 Å, spectral powers can be determined to within 15%, densities are accurate at the 20% level, and instrumental broadening can be measured with 20% accuracy. Fulfilling these stringent requirements enables WDPE experimental data to provide physically meaningful conclusions about line broadening at DB conditions.

White Dwarf (WD) stars are the end point of stellar evolution for nearly 98% of all stars including our sun.1 About 80% of all known WDs have almost pure hydrogen atmospheres and are labeled as DAs within the WD classification system.2 The remaining 20% of these stellar objects2 are helium-atmosphere WDs, and they are classified as DBs. Their origins are not well understood. Since their discovery,3,4 many WDs have been found, and significant progress in understanding the DA evolutionary origin has been made.1,5–10 However, stellar evolution models based on our understanding of DA evolution do not predict DB formation. Additional mechanisms must be included to explain DB origins. Many different DB evolutionary mechanisms have been suggested over the years.11–17 Each DB evolutionary hypothesis predicts a unique DB mass distribution. Determining DB masses is thus crucial for understanding the evolutionary origins of these objects.

DB evolutionary origins are directly tied to many unsolved problems in stellar and galactic astronomy. In stellar astronomy, DBs can be used to study the mass ejected from a star's surface before the WD evolutionary stage. This process is poorly constrained,18,19 and accurate DB masses would enable us to study the amount of ejected mass. It is currently believed that details of this mass ejection process determine whether a star develops into a DA or DB. The chemical makeup of the ejected stellar mass also contributes to galactic chemical evolution.20,21 Directly observing the ejected stellar mass is currently not possible. However, detailed DB chemical composition studies allow us to draw important conclusions about the chemical makeup of the mass ejected prior to the WD evolutionary stage. Such studies require reliable DB masses. Accurate DB mass measurements are therefore needed to address outstanding galactic and stellar evolution questions.

A common DB mass determination technique is called the spectroscopic method, which uses observed line shapes to derive a stellar mass.22 Accurate helium line shape calculations are critical to this method. Existing He theoretical models have unfortunately never been tested at the needed parameters. Furthermore, masses derived from the spectroscopic method do not agree with those of other techniques.17,23 Helium line shape models have been identified as a potential source of weakness for the spectroscopic method due to the complex physics involved in their calculation.16,17,24

van der Waals and Stark broadening dominate He line shapes at DB atmospheric conditions. Since the associated models have never been tested at DB atmospheric parameters, benchmark experimental validation data are needed. The relative importance of each mechanism depends on the stellar atmospheric temperature. At lower temperatures (T <16 000 K), van der Waals broadening is thought to dominate Stark broadening due to a lower plasma ionization fraction and a higher neutral density. An unexpected increase in spectroscopic DB masses is observed at these lower temperatures (e.g., Fig. 1 in Ref. 24). It is currently believed that this mass increase is an artifact of an incomplete van der Waals broadening model.16,17,24 No benchmark experiments are currently available to test van der Waals broadening models.56,57 Stark broadening experiments are only available at ne higher and lower than those found in a DB atmosphere (see Fig. 1). Missing benchmark data therefore limit our progress in understanding DB evolution.

FIG. 1.

Comparison of previous experiments to theory for the 2s3p He I 5015 transition (upper panel) and the 2p3d 5875 Å transition (lower panel). ne ranges applicable to DB atmospheres are given in gray. Stark broadening theories25,26 are given as dashed lines and experiments27–32 are given as individual data points. A plot comparing van der Waals theory and experiment cannot be made at this time since no experimental validation data exist.

FIG. 1.

Comparison of previous experiments to theory for the 2s3p He I 5015 transition (upper panel) and the 2p3d 5875 Å transition (lower panel). ne ranges applicable to DB atmospheres are given in gray. Stark broadening theories25,26 are given as dashed lines and experiments27–32 are given as individual data points. A plot comparing van der Waals theory and experiment cannot be made at this time since no experimental validation data exist.

Close modal

Hydrogen line shape experiments that satisfy challenging benchmark requirements were developed at Sandia National Laboratories' Z-machine using the White Dwarf Photosphere Experiment (WDPE) platform.33–38 These experiments were aimed at DAs. The hydrogen gas cell was heated by z-pinch x-rays to temperatures and densities found in a DA atmosphere.39 The experimental platform exhibits much higher uniformity compared to a DA atmosphere and due to continuous heating, ne increases as a function of experiment time. We measured hydrogen emission, absorption, and backlight spectra simultaneously in a single experiment. These measurements were then combined to provide hydrogen Balmer line shapes as a function of ne. We controlled the plasma temperature and ne through the initial hydrogen gas fill pressure and distance of our line of sight (LOS) from the plasma heating source. This platform provides a unique environment to study the dependence of hydrogen line shapes on temperatures and densities encountered within a DA atmosphere. Such detailed studies cannot be performed using DA spectra since those data are integrated over significant temperature and ne gradients in the atmosphere.

