The fusion Z-pinch experiment (FuZE) is a sheared-flow stabilized Z-pinch designed to study the effects of flow stabilization on deuterium plasmas with densities and temperatures high enough to drive nuclear fusion reactions. Results from FuZE show high pinch currents and neutron emission durations thousands of times longer than instability growth times. While these results are consistent with thermonuclear neutron emission, energetically resolved neutron measurements are a stronger constraint on the origin of the fusion production. This stems from the strong anisotropy in energy created in beam-target fusion, compared to the relatively isotropic emission in thermonuclear fusion. In dense Z-pinch plasmas, a potential and undesirable cause of beam-target fusion reactions is the presence of fast-growing, “sausage” instabilities. This work introduces a new method for characterizing beam instabilities by recording individual neutron interactions in plastic scintillator detectors positioned at two different angles around the device chamber. Histograms of the pulse-integral spectra from the two locations are compared using detailed Monte Carlo simulations. These models infer the deuteron beam energy based on differences in the measured neutron spectra at the two angles, thereby discriminating beam-target from thermonuclear production. An analysis of neutron emission profiles from FuZE precludes the presence of deuteron beams with energies greater than 4.65 keV with a statistical uncertainty of 4.15 keV and a systematic uncertainty of 0.53 keV. This analysis demonstrates that axial, beam-target fusion reactions are not the dominant source of neutron emission from FuZE. These data are promising for scaling FuZE up to fusion reactor conditions.

Designs for positive net-energy fusion reactors must satisfy the Lawson criterion,1 which states that hot and dense plasmas need be confined for long timescales. The Z-pinch is one of the simplest concepts for plasma confinement. It consists of axial current flowing between two electrodes, forming a cylindrical plasma volume. A static equilibrium for plasma confinement in Z-pinches is achieved when an inward jz×Bθ force is balanced with an outward pressure gradient, p,

ddR(p+Bθ22μ0)+Bθ2μ0R=0,
(1)

where R is the radial coordinate, jz is the axial current, Bθ is the azimuthal magnetic field, p is the plasma pressure, and μ0 is the magnetic permeability in vacuum. If long confinement timescales can be maintained for Z-pinch plasmas, then increasing the pinch current will result in plasma densities and temperatures high enough for positive net energy fusion reactor conditions.2,3

Although Z-pinch equilibria were discovered as early as 1934,4 subsequent theoretical and experimental results5–8 demonstrated large-scale, fast-growing plasma instabilities. Two of the most virulent instabilities in Z-pinch plasmas are the m =0 sausage instability and the m =1 “kink” instability, where m is the azimuthal mode number. Fast-growing instabilities in Z-pinch plasmas lower plasma confinement timescales by disrupting current flow. To mitigate the destabilizing effects of the sausage and kink instabilities, fusion research devices have largely moved away from the linear Z-pinch design.

Theoretical9 and experimental2,10 results show that the application of radially sheared, axial plasma flows can mitigate the growth of plasma instabilities in linear Z-pinch plasmas, resulting in plasma lifetimes significantly larger than instability growth times. Shumlak and Hartman9 calculated that plasma flows with sub-Alfvenic flow velocities can stabilize Z-pinches against m =1 instabilities. Shumlak and Hartman's calculations do not assume the presence of a close-fitting conducting wall and are performed in a Kadomtsev equilibrium,11 which is marginally stable to m =0 instabilities. Subsequent measurements12 in a first-generation sheared-flow stabilized (SFS) Z-pinch device, demonstrated stabilization against m =1 instabilities in Z-pinch plasma columns. Z-pinch plasmas with pinch currents of up to ∼50 kA and plasma durations of up to ∼30 μs were observed.

The fusion Z-pinch experiment (FuZE) is a second-generation SFS Z-pinch that is designed to study the effects of sheared-flow stabilization on Z-pinch plasmas in a fusion regime. FuZE uses a faster, higher-energy capacitor bank to deliver more power to the device chamber, resulting in Z-pinch plasmas with higher pinch currents (>200 kA), plasma densities (∼1017 cm−3), and ion temperatures (∼1 keV). Plasma densities and temperatures are high enough to drive measurable fusion reactions when D2 fill gas mixtures are injected into the device chamber.

To lowest order, fusion of two deuterons (“DD fusion”) at rest will produce a neutron with a kinetic energy of 2.45 MeV and a 3He ion with a kinetic energy of 0.82 MeV. Recent results13,14 from FuZE show neutron production with yields of ∼105 neutrons per discharge, long emission durations of ∼5 μs, and neutron emission regions (33 cm length) comparable to the plasma volume (50 cm length). Neutron emission durations of 5 μs are >1000 times greater than instability growth times and are therefore consistent with thermonuclear neutron production.

