Computation of Neutron Multiplicity Statistics Using Deterministic Transport

314 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 59, NO. 2, APRIL 2012

Computation of Neutron Multiplicity Statistics UsingDeterministic Transport

John Mattingly, Member, IEEE

Abstract—Nuclear nonproliferation efforts are supported bymeasurements that are capable of rapidly characterizing specialnuclear materials (SNM). Neutron multiplicity counting is fre-quently used to estimate properties of SNM, including neutronsource strength, multiplication, and generation time. Differentclasses of models have been used to estimate these and otherproperties from the measured neutron counting distribution andits statistics. This paper describes a technique to compute statisticsof the neutron counting distribution using deterministic neutrontransport models. This approach can be applied to rapidly analyzeneutron multiplicity counting measurements without relying onthe point reactor kinetics model.

Index Terms—Coincidence techniques, neutron detectors, in-verse problems, simulation.

I. INTRODUCTION

N UCLEAR material accountability and safeguards mea-surements for nuclear nonproliferation frequently employ

neutron multiplicity counting to estimate integral properties ofSNM, including:

• Neutron source strength: the rate of spontaneous neutronemission via processes like spontaneous fission and ( , n)reactions.

• Neutron multiplication: the mean number of neutrons pro-duced by induced fission chain-reactions per source neu-tron.

• Neutron generation time: the mean time between succes-sive generations in fission chain-reactions.

These properties are generally referred to as kinetics parametersof the fissile system.

Neutron multiplicity counting measures the frequency of neu-tron detection versus:

• The number of coincident neutrons detected (a.k.a., the“multiplicity” of the detection event).

• The duration of the counting time (a.k.a., the “coincidencegate” width).

Consequently, neutron multiplicity counting measures the dis-tribution of neutron detection events over the multiplicity of theevent and the width of the coincidence gate, as Fig. 1 illustrates[1].

The shape of the counting distribution over the multiplicityvariable is dictated in part by neutron multiplication. If the ma-terial is non-fissile and the source produces at most one neutron

Manuscript received August 24, 2011; revised December 22, 2011; acceptedJanuary 13, 2012. Date of publication February 16, 2012; date of current versionApril 13, 2012.

The author is with North Carolina State University, Raleigh, NC 27695 USA(e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TNS.2012.2185060

Fig. 1. The neutron count distribution measured from a 4.4-kg sphere ofweapons-grade plutonium metal, reflected by a 76.2-mm-thick spherical shellof polyethylene. The horizontal axis labeled “multiplicity” denotes the numberof coincident neutron counts. The horizontal axis labeled “coincidence gate(us)” denotes the width of the coincidence counting gate in �s. The vertical axisis the base-10 logarithm of the number of events with the given multiplicity inthe given coincidence gate.

in any given emission (as ( , n) sources do), the counting dis-tribution will be a Poisson distribution, and the variance will beidentical to the mean. In contrast, if a source contains fissile ma-terial, the multiplication of neutrons by fission chain-reactionswill cause the distribution to be wider than the correspondingPoisson distribution with the same mean. This behavior is illus-trated in Fig. 2.

In addition, the shape of the distribution over the coincidencegate width variable is dictated in part by the neutron generationtime. In systems where the mean neutron energy (and conse-quently the mean speed) is high, the distribution will tend to ap-proach its asymptotic shape rapidly. Such “fast” systems tendto be composed largely of fissile metals. In contrast, in systemswhere the mean neutron energy is relatively lower, the distribu-tion approaches its asymptotic shape less rapidly. Fissile liquidsand fissile metals reflected by hydrogenous materials tend to ex-hibit this “slow” behavior.

There is no single distinct criterion for classifying a fissilesystem as “fast” or “slow,” but fast systems typically exhibitneutron generation times in the range from tens of nanosecondsto a few microseconds, while slow systems exhibit a neutrongeneration times of tens to hundreds of microseconds.

