ARTICLE IN PRESS

Nuclear Instruments and Methods in Physics Research A 607 (2009) 451–457

Contents lists available at ScienceDirect

Nuclear Instruments and Methods inPhysics Research A

0168-90

doi:10.1

� Corr

E-m

pozzisa1 In

multipl

momen

doubles

journal homepage: www.elsevier.com/locate/nima

The detection statistics of neutrons and photons emitted from a fissile sample

Andreas Enqvist a,�, Sara A. Pozzi b, Imre Pazsit a,b

a Department of Nuclear Engineering, Chalmers University of Technology, SE-412 96 Goteborg, Swedenb Department of Nuclear Engineering and Radiological Sciences, University of Michigan, Ann Arbor, MI 48109, USA

a r t i c l e i n f o

Article history:

Received 5 March 2009

Received in revised form

13 May 2009

Accepted 19 May 2009Available online 29 May 2009

Keywords:

Nuclear safeguards

Neutron and photon numbers

Number distributions

Generating functions

Master equations

Multiplicity

Joint distribution

02/$ - see front matter & 2009 Elsevier B.V. A

016/j.nima.2009.05.131

esponding author.

ail addresses: andreas@nephy.chalmers.se (A.

@umich.edu (S.A. Pozzi), imre@chalmers.se (I

accordance with the literature, the word ‘‘mu

e production of particles in a fission reaction

ts of the emitted/detected particles, or the

and triples.

a b s t r a c t

The purpose of this paper is to present new analytical derivations to describe the statistical properties of

the emission and detection of neutrons and photons generated in and emitted from fissile samples, with

absorption included. The results of the analytical approach are compared with and validated by

corresponding Monte Carlo simulations. The joint detection probability of the generated and detected

neutrons and photons are also described. The analytical model described in this paper accounts for

absorption and detection, thus extending the model presented in previous studies. By using this new,

more realistic model, one can investigate the relative feasibilities of measuring neutrons, gamma

photons or combinations thereof, for the analysis of a specific fissile sample. It is seen that, in contrast to

the case of neutrons, for an increasing mass the absorption in the sample decreases the probability of

multiple photon emissions. Hence we confirm previous observations that although photons have a

larger initial (source) multiplication, neutrons might be more favourable to measure in the case of large

samples because of the increasing self-shielding effect for gamma photons.

& 2009 Elsevier B.V. All rights reserved.

1. Introduction

In non-destructive assay of nuclear materials, the statisticalproperties of the number distribution of neutrons and gammarays emitted by fissile samples are of high importance. Themultiplicities of neutrons [1] and photons [2] generated in fissilesamples with internal multiplication have been investigated inthe past.1 Investigations of the effect of including absorption inthe model have also recently been made. The implicit masterequations are the starting point for the generating functions of theneutron and photon number distribution. These equations havealso been used in the past for calculating factorial moments. Forpractical reasons, factorial moments are usually only interestingup to the third or fourth order. Using multiplicity or coincidencemeasurements, and information on isotopic composition one canestimate the sample mass and isotopic composition of a certainsample.

A characteristic of the factorial moments is that, experimen-tally, they are difficult to measure beyond the third moment(triples), whereas from an analytical or calculational point of view,

ll rights reserved.

Enqvist),

. Pazsit).

ltiplicity’’ is used both for the

, as well as for the factorial

detection rates of singles,

the high order moments beyond triples become increasingly longand tedious to derive by hand. The present work describes adevelopment in the latter area where calculations of the numberdistribution of fissile samples also yields results for the factorialmoments.

In contrast to the factorial moments, the probabilities PðnÞ andFðnÞ of emitting n neutrons or gamma photons, respectively, areinteresting up to large values of n. The necessary number of termsof PðnÞ or FðnÞ that need to be calculated is determined from thecondition that the cumulative probability should be sufficientlyclose to unity (within a few percent or lower). In some cases thevalue of n necessary to achieve this, can exceed 50 for bothneutrons and photons.

