Neutron resonance spectroscopy (NRS) has been used extensively to make temperature measurements that are accurate, absolute, and nonperturbative within the interior of material samples under extreme conditions applied quasistatically. Yet NRS has seldom been used in dynamic experiments. There is a compelling incentive to do so because of the significant shortcomings of alternative techniques. An important barrier to adopting dynamic NRS thermometry is the difficulty in fielding it with conventional spallation neutron sources. To enable time-dependent and spatially resolved temperature measurements in dynamic environments, more compact neutron sources that can be used at user facilities in conjunction with other diagnostic probes (such as x-ray light sources) are required. Such sources may be available using ultrafast high-intensity optical lasers. We evaluate such possibilities by determining the sensitivities of the temperature estimate on neutron-beam and diagnostic parameters. Based on that evaluation, requirements are set on a pulsed neutron-source and diagnostics to make a meaningful dynamic temperature measurement. Dynamic thermometry measurements are examined in this context when driven by two alternative fast-neutron sources: the Los Alamos Neutron Scattering Center (LANSCE) proton accelerator driving isotropic spallation neutrons as a baseline and a laser-plasma ion accelerator driving a neutron beam from deuterium breakup. Strategies to close the gap between the required and demonstrated performance of laser-based fast-neutron sources are presented. A short-pulse high-intensity laser with state-of-the-art pulse contrast and an energy of a few hundred Joules would drive a compact neutron source suitable for NRS thermometry that could transform the dynamic study of materials.

## I. INTRODUCTION AND REQUIREMENTS

Bulk thermometry is a critical and unmet scientific need for the study of the dynamics of materials subjected to transient extreme conditions.^{1–3} The temperature of a material is an independent thermodynamic variable in the equations of state. Thermometry is thus needed for full validation of theoretical models. It is not just the surface temperature, but the temperature and its variation through the entire sample that are desired.

As an example of what is needed, requirements for temperature measurements to meet the established mission need of the MaRIE project have been analyzed.^{4} The expected range of material temperatures is 300–3000 K, and the desired accuracy is ≈25 K or 2% (whichever is greater). While faster and better resolution helps, the threshold requirement for the project would be internal temperature measurements with ∼100 *μ*m spatial and ∼100 ns temporal resolution. With such a measurement coupled with faster surface thermometry and modeling, time-dependent volumetric temperature can be established to the extent necessary to challenge theoretical models.

Current dynamic temperature measurements have many drawbacks.^{5} They may be able to measure only the surface temperature of a material opaque to optical wavelengths, as in the case of pyrometry.^{1,6,7} They may be perturbative as with thermocouples.^{8,9} With techniques based on x-ray scattering, such as Thomson scattering,^{10} the beam may not penetrate sufficiently into samples that are thick or have high atomic number. In other cases, such as with the Debye-Waller effect,^{11,12} the technique is only applicable to crystalline samples.

Volumetric temperature measurements using neutron resonance scattering or spectroscopy (NRS) have been demonstrated for static^{13–18} as well as for shock-loaded samples,^{19} where the high penetration of thermal and epithermal (0.025 eV to ≈1 keV) neutrons and mature neutron time-of-flight diagnostic techniques have been exploited. Because of previous success, such a technique is being considered as a diagnostic to meet MaRIE requirements, possibly in concert with other techniques utilized simultaneously, such as pyrometry. For example, a concern with pyrometry is the modeling necessary to relate the measured surface emission to an interior temperature. NRS thermometry could validate such modeling to simplify further experiments in a campaign. Moreover, the measurement is robust. Swift *et al.*^{20} pointed out that “NRS is inherently far less sensitive than emission spectrometry to heterogeneities in the temperature, because the width of peak is proportional to *T*, whereas thermal emission is proportional to *T*^{4}.”

A neutron source driven by a linear accelerator is unsuitable for the flexible multiple-probe, multidriver facility concept for MaRIE and for the thermometry requirements therein due to practical and fundamental considerations. Practical considerations include cost, shielding requirements, size, and diagnostic/probe access. Fundamental considerations include (depending on the specific facility) a neutron-pulse duration that is too long or insufficient moderated neutron flux. Much more compact sources of neutrons that can be used at a light source beamline are needed. In this paper, the requirements to make a NRS measurement are analyzed. The sensitivity of that measurement to the source and diagnostic parameters is determined. The requirements for a source of neutrons generated by an optical laser are determined.

### A. The principle of NRS

Neutrons are absorbed by many isotopes at well-defined neutron energies (absorption resonances). The lowest resonance energies for a given nucleus are generally higher for the lightest elements and typically lie in the epithermal energy range for metals such as Zr and heavier. The natural resonance width increases by Doppler broadening, i.e., the convolved shifts from all the individual atoms in thermal motion. If the natural width is sufficiently narrow relative to the Doppler broadening, which is usually the case except for the lightest elements, the increase in the width may be used to measure temperature, as demonstrated by several groups.^{13–19} Moreover, if the sample as a whole is moving sufficiently fast along the probe direction (e.g., when shocked), that speed can also be measured by the overall shift in the resonance. The thermal motion depends on the atomic environment of the nucleus. In a gas, it is straightforward to account for that environment with a Maxwell-Boltzmann (MB) distribution, but in the crystalline solid, it is an overlay of phonon modes. Therefore, the resonance may appear differently in metallic uranium, uranium oxide, or UF_{6} gas, and therefore, it is sensitive to phase transformation. In general, the higher the sample temperature is relative to the Debye temperature *θ*_{D} of the solid (see definition and typical values below), the less sensitive the resonance profile is to these effects. Since epithermal neutrons penetrate centimeter-scale lengths in all materials, NRS is truly a bulk temperature measurement method for sample sizes up to that scale.

The NRS measurement is done by directing a source of neutrons with a broad energy distribution at a sample. In order to accurately establish the thermometry location within the sample, a localized suitable dopant is chosen and one of its broadened resonances is observed. The probe neutron flux, after traversing through the sample, is detected with a time of flight (TOF) diagnostic that resolves the neutron energies. In the simplest case, the neutron pulse is created within a negligibly small window in space and in time relative to its flight, i.e., a point-source time spike. In practice, if neutron moderation is required, it introduces both a lag and an uncertainty in the time when the neutrons traverse the sample and therefore in neutron energy as inferred from the arrival time at the detector. After its birth, the polychromatic neutron pulse spreads along the propagation direction as it flies and becomes chirped (orderly spread in time, with the highest energy leading). At the TOF diagnostic, the neutrons hit a scintillator and the resulting light is detected over time (which scales with neutron kinetic energy) by a photomultiplier tube (PMT). Typically, there are enough neutrons to operate in current mode (rather than single-pulse mode) so that the PMT current is proportional to the neutron flux on the converter.

If the neutron source is observed directly, then the PMT signal is a relatively smooth trace that represents the neutron energy distribution (transformed into time) multiplied by instrumental factors. With a material in their path, neutrons with kinetic energy near the resonance value *E*_{R} scatter, creating a dip in the signal. The width of that dip (the Doppler broadening) can be measured and a temperature can be deduced from it.