The above-described platform developed for benchmark hydrogen line shape experiments can be extended to test van der Waals and Stark broadening models and thereby shed light on DB origins and associated astrophysical problems. For these benchmark experiments, a hydrogen–helium gas mixture is used. At our experimental conditions, hydrogen atoms are mostly ionized and donate electrons to the plasma, while helium stays largely neutral. The relative importance of van der Waals broadening (caused by neutral helium) and Stark broadening (caused by electrons and protons from ionized hydrogen) can be controlled by altering the hydrogen to helium concentration in the experiment.

Data calibration is the remaining challenge for performing He benchmark experiments. This requires accurate wavelength and spectral power calibrations and reliable electron and neutral density measurements. Furthermore, the instrument broadening should be small compared to the measured line widths. Two important helium lines are 2s3p 5015 and 2p3d 5875 Å He I, which have widths of 8.5 Å at the densities achieved in the WDPE. Consequently, the instrumental broadening should be no more than 5 Å with a 20% accuracy. To fulfill the benchmark experiment requirements, wavelength calibrations need to be accurate to within 1 Å, spectral powers must be determined at the 10%–15% level, and plasma densities should have a <20% uncertainty. The WDPE measures the 2p3d 5875 Å He I transition in both emission and absorption. This spectral line is strong,27,40 and the WDPE absorption backlighting surface is not bright enough to overcome 2p3d 5875 Å He I self-emission effects. Such effects have been identified as the potentially significant source of uncertainty in experimental studies for this line.27,40–42 We measure simultaneous emission and absorption spectra to correct for this effect. Accurate cross calibration between emission and absorption data is therefore required. We also recently switched our detector from film33,43 to CCD due to increased sensitivity, better linearity, and lower noise of digital systems. Streak tube electron optics effects cause distortions in the experimental data that also have to be accounted for. Developing experimental and data reduction methods that satisfy all these criteria is an essential step for benchmarking DB line shapes and resolving associated astrophysical problems.

In this paper, we report experimental platform and data calibration updates to the WDPE that enable it to deliver benchmark He linewidth and line shape data. Updates to the WDPE platform and data collection methods are given in Sec. II. Our distortion correction, time, wavelength, and spectral power calibration techniques are discussed in Sec. III. Benchmark experiments also require an accurate assessment of errors associated each data processing step. This discussion is presented in Sec. IV. At the end of our data calibration process, we are left with benchmark experimental data that allows us to validate van der Waals and Stark broadening models at DB atmosphere conditions. We conclude this paper with final remarks in Sec. V.

We perform He atomic model validation experiments using the WDPE at the Sandia National Laboratories' Z-machine. The Z-machine is currently the most energetic pulsed x-ray source on earth. It is a pulsed power driver that converts 26 MA current into an x-ray drive using a Z-pinch dynamic hohlraum.39,44–47

The Z-machine x-rays enable the reproduction of a DB atmosphere in the 120 mm long WDPE gas cell using a hydrogen–helium mix plasma (see Fig. 2). To uniformly heat such a large plasma, the WDPE gas cell is placed 324 mm from the Z-machine x-ray source. The Z-pinch x-rays travel through hydrogen–helium gas and heat the gold back wall and backlighting surface. The WDPE collects two absorption (low- and high spectral resolution) and one emission (high spectral resolution) spectra of the experimental H/He plasma during a single experiment as indicated in Fig. 2. Every WDPE shot also collects in situ gas cell pressure data using an Omega PX72–1.5GV piezoresistive sensor. This pressure sensor records the final gas cell fill pressure right before the experiment is executed. Due to the short duration of our experiment ( 100 ns), this final pressure measurement is sufficient to determine the hydrogen and helium particle density within the gas cell. No significant gas expansion is expected during the experiment.34 

FIG. 2.

WDPE gas cell cross section. Major hardware components and distances are identified. The absorption LOS path is given in red, and the emission LOS path is plotted in blue.

FIG. 2.

WDPE gas cell cross section. Major hardware components and distances are identified. The absorption LOS path is given in red, and the emission LOS path is plotted in blue.

Close modal

The gold back wall is the dominant heating element in the WDPE platform and re-radiates the imparted energy as a Planckian with a temperature of a few eVs.34 The Planckian spectrum significantly ionizes hydrogen, but does not noticeably ionize helium. Hydrogen is therefore the major plasma electron donor while helium stays largely neutral throughout the experiment. This unique experimental environment enables studying Stark and van der Waals broadening simultaneously in a single experiment. We can also adjust the relative importance of van der Waals and Stark broadening in our experimental spectra by adjusting the hydrogen to helium experimental gas concentration. For example, neutrals influence experimental spectra more in a 50:50 H/He gas mixture experiment than in a 75:25 H/He gas mixture experiment.