Including neutron and photon absorption in the model affectsthe characteristics of the number distributions in several ways.For neutrons the process of radiative absorption eliminates themfrom the fission chain and causes the observable (leaked)neutrons to reduce in number. For photons the dependence ismore involved, because both the absorbtion of neutrons and thatof the photons themselves will affect the photon distribution.All of these effects will vary with the main parameter of thesample being investigated, i.e. the sample mass. An increase inmass affects the probability of both induced fission, as well asabsorption of both neutrons and gamma photons for largesamples in the few kg mass range.

The process of detection allows us to observe the statistics ofemission from a given sample. There is always a o100% detectorefficiency involved in the process of detection. One way of

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modelling this effect was investigated in Ref. [3], in which thedetectors are assumed to be embedded into the sample. Thismodel is not realistic since with this assumption the detectorcontributes to the absorption inside the sample, and hence itinfluences the internal multiplication. In particular, a 100%detector efficiency means that the detector material completelysuppresses that of the sample and hence the measurement yieldsinformation on the detector properties only (pure absorptionwithout multiplication).

In this work we will present a different way of accounting forabsorption and detection, with the latter only pertaining tothe particles that leaked out from the sample. In this way thedetection process can be described more realistically. The resultsof the analytical calculations are compared to those from MonteCarlo simulations.

2. Theory

The master equations, or Chapman–Kolmogorov equations, forthe generating functions of the number of neutrons and photonsin a sample with both spontaneous and induced fission have beenderived in Refs. [1,2]. Those models are based on the point model,where the probability to induce fission and to be absorbed orundergo other processes is uniform throughout the sample. Inboth cases, it is assumed in these calculations that the probabilityof a first collision before escape from the sample of an arbitraryneutron, p, is known. That model is extended here to account forabsorption and detection, and the probabilities for these eventsare included explicitly in the equations. With these extensions,the equations give the detection statistics in an analytical way,and can be used to plan experiments, or as an indicator whencomparing it to numerical results usually obtained using MonteCarlo codes. Effects of the dead time on the factorial moments areneglected throughout. This is partly due to the fact that the theoryof the effect of dead time on the higher order moments is not yetelaborated. The dead time effect will be treated in forthcomingwork.

2.1. Neutron distributions

In earlier models of the statistics of neutron and photonemission from a fissile sample [1,2,4], the absorption of neutronsand gamma photons was not accounted for. It was assumed thateach neutron has a probability p of inducing fission, or failing todo so and hence escaping with a probability 1� p.

The probability generating functions (PGFs) hðzÞ and HðzÞ ofp1ðnÞ and PðnÞ describing the number of neutrons generated byone initial neutron or one initial neutron event (spontaneousfission), are defined as

hðzÞ ¼X

n

p1ðnÞzn and HðzÞ ¼

Xn

PðnÞzn (1)

respectively. These are used in the coupled backward-type (first-event-based) master equations for neutrons [1]:

hðzÞ ¼ ð1� pÞzþ pqi½hðzÞ� (2)

and

HðzÞ ¼ qs½hðzÞ�. (3)

Here,

qsðzÞ ¼X

n

psðnÞzn and qiðzÞ ¼

Xn

piðnÞzn (4)

stand for the generating functions of the number of neutronsgenerated in a spontaneous or an induced fission, respectively. Forfinding the distributions p1ðnÞ and PðnÞ one needs to observe that

they are the Taylor expansion coefficients of hðzÞ and HðzÞ,respectively [4]:

p1ðnÞ ¼1

n!

dnhðzÞ

dzn

����z¼0

and PðnÞ ¼1

n!

dnHðzÞ

dzn

����z¼0

. (5)

As can be noted, the expressions are evaluated at z ¼ 0, in contrastto taking them at z ¼ 1, which is the case when searching for thefactorial moments of the neutrons generated in spontaneous orinduced fission ns, ni. In order to have compact expressions, themodified moments nan were introduced in Ref. [4]. They differ invalue from the traditional nuclear factorial moments nan, and aredefined as

dnqaðhÞ

dhn

����z¼0

¼ qðnÞa ðhÞjz¼0 ¼ qðnÞa ½p1ð0Þ� ¼ nan; a ¼ s; i (6)

