Neutron time-of-flight diagnostics have long been used to characterize the neutron spectrum produced by inertial confinement fusion experiments. The primary diagnostic goals are to extract the *d* + *t* → *n* + *α* (DT) and *d* + *d* → *n* + ^{3}He (DD) neutron yields and peak widths, and the amount DT scattering relative to its unscattered yield, also known as the down-scatter ratio (*DSR*). These quantities are used to infer yield weighted plasma conditions, such as ion temperature (*T _{ion}*) and cold fuel areal density. We report on novel methodologies used to determine neutron yield, apparent

*T*, and

_{ion}*DSR*. These methods invoke a single temperature, static fluid model to describe the neutron peaks from DD and DT reactions and a spline description of the DT spectrum to determine the

*DSR*. Both measurements are performed using a forward modeling technique that includes corrections for line-of-sight attenuation and impulse response of the detection system. These methods produce typical uncertainties for DT

*T*of 250 eV, 7% for

_{ion}*DSR*, and 9% for the DT neutron yield. For the DD values, the uncertainties are 290 eV for

*T*and 10% for the neutron yield.

_{ion}## I. INTRODUCTION

The National Ignition Facility (NIF) delivers up to 2 MJ of 3*ω* (351 nm) laser light to a hohlraum target containing a spherical capsule, typically filled with deuterium and tritium or deuterium and ^{3}He fuels,^{1,2} with the goal of imploding that capsule and creating thermonuclear fusion. Measurement of neutron spectral details for all implosions which create measurable neutron yield is a goal of the neutron Time-of-Flight (nToF) diagnostic program.

Previous^{3–5} descriptions of nToF data analysis from DT fusion neutrons typically attempted to deconvolve (unfold) the instrument response from the data or used the assumption that the spectrum is a Gaussian in neutron time-of-flight space convolved with a response of instant rise followed by a sum of exponential decays.^{5} We found that deconvolution introduces unphysical artifacts into the data, while the correct interpretation of the nToF signals at NIF requires a more detailed description of the instrument response than a simple model of instant rise with exponential decay can provide. Additionally, we incorporate corrections for the neutron energy dependence of the beam-line attenuation and the scintillator sensitivity. None of these effects were taken into account in above cited work, but we found them necessary to obtain an accurate estimate of the down-scattered neutron spectrum. In order to eliminate possible errors introduced by a two-step process, such as a deconvolution followed by a fit to deconvolved data, we incorporate the convolution into the fit procedure and fit the nToF data directly. For cases where one attempts to get a (physics) model independent description, we offer a spline fit as an alternative to a deconvolution. A comparison of the spline fit to the Richardson Lucy deconvolution algorithm is also provided.

In order to characterize the neutron spectrum of inertial confinement fusion (ICF) implosions, we return several characteristic quantities which depend on the capsule fill. For DT filled capsules, these are the neutron yield from 13 MeV to 15 MeV, the DD neutron yield from 2.2 MeV to 2.7 MeV, the spectral width of both peaks, which depend on the ion temperature and flow^{6} conditions of the plasma. The analysis presented here does not separate these effects and instead reports an apparent temperature labeled *T _{ion}*, which is identical to what previous

^{3–5}methods did. However, it improves on these methods by including the effects of down-scattered neutrons to the peak shape, which reduces the inferred

*T*on high yield shots. It is also important to point out that the analysis technique described here is the first method to obtain DD yield and

_{ion}*T*on cryogenic DT implosions. Another important quantity is the amount of down-scattered neutrons from 10 MeV to 12 MeV relative to the DT yield (the down-scattered ratio,

_{ion}*DSR*) as it relates to the areal density of the cold fuel and is used to diagnose the performance of an ICF experiment. For DD filled capsules, we measure the primary DD neutron yield from 2.2 MeV to 2.7 MeV, and the spectral width of that peak.

The structure of the paper is as follows: First, a brief overview of the nToF data acquisition is given and then the nToF data models are explained, which includes details on components of the fit like the instrument response function (IRF) and the sensitivity of the instrument. Finally, two different models are presented: a physics-based single temperature model allowing to extract apparent *T _{ion}* values and neutron yields for DT and DD reactions and a generic spline model used to extract DT neutron yield and

*DSR*. Example fits for both approaches are shown. Lastly, an overview of how the uncertainties of the deduced values were determined is given.

