In inertial confinement fusion implosion experiments, the presence of residual anisotropic fluid motion within the stagnating hot spot leads to significant variations in ion-temperature measurements using neutron time-of-flight detectors along different lines of sight. The minimum ion-temperature measurement is typically used as representative of the thermal temperature. In the presence of isotropic flows, however, even the minimum Deuterium–Tritium (DT) neutron-inferred ion temperature can be well above the plasma thermal temperature. Using both Deuterium–Deuterium (DD) and DT neutron-inferred ion-temperature measurements, we show that it is possible to determine the contribution of isotropic flows and infer the DT burn-averaged thermal ion temperature. The contribution of large isotropic flows on driving the ratio of DD to DT neutron-inferred ion temperatures well below unity and approaching the lower bound of 0.8 is demonstrated in multimode simulations. The minimum DD neutron-inferred ion temperature is determined from the velocity variance analysis, accounting for the presence of isotropic flows. Being close to the DT burn-averaged thermal ion temperature, the inferred DD minimum ion temperatures demonstrate a strong correlation with the experimental yields in the OMEGA implosion database. An analytical expression is also derived to explain the effect of mode =1 ion-temperature measurement asymmetry on yield degradations caused by the anisotropic flows.

For inertially confined imploding capsules, neutrons are produced by Deuterium–Deuterium (D-D) and Deuterium–Tritium (D-T) nuclear fusion reactions within a three-dimensionally (3D) distorted hot spot. The neutron production rate is determined by the fusion reactivity averaged over an approximate Maxwellian distribution.1 The width of the distribution measures the ion thermal temperature2 or the ion thermal velocity in the center-of-mass (CM) frame of a nuclear fusion reaction. Since D and T ions are not in a complete thermal equilibrium, neutron yields produced by D-D and D-T reactions depend on slightly different ion thermal temperatures. The apparent ion temperature inferred from the width of the neutron energy spectrum is averaged over all burn distributions within each fluid element with varying ion thermal temperatures and density profiles in space and time. In the presence of flow effects,1,3 broadening and deviations from Maxwellian distribution occur, leading to varying apparent ion temperatures along with different lines of sight (LOSs). The temperature to describe neutron yields, however, is the ion thermal temperature or Brysk2 ion temperature, inferred from the width of the neutron energy spectrum averaged over burn distributions within each fluid element without the effects of the Doppler shift1,3 in neutron velocities.

In National Ignition Facility implosion experiments, DD ion temperatures were inferred4 with values well below those of DT. These experiments exhibited ratios of DD to DT ion temperatures between 0.8 and 1 and small variations in ion-temperature measurements among different lines of sight. These results indicate the presence of residual fluid motion within the hot spot at peak compression.4,5 Consequently, apparent ion temperatures, which are inferred from the width of neutron energy spectra,2 are larger than the real thermal ion temperatures. This leads to underestimating the inferred hot-spot pressures6 used as a performance metric for inertial confinement fusion (ICF) implosions.

For a non-stationary fusion plasma, neutron velocities are Doppler shifted1,3,7 by fluid motions along a given LOS. The resulting width of the neutron energy spectrum is broadened. Since the fluid motion varies in space, the apparent ion temperatures, which are inferred from the width of broadened neutron energy spectra, not only are larger than the real thermal ion temperature but also vary among different LOSs. This ion temperature asymmetry is uniquely determined by the behavior of the variance of hot-spot fluid velocities.1,3 Signatures of ion-temperature asymmetry were observed in both 3D simulations5,8,9 and experiments4 dominated by low modes. The velocity variance caused by fully turbulent flows3 is homogeneous in space. This isotropy gives rise to turbulent residual kinetic energy (RKE), which consequently inflates apparent ion temperatures uniformly in 4π.

In this work, the velocity variance analysis9 is applied to examine the 3D effects of large isotropic flows. Strong correlations between the isotropic velocity variance and the ratio of DD to DT minimum ion temperatures are observed. (The terminology of minimum ion temperature in this work refers to the global minimum of ion temperatures measured by any detector, unless stated otherwise.) Two applications to derive the DD minimum neutron-inferred ion temperature and the DT burn-averaged thermal ion temperature are presented. The DD minimum ion temperature is derived by removing the anisotropic velocity variance from a simultaneous DD and DT ion-temperature measurement along the same single LOS. The resulting DD minimum ion temperatures demonstrate a strong correlation with the experimental fusion yields in the OMEGA implosion database. A hot spot of an implosion subject to multi-mode velocity perturbations is shown to exhibit large isotropic flows, resulting in a neutron-inferred ion temperature that is smaller for DD neutrons than that for DT neutrons. Ti asymmetries for mode =2 are shown to depend on the competition between residual kinetic energies driven by the converging Rayleigh–Taylor (RT) spikes along the poles and the radially outward expanding bubble at the equator. The correlations between low-mode ion-temperature measurement asymmetries and residual kinetic energies are investigated in detail. The yield-over-clean (YOC, defined by the ratio of 3D to 1D fusion yields) is derived through an analytic relation depending on the ratio of maximum to minimum DT neutron-inferred ion temperatures for mode =1.

This paper is organized as follows: In Sec. II, the properties of the neutron energy spectra are discussed, including the Doppler velocity broadening and the treatment of flow effects. In Sec. III, the properties of the isotropic velocity variance are discussed, including the methods to derive the DD minimum ion temperature, the DT thermal ion temperatures, and the analytic relation that describes the yield degradation in terms of mode =1 Ti asymmetry. Section IV summarizes our conclusions.

In ICF experiments, hot-spot ion temperatures are inferred from the width of neutron energy spectra measured along a given LOS. The center-of-mass (CM) motion of a DT ion pair and the relative kinetic energy of DT ions in the CM frame produce the thermal velocity broadening centered at E0=14.1 MeV, the birth energy of a neutron in a DT fusion reaction,2 

D+T24He(3.5MeV)+n(14.1MeV).
(1)

An alpha particle gains a birth energy of 3.5 MeV from the total nuclear energy release of Q = 17.6 MeV. For a stationary fusion plasma, the shape of a neutron energy spectrum is a Gaussian distribution.2 The width is given by the energy variance σB2, which is proportional to the DT thermal ion temperature Tithermal,

σB2=2mnTithermalE0mn+mα,
(2)

where mn and mα are neutron and alpha-particle rest masses, respectively. For a non-stationary fusion plasma, the CM frame velocity of DT ions is Doppler shifted or boosted by the fluid velocity v. Non-relativistically,2 the amount of the Doppler shift in the mean neutron energy μ is given by μE0=v·d̂2mnE0. The angles θ and ϕ for the LOS unit vector d̂=sinθcosϕx̂+sinθsinϕŷ+cosθẑ are measured in the laboratory frame.

The effects of the Doppler shift in neutron velocities are manifested in the shape of neutron energy spectra fLOS(En) observed along a given LOS. Since DT primary neutrons encounter only negligible scatterings with the cold DT shell, the shape of spectra fLOS(En)Exp[(Enμ)2/(2σB2)]dN is effectively described by the superposition9,10 of Doppler-shifted Gaussian energy spectra, produced by all fluid elements. The burn distribution d N(En) measures the number of neutrons with kinetic energy En, and N=dN is the total number of neutrons. This approach readily captures the formation of non-Gaussian spectra driven by flow effects without expensive efforts of direct numerical neutron transport modelings.11 In experiments,12 the apparent ion temperatures Tiinferred=EFWHM2(mn+mα)/(E0mn16ln2) measured along a given LOS are inferred from the full width at half maximum (FWHM)2EFWHM=σB8ln2 of Gaussian-fitted neutron energy spectra from Eq. (2).