In Z-pinch plasmas, m =0 instabilities can result in electric fields large enough to drive energetic ion beams into the plasma column. If a deuterium fill gas is used, this results in neutron emission through beam-target fusion reactions.15,16 Due to the kinetics of the fusion reaction, neutron energies emitted from beam-target fusion reactions are significantly upshifted parallel to the beam direction and downshifted opposite the beam direction. Conversely, given the isotropic nature of a thermal Maxwellian plasma, neutron energies emitted from thermonuclear reactions are relatively spatially isotropic in the lab frame. To determine whether neutrons emitted from FuZE originate from thermonuclear or beam-target fusion reactions, relative neutron energies are inferred from detectors placed at multiple angular locations around the fuzing plasma column along the direction of potential beam acceleration from m = 0 instabilities.

A method for energetically resolved neutron measurements is demonstrated for diagnosing neutron production from fusion plasmas with neutron emission durations longer than a microsecond. Neutron emission is measured with fast plastic, organic scintillators coupled to photomultiplier tubes (PMTs) operating in pulse counting mode. The PMT response is measured for the neutron emission duration of the fuzing plasma. Individual pulses are integrated using a template fitting approach to calculate the light output observed for a single neutron event. The light output is proportional to the energy of the recoil proton from neutron interactions in the detection volume. Additionally, a collection of neutron interactions from a mono-energetic neutron flux has unique statistical distribution of recoil protons. The recoil proton distribution can be numerically modeled and allows inference between the measured pulses and the incident neutron energy. With several detectors placed around the fuzing plasma, this detection methodology allows for a calculation of spatially resolved neutron emission profiles.

This manuscript is organized as follows: Section II briefly describes operating details of the FuZE device and the neutron diagnostics suite. Section III provides the physical basis for neutron energy anisotropies from beam-target fusion reactions. Results from energetically resolved neutron measurements on the FuZE device are described in Sec. IV. A discussion of the results is presented in Sec. V. Concluding remarks are given in Sec. VI.

Operating parameters and device specifications for FuZE are described previously.13,14,17 A drawing of the FuZE chamber is shown in Fig. 1. Helium fill gas mixtures with 20% D2 by pressure are injected in an acceleration region between two cylindrical electrodes and ionized by an applied voltage. Voltage is supplied by a capacitor bank consisting of twelve, 830 μF capacitors. For the data presented in this manuscript, the capacitor bank is charged to −6 kV. In the acceleration region, neutral gas is injected, ionized, and accelerated (left to right in Fig. 1) along the coaxial electrodes through a deflagration process.17,18 The plasma subsequently assembles into a 50-cm-long Z-pinch plasma column in the assembly region (Fig. 1). Continual gas injection and plasma acceleration along the coaxial electrodes result in radially sheared, axial plasma flows acting to stabilize the Z-pinch against fast-growing instabilities. Measured plasma lifetimes are 10–20 μs, or >1000 times greater than magnetohydrodynamic (MHD) instability growth times.

FIG. 1.

(a) Drawing of FuZE device chamber. Neutral gas is injected and ionized in the 1-m-long acceleration region. The ionized gas runs down the coaxial electrode arrangement and assembles into a Z-pinch plasma column in the 50-cm-long assembly region. (b) Zoomed-in view of the assembly region, showing the cathode tip and anode end wall. The length of the neutron emitting region (green) is 33 cm.14 

FIG. 1.

(a) Drawing of FuZE device chamber. Neutral gas is injected and ionized in the 1-m-long acceleration region. The ionized gas runs down the coaxial electrode arrangement and assembles into a Z-pinch plasma column in the 50-cm-long assembly region. (b) Zoomed-in view of the assembly region, showing the cathode tip and anode end wall. The length of the neutron emitting region (green) is 33 cm.14 

Close modal

Previous results from FuZE13 show pinch currents of >200 kA persisting for up to 15 μs, concurrent with measured ion densities and temperatures of 1017 cm−3 and ∼1 keV, respectively. With fill gas mixtures of 20% D2 in He, neutron yields and emission durations are ∼105 neutrons per discharge and ∼5 μs, respectively. The neutron emitting region has a length of 33 cm (Fig. 1). The centroid of the neutron emitting region is located approximately 17 cm from the end of the cathode tip.

Previous work13,14 used gas mixtures with different deuterium concentrations to show that all of the pulses observed during the neutron emission duration correspond to neutrons and not x-rays. This is an important conclusion, as the fast plastic scintillators used for measurements on FuZE are not capable of discriminating between x-rays and neutrons based on pulse shape alone.

