Three-dimensional (3-D) implosion asymmetries lead to significant variations in ion-temperature measurements in inertial confinement fusion experiments. We present an analytical method to generalize the physical properties of velocity variance in the Brysk ion-temperature model. This analysis provides a consistent explanation for the 3-D effects of inferred ion-temperature variations for various single modes and multimodes modeled by the deceleration-phase hydrocode *DEC3D* and the neutron transport code *IRIS3D*. The effect of the hot-spot flow asymmetry on variations in ion-temperature measurements is shown to be uniquely determined by a complete set of six hot-spot flow parameters. An approximated solution to the minimum inferred ion temperature is derived and shown to be close to the thermal ion temperature for low mode *ℓ* = 1, which exhibits the largest anisotropic velocity variance in the single-mode spectrum. The isotropic velocity variance for low mode *ℓ* = 2 is shown to result in the minimum inferred ion temperatures being well above the thermal ion temperature.

## I. INTRODUCTION

In inertial confinement fusion (ICF), 3.5-MeV alpha particles and 14-MeV neutrons are produced by deuterium and tritium (DT) nuclear fusion reactions within the high-temperature, low-density central hot spot. When the rate of the alpha particle's energy deposition exceeds the heat conduction and radiation losses, the hot spot ignites^{3,4} for hot-spot areal densities of 0.2 to 0.3 g/cm^{2}, large enough to confine the alpha particles. The run-away self-heating process of DT fuel releases considerable fusion energy, larger than the total input energy. Neutrons escape the hot spot and carry valuable information about the hot-spot properties including the mean hot-spot ion temperature from the width of the neutron energy spectra^{5–9} and the mean hot-spot fluid velocities from the shift of the peak neutron energy.^{8–12} The presence of implosion asymmetries leads to the growth of Rayleigh–Taylor (RT) instabilities^{13–17} at the inner and outer shell surfaces, preventing the full conversion of the shell's kinetic energy (KE) into the hot-spot internal energy.^{18,19} The unconverted kinetic energy remains as the residual kinetic energy of RT spikes and bubbles, resulting in the degradation of the neutron yield and hot-spot pressure.^{20,21} The magnitude of the hot-spot residual kinetic energy of low modes is large enough to cause significant variations in ion-temperature measurements along different lines of sight^{18,22} (LOSs). These measurement variations are caused by the anisotropic source of the hot-spot fluid velocity distribution,^{6} whereas the isotropic source of the hot-spot residual kinetic energy causes higher apparent DT ion temperatures^{5,8,9,23,24} than the thermal ion temperatures. Consequently, the implosion performance metrics including the inferred generalized Lawson criterion^{4} and the inferred hot-spot central pressure^{25,26} are altered when the higher apparent DT ion temperatures are used. It is crucial to understand the effects of hot-spot nonstagnating fluid motions on the broadening of the neutron energy spectra in ICF experiments.

In this work, we present an analytic method to quantify the effect of the hot-spot flow asymmetry on variations in ion-temperature measurements. A matrix representation is developed to study the three-dimensional (3-D) effects of the velocity variance in the Brysk ion-temperature model.^{5,6} We show that the velocity variance can be uniquely determined by a complete set of six hot-spot flow parameters including three directional variances and three covariances of the hot-spot fluid velocity distribution. The asymmetry of inferred ion temperature profiles simulated by Monte Carlo codes or observed in experiments can be explained by the analytic model in terms of the isotropy and anisotropy of the hot-spot fluid velocity distribution. An approximated solution to the true minimum of inferred ion temperature is derived using ion temperature measurements along six LOSs. This result is verified numerically using the deceleration-phase single-mode database from the radiation–hydrodynamic code *DEC3D*^{20} and the Monte Carlo–based neutron transport code *IRIS3D*.^{2} In particular, the analytic results facilitate the extrapolation of the minimum inferred ion temperature and the magnitude of single-mode isotropic velocity variances through utilizing the matrix properties. The isotropic source of the hot-spot velocity variance is shown to be nonseparable from the minimum apparent ion temperature. The dominant 3-D radial flow configuration of low mode *ℓ* = 2 is shown to exhibit a large isotropic velocity variance.

The organization of this paper is as follows: Sec. II discusses the Doppler shift and velocity broadening of neutron energy spectra for various single modes *ℓ* = 1 to 12. The result of the neutron energy spectrum model^{1} is in good agreement with *IRIS3D* simulations. The analytic treatments for the single-mode velocity variance and the multimode velocity variance are presented in Secs. III and IV, respectively. Section V derives the relationship between the hot-spot residual kinetic energy and the variations in ion temperature measurements. Section VI summarizes our conclusions.

## II. DOPPLER SHIFT AND BROADENING OF THE NEUTRON ENERGY SPECTRUM

The single-mode database is generated by the 3-D deceleration-phase hydrodynamic code *DEC3D*.^{20} Initial radial velocity perturbations are seeded on the inner shell's surface for OMEGA shot 77068 at the beginning of the deceleration phase, when the shell reaches the maximum implosion velocity. Implosion 77068 is described in detail in Ref. 27 and is considered here because it is one of the highest-performing implosions to date on OMEGA with ∼50 GBar of inferred hot-spot pressure.^{28} The initial radial velocity perturbation is in the form of spherical harmonic $Y\u2113m(\theta ,\varphi )$ modes and zero initial transverse velocities in the *θ* and *ϕ* directions. The initial unperturbed hydrodynamic profiles at the beginning of deceleration phase are obtained from the one-dimensional (1-D) simulation using *LILAC*, a 1-D Lagrangian radiation–hydrodynamic code^{29} routinely used for target designs at the Laboratory of Laser Energetics.