The NRS measurement concept using a laser-driven neutron source is illustrated in Fig. 1. The experimental configuration for the NRS measurement of the dynamic material experiment reported in Ref. 19 is sketched in Fig. 2. The actual PMT signals vs time for shocked and unshocked W samples for a 21-eV resonance are shown in Fig. 3. Other resonances have been used, for example, the paper by LeGodec *et al.*^{21} features the analysis of the 10.34-eV resonance of $\u2009\u200973181$Ta.

## II. THEORY BACKGROUND

The slow-neutron scattering resonance cross section for a static nucleus (i.e., describing purely the nuclear effects) has a Lorentzian energy dependence and is given by the Breit-Wigner formula^{13,17,22}

where the cross section at resonance *σ*_{0} is defined, e.g., in Refs. 22 and 17, and Γ is the resonance width which is inversely proportional to the lifetime of the virtual level. Values for that resonance cross section can be calculated from values for Γ and *g*Γ_{n} for resonances in nuclei of atomic number *A* that are tabulated in Ref. 23 or 24 using the expression

where $\lambda \xaf0$ is the de Broglie wavelength of the relative neutron velocity. In Eq. (1), *E*_{R} is the kinetic energy of a neutron in exact resonance and *E* is the kinetic energy of the incident neutron. *E*_{R} is given by the virtual energy level of the compound nucleus (neutron + target) which is responsible for the occurrence of the resonance plus the energy transferred to the compound nucleus, *E*_{t}. In the simplest case, where the target nucleus is free and at rest, *E*_{t} is the recoil energy *E*_{t} = *mE*/(*M* + *m*) = *E*/(*A* + 1), where *m* and *M* are the masses of the neutron and the target nuclei of atomic number *A*, respectively. In practice, only in the cross sections at the broad resonances characteristic of light nuclei may the thermal modification of the resonance shape be neglected. That complication can be turned into an advantage by exploiting it for thermometry.

In reality, the scattering involves a neutron moving at velocity $v\u2192$ in the direction $z^$ colliding with a nucleus moving with velocity $V\u2192$. In order to calculate the effective, Doppler-broadened (temperature-modified) cross section, it is useful to define a relative kinetic energy between the neutron and the nucleus, *E*_{r}. If we define $Vz\u2261V\u2192\u22c5z^$ while $v\u2261v\u2192\u22c5z^$ and assume nonrelativistic velocities, then^{22}

where terms of order $(Vz/v)2$ are ignored. The effective Doppler-broadened cross section *σ*_{eff} for a neutron of energy *E* is given by the convolution of the nuclear cross section with the probability distribution function for *V*. This distribution can be expressed, in general, as *S*(*E*_{r})*dE*_{r} (which integrates to unity). Inserting the relative energy in *σ*_{n}, the general expression for the effective cross section is

for fixed *E*. The probability distribution function depends on the environment of the nuclei. Moreover, in the case of dynamic materials, for example, the distribution can have a net velocity ($\u27e8V\u2192\u27e9\u22600$), which can be accounted for in the data analysis. Hereon, for clarity of exposition in this discussion of measurement accuracy and requirements, we assume that the nuclei are on average stationary.

The simplest nuclear environments are either a perfect gas or a classical solid (nuclei treated as harmonic oscillators) bound by Boltzmann statistics, treated in Ref. 22. In that case, the centered MB velocity distribution (no net flow) can be cast as

where

is the Doppler width near the resonance energy and *k*_{B} is the Boltzmann constant. Then, the effective cross section is

where

*ψ*($\epsilon $, *x*) is, in general, a complicated function of relative energy *x* with various limits. We highlight a couple of useful limits of this expression:

For $\epsilon $ very large (pure natural width),

- (b)
For $\epsilon $ very small (pure Doppler width) and

*x*≪ $\epsilon $^{−2},

where $K=12\pi 1/2\epsilon $. The latter approximation is the one relevant for most experiments trying to measure the sample temperature and the one used in the pioneering measurement in Ref. 19.

The case of nuclei bound in a quantum-mechanical crystal was treated by Lamb many years ago.^{25} The quantum-mechanical behavior of a crystal affects the resonance cross section in ways that are not apparent from a study of classical systems. Lamb derived expressions for the probability of creation or annihilation of quanta (phonons) in the various modes of oscillation and used these in an explicit calculation of the resonance shape to be expected if the target nuclei were bound in a Debye crystal. Note that although the Debye model (phonons in a box) is the solid-state equivalent of Planck’s law of black body radiation (photons in a box), the energy integral does not diverge because there is a maximum possible phonon frequency *ν*_{m} given by a minimum wavelength of twice the atomic separation. The detailed functional dependence of *σ*_{eff} in this case depends on the temperature relative to the Debye temperature of the crystal, *θ*_{D}, which is roughly the temperature at which the mode with the highest-frequency (*ν*_{m}) can be excited. To understand the practical implications of these limits, materials with relatively high *θ*_{D} include C (2230 K), Be (1440 K), sapphire (1047 K), Si (645 K), and Cr (630 K). On the other end lie Pb (105 K), Au (170 K), Sn (200 K), Pt and Ta (240 K), W (400 K), Ni (450 K), Al (428 K), and Fe (470 K). For a cold crystal (relative to the Debye temperature *θ*_{D}), and for a sufficiently small neutron resonance width and a fairly low recoil energy (both compared with *kθ*_{D}), Lamb showed that a recoilless Breit-Wigner peak could be expected, as well as some indistinct structure at higher energies. At higher crystal temperatures, the usual result is the classical resonance broadening, except that the classical mean energy per degree of freedom, *k*_{B}*T*, in Eq. (5) must be replaced by the quantum-mechanical mean energy^{13,17} given by

where *g*(*ν*) is the phonon density of states. $E\xafp$ is greater than *k*_{B}*T* but approaches it asymptotically in the high-temperature limit. This justifies the approximation made in Ref. 19.

Experimentally, one measures the neutron-probe flux through the sample with a thickness *l*_{0}, a density *ρ*(*l*), and a fraction *f*(*l*) of the dopant material with the desired resonance. The location and extent of the doped region determines the degree of spatial discrimination of the measurement within the sample. Neutrons with energy near *E*_{R} are scattered, depleting the PMT current around the time corresponding to the resonant energy. In practice, one measures the “attenuation” *α*(*E*) of the neutron current around *E*_{R}, given by^{26}

Hereon, the high-temperature limit ($E\xafp\u2192kBT$) is assumed, and therefore, *σ*_{eff} in Eq. (14) and the attenuation take a Gaussian form.

This Gaussian-shaped attenuation [Eq. (14)] is used to estimate the requirements for an accurate measurement. Specifically, one wishes to determine *k*_{B}*T* to a given level of accuracy, which may be a specified maximum allowed value of the fractional value *δT*/*T* or perhaps a specified maximum uncertainty value *δT* determined over a specified time interval *δt*. The NRS thermometry-accuracy requirement reduces then to detecting the absence (from scattering) of a sufficiently high number of neutrons (*N*_{2Δ}) that traverse a volume of doped and homogeneous sample that lies within the energy spread of the broadened resonance (2Δ) and within the specified period *δt*. A sufficiently high value of *N*_{2Δ} allows fitting the resonance shape with the accuracy needed to satisfy a given thermometry accuracy requirement.