## II. DATA ACQUISITION

The above described quantities can be measured with high precision nToF spectrometers. Four nToF detectors are located at about 20 m from the Target Chamber Center (TCC) of NIF to measure the neutron spectrum on their respective lines-of-sight. Those detectors use a fast organic scintillator to detect neutrons and the scintillation light is detected with Photomultiplier Tubes (PMT) and Photodiodes (PD). The detector setups have been described in Refs. 7–10. To measure the yield and ion temperatures from the neutrons associated with the DD fusions that take place during DT fusion shots, three of the nToF detectors have higher gain, gated PMTs. To insure the linear response of the photo detectors, the total charge drawn was limited to 5 nC by a combination of timing gates and scintillator light attenuators. The 5 nC limit has been established by laboratory tests of the PMTs as well as comparison of shot data recorded from different PMTs.

Each PMT/PD signal is recorded on a high bandwidth digital oscilloscope (Tektronix DPO7104) with four channels and each has an 8 bit digitizer (≈6 effective bits). To obtain a reliable *DSR* measurement, it is necessary to combine two channels recorded with different sensitivities in order to obtain higher sensitivity in the down-scattered region of the signal while retaining the unclipped peak of the less sensitive channel. To accomplish this, each signal goes through a signal splitter and is recorded with different sensitivities on the four scope channels. The algorithm that stitches those channels together first identifies the two channels to be combined as the most sensitive channel that does not contain any saturated data point and the next more sensitive channel. The uncertainties of each data point are dominated by the scope noise. They are determined by building average and standard deviation over a part of the spectrum containing only the baseline. The average is subtracted from each data point to set the baseline of the combined trace to zero. Both traces are then scaled so that their values correspond to the output voltage of the PMT/PD. Next, the two channels are aligned in time by matching the leading edge of the DT neutron peak. To allow for sub sample shifts, a linear interpolation between data points on the more sensitive channel is used. Finally, each data point of the combined trace is calculated as weighted average of both traces, except for the region in which the more sensitive channel is saturated. It has been observed that after a scope trace clips, it overestimates the signal and requires some time to approach the value of the unclipped trace. Therefore, the clipped scope trace is not used until it agrees within one standard deviation of the less sensitive channel.

## III. FORWARD FITTING nToF SPECTRA

Removing the detector response from the nToF data would require a deconvolution, which is mathematically not a well posed problem. Therefore, a *χ*^{2}-fit was chosen for the nToF scope signals in time with a function determined by parameterizing the neutron spectrum with a model, taking into account the detector sensitivity and the beam line attenuation, and convolving it with the IRF. The fit function takes the following form:

Here, *I*(*E*) denotes the model of the neutron spectrum, *s*(*E*) is the sensitivity of the detector to the given neutron energy, *a*(*E*) is the beam line attenuation factor, *E*(*t* − *t*_{0}) and $dEdt$ are used to convert the energy spectrum into the scope time space, *R*(*t*) is the instrument response, and *t*_{0} is the time offset needed to convert the scope time axis into time-of-flight. The kinetic energy of the neutron *E*(*t*) is given by

with $\gamma =(1\u2212\beta 2)\u221212,\u2009\beta =tt\gamma ,\u2009t\gamma $ being the time-of-flight of a photon from TCC to the detector (given as distance over speed of light) and *m _{n}* is the energy of the neutron at rest due to its mass. Then $dEdt$ is given by

### A. Instrument response function (IRF)

To determine the neutron spectrum accurately, the IRF of the detector has to be known, which is most important in understanding *T _{ion}* and

*DSR*of the DT neutron spectrum. This is due to the width of the IRF being a sizeable fraction of the DT peak width and the tail of the IRF causes a background in the down-scattered part of the neutron spectrum. Therefore, it is important to know the IRF with reasonable precision in order to get low uncertainties for the values of DT

*T*and

_{ion}*DSR*. It is also important to note that due to the asymmetry in the IRF, a direct conversion of the raw nToF signal scope time axis into neutron energy is not possible, i.e., the position of the peak on the scope trace depends not only on the position of the peak in the source spectrum but also on the shape of the source spectrum as well.