We begin by studying the effects of hot-spot fluid motion on broadening of the neutron energy spectra using the nonrelativistic neutron energy spectrum model.^{1} We substitute the mean neutron energy of Murphy^{6} Eq. (3)

into Brysk^{5} Eq. (36), where $E0=m\alpha *Q/(mn+m\alpha )=14.06$ MeV with the DT nuclear energy release *Q *=* *17.6 MeV, *m*_{n} is the mass of the neutron, *m _{α}* is the mass of the alpha particle, $v\u2192$ is the fluid velocity in the laboratory frame of D-T ion pairs inside the distorted hot spot, and the unit vector $d\u0302=sin\u2009\theta \u2009cos\u2009\varphi i\u0302+sin\u2009\theta \u2009sin\u2009\varphi j\u0302+cos\u2009\theta k\u0302$ is the direction along the LOS where neutrons are detected. The term $\u27e8v\u2192\xb7d\u0302\u27e9$ is calculated by the spatial neutron averaging of the hot-spot fluid velocity components parallel to the LOS unit vector $d\u0302$. Neutron-averaged quantities are defined by $\u27e8Q(t)\u27e9=\u222bQ(x\u2192,t)nDnT\u27e8\sigma v\u27e9DTd3x/\u222bnDnT\u27e8\sigma v\u27e9DTd3x$, where $\u27e8\sigma v\u27e9DT$ is the D–T fusion reactivity calculated by the Bosch and Hale model.

^{30}

*n*

_{D}and

*n*

_{T}are the D and T hot-spot ion number densities, respectively. The neutron energy spectrum

*f*(

*E*

_{n}) in the laboratory frame of D–T ion pairs is characterized by the Gaussian distribution with a nonrelativistic Doppler shift term $\u27e8v\u2192\xb7d\u0302\u27e92mnE0$ in the mean neutron energy

where $Tithermal$ is the neutron-averaged hot-spot thermal ion temperature. The flow effect is excluded in the definition of neutron-averaged thermal ion temperatures.

The Doppler broadening of the neutron energy spectra observed in experiments is a result of averaging all neutron energy spectra $\u27e8f(En)\u27e9avg$ produced by different Doppler shifts over the burn distribution within the 3-D distorted hot spot

In the neutron energy spectrum model, hydrodynamic data from *DEC3D* including the mass density, the fluid velocity, and the thermal ion temperature are post-processed according to Eq. (2) to generate the neutron energy spectra for different fluid elements, followed by the spatial neutron-averaging defined by Eq. (3) to obtain the final shape of the neutron energy spectrum to account for the collective effect of bulk fluid motion. *IRIS3D* is applied to post-process *DEC3D* hydrodynamic data to benchmark the result of the neutron energy spectrum model. *IRIS3D* performs Monte Carlo simulations to track different types of scattering events including neutron-Deuterium and neutron-Tritium. The ion temperature inferred from *IRIS3D* is fitted with a Gaussian at the full-width-half-maximum (FWHM) of the neutron energy spectrum contributed by all fluid elements. The neutron energy spectrum model has the advantage to visualize the Doppler-shifted neutron energy spectra produced by different fluid elements such as the cold bubbles or the high temperature hot-spot core. In contrast, *IRIS3D* only provides the final shape of the neutron energy spectrum at a particular LOS but does not output the neutron energy spectrum for any chosen fluid element. Since low-mode perturbations exhibit larger neutron-averaged hot-spot residual kinetic energies than high modes, the presence of low-mode asymmetries in ICF experiments can significantly modify the shape of the neutron energy spectra.

To benchmark the result of the neutron energy spectrum model, inferred ion temperatures are compared with *IRIS3D* simulations at varying LOS angles *θ* from the north to the south poles for different single modes *ℓ* = 1 to 12. Figure 1(a) shows the *DEC3D* profiles of the mass density and the velocity vectors for low mode *ℓ* = 1. The neutron-averaged hot-spot flow velocity $\u27e8vz\u27e9=\u2212176\u2009km/s$ for mode *ℓ* = 1 is large enough to produce Doppler shift by a magnitude of $\u27e8v\u2192z\xb7d\u0302\u27e92mnE0=\u221297z\u0302\xb7d\u0302$ keV. The sign depends on the position of the detector in the $z\u0302\xb7d\u0302$ term. In OMEGA experiments, the typical neutron-inferred hot-spot flow velocity^{12} is ∼40 km/s and reaches ∼100 km/s in shots with large variations in ion temperature measurements. Figure 1(b) shows the agreement of inferred ion temperatures for mode *ℓ* = 1 simulated by the neutron energy spectrum model and *IRIS3D*. The Doppler shift term is shown vanishing when the LOS is located at the equator, resulting in the minimum inferred ion temperature.

