Characterizing Subcritical Assemblies with Time of Flight Fixed by Energy Estimation Distributions
CCharacterizing Subcritical Assemblies with Time of Flight Fixed by EnergyEstimation Distributions
Mateusz Monterial a, ∗ , Peter Marleau b , Sara Pozzi a a Department of Nuclear Engineering and Radiological Sciences, University of Michigan, Ann Arbor, MI 48109, USA b Radiation and Nuclear Detection Systems Division, Sandia National Laboratories, Livermore, CA 94551, USA
Abstract
We present the Time of Flight Fixed by Energy Estimation (TOFFEE) as a measure of the fission chain dynamics insubcritical assemblies. TOFFEE is the time between correlated gamma rays and neutrons, subtracted by the estimatedtravel time of the incident neutron from its proton recoil. The measured subcritical assembly was the BeRP ball, a 4.482kg sphere of α -phase weapons grade plutonium metal, which came in five configurations: bare, 0.5, 1, and 1.5 in iron, and1 in nickel closed fitting shell reflectors. We extend the measurement with MCNPX-PoliMi simulations of shells rangingup to 6 inches in thickness, and two new reflector materials: aluminum and tungsten. We also simulated the BeRP ballwith different masses ranging from 1 to 8 kg. A two-region and single-region point kinetics models were used to model thebehavior of the positive side of the TOFFEE distribution from 0 to 100 ns. The single region model of the bare cases gavepositive linear correlations between estimated and expected neutron decay constants and leakage multiplications. Thetwo-region model provided a way to estimate neutron multiplication for the reflected cases, which correlated positivelywith expected multiplication, but the nature of the correlation (sub or super linear) changed between material types.Finally, we found that the areal density of the reflector shells had a linear correlation with the integral of the two-regionmodel fit. Therefore, we expect that with knowledge of reflector composition, one could determine the shell thickness,or vice versa. Furthermore, up to a certain amount and thickness of the reflector, the two-region model provides a wayof distinguishing bare and reflected plutonium assemblies. Keywords:
Fission chain, Organic scintillator, Neutron noise, Non-destructive assay
1. Introduction
Multiplicity analysis has been the staple of non-destructiveassay (NDA) and accountancy of fissile nuclear materialfor decades [1] [2] [3] [4]. This NDA technique relies onthe counting of single, double and higher-order multiplic-ities in a pre-defined time window. Fissile material hastwo properties that affect the multiplicity distribution: (1)multiple neutrons are emitted in coincidence from fissionand (2) the subsequent fissions may produce more coinci-dence neutrons. Traditionally, analog coincidence circuitscoupled with He-3 proportional counters are used to recordthis signature.Recent innovation in neutron coincidence counting hasfocused on replacing He-3 tubes altogether in favor of fastorganic scintillators. Development of Fast Neutron Mul-tiplicity Counters (FNMC) has been motivated by thepromise of greater precision at lower dwell times due to in-trinsically lower die-away times of these detectors leadingto lower accidental coincidence rates [5][6]. Additionally,gamma rays can be used as additional signature in pulseshape discrimination capable scintillators [7] [8]. ∗ Corresponding author
Email address: [email protected] (Mateusz Monterial)
Research and development is generally geared towardsaccounting for the differences in thermal capture He-3-based systems and FNMCs [9]. As a result, the devel-opment of new measurement systems is underpinned bymore or less the same multiplicity analysis developed overthree decades ago. Therefore, data analysis using newmeasurement systems continue to require both high effi-ciency and accurate knowledge of the efficiency of the sys-tem. This creates A design principle that drives systemstoward larger sizes and geometries that limit applicationsand portability. However, portability is often a featuredemanded by field applications such as treaty verificationand nuclear emergency response.Instead of optimizing fast organic scintillator-based sys-tems to the design principles of multiplicity counting, wehave leveraged a few key additional signatures available tothese detectors while investigating new analysis method-ologies:1. Prompt gamma rays released from fission, which aredistinguished from neutrons with pulse shape dis-crimination (PSD) [10] [11]2. Energy imparted by incident neutrons measured throughproton recoil.3. Timing between these detected events, measured with a r X i v : . [ phy s i c s . d a t a - a n ] J a n anosecond and sub-nanosecond timing resolutioncapable of resolving the timing between individualfission events in a chain.In this paper we will demonstrate how these signa-tures can be combined to measure Time of Flight Fixed byEnergy Estimation (TOFFEE) distributions. The TOF-FEE distribution is sensitive to the timing between fis-sion events present in a measured medium. The dynamicsof fission chain timing are driven by the physics of thesubcritical system, mainly the multiplication of fissile ma-terial and presence of neutron moderators and reflectors.To illustrate this, we developed a two-region point kineticsmodel of a reflected fissile assembly. The solution to thismodel was then used to fit the measured TOFFEE distri-butions in an effort to extract physical system parameterssuch as multiplication and the presence of and coupling toa reflector from an inter-event timing distribution alone.We tested our approach by fitting TOFFEE distribu-tions from measurements of Beryllium Reflected Pluto-nium (BeRP) ball [12] in a bare configuration and withiron and nickel shell reflectors. All measurements wereperformed with a hand-held array of eight 2” ×
2” Stilbenedetectors. However, to draw any broader conclusions itwas necessary for us to expand on the measured configu-rations with additional simulations. In particular, we ex-tended the measurements by simulating thicker shell reflec-tors and two different materials: aluminum and tungsten.It was also necessary to test changing neutron multipli-cation independent of presence of reflectors by simulatingbare BeRP balls with different masses.We found that fits to the TOFFEE distributions of thebare systems predicted the rate of change of the neutronpopulation quite well. For each of the reflected configura-tions, a strong correlation between the fit and parametersand shell thickness was apparent. With some assumptionsabout the fissile core, we found that fits to the reflectedconfigurations estimated the system multiplication. In ad-dition, each reflector material had a distinct positive cor-relation between simulated and estimated multiplication.
