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Sample characterization using both neutron and gamma multiplicities
ARTICLE IN PRESS Nuclear Instruments and Methods in Physics Research A 615 (2010) 62–69
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Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Sample characterization using both neutron and gamma multiplicities Andreas Enqvist a,, Imre Pa´zsit a, Senada Avdic a,b a b
Department of Nuclear Engineering, Chalmers University of Technology, SE-412 96 G¨ oteborg, Sweden Faculty of Science, Department of Physics, University of Tuzla, 75000 Tuzla, Bosnia and Herzegovina
a r t i c l e in f o
a b s t r a c t
Article history: Received 8 December 2009 Received in revised form 28 December 2009 Accepted 8 January 2010 Available online 18 January 2010
Formulas for multiplicity counting rates (singles, doubles, etc.), used for the unfolding of parameters of an unknown sample, can be derived from those for the corresponding factorial moments. So far such rates were derived only for neutrons. The novelty of this paper is related to the derivation of the individual gamma and mixed neutron–gamma detection rates as well as to investigation of the possibilities and actual algorithms for the sample parameter unfolding. Taking the individual gamma and mixed neutron–gamma moments up to third order, together with the neutron moments, there will be nine auto- and cross-factorial moments and corresponding multiplicity rates, as well as a larger number of unknowns than for pure neutron detection. The total number of measurable multiplicities exceeds the number of unknowns, but on the other hand the structure of the additional equations is substantially more complicated than that of the neutron moments. Since an analytical inversion of the highly non-linear system of over-determined equations is not possible, the use of artificial neural networks (ANNs) is suggested, which can handle both the nonlinearity and the redundance in the measured quantities in an effective and accurate way. The use of ANNs is demonstrated with good results on the unfolding of various combination of multiplicities for certain combinations of the unknown parameters, including the sample fission rate, the leakage multiplication and the a and g ratios. & 2010 Elsevier B.V. All rights reserved.
Keywords: Nuclear safeguards Neutron and gamma multiplicities Joint moments Materials control and accounting Neural networks
1. Introduction Multiplicity detection rates, based on higher order factorial moments of the neutron counts from an unknown sample, can be used to determine sample parameters [1–3]. The factorial moments here refer to those of the total number of neutrons generated in the sample by one initial source event (spontaneous fission or ða; nÞ reaction). Due to internal multiplication through induced fission, the probability distribution of the total number of generated neutrons will deviate from that by the initial source event, the deviation being a function of the sample mass via the first collision probability of the initial neutrons. From here the detection rates can be calculated by accounting for the initial source event rates, which further corroborates the dependence of the multiplicity rates on the sample mass since source event rate (sample fission rate) depends linearly on the sample mass. This gives a possibility to determine the sample mass. Multiplicity rate measurements of the first three multiples enable the recovery of three unknowns, which are usually taken as the sample leakage multiplication (related to the first collision
Corresponding author. Tel.: + 46 31 772 30 86; fax: + 46 31 772 30 79.
E-mail addresses: [email protected] (A. Enqvist), [email protected] (I. Pa´zsit), [email protected] (S. Avdic). 0168-9002/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2010.01.022
probability, itself being a function of sample mass), the ratio a of the intensity of neutron production via (a; nÞ reactions to spontaneous fission, and the spontaneous fission rate, the latter being the most important parameter. This approach is forced to leave the detector efficiency undetermined which needs to be predetermined experimentally. Alternative approaches, such as assuming the sample multiplication to be known, could allow for the detector efficiency to be unfolded. Recently it was suggested that in addition to neutron multiplicity counting, gamma multiplicities be also taken into account [4–6]. The motivation for using gamma counting is manifold: higher gamma multiplicity per fission, larger penetration through most of the strong neutron absorbers, and the relatively easy detection by organic scintillation detectors, whose potential use in nuclear safeguards have been recently investigated [7,8]. The goal is still the same, i.e. to determine the above factors, plus the further unknowns introduced, such as the gamma leakage multiplicity, the ratio g of single gamma to fission gamma intensity and gamma detector efficiency. These can though be handled since three neutron and three gamma multiplicities can be measured automatically, so one has still as many unknowns as measured quantities. In addition, there exists the further possibility of using the joint moments of the neutron and gamma counts, which supply further independent measured data to determine still the same
ARTICLE IN PRESS A. Enqvist et al. / Nuclear Instruments and Methods in Physics Research A 615 (2010) 62–69
number of unknowns. Accounting also for the joint moments up to third order, there are altogether nine factorial moments. Hence the problem becomes overdetermined. The multiplicity expressions depend highly non-linearly on most of the searched parameters and hence constitute a system of non-linear equations of order up to 9. The consequences of this fact on the possibility of unfolding will be discussed in some detail in the paper. To make maximal use of the redundant information from the measurement when also the joint moments are used, the unfolding of the parameters has to be performed by least-square type unfolding methods. Actually, there is a conceptually simple non-parametric unfolding method for such a purpose, the artificial neural networks (ANNs), whose use will be demonstrated here. In this paper we give the definitions of the quantities used and list all nine factorial moments. To give insight into the information contained in the joint moments, the dependence of the lowest order joint neutron–gamma moment on the non-leakage probability will be given quantitatively. In addition to the factorial moments, the multiplicity detection rates for the gamma photons and for mixed neutron–gamma detections will also be given. These have not been given previously. The derivation of converting the factorial moments into multiplicity detection rates follows the method described in a recent clarifying note on the relationship between factorial moments and the single, double and triple coincidence rates [9]. This will clearly show the relation between the measurable multiplicity rates and the factorial moments, when accounting for both multiple emission (spontaneous fission) and single emission ða; nÞ source events. In the last section a test and an analysis of the unfolding procedure is given for various combinations of the multiplicity rates and the parameters taken as unknown.