To study signatures of different single modes in ion temperature measurement variations, a direct comparison of the inferred ion temperature ratio $Ti,maxinferred/Ti,mininferred$ for the single mode spectrum *ℓ* = 1–12 simulated by the same magnitude of 7% initial velocity perturbation is investigated in Fig. 2 using the neutron energy spectrum model and *IRIS3D*. In OMEGA experiments, the averaged value of the experimental ion temperature ratio is ∼1.18 and the standard deviation of $Tmax\u2009exp\u2009/Tmin\u2009exp\u2009$ is ∼0.14. The study of single mode simulations at 7% initial velocity perturbation corresponds to the inferred ion temperature ratio ∼1.4 for mode *ℓ* = 1 in Fig. 2(a), which is close to observed values in shots with large ion temperature asymmetries exhibiting one standard deviation higher than the averaged experimental ion temperature ratio. Figure 2(a) shows that mode *ℓ* = 1 has the largest inferred ion temperature variation in the single-mode spectrum, and the inferred ion temperature variation is shown decreasing with Legendre mode number. The unique characteristics of the flow structure for different single modes are studied in Fig. 2(b) by comparing the maximum with the minimum inferred ion temperatures. The minimum inferred ion temperature shows the magnitude of isotropic velocity variance, while the difference in the maximum and minimum inferred ion temperatures shows the magnitude of anisotropic velocity variance. For mode *ℓ* = 1, the highly directional jet contributes to a large Doppler shift of the mean neutron energy while the vortex structure contributes to a large anisotropic velocity variance. The magnitude of $Ti,maxinferred\u2212Ti,mininferred$ decreases with Legendre mode number, showing that low modes have a large source of anisotropic velocity variance. Figure 2(b) shows that mode *ℓ* = 2 has a large minimum inferred ion temperature caused by the presence of a large radial flow structure.

The growth of symmetric converging RT spikes in the single-mode perturbation traps the vortices of high modes inside the cold bubbles, where the rate of neutron production is low. After averaging the Doppler-shifted neutron energy spectra contributed from the high-velocity vortices, the overall effect of Doppler velocity broadening remains weak because the amount of Doppler-shifted neutrons produced inside the cold bubbles is low and makes only a small contribution to the averaged neutron energy spectrum $\u27e8f(En)\u27e9avg$. Figure 3(a) shows that the *DEC3D* residual kinetic energy density of vorticity at stagnation is highly localized inside the cold bubbles for high mode *ℓ* = 40 and *m *=* *20 and vanishes inside the clean volume enclosed by *T*_{e} = 1-keV contour surface. Figure 3(b) is the *DEC3D* mass density profile of mode *ℓ* = 40, showing the symmetric converging RT spikes for the large *ℓ* mode in the single-mode simulation.

Figures 4(a) and 4(b) show the different behaviors of the Doppler shift term $\u27e8v\u2192\xb7d\u0302\u27e92mnE0$ in the neutron energy spectrum model between the low mode *ℓ* = 1 and high mode $\u2113=40,m=20$. For mode *ℓ* = 1, the jet flows along the negative *z* direction, and the sign of the $\u27e8vz\u27e9z\u0302\xb7d\u0302$ term explains the downward-shifted red curve in Fig. 4(a) when the detector is located at the north pole $d\u0302=z\u0302$ and the upward-shifted blue curve when the detector is located at the south pole $d\u0302=\u2212z\u0302$. The black curve with the peak neutron mean energy at *E*_{0} in Fig. 4(a) is the neutron energy spectrum without modeling the Doppler shift term in Eq. (3). Additionally, the $\xb1z$ spectra are broadened. Although the high-velocity vortices produce a large Doppler shift for neutrons produced inside the cold bubbles shown by the blue curve in Fig. 4(b), the overall effect of Doppler shift remains dominant by the large number of neutrons produced inside the high-temperature clean volume with negligible neutron-averaged hot-spot fluid velocities $\u27e8v\u2192\xb7d\u0302\u27e9large\u2009\u2113single\u2212mode\u22430$, which is shown by the red curve in Fig. 4(b). As a result, a vanishing effect of Doppler shift of the mean neutron energy is observed for large *ℓ* single modes, which explains the unshifted overall neutron energy spectrum shown by the black curve for mode $\u2113=40$ in Fig. 4(b). The property of symmetric converging RT spikes is lost in multimode simulations. The high-velocity vortices near the base of the bubbles experience a higher temperature than those located at the tip of bubbles. The nonvanishing contribution to $\u27e8v\u2192\xb7d\u0302\u27e9large\u2009\u2113multimode>0$ in the multimode asymmetry is described in Sec. IV.

## III. SINGLE-MODE VELOCITY VARIANCE

To quantify the effect of hot-spot flow asymmetry on variations in ion temperature measurement, the properties of velocity variance in the Brysk ion-temperature model^{5} are analyzed through a systematic vector decomposition of the $v\u2192\xb7d\u0302$ term. We define the normalized ion temperature $T\u0302i\u2261Ti/(mn+m\alpha )$ and rewrite Brysk ion temperature in Murphy^{6} Eq. (20) as

where the directional-dependent velocity variance is

The hot-spot fluid velocity vector $v\u2192$ parallel to the LOS unit vector $d\u0302$ is decomposed into three different velocity components along three Cartesian orthogonal axes. The inner product of the $v\u2192\xb7d\u0302$ term is expanded into

where the LOS unit vector $d\u0302=gxx\u0302+gyy\u0302+gzz\u0302$ is determined by three geometrical factors. $gx=sin\u2009\theta \u2009cos\u2009\varphi ,gy=sin\u2009\theta \u2009sin\u2009\varphi $, and $gz=cos\u2009\theta $ as described in Eq. (1). The $(v\u2192\xb7d\u0302)2$ term is expanded into

Substituting the expansions of vector decomposition in Eqs. (6) and (7) into the neutron-averaged brackets in Eq. (5), the velocity variance is