For non-stationary fusion plasmas, the exact relation between thermal ion temperatures and flow effects is governed by the variance1,3 of the neutron-production spectra in velocity or energy space. A brief summary of relativistic1 neutron kinematics is given in the  Appendix. By introducing the normalized burn-averaged bracket ()=()fLOS(vn)dvn/fLOS(vn)dvn, the statistics of neutron velocities are obtained from Eqs. (A8) and (A9). Non-relativistically, a beam of neutrons arriving at a detector, parallel to a given LOS unit vector d̂, has the mean anisotropic neutron velocity,

vn=v0+κ+v·d̂.
(3)

This result is obtained from the first moment of the neutron energy spectrum. Since the DT CM velocity vcmDT is isotropic in space, its mean is zero, i.e., vcmDT=0. The shift due to DT relative kinetic energies in Eq. (A4) is small, i.e., κ=v0K/(2Q)1.47km/s·keV 1×5Ti,keVthermal, where K5Tithermal is used.1,2

The general treatment of flow effects for an arbitrary shape of the neutron-production spectrum in the velocity space is given by the nth moment mn=(ww¯)n with respect to a velocity variable wvnv0 along the direction of d̂. The mean is w¯w=κ+v·d̂. The second moment is the velocity variance,

var[w]=var[vcmDT]+var[κ]+var[v·d̂]+2{cov[vcmDTκ]+cov[κ(v·d̂)]+cov[(v·d̂)vcmDT]}.
(4)

Definitions for the covariance and variance between any two scalars given by f and g are cov[fg]=fgfg and var[f]=f2f2, respectively. Three covariances in Eq. (4) can be neglected because the CM frame velocity, the small shift κ, and the fluid velocity are independent of each other. Flow effects caused by non-Gaussian distributions are characterized by higher moments such as the skew m3/var[w]3/2 and the kurtosis m4/var[w]2.

Multiplying both sides of Eq. (4) with the DT total reactant mass MDT=(mD+mT), the neutron-inferred ion temperature Tiinferred=MDT·var[w] is shown to depend on the sum of the thermal ion temperature Tithermal=MDT·var[vcmDT] and the velocity variance,

Tiinferred=Tithermal+MDT·var[v·d̂].
(5)

The magnitude of var[κ] is negligible compared with the variance of the ion thermal velocity and the variance of the hot-spot flow velocity. The effect of DT relative motion1,13 mainly shifts the neutron velocity spectrum in the w-space by a small amount of κ along the direction of the LOS, without altering the shape of the spectrum significantly. Microscopically, ion temperatures, which measure the collective random motion of ions, are represented by the variance operation. Equation (5) indicates that the variations in apparent ion temperatures are uniquely determined by the behavior of the variance of the hot-spot fluid velocities. By replacing MDT with the total reactant mass MDD in D–D fusion reactions, the same form is valid to describe DD inferred ion temperatures. The Brysk thermal ion temperature2 is recovered in the limit of zero velocity variance, whereas the Murphy fully turbulent flows3 are recovered in the limit of zero anisotropic part of var[v·d̂].

The influence of 3D flow effects on apparent ion temperatures is governed by the properties of velocity variance, contributed by both isotopic and anisotropic flows. For instance, isotropic flows lead to minimum apparent ion temperatures well above thermal ion temperatures. To describe this phenomenon, the method of velocity variance decomposition9 is applied.

The fluid velocity vector v=i=13viêi and the LOS unit vector d̂=i=13giêi are substituted into the velocity variance, followed by an expansion into six components. The resulting apparent ion temperatures in Eq. (5) can be rewritten as9 

Tiinferred=Tithermal+MDTi,j=13gigjσij.
(6)

The indices correspond to Cartesian coordinates: 1x,2y, and 3z, respectively; êi is an orthonormal unit vector. Three geometrical factors, g1=sinθcosϕ,g2=sinθsinϕ, and g3=cosθ, specify the polar θ and azimuthal ϕ angles for a given LOS. The six components of velocity variance σij=vivjvivj measure the flow structure. For indices i = j, σ11, σ22, and σ33 are called “directional variances.” For indices ij, σ12, σ23, and σ31 are called “covariances.” For DD temperatures, the DD total reactant mass MDD is used in Eq. (6). The burn-averaged brackets for σijDD are calculated by DD burn distributions fLOSDD.

The interpretation for directional variance and covariance is as follows: The fluid velocity is decomposed into a burn-averaged component v(t) representing the mean flow and a variation component v(x,t) representing the perturbed flow,

v(x,t)=v(t)+v(x,t).
(7)

With the azimuthal symmetry, even-m single modes have zero covariances in order to conserve vanishing total translational momenta of the whole imploding capsule on the plane P orthogonal to the rotation axis. Without the azimuthal symmetry, capsules for odd-m single modes translate on the plane P. Magnitudes of covariances for odd-m modes decrease with the azimuthal mode number because the azimuthal asymmetric flows are located within the cold bubbles, where neutron-production rates are low. These properties reveal that covariance terms σ12, σ23, and σ31 measure the degree of azimuthal asymmetry. For 1D spherical symmetric implosions, capsules are centered at the origin. The 1D radial flow has zero covariances.

Since v is the perturbed component in the background of a translational mean flow v, the directional variance, from its definition σii=vivi, is proportional to the nontranslational component of the hot-spot fluid kinetic energy KEhs,inontrans=Mhsvi2/2 along the directions of three Cartesian axes i=x,y,z,

σii=vi2=2KEhs,inontrans/Mhs,
(8)

where Mhs is the hot-spot mass. For turbulent flows, the terms vi can be treated as random variables with zero mean flows vi=0. The covariances, which also measure the correlations among all ij components vivj=vivj, asymptotically approach zero. The flow is homogenous with respect to all LOSs. The latter are characterized by unit vectors along the radial direction originated from target chamber centers. Apparent ion temperatures TLOS=Tth+KEhs,rnontrans·(2MDT/Mhs) are therefore inflated, uniformly in 4π, by the isotropic hot-spot fluid kinetic energies from the radial component of the flow. The velocity variance in Eq. (5) is essentially reduced to the form of isotropic velocity variance discussed in Ref. 3.

For azimuthal symmetric flows such as 2D distorted implosions, covariances are zero. The variation in apparent ion temperatures TLOS=Tth+i=13KEhs,inontransgi2·(2MDT/Mhs) depends on the competition of nontranslational fluid kinetic energies among three orthogonal directions.

In general, as shown in Fig. 1, the 3D hot spot contains isotropic flows originated from counterflows along the radial direction. They are not measurable in the first moment of the neutron-production spectra but broaden the width of spectra uniformly in 4π, causing Tmin>Tth. The residue vaniso=vviso is the anisotropic flow, represented by an anisotropic hot-spot flow vector, as shown by the large black arrow in Fig. 1. The maximum and minimum inferred ion temperatures occur when conditions d̂LOSvaniso and d̂LOSvaniso are satisfied, respectively.

FIG. 1.

A sketch for the hot-spot flow structure and the configuration14 of seven nTOF detectors in OMEGA: six nTOF for DTs and one nTOF for DDs, to infer hot-spot ion temperatures. The petal detector for DT temperatures and the 13.4 m nTOF for DD temperatures are located at the same line of sight.

FIG. 1.

A sketch for the hot-spot flow structure and the configuration14 of seven nTOF detectors in OMEGA: six nTOF for DTs and one nTOF for DDs, to infer hot-spot ion temperatures. The petal detector for DT temperatures and the 13.4 m nTOF for DD temperatures are located at the same line of sight.