## III. DERIVATION OF THERMOMETRY UNCERTAINTY

### A. Mathematical idealization of the measurement

Several authors have published data analysis methods and uncertainty analysis with varying levels of detail for NRS thermometry.^{27–29} Our discussion here goes into some detail to highlight the choices (experimental and data analysis) that can be made and to derive the uncertainties from the viewpoint of requirements to achieve a desired level of thermometry accuracy.

In general, the neutron source is not spectrally constant. However, we are concerned here with the source characteristics very near and around the resonance energy, over which the spectrum is nearly constant. Although the TOF diagnostic yields a trace of neutron current vs time, for now we work in neutron flux vs neutron energy (*E*) coordinates, assuming that the proper instrumental conversion constants are known and suitably applied.

Per the discussion above, we idealize the TOF PMT detector signal (transformed into a trace vs *E*) as a constant (proportional to the source neutron flux) minus a Gaussian-shaped function representing the resonant neutron scatter. For the sake of simplicity, we assume that all instrumental conversions are divided out and are constant over the width of the resonance so that without loss of generality, we can set the constant equal to the spectral neutron fluence *detected*, defined as *a*_{i}. We cast the scattered spectral neutron fluence [proportional to *α*(*E*) of Eq. (14)] in the simplified form *a* exp((*E* − *b*)^{2}/(2*c*^{2})), where *a* is the scattered spectral neutron fluence at resonance times the detector efficiency. To establish the requirements for an optimal (best) case with the minimum neutrons required for a given desired accuracy, we assume that *a* is a constant known precisely, and the uncertainty in the scattered component is dominated by neutron-counting (Poisson) statistics. These approximations are illustrated in Fig. 4. We further assume that the experimental setup can be optimized to make *a* → *a*_{i}. The observed signal *o*(*E*) then simplifies to

where all the uncertainty in *o*(*E*) is assumed to lie in the Gaussian term and therefore determined by our ability to fit that term with our observations.

We assume that all the conditions necessary for an accurate fit and for the analytic expressions to be used below are met. These reasonable assumptions are as follows:

The sampling is complete, i.e., we sample an energy interval that encompasses at least 90% of the area under the Gaussian. Unless the resonance lies at very high energy (early time) and is affected by the so-called “gamma flash” (the prompt gamma-ray photon burst), this condition is easy to satisfy in a TOF diagnostic.

The sampling rate is constant. That simply requires grouping the data into nonuniform time intervals corresponding to constant energy intervals.

The sampling interval is near optimal. That means it is large enough to resolve the Gaussian shape, but small enough to keep the statistical uncertainty from Poisson statistics as the dominant one. As an aside, in the case where there is a given constant uncertainty per sample, e.g., an external random noise source of constant amplitude within the resonance energy interval, the variances in the fit parameter are inversely proportional to the sampling interval, i.e., the accuracy gets

*worse*if the data are binned into more samples.^{30,31}In practice, it is optimal to group the data into a number of bins*B*so that the energy interval containing 90% of the Gaussian area is covered by*B*≈ 5. Besides, since neutron moderation will smear to some extent the time-energy correlation, there is no point in having excessively small energy bins with a large fractional variation around the mean energy.The maximum likelihood estimate (MLE) formalism

^{32}is used to fit the Gaussian so that the Cramer-Rao analytical bounding expressions for the variances in the Gaussian fitting parameters are in fact accurate estimations of the said variances.All three Gaussian parameters are fit. Although in principle it is possible to regard both

*a*and*b*as known, in practice it is very hard to know them with sufficient accuracy.For a lower bound on the requirements, we are ignoring the uncertainty in determining the neutron energy (the abscissa uncertainty).

### B. Derivation of thermometry accuracy vs neutron count

Under the conditions stated above, the variance in the Gaussian fit parameters are^{30,31,33}

where we note that the variances of *b* and *c* are the same, and the resonance Doppler width as defined above is given by Δ^{2} = 2*c*^{2}. We note that the number of neutrons detected within the resonance *N*_{2Δ} (i.e., within the resonant energy interval *δE* = 2Δ) is

Taking the derivative of Δ vs *T* and substituting into Eq. (18) yield the fractional uncertainty in temperature

where, as expected with Poisson statistics, the temperature uncertainty scales as the inverse square root of the neutrons within the resonance. Therefore, 2% fractional temperature accuracy requires >5.6 × 10^{3} neutrons detected within the resonance energy width. To gain a quantitative sense, the full Doppler width of the 21-eV resonance in Fig. 3 at 1000 K is 2Δ ≈ 0.34 eV.

## IV. MEASUREMENT SETUP AND CONSTRAINTS

### A. Time-based constraints

The desired thermometry time resolution *τ*_{rsl} places a limit on the separation *L* of the sample from the moderated neutron source. As the neutron bunch flies, it spreads out due to the finite kinetic-energy spread. Therefore, the neutrons with the energy spread necessary to cover the resonance take a certain time to reach the sample that varies by a corresponding interval *δt*_{s}. Hereon, we neglect the dopant length relative to the flight distance *L*, to be justified *a posteriori*. We want *δt*_{s} ≤ *τ*_{rsl}.

To derive the condition for the inequality above to hold, in the expression for the neutron kinetic energy *E* = (1/2)*m*(*L*/*t*)^{2}, where *t* is the delay time for the neutron to go from the moderator to sample, we can solve for time and differentiate vs *E* to obtain the desired time-resolution condition

where *δE* = 2Δ. So to reach a *τ*_{rsl} = 200 ns time resolution for a generic resonance at *E*_{R} = 21 eV broadened to a *δE* = 0.34 eV, *L* < 1.57 m is needed. For later reference, the neutron flight time for this generic sample resonance energy is 11.8 *μ*s. Since the sample size is ∼1 cm, neglecting the sample size (and the smaller doped volume) is justified. To restate, even if the energy resolution of our detector is arbitrarily accurate, this separation requirement from TOF stems from the neutrons being able to trace the resonance sufficiently fast, i.e., by enabling the neutrons within the relevant energy spread to sample the dopant before the sample changes due to some external condition, such as a decaying shock. Here, there is no uncertainty in the measurement *per se*, just in the presumed homogeneity of the sample.