The factor of 2 is included by the symmetric summation in the second term of Eq. (8). The summation indices that define the Cartesian coordinates $1=x,\u20092=y$, and $3=z$ will be used interchangeably. The shorthand notation for the directional dependent variance *σ _{ii}* (with

*i*=

*j*) and covariance

*σ*(with $i\u2260j$) is defined as

_{ij}Therefore, the Brysk ion temperature or the inferred ion temperature in Eq. (4) is well defined by a complete set of six hot-spot flow parameters in terms of three directional variances (*σ _{xx}*,

*σ*, and

_{yy}*σ*) and three covariances (

_{zz}*σ*,

_{xy}*σ*, and

_{yz}*σ*), while the LOS effects enter through three geometrical factors

_{zx}*g*,

_{x}*g*, and

_{y}*g*. The terminology of directional variances with subscripts

_{z}*xx*,

*yy*, and

*zz*is used to distinguish them from the velocity variance defined in Eq. (5). The compact form of Eq. (4) is

Equation (11) is studied in Fig. 5 by comparing the ratio of the maximum to the minimum inferred ion temperatures simulated by *IRIS3D* with the Brysk ion-temperature model using the direct computation of the variance and covariance of the hot-spot fluid velocity properties defined by Eq. (11). The ratio of $Ti,maxinferred/Ti,mininferred$ for midmodes *ℓ* = 5 to 12 is shown smaller than that for low modes *ℓ* = 1, 3, and 4 due to small magnitudes of neutron-averaged hot-spot fluid velocities. When the hot spot is significantly distorted, different warm bubbles exhibit different thermal ion temperatures. As a result, Fig. 5 shows an increasing discrepancy of Brysk ion temperature ratio from that of *IRIS3D* measurement. A single neutron-averaged thermal ion temperature is assumed in the analysis of Brysk ion-temperature model. At ion temperature ratio $Tmaxinferred/Tmininferred=1.4$ in Fig. 5, which is comparable to that of OMEGA experiments with large ion temperature variations, the data for mode $\u2113=4$ are about 5% higher than the *Y *=* X* curve. The analytic model shows good accuracy relative to the typical ion temperature variations in experiments.

The resulting inferred ion-temperature measurement variation denoted by $\u25b3T\u0302i=T\u0302iinferred\u2212T\u0302ithermal$ is a collective effect from the six hot-spot flow parameters through a linear superposition relation

Equation (12) can be generalized into a system of linear equations that connects the complete set of six hot-spot flow parameters $\sigma \u2192=(\sigma xx,\sigma yy,\sigma zz,2\sigma xy,2\sigma yz,2\sigma zx)$ with ion-temperature measurement variations $\u25b3T\u2192LOS$ observed at six LOSs through a 6 × 6 invertible line-of-sight matrix $M\u0302LOS3D$, which is determined by six pairs of LOS angles $\theta LOS$ and $\varphi LOS$

For two-dimensional (2-D) planar flows with translational symmetry in the *z* direction, $\sigma zz=\sigma zx=\sigma yz=0$ and the linear system is reduced to an invertible 3 × 3 matrix

with the 2-D determinant (det) of $M\u0302LOS2D$

where $f=sin2\theta 1\u2009sin2\theta 2\u2009sin2\theta 3,\u2009\u25b3\varphi 12=\varphi 1\u2212\varphi 2,\u2009\u25b3\varphi 23=\varphi 2\u2212\varphi 3$, and $\u25b3\varphi 13=\varphi 1\u2212\varphi 3$. The determinant $det2D$ is nonzero for any two pairs of nonparallel LOSs, i.e., $\varphi i\u2212\varphi j\u22600,\pi $. In 3-D, the six LOSs cannot be parallel with each other to avoid forming a singular LOS matrix and should be chosen to maintain a small condition number for the matrix $M\u0302LOS3D$ to minimize the propagation of errors in ion-temperature measurements. The matrix representation of velocity variance in the Brysk ion-temperature model can be utilized to approximate the true minimum of inferred ion temperature through six ion-temperature measurements. The terminology of Brysk or matrix models will be used interchangeably. We define the 6 × 1 column vector $T\u2192i$ of six inferred ion-temperature measurements, the 6 × 1 column vector $T\u2192th$ of the thermal ion temperature, and the 6 × 6 LOS matrix $M\u03020\u2261M\u0302LOS3D$

where ${e\u0302j=1,\u2026,6}$ is the six orthonormal base vectors. Let $T\u0302p$ be the prediction of the inferred ion temperature at arbitrary LOS angles $\theta p$ and $\varphi p$, which uniquely specifies the LOS matrix $M\u0302p$ and matrix elements $gigj(\theta p,\varphi p)$

The solution of directional variances and covariances vector $\sigma \u2192$ is first obtained by multiplying the inverse of the old LOS matrix with the vector of the old six ion-temperature measurements

The prediction of inferred ion temperature at the new LOS is

The first term on the right-hand side of Eq. (19) is denoted by the matrix $\delta \u0302p,0$ which characterizes the departure of the new LOS from the old one

The exact relation predicting the inferred ion temperature at the new LOS is written as

When predictions of inferred ion temperatures are near the vicinity of the old LOS, the $\delta \u0302p,0\xb7T\u2192th$ term approaches zero. Therefore, an approximated solution of the inferred ion temperature in the neighborhood of the old LOS is

and the minimum inferred ion temperature in the 4*π* Hammer map for all angles $\theta p$ and $\varphi p$ is