Close modal

The challenge to infer the hot-spot real thermal temperature requires separating the isotropic flows and thermal temperatures. As mentioned in Ref. 9, six DT ion-temperature measurements complete the reconstruction of the six components of the velocity variance. However, in this work, we show that the thermal ion temperature can be extracted by introducing a seventh measurement for the DD ion temperature along the same LOS of a DT neutron time-of-flight (nTOF) detector. The steps in Secs. III CIII E for the overall method to determine the ion thermal temperature are summarized as follows: (1) determining the anisotropic portion of the apparent DT ion temperature in the 4π angular variation from six apparent DT ion temperature observations, (2) determining the isotropic contribution from isotropic flows by comparing the DD and DT apparent ion temperatures along a single line of site, and (3) determining the ion thermal temperature by removing the effect of the isotropic contribution from the minimum apparent DT ion temperature.

The isotropic velocity variance is the global minimum of the velocity variance in Eq. (5). For single modes, it has a simpler form9 given by

σiso(t)=Min[σxx(t),σyy(t),σzz(t)].
(9)

This result is obtained by decomposing the directional variance into a constant part σiso and a varying part σiiσiiσiso with respect to all LOSs. As discussed in Sec. III A, covariances for single modes approach zero. Apparent ion temperatures in Eq. (6) become TLOS=Tth+MDT(σiso+i=13σiigi2), where the unit vector property i=13gi2=1 is used. Because of azimuthal symmetry, the form of isotropic velocity variance for 2D modes is σiso(t)=Min[σxx(t),σzz(t)].

The time evolution of directional variances in Eq. (9) affects the temperature measurements. A particular case is the mode =2 perturbation, which exhibits time-varying z-directional variances. In the rest of this work, the 3D hydro code DEC3D15 was applied to simulate the flow effects on ion-temperature measurement asymmetries caused by deceleration-phase Rayleigh-Taylor hydrodynamic instabilities. Initial radial velocity perturbations are seeded at the inner shell surface at the time of peak implosion velocity in 1D LILAC16 simulations. The effects of single modes are presented in Secs. III B and III F, whereas the effects of multimode are presented in Secs. III C and III E. Different DD or DT ion thermal temperatures and velocity variances are calculated by the time-integrated burn-averaged measurements in DEC3D simulations. As shown in Fig. 2, before the pair of RT spikes reaches the center, perturbations along the poles dominate over perturbations on the equatorial plane. The latter are driven by the expanding bubble. After the RT spikes reach the center, the burn volume of the hot core along the poles is reduced significantly, leading to decreasing burn-averaged z-directional variances. Perturbations of radial flows on the equatorial plane continue to grow regardless of the initial perturbation amplitudes. As a result, the isotropic velocity variance σiso=2(t)=Min[σxxbubble(t),σzzspike(t)] is governed by the competition between σxxbubble(t) and σzzspike(t) over time. For modest =2 perturbations, the time-integrated, burn-averaged z-directional variance is larger than that in the x direction and vice versa in strongly perturbed =2 distorted implosions.

FIG. 2.

Flow patterns for modes =1 and 2 at 7% implosion velocity perturbations. The distorted cold shell is filled with the blue color. Only flow patterns are drawn inside the hot spot bounded by the contour of Te= 0.5 keV.

FIG. 2.

Flow patterns for modes =1 and 2 at 7% implosion velocity perturbations. The distorted cold shell is filled with the blue color. Only flow patterns are drawn inside the hot spot bounded by the contour of Te= 0.5 keV.

Close modal

Figure 3 compares the time-integrated burn-averaged σxx and σzz between modes =1 and 2 against different initial velocity perturbations. The blue and red curves represent apparent ion temperatures measured at LOS ẑ and x̂, respectively. Only mode =2 exhibits a transition from σzz>σxx to σzz<σxx at increasing initial velocity perturbations. This is caused by the increasingly large donut-shaped warm bubble. At a large perturbation level of 14%, the flow of =2 transits to anisotropic as discussed in Ref. 8. Mode =1 has a vanishing isotropic velocity variance caused by negligible σxx=1 or σyy=1, whereas its large ion-temperature measurement asymmetry is caused by growing σzz=1. Since modest perturbations are more frequent than large perturbations in high-performance ICF implosion experiments, mode =2 is expected to contribute significantly to the isotropic velocity variance with only a small anisotropic contribution.

FIG. 3.

Comparison of time-integrated burn-averaged directional variance σxx and σzz between modes =1 and 2 against different initial velocity perturbations.

FIG. 3.

Comparison of time-integrated burn-averaged directional variance σxx and σzz between modes =1 and 2 against different initial velocity perturbations.

Close modal

Figure 4 compares the ratio of σzz/σxx for single modes =3 to 12 simulated by 1%–14% implosion velocity perturbations using DEC3D. The x-directional variance is shown to not overtake the z-directional variance. Two-dimensional m = 0 modes are shown to be more anisotropic than 3D m0 modes with σzz/σxx>2. This occurs because RT spikes along the poles grow faster than spikes on the 2D rings. As a result, velocity fluctuations in 2D modes are slightly more anisotropic than those in 3D modes. The latter have velocity fluctuations that are about uniform in 4π. Three-dimensional modes are shown to lie inside the range R: 1σzz/σxx2, in which the flows are close to isotropic because σxx and σzz are about equal. Two-dimensional modes =5 and 6 are shown to approach the range R at large perturbations.

FIG. 4.

The ratio of σzz/σxx for single modes =3 to 12 characterizes the magnitudes of isotropic velocity variance.

FIG. 4.

The ratio of σzz/σxx for single modes =3 to 12 characterizes the magnitudes of isotropic velocity variance.

Close modal

The maximum isotropic velocity variance in Eq. (9) occurs when all directional variances are equal and large σisomax=Min[σii0]; simultaneously, the anisotropic parts vanish σii=0. This results in small variations in ion-temperature measurements along different LOSs. At the transition, defined by σzz/σxx=1, from modest to large perturbations, mode =2 satisfies the condition to maximize the isotropic velocity variance according to Eq. (9). Mode =2 is a special case due to the competition of nontranslational hot-spot fluid kinetic energies between σzz and σxx. The mechanism to produce a large isotropic velocity variance requires breaking the azimuthal symmetry to form 3D radial flow structures within the hot core, thus increasing the overall magnitudes of directional variances.

Isotropic flows are keys to account for the difference between DD and DT minimum ion temperatures,

TminDD=TDDthermal+MDDσisoDD,TminDT=TDTthermal+MDTσisoDT,
(10)

where quantities without the label “(t)” are referred to as “time-integrated, burn-averaged.” Labels of ()iinferred are omitted. The ratio of DD to DT minimum ion temperatures from Eq. (10) is

TminDDTminDT=(α+MDDσisoDDTDTthermal)(1+MDTσisoDTTDTthermal)1,
(11)

where α=TDDthermal/TDTthermal is the ratio of DD to DT burn-averaged thermal ion temperatures. The fraction of residual kinetic energy frke is defined as the ratio of the isotropic velocity variance to the DT burn-averaged thermal ion temperature,

frke=MDTσisoDT/TDTthermal.
(12)