The procedure for fielding plastic scintillator detectors on the FuZE device is described previously.14 The basic principles of fast neutron detection using organic scintillators are described in Klein and Brooks,19 and a picture of one of the scintillator detectors is shown in Fig. 2. Neutron detectors consist of 5 cm length, 5 cm diameter, and fast plastic scintillators (Eljen Technology, EJ-204, 0.7 ns rise time, and 1.8 ns decay time) that are coupled to fast PMTs (Hamamatsu R7724, 2.1 ns rise time). The PMT output is digitized using a 1 GHz bandwidth, deep-memory oscilloscope (Tektronix, DPO4104C). A characteristic waveform from a FuZE discharge is shown in Fig. 3.

FIG. 2.

(Top) Picture of one of the scintillator detectors used for neutron measurements on the FuZE device. (Bottom) Simplified schematic of the physical mechanism for pulse generation in the detector, where recoil protons produced by fast neutron interactions generate light via excitation and ionization of the scintillating medium. The scintillation light is converted to an electric signal using a photomultiplier tube (PMT).

FIG. 2.

(Top) Picture of one of the scintillator detectors used for neutron measurements on the FuZE device. (Bottom) Simplified schematic of the physical mechanism for pulse generation in the detector, where recoil protons produced by fast neutron interactions generate light via excitation and ionization of the scintillating medium. The scintillation light is converted to an electric signal using a photomultiplier tube (PMT).

Close modal
FIG. 3.

Digitized trace from a FuZE discharge. The detector is operating in pulse-counting mode. Individual pulses correspond to light output from recoil protons in the plastic scintillator.

FIG. 3.

Digitized trace from a FuZE discharge. The detector is operating in pulse-counting mode. Individual pulses correspond to light output from recoil protons in the plastic scintillator.

Close modal

The total light observed in an organic scintillator is related to the energy deposited by recoiling particles in the scintillator. For a given incident neutron energy, the distribution of recoiling particles can be modeled using the Geant4 toolkit.20 The light generation from these recoil distributions is largely from proton energy deposition, which can be converted to light output using the measured non-linear relation between proton energy deposition and light production, the proton light yield. The conversion for this work was done using the EJ-204 proton light yield relation previously measured by Laplace et al.21 By using the proton light yield in conjunction with the detector resolution function,22 modeled neutron response functions can be generated for the detectors in a variety of neutron fields.

Neutron emission durations from FuZE are microseconds long (Fig. 3), which is significantly longer than neutron emission durations from inertial confinement fusion (ICF) plasmas (typically <1 ns) and significantly shorter than neutron emission durations from magnetic confinement fusion (MCF) plasmas (typically >1 ms). Consequently, neutron measurement techniques that are applied to ICF or MCF plasmas are not readily applicable to diagnosing neutron emission from FuZE. The neutron time-of-flight (nToF) method is commonly used to infer neutron energies from experiments with sub-microsecond neutron emission durations, such as laser-driven ICF experiments and dense Z-pinch experiments.23–25 However, nToF measurements are difficult to apply on FuZE due to the microsecond long neutron emission durations. Magnetic recoil neutron spectrometers26 are applied for absolute neutron energy measurements on the National Ignition Facility (NIF), but the efficiency of the magnetic recoil spectrometer is ∼10–9, which is too low to measure neutron emission on FuZE. Methods for measuring fusion neutron energies on MCF experiments, such as segmented spectrometers or 3He spectrometers,27,28 often rely on neutrons thermalizing before interacting with a low-Z medium (e.g., 3He, 6Li). Consequently, the temporal responses of these methods are typically greater than 1 μs, which are too long with respect to neutron emission durations from the FuZE device.

Scintillation pulses were integrated using a functional form, y(t), adapted from Friesen and Howell29 

y(t)=AP[RP(t)·DP(t)+r·RP(t+tR)·DP(t+tR)],
(2)

where AP is the pulse amplitude, and RP(t) and DP(t) parametrize the pulse rise and decay, respectively. A signal reflection was observed in the measured waveforms at time tR from the scintillation pulse with relative amplitude r, accounting for the second term. The pulse rise, RP(t), is given by

RP(t)=1exp(tt0τr)+1,
(3)

where t0 is the pulse arrival time and τr is the rise time constant. Similarly, the pulse decay, DP(t), is given by

DP(t)=1exp(tt0τd)+1,
(4)

where τd is the decay time constant. Figure 4(a) shows an example fit for an isolated scintillation pulse. Piled-up pulses (i.e., pulses that had not reached baseline before the start of the next pulse, which represented ∼19% of the events in the measured traces) were fit using parameters from isolated pulse fits as initial seeds. Figure 4(b) shows an example of a successful fit of overlapping scintillation pulses. The goodness of fit was estimated by calculating the reduced chi-square, χν2, which can be defined as

χν2=1Nmi(yif(xi)σi)2.
(5)
FIG. 4.

(a) Example of fit for isolated scintillation pulse. (b) Example of successful fit for overlapping scintillation pulse.

