In this paper, we present a theoretical framework for interpreting the hot-spot electron temperature (*T*_{e}) inferred from hard (10- to 20-keV) x-ray continuum emission for inertial confinement fusion implosions on OMEGA. We first show that the inferred *T*_{e} represents the emission-weighted, harmonic mean of the hot-spot *T*_{e} distribution, both spatially and temporally. A scheme is then provided for selecting a photon energy of which the emission weighting approximates neutron weighting. Simulations are then used to quantify the predicted relationship between the inferred *T*_{e}, neutron-weighted *T*_{i}, and implosion performance on OMEGA. In an ensemble of 1-D simulations, it was observed that hot-spot thermal nonequilibrium precluded a sufficiently unique mapping between the inferred *T*_{e} and neutron-weighted *T*_{i}. The inferred *T*_{e} and hard x-ray yield's sensitivity to implosion asymmetry was studied using a 3-D simulation case study with low-harmonic-mode perturbations (i.e., laser beam power imbalance, target offset, and beam port geometry departures from spherical symmetry) and laser imprint (*l*_{max} = 200).

## I. INTRODUCTION

Evaluating the performance of inertial confinement fusion (ICF) implosions requires metrics that quantify the hot-spot internal energy,^{1–4} in addition to density and volume information. This is commonly done by inferring the neutron-weighted ion temperature from neutron time-of-flight (nTOF) diagnostics.^{5,6} These nTOF devices collect neutrons a known distance away from a neutron source [i.e., the implosion hot spot], and from the spread of their arrival times, the energy distribution of the neutrons can be determined. From this spectrum, the hot spot's neutron-weighted ion temperature is inferred from the thermally broadened width of the neutron energy peak centered at 14.1 MeV [i.e., the mean energy of neutrons produced by the deuterium–tritium (D–T) fusion reaction].

Hot-spot internal energy measurements using the neutron-weighted ion temperature measurements can be biased because of fluid-motion effects.^{7,8} Turbulent fluid motion can impart an additional energy spread, on top of thermal effects, to the emitted neutrons. This can distort the neutron spectra, resulting in the inferred ion temperature values always being higher than the true thermal temperature by an amount that depends on the view direction relative to the fluid-flow direction. Variations as high as 50% in ion temperatures inferred from different nTOF detectors have been observed from direct-drive implosions performed at the Omega Laser Facility.^{9} Even with little ion-temperature variation, hot-spot fluid motion has still been hypothesized to be responsible for measured values being greater than expected for indirect-drive implosions^{10,11} at the National Ignition Facility^{12} (NIF). According to calculations by Murphy^{8} for NIF shot N130812, turbulent fluid velocities of 1.55 ± 0.1 × 10^{7} cm/s (or ≈1% of a 14.1-MeV neutron's velocity) can explain the differences between simulated and inferred DT temperatures of 1.25 ± 0.16 keV. Given this effect of fluid-motion bias, there exists a need to develop an alternate robust method of obtaining the hot-spot internal extra energy.

The hard x-ray continuum emission provides an alternative means to infer a hot-spot temperature. Measurements of the hot-spot electron temperature using x rays have been performed before at the NIF,^{13–15} with the benefit that they are not biased by hot-spot fluid motion. Electrons within the hot spot emit x rays primarily through free–free bremsstrahlung, and the resulting x-ray spectrum encodes information about their temperature. Since hot-spot fluid motion is small compared to the speed of light (the turbulent fluid velocities of 1.55 × 10^{7} cm/s as inferred by Murphy^{8} would be ≈0.05% the speed of light), the x-ray emission will not be significantly affected by Doppler shift from macroscopic flow and therefore more resistant to fluid-motion bias compared to neutrons.