In Eq. (11), about 3% small difference between TDDthermal and TDTthermal in DEC3D multi-mode simulations is observed in Sec. III E. This observation is in close agreement with 1D and 2D HYDRA simulations for NIF high-foot implosions reported by Gatu et al.4 and Kritcher et al.,17 respectively. The profile effect has a non-negligible impact on deriving the thermal ion temperature in this work. Since the ratio TDDthermal/TDTthermal is shown to be about constant in Sec. III E, we account for the profile effect by treating the ratio as the averaged value obtained from simulations, i.e., α0.97. For hot-spot temperatures at a few keV, the diffusion mean free path of thermal ions is much less than the hot-spot radius, so DT ions are rapidly thermalized through collisions. The ion-loss effect18 on reducing hot-spot fusion reactivities is neglected because the escaping of faster ions tends to occur at higher temperatures at 10 keV for ignition-relevant implosions. Under these conditions, ion distributions are close to the Maxwellian. At subignition temperatures of 15 keV, the burn-averaged velocity variances for D–D and D–T reactions are shown to be closed in Fig. 5. The ratio of DD to DT minimum inferred ion temperatures in Eq. (11) is

TminDD/TminDT=(α+frke·MDDMDT·σisoDDσisoDT)(1+frke)1.
(13)

Substituting Eq. (8) into Eq. (12), we obtain

frke=(61Pe/Phs)·KEhs,i,DTnontransIEhs,
(14)

where Phs=Pi+Pe is the total hot-spot pressure, Pe/i is the electron/ion pressure, IEhs=32PhsVhs is the total hot-spot internal energy, and Vhs is the hot-spot volume. The value within the bracket in Eq. (14) equals 12, assuming thermal equilibrium between electrons and ions, and neglects the small mass deficit in fusion reactions. In Eq. (14), frke measures the fraction of nontranslational isotropic hot-spot fluid kinetic energy to the hot-spot internal energy.

FIG. 5.

Comparison of isotropic velocity variance between DDs and DTs in multimode simulations. D–D and D–T fusion reactivities are used to calculate the burn-averaged bracket for the velocity variance.

FIG. 5.

Comparison of isotropic velocity variance between DDs and DTs in multimode simulations. D–D and D–T fusion reactivities are used to calculate the burn-averaged bracket for the velocity variance.

Close modal

Murphy3 defined the kinetic energy fraction frkeM=EkM/(EkM+EthM) as the ratio of the hot-spot fluid kinetic energy EkM to the total hot-spot fluid energy EkM+EthM such that EkM/EthM=frke/4. Here, EkM=32MhsσisoDT and EthM=32(nD+nT+ne)TDTthermalVhs. The hot-spot mass is related to its volume by Mhs=12(mD+mT)niVhs. The total hot-spot ion number density ni=nD+nT is assumed to be equal to the electron number density ne for a fully ionized DT plasma. A factor of 3 is required to express Murphy's total hot-spot kinetic energy EkM=3KEhs,i,DTnontrans because our definition of σiso in Eq. (9) accounts for only nontranslational hot-spot fluid kinetic energy in one direction. The ratio in Eq. (13) has a lower bound of 0.8 in the limit of large frke. At the first-order expansion with respect to small frke in Eq. (13),

TminDD/TminDTα+frke(0.8α),
(15)

the ratio drops linearly with the fraction of isotropic residual kinetic energy.

To approach the limit of TminDD/TminDT0.8, more isotropic flows and lower thermal ion temperatures are required to increase frke in Eq. (12). Multimode perturbations satisfy this requirement because the superposition of isotropic velocity variance from modest =2 and 3D m0 modes can lead to large isotropic flows within the hot spot. Simultaneously, large-amplitude multimode perturbations can degrade the thermal ion temperature by converting less of the shell's kinetic energies into hot-spot internal energies.15 

A set of multimode simulations by the superposition of the same single mode with random phases is carried out. The initial multimode velocity perturbation spectrum is given by Am=i=1N(v/v0)Ym(θ+θi,ϕ+ϕi), where θi and ϕi are random phases for the ith single mode, N = 20 is the total number of random phases assigned for a given Ym single mode, and v/v0 is the initial velocity perturbation.

Figure 5 shows similar isotropic velocity variance between DDs and DTs due to their small difference in temperature dependence between D-D and D-T fusion reactivities in simulations. Figure 6 shows a significant reduction of the DD/DT minimum ion-temperature ratio to the level of 0.9 by a rapid increase in kinetic energy fraction frkeM. Single-mode perturbations were observed to exhibit small values of frke<0.3, which is insufficient to degrade the DD/DT minimum ion-temperature ratio. The hot spot in a multimode perturbation, however, is filled with enhanced isotropic flows, and simultaneously, the hot-spot thermal ion temperature is degraded to give a large value of frke0.3 to 1 or frkeM0.1. A good agreement is obtained between the simulation result and the black analytic curve given by Eq. (13). The global minimum of the velocity variance var[v·d̂] behaves in the same way as Murphy's velocity variance in fully turbulent flows, whereas the variation part of var[v·d̂] contributes to ion-temperature asymmetries.

FIG. 6.

Comparison of the DD/DT minimum neutron-inferred ion-temperature ratio with the kinetic energy fraction frkeM in multimode simulations. The black line is the analytic curve using Murphy's definition frkeM=EkM/(EkM+EthM) in Eq. (13).

FIG. 6.

Comparison of the DD/DT minimum neutron-inferred ion-temperature ratio with the kinetic energy fraction frkeM in multimode simulations. The black line is the analytic curve using Murphy's definition frkeM=EkM/(EkM+EthM) in Eq. (13).

Close modal

Equation (10) indicates that the DD minimum ion temperature is closer to the thermal ion temperature than DTs due to a smaller total fusion reactant mass MDD/MDT=0.8. Consider a simultaneous ion-temperature measurement for DD and DT along the same single LOS,

TDTLOS=TminDT+MDTσanisoDT,
(16)
TDDLOS=TminDD+MDDσanisoDD.
(17)

As urged in Sec. III C, DD and DT anisotropic velocity variances are approximately the same, i.e., σanisoDDσanisoDT. The DD minimum ion temperature can be derived by removing the common anisotropic part,

TminDD=TDDLOS(TDTLOSTminDT)(MDDMDT).
(18)

Here, the DT minimum ion temperature TminDT can be either extracted from the six-LOS method9 by simply taking the global minimum of all available DT ion temperatures. The error propagation is about 0.3 keV for the derived TminDD in Eq. (18).

Figure 7(a) shows a poor correlation with the DD ion temperatures measured by the 13.4 m nTOF. Figure 7(b) shows a strong correlation between the derived DD minimum ion temperatures using Eq. (18) and the experimental yield in OMEGA implosion database. In this plot, the minimum DT ion temperatures are taken as the global minimum among all available DT temperatures. The application of the six-LOS method is not considered here because the current OMEGA configuration of six nTOFs for DT temperatures, as shown in Fig. 1, produces about 1-keV error. The propagated error is larger than the measurement error, which is about 0.2 keV, from the minimum measured value. To minimize the error propagation in the six-LOS method, a new nTOF at a new line of sight has been proposed in OMEGA. The observed strong scaling T4 of the experimental yields with the derived DD minimum ion temperatures is in agreement with the temperature dependence for the DT fusion reactivity.19 Since the D–D reaction has a smaller total nuclear reactant mass than DTs in Eq. (10), the derived DD minimum ion temperature is closer to the burn-averaged thermal ion temperature. This observation agrees with 2D HYDRA simulations for NIF high-foot implosions reported by Kritcher et al.17 In Fig. 7(c), the experimental yields were shown to scale as T3.64 with the measured DT minimum ion temperatures, which is slightly worse than the correlation with the derived DD minimum ion temperatures. The DT's standard deviation for the relative fitting error is shown to be slightly larger than DD's. It is expected that the correlation with the thermal ion temperature is even better than either DT or DD minimum ion temperatures due to the presence of isotropic flows.

FIG. 7.

Inferring for the DD minimum neutron-inferred ion temperatures in the OMEGA experiments using Eq. (18). Comparison of correlation between experimental yields and (a) the DD ion temperatures measured by the 13.4 m nTOF, (b) the derived DD minimum ion temperatures, and (c) the minimum of all measured DT ion temperatures.