Since it is not feasible to create a short impulse of 14 MeV neutrons, which would be necessary to measure the IRF directly, a two step process is used to construct the IRF. First, an X-ray impulse of a timing shot, which has a FWHM of typically ≈100 ps, is used as an approximation of an impulse response for photons. The second step to obtain a neutron IRF is to include the time it takes for a neutron to traverse the scintillator. The IRF consists of many components, which are difficult to model accurately (e.g., scintillator rise and decay, light propagation to the PMT, cable response) and all of those components are included in the X-ray impulse recorded, which has a FWHM of ≈6 ns, where as the neutron propagation time for a DT neutron modeled in MCNP (Monte Carlo N-Particle code) is only 0.5 ns (see Fig. 1). To model this part of the IRF, MCNP simulations of the time dependent neutron energy deposition in the scintillator were performed. The MCNP model included detector housing and any surrounding structures to account for neutrons scattering back into the scintillator. The simulation was done for DD and DT neutrons separately, because of their different transit times through the scintillator. Separate IRFs were created for DD and DT neutrons by convolving the X-ray impulse data with the respective MCNP simulation output. Fig. 2 shows the obtained IRF for 14 MeV neutrons of one of the nToF detectors.

### B. Scintillator sensitivity

The scintillator sensitivity as a function of neutron energy was simulated by a program based on the Stanton^{11} code. The code was modified to tabulate light yield instead of efficiency and pulse height distribution. This code uses the Monte Carlo method to track neutrons traversing the scintillator, taking into account the scintillator geometry. It uses the neutron interaction cross sections to determine the charged particles produced and their light output. The reactions included are elastic proton and carbon scattering and inelastic carbon scattering, including the reactions ^{12}C(*n*, *α*)^{9}Be and ^{12}C(*n*, 3*α*). The cross sections for these calculations were taken from the ENDF/B-VI evaluation. In the code, scattered neutrons are being tracked for secondary interactions until their energy falls below 0.1 MeV. The light output for each charged particle is determined separately using fits to the data given by Verbinski *et al*.^{12} Because the use of a model based on a liquid scintillator light output is not ideal, methods to measure the light output of the Bibenzyl scintillator are currently under investigation (see, e.g., Ref. 13). The sensitivity dependence of the neutron energy is only significant when a large neutron energy range needs to be analyzed as is the case for *DSR* analysis and the *DSR* uncertainty includes a contribution from the scintillator sensitivity model. Fig. 3 shows the energy dependence of the scintillator sensitivity in the form it is used in the data analysis.

### C. Beamline attenuation

The beam line attenuation has been modeled with MCNP to get the fraction of the neutrons which hit the scintillator relative to those started at the TCC. An alternative model which computes the losses from the total interaction cross section of all materials in the beam line was also computed and found to be in good agreement with the MCNP output. The model included the port covers on the target chamber, the port, and clean up collimator, the tungsten plate acting as a photon shield, and the air between target chamber and detector. Fig. 4 shows a basic schematic of the beam line for a detector and Fig. 5 shows the resulting transmission factor as a function of neutron energy.

## IV. MODEL OF THE NEUTRON SPECTRUM

Two different approaches are taken to describe the spectrum. One approach seeks to describe the neutron spectrum in a physics independent form using a spline. The other approach extracts an apparent temperature using models for the neutron peak^{14} and scattering background. These two approaches are being used to extract different quantities: only the second approach allows the extraction of an ion temperature, whereas the spline description only extracts quantities like yield *Y _{n}*,

*DSR*, or

*FWHM*which do not need a physics-based model for extraction.

### A. P-spline fit of the data

In order to extract the neutron spectrum in a generic way, a spline can be used as a model. A spline is a piecewise polynomial function. Different polynomial pieces connect with each other at the so called knot points, where continuity and smoothness conditions are applied. Any spline function used as the model *I*(*E*) can be expressed as a sum of basis splines (B-Splines)

Here, *k* indicates the order of polynomials used with *k* = 4 corresponding to cubic polynomials. Given a set of knots $x0=\cdots =xk<\u2026<xn\u2212k=\cdots =xn$, the B-Splines can be calculated recursively using