For the past several years, Sandia National Laborato-ries and the University of Michigan have been collaborat-ing on a new technique that uses both the timing betweencorrelated gamma rays and neutrons in conjunction withthe energy deposited by correlated neutrons. These sig-natures were first combined into Time-Correlated PulseHeight (TCPH) distributions which were shown to be sen-sitive to multiplication of neutrons within fissile materialand by extension the material mass and presence of inter-vening moderators or reflectors [13, 14, 15, 16, 17]. TheTCPH distribution is a raw representation of the mea-sured signature in a bi-variate histogram of the neutrondeposited energy and the time to correlated gamma ray.Therefore, it is difficult to interpret and challenging tomodel for the purpose of extracting physical parameters [18]. For this reason we re-combined the measured signa-tures into a single one-dimensional distribution we call theTime of Flight Fixed by Energy Estimation (TOFFEE).The time-of-flight (TOF) in TOFFEE is the measuredtime between correlated gamma rays and neutrons. In or-der to correct this TOF by the expected TOF of the neu-tron from the point of emission to the detector, we estimateits energy, E n , as the energy it deposits in the detector byelastic scatter on a proton, E p . Because the neutron typi-cally deposits only a fraction of its energy in this interac-tion, the estimated energy will be systematically low andthus the estimated TOF will be systematically too large.With a known source-to-detector distance d , it is possibleto estimate the travel time difference between a neutronand gamma ray emitted simultaneously: t p = d (cid:18)(cid:114) m E p − c (cid:19) (1)where c is the speed of light and m is neutron’s mass. Thecalculated quantity t p is therefore the estimated differencein neutron and gamma-ray time of flight difference fromthe proton recoil energy. Since E p is systematically smallerthan the true incident neutron energy, t p will overestimatethe true time of flight difference between the neutron andgamma way. Finally, the “fixed” in TOFFEE refers tosubtracting this quantity from the measured time betweengamma ray and neutron pair, t n,γ .For non-multiplying sources (e.g. spontaneous fission,( α , n)), the actual travel time difference between gammaray and neutron, T n,γ , will equal the measured t n,γ , asshown in Figure 1(a). Therefore, TOFFEE for non-multiplyingsources will be less than or equal to zero t n,γ − t p ≤ . (2)In contrast, for multiplying sources the measured time dif-ference between correlated gamma rays and neutrons willinclude the difference in generation time, ∆ T g , as shownin Figure 1(b) and (c). As a result TOFFEE for sourceswith present fission chains will be less than or equal tothe times between fission events that gave birth to eachparticle t n,γ − t p ≤ ∆ T g . (3)There are three important implications from Eqs. 2and 3 on the relationship between TOFFEE and the typeof source measured. First, there is a sharp distinctionbetween non-multiplying and multiplying sources becausethe former should have a steep drop in counts on the pos-itive side of the TOFFEE distribution. The TOFFEEdistribution of a multiplying source will, in contrast, be“smeared” in both negative and positive time directionsby ∆ T g . The bi-directional smearing is exemplified in Fig-ures 1 (b) and (c), and is the consequence of correlating2 a) Non-multiplying(b) Multiplying, gamma ray born first (c) Multiplying, neutron born first Figure 1: Space-time diagrams of gamma ray (green) and neutron (red) particle paths from birth to detection (dashed blue line). The (a)non-multiplying diagram depicts the simultaneous birth of particles, and the (b) and (c) multiplying diagrams depict a fission chain whereeach fission is separated by generation time ∆ T g . The measured time-of-flight difference, t n,γ , is equivalent to the true time-of-flight difference T n,γ in the non-multiplying case, but it includes the generation time in the multiplying case. The dashed red lines depict possible estimatesof the neutron’s velocity from proton recoil. The end-points of those dashed lines on the time-axis at the assumed source distance make upthe TOFFEE distribution. gamma-neutron pairs where either the gamma ray or theneutron were born first.Second, the TOFFEE distribution is quite sensitive tothe level of neutron multiplication within a source. If k isdefined as the ratio of the number of neutrons born in onegeneration to those in the previous generation, then thesubcritical neutron multiplication is M = 11 − k . (4)Neutron multiplication is equivalent to the average numberof neutrons produced per starting neutron or the averagelength of a fission chain [19]. The probability of detect-ing particles from the same fission grows linearly with M ,which will be distributed according to Eq. 2. Whereas,the probability of detecting particles from two differentfissions in a chain increases factorially with M and will bedistributed according to Eq. 3.The contributions of the particles correlated in thesame generation and different generation from a simula-tion of the bare BeRP ball is shown in Figure 2. In this example, generations are used to distinguish corre-lated events, because MCNPX-PoliMi output provides thegeneration number of a fission that originated a detectedparticle, but not a unique identifier of the fission event it-self. Multiple fissions can belong to the same generation,because of branching in a fission chain, therefore this ex-ample is an approximation to TOFFEE distributions fromsame and different fissions. As a consequence, the samegeneration TOFFEE distribution, shown in Figure 2, willsometimes include the time between fissions of the samegeneration and will therefore also include a ∆ T g smearingterm. The different generation TOFFEE distribution isnot only smeared out due to the addition ∆ T g , but alsohas noticeably more counts due to the greater probabil-ity of detecting particles that are correlated from separatefission events.Finally, the influence of ∆ T g , as shown in Eq. 3, meansthat the TOFFEE distribution is simultaneously a mea-sure of a length of a fission chain and the timing distribu-tion of fissions within that chain. The characteristic timebetween fission events in a chain is indicative of the prob-3 TOFFEE (ns) -1 C o un t s Same GenerationDifferent Generation
Figure 2: TOFFEE distributions of simulation of the BeRP ballconstructed from gammas and neutrons originating from the samegeneration and different generations of fissions. There are more cor-relations from different generations due to neutron multiplication ofthe BeRP ball ( M = 4 . ± . ability of fission, and the average neutron energy betweenfissions. In addition, for assemblies that are coupled to areflector, the time between fission events also depends onthe probability and the time for a neutron to return tothe fissile material. In this paper we demonstrate that theTOFFEE distribution relates to these physical properties.