The absorption LOS (red in Fig. 2) traverses through the buffer (standoff needed to separate collection optics from hot plasma in gas cell) enters the gas cell parallel to gold wall and points at the backlighting surface. This surface is heated to similar temperatures as the gold back wall and thus provides the backlight radiation needed for an absorption measurement. WDPE absorption spectra are collected in low- and high-spectral resolution. The low-resolution absorption spectra contain Hβ, the plasma Te and ne diagnostic,35 the 2s3p He I 5015 and 2p3d 5875 Å He I line shapes. This spectral range is obtained using a McPherson 2061 1-m f/7 Czerny–Turner spectrometer with a 150 g/mm (grooves per mm) grating and an EG&G model L-CA-24 streak camera equipped with an MCP intensifier. This system records the dispersed light on a Kodak T-MAX 400 film and provides a spectral resolution (λ/Δλ) of 500 in the wavelength region of interest (∼5000 Å). The high-resolution absorption spectra contain 2p3d 5875 Å He I data. These spectra are collected using a McPherson 207 0.67-m f/4.7 Czerny–Turner spectrometer with a 300 g/mm grating and a Sydor Instruments ROSS 5100 streak camera containing a Spectral Instrument Series 800 TE cooled CCD camera. This CCD system configuration produces λ/Δλ 1000 at 5875 Å. Panels (a) and (b) in Fig. 3 show low- and high-resolution WDPE absorption data from shot z3195. The data from this experiment are used in this paper as an example for all WDPE results.

FIG. 3.

(a) Low-resolution film absorption data captured on shot z3195. (b) High-resolution CCD absorption data captured on shot z3195. (c) High-resolution CCD emission data captured on shot z3195. Major spectral features as well as timing and wavelength fiducials are identified in each panel.

FIG. 3.

(a) Low-resolution film absorption data captured on shot z3195. (b) High-resolution CCD absorption data captured on shot z3195. (c) High-resolution CCD emission data captured on shot z3195. Major spectral features as well as timing and wavelength fiducials are identified in each panel.

Close modal

Timing combs, timing impulses, and laser fiducials are also identified in Fig. 3. The timing features are used to time calibrate the streaked film and CCD data. Film laser fiducials enable derivation of a wavelength dispersion for those data. Laser fiducials for the CCD data shown in Fig. 3 are captured before the experimental data on a different CCD exposure. Procedures and uncertainties for CCD time and wavelength calibration procedures are discussed in Secs. III and IV.

Emission from the cell interior is collected with a LOS that avoids any hot components that could compete with the signal (blue in Fig. 2). The emission LOS starts in the buffer opposite the absorption LOS and runs parallel to the gold wall. It is not pointed at a hot surface, and thus only plasma self-emission is recorded. Panel (c) in Fig. 3 shows the emission data for shot z3195. It contains 2p3d 5875 Å He I data. The same spectrometer, streak camera, and CCD components as in the high-resolution absorption data [panel (b) in Fig. 3] are used to collect the emission spectra. These emission spectra are used to self-emission correct the absorption data (see Sec. I). This process is further described in Sec. III.

WDPE CCD are processed in three steps. In the first step, basic corrections are applied to the CCD data. Accounting for geometric distortions caused by the streak camera is the initial basic corrections step. Next, we derive a timing and wavelength correction for the CCD data. Finally, photon arrival delays introduced by the fiber used to transport the data from the WDPE gas cell to the spectrometers are calculated and applied to the data.48 

In the second WDPE CCD processing step, we convert the emission and absorption CCD data to spectral power units. This requires deriving a spectral power calibration factor as well as correcting for plasma volume effects so that the emission spectra can be used to correct the absorption spectra for self-emission effects. The self-emission corrected absorption spectra can then be converted to absolute intensity. In the third data processing step, we derive instrumental broadening for the absorption and emission CCD systems. These values are determined using laser line fits. We discuss the basic corrections in Sec. III A. Section III B details spectral power and intensity calibration methods, and Sec. III C describes the instrumental broadening measurements.

This section describes the basic corrections applied to the CCD data. The data shown in panels (b) and (c) of Fig. 3 will be used to illustrate these methods. The CCD data are corrected for geometric distortions in the first basic data correction step. This distortion mainly arises in the streak camera electron optics. Geometric distortion affects timing and spectral dispersion accuracy, and thus geometric distortion corrections are integral to the WDPE data calibration method.

The geometric distortion correction procedure relies on a streak camera reticle image. This reticle is chrome-on-glass with a lithographically imprinted pattern. It is illuminated by a flat field lamp that produces a uniform and flat output. This lamp is built in to the streak camera system. The reticle has equally spaced slits that, in the absence of geometric distortions, produce a CCD image with equidistant points that form vertices of squares. Measurements show that the WDPE CCD systems exhibit distortions in both the vertical and horizontal CCD directions. Vertical (temporal) distortions are 600 μm, and horizontal (wavelength) distortions amount to 50 μm. Identifying the vertices in the reticle CCD image and transforming (i.e., stretching and rotating) them to a coordinate system that results in equidistant points forming vertices of squares provides a geometric distortion correction.49 The resulting coordinate transformation is then applied to the experimental data and laser fiducial images.