Aside from fluid-motion effects, x-ray–inferred hot-spot temperatures will also differ from neutron-based values as a result of different temperature weighting and averaging. This was recently illustrated by Jarrott *et al.*^{15} using a *HYDRA* simulation ensemble of indirect-drive ICF implosions at the NIF scale, which included both drive amplitude and low-mode asymmetry variations. Upon comparing the calculated x-ray–inferred *T*_{e} with the neutron-weighted *T*_{e}, the authors point out “relatively higher emissivity-weighted temperatures at lower peak temperatures and higher neutron-weighted temperatures as the peak temperature is increased.” The electron temperature average will also be dependent on photon energy. Whereas neutrons are emitted with an average energy of 14.1 MeV for D–T reactions, x rays emitted from bremsstrahlung have an entire spectrum of energies that vary with the electron temperature itself. This means multiple x-ray energies can be used to return different temperature averages, unlike the one for neutrons.

In this paper, we present findings that will form the interpretation basis of x-ray–inferred *T*_{e} from direct-drive ICF implosions on OMEGA: (1) an analytical derivation of the electron temperature as the emission-weighted, harmonic mean temperature; (2) an analytical derivation of x-ray energy that gives emission weighting closest to neutron weighting; (3) simulation results showing disparity between hot-spot electron temperature and ion temperature, even without fluid-motion biasing for OMEGA-scale implosions; and (4) simulation results showing correlation of the implosion degradation with the hot-spot electron temperature and x-ray yield. The role of kinetic effects on inferring *T*_{e} is also addressed and believed small for OMEGA implosions based on results from Kagan *et al.*^{16}

## II. ANALYTICAL DESCRIPTION OF ELECTRON TEMPERATURE FROM X-RAY CONTINUUM

In this section, we derive the physical meaning behind the hot-spot electron temperature inferred from x-ray continuum emission. The hot-spot ion temperature inferred from nTOF measurements represents a “neutron-weighted” quantity. To derive the averaging form of the complementary electron temperature, we begin with the emissivity relevant for hot-spot conditions.

The hot-spot x-ray emission can be primarily described by the free–free bremsstrahlung emissivity^{17} $\epsilon \nu FF.$ Assuming full ionization, the instantaneous and spatially integrated spectrum from a radiating volume is given by^{18,19}

where *α* is the fine-structure constant; *χ*_{H} is the ionization energy of hydrogen; *a*_{0} is the Bohr radius; $\u27e8Z2\u27e9$ is the average nuclear charge state squared; *hν* is the photon energy; *k* is the Boltzmann constant; *T*_{e} is the electron temperature; *n*_{e} and *n*_{i} are the electron and ion number densities, respectively; *g*_{FF} is the Gaunt factor^{20} (a quantum correction that is a function of both photon energy and temperature); and *V* is the volume of the radiating body. The units of *I _{ν}* are energy per time, steradian, and frequency. For illustrative purposes, this quantity is shown in log space in Fig. 1(b) for the hot-spot temperature, and density profiles are shown in Fig. 1(a).

Assuming a constant temperature body and *g*_{FF} = 1, *I _{ν}* reduces to a single exponential and ln

*I*will simply be proportional to –

_{ν}*hν*/

*kT*

_{e}. Therefore, when applying a linear fit to –ln

*I*in spectral space, the slope of that line fit [i.e., Eq. (2) above] will be equal to −1/

_{ν}*kT*

_{e}. For a multitemperature body, the spectrum

*I*itself can no longer be fully described by a single exponential. Performing the same linear fit as before but local to specific spectral energies, one will obtain a temperature that is a function of the spectral energy, as seen in Fig. 1(c), with the inferred temperature increasing with the measured photon energy. Applying a one-temperature fit (i.e., linear fit) to Eq. (2) is equivalent to the expression

_{ν}where the left side was obtained by taking the spectral log derivative of Eq. (1), but assuming *T*_{e} is some yet-to-be-determined constant temperature *T*_{fit}. From here, the goal is to solve for *T*_{fit} and determine its dependence on *hν.* Realizing that *g*_{FF} (*hν*, *T*_{fit}) and 1/*kT*_{fit} are both explicitly independent of position, both terms in parentheses can be pulled out of the volume integral, leaving it to simply become $\u222b\epsilon \nu ,fitFFdV.$ Then, knowing that $Ih\nu ,fit=\u222b\epsilon \nu ,fitFFdV$ as defined by Eq. (1), we can cancel the volume integral entirely on the left-hand side and solve for *T*_{fit},