FIG. 7.

Inferring for the DD minimum neutron-inferred ion temperatures in the OMEGA experiments using Eq. (18). Comparison of correlation between experimental yields and (a) the DD ion temperatures measured by the 13.4 m nTOF, (b) the derived DD minimum ion temperatures, and (c) the minimum of all measured DT ion temperatures.

Close modal

By performing DD and DT minimum ion-temperature measurements, the degeneracy between thermal ion temperatures and isotropic flows can be recovered because the isotropic velocity variance is multiplied with different DD and DT total reactant masses. Consequently, thermal ion temperatures can be solved from Eq. (10). First, the solution of frke is expressed in terms of DD/DT minimum ion-temperature ratios using Eq. (13),

frke=αTminDD/TminDTTminDD/TminDTRMDD/DTRσDD/DT,
(19)

where RMDD/DTMDD/MDT and RσDD/DTσisoDD/σisoDT are ratios of DD to DT total reactant masses and isotropic velocity variances, respectively. The DT minimum ion temperature in Eq. (10),

TminDT=TDTthermal(1+frke),
(20)

can be inverted to solve for the DT burn-averaged thermal ion temperature TDTthermal=TminDT/(1+frke) using Eq. (19). The required minimum ion temperatures can be obtained from the global minimum for DT ion temperatures reconstructed from the six-LOS method9 and the derived DD minimum ion temperatures in Sec. III D. However, the derived TDTthermal has a 4× larger error propagation than the derived TminDD in Eq. (18).

To solve for the thermal temperatures, the seventh temperature that should be added to the original six-LOS ion-temperature measurements9 is DD temperature. It forms an invertible LOS matrix M̂LOS from the exact relation of Eq. (6), which describes six DT and one DD apparent ion temperatures,

T7=M̂LOS·X7.
(21)

The column vector T7=[{Ti=1,..,6DT,T7DD}] contains the seven ion-temperature measurements, whereas the state vector X7=[{T̂DTth,σ11,σ22,σ33,2σ12,2σ23,2σ31}]MDT contains the DT thermal ion temperature and the six hot-spot flow parameters. T̂DTth=TDTthermal/MDT is the normalized DT thermal ion temperature. The explicit form for M̂LOS can be written as

[1g11g11g21g21g31g31g11g21g21g31g31g111g12g12g22g22g32g32g12g22g22g32g32g121g13g13g23g23g33g33g13g23g23g33g33g131g14g14g24g24g34g34g14g24g24g34g34g141g15g15g25g25g35g35g15g25g25g35g35g151g16g16g26g26g36g36g16g26g26g36g36g16αβg17g17βg27g27βg37g37βg17g27βg27g37βg37g17].
(22)

The matrix elements are determined by matching Eq. (6) with the state vector X7, where β=MDD/MDT. The solution for the thermal temperature and six components of velocity variance is given by X7=M̂LOS1·T7. In a special case, when the seventh temperature is chosen as DT temperature with α=β=1 in Eq. (22), M̂LOS is not invertible. This is because in a hot spot filled with turbulent flows, all seven DT temperatures are reduced to one single apparent ion temperature TDT=TDTthermal+MDTσiso with two unknowns: thermal ion temperatures and isotropic flows. Equation (22) represents the least number of LOSs for multiple DT and DD ion-temperature measurements to separate the real thermal ion temperature from isotropic flows and the six components of velocity variance.

The existing nTOF configuration in OMEGA as shown in Fig. 1 can form the invertible LOS matrix to solve for thermal ion temperatures with about 0.8-keV errors. As a result, the correlation between experimental yields and the derived thermal ion temperatures according to the matrix inversion given by Eq. (22) is not as robust as that with the derived DD minimum ion temperatures as shown in Fig. 7(b). Error propagations in the matrix inversion from Eq. (22) are suppressed by adding more LOSs or reallocating the positions of LOSs.

The viability of the method to infer the thermal ion temperature is demonstrated by a DEC3D simulation database for strongly perturbed multimode distorted implosions as shown in Fig. 8. The spectrum of an initial multimode perturbation contains Legendre modes =1 to 12 including 2D modes m = 0 and 3D modes m=even/2. The initial velocity perturbation is uniform for the whole spectrum. The database of multimode simulations in Fig. 8 shows that the burn-averaged TDDthermal is about 3% smaller than the burn-averaged TDTthermal due to the profile effect as shown in Fig. 9. This general trend can be applied to set α=0.97 in Eq. (22) to obtain a better agreement between the derived and the DT burn-averaged thermal ion temperatures. However, a few percent deviations from the general trend are observed. The deviations are shown to grow with perturbations, caused by different DD and DT time-integrated neutron production profiles within a 3D hot spot.

FIG. 8.

Comparison of the derived and the DT burn-averaged thermal ion temperatures in DEC3D multimode simulation for a strongly perturbed hot spot. The inset is the 2D xz plane for the hot-spot temperature at stagnation.

FIG. 8.

Comparison of the derived and the DT burn-averaged thermal ion temperatures in DEC3D multimode simulation for a strongly perturbed hot spot. The inset is the 2D xz plane for the hot-spot temperature at stagnation.

Close modal
FIG. 9.

Comparison of DD and DT burn-averaged thermal ion temperatures in DEC3D multi-mode simulations in Fig. 8.

FIG. 9.

Comparison of DD and DT burn-averaged thermal ion temperatures in DEC3D multi-mode simulations in Fig. 8.

Close modal

The 2D temperature profile for a perturbed hot spot in a sample of multimode simulations is given in Fig. 8. In this simulation, the large hot-spot flow isotropy leads to a DD/DT minimum ion-temperature ratio of TminDD/TminDT=0.934 and a moderate maximum to minimum DT ion-temperature ratio of TmaxDT/TminDT=1.13. Table I summarizes the thermal, maximum, and minimum inferred ion temperatures, as well as the six hot-spot flow parameters in this simulation. The square roots of three directional variances give the magnitudes of nontranslational velocity fluctuations vivi100 km/s, in which the magnitude is close to the DT ions' thermal velocity given by Ti,DTthermal/(mD+mT) = 238 km/s. Differences between DD and DT directional variances and covariances are observed to be small, as discussed in Sec. III C.

TABLE I.

DEC3D multimode perturbation (unit for temperatures is keV and σij is km/s). The imaginary number i=1 is used to represent the square root of negative covariances.

VariablesTionthermalTmininferredTmaxinferredσ11σ22σ33σ12σ23σ31
DD 2.86 3.47 3.83 134 152 122 34.9i 32.6i 17.4i 
DT 2.95 3.72 4.20 135 154 122 36.1i 34.5i 8.86i 
VariablesTionthermalTmininferredTmaxinferredσ11σ22σ33σ12σ23σ31
DD 2.86 3.47 3.83 134 152 122 34.9i 32.6i 17.4i 
DT 2.95 3.72 4.20 135 154 122 36.1i 34.5i 8.86i 

Another application of Ti asymmetry is to describe yield degradations in the presence of large anisotropic flows. Previous studies15,20 showed that the YOC is a strong function of the RKE,

YOC(1RKEtot)μ,
(23)

for implosions perturbed by low modes. The normalized total residual kinetic energy at stagnation is RKEtot=RKEhs+RKEsh and μ=4.4–5.5. The normalized hot-spot and shell residual kinetic energies are defined as RKEhs=(KEhs3DKEhs1D)stag/KEmax1D and RKEsh=(KEsh3DKEsh1D)stag/KEmax1D, respectively. Here, KEhs3D/1D and KEsh3D/1D are the 3D/1D hot-spot fluid kinetic energy and the 3D/1D total shell kinetic energy, respectively. KEmax1D is the maximum 1D in-flight shell's kinetic energy. To apply this result in experiments, RKE must be interpreted in terms of observables of implosion asymmetries.