The distortion corrected images are used to derive timing corrections. Experimental time scale accuracy is important since the time dependent data from the film and CCD systems are combined to extract the final WDPE results. The film data provide the electron densities from Hβ line broadening35 and lower resolution He I line shape measurements. The CCD data record the higher spectral resolution He I line shapes. Combining measured timing comb locations [see panels (b) and (c) in Fig. 3] with their known 28.5 ns intervals enables derivation of the experimental data time scale. We define t = 0 ns at the data onset 50% rising edge location in lineouts taken at ∼5600 Å in each dataset (see Fig. 3).

Once geometric distortion and timing corrections are applied to the data, we use the laser fiducials to calibrate the spectral axis. An accurate wavelength dispersion is important for the WDPE since line shapes are adversely affected by inaccuracies in this measurement. Four He–Ne laser fiducial images are collected for an experiment: green (5433.651 Å), yellow (5939.315 Å), orange (6046.135 Å), and orange–red (6118.019 Å). The green laser line is from a Melles Griot 25 LGR 193 laser, and all others are produced by a Research Electro Optics 30602 line tunable laser system.

WDPE data are extracted by integrating over 20 ns intervals and processing the resulting spectra. Since the laser fiducial images are on the same time scale as the experimental data, a wavelength scale for each time integral is derived by fitting the laser line locations with a second order polynomial. The laser line locations in each 20 ns window are obtained by fitting Gaussian profiles to the fiducials and extracting their centers. Laser wavelengths are known, and the combination of these two datasets then leads to a wavelength dispersion across the CCD.

The time and wavelength calibrated CCD data are corrected for time delays caused by the fiber used to transport the signal from the WDPE gas cell to the spectrometers. These fibers are 71.2 m long with fused silica cores. Transit time values for this correction are taken from Ref. 48. Reference 50 supplies the wavelength dependent fused silica index of refraction. The timing comb signals [see panels (b) and (c) in Fig. 3] used during the temporal calibration are produced by the streak camera system and therefore not subject to transit delay effects.

Converting the WDPE absorption spectra to spectral power units is the second data calibration step. Two motivations exist for this calibration. First, the strength of the 2p3d 5875 Å He I line requires the WDPE absorption spectra to be corrected for self-emission before extracting accurate line shapes from the experimental data. Emission and absorption spectra that share the same power scale are needed for this self-emission correction process. Second, self-emission corrected absorption spectra can be converted to absolute intensity. Such data enable backlighter brightness and backlighter/back wall temperature (see Fig. 2) determination. These measurements contribute to our understanding of the experimental plasma in the gas cell.

Figure 4 shows a schematic representation of the WDPE absorption LOS and demonstrates that the recorded data arise from a single location. This location is defined by the solid angle subtended by the collection optic on the backlighter (Ω) and a single location within collection beam. Converting from raw CCD data (in scaled photon count units) to spectral power (W/Å) to spectral radiance (W/sr/cm2/Å) is therefore a straightforward process.

FIG. 4.

Schematic WDPE absorption LOS representation. We identify the solid angle subtended by the collection optic at d (Ω) and the collection beam.

FIG. 4.

Schematic WDPE absorption LOS representation. We identify the solid angle subtended by the collection optic at d (Ω) and the collection beam.

Close modal

The WDPE self-emission requirement adds complexity to the above procedure. Figure 5 depicts the WDPE emission LOS. Each plasma element (yellow disks) has its own Ω and location within the collection beam that contributes to the total recorded emission signal. Ω and the location within the collection beam for emission are not defined by a single number. Plasma volume and collection efficiency therefore vary along the emission LOS path making it difficult to convert WDPE emission data to spectral radiance. Fortunately, the self-emission correction for the absorption data only requires knowledge of the emission contribution within the absorption LOS. Performing the self-emission correction in W/Å and accounting for solid angle weighted plasma volume difference between the emission and absorption LOS allow for an accurate self-emission correction.

FIG. 5.

Schematic WDPE emission LOS representation. We identify the solid angle subtended by the collection optic at x (Ω), and locations within the collection beam.

FIG. 5.

Schematic WDPE emission LOS representation. We identify the solid angle subtended by the collection optic at x (Ω), and locations within the collection beam.