2. Experiments
To demonstrate the sensitivity of the TOFFEE distri-bution to the physical configuration of a fissile core andsurrounding reflective materials, a series of measurementsof the BeRP ball were made. The BeRP ball is a 4.482 kgsphere of α -phase weapons-grade plutonium metal (93.3 wt % Pu-239, 5.9 wt % Pu-240), originally manufacturedin October 1980 by Los Alamos National Laboratory [12].This sphere has a mean radius of 3.7938 cm, and is encasedin a 304 stainless steel shell that is 0.0305 cm thick. Themeasurements were conducted at the Nevada National Se-curity Site (NNSS), with five distinct reflector configura-tions shown in Table 1. The reflectors were made fromclose fitting sets hemispherical shells made of iron andnickel, with a single 4.509 cm diameter hole going throughthem.In addition to a multiplying source, we measured a 21 µ Ci Cf-252 source at a source to detector distance of 35cm. This measurement was performed independently atSandia National Laboratories. These measurements serveto validate Monte Carlo simulations of the detection sys-tem and fissile assembly.All measurements were performed with a purpose-builtportable array of eight 2” by 2” cylindrical stilbene crys-tals. Each stilbene crystal was coupled to H1949-50 Ham-mamtsu photomultiplier tube (PMT) with a custom lowvoltage to high voltage bias converter. Quarter inch thickpucks of lead were attached to the front of the detec-tors in order to minimize count rate from uncorrelated decay gamma rays emitted by the plutonium and ameri-cium in the BeRP ball. These pucks were used for allsubsequent Cf-252 and calibration experiments. A pho-tograph of the instrument is shown in Figure 3. Theanode outputs were digitized using CAEN DT5730 digi-tizer, capable of 14-bit vertical resolution and a 500 MHzsampling rate. The acquisition threshold was approxi-mately 20 keVee (keV electron-equivalent), and the post-processing threshold was set to be 100 keVee.
Figure 3: The front of purpose-built stilbene array used for all mea-surements.
Energy calibration measurements were performed witha Na-22 source. Calibration constants were estimated bymatching measurements to an MCNPX-PoliMi simulation[20]. The measured pulse heights (PH) were shifted by alinear calibration formula L = a ∗ P H + b (5)where L is the calibrated light output and a and b arecalibration parameters, while the simulated results werebroadened by the approximate energy resolution of eachdetector: ∆ LL = (cid:114) α + β L + γ L (6)where α , β and γ parameters include contributions fromlight transmission within detector cell, statistical fluctua-tions of light production and electronic noise, respectively[21].Levenberg-Marquardt algorithm was employed to findthe optimum calibration and resolution parameters andan example of the results are shown in Figure 4. Extraweight was given to the regions of the spectra around theCompton edges, and the back-scatter peak was ignored.4 able 1: Measurement details of the various configurations of the BeRP ball with iron and nickel reflectors. The neutron multiplication wascalculated from MCNP6 k-code simulation. case measurement rate of gamma ray multiplicationtime (sec) neutron pairs (Bq)bare 1968 136 4.433 ± ± ± ± ± P r o b a b ili t y D e n s i t y MeasurementSimulation w/ resolutionSimulation w/o resolution
Figure 4: Measured and simulated spectra of Na-22 source matchedwith optimum resolution and calibration parameters.
The neutron light output yield for 2” stilbene crystalswas measured by Bourne et al. in a separate set of ex-periments [22]. The set of proton recoil energies and cor-responding calibrated light outputs were fitted to Birks’formula L ( E p ) = (cid:90) a b ( dE/dx ) dE (7)where dE/dx is the proton stopping power in stilbene [23].The fitted parameters of a and b were found to be 1.63(MeVee/MeV) and 27.83 (mg/(cm MeV)), respectively.The integrand in Eq. 7 was evaluated for deposited ener-gies ranging from 1 keV to 250 MeV, and the results weresaved in a lookup table. Linear interpolation of the valuesin the table was used to calculate light output from sim-ulations and approximate proton recoil energy from lightoutput in measurement.We employed a Bayesian approach to pulse shape dis-crimination (PSD) [24], in order to separate neutrons fromgamma rays and give each prospective particle detectionan appropriate weight. This approach allowed us to studythe effect PSD cuts have on the outcome of the final TOF-FEE distribution. As expected the “harder” PSD cuts areakin to raising the energy threshold, due to the overlapbetween neutrons and gamma rays PSD at lower energies.We used the time differences between 90% and 10% of the pulse cumulative trapezoidal integral as the PSD parame-ter.