Models of the neutron counting distribution or its statisticscan be applied to estimate the kinetics parameters from a mea-

0018-9499/$31.00 © 2012 IEEE

MATTINGLY: COMPUTATION OF NEUTRON MULTIPLICITY STATISTICS 315

Fig. 2. The measured neutron count distribution shown in Fig. 1 for a fixedcoincidence gate width of 1024 �s. The red circles show the measured distri-bution. The black line is a Poisson distribution with the same mean. The broad-ening evident in the measured distribution is the result of neutron multiplicationby induced fission chain-reactions.

sured counting distribution. This paper describes an approach tocompute neutron multiplicity counting statistics using determin-istic transport models, and it demonstrates an implementation ofthat approach.

II. MODELING METHODS

Estimating the kinetics parameters requires a model of thecounting distribution or its statistics (e.g., its mean and vari-ance). Several classes of models can be applied to analyze thecounting distribution for the kinetics parameters.

Generally speaking, each class of model embodies somecompromise between accuracy and speed of execution. Pointmodels can be solved essentially instantaneously, but they in-volve the greatest number of assumptions and approximations.Monte Carlo models make few assumptions or approximations,but they typically require substantial computational time. Deter-ministic models employ fewer, less-gross approximations thanpoint models, and they tend to require much less computationaltime than Monte Carlo models.

Nuclear material accountability, safeguards, and other nu-clear security applications of neutron multiplicity countinggenerally require rapid estimation of kinetics parametersusing modest computational resources. The measurementsare typically conducted under field conditions using portableinstruments and computers, and in many cases a large numberof samples must be analyzed in a short time.

Consequently, almost all current applications of neutron mul-tiplicity counting are analyzed using point models [2]. The com-putational time and resources required for Monte Carlo modelsare too substantial for many field applications of neutron multi-plicity counting. However, deterministic models may provide areasonable compromise between speed and accuracy even usingmodest computational resources.

A. Point Models

Historically, the point reactor kinetics model has been em-ployed to analyze neutron multiplicity counting measurements[2]–[7]. The point kinetics model represents only the temporal

behavior of the neutron population. It neglects the neutron pop-ulation’s energy and directional dependence, and it treats theneutron population spatial and temporal distribution as sepa-rable. These approximations affect the interpretation of kineticsparameters derived from measurements, particularly when thefissile system violates the assumptions underlying the point re-actor kinetics model. Point models of neutron counting mo-ments (i.e., the mean, variance, etc.) can generally be expressedin closed form and can usually be solved algebraically. Conse-quently, analysis of neutron counting moments using the pointmodel is essentially instantaneous.

Point models of the counting distribution itself (instead of itsmoments) have also been applied to estimate the kinetics param-eters [8]. These models cannot be expressed in closed form, andthey cannot be algebraically solved for the kinetics parameters.However, they can be used to recursively generate an estimateof the counting distribution given the kinetics parameters, andthe computational time required to synthesize the distribution ismodest (usually less than a minute). In theory, these recursivemodels could be used in a nonlinear regression solver to esti-mate the kinetics parameters. However, in practice the modelsare typically used to generate lookup tables that can be searchedfor the set of kinetics parameters that yield the best estimate ofthe measured counting distribution [9].

The simplifications in the detailed transport model that areemployed to derive the point reactor model impose limitationson the model’s applicability. In particular, the assumption thatthe neutron population’s spatial distribution is separable from itstemporal distribution following a perturbation is often invalid,particularly in systems where the neutron population’s spectralshape varies substantially with position, e.g., in reflected metalsystems. In such systems, the temporal evolution of the neutronpopulation’s spatial shape can vary significantly within differentregions of the system, rendering the point model invalid.

B. Monte Carlo Models

Monte Carlo transport modeling methods have been appliedto computationally synthesize the neutron counting distribution.Monte Carlo radiation transport codes have been developed thatsimulate not only the average of the neutron population, butalso the fluctuations in the population that result from the multi-plicity of neutrons emitted during spontaneous and induced fis-sion [10]–[12].

Such implementations of Monte Carlo transport models at-tempt to accurately simulate the distribution of the neutrons overposition, energy, direction, time, and multiplicity. Therefore,they are in theory capable of producing extremely accurate es-timates of the neutron counting distribution.