Close modal

The following spectral power calibration procedure addresses the above issues with the emission LOS: first, we derive a conversion coefficient to translate the CCD scaled photon counts to spectral power (W/Å). Deriving this conversion coefficient requires the collection of two datasets. The first is that of a NIST traceable calibrated tungsten spectral lamp. This calibration step allows us to derive the sensitivity of our streaked spectrometer setup. Due to the relative weakness of the tungsten lamp, this dataset has to be recorded using a slow (8 s) sweep duration. The second dataset consists of laser driven light source (LDLS) images that are used to measure the sensitivity difference between the fast (500 ns) sweep used to record experimental data and the slow sweep employed for the tungsten lamp calibration. The LDLS is a broadband white light source covering a spectral range from about 3200 to 7000 Å.37 Each calibration data collection setup is shown in Fig. 6. The tungsten lamp images are collected using the collection optics and the same fiber run as the experimental data. This collection optic contains the optical fiber, a lens, and a 3 mm diameter aperture. Since calibration and experimental data are collected using the same equipment, individual component calibration (e.g., fibers, fiber couplers, lenses, and streak camera) is not necessary. The tungsten lamp data and LDLS data are combined to derive the spectral power conversion coefficient. These calibration measurements are performed for both the emission and absorption CCD systems and fibers.

FIG. 6.

Block diagram of the WDPE data collection setup during the streaked spectrometer sensitivity calibration procedure (top panel) and the fast/slow streak duration calibration (bottom panel). All major hardware components are labeled. Further description of each calibration step is given in the text.

FIG. 6.

Block diagram of the WDPE data collection setup during the streaked spectrometer sensitivity calibration procedure (top panel) and the fast/slow streak duration calibration (bottom panel). All major hardware components are labeled. Further description of each calibration step is given in the text.

Close modal

Using the conversion coefficient, the emission and absorption data are converted to spectral power (W/Å) incident on the collection optic. Next, we derive a solid angle weighted plasma volume scaling factor for the emission data. We then subtract the scaled emission data from the absorption and thereby obtain self-emission corrected absorption spectra that can be used to derive the 2p3d 5875 Å He I line shape. These self-emission corrected absorption data can be converted to absolute intensity (W/sr/cm2/Å) using the solid angle and collection beam shown in Fig. 4. Having given a broad overview of our data calibration method, we will now discuss each step in more detail.

The tungsten lamp is a Gooch & Housego OL Series 455 with a 6-in. integrating sphere attached. The integrating sphere ensures that the collection optics are completely filled with a known irradiance. Figure 6 and the associated discussion show that we also collect two LDLS datasets as part of the tungsten lamp measurements. The tungsten lamp is an inherently weak source and must be collected using the 8 s streak duration on the CCD camera. The experimental data, however, are collected at 500 ns. To correct for CCD pixel sensitivity differences between the 500 ns (experimental data) and 8 s sweep (tungsten lamp) durations, we then perform two LDLS measurements (bottom panel Fig. 6) at each sweep speed. The make and model of the LDLS is Energetiq EQ-99FC.51 Dividing the two LDLS datasets allows for the derivation of a weakly wavelength and time-dependent scaling factor between the 500 ns and 8 s sweep speeds. The LDLS spectral shape is stable over several months,37 making it a good instrument for this measurement. An ND4 filter that reduces the incoming signal by a factor of 10 000 is included in the beam path for the 8 s LDLS measurement to prevent CCD saturation. The wavelength dependence of the ND4 filter is accounted for when deriving the final 500 ns/8 s scaling factor. All LDLS images are collected using almost the entire experimental fiber run including all fiber couplers. The 2.1 m fiber length difference and the removal of the collection optics between the tungsten lamp and LDLS spectra are inconsequential since LDLS data are only used to derive the 500 ns/8 s scaling factor. This scaling factor is applied to the 8 s tungsten spectrum so that it can be converted to the 500 ns experimental sweep duration.

Once sweep duration differences have been corrected, we determine the W/Å conversion factor for the raw emission and absorption CCD data. Figure 7 demonstrates how the WDPE optics interact with the tungsten lamp integration sphere. The optics and fibers used during these calibration measurements are the same as those used to collect the experimental data. Gooch & Housego provides the integrating sphere inner surface spectral radiance in W/sr/cm2/Å. To derive the W/Å conversion factor, the solid angle and collection beam area effects must be removed from the tungsten lamp calibration data. These effects are corrected for by performing ray tracing simulations similar to those described in Ref. 52. The simulated solid angle and collection beam areas for the emission and absorption LOS are nearly identical since the fibers used to collect the experimental and calibration data for these LOS share the same characteristics. They both have 200 μm diameter fibers cores and have the same optics attached.

FIG. 7.

Optical setup used for tungsten lamp measurements. We identify all relevant distances, solid angles, and collection beams. Further details are given in text.

FIG. 7.

Optical setup used for tungsten lamp measurements. We identify all relevant distances, solid angles, and collection beams. Further details are given in text.