3. Simulation Validations
The measured configurations of the BeRP ball, shownin Table 1, include three sets of shell thickness and twotypes of shielding material. We used simulations to ex-pand the range of shell thicknesses, explore other reflectormaterials, and vary the mass of the BeRP ball by changingthe diameter of the sphere. The simulations of the TOF-FEE distributions were performed with MCNPX-PoliMi,and details simulated materials are provided in Table 2.The k-code calulcations performed with MCNP6 used thesame materials, and k eff was estimated over 50 cycles with1 million neutrons per cycle. In order to have confidencein these results it was necessary to validate the simulationsby comparing them with measurements. First, we compared the neutron pulse height distribu-tions (PHDs), which should test the energy calibration andneutron light output function. We limited the neutrons tothose that were correlated with gamma rays inside a 2 µ swindow. The measured and simulated PHDs shown in Fig-ure 5 overlap with the entire range of measured energies,without any noticeable systematic bias and within the sta-tistical error. The statistical fluctuations are reflected inthe relative error, which oscillates around zero.In contrast to the PHD, which is relatively featureless,the TOFFEE distributions shown in Figure 6 have sev-eral features whose shape depend on the detector response.The most significant feature is the bell-like curve between-10 and 5 ns which includes the vast majority of correlatedcounts. The width of these curves line up with each other,indicating that the energy calibration and correspondingthresholds are matched, and that time resolution is prop-erly applied. In addition, the width is affected by thesource-to-detector distance, which in both the simulationand measurement was 35 cm.The higher counts in the measurement in the regionbetween -20 and -10 ns is partly due to PSD misclassifi-cation, where gamma-gamma correlations are mistakenlyclassified as gamma-neutron. There is also good agreementin the region beyond 60 ns, where the effects of scattering5 able 2: Specifications for the materials used in simulation of the BeRP ball with various reflectors. material isotopic concentration cross-section densitymass fraction (wt%) library name (g/cc)Bare (Pu) 93.27 Pu, 5.91
Pu endf70j 19.60.45
U, 0.25
PuNickel 67.20 Ni, 26.78 Ni, 3.84 Ni rmccs 8.9091.18 Ni, 1.01 NiIron 91.90 Fe, 5.65 Fe rmccs 7.8742.16 Fe, 0.29 FeAluminum 100 Al endf71x 2.7Tungsten 30.69
W, 28.79
W, 26.26
W rmccs 19.314.26
W, 0.12 W Neutron Light Output (MeVee) C o un t s SimulationMeasurement (a)
Neutron Light Output (MeVee) R e l a t i v e E rr o r ( % ) (b) Figure 5: Measurement and simulation comparison of the Cf-252source (a) pulse height distribution of gamma ray correlated neutronsand (b) corresponding relative error of the simulation. from the floor is evident. There are not many counts inthat region, which contributes to the erratic relative error,but the simulation and measurement match within the sta-tistical error. The region of the largest notable error liesroughly between 5 and 25 ns, right around the steep dropin counts expected from a non-multiplying source.Finally, there is the rate of “accidental” correlationsthat depend on the source strength and appear as a flatbackground in the TOFFEE distribution. The contribu-tion from accidentals is estimated by averaging counts ineach bin of a region offset by 1000 to 1500 ns from eachcoincidence trigger. This is then subtracted from the TOF-FEE distribution. In simulation this can be varied by ma-nipulating the effective equivalent “measurement” time.
The BeRP ball is a much more complicated source com-pared to Cf-252. It’s a distributed spherical source havinga diameter of 7.59 cm and a multiplication of 4.4, andtherefore cannot be treated as a non-multiplying source.For our simulation we evenly distributed spontaneous fis-sions of Pu-240 and ignored the more complicated mix ofisotopes that become ingrown over time.The bare configuration comparison, shown in Figure7, shows that the measured TOFFEE distribution is justslightly wider. The larger source of discrepancy is in theregion between 40 and 100 ns, which is dominated by re-flection from the floor. There are many time bins thatare within statistical agreement in that region, but also ahandful that have no counts at all. The problem is thatthe accidental background rate is much lower in the simu-lation (1 per ns) compared to the measurement (62 per ns)because of the lack of ingrown isotope sources in the for-mer. The higher accidental background competes with theeffect of room return and is statistically significant whenthe two are subtracted, which is apparent from the largeuncertainties.We found that the overall agreement between simu-lated and measured TOFFEE distributions improves asshielding material is added on. The BeRP ball with 1 inchiron is shown in Figure 8 as a representative example of6
TOFFEE (ns) -1 C o un t s SimulationMeasurement (a)
20 0 20 40 60 80 100
TOFFEE (ns) R e l a t i v e E rr o r ( % ) (b) Figure 6: Measurement and simulation comparison of the Cf-252source (a) TOFFEE distribution and (b) corresponding relative errorof the simulation.
20 0 20 40 60 80 100
TOFFEE (ns) -1 C o un t s SimulationMeasurement (a)
20 0 20 40 60 80 100
TOFFEE (ns) R e l a t i v e E rr o r ( % ) (b) Figure 7: Measurement and simulation comparison of the bare BeRPball (a) TOFFEE distribution and (b) corresponding relative errorof the simulation. the improvement. It appears that the time smearing asso-ciated with longer fission chains sweeps up some discrep-ancies caused by room return, and accidental background.
4. Exponential Fitting
In this work, TOFFEE distributions were character-ized by fitting the time range between zero and 100 nswith a double exponential function. We first motivate thedouble exponential distribution as a plausible physical re-sponse of reflected fissile material. As described in Section1.1, the spread in the TOFFEE distribution of a multiply-ing source is driven by the generation time, ∆ T g , betweenthe detected gamma rays and neutrons. The probabilityof detecting these particles is governed by the time depen-dent population of fissions, or the corresponding neutronsthat propagate fission chains. Point kinetics equations area well established method for studying the time-dependentneutron populations in a nuclear reactor. However, mod-eling neutron behavior inside reflected assemblies requireda two-region kinetic model [25, 26].7 TOFFEE (ns) C o un t s SimulationMeasurement (a)
20 0 20 40 60 80 100
TOFFEE (ns) R e l a t i v e E rr o r ( % ) (b) Figure 8: Measurement and simulation comparison of the BeRP ballwith 1 inch iron shielding (a) TOFFEE distribution and (b) corre-sponding relative error of the simulation.