where p1ð0Þ is the probability of having zero neutrons generatedwhen starting with one initial neutron and occurs due toevaluating the corresponding expressions at z ¼ 0. This initialprobability needs to be found, since higher order terms of p1ðnÞ

and PðnÞ all depend on it, and it appears also in the modifiedfactorial moments. Note can be made of the fact that thisprobability is highly dependent on the sample mass, and whetheror not absorption and detection are included into the model. Oneexpects this probability to increase with the inclusion ofabsorption and detection probability. Absorption can remove theinitial neutron or all neutrons generated in the short chainsstarted by the first neutron. Likewise, one can end up with zeroneutrons detected, even if the single neutron started a long chainof fissions, when the detection probability is low.

In the neutron probability balance equation, the event ofabsorption can be included into the fission distribution, becausewhen looking at the progeny of neutrons it is the same as a fissionevent with zero neutrons generated. Therefore, one can includethe absorption by a suitable increase of the first collisionprobability from a value p to p0. The new first collision probabilitythen accounts for both fission and absorption, and the comple-mentary probability 1� p0 now properly describes the probabilityfor a neutron to escape the sample and become available forexternal detectors to register. To maintain normalization, theprobabilities piðnÞ for n40 will need to be decreased according tothe following formula:

epiðnÞ ¼p0 � p

p0dn;0 þ

p

p0piðnÞ. (7)

The generating function of epiðnÞ, Eq. (4), will also changeaccordingly into eqiðzÞ. The first master equation (2) will now read:

hðzÞ ¼ ð1� p0Þzþ p0eqi½hðzÞ� (8)

and describes the leaked out neutrons. In the numerical work, thevalues of p, p0 and the probabilities piðnÞ are taken from thecode MCNP-PoliMi [5], which contains extensive nuclear datatables. This will have the advantage that in the comparisonswith Monte Carlo simulations, it is assured that the same nucleardata are used in both the simulations and the analyticalcalculations.

The process of detection can be added in a straightforwardmanner by considering only the neutrons that have already leakedout of the sample, since those are the ones available for detectionby external detectors. Using a detection probability for neutrons, �,we can create a generating function eðzÞ of the binary probabilitydistribution (i.e. either zero or one neutron) of the number ofneutrons detected per neutron emitted from the sample:

eðzÞ ¼ �zþ ð1� �Þ. (9)

Using a constant � assumes no space dependence or alternativelythat neutrons from different events have the same energy

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distribution. The new generating functions that also include thedetection process are given by

hdðzÞ ¼ h½eðzÞ�; HdðzÞ ¼ H½eðzÞ�. (10)

The derivatives needed for finding the factorial moments and thenumber distributions change in a simple way

dnhdðzÞ

dzn ¼dnhðzÞ

dzn � ð�Þn;

dnHdðzÞ

dzn ¼dnHðzÞ

dzn � ð�Þn. (11)

For the factorial moments the change is

end;n ¼ ð�Þn� enn. (12)

This is due to the factorial moments (multiplicities) beingevaluated at z ¼ 1. For the number distributions, evaluation ofthe expressions at z ¼ 0 result in modified moments whichdepend on the probability p1ð0Þ. This probability changes whenabsorption and detection are accounted for, so even if thederivatives change formally in the same manner as for thefactorial moments, the modified moments will also obtain newnumerical values.

In our previous work [4], i.e. with no absorption and detection,the modified moments were given by

dneqaðhÞdhn

����z¼0

¼ eqðnÞa ðhÞjz¼0 ¼ eqðnÞa ½p1ð0Þ� ¼ nan; a ¼ fs; ig. (13)

In the case of detection, pdð0Þ will replace p1ð0Þ, which is obtainedfrom the solution of the N-th order polynomial

pdð0Þ ¼ ð1� p0Þð1� �Þ þ peqi½pdð0Þ�

¼ ð1� p0Þð1� �Þ þ p0XN

n¼0

epiðnÞ½pdð0Þ�n (14)

where N is the largest neutron multiplicity in an induced fission(set to N ¼ 8 in the case of plutonium).