However, the stochastic nature of the Monte Carlo approachgenerally requires a substantial number of neutron historiesto be simulated to estimate the neutron counting distributionwith acceptable certainty—this is particularly true in situationswhere the neutron leakage probability, detector solid angle, ordetector efficiency is small. Furthermore, variance reductiontechniques commonly used to accelerate the calculation ofquantities based on an average response to the neutron fieldcannot be applied to estimate the distribution of a response overneutron multiplicity. Variance reduction techniques by defini-tion alter the shape of the calculated response’s distribution.


Consequently, Monte Carlo synthesis of the neutron countingdistribution can require many minutes to hours or even days ofcomputational time, even when the simulation runs many his-tories in parallel. Therefore, Monte Carlo models are currentlyimpractical for use in an approach that attempts to iteratively fita model to the measured neutron counting distribution.

C. Deterministic Models

Deterministic transport methods estimate the neutron popu-lation versus position, energy, direction, and time by explicitlysolving the Boltzmann transport equation discretized over thoseindependent variables.

For a deterministic solution, computational accuracy andspeed generally run counter to one another. The solution can beaccelerated by employing a coarser discretization of the inde-pendent variables, but that approach can degrade the accuracyof the solution. The accuracy of the solution can generally beimproved by using a more granular discretization of the inde-pendent variables, but that accuracy is usually obtained at thecost of computational speed. However, numerous techniqueshave been developed to improve the accuracy of approxima-tions for the spatial, spectral, and directional dependence of thetransport medium properties and the neutron population andaccelerate convergence of the solution. Consequently, relativeto Monte Carlo transport methods, deterministic transportmethods are generally faster, particularly for problems wherethe detection probability is low due to low neutron leakageprobability, detector solid angle, or detector efficiency.

Muñoz-Cobo, et al., demonstrated that the linear Boltzmanntransport equation can be derived as the first moment (i.e., themean) of a more general equation that models the moment gen-erating function for the multiplicity distribution of the neutronpopulation [13]. Furthermore, they demonstrated that any arbi-trary order moment of the neutron population’s multiplicity dis-tribution can be obtained as a solution of the linear Boltzmanntransport equation by merely changing the source term.

Consequently, it is possible to estimate the moments of thecounting distribution by explicit solution of the Boltzmanntransport equation. Subsequently, this paper describes animplementation of Muñoz-Cobo’s technique to compute themean and variance of the neutron counting distribution usingdeterministic transport calculations. Experiments conductedto validate the implementation are presented at the end of thepaper.

III. IMPLEMENTATION

The implementation described in this paper was developed tocompute the Feynman-Y counting statistic, defined by the ratioof the count distribution’s variance to its mean :

(1)

Observe that vanishes if the count distribution is Poisson.Consequently, the Feynman-Y represents the “excess variance”in the count distribution relative to the variance characteristic ofa Poisson distribution with the same mean.

Fig. 3 shows the Feynman-Y for the measured counting dis-tribution shown in Figs. 1 and 2. Note that the Feynman-Y ex-hibits two notional features:

Fig. 3. The Feynman-Y of the measured neutron count distribution shown inFig. 1. The Feynman-Y was accumulated using Eq. (1) for varying coincidencegate widths. The asymptotic value tends to increase with the square of neutronmultiplication, and the rise-time is dictated by the neutron generation time. Theerror bars represent one standard deviation.

1) The asymptotic value, i.e., the value of for infinitelywide coincidence gates. The asymptote is reached as thecoincidence gate width exceeds the duration of the longestfission chain-reactions. The asymptotic value tends to in-crease with the square of multiplication [2].

2) A monotonic rise from a value of zero (for a zero-widthcoincidence gate) to the asymptotic value. For very narrowcoincidence gates, the distribution is accumulated fromincomplete chain-reactions, so the variance is relativelysmaller than it is for very wide coincidence gates. Therise-time is dictated by the neutron generation time [2].