There is a concern of whether the duration of the moderated neutron pulse at birth may smear the measurement time-base (and therefore the neutron energy) enough to impact the measurement accuracy. We start by estimating analytically the effective pulse length of the moderated neutron pulse, *τ*_{n}. It is energy dependent, and a good estimate of it is the time that it takes a neutron with a kinetic energy equal to the resonance energy *E*_{R} to transit within the moderator a collision mean free path at that energy. The reason is that neutrons of that energy that are further in from the moderator surface do not reach the detector. It does not matter that the moderation time is much longer, although this latency time must be accounted for in the timing of the experiment. Assuming a solid polyethylene moderator, the neutron mean free path is 0.5 cm for energies up to 20 keV, so for our generic resonance at *E*_{R} = 21 eV, *τ*_{n} = 80 ns. That estimate has been cross checked with the analytic expression provided in Ref. 19 for the energy-dependent moderated-neutron pulse shape obtained from simulations of their moderator response. This effect represents true measurement uncertainty because the same energy neutrons may be born within the finite time spread of *τ*_{n} and thus represent an irreducible time jitter of the same magnitude at the detector, which translates into an uncertainty in the determination of *E*. It may seem counterintuitive to seek a 200 ns time accuracy when *τ*_{n} may be similar. However, although the pulse may be smeared in time, the scattering remains precisely tuned to *E*, regardless of the actual emission time of the neutron from the moderator. As long as *τ*_{n} ≤ *τ*_{rsl} is satisfied, the moderation energy smearing does not impact the measurement much (see below). If *τ*_{n} > *τ*_{rsl}, the problem is that the resonance will be sampled outside the desired time-resolution window, possibly adding data from changed sample conditions. Such a situation would require additional remedies, such as chopping the neutron beam, which may be possible, but likely to be bulky (constraining dimensions), finicky, and expensive. Since the source-detector distance *L*_{d} is different (and larger than *L* so that the resonance can be spread further in time relative to the fixed *τ*_{n}), the finite *τ*_{n} imposes a requirement on the minimum value of *L*_{d}. This is the reason why the neutron detector cannot be placed right behind the sample to maximize the neutron flux on it.

To estimate the minimum detector separation to meet a given *τ*_{rsl}, we use the same derivation as above, this time for *L*_{d} and *τ*_{n}, and assume that we divide the resonance in *B* bins and require the difference in transit time within a bin be larger than *τ*_{n}. That yields the minimum separation *L*_{d} condition

where again *δE* = 2Δ. For the 21 eV resonance discussed above, *τ*_{n} = 80 ns and *B* = 5, and the detector placement must be *L*_{d} ≥ 3.1 m. It is important to keep *L*_{d} as close as possible to this limit to maximize the neutron detector signal. Note the penalties in *L*_{d} (linear) and necessary neutron yield (quadratic) incurred by overbinning the resonance.

Besides minimizing *L*, time resolution can be shortened up to a point by going to higher energy resonances. While Fowler and Taylor tabulated only the 4.16 eV resonance for making their very-long-beamline near-static measurements, the 21 eV resonance chosen by Yuan comparatively reduces the transit-time contribution to the thermometry time resolution in their dynamic experiment. This choice becomes a trade-off of relative neutron cross section, spectral neutron fluence, background, and detector efficiency while optimizing the time response. Higher energy may only mean by a factor of ten, i.e., a factor of three in velocity and hence improvement in time resolution. For example, the $\u2009\u200974182$W has resonances at 114 eV (with a huge natural width) and 213 eV (above which the resonances get many and close together).^{24} The 213 eV resonance^{24} has a *g*Γ_{n} = 2.6 meV which would make it very sensitive. However, the moderated neutron flux at higher energies is lower and may be insufficient for the available cross section.

### B. Geometric setup and constraints

Consider a source of fast neutrons of yield *Y*_{f}. Assume a fraction *η*_{f} of those are incident on the moderator, resulting in a total moderated-neutron yield *Y*_{m}, a portion of which is used for the thermometry measurement. (Figs. 1 and 2 illustrate the setups with laser-driven and accelerator-driven neutron sources, respectively). *Y*_{m} is emitted over the surface area of the moderator *A*_{m}. From any differential areal element, neutrons are emitted over a range of solid angle Ω ≈ 2*π* and over a spectrum of neutron kinetic energy *E*. (In general, neutron emission is not uniform over energy or the moderator area.) The differential yield of moderated neutrons is defined as

The normalized differential yield

depends solely on the moderator for a given fast spectrum.

The moderator is observed by a detector (or an array of detectors) with total area *A*_{d}, placed a distance *L*_{d} away, and the detector surface is assumed to be normal to the moderator line of sight. From the moderator surface viewed, the detector subtends a solid angle $\delta \Omega dm=Ad/Ld2$. If there is a collimator between the two, it may constrain the detector view of the moderator to a portion of its area *δA*_{m} or, equivalently, to a solid angle $\delta \Omega md=\delta Am/Ld2$.

For a given moderator line of sight, the number of moderated neutrons detected over a given energy range is given by

Once the nuclear resonance is chosen (and therefore *δE*) and the moderator designed (presumably to maximize the differential yield at *E*_{R}), it is clear that within the thermometry accuracy constraints it is advantageous to maximize the detector solid angle and the viewed area of the moderator. From Eq. (23), the thermometry accuracy constraint becomes

where *N*_{2Δ} is related to *δT*/*T* in Eq. (19).

It is assumed that any temperature variation within the doped sample volume being probed by neutrons that reach the detector is smaller than the accuracy specified. It is also assumed the setup is well matched in the sense that the solid angle *δ*Ω_{s} subtended by the dopant when viewed from the moderator is such that *δ*Ω_{s} > *δ*Ω_{md}. Otherwise, the detection area is being wasted by observing the moderated neutron source directly.

## V. DYNAMIC THERMOMETRY EXPERIMENTS

In this section, we examine the two dynamic thermometry cases of interest here in light of the requirements and constraints discussed above. The first case is the one in Ref. 19 where the Los Alamos National Laboratory Neutron Scattering Center (LANSCE) proton accelerator drives a spallation-neutron source. As a realized and published dynamic thermometry measurement, it serves as a check on our methodology and as a baseline to compare directly to an alternative case. This alternative features the same dynamic experiment probed instead in a thermometry configuration using a laser-driven neutron source with a performance level to be specified as a result of our analysis. For this latter case, we specify a thermometry measurement accuracy of 2%, implemented in a miniaturized configuration (relative to the former) with straightforward optimization to decrease greatly the required fast-neutron yield, thus making it feasible to use a laser-driven source.

### A. Thermometry at LANSCE with spallation neutrons

To examine this case, first we discuss the key elements of the experimental setup. That is followed by a discussion of our simplified transport calculations of this setup done with version 6.2 of the MCNP Monte Carlo transport code^{34} to extract key parameters necessary both to establish the upper bound on the accuracy of that measurement based on the criteria presented in this article and to evaluate the laser-based case.

#### 1. LANSCE setup

A custom setup (illustrated in Fig. 2) in the LANSCE Blue Room was used, which allowed a moderator-sample standoff as small as 1 m. That provides a time resolution of 170 ns for the 21 eV resonance, limited by the neutron transit time through the sample plane [see Eq. (20)]. Otherwise, the minimum standoff possible at the Lujan Center would have made the time resolution ≈5× longer.

To generate a neutron pulse at LANSCE, a set of 800-MeV proton micropulses are accumulated in the proton storage ring (PSR). The PSR beam has up to 5 *μ*C, or *N*_{p} = 2.5 × 10^{13} protons, delivered in a pulse approximately triangular in shape with a zero-to-zero duration of 270 ns and 125 ns FWHM.

At the desired time, the PSR proton beam is directed at the spallation target in the Blue Room. The geometry of the spallation target and moderator assembly is shown in Fig. 5. The proton beam drives a nuclear spallation process resulting in a nearly isotropic fast-neutron source with a total yield of *Y*_{f} = 5 × 10^{14} neutrons. The multiplication factor of *Y*_{f}/*N*_{p} ≈ 20 spallation neutrons per incident proton in this energy range is very well established. The resulting fast-neutron spectrum in that custom setup was not reported, as it bears on the moderator design but not directly on thermometry. However, it likely resembled the one from the Weapons Neutron Research (WNR) facility observed at 90° presented in Fig. 2 of Ref. 35. That spectrum has an average neutron energy E = 8 MeV.