FIG. 4.

(a) Example of fit for isolated scintillation pulse. (b) Example of successful fit for overlapping scintillation pulse.

Close modal

When χν2 is calculated for fits of scintillation pulses, yi corresponds to the Volts for the measured pulses, f(xi) corresponds to the functional form [Eq. (2)] of the fitted pulses, σi corresponds to the bit noise for the measured trace, N is the number of points used to fit the pulses, and m is the number of fit parameters. The distribution of χν2 values is shown in Fig. 5. An arbitrary constraint was applied to discard fits with χν2>5. This provided a recovery of 40% of piled-up pulses.

FIG. 5.

Distribution of χν2 for scintillation pulse fits.

FIG. 5.

Distribution of χν2 for scintillation pulse fits.

Close modal

Measuring neutron energy emission profiles from fuzing plasmas yields information about the origin of neutron production within the plasma. Neutron production from thermonuclear fusion reactions results in spatially isotropic neutron yield and energy profiles. Conversely, neutron production from non-thermal, beam-target fusion reactions results in strongly anisotropic neutron yield and energy emission profiles in the lab frame. Neutron energy emission profiles can be described by the following equation:30 

En(θB)=mn2(vcmcosθB±m4mn(mn+m4)(2Q+mabvab2)vcm2sin2θB),
(6)

where En is the emitted neutron energy as a function of the beam angle, θB; mn, m4, and mab are the neutron mass, helium-3 ion mass, and reduced mass of two deuterons, respectively; Q =3.27 MeV is the net reaction energy of the DD fusion reaction; and vcm and vab are the center-of-mass and relative velocities of two fuzing deuterons, respectively. The term mabvab2=Ed describes the energy of a mono-energetic deuteron beam incident on a thin, cold, stationary deuterium target. For deuteron beams with energies <100 keV, this term is much smaller than Q. However, because emitted neutron energies are dependent on the sum of the deuteron beam energy and the net reaction energy of the DD fusion reaction, the presence of deuteron beams with relatively low energies can cause relatively larger anisotropies in emitted neutron energies.

The magnitude of the anisotropy in emitted neutron energies is much larger than the actual deuteron beam energy. Figure 6 shows neutron energy emission profiles as a function of three deuteron beam energies, Ed = {0, 10, 50} keV. The calculations in Fig. 6 assume neutron emission from DD beam-target fusion reactions with mono-energetic deuteron beams and thin deuterium targets; here, thin refers to the fact that deuterons are not slowed significantly in the plasma, which is true in this case. The null case (Ed = 0 keV) shows neutron energy emission is isotropic in the absence of a deuteron beam. A deuteron beam with an energy as low as Ed = 10 keV can cause significant anisotropies in emitted neutron energies, as the difference in neutron energies emitted at θB = 45° and θB = 135°, or δEn, is 157 keV. Larger deuteron beam energies of Ed = 50 keV will cause greater anisotropies in emitted neutron energies, as δEn increases to 351 keV.

FIG. 6.

Modeled neutron energies emission profiles [Eq. (6)] for mono-energetic deuteron beams with energies of Ed = 0, 10, and 50 keV. E0 is equal to 2.45 MeV and δEn refers to the difference between neutron energies emitted at a beam angle of 45° and neutron energies emitted at a beam angle of 135°. Note that δEn is significantly larger than the deuteron beam energy.

FIG. 6.

Modeled neutron energies emission profiles [Eq. (6)] for mono-energetic deuteron beams with energies of Ed = 0, 10, and 50 keV. E0 is equal to 2.45 MeV and δEn refers to the difference between neutron energies emitted at a beam angle of 45° and neutron energies emitted at a beam angle of 135°. Note that δEn is significantly larger than the deuteron beam energy.

Close modal

Figure 7 shows the relative locations of two scintillation detectors, denoted as detector #1 and detector #2, around the fuzing plasma column. Measured pulse integral spectra for each detector at both detector locations are shown in Fig. 8. For one set of 58 discharges, detector #1 was positioned at a location of (R, Z) = (45, 61) cm, and detector #2 was positioned at a location of (R, Z) = (45, −45) cm, where R and Z refer to radial and axial distances from the cathode tip, respectively. For a subsequent set of 58 discharges, the relative locations of the scintillation detectors were swapped. Detector #1 was moved to a location of (R, Z) = (45, −45) cm, and detector #2 was moved to a location of (R, Z) = (45, 61) cm. The absolute detector locations of (R, Z) = (45, −45) cm and (45, 61) cm were chosen to observe differences in measured spectra due to axial beam-target neutron production while minimizing artifacts of neutron attenuation and down-scattering from the FuZE chamber geometry.

FIG. 7.