Further simplification of Eq. (4) depends on the final expression of *g*_{FF}. A variety of expressions for *g*_{FF} have been derived, the most well-known standard of which was done by Karzas and Latter^{22} (KL) for nondegenerate conditions. Here, we will use an asymptotic approximation of their expression provided by Kulsrud^{23} that is valid when *hν* > *kT*_{e} and when $kTe\u226b\chi H$,

where *β* represents the square root of the average ratio between the electron kinetic energy and the thermal energy. In this work, we use *β* = 0.87, which Kulsrud^{23} had found well approximate results from a more-accurate quantum model developed by Sommerfield.^{24}

The percent difference between this form of the Gaunt factor and the KL formula was calculated to be ≈0.2% in the phase space, where 100 eV ≤ *kT* ≤ 9 keV and 1 keV ≤ *hν* ≤ 60 keV. Using Eq. (5) to evaluate *g*_{FF} (*hν*, *T*_{fit}) and its derivatives in Eq. (4) results in the following formula for *T*_{fit}:

The above equation shows that by applying a single-temperature fit to the log spectrum, *T*_{fit} (i.e., the inferred temperature) will represent a *harmonic* mean temperature, weighted by the emission yield. Note in Eq. (6) that *T*_{fit} will be a function of the photon energy, a consequence of its value being emission weighted so that different components of a multitemperature source will dominate the spectrum at different spectral energies. Therefore, consistent with expectations, as one increases the selected photon energy, the spectral weighting of the temperature average favors hotter regions of the hot spot as illustrated in Fig. 1(c). Since the electron temperature will be a harmonic average, it will be generally lower than an emission-weighted, arithmetic average as shown in Fig. 1(c) by ≈100 eV. One can see that Eq. (6) reduces to the actual temperature of an isothermal object. Equations (1)–(6) apply to a single instant in time. For time-integrated measurements, this simply adds time integration to Eq. (6) in conjunction with a time-integrated *I _{ν}* to make the harmonic mean also time averaged.

## III. DETERMINING OPTIMAL EMISSION WEIGHTING

With a physical understanding of the inferred temperature and its weighting on photon energy, it is next important to know the photon energy most optimal for inferring the hot-spot electron temperature. At a minimum, one should look at photon energies high enough where the surrounding material will be optically thin. This threshold photon energy is predicted to be ∼4 keV for OMEGA-scale implosions (see Fig. 2) and was obtained by postprocessing^{21} 1-D *LILAC*^{25} simulation results of shot 77066 as a representative OMEGA cryogenic implosion (achieved ∼50 Gbar of hot-spot pressure and a DT areal density of ∼200 mg/cm^{2} in experiment^{26}). The *LILAC* 1-D code includes an inverse bremsstrahlung model with ray tracing, a nonlocal thermal transport model based on a simplified Boltzmann equation with a Krook collision term,^{27} and a first-principles equation-of-state (FPEOS) model^{28} based on molecular dynamics methods. As a point of comparison, NIF-scale indirect-drive implosions currently have a higher-threshold photon energy of 20 keV (Ref. 15) as a result of higher absorption in shells with areal densities upward of 1000 mg/cm^{2}. Past this minimum threshold, the photon energy measured depends on what spatial region of the hot spot is of most interest. Given that ion temperature values are neutron weighted, it would be most meaningful for the electron temperature to be weighted by the same spatial distribution as the neutron emission for purposes of assessing implosion performance. Unfortunately, the emission weighting can never be made precisely identical to the neutron weighting because while both x-ray and neutron production scale with $ni2$ for a quasineutral plasma, they have fundamentally different temperature dependencies. One can find, however, an optimal emission weighting that minimizes this temperature-dependency difference.