For mode =1, the ratio of maximum to minimum neutron-inferred ion temperature is dominated by the nontranslational hot-spot fluid kinetic energy along the direction of the jet. For initial velocity perturbations in the form of spherical harmonics, the jet of mode =1 without any phase shift in DEC3D15 simulations is parallel to the z axis. For single-mode perturbations, the covariances are zero so that the ratio of maximum T̂maxDT(θ=0) to minimum T̂minDT(θ=π/2) neutron-inferred ion temperature is

(TmaxDTTminDT)=1=T̂ithermal+σisoDT+σ33DTT̂ithermal+σisoDT.
(24)

Since mode =1 has negligible nontranslational fluid kinetic energies σ11=σ220 along x, y directions, the isotropic velocity variance in Eq. (9) is approximately zero σisoDT0. The anisotropic velocity variance is reduced to the z-directional variance, i.e., σ33DT=σ33DTσisoDTσ33DT, such that (TmaxDT/TminDT)=11+σ33DT/T̂ithermal. We define the fraction of the total nontranslational hot-spot fluid kinetic energy as frketotal=frke+frkeaniso, where the isotropic part frke is given by Eq. (12) and the anisotropic part is frkeaniso=σii/T̂ithermal. By substituting frketotal=σ33DT/T̂ithermal, the Ti ratio in Eq. (24) is rewritten as

(TmaxDT/TminDT)=11+frketotal.
(25)

Two approximate hot-spot fluid properties for mode =1, observed from large ensembles of DEC3D deceleration-phase simulations, are used in the following derivations: The first property is

Mhs3Dσ33DT/2KEhs3D/3,
(26)

such that the flow structure of mode =1 approximately satisfies v32v32/3. This relation is shown in Fig. 10(a). The expression for frketotal is obtained by replacing the isotropic nontranslational hot-spot fluid kinetic energy with Mhs3Dσ33/2 in Eq. (14). The latter is one third of the total hot-spot residual kinetic energy by Eq. (26) so that frketotal4KEhs3D/IEhs3D, where IEhs3D is the 3D hot-spot internal energy. The ratio of 3D hot-spot kinetic energy to internal energy is rewritten as

KEhs3D/IEhs3D=(IEhs1DIEhs3D)(KEmax1DIEhs1D)(KEhs3DKEmax1D).
(27)
FIG. 10.

Survey of fluid properties for mode =1: (a) for the first property of v32v32/3 in Eq. (26) and (b) for the consequence of the second property in Eq. (28).

FIG. 10.

Survey of fluid properties for mode =1: (a) for the first property of v32v32/3 in Eq. (26) and (b) for the consequence of the second property in Eq. (28).

Close modal

Three different terms on the right-hand side of Eq. (27) are (1) IEhs3D/IEhs1D=1RKEtot, which results from the conservation of total energy at stagnation;15 (2) IEhs1D/KEmax1D1/2, which is a good approximation for 1D implosions because typically about a half of the shell maximum kinetic energy is converted into the hot-spot internal energy at stagnations; and (3) KEhs3D/KEmax1DRKEtot/2 using the second property RKEhsRKEsh. The resulting Eq. (27) is then reduced to

KEhs3D/IEhs3DRKEtot/(1RKEtot).
(28)

However, the correlation of Eq. (28) is not strong as shown in Fig. 10(b) with an increasing number of outliers when =1 perturbations are intensified. Equation (25) becomes a unique function of the total residual kinetic energy,

(TmaxDT/TminDT)=11+4RKEtot/(1RKEtot),
(29)

which is inverted to give

RKEtot=ξ/(1+ξ).
(30)

The ion-temperature measurement asymmetry parameter ξ=(1/4)(RT1)0 is a function of the ion-temperature ratio RT=TmaxDT/TminDT. Therefore, the yield degradation through Eq. (23) is an expression of the neutron-inferred ion-temperature measurement asymmetry parameter ξ,

YOC=[1ξ/(1+ξ)]μ.
(31)

Equation (31) is simplified by fitting the YOC against the ion-temperature ratio in terms of a simple power law YOC=(RT)bfit. For the value of μ = 5, the best fit is

YOC(TmaxDTTminDT)1.53.
(32)

For mid/high modes or fully turbulent flows, the anisotropic velocity variance decreases significantly, leading to small ion-temperature measurement variations among different LOSs. Simultaneously, the hot spot contains large isotropic flows, resulting in larger minimum neutron-inferred ion temperatures than the true thermal ion temperatures. The two fluid properties for mode =1 are not held by other modes in simulations because different flow structures have their own scalings for Eqs. (26) and (28).

To study Eqs. (31) and (32), DEC3D hydrodynamic data at stagnation, simulated by 1%–14% initial velocity perturbations, are post-processed by a Monte Carlo neutron transport code IRIS3D.11 The ion temperatures are inferred from the width of neutron energy spectra using 16 detectors at LOSs distributed from the north to south poles uniformly at a fixed azimuthal angle ϕ=0 in IRIS3D. Figure 11 shows the yield degradation vs the neutron-inferred ion-temperature ratio. The results of mode =1 are shown accurately lying on the fitting curve given by Eq. (32). This is the first result to explain the yield degradation caused by mode 1 ion-temperature measurement asymmetries in terms of analytic models of residual kinetic energies and velocity variance analysis. In OMEGA implosion database, A. Lees showed that the fitting exponent was close to −1.3 by performing machine-learning data analysis. The close match of the fitting exponents implies the existence of mode =1 asymmetry in OMEGA database for implosions with the ion temperature ratio Tmax/Tmin>1.1. In general, the correlation between yield degradation and Ti asymmetry decreases with increasing isotropic flows. For instance, 2D modes =3–4 exhibit a weaker Ti asymmetry because of decreasing anisotropic flows. Mode =2 and other mid/high modes >4 exhibit a much weaker Ti asymmetry because their high-velocity fluid motions driven by vorticity are located within cold bubbles, which contribute only negligible Doppler velocity broadening.

FIG. 11.

Yield degradations vs ion-temperature measurement asymmetries for single modes. The black curve is given by Eq. (32).

FIG. 11.

Yield degradations vs ion-temperature measurement asymmetries for single modes. The black curve is given by Eq. (32).

Close modal

In conclusion, a systematic analysis of 3D effects of large isotropic flows on DD and DT neutron–inferred ion temperatures is presented. Strongly perturbed multimode perturbations are shown to produce a large content of isotropic flows within the hot spot, resulting in smaller DD than DT minimum neutron–inferred ion temperatures. The presence of large isotropic flows leads to the ratio of DD to DT minimum ion temperatures approaching the lower bound of 0.8. The method to infer the DD minimum ion temperature through the removal of the anisotropic velocity variance is derived. The resulting DD minimum ion temperature is shown to demonstrate a strong correlation with the experimental fusion yields. A method to infer the DT thermal ion temperature through simultaneous DD and DT ion-temperature measurements at different LOSs is described. Reasonable agreement is observed with strongly perturbed multimode simulations. An analytic expression is derived to explain the anisotropic flow effect of ion-temperature measurement asymmetry on yield degradations for mode =1.

This material is based upon work supported by the Department of Energy National Nuclear Security Administration under Award No. DE-NA0003856. 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.