In this implementation, the Feynman-Y is synthesized in twocomputationally distinct steps. In the first step, the asymptoticvalue of the is estimated from a pair of static (time-indepen-dent) forward and adjoint neutron transport calculations. In thesecond step, the shape of the is synthesized from a dynamic(time-dependent) calculation of the system’s step response.

A. Calculation of the Feynman-Y Asymptote

The mean value of the count distribution is calculated fromthe inner product

(2)

where is the neutron flux obtained by solving the usualfixed source formulation of the time-independent forwardneutron transport equation, given in the Appendix as (A.3). Thesource term in (A.3) is just the natural source term arisingfrom spontaneous fission and ( , n) reactions in the transportmedium. is the detector cross section. Note that as expressedabove, is the mean detector count rate.

In this implementation, the transport medium is modeledusing a single spatial dimension, such that the neutron leakageis the same at every point on the transport medium’s externalboundary. The detector is treated as a point external to the trans-port medium with an energy-dependent efficiency , whichrepresents the probability that a neutron escaping the transportmedium will be counted. Note that the detector efficiency in-cludes both the detector solid angle (dictated by the detector’s


size and distance from the source) and the intrinsic efficiencyversus energy, i.e., the probability that a neutron entering thedetector will be counted.

Consequently, in this implementation, the mean is calculatedas the inner product of the efficiency and the neutron current

for neutrons exiting the external boundary of the transportmedium:

(3)

where and respectively denote the external boundary ofthe transport medium and the surface normal at the externalboundary. In other words, because , the detector crosssection is treated as

(4)

The detector efficiency can be estimated a priori eitherby Monte Carlo modeling or by calibration measurementsperformed with known neutron sources [14].

For a detector that operates on the principal of neutron cap-ture, Muñoz-Cobo showed that the variance of the count dis-tribution is

(5)

where is excess variance contributed by the source mul-tiplicity distribution, and is excess variance contributed byinduced fission chain-reactions.

Excess variance due to the source is calculated from innerproduct

(6)

where is the forward fixed source term, and is the innerproduct:

(7)

Calculation of the adjoint flux is discussed below.Above, denotes the source neutron spectrum, and and

respectively represent the first and second momentsof the source neutron multiplicity distribution. Note that if thesource emits at most one neutron at a time, e.g., if it is an ( ,n) source, then the second moment vanishes and thesource contributes no excess variance.

Excess variance due to induced fission is calculated frominner product

(8)

where is the fission neutron production cross section, andis the inner product:

(9)

The adjoint flux is the same as in (7); its calculation is dis-cussed below.

Above, denotes the induced fission neutron spectrum, andand respectively represent the first and second mo-

ments of the induced fission neutron multiplicity distribution.In (7) and (9), the adjoint flux is calculated by solving

the fixed source form of the time-independent adjoint neutrontransport equation, which is given in the Appendix as (A.12). In(A.12), the adjoint source term is the detector cross section

. Consequently, the quantities and defined in (7) and(9) respectively represent the importance of source and inducedfission neutrons to detection.

Given this implementation’s treatment of the detector as apoint external to a one-dimensional transport medium, as shownin (4), the adjoint source term is simply a surface source atthe external boundary of the transport medium with a spectrumequal to the detector efficiency .

Note that the approach described in this paper can be imple-mented without the specialization to a one-dimensional trans-port medium measured by a point detector. A more general im-plementation would have to employ the general form of the de-tector cross section instead of the specialized definition pro-vided in (4). Whatever model is chosen for the detector crosssection must be used as the adjoint source term. For example, Yiet al. have implemented calculations of the Feynman-Y asymp-tote (but not its shape) using three-dimensional discrete ordi-nates [15].

Subsequently, the asymptotic value of the Feynman-Y is cal-culated from the mean, source excess variance, and fission ex-cess variance:

(10)

The preceding approach for calculating the asymptote is deriveddirectly from the method originally prescribed by Muñoz-Cobo.