It can be shown that by selecting photons with energies near 4 k*T*_{0}, where *T*_{0} is the characteristic hot-spot temperature (e.g., neutron-weighted temperature), the emission weighting will be closest to that of neutron production. Work by Epstein *et al.*^{19} showed using a power-law approximation that the emissivity of x rays with energies *hν* ≈ 2 *kT*_{0} followed a *T*^{2} temperature dependence. That same power-law approximation can be exploited to find the optimal photon energy that gives *T*^{4} emissivity scaling like the DT neutron-production cross section.^{29} First, the Gaunt factor approximation from Eq. (5) is applied to $\epsilon \nu FF$ to give

Note that the explicit temperature dependence of $\epsilon \nu FF$ is now only in the exponential. To approximate it as a power law at some photon energy *hν*_{0}, the magnitude of the exponential is first made equal to *x ^{η}*,

where *x* = *kT*/*hν*_{0} and *η* is a yet-to-be-determined constant. This is equivalent to stating that $\epsilon \nu FF\u221d(kT)\eta .$ Second, the derivatives of each side of Eq. (8) with respect to *x* are also made equal,

The relation in Eq. (10) says that for photon energies near *hν*_{0}, we can expect $\epsilon \nu FF\u221d(kT)\eta $ at temperatures where $kT=h\nu 0/\eta .$ Put another way, if we expect a radiating body to mostly have a temperature near value *kT*_{0}, then we can expect the emission at photon energy *hν*_{0} to scale approximately as (*kT*)^{η}, where $\eta =h\nu 0/kT0.$ Direct-drive ICF implosions on OMEGA typically have hot-spot temperatures of the order of 3.75 keV. This means if *kT*_{0} were set to a value of 3.75 keV, it would give a photon energy of 15 keV, where the emission will scale approximately as *T*^{4}. At this photon energy, the inferred electron temperature can be said to have an emission weighting that closely, but not equally, resembles neutron weighting. This can be seen in Fig. 3(a), which compares the normalized x-ray emission and neutron production for an isobaric temperature and density profile.^{30}

In addition, this hot-spot temperature and photon energy combination (∼4-keV hot-spot temperature and 15-keV photon energy) is believed to have small non-Maxwellian errors based on work by Kagan *et al.*^{16} In that paper, the authors calculate that implosions with Knudson numbers (i.e., the ratio of a particle mean free path to a characteristic system scale length) near *N*_{K} ∼ 0.01 can underpredict the true electron temperature by up to 5% when 3 < *hν*/*kT*_{e} < 5 (the range of interest in this paper). Simulations of ignition-scalable, direct-drive implosions performed on OMEGA have predicted hot-spot *N*_{K} ∼ 0.003; the same simulations were determined by Forrest *et al.*^{32} to give the same D–T to D–D yield ratios (a measure of multifluid effects) as those observed in experiment. Therefore, the expected temperature error from non-Maxwellian effects is expected to be even less than 5%. In addition, the calculations done by Kagan *et al.*^{16} were based on an ice-block model of the hot spot. It is possible that the temperature error could drop even farther for more-realistic hot-spot profiles, where the mass density increases near the hot-spot boundary instead of remaining constant. The increasing density would imply less escaping electrons from the distribution tail and therefore an overall more-valid Maxwellian approximation. Calculations using more-relevant hot-spot profiles would be beneficial to better quantify non-Maxwellian effects.

We have numerically checked the approximate surrogacy of the 15-keV emission weighting to neutron weighting by plotting the x-ray–inferred electron temperature at that photon energy against the calculated neutron-weighted electron temperature for a variety of simulated implosions. This is done in Fig. 3(b) using *LILAC*^{25} postshot simulations of all past DT cryogenic shots performed on OMEGA that are stored in the simulation database.^{31} The neutron weighting is calculated assuming *T*_{e} = *T*_{i} when computing $\sigma vDT$. In Fig. 3(b), one sees the x-ray (*hν* = 15 keV) and neutron-weighted electron temperatures closely following the *y *=* x* trend line, indicating a near 1:1 correspondence. Some divergence is expected given the power-law approximation breaking away from the *T*^{4} scaling. For example, using a photon energy that is too high above 4*kT*_{0} will give an electron temperature that is biased to regions hotter than the neutron-producing region and vice versa for photon energies below 4*kT*_{0}. Some divergence can also be attributed to differences between a harmonic mean and an arithmetic mean, with the harmonic mean being shifted from the arithmetic mean by the amount shown in Fig. 1(c). Regardless, the divergence is slight for the implosions simulated and shows the approximation strength of using *hν* ∼15-keV photons to get an emission weighting close to neutron weighting. Assuming *T*_{e} = *T*_{i}, this photon energy then suggests a possible pathway for the electron temperature to be a surrogate for estimating the neutron-weighted ion temperature. As will be discussed in Sec. IV, however, this direct surrogacy has been shown to be unreliable for OMEGA-scale implosions.