In this work we deviate from reactor point kinetics byignoring delayed neutron precursors that result from thecascade of decays of fission product isotopes since the cor-relation times under consideration are on the order of onlya hundred nanoseconds [27]. These isotopes are typicallyorganized into six groups with half-lives ranging from hun-dreds of milliseconds to tens of seconds. In this work weignore the source of these delayed neutrons, since the TOF-FEE correlation window of interest is only on the order ofa hundred nanoseconds. Ignoring delayed precursors, thetime-dependent neutron population of prompt neutrons ina reflected assembly can be approximated by: dN c dt = k c − l c N c + f rc N r l r (8) dN r dt = f cr N c l c − N r l r (9)where: N c is the number of neutrons in the fissile core region N r is the number of neutrons in the reflector k c is the multiplication factor in the fissile core region l c is the neutron lifetime in the fissile core region l r is the neutron lifetime in the reflector region f cr is the fraction of neutrons that leak from thefissile core region into the reflector f rc is the fraction of neutrons that leak from thereflector back into the coreNote that the k c is different from the multiplication factor k in Eq. 4. The former is the property of only the core,while the latter is the property of the whole system (i.e.the core and reflector assembly).The system of equations in Eqs. 8 and 9 can be solvedby converting them to a second order differential equation: l r l c d N c dt + ( l c − l r ( k c − dN c dt − ( f + k c − N c = 0(10)The new variable f is the fraction of neutrons that leakout of the core and are reflected back, which is just theproduct of two previously defined terms f = f rc f cr (11)In order to fully solve this problem, we enforce twoinitial conditions: N c (0) = N o (12) N r (0) = 0 (13)at t = 0 the neutron population in the core is N o andno neutrons are present in the reflector. The solution toEq. 10, given these initial conditions is a familiar doubleexponential: N c ( t ) = N o (cid:2) (1 − R ) e tr + Re tr (cid:3) (14)where the roots to the characteristic polynomial are r = − (cid:112) l c l r ( f + k c −
1) + ( l c − l r ( k c − − l c + l r ( k c − l c l r (15) r = (cid:112) l c l r ( f + k c −
1) + ( l c − l r ( k c − − l c + l r ( k c − l c l r (16)and scaling ratio R is R = r − αr − r (17)where α = k c − l c (18)(19) f and k c are constrained to be less than 1. We have foundthat R falls within the range 0 to 1 for all plausible com-binations of these variables.8 .2. Single-region Point Kinetics The parameter α from Eq. 18 represents the rate ofloss of neutrons in a bare system (i.e. f = 0) in which casethe time dependent neutron population is simply N c ( t ) = N o e αt . (20)This solution to the neutron population behaviour in anunreflected assembly is a starting point to Rossi-alphaanalysis [28]. We will use it to fit the TOFFEE distri-butions of BeRP balls with various fissile material masses.
5. Results and Discussion
Neutron multiplication and reflector thickness are cor-related, since neutron reflection increases the neutron pop-ulation and average length of fission chains. In order tostudy the effect of multiplication independently of the ef-fects of reflector,we simulated bare BeRP balls with var-ious masses, ranging from 1 to 8 kg. Because of the lackof reflection, we fit the resultant TOFFEE distributionsto Eq. 20. The fits are made to the positive side of theTOFFEE distribution ranging from 0 to 100 ns.A comparison of the measured and simulated BeRPball is shown in Figure 9. As explained in Section 3.2,there is some disagreement at later times due to compet-ing effects of floor reflection and accidental correlations.However, the fits and resulting α parameters for measure-ment (0 . ± . . ± . C o un t s Measurement FitSimulation Fit
Figure 9: Comparison of the measured and simulated bare BeRPball TOFFEE distributions and exponential fits from Eq. 20.