The moderator (see Fig. 5) was a polyethylene block placed on top of the spallation source. As determined from our simplified transport simulations of this LANSCE setup done for this manuscript, the moderator volume intercepts 6.5% of the spallation neutrons, i.e., *η*_{f} = 0.065. That determines the effective solid angle *δ*Ω_{ms} subtended by the moderator from the spallation block center (Fig. 2): *δ*Ω_{ms} = 0.065 × 4*π* = 0.8 sr.

The collimator (clear hole with 2 in diameter, 3 in length) limiting the field of view (FOV) of the moderator from the detectors was essential in order to avoid any view nearby the spallation target because of its copious production of *γ* rays, to which the TOF detectors are sensitive also. Too large a gamma flash would saturate strongly the TOF detector and electronics which would take unacceptably long to recover. The detectors were placed 23 m away from the moderator. This is much larger than the minimum distance required to properly fit mathematically the 21 eV resonance probed by the 125 ns neutron pulse [see Eq. (21)]. The 2-cm collimator subtends a solid angle of *δ*Ω_{md} = 6.5 × 10^{−7} sr from the detector plane, which sets the visible area of the moderator to *δA*_{m} = 3.45 cm^{2} (2.1 cm spot diameter), which is only 6.1% of the area of the moderator-face viewed.

As reported in Ref. 19, the detector array consisted of 11 1-cm thick, 12.7-cm diameter, ^{6}Li-loaded glass scintillators coupled to photomultiplier tubes (PMTs). That sets upper bounds for the detector surface *A*_{d} = 1393 cm^{2} and *δ*Ω_{dm} = 2.63 × 10^{−4}. (These are slight upper bounds because the active area of the PMT has a slightly smaller diameter than the scintillator.^{36}) The collimator aperture is well matched to the detector solid angle, i.e., it subtends the same solid angle.

Important details relating to the detectors used in Ref. 19 which enable an estimation of the neutron-detection efficiency are presented in Ref. 36 and in the data sheet of the Hamamatsu R1513 PMT used. The measured neutron-detection efficiency of the scintillator glass used is 0.5 at 11 eV. Using the known dependence of the ^{6}Li(n, t)^{4}He cross section on the inverse of the neutron velocity at these epithermal energies, we estimate a 0.35 efficiency at 22 eV. About half of the scintillator photons are emitted backwards into the PMT, which has a quantum efficiency of 0.2. The product of all these factors yields an overall neutron detection efficiency *ϵ*_{d} ≈ 0.035.

#### 2. Simulation of the LANSCE setup

To ascertain how does the LANSCE spallation neutron source in this dynamic setup compare to the neutron-yield criterion derived above [Eq. (19)], we have run a transport simulation of that setup simplified by excluding the Be reflector, using an idealized collimator, omitting the dynamic sample as well as any walls or objects that would scatter neutrons, etc.

According to the simulation, for each of the 800 MeV protons, *Y*_{f}/*N*_{p} = 20 spallation neutrons are made nearly isotropically. Of those, *η*_{f}*Y*_{f}/*N*_{p} = 1.3 fast neutrons are intercepted by the moderator. For a small representative 2-cm FOV of the moderator (i.e., *δA*_{m} = (*π*/4)2^{2} cm^{2}) and *L*_{d} = 2300 cm, the differential moderated neutron fluence at the detector plane normalized to *N*_{p} over the resonance width is

i.e., 3.6 × 10^{−6} neutrons/sr/proton. Therefore, in the limit of a small FOV of this moderator (at the center of the face), we find

for the number of moderated neutrons/detector solid angle/moderator area/fast neutron into the moderator.

To determine the number of neutrons actually measured in the experiment in Ref. 19, the first step is to estimate the number of moderated neutrons arriving at the detector plane, *Y*_{md}. For that, we multiply the normalized differential moderated neutron yield in Eq. (25) by the product of the parameters for this setup: the number of fast neutrons into the moderator [*N*_{p}*η*_{f}(*Y*_{f}/*N*_{p})], *δ*Ω_{dm}, and *δA*_{m}. Thus, we obtain *δY*_{md} = 2.6 × 10^{4} moderated neutrons within the resonance arriving at the detector array. Multiplying that number by *ϵ*_{d} means that about *ϵ*_{d}*δY*_{md} = 910 neutrons were detected in that experiment. According to Eq. (19), that corresponds to an accuracy of at best 5%. This is similar to the error bar in the temperature measurement shown in Ref. 19. The relevant parameters to compute these values are summarized in the third column of Table I for the LANSCE case.

Item . | Expression . | Units . | LANSCE . | Laser, scaled . | Laser, MCNP6 . |
---|---|---|---|---|---|

Fast neutrons | Y_{f} | 5 × 10^{14} | 2.25 × 10^{11} | 2.9 × 10^{11} | |

Fraction into moderator | η_{f} | 0.065 | 1 | 1 | |

Fast neutrons into moderator | η_{f}Y_{f} | 3.25 × 10^{13} | 2.25 × 10^{11} | 2.9 × 10^{11} | |

Normalized differential neutron yield at 21 eV | M2Δ | 1/(sr cm^{2}) | 8.8 × 10^{−7} | 8.8 × 10^{−7} | 6.9 × 10^{−7} |

Moderator-sample distance | L | cm | 96 | 96 | 96 |

Moderator-detector distance | L_{d} | cm | 2300 | 310 | 310 |

Detector solid angle | δΩ_{dm} | Sr | 2.6 × 10^{−4} | 1.45 × 10^{−2} | 1.45 × 10^{−2} |

Moderator area viewed | δA_{m} | cm^{2} | 3.45 | 56.25 | 56.25 |

Moderated neutrons at detector plane | δY_{md} | 2.6 × 10^{4} | 1.6 × 10^{5} | 1.6 × 10^{5} | |

Detector efficiency | ϵ_{d} | 0.035 | 0.035 | 0.035 | |

Moderated neutrons measured | ϵ_{d}δY_{md} | 910 | 5.65 × 10^{3} | 5.7 × 10^{3} | |

Thermometry accuracy | δT/T | % | 5 | 2 | 2 |

Item . | Expression . | Units . | LANSCE . | Laser, scaled . | Laser, MCNP6 . |
---|---|---|---|---|---|

Fast neutrons | Y_{f} | 5 × 10^{14} | 2.25 × 10^{11} | 2.9 × 10^{11} | |

Fraction into moderator | η_{f} | 0.065 | 1 | 1 | |

Fast neutrons into moderator | η_{f}Y_{f} | 3.25 × 10^{13} | 2.25 × 10^{11} | 2.9 × 10^{11} | |

Normalized differential neutron yield at 21 eV | M2Δ | 1/(sr cm^{2}) | 8.8 × 10^{−7} | 8.8 × 10^{−7} | 6.9 × 10^{−7} |