Using these properties we can now derive the detectionstatistics from the Taylor expansion

pdðnÞ ¼1

n!

dnhdðzÞ

dzn

����z¼0

and PdðnÞ ¼1

n!

dnHdðzÞ

dzn

����z¼0

. (15)

The terms in the probability distribution can now be calculatedrecursively because the starting master equation is in implicitform. This fact means that the probabilities PðnÞ will occur in theexpressions for PðmÞ, where m4n. This makes it computationallyfavourable to use the symbolic manipulation code Mathematica[6], which was used to derive the higher order terms.

The analytic model allows us to readily analyze the depen-dence of the measured quantities on simple parameters:

�

the probability to induce fission, which is a parameter thatincreases with mass, and can be calculated from the mass andgeometry of the sample, provided the density is known; � the absorption probability which also depends on sample size

and composition; and

� finally the detection efficiency, which can be changed to reflect

what type of detector is used, such as fast scintillationdetectors or large arrays of helium tubes in the form ofmultiplicity counters.

2.2. Photon distributions

As in the case of neutrons, a set of coupled backwardsmaster equations have earlier been derived and used to find thestatistics of the generated photons [2,4]. The starting equationswere

gðzÞ ¼ ð1� pÞ þ priðzÞqi½gðzÞ� (16)

and

GðzÞ ¼ rsðzÞqs½gðzÞ� (17)

where gðzÞ and GðzÞ are the probability generating functions off 1ðnÞ and FðnÞ, describing the number of generated photons whenstarting with one neutron or one source event, respectively:

gðzÞ ¼X

n

f 1ðnÞzn; GðzÞ ¼

Xn

FðnÞzn. (18)

The nuclear data for the distribution of photons, generated in oneinduced or spontaneous fission event, respectively, are neededand their PGFs are defined as

riðzÞ ¼X

n

f iðnÞzn; rsðzÞ ¼

Xn

f sðnÞzn. (19)

When performing the differentiations to find the probabilitydistributions f 1ðnÞ and FðnÞ, one will, just as in the case ofneutrons, encounter modified moments defined as

dnraðzÞ

dzn

����z¼0

¼ n!f aðnÞ � man; a ¼ fs; ig. (20)

In the calculation of these modified moments, which refer to thatof the photons, also the modified moments of neutrons willappear, due to the structure of Eqs. (16) and (17). Physically, this isdue to the fact that the internal multiplication of photons is aresult from the neutron branching, hence the correspondingneutron moments will also appear. However, these neutronmoments will not be the same as the ones that are derived forthe ‘‘pure’’ neutron distribution, Eqs. (2) and (3). The reason is thatfor the calculation of these modified neutron moments thatappear in the gamma photon distributions, the factor p1ð0Þ inEq. (6) will be replaced by the factor f 1ð0Þ in the case of photons inthe corresponding expressions.

To further extend this model to account for detection statisticsinstead of merely the number of generated particles [4], one needsto include both absorption and the process of detection, whichtakes place with a certain probability, referred to often as thedetection efficiency, �.

In the case of neutrons, the absorption was taken into accountby modifying the probability distribution epiðnÞ and changing thefirst collision probability from p to p0. For photons the situationwill be slightly different, because leakage and internal absorptionof a neutron is assumed to lead to zero generated photons. Hencethe parameter p in Eq. (16) remains that of the probability toinduce a fission. The generating function eqiðzÞ will still be used,since with zero neutrons generated the branching process willstop, and no more photons can be generated.

Gamma absorption will be accounted for by the probability lgthat describes the leakage probability for one single photon,likewise ð1� lgÞ is the probability for a created photon to beabsorbed and not escape the sample. The gamma capture will beaccounted for by an additional generating function ‘gðzÞ,

‘gðzÞ ¼ lgzþ ð1� lgÞ. (21)

Here ‘gðzÞ is the generating function of the binary probabilitydistribution of gamma photons leaving the sample per initialphoton. Due to the simple form of this relationship, the factorialmoments of the leaked out photons are simply the factorialmoments for the generated photons times the leakage probability:

eml;n ¼ ðlgÞn� emn. (22)

The master equations for the leaked out photons are then given as

glðzÞ ¼ g½‘gðzÞ�; GlðzÞ ¼ G½‘gðzÞ�. (23)