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

The relativistic motion of a single neutron along a straight line transport parallel to the LOS unit vector d̂ is described as follows: Consider a Lorentz boost of a neutron momentum pn in the CM frame of a DT ion pair by its CM frame velocity vcmDT relative to the neutron momentum pn observed in the fluid rest frame. Both pn and vcmDT are components parallel to d̂,

pn=γcm(pn+vcmDTEn/c2),
(A1)

where En=mnc2+Kn is the total mass energy of the neutron in the CM frame, c is the speed of light, and mn and Kn are the rest mass and the relativistic kinetic energy of the neutron in the CM frame, respectively. The Lorentz factor γcm=(1βcm2)1/2 is a function of the DT ion-pair CM frame velocity through the factor of βcm=vcmDT/c. The kinetic energy of the neutron in the CM frame is obtained from the DT nuclear fusion energy release Q and the relative kinetic energy K of DT ions in their CM frame,21 

pn22mn=K0+μmnK,
(A2)

where K0=mαQ/(mn+mα)=mnv02/2 is the neutron birth energy and μ=mnmα/(mn+mα) is the reduced mass of DT fusion products. Let p0=mnv0 be the neutron momentum at the zero DT relative kinetic energy limit K = 0 and substitute p0=2μQ into Eq. (A2) to expand the neutron momentum pn=p01+K/Q with the relative kinetic energy K,

pn(1+K2Q)p0.
(A3)

Substitute Eq. (A3) into Eq. (A1) and expand the Lorentz factor γcm1+βcm2/2 to first order to obtain the neutron momentum in the fluid rest frame,

pn=p0+K2Qp0+mnvcmDT+1+2,
(A4)

where 1=βcmKn/c and 2=βcm2(pn+vcmDTEn/c2)/2 are two relativistic correction terms. The neutron velocity pn observed in the laboratory frame parallel to d̂ is obtained by the second Lorentz boost by the fluid velocity v,

pn·v̂=γv(pn·v̂+vEn/c2),
(A5)
pn·v̂=pn·v̂,
(A6)

where En=γvmnc2 is the total mass energy of the neutron in the fluid rest frame and γv=(1βv2)1/2 is the Lorentz factor as a function of the fluid velocity through βv=v/c. Here, v̂ is a unit vector perpendicular to the direction of the fluid velocity unit vector v̂=v/v. Only the component of the neutron momentum pn·v̂ parallel to the fluid velocity is Lorentz boosted. Expand the Lorentz factor γv1+βv2/2 and add Eqs. (A5) and (A6) together to obtain the neutron momentum vector pn=(pn·v̂)v̂+(pn·v̂)v̂ in the laboratory frame,

pn=pn+mnv+βv22[mnv+pn·v̂+vEn/c2]v̂.
(A7)

Magnitudes of neutron momenta pn=pn·d̂ and pn=pn·d̂ parallel to d̂ are obtained by taking a dot product on both sides of Eq. (A7),

pn=pn+mnv·d̂+βv22[mnv+pn·v̂+vEn/c2]v̂·d̂.
(A8)

Non-relativistically with βv0 and 1=20 in Eq. (A4), Eq. (A8) is reduced to a simple momentum addition pn=pn+mnv·d̂, where pn is given by Eq. (A4). The non-relativistic neutron velocity vn=pn/mn detected along d̂

vn=v0+vcmDT+κ+v·d̂
(A9)

is the sum of the neutron birth velocity v0=2K0/mn, the DT center-of-mass velocity vcmDT in the fluid rest frame that contributes to the primary thermal ion temperature, a small positive velocity shift κ=v0K/(2Q)>0 due to the DT relative kinetic energy, and a Doppler velocity shift v·d̂ due to the fluid velocity in the non-stationary fusion plasma.