Moderator-sample distance | L | cm | 96 | 96 | 96 |

Moderator-detector distance | L_{d} | cm | 2300 | 310 | 310 |

Detector solid angle | δΩ_{dm} | Sr | 2.6 × 10^{−4} | 1.45 × 10^{−2} | 1.45 × 10^{−2} |

Moderator area viewed | δA_{m} | cm^{2} | 3.45 | 56.25 | 56.25 |

Moderated neutrons at detector plane | δY_{md} | 2.6 × 10^{4} | 1.6 × 10^{5} | 1.6 × 10^{5} | |

Detector efficiency | ϵ_{d} | 0.035 | 0.035 | 0.035 | |

Moderated neutrons measured | ϵ_{d}δY_{md} | 910 | 5.65 × 10^{3} | 5.7 × 10^{3} | |

Thermometry accuracy | δT/T | % | 5 | 2 | 2 |

It would be expensive (∼$1B) and difficult to reproduce the LANSCE NRS thermometry setup on a light-source facility. A spallation neutron source so close to an experimental setup presents significant safety issues, diagnostic-noise issues, and geometrical-constraint issues in dynamic mesoscale material experiments, which have much smaller samples and a more compact setup than the high-explosive-driven dynamic material experiments in Ref. 19. Moreover, that setup would have to be improved to reach the desired 2% accuracy. Those issues motivate consideration of an alternative neutron source using intense short-pulse lasers.

### B. Thermometry with laser-driven neutrons

At present, laser-driven beam sources deliver neutron yields in a single shot that are orders of magnitude below the spallation neutron yields at LANSCE. Therefore, a dynamic-thermometry measurement is not possible by simply substituting the spallation neutron source in the same exact setup in Ref. 19 with a single-shot laser-driven source. However, as explained above, the use of a spallation source imposes constraints which can be relaxed with a laser source. As a result, the setup can be greatly shrunk in size and modified in other ways to enable a much more efficient neutron utilization, making dynamic thermometry feasible with such a source. Our evaluation of such a source goes as follows. In Subsection V B 1, we describe the state of the art in laser-driven neutron sources and summarize their performance. In Subsection V B 2, we explain simple improvements to the thermometry setup in Ref. 19 within the constraints from the desired thermometry accuracy and use simple analytical estimates to tally up the resulting increased efficiency in neutron utilization. (The dynamic material experiment itself is unchanged.) The required fast-neutron yield with this setup is estimated. In Subsection V B 3, a neutron-transport and moderation simulation to refine our analytical estimate of the required yield is presented. Finally, possible measures to increase the neutron yield of laser-driven neutron sources to meet the requirements are discussed.

#### 1. Laser-driven neutron-beam sources

As reported in Refs. 37 and 38, experiments at the Los Alamos National Laboratory (LANL) Trident laser have utilized a laser pulse of energy *E*_{L} ≈ 70 J and FWHM duration *τ*_{L} = 0.6 ps, typical of short-pulse lasers with glass amplifiers. That pulse drives a deuterium (d^{+}) beam used in turn to produce a forward directed beam (1 sr) of fast neutrons (average energy ∼1 MeV) of *Y*_{f} ≈ 2 × 10^{10} born in ≈1 ns. As shown in Fig. 1, the laser-driven d^{+} beam is directed at a Be disk (the converter) where the neutron beam is made by deuteron breakup. The converter intercepts the full d^{+} beam because it is placed ≈1 cm from the laser target.

Neutron-beam yields on the Trident laser are ≈100× higher than prior laser-based efforts using other laser-driven ion acceleration mechanisms and laser targets. Such performance at Trident has required the implementation of pulse-cleaning techniques^{39,40} to achieve exceptionally high contrast: intensity down by a ratio of 10^{8} at 12 ps prior to the peak and 10^{9} at 33 ps. (See supplementary Fig. 2 in Ref. 41 for the measured contrast compared to an ideal pulse.) Such a high-contrast laser pulse in turn allows the utilization of submicron thick deuterated plastic-foil laser targets to make d^{+} beams with higher efficiency and higher average energy. The neutron spectrum from the initial Trident experiments^{37} has been measured, and the resulting average neutron energy is ⟨*E*⟩ = 2.7 MeV.^{38} That has been decreased subsequently to as little as 0.5 MeV by modifying the converter dimensions. These energies are significantly smaller than in spallation sources, suggesting that in the detailed design of an actual thermometry experiment in the future, it would be desirable to optimize the neutron moderator rather than just using the one in Ref. 19 as we do here.

The Trident neutron generation performance has been matched using the PHELIX glass-laser system at GSI in Darmstadt. Neutron fluences within a factor-of-two of Trident have been obtained on the Texas Petawatt Laser.^{42} Both laser systems have been able to deliver those results following their successful implementation of comparably high-contrast in their laser pulses,^{43,44} enabling the use of similar deuterated nanofoil laser targets.

The deuteron-beam diameter at birth is ≈10 *μ*m (compared to ≈2 cm at LANSCE), but it diverges in a cone approaching 1 sr. Thus, the neutron-source size demonstrated so far on Trident^{45} is ≈1 mm, limited by the laser target-converter separation used. A desire to protect the converter to avoid Be dispersal has led to a conservatively large target-converter separation limiting the neutron-source size. The source size is an issue for neutron imaging applications, but it is sufficiently small for accurate thermometry.

The fast-neutron yield demonstrated at Trident (not significantly optimized) is comparable to a first-generation neutron-spallation source. While not insignificant, the shielding requirements for this miniaturized laser-driven neutron source are much smaller than in Ref. 19, a fact that can be exploited for a more efficient utilization of neutrons.

#### 2. Modified thermometry setup

For the sake of a direct comparison between the spallation and the laser neutron sources, we limit our discussion here to a thermometry measurement based on the same 21 eV resonance and similar temperature as in Ref. 19. We assume that the same moderator is used, despite its clearly nonoptimal dimensions (too deep and wide) for the lower fast-neutron energies from this source. Since the laser neutron source is beamed rather than isotropic, it is assumed that the moderator is placed right up against the Be neutron converter to ensure the intercept of the full cone of the fast-neutron beam, i.e., *η*_{f} = 1. For a given fast neutron yield, this modification alone increases the neutron fluence on the detector by a factor of 1/0.065 = 15× relative to the geometry in Ref. 19. We also assume that the same detector array is used, albeit located closer to the moderator. We set a thermometry-accuracy requirement of *δT*/*T* = 2% (better than achieved in LANSCE) and a time-resolution requirement of *τ*_{rsl} = 200 ns.

The first basic parameter examined is the standoff between the moderator and the dynamic sample (*L*). From Eq. (20), *L* ≤ 1.57 m is required for *E*_{R} = 21 eV. The value of *L* = 0.96 m used in Ref. 19 is perfectly adequate, so it can be left unchanged.

The second basic parameter examined is *L*_{d}. With the neutron-pulse width of *τ*_{n} = 80 ns from our moderator, Eq. (21) indicates that there is no need to place the detector any further than *L*_{d} = 3.1 m, which we do here. That represents a large improvement relative to the 23 m in Ref. 19 as the neutron flux on the detector scales as $Ld2$ and provides a sensitivity gain by a factor of (23/3.1)^{2} = 55×. The revised value of *L*_{d} results in a greatly increased detector solid angle *δ*Ω_{dm} = 1.45 × 10^{−2}.