Orientation of scintillator detectors with respect to fuzing plasma. (The cathode, anode, and device chamber are not shown.) Scintillator detectors were placed upstream and downstream of the fuzing plasma to detect neutron energy anisotropies from axial, beam-target fusion reactions.

FIG. 7.

Orientation of scintillator detectors with respect to fuzing plasma. (The cathode, anode, and device chamber are not shown.) Scintillator detectors were placed upstream and downstream of the fuzing plasma to detect neutron energy anisotropies from axial, beam-target fusion reactions.

Close modal
FIG. 8.

Pulse integral spectra measured with detector #1 (left) and detector #2 (right) at two locations around the FuZE plasma column. In the figure legend, N refers to the total number of pulses in each spectrum. The scintillator light output corresponds to the integrated charge in arbitrary units.

FIG. 8.

Pulse integral spectra measured with detector #1 (left) and detector #2 (right) at two locations around the FuZE plasma column. In the figure legend, N refers to the total number of pulses in each spectrum. The scintillator light output corresponds to the integrated charge in arbitrary units.

Close modal

A strong axial, deuteron beam traveling from the anode to the cathode (right to left, in Figs. 1 and 7) would cause an upshift in neutron energies incident on the left-most detector location, (R, Z) = (45, −45) cm, and a downshift in neutron energies incident on the right-most detector location, (R, Z) = (45, 61) cm.

A novel, forward-fit method is applied to determine whether neutrons are emitted from thermonuclear or beam-target fusion reactions. This approach assumes that beam-target fusion reactions are driven by an axially propagating, mono-energetic deuteron beam with energy Ed. The forward-fit method is used to determine a value for Ed that results in a best fit between measured (Fig. 8) and simulated pulse integral spectra.

The forward fit requires a prediction of the observed spectra for each detector as a function of Ed. To accomplish this, the Geant4 toolkit is used to simulate neutrons incident on each of the scintillation detectors (Fig. 9) for a given Ed. The yield and energy emission profiles for a neutron line source are parametrized as a function of Ed. Equation (64) in Higginson et al.30 is used to define a spatially resolved neutron yield profile, and Eq. (6) is used to define a spatially resolved energy emission profile. The length of the neutron line source is 33 cm and is centered 16.5 cm from the cathode tip.14 The characteristics of the neutron line source, locations of each scintillator detector, and materials surrounding the plastic scintillators (including the PMT, a thin aluminum case around the scintillator, and mu-metal around the PMT) are representative of experimental conditions (Fig. 9). Simulations were performed where the initial value of Ed for an event was sampled from a uniform distribution from −50 to 50 keV. Negative values of Ed describe a deuteron beam propagating from anode to cathode, or right to left in Fig. 9, and in the opposite direction for positive values of Ed.

FIG. 9.

Geometry showing locations of scintillator detectors with respect to a neutron line source in Geant4. Gray lines show simulated neutron tracks.

FIG. 9.

Geometry showing locations of scintillator detectors with respect to a neutron line source in Geant4. Gray lines show simulated neutron tracks.

Close modal

The energy deposited from recoiling protons in the scintillator and the initially sampled value of Ed was recorded on an event-by-event basis. Using the previously measured EJ-204 proton light yield relation from Laplace et al.,21 recoil proton energies were converted to relative scintillator light output. The units of relative light output are defined with respect to the light that would be observed from an electron deposition of equivalent energy, or MeVee. A total of 1.2×1010 source particles were transported in the simulation. Figure 10 shows the results of accumulating the calculated light for each event into a two-dimensional histogram where the abscissa represents the initially selected deuteron energy, Ed; the ordinate axis represents the total light, Lν, observed for a given event; and the z color bar represents the number of events observed per 0.05 MeVee per 5 keV Ed for 1.2×1010 simulated particles. These histograms represent the light observed by a detector with perfect light resolution and will be referred to as Λj(Ed,Lν) where the jth subscript denotes the detector location index, that is, j={1,2}. Detector locations of (R, Z) = (45, −45) cm and (R, Z) = (45, 61) cm are shown in Fig. 7.

FIG. 10.

Simulated scintillator light output as a function of deuteron beam energy, Ed. The two figures show the simulated pulse integral spectra for detector locations of (R, Z) = (45, −45) cm and (R, Z) = (45, 61) cm as a function of deuteron beam energy. Values of Ed varied from 0 (no beam present in plasma) to ±50 keV. For units used in this numerical calculation, negative values of Ed describe a deuteron beam propagating from the anode to cathode (right to left in Fig. 7).

FIG. 10.

Simulated scintillator light output as a function of deuteron beam energy, Ed. The two figures show the simulated pulse integral spectra for detector locations of (R, Z) = (45, −45) cm and (R, Z) = (45, 61) cm as a function of deuteron beam energy. Values of Ed varied from 0 (no beam present in plasma) to ±50 keV. For units used in this numerical calculation, negative values of Ed describe a deuteron beam propagating from the anode to cathode (right to left in Fig. 7).