To validate the vanishing contribution of the $\delta \u0302p,0\xb7T\u2192th$ term, six LOSs at the same neutron time-of-flight (nTOF) locations in OMEGA are chosen. Note the red dots on the Hammer map in Fig. 6. The positions of these six LOS angles complete all matrix elements for $M\u0302LOS3D$ or its inverse $M\u03020\u22121$. In this configuration, the following vector can be directly computed:

with vector elements

By neglecting the two small constants $\alpha j=8.9\xd710\u221216$ and $\beta j=\u22122.2\xd710\u221216$ in Eq. (25), we have $aj\u22431$ in Eq. (24), resulting in well-approximated zero vector components of $\delta \u0302p,0\xb7T\u2192th=(I\u0302\u2212M\u0302p\xb7M\u03020\u22121)\xb7T\u2192th\u22430\u2192$. Figure 6 shows the numerical test to estimate $\delta \u0302p,0(\theta p,\varphi p)\xb7T\u2192th$ at all angles $\theta p$ and $\varphi p$ by directly computing the matrix $\delta \u0302p,0$ using Eq. (20) and indicates that the magnitudes of the vector components for $\delta \u0302p,0(\theta p,\varphi p)\xb7T\u2192th\u223c10\u221215$ keV are negligibly small. In the 2-D numerical test, however, the vector components are observed to be nonzero. For example, by using fixed LOS angles $\theta 1=\theta 2=\theta 3=\pi /2$ in $M\u0302LOS2D$, the nonzero contribution is observed to be $\delta \u0302p,0(\theta p,\varphi p)\xb7T\u2192th=(1,1,1)\u2009cos2\theta pTithermal$ and remains nonzero for other choices of three LOS.

Figure 7(a) shows the reconstruction of the 4*π* approximated prediction of inferred ion temperature using Eq. (22) based on six inferred ion temperatures from six LOSs for mode *ℓ* = 1 perturbation obtained from an *IRIS3D* simulation. The neutron-averaged thermal ion temperature $Tithermal=3.551$ keV for mode *ℓ* = 1 is shown to be in agreement with the prediction of minimum inferred ion temperature $Ti,minappro.inf=3.525$ keV obtained by Eq. (23). Figure 7(b) shows that the predicted inferred ion measurement variations using Eq. (21) from the north to south poles for mode *ℓ* = 1 at the fixed angle *ϕ* = 0 agree with those simulated by *IRIS3D* and the neutron energy spectrum model, both using 16 LOSs uniformly distributed over the azimuthal angle $\theta \u2208[0,\pi ]$. However, the neutron energy spectrum model exhibits some discrepancy at the north and south poles due to its simple approach of superposition neutron energy spectra from different fluid elements to model the bulk velocity broadening phenomenon. Techniques described in Refs. 7–9 provide the exact shape of the neutron energy spectrum.

Equation (12) can be used to explain the symmetric variation of inferred ion temperature for mode *ℓ* = 1 from the north to the south poles. From the single-mode database, the magnitudes of single-mode covariances are shown to be negligibly small. The 3-D single-mode variation in inferred ion temperature is, therefore, dominated by the three directional variance terms according to Eq. (12)

The LOS dependence enters through the square of the three geometrical factors $gx2=sin2\theta \u2009cos2\varphi ,\u2009gy2=sin2\theta \u2009sin2\varphi $, and $gz2=cos2\theta $, while three directional variances *σ _{xx}*,

*σ*, and

_{yy}*σ*of the hot-spot fluid velocity distribution determine the relative amplitudes of variations in ion temperature measurement. For rotational symmetric modes defined by

_{zz}*m*=

*0, the property of directional variance*

*σ*=

_{xx}*σ*further leads to a simplified approximation of 2-D single-mode inferred ion-temperature variation

_{yy}Figure 8(a) investigates the magnitudes of directional variances in the mode spectrum. For mode *ℓ* = 1, the property of $\sigma xx+\sigma yy\u226a\sigma zz$ indicates that the last term $\sigma zz\u2009cos2\theta $ in Eq. (27) dominates. Therefore, the inferred ion-temperature variation for mode *ℓ* = 1 is approximated by $\u25b3T\u0302i\u2113=1\u2243\sigma zz\u2009cos2\theta $, which is in agreement with the sinusoidal curve shown in Fig. 7(b). Although the magnitudes of *σ _{zz}* for modes

*ℓ*= 1, 3, and 4 are close, the vortex structure of mode

*ℓ*= 1 has the least non-translational fluid motion on the

*x*–

*y*plane, resulting in the largest anisotropic velocity variance $\sigma zz\u2212\sigma xx$ or $\sigma zz\u2212\sigma yy$ as shown in Fig. 8(a).

Mode *ℓ* = 2 is shown to exhibit the largest total directional variance $\sigma x2+\sigma y2$ in Fig. 8(a) because of the high degree of radial flow structure of mode *ℓ* = 2 on the *x* – *y* plane. Figure 9 shows that mode *ℓ* = 4, *m *=* *2 has variations in the sinusoidal ion-temperature measurement in the azimuthal angle *ϕ*. This feature is caused by the unequal directional variances $\sigma xx\u2260\sigma yy$, and the period of 2 is due to $\u2009sin2\varphi $ and $\u2009cos2\varphi $ terms in Eq. (26). Two-dimensional *m *=* *0 modes are shown to have larger directional variances in the *z* direction than in the *x* and *y* directions. Since the central spike for 2-D *m *=* *0 modes along the *z* axis behaves as a 3-D spike because of the azimuthal rotational symmetry, the central spike grows faster than the 2-D RT spike, resulting in a higher degree of fluid motion in the *z* direction.