The third parameter examined is the aperture limiting the detector view of the moderator. The detector view in Ref. 19 was limited to 9% of the moderator area facing it (3.45 out of 56.3 cm^{2}) to avoid any sight of the spallation target. Here, we envision a different collimation and shielding scheme. As we have verified, if needed the neutron converter can be made visible to the detectors without blinding them with prompt radiation, so there is no need to restrict their view of the moderator. Relaxing that constraint gains another factor of 56.3/3.45 = 16× in sensitivity. The moderator solid angle becomes *δ*Ω_{md} = 6.5 × 10^{−7} sr.

The product of all these geometric increases in sensitivity for a given fast-neutron source is 1.3 × 10^{4}. With this modified setup, working backward from the 2% thermometry accuracy requirement, the fast-neutron yield required can be determined to be *Y*_{f} = 2.25 × 10^{11}. The breakdown for this calculation is shown in the fourth column of Table I. It assumes that the moderator performs identically with the laser neutron source spectrum and with the wider FOV, i.e., the normalized differential neutron yield remains the same. This assumption is examined next.

#### 3. Simulation with laser-driven neutrons

A key assumption in the analytical estimate above is that the normalized differential moderated-neutron yield remains invariant with the different laser-based spectrum and the increased FOV. The specific concern on the latter is that when injecting fast neutrons from only one side, especially into a large moderator, one would expect the moderated neutron flux to decrease toward the far edges. If that decrease is significant, increasing the FOV would increase the neutron count at the detector less than linearly with *δA*_{m}, impacting the geometric scaling discussed above. To investigate these points, a simplified transport simulation of a laser-based case has been done with MCNP6, as for the LANSCE case above. It assumes a fast-neutron source with the published spectrum^{38} emanating from a point as a cone with a divergence half-angle of 10°, directed with normal incidence into the center of a narrow face of the same moderator in Ref. 19 shown in Fig. 5. The moderator easily intercepts the full fast-neutron beam (*η*_{f} = 1). In this simulation, as in the LANSCE case, a virtual detector has a view along the normal from the center of one of the two wide faces of the moderator.

To investigate the properties of this moderator with the laser source, the moderated-neutron flux on the virtual detector as a function of *δA*_{m} was determined by varying the diameter of a FOV circle on the moderator face, specifically considering 2, 4, and 7.5 cm. From that, the value of the normalized differential yield averaged over the FOV is computed and compared to the value in Eq. (25) shown as well in the third and fourth columns of Table I. As the FOV increasingly includes the area near the moderator edges, a decreasing normalized differential neutron yield is expected.

With the smallest FOV (3.14 cm^{2}, 2-cm spot), the differential yield is the same as before [Eq. (25)]. That eliminates the concern of spectral sensitivity of the moderator in this regime.

As for increasing the FOV, the numerical value of the FOV-averaged differential neutron yield in Eq. (25) is taken as the limit when *δA*_{m} → 0, and the FOV-averaged differential yield derived from the simulation is renormalized to that value as the FOV is increased. The renormalized differential yield is *Ŷ*_{m} = 1 for *δA*_{m} = 0 and *Ŷ*_{m} = 0 for *δA*_{m} → ∞. The simulation results are fit very well (correlation coefficient of 1.0) by the expression $\u0176m=1.003\u22121.022\xd710\u22123\delta Am\u22124.916\xd710\u22125(\delta Am)2$. With a small extrapolation to the full area of the moderator face, we find *Ŷ*_{m} = 0.79 and

for *δA*_{m} = 56.3 cm^{2}.

The consequences of this revised value for the normalized differential yield are shown in the fifth column of Table I. The main result is that *Y*_{f} = 2.9 × 10^{11} neutrons are required with this setup for thermometry accurate to 2%. We note that it would take only *Y*_{f} = 4.6 × 10^{10} to make a dynamic thermometry measurement as accurate as in Ref. 19 with this laser-based setup. The fast-neutron yield *Y*_{f} = 2 × 10^{10} demonstrated on Trident would enable a measurement only accurate to 8%.

## VI. IMPROVING LASER-DRIVEN FAST-NEUTRON YIELDS

The *Y*_{f} = 2.9 × 10^{11} required for thermometry accurate to 2% is higher than the value demonstrated on Trident by a factor of 15×. In this section, feasible strategies that may be combined for improving laser-based neutron yields are discussed below. To summarize these improvements, the recommended increase in laser energy would increase the neutron yield by 5.6× and the improved laser-target design would do so by 12×, for a total increase of 65×. This provides plenty of margin to reach and to exceed the requirement.

### A. Increased laser energy and intensity

Increased laser energy is a straightforward strategy to increase the fast-neutron yield. There is no reason to restrict the laser energy to the 70 J used at Trident. In order to keep the size and cost manageable and accessible with present laser technology, we envision utilizing a glass laser (*τ*_{L} ≈ 0.5 ps pulse) with *E*_{L} = 300 J and a relatively high repetition rate (>1 shot/min) for operational stability and rapid prototyping. (In the context of DOE critical decision 0 for a MaRIE project, we have found that such a laser would have an equipment cost ∼$10 M.) The question of how best to utilize that extra 4× factor in laser energy is examined below.

The straightforward way to utilize the 4× higher laser energy is to keep the laser intensity constant by increasing the laser-spot area proportionately and to use the same deuterated-plastic target. That simply increases proportionately the volume of the laser-target material driven by the laser and correspondingly the d^{+} ion yield. Thus, we expect *Y*_{f} to be 4× higher in that scenario.

A different strategy is to keep the same laser spot size in order to increase the laser intensity *I*_{L} by 4×. In this regime of laser-driven ion acceleration, optimal laser-target thickness scales as $(IL)1/2$.^{46} If that 2× adjustment in target thickness is made, the target volume driven by the laser and therefore the d^{+} yield and *Y*_{f} would increase by that same factor. The d^{+} energy *E*_{d} for an optimum target also scales as $(IL)1/2$ in this regime.^{46} That can be exploited via the steep dependence of *Y*_{f} from d^{+} breakup on *E*_{d}. The empirical scaling is $Yf\u223c(Ed)3/2$, as shown in Fig. 6, reproduced from Ref. 47. Therefore, we get an additional factor of 2^{3/2} = 2.8 from the higher *E*_{d}, for a total 5.6× higher *Y*_{f} from the increase in *E*_{L}. We therefore pick this strategy as the preferred alternative.

### B. Improved laser targets

One could increase the average *E*_{d} observed on Trident by a significant factor at the same *E*_{L} by using more sophisticated multilayered targets. There are two problems with the present deuterated-plastic laser-target nanofoils. The first is a higher sensitivity to laser prepulse that promotes premature hydrodynamic disassembly. The second is a relatively low initial electron density which decreases the volumetric laser-plasma coupling in the relativistically induced transparency regime and hastens the point in time when the target becomes classically underdense and the laser-plasma coupling effectively vanishes. We envision getting around these problems by using more sophisticated multilayer nanofoil laser targets to reproduce the 18 MeV/nucleon demonstrated with C^{6+} ions (same charge/mass as d^{+}) on Trident^{48} using diamond nanofoil targets. Basically, rather than using a single diamond nanofoil, a layer of deuterated plastic would be deposited either on the rear side of the nanofoil (away from the laser) or sandwiched between two nanofoils, so that the laser-plasma interaction is still dominated by the nanofoils, but the d^{+} would undergo the same acceleration as the dominant C^{6+}.