Close modal

To calculate the likelihood that the observed data corresponds to a given Ed, projections of Λj(Ed,Lν) are generated over Ed in 5 keV steps ranging from –50 to 50 keV yielding a series of light yield spectra, Λj(Lν;Ed), for each 5-keV bin. The projections undergo several transformations to account for overall event normalization, the detector resolution, and the calibration from the measured integrated charge to the simulated MeVee relative light unit. The prediction of the measurement for the jth detector location, μj, is given by

μj(C;A,α,β,γ,Sj,Ed)=Rν(C,α,β,γ)*(Sj×Λj(A×C;Ed)).
(7)

This introduces six parameters to the forward model: A, α, β, γ, S1, and S2. Here, * is the convolution operator; A is an amplitude scalar used to convert the measured integrated charge, C, into MeVee; α, β, and γ are parameters describing the amplitude-dependent, detector resolution as a function of the integrated charge, Rν(C,α,β,γ) as described in Dietze and Klein;22 and S1 and S2 represent normalization scalars.

The six parameters are determined by minimizing a loss function, defined as the negative log-likelihood, between the measured and simulated spectra. The log-likelihood assumes that measurement errors are independent and can be described by Poisson statistics. The loss function is defined as

(8)
lnL=i,jlnρ(yi,j,μi,j),
(8a)
=i,jlnμi,jyi,jeμi,jyi,j!,
(8b)
where ρ is a Poisson distribution, i is the bin index, j is the detector location index, yi,j is the number of counts in the ith bin of the measured spectrum of the jth detector position, and μi,j is the predicted counts for the ith bin of the simulated spectrum for detector location j given by Eq. (7). The minus sign in front of lnL is used so that best-fit values of fit parameters in μ can be calculated with an existing numerical minimizer.31 Therefore, the loss Eq. (8) is minimized for all parameters of μ for each Ed in the range.

For illustrative purposes, Fig. 11 shows simulated spectra overlaid with measured spectra for detector #1 at both detector locations. Values of Ed = 0, 10, and 20 keV are used for calculating the simulated spectra. Each spectrum is transformed by the six parameters, A, α, β, γ, S1, and S2, as described previously [Eq. (7)]. Values of the six parameters are determined for each of the three simulated spectra in Fig. 11 by minimizing the loss function [Eq. (8)] between the measured and simulated spectra. Values calculated by the loss function between the measured and simulated pulse integral spectra are 61.4 for Ed = 0 keV, 64.1 for Ed = 10 keV, and 68.6 for Ed = 20 keV. These values between the measured and simulated spectra are calculated over a range of values of 0.2–1.3 MeVee (Fig. 10). A total of 125 bins were used for constructing the measured and simulated spectra in Fig. 11 

FIG. 11.

Illustrative plot showing measured and simulated spectra for detector #1, for scintillator detector locations of (R, Z) = (45, −45) cm (left) and (R, Z) = (45, 61) cm (right). In the plots above, simulated spectra are calculated for fixed values of Ed = 0, 10, and 20 keV.

FIG. 11.

Illustrative plot showing measured and simulated spectra for detector #1, for scintillator detector locations of (R, Z) = (45, −45) cm (left) and (R, Z) = (45, 61) cm (right). In the plots above, simulated spectra are calculated for fixed values of Ed = 0, 10, and 20 keV.

Close modal

The best fit of Ed is determined by calculating the loss [Eq. (8)] over the full grid of values of Ed in Fig. 10. A second-degree polynomial function

LFit=A0+A1×Ed+A2×Ed2,
(9)

is fitted to the loss calculated as a function of Ed. The best fit value of Ed is found at the minimum of the fitted polynomial curve, Ed=0.5A1/A2. The statistical uncertainty32 corresponds to the intercept between the loss curve and loss values equal to Lmin + 0.5, where Lmin is the value of the fitted polynomial calculated at Ed.

The systematic uncertainty of Ed is determined by scanning over user-specified parameters of the forward-fit method. User-specified parameters include initial conditions for the minimizer, the fit range, and the number of bins in each spectrum. The initial conditions for the minimizer correspond to projections of the simulated spectra onto the measured spectra for a fixed value of Ed. Values of Ed equal to 0, +25, and −25 keV were used to generate each of the three initial conditions for the minimizer. The fit range corresponds to the range of recoil proton energies over which the measured and simulated spectra are compared. The lower and upper values for the fit range varied from 0.17–0.23 and 1.2–1.4 MeVee, respectively. Constructing spectra with 55, 75, 105, or 125 bins has no statistically significant effect on the inferred value or uncertainty of Ed. The number of bins was chosen to resolve the edge of the spectra. Varying the initial conditions of the minimizer, fit range, or number of bins has no statistically significant impact on the inferred value of Ed.