B. Calculation of the Feynman-Y Shape

Muñoz-Cobo’s original work described a technique forcalculating the Feynman-Y versus coincidence gate widththat required the time-dependent adjoint transport equation tobe solved in response to a transient adjoint source term. Theadjoint source term’s energy dependence was dictated by thedetector cross section, and its time-dependence was dictated bythe detector’s impulse response. However, during developmentof this implementation, it was observed that the deterministictransport solver PARTISN exhibited slow convergence for thiskind of adjoint transport problem [16].

Consequently, an alternative equivalent approach was devel-oped based upon solution of the time-dependent forward trans-port equation, which converged more rapidly. The shape of theFeynman-Y is calculated from the convolution

(11)

where is the coincidence gate width, is the detector’s im-pulse response, and is the time-dependent neutron flux. Note


that the inner integral over is merely the convolution of thedetector response function with the neutron flux , whichpredicts the detector’s output at time . The outer integral over

accumulates the detector’s output over the coincidence gatewidth .

The time-dependent flux is calculated by solving the fixedsource form of time-dependent forward neutron transport equa-tion, given in the Appendix as (A.1). Specifically, the flux iscalculated as the transport medium’s response to a step changein the fixed source term, i.e.,

(12)

where is the steady-state value of the source intensity andis the Heaviside step function

(13)

Note that the implementation of (11) does not require a sepa-rate solution of the time-dependent transport equation for eachvalue of the coincidence gate width . When solving time-dependent transport problems, PARTISN records the solutionat time-steps intermediate to the maximum time specified bythe user. Consequently, the Feynman-Y shape can be computedfrom a single PARTISN calculation, where the maximum timecorresponds to the maximum coincidence gate width. For mostsystems, even extremely slow ones, a maximum time of severalmilliseconds is sufficient for the flux to reach its steady statevalue in response to the step change in the fixed source.

For many multiplicity counting systems, which are often con-structed using moderated proportional counters, the de-tector impulse response can be approximated as:

(14)

where the time-constant is primarily dictated by the neutronslowing-down time in the moderator. The detector time-constantcan be measured relatively easily by accumulating the correla-tion function , which is just distribution of coincident countpairs as a function of the time delay between each count inthe pair. If it is measured using a spontaneous fission source, orpreferably an ( , n) source, the correlation function will gener-ally have the shape

(15)

where and are constants. The value of the time-constantcan be estimated by fitting the correlation function for .

IV. EXPERIMENTAL VALIDATION

In January 2009, experiments with plutonium metal reflectedby polyethylene were conducted at Nevada Test Site to vali-date the accuracy of the preceding computational technique. Thesource was a 4.4 kg sphere of weapons-grade plutonium metal.It was measured bare and reflected by spherical polyethyleneshells between 12.7 and 152.4 mm thick. A neutron multiplicitycounter constructed from proportional counters embeddedin a polyethylene moderator was used to accumulate the neutronmultiplicity distribution.

Fig. 4 shows the plutonium source and polyethylene reflec-tors. The plutonium metal is alpha-phase, with a density of 19.6

Fig. 4. The plutonium source and polyethylene reflectors. The source is 4.4 kgof weapons-grade, alpha-phase plutonium metal in a thin stainless steel shell.The reflectors are a series of nesting high density polyethylene. The reflectorthickness ranged from 12.7 mm to 152.4 mm.

TABLE IPLUTONIUM ISOTOPIC COMPOSITION

. The total plutonium mass is 4438 g. The plutonium iso-topics are listed in Table I. The source was clad in a 0.3-mm-thick stainless steel shell.

The neutron source term was predominantly spontaneous fis-sion of , and the vast majority of induced fission occurredin . Simulations of the experiments used Terrell’s modelof the neutron multiplicity distribution for spontaneous fissionof [17]. The simulations also used a model the neutronmultiplicity distribution for induced fission of based onthe measurements of Zucker and Holden [18].

The reflectors were fabricated from high-density poly-ethylene with a nominal density of 0.95 . They wereconstructed as a set of 5 nesting spherical shells. The shellswere assembled to provide total reflector thicknesses of 12.7,25.4, 38.1, 76.2, and 152.4 mm.