One issue with these B-Splines is the choosing of the knots: if there are too many over a given region, then the fitted curve will show more variation than the data justify, whereas too few knots do not allow the curve to change fast enough to fit the data well. To overcome this problem, the P-spline method^{15} uses a dense set of knots and adds a penalty to the *χ*^{2} enforcing a certain smoothness to the fit. The penalty term takes the form

with the penalty matrix *P* defined as

The penalty magnitude *λ* is a positive number that needs to be chosen large enough to suppress oscillations of the solution but low enough to receive a good fit to the data. To choose the optimal *λ*, the penalty term is plotted as function of the residual *χ*^{2} on a log-log-plot for different values of *λ*; this plot is called the L-curve. The *λ* corresponding to the highest curvature of the L-curve is chosen for the best solution. Fig. 6 shows the P-spline fit and the resulting neutron spectrum, while Fig. 7 shows the L-curve for this fit, illustrating how the optimal value for *λ* is found. As can be seen in the energy spectrum of Fig. 6, the spline fit shows artifacts in the low energy part similar to the Richardson Lucy deconvolution. The spline fit does however have several advantages over a deconvolution algorithm:

Most time-domain deconvolution algorithms assume non-negativity of the data and IRF, but cable reflections can make the IRF go negative.

The Richardson Lucy algorithm, for example, assumes the noise to be Poisson distributed, which the scope noise of the nToF data is not.

The definition of the spline as a functional form makes the determination of a FWHM a better defined quantity compared to a deconvolution, which returns a noisy dataset.

The spline fit can model high-frequency regions in a signal (due to flexibility in knot locations) that pose a challenge for deconvolution methods.

Since the P-Spline is described in the neutron energy domain and has no characteristic energy associated with it, it is necessary to know the bang time (defined as the point in time at which peak neutron production occurs) as well as the detector timing to be able to determine *t*_{0} of Eq. (1) in order to convert neutron time-of-flight into neutron energy. The bang time has to be taken from a different target diagnostic at NIF (e.g., SPIDER,^{16} GRH (Ref. 17)) and the time-offset from the fiducial signal has to be determined from an timing shot, which utilizes a gold disk target that produces X-rays as soon as the laser hits the target and can therefore be used to establish absolute timing. It is also advantageous that this spline fit is linear, i.e., the minimum of the *χ*^{2} can be determined by a single matrix inversion.

### B. Physics model of a single temperature plasma with single scatter addition

To extract an ion temperature, the birth spectrum can be modeled as a homogeneous and static, neutron emitting plasma at a single temperature.^{14,18} The neutron energy of a fusion reaction at zero temperature can be calculated from the mass of the neutron *m _{n}*, the mass of the product

*m*, and the

_{r}*Q*-value as

The distribution of the neutron energies assumed has the form

In this model, the mean energy of the distribution and the second central moment are calculated from the *T _{ion}* of the plasma as

and

Both Δ*E _{th}* and

*δ*are calculated with the same expression

_{w}where the values for *α*_{1},…,*α*_{4} are taken from Table III in Ref. 14.

This model uses a total of three parameters: the amplitude of the signal, the ion temperature, and the time offset. For shots, where there is significant scattering, a model for the scattered spectrum has been added, with a fourth parameter, which assumes a single down-scatter of a neutron (multiple down-scatters are insignificant in the neutron spectrum from 15 MeV to 10 MeV). The shape of the scatter spectrum is computed as

where $d\sigma d\Omega $ is the differential cross section and $cos\u2009\theta (CM)$ can be calculated from the initial and final energies of the scattered neutron

with $Ei(lab)$ the initial neutron energy in lab system; $Ef(lab)$ the final neutron energy in lab system; $Et(lab)$ the total energy in lab system ($Ei(lab)+mn+mS$); $Ef(CM)$ the final neutron energy in C.M. system; $Et(CM)$ the total energy in C.M. system; $pi(lab)$ the initial momentum in lab system; $pf(CM)$ the final momentum in C.M. system; *m _{n}* the mass of neutron;

*m*the mass of nucleus that is being scattered on;

_{S}*m*mass of recoiling nucleus (

_{R}*m*+ excitation energy).

_{S}For the calculation of Eq. (12), the differential cross sections of the elastic scattering of neutrons on tritium and deuterium have been fitted with a series of Legendre polynomials in $cos\u2009\theta $

For scattering on deuterium, a fit with nine coefficients (*a*_{0}…*a*_{8}) was chosen to parameterize the differential scattering cross section: 51.6 mb/sr, 52.5 mb/sr, 59.5 mb/sr, −17.4 mb/sr, 20.8 mb/sr, −14.8 mb/sr, 7.73 mb/sr, −4.92 mb/sr, 3.11 mb/sr. The differential scattering cross section on tritium needed only five coefficients: 79.2 mb/sr, 116 mb/sr, 118 mb/sr, 14.8 mb/sr, 14.8 mb/sr. The scattering model used assumed 50% deuterium and 50% tritium mixture as scattering source and the total neutron spectrum hitting the detector are then given as a linear combination of these

The scattering model is assuming a point source of neutrons inside a thin spherical shell of DT fuel. This approximation, neglecting scattering on different materials than DT as well as a simplified geometry, was chosen since it can be computed during the minimization process and the spectral shape it produces was found to be in reasonable agreement with that of simulated spectra.