Figures 12 and 13 show results from scanning over user-specified parameters. Figure 12 shows individual loss curves calculated for each detector as a function of Ed. Values for the loss shown in Figs. 12 and 13 correspond to negative log-likelihood values [Eq. (8)]. The individual loss curves (Fig. 12) are calculated with different values of user-specified parameters, including the fit range and initial conditions for the minimizer. For example, one of the loss curves in Fig. 12 is calculated with a fit range of 0.17–1.3 MeVee and one set of initial conditions. A different loss curve in Fig. 12 is calculated with a fit range of 0.2–1.4 MeVee and a different set of initial conditions. The number of bins in the measured and simulated spectra for each of the individual loss curves in Fig. 12 is fixed at 125. Individual loss curves with the same number of bins can be averaged to calculate an average loss curve as a function of Ed.

FIG. 12.

Results of scanning over user-specified parameters for detectors #1 (left) and #2 (right). Each individual loss curve is calculated with different values for the set of user-specified parameters of the forward-fit model. The loss corresponds to negative log-likelihood values, as defined in Eq. (8).

FIG. 12.

Results of scanning over user-specified parameters for detectors #1 (left) and #2 (right). Each individual loss curve is calculated with different values for the set of user-specified parameters of the forward-fit model. The loss corresponds to negative log-likelihood values, as defined in Eq. (8).

Close modal
FIG. 13.

Results for second-degree polynomial fits (orange curve) to loss curves (green points) for detectors #1 (left) and #2 (right). For detector #1, the fitted value of Ed is 10.96 keV, with a statistical uncertainty of 5.65 keV and a systematic uncertainty of 0.60 keV. For detector #2, the fitted value of Ed is −3.81 keV, with statistical and systematic uncertainties of 6.13 and 1.11 keV, respectively. The loss corresponds to negative log-likelihood values, as defined in Eq. (8).

FIG. 13.

Results for second-degree polynomial fits (orange curve) to loss curves (green points) for detectors #1 (left) and #2 (right). For detector #1, the fitted value of Ed is 10.96 keV, with a statistical uncertainty of 5.65 keV and a systematic uncertainty of 0.60 keV. For detector #2, the fitted value of Ed is −3.81 keV, with statistical and systematic uncertainties of 6.13 and 1.11 keV, respectively. The loss corresponds to negative log-likelihood values, as defined in Eq. (8).

Close modal

Figure 13 shows average loss curves for each detector. The green data points in Fig. 13 correspond to the mean and standard deviation of individual loss curves in Fig. 12. Chi-square minimization [Eq. (5)] is used to fit a second-degree polynomial curve, LFit(Ed,i) [Eq. (9)], to the data points. Values of yi and σi correspond to the mean and standard deviation of the loss curves. Fit results are shown in Table I. For detector #1, the value of Ed that best fits the data, Ed, is 10.96 keV with a statistical uncertainty of 5.65 keV and a systematic uncertainty of 0.60 keV. For detector #2, Ed has a value of −3.8 keV with a statistical uncertainty of 6.13 keV and a systematic uncertainty of 1.11 keV. The systematic uncertainty of Ed is found from the uncertainty of the fit parameters, Ai. The uncertainty of the fit parameters is determined by calculating the covariance matrix of the fit. The formula used to calculate the systematic uncertainty is

USys=|0.5Ed|×(σA1A1)2+(σA2A2)2σA1,A2A1A2,
(10)

where USys is the systematic uncertainty, σAi is the uncertainty in the fit parameter Ai(i=1,2), and σA1,A2 is the covariance between A1 and A2. Using inverse-variance weighting, the weighted average of Ed is 4.65 keV, with an average statistical uncertainty of 4.15 keV and an average systematic uncertainty of 0.53 keV.

TABLE I.

Results from fits to data in Fig. 13. Values for Ai(i=0,1,2) correspond to fit parameters in a second-degree polynomial function and Ed corresponds to the value of Ed that best fits the data. Values and uncertainties of Ed are in units of keV.