Close modal

Next, we derive the solid angle weighted volume scaling factor between the emission and absorption LOS. Inspection of Fig. 5 shows that the recorded emission intensity can be described as an integral over the emission LOS length (xem),

(1)
Iνem=0xemεν(x)×EΩ(x)×Ω(x)dx
(1a)
=εν×VCBem,
(1b)
where

VCBem=0xemEΩ(x)×Ω(x)dx
(2)

is the solid angle weighted emission volume within the collection beam. εν(x) represents the emissivity from each plasma element in the emission LOS (yellow disks in Fig. 5) in Eq. (1a), while εν is the total plasma emissivity in Eq. (1b). Converting from Eq. (1a) to (1b) requires assuming an optically thin and uniform plasma, reasonable for WDPE emission spectra. EΩ(x) in Eqs. (1) represents the solid angle efficiency factor for Ω(x). This factor is derived from raytracing simulations52 and accounts for on- and off-axis effects as well as light acceptance differences between the collection optics lens and fiber. According to Fig. 4 and associated discussion, the absorption LOS intensity can be described as

Iνabs=(IνBL×Tνabs×EΩ(xabs)×Ω(xabs))+(εν×VCBabs),
(3)

where

VCBabs=0xabsEΩ(x)Ω(x)dx
(4)

represents a solid angle weighted plasma volume integrated over the absorption LOS distance (xabs). The first parenthesis on the right-hand side in Eq. (3) represents the absorption contribution to the recorded signal. Tνabs in this first parenthesis refers to the plasma transmission. The second parenthesis gives the self-emission contribution of the plasma to the recorded absorption LOS signal.

Comparing Eqs. (1) and (3) reveals that accounting for self-emission in the absorption LOS requires a relation between VCBabs and VCBem. The raytracing simulations indicate that these solid angle weighted LOS volumes differ slightly between the emission and absorption data. This behavior can be traced back to slight differences in LOS length (xabs = 17.08 cm, xem = 18.00 cm). A solid angle weighted volume correction factor must therefore be derived. The emissivities (εν) given in Eqs. (2) and (3) refer to the same quantity. The self-emission contamination in the absorption LOS [εν×VCBabs in Eq. (3)] can therefore be extracted from the recorded emission signal as follows:

εν×VCBabs=Iνem(VCBabsVCBem),
(5)

where VCBabs/VCBem is the solid angle weighted volume scaling factor. We derive this factor from the ray tracing simulations and apply it to the emission data. We can now perform the self-emission correction of the absorption data.

As a final step, we can convert the self-emission corrected absorption spectra from spectral power (W/Å) to radiance (W/sr/cm2/Å) units. This conversion will not affect extracted line shapes, but it will allow us to derive backlighter brightness, backlighter temperature, and back wall temperature (under certain assumptions). To perform this final correction step, we simply divide the self-emission corrected absorption spectral power by the collection beam diameter and solid angle given by our raytracing simulations.

The spectral lines measured in the WDPE are mainly broadened by plasma density effects and by the instrument. Determining the WDPE instrumental profile is therefore crucial to perform accurate model-data comparisons. Once determined, instrumental broadening can be corrected for in two ways: deconvolving the instrumental profile from the experimental data or convolving theoretical line shapes with the instrumental profile. The former method can be numerically challenging and may not result in a unique solution. We therefore adopt the latter approach.

The four laser lines collected for WDPE wavelength calibration purposes are used for determining the instrumental profile. Two achromat lenses placed at the spectrometer entrance slit ensure proper illumination of the grating during the instrumental broadening measurement. Laser line widths are negligible compared to the instrumental broadening. A Gaussian line shape fit to the laser lines is used to infer the instrumental function. Figures 8 and 9 show sample fits to laser lines used to derive the instrumental broadening for the emission and absorption CCD systems, respectively.

FIG. 8.

Gaussian fits to the emission CCD system laser lines. The data (dots) are reproduced by the fits (black lines).

FIG. 8.

Gaussian fits to the emission CCD system laser lines. The data (dots) are reproduced by the fits (black lines).

Close modal
FIG. 9.

Gaussian fits to the absorption CCD system laser lines. The data (dots) are well reproduced by the fits (black lines).

FIG. 9.

Gaussian fits to the absorption CCD system laser lines. The data (dots) are well reproduced by the fits (black lines).