2. Definitions The following definitions and conventions will be used. Random variables and their moments referring to neutrons will be denoted by n, and those for gamma photons by m. Variables referring to spontaneous fission will have a subscript sf, and those referring to induced fission a subscript i. For the factorial moments, there will always be a second index, giving the order of the moment. Hence, nsf ;2 will stand for /nðn1ÞS in case of spontaneous fission. In addition, we will distinguish between two sets of variables for both neutrons and photons, depending on whether they belong to a source event (all processes, i.e. spontaneous fission and ða; nÞ processes included), or to the total number of generated neutrons, which accounts for the internal multiplication (superfission). The parameters belonging to the first set will be written with a subscript indication, such as ns;2 ; ni;3 (the subscript s alluding to ‘‘source’’), whereas those of the second set will be denoted with just a numerical subscript indicating their order such as n1 ; m3 . The factorial moments corresponding to the distribution of neutrons or gammas emitted in fission, whether induced or spontaneous, are nuclear constants and are known in advance. However, as is usual in such work, it is practical to include the (unknown) contribution from generation of single neutrons and photons, such as by ða; nÞ processes for the neutrons, into the moments related to spontaneous fission, thus creating a source moment with a clear dependence on the different types of source events. Inclusion of single neutron generating processes in the calculations has long been applied. However there is a need for introducing a similar correction for gamma photons, since they also can be produced either in bunches (in the spontaneous fission process) or as singular gamma photons, for example in the same
63
ða; nÞ-reactions which lead to the emission of single neutrons, in radiative capture of neutrons, and finally from independent ‘‘background’’ sources such as beta-decay. The need for accounting for such single photon generating processes was suggested recently by Sanchez [10]. Further, there is also the possibility of producing individual gamma photons through inelastic scattering of the neutrons. Accounting for this possibility as well as for radiative capture is though deferred to later work. To account for the presence of single neutron producing events, one introduces the statistics of the total source events as a weighted average of the two processes of spontaneous fission and ða; nÞ reactions [1,6]. As mentioned earlier, quantities belonging to such a generalized source event will be denoted by a subscript s. Hence, we will use
ns;n ¼
nsf ;n ð1þ ad1;n Þ 1 þ ansf ;1
ð1Þ
as source moments for neutrons. The factor a is defined as
a¼
Qa : Qf nsf ;1
Here, Qf and Qa are the intensities of spontaneous fission and ða; nÞ processes, respectively, and hence a stands for the ratios of the mean neutron production intensity of ða; nÞ and spontaneous fission processes. For gamma photons produced also in connection to ða; nÞ-reactions, the source distribution changes and leads to the following modified source moments:
ms;n ¼
msf ;n þ d1;n ansf ;1 : 1þ ansf ;1
ð2Þ
Here, nsf ;n and msf ;n are the true moments of spontaneous fission (i.e. nuclear constants), whereas ns;n and ms;n are the ones corrected for the inclusion of production of neutrons and gammas by reactions other than fission. The moments relating to induced fission remains unchanged for neutrons and photons (ni;n and mi;n respectively). The second set of moments and random variables concerns the distribution of the total number of generated or detected neutrons and gammas due to one initial source event, with internal ¨ multiplication included (‘‘superfission’’ in Bohnels terminology [1]). These will not be denoted by any lettered subscript, only with a single number, expressing the order of the moment. The purpose of the calculations is to express these latter type of variables with the ones given by the nuclear constants, based on the distributions from spontaneous and induced fission, and the parameter a, describing the (unknown) relative intensity of production of single neutrons. To obtain this relationship we need also the first collision probability p. For simplicity of the description, absorption will be neglected; however, detection efficiency will later be taken into account by the factors en and eg for the neutrons and gamma photons, respectively. Absorption for the gamma photons can actually be included into the efficiency factor eg in an exact way, due to the fact that gamma photons do not take part in the internal multiplication. This fact has the significance that regarding a fissile sample itself (any possible shielding disregarded), the internal absorption in the sample is only significant for the gamma photons but not for the neutrons.