Fig. 5 shows the experiment setup. The source was measuredusing a portable multiplicity counter composed of 15 pro-portional counters embedded in a polyethylene moderator. Eachproportional counter was 25.4 mm in diameter, had a sensitivelength of 381 mm, and contained 10 atm of . The moder-ator was 421.6 mm tall, 430.2 mm wide, and 10.2 mm deep. Tominimize the instrument’s sensitivity to neutrons reflected fromthe floor and walls, the moderator was wrapped on all 6 sideswith a 0.8 mm thick cadmium sheet. The multiplicity counterwas positioned with its front face 500 mm from the center ofthe plutonium source.

The multiplicity counter was operated in list mode, whichrecorded the time of each detection event with 1 s resolution.It also recorded the individual proportional counters that reg-istered a neutron count. The list mode data were processed toaccumulate the count distribution versus multiplicity and co-incidence gate width. For example, the distributions shown in


Fig. 5. A photograph of the experiment setup. The plutonium source is shownin a 76.2 mm thick reflector. The detector contained 15 �� proportional coun-ters in a polyethylene moderator. The moderator was wrapped in cadmium. Thedetector’s front face was positioned 500 mm from the center of the plutoniumsource.

TABLE IITEST RESULTS

Figs. 1 and 2 were accumulated from the measurement of theplutonium source reflected by the 76.2-mm-thick reflector.

For each measurement, the asymptotic value of theFeynman-Y was accumulated from the count distributionusing (1) with a gate width of 4096 s. The measured values ofthe Feynman-Y asymptote are listed in Table II.

The Feynman-Y was also calculated using the deterministictransport method described in Section III. One-dimensionalmodels of the source were used to compute the Feynman-Yasymptote and shape. The calculated Feynman-Y asymptotesare compared to the measured values in Table II. The calculatedasymptotes are all within 10% of the measured values.

Table II also lists the calculation time required to estimate theFeynman-Y asymptote and shape using one-dimensional deter-ministic transport. The calculation times shown were obtainedusing a single core on a desktop computer. By comparison, theequivalent calculations shown in reference [10] took hours tocomplete using a 64-core high performance computing cluster.

Figs. 6 and 7 compare the calculated Feynman-Y shape tothe measurement for two cases: the plutonium source reflectedby 12.7 and 76.2 mm of polyethylene. The calculations fit theFeynman-Y shape and asymptote accurately, though some smallsystematic error is evident in the estimated step response of thefissile system. This error could arise from small inaccuracies inseveral components of the transport model, including:

• The model of the source term, particularly the multiplicitydistribution of spontaneous fission neutrons.

Fig. 6. The measured and calculated Feynman-Y for the plutonium source re-flected by 12.7 mm of polyethylene. The black circles show the measurement,and the red line shows the calculation. The error bars represent one standarddeviation in the measurement.

Fig. 7. The measured and calculated Feynman-Y for the plutonium source re-flected by 76.2 mm of polyethylene. The black circles show the measurement,and the red line shows the calculation. The error bars represent one standarddeviation in the measurement.

• The model of the transport medium, particularly the in-duced fission cross section, the multiplicity distribution ofinduced fission neutrons, and the effective density of thepolyethylene reflectors.

• The model of the multiplicity counter’s detector efficiency.Since the calculated asymptote lies within the measured uncer-tainty, the small errors evident in Figs. 6 and 7 are probably dueto errors in modeling neutron slowing down in the polyethylenereflector and induced fission in the plutonium, which could re-sult from errors in the respective scatter and fission cross sec-tions or from errors in the material and geometry model itself.

Fig. 8 shows the detector efficiency used to calculate the neu-tron count rate and its variance. This efficiency was estimatedusing a three-dimensional Monte Carlo model of the neutronmultiplicity counter. The lower threshold evident in the detectorefficiency results from the cadmium cutoff near 0.2 eV. The dipsin the detector efficiency between 10 eV and 1 keV correspondto resonances in the cadmium absorption cross section betweenthose same energies. The decline in efficiency above 1 MeV isdue to the multiplicity counter moderator. The moderator isn’tthick enough to slow very high energy neutrons down to speedswhere the (n, p) reaction cross section is large. Conse-quently, the efficiency drops off precipitously above 1 MeV.