To separate the photons that undergo detection compared to thelarger numbers of photons that escape the sample, an extraequation is added that describes the probability for one photon to

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undergo detection or not. The detection efficiency, �g, is used forthe generating function egðzÞ of the binary probability of detectionof one photon out of one escaped photon. This generating functionis given by

egðzÞ ¼ �gzþ ð1� �gÞ. (24)

The probability distribution can now be extracted by using themaster equations

gdðzÞ ¼ g½‘gfegðzÞg� ¼ ð1� pÞ þ priðzÞeqi½gdðzÞ� (25)

and

GdðzÞ ¼ G½‘gfegðzÞg� ¼ rsðzÞqs½gdðzÞ�. (26)

Since one evaluates the expressions at z ¼ 0 to get the probabilitydistribution, modified moments are created as earlier. In thecase of detected photons these modified moments will depend onf dð0Þ instead of on f 1ð0Þ, i.e. the probability of having zeroneutrons detected when starting with one initial neutron in thesample. This quantity can be found by setting z ¼ 0 in Eq. (25),and finding the root of the finite degree polynomial thatarises:

f dð0Þ ¼ ð1� pÞ þ prf ½‘gfegð0Þg�qf ½f dð0Þ�

¼ ð1� pÞ þ pX�24

n¼0

f f ðnÞ½‘gfegð0Þg�n !

�X8

n¼0

pf ðnÞ½f dð0Þ�n. (27)

The modified moments using this new initial term are quantita-tively different compared to the neutron case and are defined asfollows when we include absorption and detection into themodel:

dnra½‘gfegðzÞg�dzn

����z¼0

� md;an; a ¼ s; i (28)

dnqaðgdÞ

dgnd

�����z¼0

� nd;an; a ¼ s; i. (29)

The factors are straightforward to calculate, but lead to expres-sions that contain sums that have an increasing number ofterms for higher order moments. The formulae obtained for thenumber distributions, both in the case of neutrons and photons,contain several quantities that are based on nuclear physicsconstants (fission neutron and gamma photon multiplicities),weighted by factors depending on the first collision probability,absorption probabilities etc. Among those quantities one alsofinds the modified moments of nan and man. When calculatinghigher order terms and getting longer expressions one notesthat nan becomes zero for n48, due to the fact that fissionneutron multiplicities are zero above that limit. In the samemanner man vanishes for a larger n (n � 24 as indicated in Eq.(27)). Considerations like these make the otherwise rapidlygrowing expressions more manageable; however, the use ofcomputer software to handle the numerical evaluations remainsa necessity.

A further fact to be mentioned concerns the formal equivalencebetween the probabilities PðnÞ and the corresponding n-th orderfactorial moments of the neutrons and photons generated in thesample, as noted in Ref. [4]. In Ref. [4] it was noted that thefactorial moments of the number of neutrons and photonsgenerated in the sample could be obtained from the probabilitiesby replacing the modified moments with the ordinary momentswhich are based on the pure nuclear data. It is seen from theexpressions derived in this paper that the same formal equiva-lence exists between the probabilities PdðnÞ of the detectedneutrons and photons and the corresponding factorial moments.Thus by finding the expressions for the probability distribution ofthe detected particles, one also receives the multiplicities as a

limiting simplified case up to the same high order as thedistribution was determined. This order is generally much higherthan the third or fourth, up to which the factorial moments areusually computed.

2.3. Joint distributions

In a more complete description of emission of neutrons andgamma photons from fissile samples, one can extend thedescription to the joint statistics (probabilities of joint detections)of neutrons and photons. For this one also would need the jointnumber distributions psðn;mÞ and piðn;mÞ of n neutrons and m

photons emitted in a spontaneous and an induced fission,respectively. Unfortunately there are no data available for suchjoint distributions in the literature. Intuitively the correlationsbetween neutrons and photons per fission event are expected tobe week, and generally, the generation of neutrons and photons isassumed to be independent. Hence, the above distributions arejust products of the individual distributions. The correspondingPGF’s are defined as

qsðz; yÞ ¼X

n

Xm

psðn;mÞznym (30)

qiðz; yÞ ¼X

n

Xm

piðn;mÞznym. (31)