Detector #1Detector #2
Ed 10.96 −3.81 
Statistical uncertainty 5.65 6.13 
Systematic uncertainty 0.60 1.11 
Detector #1Detector #2
Ed 10.96 −3.81 
Statistical uncertainty 5.65 6.13 
Systematic uncertainty 0.60 1.11 

The results shown in Fig. 13 and Table I facilitate determination of the primary mechanism for neutron emission from FuZE (i.e., beam-target fusion vs thermonuclear). Based on the measured pulse integral spectra from detectors #1 and #2, the results indicate the presence of a 4.65 ± 4.15 ± 0.53 keV deuteron beam propagating from the cathode to the anode. Based on electrode polarity, beam-target fusion reactions are typically driven by an energetic deuteron beam propagating from the anode to cathode.33 In Z-pinch plasmas, beam-target fusion reactions are typically caused by m =0 sausage instabilities, which creates large, transient electric fields. The electric fields will drive energetic deuteron beams into a deuterium target,15,16 resulting in neutron production through beam-target fusion reactions. Previous measurements from dense Z-pinch devices have observed production of deuteron beams with energies in excess of 100 keV.34–36 Given that neutron energy emission from the FuZE device has not shown significant energy anisotropies, thermonuclear fusion reactions are likely to be the dominant source of neutron production.

Note that the FuZE device generates radially sheared, axial plasma flows along the pinch axis. Axial flow velocities of ∼120 km/s have been inferred from measurements on the FuZE device,17 which would correspond to a mono-energetic, deuteron beam energy of ∼0.15 keV. The direction and low energy (100 keV) of the inferred deuteron beam is consistent with thermonuclear neutron emission from a quiescent plasma in the FuZE device.

Future work will involve fielding additional scintillation detectors around the FuZE device chamber to rule out non-thermal neutron emission from a greater range of instabilities and increase measurement statistics. The results in Fig. 13 and Table I were obtained by measuring relative neutron energies at two detector locations (Fig. 7) around the fuzing plasma. With two detector locations, the presence of energetic, axial deuteron beams from m =0 instabilities can be precluded. Non-thermal neutron production driven by m =0 instabilities is the dominant source of neutron emission from several dense Z-pinch devices. Observing isotropic neutron energy emission at more detector locations can preclude the presence of non-thermal fusion reactions from other types of plasma instabilities.

Fielding more scintillator detectors around the FuZE device chamber will also allow for increased measurement statistics over fewer discharges. The pulse integral spectra shown in Fig. 8 were integrated over 58 discharges per spectra. The average number of pulses measured for an individual discharge on a single detector must be kept small in order to minimize the effects of pulse pileup. Neutron emission durations from the FuZE device are ∼2–5 μs. The functional form [Eq. (2)] used for pulse integrals was fitted over a 56 ns integration path length. Fielding multiple detectors at each detector location would allow for pulse integral spectra with more statistics and/or pulse integral spectra constructed over fewer discharges.

Future work will also include inferring absolute neutron energies from pulse integral spectra constructed using calibrated detectors. Plastic scintillator detectors can be calibrated by measuring Compton spectra from gamma sources with known emission energies, such as 137Cs or 60Co. Photons emitted from gamma sources undergo Compton scattering with electrons in the scintillator. The resulting scintillator light output spectrum can be used to determine parameters for detector resolution and provide a reference standard for the scintillator light output. Light output units for recoil proton energies can then be expressed in units relative to the light output associated with a given electron energy (e.g., Compton edge of a 137Cs spectrum). Using a measured Compton spectrum from a gamma source to calibrate scintillator light output units along with the measured proton light yield relations could also enable absolute neutron energy measurements with a single scintillator detector. Additionally, the three parameters [α, β, and γ in Eq. (7)] that define detector resolution can be derived from a calibration with a gamma source. Because the scintillator detectors used in the present work were not calibrated, the three parameters were treated as free parameters.

Results presented in this manuscript preclude the possibility of axial, beam-target fusion reactions as the dominant source of neutron emission from the FuZE device. Together with long neutron emission durations,13 these results indicate thermonuclear neutron production from FuZE, which is a necessary condition for scaling the FuZE device up to fusion reactor conditions. Future work will involve refining the measurement technique to monitor for anisotropies in emitted neutron energies as SFS Z-pinch devices operate with greater device powers and higher pinch currents.

The authors would like to thank Bob Geer and Daniel Behne for technical assistance, as well as Amanda Youmans, Christopher Cooper, and Clément Goyon for advice and discussions. The authors would also like to thank Phil Kerr and Vladimir Mozin for the use of their Thermo Fisher P385 neutron generator, which was important in verifying the ability to measure neutron energy shifts via the pulse integral technique. The information, data, or work presented herein was funded in part by the Advanced Research Projects Agency—Energy (ARPA-E), U.S. Department of Energy, under Award Nos. DE-AR-0000571, 18/CJ000/05/05, and DE-AR-0001160. 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 and Lawrence Berkeley National Laboratory under Contract No. DE-AC02-05CH11231. U. Shumlak gratefully acknowledges support of the Erna and Jakob Michael Visiting Professorship at the Weizmann Institute of Science and as a Faculty Scholar at the Lawrence Livermore National Laboratory.

The authors have no conflicts to disclose.

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

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