## IV. RELATIONSHIP BETWEEN HOT-SPOT ELECTRON TEMPERATURE AND ION TEMPERATURE

For OMEGA-scale implosions, simulations show that the neutron-weighted ion temperature is not well approximated by the electron temperature, regardless of the photon energy used. The clearest way to illustrate this is to plot the neutron-weighted ion temperature against the x-ray–inferred electron temperature for a variety of implosions. This is shown in Fig. 4 using the same *LILAC* simulation database as mentioned before. At all photon energies, the functional mapping between the electron temperature and the neutron-weighted ion temperature does not follow a clear *y *=* x* trend. Moreover, the scatter of the mapping is too large to reliably estimate an ion temperature from the electron temperature with an uncertainty below that inferred from nTOF diagnostics. Given an expected inferred electron temperature uncertainty of ∼100 eV, the consequent mapping uncertainty in ion temperature can be as large as ∼400 eV according to scatter in Fig. 4, compared to the precision error of ≈130 eV from current nTOF diagnostics on OMEGA.

This imprecise surrogacy between the ion and electron temperatures is because of the hot spot's thermal nonequilibrium state for the simulated OMEGA implosions. This can be seen by comparing the neutron-weighted electron and ion temperatures as functions of time during neutron production as done in Fig. 5(a) for the previously mentioned OMEGA shot 77066. At the start of stagnation, the ion temperature starts off several kilo-electron-volts above the electron temperature as a result of the strong ion shock heating occurring earlier from shock convergence. After that time, the ions lose their thermal energy to the electrons primarily through thermal equilibration. The two temperatures differ by approximately several hundred electron volts at peak neutron-production time. Toward the end of the neutron-production time, the electrons and ions start coming closer to equilibrium. However, since the neutron-weighted ion temperature is a time-integrated quantity in experiment, it should not be expected to be equal to the electron temperature if both species spend most of their time out of thermal equilibrium during burn time. It should be mentioned here that this does not immediately negate quantities that assume *T*_{i} = *T*_{e}. The assumption's error contribution can be small for some quantities, such as the inferred hot-spot pressure, which depends on the size of an emission volume rather than its absolute magnitude. Thermal nonequilibrium should only be accounted for quantities that absolutely depend in the magnitude of *T*_{e} vs *T*_{i} (e.g., inferred hot-spot mix).

The persistence of this thermal nonequilibrium can be surprising, considering that the equilibration time, which scales^{18} as *τ*_{ei} ∼ *T*^{3/2}*ρ*^{−1}, is of the order of 10 ps or only about 10% of the burnwidth full width at half maximum (FWHM). This discrepancy can be explained by examining the electron thermal conduction during the same time, which is seen to prevent thermal equilibration from being reached so soon. This is shown in Fig. 5(b) for the same shot. The heating and cooling rates are comparable in magnitude at the hot spot and generally suggest that equilibration heating is not fast enough when including heat losses from thermal conduction to bring the ions and electron temperatures into thermal equilibrium within a burnwidth time. Figure 5 was generated for an OMEGA-scale implosion with a convergence ratio (CR) of ≈20, and it is expected that thermal equilibration will be more quickly achieved for higher CR implosions. The reason is that as the CR increases, so too does the hot-spot density, causing *τ*_{ei} to decrease.