In total, the fit function has four parameters: a time offset *t*_{0}, an amplitude of the spectrum *A*, the ion temperature *T _{ion}*, and scattering amplitude

*f*. Since the energy spectrum of the neutron is determined by the model, the time offset can be obtained by the fit which allows this analysis to be performed without knowledge of the shot bang time.

_{scat}## V. DATA ANALYSIS

In this section the application of the forward fitting method to actual shot data from the NIF will be examined in detail. The analysis of the nToF data depends on the type of shot being analyzed. For the DD neutron peak on shots without tritium, the ion temperature and neutron yield can be determined from a three parameter fit using Eq. (9). For high purity tritium shots, the TT yield (experimentally defined as neutrons from 5 to 10 MeV) can be determined either using a spline fit or by using an R-matrix model of the TT neutron spectrum as described in Ref. 19. Shots with significant fraction of both tritium and deuterium show both DD and DT neutron peaks which can be fitted as described below.

### A. Analysis of the primary DT fusion neutron peak

Fig. 8 shows an example fit of the model described by Eqs. (9)–(19). In this fit, the ion temperature is extracted directly as a fit parameter, whereas the primary neutron yield is obtained by numerical integration of the fit result from 13 to 15 MeV neutron energy. The uncertainties of the individual data points are dominated by the scope noise which can be seen by the fact that the *χ*^{2} per degree of freedom is typically between 1 and 1.4. To determine the primary neutron yield, it is necessary to obtain a calibration constant in order to be able to convert the charge of the nToF detector into neutron yield. This constant was determined by calibrating against a neutron activation diagnostic^{21} on an exploding pusher^{22,23} shot. Since these shots have almost no compression, the low areal density of the fuel ensures very little down scatter and therefore an isotropic neutron yield making them ideal for calibration purposes. This model fit is used to extract *T _{ion}* and

*Y*, whereas the

_{n}*DSR*is obtained from the spline fit described above. While one can obtain a

*DSR*value from this model fit, it has the disadvantage that the result is sensitive to how well shot data are described by the model (e.g., highly deformed or asymmetric implosions are not well described), whereas the spline fit can adapt to any spectrum. However, the model is the only way to account scattering contributions in the deduction of the apparent ion temperature. It is also the only option to separate the DD peak on DT shots from its background as is described below.

### B. Analysis of DD peak

The peak of the DD neutrons on a DT shot can be recorded using a gated PMT. Fig. 9 shows the spectrum recorded by the gated PMT. The analysis of this data is complicated due to the fact that there are several more components to the signal. The individual components are the neutrons from DD reactions, background from previously recorded signal due to the IRF of the detector, signal from lower energy neutrons, such as down-scattered DT neutrons and neutrons from TT reactions as well as down-scattered DD neutrons.

By comparing the scope trace of a low *ρR* exploding pusher shot with that of a cryogenic layered shot, it can be concluded that the background under the DD peak is dominated by the neutron produced signal rather than scintillator decay from the gated off DT peak. The model for the fit has therefore an additional parabolic component to model the neutron induced background underneath the DD peak in the fit region from *E _{n}* = 2–3 MeV.

Due to the nature of this parabolic background, it is not possible to determine the amount of down scattered DD neutrons directly from the fit. To include the scatter component, an MCNP5 simulation of a capsule was constructed in order to determine the DD scattering contribution and yield correction as a function of DT *DSR*. A spherical hot spot of radius 25 *μ*m was placed inside a spherically concentric capsule with four regions:

low density DT hot spot (

*T = T*), ρ = 20 g/cm_{ion}^{3}, r < 25 μm,low density DT cold plasma (

*T = T*), 25 μm < r< 50 μm,_{rad}high density DT cold plasma (

*T = T*), 50 μm < r< 60 μm,_{rad}high density CH cold plasma (

*T = T*), 60 μm < r< 70 μm,_{rad}

where *T _{ion}* is the ion temperature, and

*T*is the radiation temperature inside the hohlraum (set to be 300 eV for all of these simulations). DT and DD source neutrons were generated using the Ballabio model and chosen to uniformly fill the inner hot spot sphere. The model was run for several values of