3. Factorial moments Here we only list the various single and mixed factorial moments without any details of the underlying derivation. The principles of derivation through master equations can be found in Refs. [1,4–6].
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being the gamma leakage multiplicity per one initial neutron.1
3.1. Neutrons
m2 ¼
The factorial moments of neutrons are given below
n1 ¼
1p ns;1 ¼ Mns;1 1pni;1
pni;1 o 1
(
) M1 M1 2 ð3ns;2 ni;2 þ ns;1 ni;3 Þ þ 3 ns;1 n2i;2 : ni;1 1 ni;1 1
ð12Þ
m3 ¼ ð4Þ
ð6Þ
In these formulae the factorial moments of the combined source events are used. In order to be able to unfold the sample parameters, including the unknown factor a, one has to re-write these formulae in terms of the factorial moments of spontaneous fission and a. This is easily achieved by using Eq. (1), the results being:
n1 ¼
M ð1 þ ansf ;1 Þ
nsf ;1 ð1 þ aÞ
M2 M1 n2 ¼ nsf ;2 þ nsf ;1 ð1 þ aÞni;2 ð1 þ ansf ;1 Þ ni;1 1 M3 M1 ½3nsf ;2 ni;2 þ nsf ;1 ð1 þ aÞni;3 n3 ¼ nsf ;3 þ ð1 þ ansf ;1 Þ ni;1 1 # 2 M1 nsf ;1 ð1þ aÞn2i;2 : þ3 ni;1 1
ðmsf ;1 þ ansf ;1 Þ nsf ;1 ð1 þ aÞ nsf ;2 nsf ;1 ð1 þ aÞ Mg þ M2 þ g2 ð1 þ ansf ;1 Þ g ð1 þ ansf ;1 Þ ð1 þ ansf ;1 Þ ð1 þ ansf ;1 Þ
and
is called the leakage multiplication and p is the first collision probability. The higher order moments are p M1 ð5Þ n2 ¼ M2 ns;2 þ ns;1 ni;2 ¼ M2 ns;2 þ ns;1 ni;2 ni;1 1 1pni;1
n3 ¼ M3 ns;3 þ
þ2
ð3Þ
where 1p ; M 1pni;1
msf ;2 1þ ansf ;1
ð7Þ
n ð1 þ aÞ Mg þ 3ðmsf ;1 þ ansf ;1 Þ m þ 3msf ;2 sf ;1 ð1þ ansf ;1 Þ sf ;3 ð1 þ ansf ;1 Þ nsf ;2 nsf ;1 ð1þ aÞ M2 þ g2 þ nsf ;3 M3g þ 3nsf ;2 g2 ð1 þ ansf ;1 Þ g ð1 þ ansf ;1 Þ þ nsf ;1 ð1 þ aÞg3 : 1
ð13Þ
In Eqs. (12) and (13) gn are the factorial moments of the photon distribution initiated by a single neutron, which are defined as [6] dn gðzÞ : ð14Þ gn ¼ n dz z ¼ 1 This leads for g2 and g3 to the following expressions: g2 ¼
M1
ni;1 1
fmi;2 þ 2mi;1 ni;1 Mg þ ni;2 M2g g
ð15Þ
and g3 ¼
M1
fm þ 3mi;2 ni;1 Mg þ 3mi;1 ½ni;2 M2g þ ni;1 g2 ni;1 1 i;3 þ ni;3 M3g þ 3ni;2 Mg g2 g:
ð16Þ
ð8Þ
ð9Þ
As discussed in Ref. [9], these formulae are incorrectly given in Ref. [11] and in several other publications. The multiplicity rates found in the literature including Ref. [11] are, however, correct.