If b1ðn;mÞ is the probability to obtain n neutrons and m photonsgenerated by one initial neutron, a backward-type masterequation can be readily derived by considering the two mutuallyexclusive events of not having or having a first collision beforeleaking out:

b1ðn;mÞ ¼ ð1� pÞdn;1dm;0

þ pX1k¼0

X1l¼0

piðk; lÞX

n1þ���þnk¼nm1þ���þmk¼m�l

Yk

i¼1

b1ðni;miÞ. (32)

Defining the generating function

cðz; yÞ ¼X

n

Xm

b1ðn;mÞznym (33)

from (32) one obtains

cðz; yÞ ¼ ð1� pÞzþ pqi½cðz; yÞ; y�. (34)

In the same manner an equation for the probability generatingfunction Bðn;mÞ of the probability to have n neutrons and m

photons generated when starting with one source event, can bederived as

Cðz; yÞ ¼ qs½cðz; yÞ; y�. (35)

Also for the joint distributions, from the practical point of view itis more interesting to determine the distribution of detectedparticles. Using the same concepts as before, one defines thegenerating functions of the leakage probability of a single neutronand photon, respectively:

‘nðzÞ ¼ lnzþ ð1� lnÞ

‘gðyÞ ¼ lgzþ ð1� lgÞ

((36)

where ln and lg are the leakage probabilities of neutronsand photons, respectively. Actually, in the earlier notations, ln issimply equal to 1� p0. The values of these parameters willnaturally vary with the main parameter of the investigatedsample, which is the sample mass. Detection can then be includedin the same manner by using the earlier defined generatingfunctions

enðzÞ ¼ �n zþ ð1� �nÞ

egðyÞ ¼ �gzþ ð1� �gÞ

((37)

ARTICLE IN PRESS

0 10 20 30 40

10−6

10−4

10−2

100

Number of neutrons

Pro

babi

lity,

P (n

)

Spontaneous fission335g (no absorption)2680g (no absorption)9047g (no absorption)Analytical model with abs.

10−2

100

(n)

A. Enqvist et al. / Nuclear Instruments and Methods in Physics Research A 607 (2009) 451–457 455

with �n and �g being the detection efficiencies of neutrons andphotons, respectively. The coupled master equations for thedetection statistics are then given as

cdðz; yÞ ¼ ð1� pÞ‘nfenðzÞg þ pqi½cdðz; yÞ; ‘gfegðyÞg� (38)

Cdðz; yÞ ¼ qs½cdðz; yÞ; ‘gfegðyÞg�. (39)

The individual distributions and corresponding defining equationsfor the neutrons or photons can now be found as special cases:y ¼ 1 in Eqs. (38) and (39) gives the neutron distributionsequivalent of Eqs. (10), while setting z ¼ 1 in the Eqs. (38) and(39) gives Eqs. (25) and (26). The sought joint probabilitydistributions are calculated as the n;m-th derivatives of cdðz; yÞ

and Cdðz; yÞ with regard to z and y, respectively.The data used in the calculations (fission probability, leakage

probability etc.) and the assumptions such as independentneutron and photon distributions in each fission event, were alltaken to be the same as used in MCNP-PoliMi simulations tofacilitate direct comparisons. In connection to a real detectionmeasurement the effect of dead time would also come into playespecially for high count rates. The model assumes perfectdetectors with no dead time or backscattering. For metallicsamples of composition and geometry such as those simulatedhere, comparisons to real data can be readily performed, whileother samples and compositions would require further computa-tions and new starting parameters simulated using MCNP orcalculated through other independent models.

0 10 20 30 40

10−6

10−4

Number of photons

Pro

babi

lity,

F

Spontaneous fission335g (no absorption)2680g (no absorption)9047g (no absorption)Analytical model with abs.

Fig. 1. Comparison of the number distribution for neutrons (a) and photons (b),

with and without accounting for absorption, both computed from the analytical

model. Coloured markers are without absorption and black lines with absorption

included. For neutrons the lines coincide, while for photons a major change is seen,

but for high numbers the trend is the same with the lightest sample having the

smallest probabilities also in the analytical model. (For interpretation of the

references to color in this figure legend, the reader is referred to the web version of

this article.)