Figure 8(b) compares the minimum inferred ion temperature given by Eq. (23) for different single modes in which the six inferred ion temperatures are simulated by *IRIS3D*. The property of large isotropic velocity variance for mode *ℓ* = 2 is shown, leading to more than 10% higher minimum inferred ion temperature than the neutron-averaged thermal ion temperature. The 3-D effect of variation in the azimuthal ion temperature measurement for mode *ℓ* = 4 and *m *=* *2 is caused by different values of directional variances in *x* and *y* directions, i.e., $\sigma x2\u2260\sigma y2$, and in the same way for all $m\u22600$ modes, and $\sigma x2=\sigma y2=\sigma z2$ for the unperturbed simulation. Based on the Brysk ion-temperature model in Eq. (4), it implies that the variation among different directional variances is the source of variation in the inferred ion temperature measurement. The directional variance in Eq. (26) can be rewritten into a sum of a directional-independent isotropic velocity variance $\sigma iso2$ and a fluctuation of directional variance $\u25b3\sigma ii$

The effect of isotropic velocity variance on the minimum inferred ion temperature can be explained by substituting Eq. (28) into Eq. (12). Here, the vector property $\u2211i=13gigi=d\u0302\xb7d\u0302=1$ is used so that the isotropic velocity variance $\sigma iso2$ exists uniformly in the three different orthogonal directions

In this analysis, the fluctuation of directional variance is shown to be the origin of inferred ion-temperature variation, while the isotropic velocity variance is shown to cause a higher minimum inferred ion temperature. Equation (12) can be written as

The isotropic velocity variance $\sigma iso2$ should be subtracted from all directional variance *σ _{ii}* in order to shift the minimum inferred ion temperature in Eq. (22) down to the level of thermal ion temperature. By neglecting the covariance terms in Eq. (30), a reasonable form of single-mode isotropic velocity variance can be assumed as the minimum value among directional variances

*σ*,

_{xx}*σ*, and

_{yy}*σ*in order to remove the uniform background of directional variance as explained by Eqs. (28)–(30)

_{zz}An improved approximation for the thermal ion temperature is obtained by separating the isotropic velocity variance using Eq. (31) from the minimum inferred ion temperature of Eq. (22)

The improved approximation for the thermal ion temperatures using Eq. (32) is validated by the blue squares in Fig. 8(a). After separating the isotropic velocity variance, the minimum inferred ion temperatures indicated by the red circles are all down to the level of neutron-averaged thermal ion temperatures for all different single modes, with 3% uncertainty. Low mode *ℓ* = 1 is shown to exhibit the least isotropic velocity variance, and its minimum inferred ion temperature is close to the neutron-averaged thermal ion temperature with 1% uncertainty, even without separating the isotropic velocity variance. When the method of isotropic velocity variance separation is applied to the 4*π* minimum of Brysk ion temperatures given by Eq. (11), the improved approximation of thermal ion temperatures is observed to shift down precisely to the level of true thermal ion temperatures.

However, the solution of isotropic velocity variance given by Eq. (31) is obtained from calculating the directional variance using the hydrodynamic data of simulations. $\sigma iso2$ cannot be determined in experiments. This limitation is caused by the fact that both isotropic velocity variance and thermal ion temperature are directionally independent. As a result, the minimum inferred ion temperature includes a nonseparable contribution from the isotropic fluid motion within the hot spot.

In experiments, when implosions are dominated by low mode *ℓ* = 1, the large variation of ion temperature measurements occurs along the same direction of the neutron-averaged hot-spot flow velocity. However, the nature of the large anisotropic velocity variance is caused by the spatially varying neutron-averaged fluid velocity distribution driven by the large vortex structures. A jet with uniform flow velocity distribution in homogenous fusing plasma has zero velocity variance. The signature of a plasma jet is manifested in the shift of the mean neutron energy in Eq. (1).

The matrix formalism can be extended to the improved Munro model to account for relativistic kinematics. The method of vector decomposition can be applied to expand each high order correction term in Munro^{8} Eq. (74), which in turn requires more lines of sights ion temperature measurements. For 14 MeV neutrons, the relativistic correction is approximately to be <5% as indicated in Refs. 8 and 9. A more accurate evaluation of Munros relativistic model is under investigation.

In applications, the true minimum inferred ion temperature provides more realistic evaluations of the true thermal temperature. The reconstruction of inferred ion temperature profile through six LOS ion temperature measurements can also be used to infer the orientation of the neutron-averaged hot-spot flow velocity. Since the latter is well correlated with the neutron-inferred areal density variations in the presence of low modes, the relation between the variations of ion temperature and areal density measurements can be used to understand the signature of low mode asymmetries in experiments.

## IV. MULTIMODE VELOCITY VARIANCE

Figure 10(a) investigates the effect of rotational asymmetry on the inferred ion-temperature variation for a two-mode simulation that has a dominant low mode *ℓ* = 1 with a small perturbation of high mode *ℓ* = 10 and *m *=* *5. The azimuthal symmetry is shown broken slightly due to the presence of odd-*m* mode, and significant 3-D variations of inferred ion temperatures on the Hammer map are observed in Fig. 10(b). The 3-D effect of non-zero covariance terms denoted by $\u25b3T\u0302icov$ in Eq. (8) is considered separately

The two bright spots observed in Fig. 10(b) are caused by the period of 2 in angles *θ* and *ϕ*. Since the covariance terms can be negative, the 4*π* minimum of the covariance terms $\u25b3T\u0302icov$ should be included inside the minimum bracket in Eq. (31) to estimate the value of isotropic velocity variance for multimode perturbations.