_{rad}*T*, DT fuel, and CH ablator densities. The model with areal density of the CH ablator equal to 250 mg/cm

_{ion}^{2}was chosen to match the “typical” reported values of CH

*ρR*from the GRH diagnostic in HiFoot experiments,

^{24}and these simulations are the basis of the values plotted in Fig. 10. While this deduction of the DD down-scatter signal is only an approximation, Fig. 9 shows that the influence of the scattering contribution to the peak shape is actually small despite the fact that a large fraction of the DD neutrons undergoes scatter reactions.

It is important to note that the signal quality of the gated PMTs suffers due to the limitations of the gating system: the gate of the PMT needs considerable time to turn on and stabilize (≈100 ns). This necessitates the use of high light attenuators in front of the PMT which increases considerably the noise in the signal due to limited photon statistics.

The fit shown in Fig. 9 allows the determination of DD *T _{ion}* and DD neutron yield, defined here as the signal coming from the DD peak and its down-scatter. The yield calibration is again obtained relative to a neutron activation diagnostic on an exploding pusher shot.

## VI. UNCERTAINTIES OF DEDUCED VALUES

There are several different sources of uncertainty for the deduced values of *Y _{n}*,

*T*, and

_{ion}*DSR*: statistical uncertainties due scope noise, neutron statistics, and photo electron statistics as well as systematic uncertainties due to calibration, uncertainties in the IRF, and the applied scatter model and the chosen fit range.

For the *Y _{n}* value, the statistical uncertainties are below 1%, whereas the calibration to the neutron activation diagnostics has a 7% uncertainty.

^{21}Another part of the

*Y*uncertainty is the light attenuators used. Currently, the attenuation factor of identical designed light attenuator can vary by as much as 15%. Calibrations are planned for individual filters to lower this uncertainty to ≈5%. Therefore, the dominating uncertainty for DT yield comes from systematics and is of the order of 9%.

_{n}The statistical uncertainties due to photo electrons are small compared to the scope noise, which gets included into the *χ*^{2} of the fit. This changes the fit to have effectively equal weighting of all data points. Comparing fits to simulated data with either equal weighted uncertainties or Poisson error bars shows that the deduced *T _{ion}* can vary by about 150 eV. This systematic uncertainty is introduced due to the scope noise, masking the actual Poisson uncertainty. Another systematic uncertainty arises from the simple scatter model in the fit which assumes a single scatter from a thin shell of DT cold fuel. Comparing fit results from different simulations this uncertainty has been determined to be 50 eV. It has also been observed that the deduced

*T*result changes depending on the chosen range over which the fit minimizes the

_{ion}*χ*

^{2}: the region for fitting the primary DT peak is from 5% of the peak height to the data point corresponding to ≈10 MeV neutron energy. The starting point was chosen to exclude any contributions from tertiary neutrons, while the end point is a good approximation to where the single scatter approximation remains valid. Changing the fit range has been observed to change the deduced

*T*by approximately 100 eV. The ion temperature depends on accuracy of the IRF shape. Since the ion temperature is defined through the second central moment in neutron energy space, but the IRF acts exclusively in the time domain and no direct relation to a single moment of IRF exists. To estimate the uncertainty introduced by the IRF, the standard deviation $\sigma Tion$ for several different PMTs (with each having its own IRF) recording the same neutron peak has been determined to be 150 eV. This part shows only statistical variations in the IRF due to construction; additional systematics of the IRF, such as systematically overestimating its width due to the finite width of the X-ray pulse used to construct it, are small and have been estimated to be 50 eV or less. The statistical uncertainty comes from the covariance matrix of the fitting routine used

_{ion}^{25}and is typically 50 eV. The fitting uncertainty needs to be multiplied by the square root of the

*χ*

^{2}per degree of freedom, which essentially renormalizes the individual data point uncertainties such that the reduced

*χ*

^{2}is 1. All errors have been added in quadrature and the result separated by statistical or systematical source is summarized in Table I.