The scattered mapping in Fig. 4 also implies that one cannot reliably use the method outlined by Jarrot *et al.*^{15} to infer residual kinetic energy for OMEGA-scale implosions. Since that method relies on a sufficiently unique mapping between *T*_{e} and the true *T*_{i}, additional uncertainty to that relationship would lead to more error for inferring the residual kinetic energy. This error may not be relatively significant if the fluid flow component to *T*_{i} is much larger than the mapping uncertainty, as could be the case for high CR implosions. Otherwise, inferring the residual kinetic energy on OMEGA per the method in Jarrot *et al.*^{15} requires the true *T*_{i} to be estimated by an alternative avenue.

## V. SIMULATION CASE STUDY OF HOT-SPOT ELECTRON TEMPERATURE SENSITIVITY TO IMPLOSION ASYMMETRY

To study the electron temperature and the absolute x-ray yield sensitivity to implosion perturbations, 3-D *ASTER*^{33} simulations were performed with different initial perturbations. One simulation represented an ideal case where the implosion was perfectly 1-D and another included relevant perturbations from target offset (Δ*r *=* *5.4-*μ*m), beam imbalance (*σ*_{rms} = 3.5%), and beam port geometry as well as laser-imprint modulations^{34} (*l*_{max} = 200). Both simulations use target parameters from OMEGA shot 89224, an *α* ∼5 implosion with an in-flight aspect ratio $(IFAR=Rshell/\Delta Rshell)$ of ∼40 and a peak implosion velocity of 480 km/s. A complete outline of the target dimensions and pulse shape is shown in Fig. 6. Between the 1-D and perturbed simulations, the neutron-weighted ion temperature dropped from 4.67 keV to 4.35 keV and the neutron yield dropped from 4 × 10^{14} to 2 × 10^{14} (compared to an experimental yield of 1.2 × 10^{14}), which was a result stemming from decreased hot-spot compression. Figure 7 compares the inferred electron temperature and x-ray yield between the two cases. In similarity to the neutron-weighted ion temperature, the electron temperature dropped by almost the same amount (≈300 eV) throughout the 10- to 20-keV emission energy range. The absolute x-ray emission in Fig. 7(b) also dropped by ≈48% on average, nearly the same amount as the neutron yield. These changes being similar are not believed a coincidence; the 10- to 20-keV emission energy range is centered around the optimal energy range at which emission weighting is closest to neutron weighting. The drop being almost consistent across the entire range suggests the weighting is robust across a wide energy space.

In addition to characterizing the hot-spot internal energy, T_{e} and absolute x-ray–emission measurements can be combined with methods published by Epstein *et al.*^{19} and Ma *et al.*^{35} to estimate hot-spot mix. With the existence of a *T*_{e} measurement, the thermal-equilibrium assumption can also be removed and thereby improve the estimate's accuracy for implosions on OMEGA. An alternate method using a parameterized physics model^{36} can also be used to infer mix distribution, with the measured x-ray spectrum and inferred T_{e} serving as constraints.

## VI. CONCLUSION

Interpretation and sensitivity analysis of the hot-spot electron temperature inferred from hard x rays have been performed. The electron temperature inferred from hard x-ray continuum emission was shown to be an emission-weighted, harmonic mean electron temperature. As this value varies with photon energy, it was shown both analytically and with simulations that the optimal photon energy for approximate neutron weighting is near 15 keV or more generally near 4× the neutron-weighted hot-spot temperature. Simulations also suggest, however, that one should not expect the hot-spot electron and ion temperatures to be equal in value for OMEGA-scale implosions caused by thermal nonequilibrium. For a 3D simulation with relevant drive perturbations, the drop in electron temperature was found to be of the same order as the drop in the ion temperature, and the x-ray yield-over-clean ratio tracked the neutron yield-over-clean ratio.

## ACKNOWLEDGMENTS

This material is based upon work supported by the Department of Energy National Nuclear Security Administration under Award No. DE-NA0003856, the University of Rochester, and the New York State Energy Research and Development Authority. This report was prepared as an account of work sponsored by an agency of the U.S. Government. Neither the U.S. Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the U.S. Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the U.S. Government or any agency thereof.