The multiplication factors and neutron lifetimes for thebare cases were tallied in MCNP6 simulations, and Eq. 18was used to calculate corresponding α parameters. Figure10 shows the comparison of these MCNP derived alphavalues with the alpha values derived from the exponentialfits. The relationship between estimated and MCNP al-phas is linear with a correlation coefficient greater than0.98, and the slope of 1.0974. E s t i m a t e d A l p h a α F = 1 . α M − . SimulationMeasurement
Figure 10: The fitted ( α F ) and calculated, from MCNP6, ( α M )alpha parameters for BeRP balls with mass ranging from 1 to 8 kg.A linear regression was performed with the resulting relationshipshown in the legend and a correlation coefficient of 0.9890. As shown in Figure 10, there is a slight deviation fromthe linear trend for the actual BeRP ball simulation andmeasurement. For all other masses we removed the thinstainless steal shell and simulated a truly bare Pu sphere.Neutron multiplication can be derived from neutrondecay constant and core lifetime by rearranging Eq 18: M = − αl c . (21)We derived neutron multiplications from fitted alpha pa-rameters by using previously tallied core neutron lifetimesfrom MCNP6. As expected there was a positive linearcorrelation between the derived and actual neutron multi-plications, with the derived values that underestimate theneutron multiplication obtained from MCNP6 k-code cal-culations. We found that our derived multiplications bet-ter aligned with leakage multiplication, as shown in Figure11, with an average relative error deviation of 10.6%.Neutron leakage is a product of the neutron multiplica-tion and probability of neutron escape from the core. Thebetter agreement makes some qualitative sense becauseneutrons that leak are the only ones that are available fordetection. Furthermore, due to self-shielding the gammarays available for detection are predominately drawn fromthe surface of the BeRP ball. As a result we dispropor-tionately detect from correlations of particles originatingfrom fission chains near the surface, which due to neutronleakage will have a shorter length than the average. Next we moved to fitting Eq. 14, derived from two-region point kinetics model in Section 4.1, to the reflectedBeRP cases. We first used the alpha parameter determinedfrom the fit of the bare BeRP ball measurement. Nextwe used the value k c derived from MCNP6 simulations tosolve for l c using Eq. 18. This fully describes the behavior9 D e r i v e d M u l t i p li c a t i o n ( ( α l c ) − ) SimulationMeasurement
Figure 11: Derived neutron multiplications from TOFFEE fits ofthe bare BeRP balls with different masses with the correspondingleakage multiplications obtained through MCNP6 simulations. Thedashed line corresponds to perfect agreement between derived andactual leakage multiplication, with the points above and below cor-responding to overestimation and underestimation, respectively of the fissile core (BeRP ball). We next fit the remainingtwo free parameters, l r and f , to the TOFFEE distributionof the reflected configurations.The double exponential fit to the measured iron casesis shown in Figure 12. In the first 60 ns time window thefit tracks quite well with the data, but undershoots thedata at later times in the 80-100 ns window. Some of thatis due to the lower statistic in that region which make itless important for the fit. There is also some effect of floorreflection that is not accounted for in the two-region pointkinetics model and therefore missing from Eq. 14. -1 N o r m a li z e d C o un t s Fe 0.5 inFe 1.0 inFe 1.5 in
Figure 12: TOFFEE distributions of the measured iron configura-tions with corresponding double exponential fits from Eq. 14.
The parameter f is related to the total system k by k = k c (1 − f ) . (22)Neutron multiplication is then calculated from Eq. 4. The comparison of this “Estimated Multiplication” with theMCNP6 equivalent for the measured and simulated casesis shown in Figure 13. As expected, there is convergencebetween the simulated and measured cases with increasingshell thickness. The average relative difference betweenestimated and expected multiplication was 10%. E s t i m a t e d M u l t i p li c a t i o n IronNickelSimulationMeasurement
Figure 13: Comparison of the estimated multiplication of the mea-sured and simulated TOFFEE distribution for the shielded configu-ration of the BeRP ball. The dashed line represent perfect agreementbetween the fit and the expectation from MCNP simulation.
As before, we expanded on the measured cases with ad-ditional simulations, and the results are shown in Figure14. The estimated multiplications for all shielding materi-als have positive correlations with the MCNP multiplica-tion, although the relationship is different between mate-rials. The trend is superlinear for aluminum and sublinearfor tungsten. Iron and nickel have a more linear trend.The average relative difference also varied from one mate-rial type to the next, with as little at 14% for aluminumand as much at 22% for iron. Unlike with the bare case cor-relation with leakage multiplication produced even worseagreement and the trends among the different materialsremained the same.The discrepancies between materials are due to the as-sumptions in the two-region kinetic model. The modelonly allows for a neutron to either fission, or leak out ofa region. There are no considerations for inelastic inter-action, such as parasitic neutron capture, which differsamong different materials. Neutron energy is also col-lapsed into one group, which works if neutron energy is notchanging much. However, a neutron will on average losemore energy scattering in a low-Z aluminum, compared toa high-Z tungsten, due to scattering kinematics. As a re-sult, a neutron reflected back into the fissile core from analuminum reflector will have relatively greater probabilityof fission, due to the energy dependence of induced fissioncross-section in Pu-239. This would in effect increase totalmultiplication, which may explain the underestimation oftrue multiplication for the aluminum reflected BeRP balls.10
10 20 30 40 50MCNP Multiplication01020304050 E s t i m a t e d M u l t i p li c a t i o n TungstenNickelIronAluminum
Figure 14: Estimated multiplication for simulated TOFFEE distri-butions of several configurations of shielded BeRP ball with differentmaterial types.
There are two derived quantities from the fitting pro-cedure outlined in the previous Section 5.2 which relateto shell thickness and material type. First, the scaling ra-tio asymptotically approaches unity with increasing shellthickness, as shown in Figure 15. As the amount of reflec-tor material goes up, so does its effect on the TOFFEEdistribution. Eventually this dominant reflector term col-lapses the double exponential fit into a single exponential.This suggests that it may be difficult to separate the ef-fects fissile material mass and presence coupled reflectorat sufficiently high amounts of said reflector. S c a li n g R a t i o ( R ) TungstenNickelIronAluminum
Figure 15: The scaling ratio from the fit of Eq. 14 to TOFFEEdistributions of the BeRP ball with various reflector shell thicknesses.