3. Results

The number distributions were calculated for neutrons andphotons separately, as well as for joint distributions, forplutonium metal spheres of varying mass. The values obtainedwere compared to simulations with the code MCNP-PoliMi [2,5,7].To this end, MCNP-PoliMi had to be modified to supplythe necessary tallies. The values of the probability to inducefission p, and that of the nuclear physics constants suchas the fission parameters psðnÞ, f iðnÞ, etc. were taken fromMCNP-PoliMi runs for the analytical model. The probabilityto induce fission varies with the mass of the sample. Thevalues of p are shown in Table 1 for the three samples forwhich we have performed calculations, the uncertainty is listed in1� s-values.

The analytical model in the case of generated particles (noabsorption included), has earlier been validated against MonteCarlo simulations [4] with very good agreement. In MCNP-PoliMia spherical encompassing idealized detector was used to measureall outgoing particles without reflecting them back to the sampleto change the statistics. Thus the result obtained refers to thenumber of emitted particles.

As can be seen in Fig. 1, for photons there is a very evidenteffect of self-shielding. The probabilities of high numbers ofphotons escaping the sample are reduced significantly when

Table 1Probability to induce fission, p, for one neutron depending on the mass of the

sample.

Sample Mass (kg) p s

1 0.335 0.0852 o0:0002

2 2.680 0.1678 o0:0002

3 9.047 0.2461 o0:0002

The metal spheres have a composition of 80 wt% Pu-239 and 20 wt% Pu-240, and a

density of 15.9 g/cc.

absorption is taken into account. The results show that forrealistic samples, which might be investigated with typical non-destructive assay (NDA) techniques, the initial advantage ofhaving high photon multiplicities diminishes, because theinternal absorption of photons is much greater than that forneutrons in materials of high atomic numbers.

Fig. 2 shows a good agreement between the analyticalmodel and the numerical simulations what regards the emittedparticles. At low probabilities, i.e. with higher numbers ofparticles emitted, the Monte Carlo data are difficult to compareto, due to poor statistics caused by the finite number of historiesrun. Photon detectors can still be very useful if we consider ascenario where the sample is heavily screened with low Z

materials, and/or materials with high neutron absorption crosssections, which do not screen gamma photons. In such casesphoton detectors can be advantageous compared to neutron

ARTICLE IN PRESS

0 10 20 30 40

10−6

10−4

10−2

100

Number of neutrons

Pro

babi

lity,

P (n

)

335g MCNP−PoliMi2680g MCNP−PoliMi9047g MCNP−PoliMiAnalytical model with abs.

0 10 20 30 40

10−6

10−4

10−2

100

Number of photons

Pro

babi

lity,

F(n

)

335g MCNP−PoliMi2680g MCNP−PoliMi9047g MCNP−PoliMiAnalytical model with abs.

Fig. 2. Comparison between numerical simulations and analytical results when

the effect of absorption is accounted for. As can be seen, the agreement is good in

general. The MCNP-PoliMi results are shown with error-estimates, both for the

neutrons (a) and photons (b).

0 5 10 15 20 25 30

10−6

10−4

10−2

100

Number of neutrons

Pro

babi

lity,

P (n

)

335g emitted2680g emitted9047g emittedDetection included

0 5 10 15 20 25 30

10−6

10−4

10−2

100

Number of photons

Pro

babi

lity,

F (n

)

335g emitted2680g emitted9047g emittedDetection included

Fig. 3. Probability distributions with detection included. The results are obtained

from the analytical models. The shape of the curves will be dependent on the

detector efficiency �x, which was taken as 10% for neutrons (a) and 20% for photons

(b) in these plots.

A. Enqvist et al. / Nuclear Instruments and Methods in Physics Research A 607 (2009) 451–457456

detectors. Different types of shielding can be accounted for in themodel by correctly changing the detection efficiency oralternatively increasing the absorption probability which mightbe easier in the case of photons.