Fig. 8. The neutron detector efficiency used in deterministic transport calcu-lations of the Feynman-Y. This efficiency was estimated using a three-dimen-sional Monte Carlo model of the detector.

Fig. 9. The measured correlation function used to estimate the detector time-constant. The black circles show the measured number of coincidences betweenpairs of neutron counts versus the delay between the counts in the pair. The redline shows a fit to the measurement using Eq. (15).

Fig. 9 shows the correlation function measured using a baresource. This measurement was used to estimate the de-

tector time-constant using (15). Based on this measurement, thedetector time-constant is 36.4 s. This time constant was usedto model the detector’s impulse response in (14) and (11).

V. CONCLUSIONS

This paper demonstrates the implementation and testing of atechnique to compute neutron multiplicity statistics using deter-ministic neutron transport calculations. The implementation isdesigned to estimate moments of the neutron counting distribu-tion from the solution of three forms of the Boltzmann neutrontransport equation:

1) The fixed source time-independent forward transportequation. The source term represents neutron emission byspontaneous fission and ( , n) reactions in the transportmedium.

2) The fixed source time-independent adjoint transport equa-tion. The adjoint source term is the detector cross sectionsuch that the adjoint flux reflects the importance of sourceneutrons to detection.

3) The fixed source time-dependent forward transport equa-tion. The source term is the same as the intrinsic fixed

source used in the time-independent forward calcula-tion, but it is instantaneously stepped from zero to itssteady-state value such that the flux reflects the transportmedium’s dynamic step response.

Neutron multiplicity counting measurements were conductedwith plutonium reflected by polyethylene to test the accuracy ofthe implementation. For these tests, the accuracy and speed ofthe deterministic transport calculations were adequate for use inan iterative solution framework like the one described in refer-ence [19].

It is possible to apply the technique described in this paper toestimate the composition and geometry of an unknown sourcefrom its measured neutron multiplicity statistics. It is also pos-sible to combine this method for neutron multiplicity analysiswith the gamma spectrometry analysis techniques described inreference [19]. Because of the complementary characteristicsof neutron and gamma transport, a multivariate analysis com-bining neutron multiplicity and gamma spectroscopy measure-ments generally constrains the solution for the source configura-tion more effectively than an analysis based on either measure-ment alone. For example, reference [20] demonstrates a multi-variate analysis using the technique presented in this paper.

APPENDIX

NEUTRON TRANSPORT EQUATIONS

For completeness, the different forms of the Boltzmann trans-port equation applied in this paper are described in the followingAppendix. The specific options used to solve the equations forthe examples in Section IV are also described in this Appendix.

A. Different Forms of the Boltzmann Transport Equation

Forward Boltzmann Transport Equation: The fixed sourceform of the time-dependent forward Boltzmann transport equa-tion is:

(A.1)

where• is the neutron flux• denotes spatial position• denotes neutron energy• denotes direction of travel• denotes time• denotes neutron speed• is the macroscopic cross section for all neutron interac-

tions (absorption, scatter, and fission)• is the macroscopic cross section for scattering from

energy to and from direction to• is the induced fission neutron spectrum• is the macroscopic cross section for fission neutron

production• is the source term


In its most general form, each of the preceding cross sectionsmay also depend on direction and time. However, this paperconsiders only problems where neutron scattering depends onlyon the change in direction , and absorption and fissionare entirely independent of neutron direction. In addition, thispaper considers only problems where the transport medium istime-invariant.

The solution to (A.1), the neutron flux , is related to theneutron population by

(A.2)

The neutron population represents the density of neutrons atphase-space location . More specifically, it reflectsthe average number of neutrons per unit volume found at thatphase-space location.