. | . | Uncertainty . | . |
---|---|---|---|

Quantity . | Statistical . | systematic . | Total . |

DT Y _{n} | <1% | 9% | 9% |

DT T _{ion} | ≈1% | 240 eV | 250 eV |

DSR | ≈5% | ≈ 5% (rel) | ≈ 7% (rel) |

and 0.003 (abs) | and 0.003 (abs) | ||

DD Y _{n} | ≈2% | 10% | 10% |

DD T _{ion} | ≈2% | 270 eV | 290 eV |

. | . | Uncertainty . | . |
---|---|---|---|

Quantity . | Statistical . | systematic . | Total . |

DT Y _{n} | <1% | 9% | 9% |

DT T _{ion} | ≈1% | 240 eV | 250 eV |

DSR | ≈5% | ≈ 5% (rel) | ≈ 7% (rel) |

and 0.003 (abs) | and 0.003 (abs) | ||

DD Y _{n} | ≈2% | 10% | 10% |

DD T _{ion} | ≈2% | 270 eV | 290 eV |

The uncertainty of the *DSR* has four components: a systematical uncertainty due to the shape uncertainty of the IRF, an uncertainty due to the noise introduced in the IRF construction, a systematic uncertainty due to the scintillator sensitivity assumed, and the statistical uncertainty from the fit itself. The influence of the IRF shape has been tested by fitting exploding pusher shots where the down-scattered signal should be negligibly small and the uncertainty on the *DSR* value was determined to be 0.002–0.004 depending on the detector looked at. The uncertainty due to the scintillator sensitivity was estimated as follows: a fit without a sensitivity correction changes the result by about 10% and the sensitivity ratio for the 13–15 MeV region relative to the 10–12 MeV region could be at most 10% off. A 1% systematical uncertainty is therefore assumed. The statistical uncertainty due to the scope noise obtained from the fit is typically 5%. Since the IRF was created from scope data, it has the same noise contribution of about 5%. These values are also summarized in Table I.

The error analysis of the obtained values for DD yield and *T _{ion}* on DT shots is also dominated by systematic effects: it has been observed that the deduced

*T*depends on the parameterization of the background shape and the chosen fitting range. Also, a slight variation in the assumed scintillator sensitivity changes the value of the deduced DD

_{ion}*T*. By comparing the results from different PMTs recording the same spectrum, this uncertainty has been determined to be 250 eV. The fit range varies the

_{ion}*T*by about 100 eV. The statistical uncertainty from the fit is about 100 eV.

_{ion}The DD neutron yield, defined here as the signal coming from the DD peak and its down-scatter has the same uncertainty constrains as the DT yield, which is ≈7% plus the uncertainty from the light attenuators used in front of the PMT. The fit uncertainty is about 2%.

## VII. SUMMARY

In the analysis of nToF spectra, we chose forward fitting of a spectrum model to the data obtained as the means to obtain consistent values for neutron yield, ion temperature, and down-scattered ratio. The fit starts with a model of the spectrum in neutron energy space and incorporates corrections for detector sensitivity, the attenuation along the line-of-sight as well as the time response of the instrument. A model has been implemented to extract a value for *T _{ion}* on DT and DD peaks in the presence of significant down-scatter. To extract spectral integrals and

*FWHM*of an underlying neutron spectrum, a spline fit was implemented as an alternative to deconvolution algorithms.

The improvements over previous works^{3–5} are as follows:

the inclusion of a detailed description of instrumental effects, such as the IRF, beam line attenuation, and scintillator sensitivity,

forward modeling prevents the inherent uncertainty in two step processes, like first deconvolving and then fitting the data,

the use of a scattering correction in the determination of

*T*_{ion},the analysis of DD neutrons on a DT implosion, allowing for simultaneous measurements of both yields and ion temperatures. This is done by adding a background model to the spectrum recorded by a gated PMT.

Future improvements of the described methods are measurement of scintillator sensitivity in order to reduce uncertainties due to the current model and the understanding and improvement of IRF related uncertainties. This work is also the basis of understanding ion temperature asymmetries currently under investigation at the NIF.

The overall uncertainties for these measurements described here are typically 9% for the DT neutron yield, 10% for the DD neutron yield, 250 eV for DT ion temperature, and 290 eV for the DD ion temperature. The uncertainty for the *DSR* is 7% relative to an 0.003 absolute added in quadrature.

## ACKNOWLEDGMENTS

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344.