Close modal

The wavelength calibration laser images are captured at the same sweep duration as the experimental data (500 ns) and cover the entire CCD in the temporal direction. These laser lines also cover a wide range in the CCD wavelength direction. The green laser (λ = 5433 Å) is located on the far left of the CCD image while the orange–red laser (λ = 6118 Å) will be on the right [compare panels (b) and (c) in Fig. 3]. Laser line fits therefore allow for the instrumental profile determination at many points on the CCD. We plot the Gaussian FWHM as a function of time in Fig. 10 for the emission and absorption CCD systems. Times relevant to the WDPE are highlighted in gray. The variations in measured instrumental broadening are <10% as a function of time, but the different lasers appear to indicate a changing instrumental profile in the horizontal CCD direction. The green laser (green line in Fig. 10) appears to be an outlier for both emission and absorption. This is most likely caused by streak camera distortion at the CCD edges that was not properly corrected for during the distortion correction procedure (see Sec. III A). Distortion effects are expected to be more severe at that location than in the CCD center. Since the green laser line is far away from the transition being captured on the CCD (|Δλ| = 442 Å, over half the data range covered by the CCD), we do not include the green laser line in the instrumental broadening determination. All other derived instrumental FWHM values agree well. Based on these data, we conclude that the WDPE instrumental profile is a Gaussian with a 5.2 Å FWHM for absorption and a 6 Å FWHM for emission. For comparison, the film systems also have a Gaussian profile with a 10 Å FWHM.35 

FIG. 10.

Measured instrumental broadening for the emission and absorption CCD systems. The 5433 Å laser line measurement is shown in green, 5939 Å is plotted in black, the 6046 Å instrumental broadening is given in orange, and the 6118 Å data are depicted in red. These colors correspond to Figs. 8 and 9. Times relevant to the WDPE are highlighted in gray.

FIG. 10.

Measured instrumental broadening for the emission and absorption CCD systems. The 5433 Å laser line measurement is shown in green, 5939 Å is plotted in black, the 6046 Å instrumental broadening is given in orange, and the 6118 Å data are depicted in red. These colors correspond to Figs. 8 and 9. Times relevant to the WDPE are highlighted in gray.

Close modal

In the final section of this paper, we discuss the uncertainties associated with the data calibration steps presented in Secs. III A and III B and the instrumental broadening determination of Sec. III C. The data calibration uncertainty has three distinct contributions: time, wavelength, and spectral power calibration. The time uncertainty has the largest effect on an electron density associated with a measured line shape.35 This is mainly because electron density is a function of time in the WDPE, and this density is not measured on the same instrument as the He I line shapes. Combining these two datasets therefore introduces timing uncertainties and thus also density uncertainties. The wavelength dispersion uncertainty mainly influences the measured line shape width and centroid wavelength. An inaccurate wavelength dispersion could lead to an artificially broadened or narrowed line shape. The spectral power uncertainty mostly affects the self-emission correction of the 2p3d 5875 Å He I absorption data and thereby the measured line shape. Instrumental broadening uncertainties can also lead to artificially broadened/narrowed lines and changes in the measured line shape.

Since the WDPE combines ne values captured on a film system with line shapes captured on CCD streak cameras, timing errors directly contribute to plasma density uncertainties. Total timing uncertainty contributions are split into two parts: one is associated with the time scale and the other arises from the time scale zero point selection. The time scale uncertainty originates from the data calibration process that converts the timing comb signals [see panels (b) and (c) in Fig. 3] into a time axis for the experimental data. The zero point uncertainty contribution stems from the zero point selection procedure in the CCD and film datasets.

The time scale uncertainty is almost entirely controlled by timing comb location accuracy. Timing comb locations are derived by fitting a Gaussian profile to each signal. The Gaussian fit center uncertainty is adopted as the comb location uncertainty. Each CCD image contains 20 timing comb signals. The comb locations and uncertainties are then fit with a second order polynomial to derive a time for each pixel on the CCD. Comb location and second order fit uncertainties combine to give a 1 ns absolute timing uncertainty.

Zero point uncertainties are derived by comparing the reproducibility of the derived onset of data for each dataset. Such a comparison is shown in Fig. 11 for reference. The curves plotted in that figure are the rising data edges for images shown in Fig. 3. The data were extracted from the 5600 Å wavelength region of shot z3195. All datasets overlap indicating that each spectrum had the same zero point applied. Tests have shown that the zero point procedure can introduce a 1 ns error in certain shots. In this case, an offset between the different dataset shown in Fig. 11 would be visible. Such an offset can most likely be attributed to the severe brightness difference between the emission and absorption data [see panels (b) and (c) in Fig. 3]. It is much easier to determine data onset for absorption than for emission. Combining absolute ( 1 ns) and relative timing ( 1 ns) contributions in quadrature yields a total uncertainty of 1.5 ns. Electron density uncertainties resulting from this 1.5 ns timing uncertainty amount to 10%. Adding this uncertainty to the inherent 10% uncertainty in the ne measurements38 results in a total plasma ne uncertainty of 15%. We assign the same uncertainty to the neutral density.

FIG. 11.

Data onset differences for emission (blue) and absorption (solid and dotted red) spectra. These data were obtained from shot z3195 at ∼5600 Å.

FIG. 11.

Data onset differences for emission (blue) and absorption (solid and dotted red) spectra. These data were obtained from shot z3195 at ∼5600 Å.