1.
D. H.
Munro
, “
Interpreting inertial fusion neutron spectra
,”
Nucl. Fusion
56
,
036001
(
2016
).
2.
H.
Brysk
, “
Fusion neutron energies and spectra
,”
Plasma Phys.
15
,
611
(
1973
).
3.
T. J.
Murphy
, “
The effect of turbulent kinetic energy on inferred ion temperature from neutron spectra
,”
Phys. Plasmas
21
,
072701
(
2014
).
4.
M.
Gatu Johnson
,
J. P.
Knauer
,
C. J.
Cerjan
,
M. J.
Eckart
,
G. P.
Grim
,
E. P.
Hartouni
,
R.
Hatarik
,
J. D.
Kilkenny
,
D. H.
Munro
,
D. B.
Sayre
,
B. K.
Spears
,
R. M.
Bionta
,
E. J.
Bond
,
J. A.
Caggiano
,
D.
Callahan
,
D. T.
Casey
,
T.
Döppner
,
J. A.
Frenje
,
V. Y.
Glebov
,
O.
Hurricane
,
A.
Kritcher
,
S.
LePape
,
T.
Ma
,
A.
Mackinnon
,
N.
Meezan
,
P.
Patel
,
R. D.
Petrasso
,
J. E.
Ralph
,
P. T.
Springer
, and
C. B.
Yeamans
, “
Indications of flow near maximum compression in layered deuterium-tritium implosions at the national ignition facility
,”
Phys. Rev. E
94
,
021202(R)
(
2016
).
5.
B. K.
Spears
,
M. J.
Edwards
,
S.
Hatchett
,
J.
Kilkenny
,
J.
Knauer
,
A.
Kritcher
,
J.
Lindl
,
D.
Munro
,
P.
Patel
,
H. F.
Robey
, and
R. P. J.
Town
, “
Mode 1 drive asymmetry in inertial confinement fusion implosions on the national ignition facility
,”
Phys. Plasmas
21
,
042702
(
2014
).
6.
B. K.
Spears
,
S.
Glenzer
,
M. J.
Edwards
,
S.
Brandon
,
D.
Clark
,
R.
Town
,
C.
Cerjan
,
R.
Dylla-Spears
,
E.
Mapoles
,
D.
Munro
,
J.
Salmonson
,
S.
Sepke
,
S.
Weber
,
S.
Hatchett
,
S.
Haan
,
P.
Springer
,
E.
Moses
,
J.
Kline
,
G.
Kyrala
, and
D.
Wilson
, “
Performance metrics for inertial confinement fusion implosions: Aspects of the technical framework for measuring progress in the national ignition campaign
,”
Phys. Plasmas
19
,
056316
(
2012
).
7.
L.
Ballabio
,
J.
KAllne
, and
G.
Gorini
, “
Relativistic calculation of fusion product spectra for thermonuclear plasmas
,”
Nucl. Fusion
38
,
1723
(
1998
).
8.
J. P.
Chittenden
,
B. D.
Appelbe
,
F.
Manke
,
K.
McGlinchey
, and
N. P. L.
Niasse
, “
Signatures of asymmetry in neutron spectra and images predicted by three-dimensional radiation hydrodynamics simulations of indirect drive implosions
,”
Phys. Plasmas
23
,
052708
(
2016
).
9.
K. M.
Woo
,
R.
Betti
,
D.
Shvarts
,
O. M.
Mannion
,
D.
Patel
,
V. N.
Goncharov
,
K. S.
Anderson
,
P. B.
Radha
,
J. P.
Knauer
,
A.
Bose
,
V.
Gopalaswamy
,
A. R.
Christopherson
,
E. M.
Campbell
,
J.
Sanz
, and
H.
Aluie
, “
Impact of three-dimensional hot-spot flow asymmetry on ion-temperature measurements in inertial confinement fusion experiments
,”
Phys. Plasmas
25
,
102710
(
2018
).
10.
V. N.
Goncharov
,
T. C.
Sangster
,
R.
Betti
,
T. R.
Boehly
,
M. J.
Bonino
,
T. J. B.
Collins
,
R. S.
Craxton
,
J. A.
Delettrez
,
D. H.
Edgell
,
R.
Epstein
,
R. K.
Follett
,
C. J.
Forrest
,
D. H.
Froula
,
V. Yu.
Glebov
,
D. R.
Harding
,
R. J.
Henchen
,
S. X.
Hu
,
I. V.
Igumenshchev
,
R.
Janezic
,
J. H.
Kelly
,
T. J.
Kessler
,
T. Z.
Kosc
,
S. J.
Loucks
,
J. A.
Marozas
,
F. J.
Marshall
,
A. V.
Maximov
,
R. L.
McCrory
,
P. W.
McKenty
,
D. D.
Meyerhofer
,
D. T.
Michel
,
J. F.
Myatt
,
R.
Nora
,
P. B.
Radha
,
S. P.
Regan
,
W.
Seka
,
W. T.
Shmayda
,
R. W.
Short
,
A.
Shvydky
,
S.
Skupsky
,
C.
Stoeckl
,
B.
Yaakobi
,
J. A.
Frenje
,
M.
Gatu-Johnson
,
R. D.
Petrasso
, and
D. T.
Casey
, “
Improving the hot-spot pressure and demonstrating ignition hydrodynamic equivalence in cryogenic deuterium-tritium implosions on omega
,”
Phys. Plasmas
21
,
056315
(
2014
).
11.
F.
Weilacher
,
P. B.
Radha
, and
C.
Forrest
, “
Three-dimensional modeling of the neutron spectrum to infer plasma conditions in cryogenic inertial confinement fusion implosions
,”
Phys. Plasmas
25
,
042704
(
2018
).
12.
M. G.
Johnson
,
J. A.
Frenje
,
D. T.
Casey
,
C. K.
Li
,
F. H.
Séguin
,
R.
Petrasso
,
R.
Ashabranner
,
R. M.
Bionta
,
D. L.
Bleuel
,
E. J.
Bond
,
J. A.
Caggiano
,
A.
Carpenter
,
C. J.
Cerjan
,
T. J.
Clancy
,
T.
Doeppner
,
M. J.
Eckart
,
M. J.
Edwards
,
S.
Friedrich
,
S. H.
Glenzer
,
S. W.
Haan
,
E. P.
Hartouni
,
R.
Hatarik
,
S. P.
Hatchett
,
O. S.
Jones
,
G.
Kyrala
,
S. Le
Pape
,
R. A.
Lerche
,
O. L.
Landen
,
T.
Ma
,
A. J.
MacKinnon
,
M. A.
McKernan
,
M. J.
Moran
,
E.
Moses
,
D. H.
Munro
,
J.
McNaney
,
H. S.
Park
,
J.
Ralph
,
B.
Remington
,
J. R.
Rygg
,
S. M.
Sepke
,
V.
Smalyuk
,
B.
Spears
,
P. T.
Springer
,
C. B.
Yeamans
,
M.
Farrell
,
D.
Jasion
,
J. D.
Kilkenny
,
A.
Nikroo
,
R.
Paguio
,
J. P.
Knauer
,
V. Yu
Glebov
,
T. C.
Sangster
,
R.
Betti
,
C.
Stoeckl
,
J.
Magoon
,
M. J.
Shoup
,
G. P.
Grim
,
J.
Kline
,
G. L.
Morgan
,
T. J.
Murphy
,
R. J.
Leeper
,
C. L.
Ruiz
,
G. W.
Cooper
, and
A. J.
Nelson
, “
Neutron spectrometry-an essential tool for diagnosing implosions at the national ignition facility (invited)
,”
Rev. Sci. Instrum.
83
,
10D308
(
2012
).
13.
D. H.
Munro
,
J. E.
Field
,
R.
Hatarik
,
J. L.
Peterson
,
E. P.
Hartouni
,
B. K.
Spears
, and
J. D.
Kilkenny
, “
Impact of temperature-velocity distribution on fusion neutron peak shape
,”
Phys. Plasmas
24
,
056301
(
2017
).
14.
O.
Mannion
,
J.
Knauer
,
V.
Glebov
,
C.
Forrest
,
A.
Liu
,
Z.
Mohamed
,
M.
Romanofsky
,
T.
Sangster
,
C.
Stoeckl
, and
S.
Regan
, “
A suite of neutron time-of-flight detectors to measure hot-spot motion in direct-drive inertial confinement fusion experiments on omega
,”
Nucl. Instrum. Methods Phys. Res. Sect. A
964
,
163774
(
2020
).
15.
K. M.
Woo
,
R.
Betti
,
D.
Shvarts
,
A.
Bose
,
D.
Patel
,
R.
Yan
,
P.-Y.
Chang
,
O. M.
Mannion
,
R.
Epstein
,
J. A.
Delettrez
,
M.
Charissis
,
K. S.
Anderson
,
P. B.
Radha
,
A.
Shvydky
,
I. V.
Igumenshchev
,
V.
Gopalaswamy
,
A. R.
Christopherson
,
J.
Sanz
, and
H.
Aluie
, “
Effects of residual kinetic energy on yield degradation and ion temperature asymmetries in inertial confinement fusion implosions
,”
Phys. Plasmas
25
,
052704
(
2018
).
16.
J.
Delettrez
,
R.
Epstein
,
M. C.
Richardson
,
P. A.
Jaanimagi
, and
B. L.
Henke
, “
Effect of laser illumination nonuniformity on the analysis of time-resolved x-ray measurements in uv spherical transport experiments
,”
Phys. Rev. A
36
,
3926
(
1987
).
17.
A. L.
Kritcher
,
D. E.
Hinkel
,
D. A.
Callahan
,
O. A.
Hurricane
,
D.
Clark
,
D. T.
Casey
,
E. L.
Dewald
,
T. R.
Dittrich
,
T.
Döppner
,
M. A.
Barrios Garcia
,
S.
Haan
,
L. F.
Berzak Hopkins
,
O.
Jones
,
O.
Landen
,
T.
Ma
,
N.
Meezan
,
J. L.
Milovich
,
A. E.
Pak
,
H.-S.
Park
,
P. K.
Patel
,
J.
Ralph
,
H. F.
Robey
,
J. D.
Salmonson
,
S.
Sepke
,
B.
Spears
,
P. T.
Springer
,
C. A.
Thomas
,
R.
Town
,
P. M.
Celliers
, and
M. J.
Edwards
, “
Integrated modeling of cryogenic layered highfoot experiments at the NIF
,”
Phys. Plasmas
23
,
052709
(
2016
).
18.
K.
Molvig
,
N. M.
Hoffman
,
B. J.
Albright
,
E. M.
Nelson
, and
R. B.
Webster
, “
Knudsen layer reduction of fusion reactivity
,”
Phys. Rev. Lett.
109
,
095001
(
2012
).
19.
H.-S.
Bosch
and
G.
Hale
, “
Improved formulas for fusion cross-sections and thermal reactivities
,”
Nucl. Fusion
32
,
611
(
1992
).
20.
A. L.
Kritcher
,
R.
Town
,
D.
Bradley
,
D.
Clark
,
B.
Spears
,
O.
Jones
,
S.
Haan
,
P. T.
Springer
,
J.
Lindl
,
R. H. H.
Scott
,
D.
Callahan
,
M. J.
Edwards
, and
O. L.
Landen
, “
Metrics for long wavelength asymmetries in inertial confinement fusion implosions on the national ignition facility
,”
Phys. Plasmas
21
,
042708
(
2014
).
21.
B.
Appelbe
and
J.
Chittenden
, “
The production spectrum in fusion plasmas
,”
Plasma Phys. Controlled Fusion
53
,
045002
(
2011
).