In the case of gamma photons one can note the somewhatunexpected result that it is more likely to detect relatively highmultiplicities, such as triples and quadruples, for a lighter samplethan for a heavy one. The reason is that, although the total amountof generated photons is higher for the heavier samples, the self-shielding effect counteracts this. The fast growing self-shielding(with increasing mass) constitutes of course a disadvantage.Note that these probabilities are per source event, and in asample of higher mass the number of spontaneous fissions will behigher.

The effect of the detection process in the model, so that thedistribution shows the number of detected particles ratherthan the emitted particles, is shown in Fig. 3. The changeof the distributions is directly linked to the detection efficiency,and the lower the detection efficiency, the more difficult

it is to see higher order multiplicities which is visible in thefigures.

Using the joint statistics and multiple detections not onlyof the same particle but also of different particles, Fig. 4shows that a combined detection of one neutron and onephoton, can be more frequent than the doublet of any oneof them. Using the joint statistics and applying realisticdetector efficiencies, one can use the analytical model todecide in a fast and relatively easy manner whether both typesof particles should be detected, and if so which combinedfactorial moments should be measured for best estimationperformance. The types of available detectors and further partialinformation of the sample will of course also influence thedecisions.

4. Conclusions

Using proper application of symbolic computation, it has beendemonstrated in previous work that high order terms of thenumber distribution can be derived and evaluated. This approach

ARTICLE IN PRESS

Fig. 4. The first graph shows data from the analytical model where we have

assumed a larger detection efficiency for neutrons compared to photons, e.g. a

sample shielded with a material of high atomic mass. The right-hand side graph is

from MCNP-PoliMi, showing the detection statistics for a detector setup using six

scintillator detectors.

A. Enqvist et al. / Nuclear Instruments and Methods in Physics Research A 607 (2009) 451–457 457

has now been used to incorporate also absorption and detectioninto the modelling of the probability distributions of neutrons andphotons in a fissile sample.

The formal equivalence between the number distribution andfactorial moments is also kept, meaning that factorial moments ofvery high orders can be easily calculated as limiting simplifiedevaluations of the formulae derived. The modified momentsoccurring in these expressions can all be readily calculated basedonly on nuclear data and characteristics such as sample mass,geometry and matrix and the detection efficiency.

The quantitative results show a good agreement withMCNP-PoliMi simulations. As can be expected, the inclusion ofabsorption does not have a significant effect on the neutrons; onthe other hand, gamma photons are heavily attenuated. This has aconsiderable effect on the number distributions and factorialmoments of the emitted photons, whereas the distributions andmoments of the neutrons are not much affected. Accounting forthe process of detection by using detection efficiencies changesthe statistics further. These studies are useful in giving indicationsas to what type of particle to focus on for assessing the samplewith greatest accuracy.

Further, the model has been extended to be able to handle jointdetection probabilities of neutrons and photons. Using jointmoments adds to the diversity of the experimental methods,and can enhance the identification and detection of samples. Thecalculations again supply indications such as multiplication andneutron and photon emission which could be used as guidelineson how to design the measurement to get maximum efficiency.Using joint detections might lead to shorter measuring times, orto get further data about the sample without having to usequadruplets.

Acknowledgements

The work of the Swedish authors was supported by theSwedish Nuclear Power Inspectorate. The work of the US authorwas supported in part by the US Department of Energy NationalNuclear Security Administration Office of Non-proliferationResearch Engineering NA-22.

References

[1] K. Bohnel, Nucl. Sci. Eng. 90 (1985) 75.[2] I. Pazsit, S.A. Pozzi, Nucl. Instr. and Meth. A 555 (2005) 340.[3] M. Lu, T. Teichmann, Nucl. Instr. and Meth. A 313 (1992) 471.[4] A. Enqvist, I. Pazsit, S.A. Pozzi, Nucl. Instr. and Meth. A 566 (2006) 598.[5] S.A. Pozzi, E. Padovani, M. Marseguerra, Nucl. Instr. and Meth. A 513 (2003)

550.[6] Wolfram Research Inc., Mathematica, Version 5.2, Champaign, IL, 2005.[7] S.A. Pozzi, J.A. Mullens, J.T. Mihalczo, Nucl. Instr. and Meth. A 524 (2004) 92.