Figure 11 compares the ratio of the maximum to the minimum inferred ion temperature and the yield-over-clean (YOC = 2-D or 3-D fusion yield/1-D clean yield) in *DEC3D* two-mode simulations: (1) a dominant low mode *ℓ* = 1 with increasing initial perturbations of high mode *ℓ* = 10, *m *=* *5; (2) a dominant high mode *ℓ* = 10, *m *=* *5 with increasing initial perturbations of low mode *ℓ* = 1. The steeper rise of inferred ion-temperature variations observed for the dominant high mode *ℓ* = 10 simulations can be explained by diagnosing the changes in directional variances from weak to strong perturbations. The steep rise of the inferred ion-temperature variation for the dominant mode *ℓ* = 10 case is caused by the rapid increasing fluid motion in the *z* direction from $\sigma zzkeV=0.44$ to $\sigma zzkeV=1.82$, and the fluid motions on the *x* – *y* plane was found to increase slightly from $\sigma xxkeV=0.294$ to $\sigma xxkeV=0.325$. The superposition of mode *ℓ* = 1 with mode *ℓ* = 10 significantly increases the flow asymmetry in the *z* direction, which is well indicated by the behavior of *σ _{zz}*. The thermal ion temperature of high-velocity vortices at the base of the cold bubbles starts to increase as RT spikes become more radially nonsymmetric in the multimode perturbations so that the high-velocity hot vortices of high modes start to contribute more Doppler-shifted neutrons and broaden the final shape of neutron energy spectrum.

Figure 12(a) shows an inferred ion-temperature profile simulated by *IRIS3D* for a strongly perturbed 3-D multi-mode simulation with YOC = 0.36. The initial perturbation is a spectrum $\u25b3v/v0\u2211\u2113=112Y\u2113m=\u2113/2$ consisting of modes *ℓ* = 1 to 12, *m *=* *0 for odd-*ℓ* modes and *m* = *ℓ*/2 for even-*ℓ* modes, with uniform initial velocity perturbations $\u25b3v/v0=0.14$. The directional variances are $\sigma xxkeV=1.2,\u2009\sigma yykeV=0.74$, and $\sigma zzkeV=0.76$, indicating the existence of a dominant jet structure as indicated by the velocity field profile shown in Fig. 12(b). The motion of the jet on the *x* – *y* plane can be read from the neutron-averaged hot-spot fluid velocities $\u27e8vx\u27e9=\u22127.6$ km/s and $\u27e8vy\u27e9=\u22125.4$ km/s, roughly indicating the orientation of the jet from $(\theta =\pi /2,\varphi 1=0.95)$ to $(\theta =\pi /2,\varphi 2=4.1)$, which accounts for the formation of the two asymmetric bright spots observed in Fig. 12(a). Figure 13(a) shows the reconstruction of a 3-D inferred ion-temperature profile by the Brysk ion-temperature model using Eq. (11) and is in good agreement with the *IRIS3D* simulation in Fig. 12(a). The magnitudes of covariance terms $\sigma xykeV=0.26,\u2009\sigma yzkeV=\u22120.048$, and $\sigma zxkeV=\u22120.089$ in the high degree of flow asymmetry in the strongly perturbed multimode simulation are large enough to cause significant 3-D effects. The variations in inferred ion-temperature measurements caused by directional variances and covariances are investigated separately in Figs. 13(b) and 13(c). They show that the formation of the asymmetric bright spot located at *θ* = *π*/2 and *ϕ* = 0 in Fig. 13(a) is driven by the large covariance terms.

## V. HOT-SPOT RESIDUAL KINETIC ENERGY

The effect of Doppler velocity broadening on neutron energy spectra can be used to infer the properties of the hot spot's nontranslational residual kinetic energy. The sum of three measurements of ion temperature at orthogonal LOSs along with $x1=x,\u2009x2=y$, and $x3=z$ directions is related to the thermal ion temperature and the total velocity variance. In this configuration, the covariance terms vanish exactly

The appearance of isotropic velocity variances causes larger averaged ion temperatures than the thermal ion temperature by the amount of $\sigma iso2$. Since geometrical factors or LOS effects do not appear in the sum of ion-temperature measurements at three orthogonal axes, the apparent ion temperature is further increased by the amount of $\u25b3\sigma xx2+\u25b3\sigma yy2+\u25b3\sigma zz2$ as a result of anisotropic velocity variances. The total velocity variance and the hot spot nontranslational fluid motion are closely related. The hot-spot fluid velocities in three orthogonal directions are decomposed into a translational component of neutron-averaged linear velocity $\u27e8vi\u27e9$ and a nontranslational component of velocity fluctuation denoted by $\u25b3vinontrans$

Substituting Eq. (35) into the definition of directional variance $\sigma ii=\u27e8(vi\u2212\u27e8vi\u27e9)2\u27e9$, the expression of total velocity variance is equivalent to the inner product of total nontranslational hot-spot fluid velocities

We introduce the total nontranslational hot-spot residual kinetic energy (KE)

where $MHSb=12nibmDTVHSb$ is the neutron-averaged hot-spot mass, $nib$ is the neutron-averaged hot-spot total ion number density, $VHSb$ is the neutron-averaged hot-spot volume, *m*_{DT} is the DT ion mass, and the superscript b denotes spatial neutron-averaging. By expressing the total velocity variance in terms of the hot-spot nontranslational residual kinetic energy using Eq. (37), Eq. (34) can be written into