Close modal

The CCD wavelength dispersion is derived in a similar manner to the CCD timing. Laser line peaks are determined using Gaussian fits (see Figs. 8 and 9). These peak locations and the associated known laser wavelengths are then fit with a second order polynomial to derive a wavelength dispersion for the 20 ns interval currently being calibrated. This process has several uncertainties associated with it: laser line wavelengths, variation in laser line peak location, and accuracy of the second order polynomial fit.

The laser line wavelength uncertainty is vanishingly small. Neon energy levels in the WDPE He–Ne lasers have been studied since at least 195053 and are thus very well constrained. The refractive index of air needed to convert the laser vacuum wavelength to air has also been well studied.54,55

Laser line peak location uncertainties are captured through the Gaussian fits to these features. These uncertainties are passed to the second order polynomial fits used to derive a wavelength dispersion across the entire CCD. Uncertainties in the wavelength dispersion fit are combined with the remaining differences between derived and fit laser line locations to derive a total wavelength dispersion uncertainty. For the emission CCD system, this amounts to 1 Å. The absorption CCD system has an uncertainty of 0.8 Å. Uncertainty in the wavelength dispersion fit is the dominant factor in the total wavelength dispersion uncertainty.

The spectral power calibration is critical to the WDPE since it allows for the self-emission correction of absorption spectra. This correction is especially important for the 2p3d 5875 Å He I transition shown in panels (a) and (b) of Fig. 3. Tests have shown the dominant contributors to the spectral power calibration uncertainties are the neutral density filters used during the LDLS image collection. Those uncertainties control the CCD pixel integration scale factor used to translate the tungsten calibration lamp image from an 8 s scale to a 500 ns scale. Preliminary data suggest that the ND used during the LDLS data calibration measurements can introduce a 10%–15% uncertainty into the spectral power calibration. Further tests are in progress to obtain a more accurate uncertainty estimate.

The instrumental broadening is measured by integrating emission and absorption laser fiducial images over 50 pixel rows ( 12.5 ns) in the vertical direction. The resulting spectra contain the four lasers discussed earlier. Each laser feature is fitted with a Gaussian, and the associated FWHM and uncertainty are recorded. FWHM trends as a function of vertical position on the CCD for each laser are shown in Fig. 10. The final instrumental broadening is calculated by averaging all laser line FWHM values, except the green. The broadening uncertainty is computed by adding the laser line FWHM standard deviation and the laser line Gaussian width uncertainties in quadrature. This translates to a 10% broadening uncertainty for most shots on both the emission and absorption CCD system. In cases where the recorded laser lines are weak, the instrumental broadening uncertainty can be as high as 20%.

We also note that Fig. 10 shows a 15% higher broadening for the emission CCD system as compared to the absorption setup. The raw measured FWHM for 2p3d 5875 Å He I in shot z3195 is 26 Å in both emission and absorption. Once deconvolved, this 15% instrumental broadening difference between the emission and absorption CCD system changes the 2p3d 5875 Å He I FWHM by less than 1 Å. This contribution is much smaller than the broadening errors introduced by effects discussed above.

The origin of He WDs plays an integral role in galactic and stellar evolution. These stellar objects may provide important constraints on galactic chemical evolution and mass loss processes in stars. However, accurate DB masses are needed to gain insight into each of these processes. Current DB mass determination methods are unreliable due to uncertain atomic input physics. The WDPE at Sandia National Laboratories' Z-machine aims to perform He atomic model benchmark experiments so that the true potential of DBs can be unlocked.

Benchmark experiments have many requirements associated with them. One of the most important is an accurate data calibration approach with well-characterized uncertainties. These uncertainties must be within certain limits for results to be physically meaningful. Wavelength uncertainties should be within 1 Å. Electron and neutral densities should be determined with uncertainties <20%. Maximum uncertainties for experimental spectral power should also be <15%. As we have demonstrated in this paper, the WDPE He experiments fulfill all the benchmark experiment requirements. Our experimental measurements are therefore accurate enough to make meaningful differentiations between different He atomic theories. In a forthcoming publication, we will present several data-model comparisons that will shed some light on outstanding DB questions. All the data presented here will also be used to perform data sensitivity analyses and derive a final uncertainty for our experimental results.

This work was performed at Sandia National Laboratories. We thank the Z-facility teams and in particular D. Scogiletti, G. Loisel, D. Begay, D. Bliss, K. Shelton, A. York, and K. MacRunnels for their support. We also thank Alan Wootton and Greg Rochau for championing our fundamental science research efforts. M.S. thanks Don Winget, Mike Montgomery, and Ross Falcon (all UT Austin) for guidance and useful discussions. The authors also thank the referees for helpful suggestions that improved the quality and clarity of this manuscript. Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy's National Nuclear Security Administration under Contract No. DE-NA-0003525. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government. M.S. and B.D. acknowledge support from the Wootton Center for Astrophysical Plasma Properties under U.S. Department of Energy Cooperative Agreement No. DE-NA-0003843.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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