To estimate the improvement from this change, a “temperature” of ≈14 MeV can be extracted from the d^{+} spectrum in Ref. 49, corresponding to an average energy ⟨*E*_{d}⟩ = 7 MeV (1 degree of freedom). Increasing ⟨*E*_{d}⟩ from that value (3.5 MeV/nucleon) to the 18 MeV/nucleon achieved with C^{6+} would increase the neutron yield by *Y*_{f} ∼ (18/3.5)^{3/2} ≈ 12×.

### C. Neutron converter

Another potential target for optimization lies in the neutron converter geometry and material composition. Both questions are ideally explored computationally, but a quantitatively credible treatment of neutron production via d^{+} has only become accessible recently. We have explored the converter material question, as described below. The bottom line is that Be remains the best material to use as a neutron converter, so there is no gain to be had here. The process leading to that conclusion is explained below.

The first step in our evaluation is to model the neutron production using the published d^{+} spectrum from Trident in Ref. 49. (In that reference, the measured spectrum is presented as a shaded range band resulting from alternative published image-plate detector calibrations for d^{+}. Here, we take a spectrum that lies in the middle of the band.) The spectrum is monotonically decaying from a maximum that lies below the 18 MeV measurement lower bound to 100 MeV and beyond. The code allows the user to control nuclear data used in the problem. Typically, the evaluated data tables are used whenever possible at low energies and the nuclear-physics models are employed otherwise. The relevant physics models in MCNP6, such as ISABEL^{50} and INCL4,^{51} are intranuclear cascade models developed primarily for the high-energy region, starting at energies 100–150 MeV/nucleon. These models, however, have traditionally been used even at intermediate energies where no evaluated data are available. Version 6.2 is the first that supports data tables for d^{+} projectiles. The nuclear data library TENDL-2015 contains data not only for deuterons but also for several other incident particles: neutron, proton, triton, ^{3}He, alpha, and gamma for most isotopes. TENDL library is the evaluated nuclear data file that provides the output of the TALYS nuclear code system by randomly varying model parameters.^{52} TALYS is a deterministic nuclear reaction program for predicting cross sections and other important physics quantities in the energy region up to 200 MeV (with some extensions up to 1 GeV). Given the energy spectrum of incident deuterons, it is therefore appropriate to use the TENDL-2015 library here and to compare the results with the performance of MCPN6 models and experimental data.

The neutron spectra in TOF detectors at 10°, 15°, 90°, and 160° from the d^{+} propagation direction have been simulated. The integrals of neutron fluxes above 1 MeV are calculated. 1 MeV is a typical threshold for measurements with scintillation and bubble detectors. For a case where the forward directed neutrons (measured using bubble detectors) from 5 × 10^{11} d^{+} was *Y*_{f} = 4.5 × 10^{9}, the three cases (TENDL, ISABEL, and INCL4) predict values ranging from 1.5 to 2.8 × 10^{9}, with the lowest value resulting from using TENDL. The differences between the observation and calculations are within the uncertainty in the d^{+} spectrum that is used. Therefore, it seems reasonable to use this formalism to evaluate candidate neutron converter materials.

Several such materials shaped like the converter in Ref. 49 have been studied: natural Li, ^{9}Be, ^{12}C, ^{27}Al, ^{59}Co, ^{89}Y, natural W, ^{197} Au, ^{209}Bi, ^{232}Th, and natural U. Be is found to perform best, followed by Li which has a 65% lower conversion efficiency than Be.

## VII. CONCLUSION

In this manuscript, the concept of bulk (volumetric) thermometry based on NRS in the context of dynamic experiments has been explained and examined quantitatively. Specifically, the theoretical framework has been summarized in Sec. II. The relationship between the accuracy of the thermometry measurement and the measured number of moderated neutrons has been derived in Sec. III. The temporal and geometric relationships and constraints on the experimental setup have been derived in Sec. IV. In Sec. V, dynamic NRS thermometry measurements have been examined and requirements established for two sources of fast neutrons: spallation-neutrons generated by the LANSCE proton accelerator and d^{+}-breakup neutrons generated by a laser-plasma ion accelerator. Advantages of the latter source have been exploited to maximize the utilization of the fast neutrons. Measures to improve the existing performance of laser sources in order to reach the desired thermometry accuracy have been discussed in Sec. VI.

In general, decreasing the moderator-to-sample distance *L* [Eq. (20)] increases the thermometry time resolution up to a point. However, the slow speed of epithermal neutrons and the intrinsic duration of moderated neutron pulses at these energies limit the time resolution to $\u223c$100 ns. It is challenging for NRS measurements to resolve the time scales of some dynamic material processes, such as temperature changes in metals from plasticity.^{26} If the plastic heating depends on the microstructure at the grain size, for typical metals with a shock traveling at ≈5 km/s and grain sizes ∼50 *μ*m, there would be a characteristic time scale of ∼10 ns for a plastic wave to cross a grain. Therefore, NRS use in dynamic experiments must be judicious. NRS can be unconditionally useful for slower temperature measurements such as during processing or manufacturing to study selected dynamic processes or as a high accuracy standard measurement to calibrate other faster indirect methods. It can provide thermometry measurements at multiple times and points within a sample in the same shot if different regions are doped with different tracer materials.

As mentioned above, temperature measurements in dynamic experiments on the equation of state of Mo using the LANSCE accelerator to produce neutrons were successfully demonstrated and reported in Ref. 19. These measurements have been examined in the context of the constraints and accuracy relationships derived here. Our analysis is consistent with the published results and thus anchors our work, based on which additional information beyond what is published has been ascertained and presented here. We have used the same analysis to examine a similar thermometry setup driven instead by a laser-driven fast-neutron source. Exploiting the properties and advantages of such as source, the dimensions of the LANSCE setup have been optimized and the required performance of the laser-driven source has been established. That performance exceeds somewhat what has been achieved with the existing lasers and targets tried so far. However, we have identified two strategies to close the gap and more: a higher laser energy and improved laser target design. Specifically, it is estimated that a short-pulse high-intensity laser with state-of-the-art pulse contrast (as on Trident) and an energy of a few hundred Joules would drive a suitable neutron source for dynamic thermometry. The requirements are very much achievable with present laser technology at a price relatively affordable for major facilities such as MaRIE. For further work, we recommend exploration of advanced laser targets such as those discussed here to improve the demonstrated performance of laser-driven d^{+} beams. That exploration, besides hastening the practical utilization of NRS thermometry, would have a significantly beneficial impact on other compelling societal and security applications of this promising neutron source.

## ACKNOWLEDGMENTS

This work was sponsored by the National Nuclear Security Administration of the U.S. Department of Energy. We gratefully acknowledge valuable discussions with V. Yuan and G. Morgan, as well as the referees for their constructive comments and suggestions to improve the manuscript.