The product of neutron-averaged hot-spot mass and the normalized thermal ion temperature can be simplified into $MHSbT\u0302ithermal\u2243IEHSb/6$. The thermodynamic equilibrium between electron and ion temperatures is assumed to give the neutron-averaged hot-spot total pressure $PHSb=2nibTithermal$, which is a typical approximation in the deceleration phase hydrodynamic models^{31} to express the ion-temperature-dependent fusion reactivity as a function of the total gas pressure. $mDT\u2243mn+m\alpha $ is assumed by neglecting the small mass deficit because of the release of nuclear fusion energy, and $IEHSb=32PHSbVHSb$ is the neutron-averaged hot-spot internal energy. Therefore, the average of three ion temperatures measured at orthogonal directions in Eq. (38) is related to the ratio of neutron-averaged nontranslational hot-spot residual kinetic energy to the neutron-averaged hot-spot internal energy

Equation (39) is validated in Fig. 14(a), in which the nontranslational hot-spot kinetic energy is replaced by the total hot-spot kinetic energy $KEHStotal$ because their magnitudes are close: $KEHStotal\u2243KEHSnontrans$. This approximation is valid for modes $\u2113\u22652$ with a small amount of translational hot-spot residual kinetic energy, but it is not valid for mode *ℓ* = 1. Figure 14(b) investigates the effect of isotropic velocity variance by comparing the minimum inferred ion temperatures obtained by Eq. (23) with the thermal ion temperatures in the mode spectrum. Mode *ℓ* = 1 is shown to exhibit the least isotropic flow so that its minimum inferred ion temperature is close to the thermal ion temperature. Mode *ℓ* = 2 is shown to exhibit a large deviation between minimum inferred temperature and thermal temperature because of a large isotropic velocity variance. The intimate relation between variations in ion temperature measurement and residual kinetic energy can be shown by considering the conservation of total energy at the time of stagnation between 3-D and 1-D implosions $IEHS,3Db/IEHS,1Db=1\u2212RKEtotal\u2212RIESH$, where the shorthand notations for normalized total residual kinetic energy RKE_{total}, the normalized neutron-averaged hot-spot residual kinetic energy $RKEHSb$, and the normalized shell internal energy RIE_{SH} are defined by

The ratio of the neutron-averaged hot-spot residual kinetic energy to the neutron-averaged hot-spot internal energy can be rewritten as $KEHS,3Db/IEHS,3Db=RKEHSb/(1\u2212RKEtotal\u2212RIESH)$. Except for low mode *ℓ* = 1, the neutron-averaged nontranslational hot-spot residual kinetic energy in Eq. (39) can be well approximated by the neutron-averaged total hot-spot residual kinetic energy since other modes contain less translational hot-spot fluid motions, i.e., $KEHSnontrans/IEHSb\u2243KEHS,3Db/KEHS,3Db$. Therefore, the summation of inferred ion temperatures at three orthogonal directions in Eq. (39) is a function of residual kinetic energies

Equation (41) shows that the increasing inferred ion temperatures are caused by the nontranslational hot-spot residual kinetic energy. Only when $KEHSnontrans=0$ in Eq. (38), the average of inferred ion temperatures measured at three orthogonal axes is equal to the thermal ion temperature. Figure 14(c) shows that Eq. (41) is valid for all modes $\u2113\u22652$. Mode *ℓ* = 1 is the outlier because it has large translational hot-spot fluid motions that violates the approximation of $KEHSnontrans\u2243KEHS,3Db$ in Eq. (41).

## VI. CONCLUSION

In summary, we present a detailed study of the velocity variance of the Brysk ion-temperature model^{5,6} for single-mode and multimode Rayleigh–Taylor instabilities in the deceleration phase of ICF implosions. The 3-D effect of hot-spot flow asymmetry on variations in ion-temperature measurements is shown to be uniquely determined by a complete set of six hot-spot flow parameters including three directional variances and three covariances, which are calculated from the distribution of hot spot fluid velocity. An approximated solution of the minimum inferred ion temperature is derived, and the approximated inferred ion-temperature profile over the full 4*π* solid angles is shown to be well reconstructed based on six ion-temperature measurements. The reconstruction of inferred ion-temperature profiles is in agreement with the result of the neutron energy spectrum model^{1} and the neutron transport code *IRIS3D*.^{2} The prediction of minimum inferred ion temperature for low mode *ℓ* = 1 is close to the thermal ion temperature because of its least isotropic velocity variance, but it also exhibits the largest variations in inferred ion temperatures. Low mode *ℓ* = 2 is shown to exhibit a large isotropic velocity variance due to its large radial flow structure on the *x* – *y* plane. The effect of large isotropic velocity variance is shown to cause the minimum inferred ion temperature well above the thermal ion temperature. An improved approximation of the thermal ion temperature is derived by separating the isotropic velocity variance. The presence of nontranslational hot-spot residual kinetic energy is shown to cause larger averaged inferred ion temperatures measured at three orthogonal directions than the thermal ion temperature.

## ACKNOWLEDGMENTS

This material is based upon work supported by the Department of Energy National Nuclear Security Administration under Award No. DE-NA0001944. Partial support was provided by the DOE Office of Fusion Energy Sciences under Grant No. DE-SC0014318. 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.