We quantified the amount of reflector material by cal-culating the effective areal density, which takes into ac-count the hole in the shells. The integral of the aforemen-tioned fits to TOFFEE distribution, defined as (cid:90) ∞ N c ( t ) dt = R − r − Rr (23) provided best linear correlation with the effective arealdensity. The results of linear least-squares regression foreach material and all of them combined is shown in Table3. Each material has a unique linear correlation, but com-bined the regression shows a strong correlation coefficientof 0.9717. The deviation between materials is likely dueto inelastic and other capture neutron interactions withineach reflector, which we are not correcting for. ( g/cm ) I n t e g r a l ( [ R − ] / r − R / r ) TungstenNickelIronAluminum
Figure 16: The integral of the fit of Eq. 14 to TOFFEE distributionsof the reflected configurations of the BeRP ball and the effective arealdensity of each of the shells.Table 3: Linear least-squared regression for the correlation betweenintegral of the fit to TOFFEE distribution and effective areal densityof the reflector material. reflector slope intercept correlationcoefficientAluminum 0.064 7.06 0.9748Iron 0.175 8.46 0.9889Nickel 0.274 9.10 0.9895Tungsten 0.237 8.91 0.9815All 0.251 6.55 0.971711 . Conclusions
We introduced the TOFFEE distribution and its re-lationship to fission chain dynamics. We then fitted thepositive side of this distribution, from 0 to 100 ns, to timedependent neutron population derived from point kinet-ics theory. A bare subcritical assembly is sufficiently de-scribed by a single exponential in Eq. 20, and introductionof a reflector yields a double exponential shown in Eq 14.We found that for a subset of bare BeRP ball masses,between 3 and 8 kg, the estimated alpha parameters andthe expected alpha values are linearly correlated. A de-rived multiplication was calculated from the estimated al-pha parameters by assuming a known l c from MCNP6 sim-ulations. This derived multiplication positively correlatedwith the leakage multiplication, with an average relativeerror of 10.6%.The TOFFEE distributions from the reflected BeRPball assemblies were fitted to the double exponential modelfrom Eq. 14. The derived multiplication from f had apositive correlation with the expected multiplication forMCNP6, although the relationship varied between mate-rial types. Furthermore, we determined that the effectiveareal density of the reflectors was positively and linearlycorrelated with the integral of those same double expo-nential fits. It is conceivable that with knowledge of eithershell thickness or material composition it would be possi-ble to determine the other property. Acknowledgments
This material is based upon work supported by the U.S.Department of Homeland Security under Grant AwardNumber, 2012-DN-130-NF0001. The views and conclu-sions contained in this document are those of the authorsand should not be interpreted as representing the officialpolicies, either expressed or implied, of the U.S. Depart-ment of Homeland Security.Sandia National Laboratories is a multimission labora-tory managed and operated by National Technology andEngineering Solutions of Sandia, LLC., a wholly ownedsubsidiary of Honeywell International, Inc., for the U.S.Department of Energys National Nuclear Security Admin-istration under contract DE-NA-0003525.This work was funded in-part by the Consortium forVerification Technology under Department of Energy Na-tional Nuclear Security Administration award number DE-NA0002534. 12 eferencesReferences [1] N. Ensslin, Principles of Neutron Coincidence Counting, LosAlamos National Laboratory, LA-13422-M (November 1998).[2] W. Hage, D. M. Cifarelli, On the factorial moments of the neu-tron multiplicity distribution of fission cascades, Nuclear Instru-ments and Methods in Physics Research A 236 (1985) 165–177. doi:10.1016/0168-9002(85)90142-1 .[3] K. Bohnel, The effects of multiplication on the quantatitativedetermination of spontaneously fissioning isotopes by neutroncorrelation analysis, Nuclear Science and Engineering 90 (1)(1985) 75 – 82. doi:http://dx.doi.org/10.13182/NSE85-2 .[4] D. Cifarelli, W. Hage, Models for a three-parameter analy-sis of neutron signal correlation measurements for fissile ma-terial assay, Nuclear Instruments and Methods in Physics Re-search Section A: Accelerators, Spectrometers, Detectors andAssociated Equipment 251 (3) (1986) 550 – 563. doi:http://dx.doi.org/10.1016/0168-9002(86)90651-0 .[5] L. F. Nakea, G. F. Chapline, A. M. Glenn, P. L. Kerr, K. S.Kim, S. A. Ouedrago, M. K. Prasad, S. A. Sheets, N. J. Sny-derman, J. M. Verbeke, R. E. Wurtz, The use of fast neutrondetection for material accountability, International Journal ofModern Physics: Conference Series 27.[6] D. L. Chichester, S. J. Thompson, M. T. Kinlaw, J. T. John-son, J. L. Dolan, M. Flaska, S. A. Pozzi, Statistical estima-tion of the performance of a fast-neutron multiplicity system fornuclear material accountancy, Nuclear Instruments and Meth-ods in Physics Research Section A: Accelerators, Spectrome-ters, Detectors and Associated Equipment 784 (2015) 448 – 454. doi:http://dx.doi.org/10.1016/j.nima.2014.09.027 .[7] A. Enqvist, M. Flaska, J. L. Dolan, D. L. Chichester, S. A.Pozzi, A combined neutron and gamma-ray multiplicity counterbased on liquid scintillation detectors, Nuclear Instruments andMethods in Physics Research Section A: Accelerators, Spec-trometers, Detectors and Associated Equipment 652 (1) (2011)48 – 51, symposium on Radiation Measurements and Applica-tions (SORMA) { XII } doi:http://dx.doi.org/10.1016/j.nima.2010.10.071 .