If the source term is also time-invariant, then (A.1) reducesto the fixed source form of the time-independent forward Boltz-mann transport equation:

(A.3)

Adjoint Boltzmann Transport Equation: Eq. (A.1) is some-times expressed in operator notation

(A.4)

where is the forward transport operator:

(A.5)

and denotes an operand that is integrable over .The adjoint to the transport operator is defined to satisfy

the equality:

(A.6)

where denotes the inner product over ,

(A.7)Consequently, the adjoint transport operator is

(A.8)

Therefore, the fixed source form of the time-dependent ad-joint Boltzmann transport equation is:

(A.9)

where• is the adjoint flux• is the adjoint source termObserve that the inner product in (A.6) can also be expressed

as:

(A.10)

such that the adjoint flux is frequently interpreted as the impor-tance of source neutrons to the response of interest embodied inthe adjoint source term.

For example, if the adjoint source term is the detector crosssection :

(A.11)

In this case, the inner product yields the detector re-sponse , and the adjoint flux may be loosely interpretedas the importance of a source neutron at to the de-tector response.

If the adjoint source term is time-invariant, the (A.9) reducesto the time-independent adjoint Boltzmann transport equation:

(A.12)

B. Solution Methods

Computation of the Feynman-Y asymptote as described inSection III-A requires the solutions to the fixed source forms ofthe time-independent forward and adjoint transport equations,respectively (A.3) and (A.12). The shape of the Feynman-Yshape is computed using the solution to the fixed source time-dependent forward transport equation, (A.1), as described inSection III-B.

A detailed description of deterministic methods to solve thepreceding forms of the transport equation is the subject of sev-eral textbooks and numerous articles. It is beyond the scope ofthis paper. Interested readers should refer to Chapter 8 of thePARTISN user manual for a thorough discussion of the solu-tion methods employed by that code and therefore used in thispaper [15].


PARTISN solves the multigroup discrete ordinates approxi-mation to the preceding transport equations, which requires dis-cretization of the problem domain over position, energy, direc-tion, and time (in the case of (A.1)). In order to permit others toduplicate the examples in Section IV, the following discussiondescribes the PARTISN solver options that were employed.

The same approach to generating the spatial mesh was ap-plied to solve all three of the preceding forms of the transportequation. Within a given region composed of a single material,the minimum spatial mesh width was set to one mean-free-path(based on the total cross section) at the boundary of the re-gion. For mesh intervals interior to the region, the width ofeach successive inner interval was allowed to grow by a factorof two relative to the preceding adjacent outer interval. Conse-quently, the spatial mesh was finest at the boundaries betweenregions, where the flux can change rapidly; the spatial mesh be-came coarser interior to each region where the flux varies moresmoothly.

Spatial derivatives were approximated using the diamond dif-ference formula with negative flux fix-up for the two precedingforms of the forward transport equation. For the adjoint form,negative flux fix-up was suppressed.

The Kynea3 cross section library was employed to solve allthree forms of the transport equation [21]. This library was de-signed be relatively problem independent. The library uses 79energy groups, and it employs a fine group structure at low en-ergy and a relatively broader group structure at high energy. Thelibrary employs scatter cross section Legendre polynomial ex-pansions with orders ranging from 3 to 7. For nuclides witha mass number , a third order expansion is used; for

, a fifth order expansion is used; and for , aseventh order expansion is used. At low energies, some scattercross sections have upscatter elements.

The eighth order Gaussian quadrature set was used to calcu-late the directional dependence of the forward and adjoint flux.Diamond differencing in time with negative flux fix-up was usedto solve the time-dependent forward transport equation. Diffu-sion synthetic acceleration was disabled when solving all threeforms of the transport equation.

All other options for solving the transport equation were set tothe recommended default values for version 4.0 of the PARTISNsolver.

REFERENCES

[1] J. Mattingly, “Polyethylene-Reflected Plutonium Metal Sphere: Sub-critical Neutron and Gamma Measurements,” Sandia National Labora-tories, SAND2009-5804/R2, Dec. 2009.

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Computation of Neutron Multiplicity Statistics Using Deterministic Transport