[8] G. Miloshevsky, A. Hassanein, Multiplicity correlation betweenneutrons and gamma-rays emitted from { SNM } and non-snmsources, Nuclear Instruments and Methods in Physics ResearchSection B: Beam Interactions with Materials and Atoms 342(2015) 277 – 285. doi:http://dx.doi.org/10.1016/j.nimb.2014.10.017 .[9] S. Li, S. Qiu, Q. Zhang, Y. Huo, H. Lin, Fast-neutron multi-plicity analysis based on liquid scintillation, Applied Radiationand Isotopes 110 (2016) 53 – 58. doi:http://dx.doi.org/10.1016/j.apradiso.2015.12.064 .[10] G. F. Knoll, Radiation Detection and Measurement, 3rd Edi-tion, John Wiley & Sons, Inc., 2000.[11] J. Adams, G. White, A versatile pulse shape discriminator forcharged particle separation and its application to fast neutrontime-of-flight spectroscopy, Nuclear Instruments and Methods156 (3) (1978) 459 – 476. doi:10.1016/0029-554X(78)90746-2 .[12] J. Mattingly, Polyethylene-reflected plutonium metal sphere:Subcritical neutron and gamma measurements, Tech. Rep.SAND2009-5804, Sandia National Laboratories (2009).[13] E. Miller, S. Clarke, A. Enqvist, S. A. Pozzi, P. Marleau, J. K.Mattingly, Characterization of special nuclear material using atime-correlated pulse-height analysis, Journal of Nuclear Mate-rials Managment XLI (1) (2012) 32–37.[14] E. Miller, J. Dolan, S. Clarke, S. Pozzi, A. Tomanin, P. Peerani,P. Marleau, J. Mattingly, Time-correlated pulse-height mea-surements of low-multiplying nuclear materials, Nuclear Instru-ments and Methods in Physics Research Section A: Accelera-tors, Spectrometers, Detectors and Associated Equipment 729(2013) 108 – 116. doi:10.1016/j.nima.2013.06.062 .[15] M. Monterial, M. Paff, S. Clarke, E. Miller, S. A. Pozzi, P. Mar-leau, A. N. S. Kiff, J. K. Mattingly, Time-correlated-pulse-height technique measurements of fissile samples at the device assembly facility., in: Proceesdings of the Institute of NuclearMaterials Management 54 th Annual Meeting, Palm Desert, Cal-ifornia, 2013.[16] P. Marleau, A. Nowack, M. Paff, M. Monterial, S. Clarke,S. Pozzi, Gamma/neutron time-correlation for special nuclearmaterial detection active stimulation of highly enriched ura-nium, Tech. Rep. SAND2013-7442, Sandia National Laborato-ries (2013).[17] M. G. Paff, M. Monterial, P. Marleau, S. Kiff, A. Nowack, S. D.Clarke, S. A. Pozzi, Gamma/neutron time-correlation for spe-cial nuclear material detection active stimulation of highly en-riched uranium, Annals of Nuclear Energy 72 (2014) 358 – 366. doi:doi.org/10.1016/j.anucene.2014.06.004 .[18] M. Monterial, P. Marleau, M. Paff, S. Clarke, S. Pozzi, Multipli-cation and presence of shielding material from time-correlatedpulse-height measurements of subcritical plutonium assemblies,Nuclear Instruments and Methods in Physics Research Sec-tion A: Accelerators, Spectrometers, Detectors and AssociatedEquipment 851 (2017) 50 – 56. doi:https://doi.org/10.1016/j.nima.2017.01.040 .[19] R. Serber, The definitions of neutron multiplication, Tech. Rep.LA-335, Los Alamos National Laboratory (1945).[20] S. A. Pozzi, E. Padovani, M. Marseguerra, MCNP-PoliMi: aMonte-Carlo code for correlation measurements, Nuclear In-struments and Methods in Physics Research Section A: Accel-erators, Spectrometers, Detectors and Associated Equipment513 (3) (2003) 550 – 558. doi:doi.org/10.1016/j.nima.2003.06.012 .[21] H. Schlermann, H. Klein, Optimizing the energy resolution ofscintillation counters at high energies, Nuclear Instruments andMethods 169 (1) (1980) 25 – 31. doi:http://dx.doi.org/10.1016/0029-554X(80)90097-X .URL [22] M. Bourne, S. Clarke, M. Paff, A. DiFulvio, M. Norsworthy,S. Pozzi, Digital pile-up rejection for plutonium experimentswith solution-grown stilbene, Nuclear Instruments and Meth-ods in Physics Research Section A: Accelerators, Spectrome-ters, Detectors and Associated Equipment 842 (2017) 1 – 6. doi:https://doi.org/10.1016/j.nima.2016.10.023 .[23] M. A. Norsworthy, A. Poitrasson-Rivire, M. L. Ruch, S. D.Clarke, S. A. Pozzi, Evaluation of neutron light output responsefunctions in ej-309 organic scintillators, Nuclear Instrumentsand Methods in Physics Research Section A: Accelerators, Spec-trometers, Detectors and Associated Equipment 842 (2017) 20– 27. doi:https://doi.org/10.1016/j.nima.2016.10.035 .[24] M. Monterial, P. Marleau, S. Clarke, S. Pozzi, Application ofbayes theorem for pulse shape discrimination, Nuclear Instru-ments and Methods in Physics Research Section A: Accelera-tors, Spectrometers, Detectors and Associated Equipment 795(2015) 318 – 324. doi:http://dx.doi.org/10.1016/j.nima.2015.06.014 .[25] C. E. Cohn, Reflected-reactor kinetics, Nuclear Science and En-gineering 13 (1) (1962) 12 – 17.[26] G. D. Spriggs, R. D. Busch, J. G. Williams, Two-region ki-netic model for reflected reactors, Annals of Nuclear Energy24 (3) (1997) 205 – 250. doi:http://dx.doi.org/10.1016/0306-4549(96)00062-X .[27] J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis,John Wiley and Sons, 1976, Ch. 1 Introductory Concepts ofNuclear Power Reactor Analysis, pp. 62–63.[28] J. D. Orndoff, Prompt neutron periods of metal critical assem-blies, Nuclear Science and Engineering 2 (1957) 450 – 460..[27] J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis,John Wiley and Sons, 1976, Ch. 1 Introductory Concepts ofNuclear Power Reactor Analysis, pp. 62–63.[28] J. D. Orndoff, Prompt neutron periods of metal critical assem-blies, Nuclear Science and Engineering 2 (1957) 450 – 460.