Since these do not contain the ns;n and ms;n , these expressions do not change for a compound source compared to a pure spontaneous fission source. From Eqs. (10) and (13) one can note that the ða; nÞ processes affect also the photon equations, in a way analogous to the neutron moment equations, due to single photon emission processes accompanying the ða; nÞ reactions. The appearance of the factor a becomes also highly non-linear, due to its occurring in multiple ways in the process. 3.3. Mixed moments
3.2. Photons The gamma processes accounted for in this paper will be as follows: (a) spontaneous fission, (b) single gamma production in ða; nÞ processes, and (c) independent generation of single gammas from radioactive processes other than fission. Out of these, the first two are needed to be considered already at the level of the factorial moments; the third one, the ‘‘background’’ gamma generation, can be conveniently considered at the level of the multiplicity rates. As mentioned, radiative capture and gamma generation in inelastic scattering will not be considered in the present work. Although these can expected to occur with a low relative frequency, they have the significance that they influence the neutron–gamma correlations. The factorial moments of the photons (which are more complicated due to the simple fact that photons do not selfmultiply, rather they depend on the multiplication of neutrons), are given below. The detailed derivation can be found in Ref. [6]. Again, the moments are re-written in terms of the true fission neutron and gamma photon factorial moments and the factor a by using Eqs. (1) and (2). This leads to the following expressions:
m1 ¼
msf ;1 þ ansf ;1 nsf ;1 ð1 þ aÞ þ Mg ð1 þ ansf ;1 Þ ð1 þ ansf ;1 Þ
ð10Þ
pmi;1 1pni;1
ð11Þ
For the mixed moments the following expressions can be derived: /mlS ¼
ðmsf ;1 þ ansf ;1 Þnsf ;1 ð1 þ aÞ M þ nsf ;2 MMg þ nsf ;1 ð1 þ aÞc1;1 ð1 þ ansf ;1 Þ ð1 þ ansf ;1 Þ 1
ð17Þ /m ðm 1ÞlS ¼
1 ð1þ ansf ;1 Þ
ðmsf ;1 þ ansf ;1 Þ
nsf ;2 nsf ;1 ð1 þ aÞ M2 þ h2 ð1 þ ansf ;1 Þ ð1 þ ansf ;1 Þ
þ nsf ;3 Mg M2 þ nsf ;2 ½2Mc1;1 þ Mg h2 þ nsf ;1 ð1 þ aÞc2;1
ð18Þ and nsf ;2 1 Mþ 2ðms;1 þ ansf ;1 Þ msf ;2 /mlðl1ÞS ¼ ð1 þ ansf ;1 Þ ð1 þ ansf ;1 Þ nsf ;2 nsf ;1 ð1 þ aÞ MMg þ c1;1 þ nsf ;3 MM2g ð1 þ ansf ;1 Þ ð1 þ ansf ;1 Þ þ nsf ;2 ½g2 Mþ 2Mg c1;1 þ nsf ;1 ð1þ aÞc1;2 :
ð19Þ
with Mg
1 In the previous literature this quantity was termed ‘‘gamma leakage multiplication’’. However since gamma photons do not multiply themselves, the term ‘‘multiplicity’’ feels much more justified.
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neutrons per one initial event. This requires the introduction of the detector efficiency and for the first few moments these factorial moments are obtained as
Covariance of neutrons and photons 106 105 Covariance, Cov (ν,μ)
65
n~ 1 ¼ en n1
ð25Þ
104
n~ 2 ¼ e2n n2
ð26Þ
103
n~ 3 ¼ e3n n3 :
ð27Þ
102
We also need to account for the fact that a measured doublet could be the result of detecting a certain number of particles from a higher order multiplet, and that the detection rates Ck , k ¼ 1; 2; 3 are also related to the total source intensity. Using Ck as the notation for the k-th order multiplet (such that C1 ¼ S, C2 ¼ D, etc.) and the total neutron source rate Qs Qf þQa it follows that
X n~ n! n~ Ck ¼ Qs ð28Þ ¼ Qs ekn PðnÞ ¼ Qs k : k! k!ðnkÞ! k n
101 100 10−1 10−2 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Reaction probability (p)
Qs ¼ F þQa ¼ Fð1 þ ansf ;1 Þ:
Fig. 1. Covariance between neutrons and photons.
In the above (as found in Ref. [6]) one has m ni;1 Mþ ni;2 MMg c1;1 ¼ p i;1 1pni;1 c2;1 ¼
S ¼ F en ð1þ ansf ;1 Þ ð20Þ
p fm ½n M2 þ ni;1 h2 þ ni;3 M2 Mg þ ni;2 ½h2 Mg þ 2Mc1;1 g 1pni;1 r;1 i;2 p fm n Mþ 2mi;1 ½ni;2 MMg þ ni;1 c1;1 þ ni;3 MM2g 1pni;1 i;2 i;1 þ ni;2 ½2Mg c1;1 þ Mg2 g:
ð22Þ
Here h2 is the second factorial moment of the total number of neutrons generated in the sample, internal multiplication included, by one single neutron. This and other moments can be found in Ref. [4] M1 n M2 : h2 ¼ ni;1 1 i;2
ð29Þ
In the case of singles for neutrons the following expression is derived as
ð21Þ c1;2 ¼
Using the a-factor, the source factor can be expressed as
ð23Þ
One reason for finding and using also the joint moments can be illustrated by the covariance between neutrons and photons. As shown in Fig. 1 there is a very strong relationship between the two particle types, for increasing values of the probability to induce fission p. ðnsf ;1 ð1þ aÞÞ 1 ðmsf ;1 þ ansf ;1 Þ Mþ nsf ;2 MMg Covfm ; lg ¼ ð1 þ ansf ;1 Þ ð1 þ ansf ;1 Þ M1 fmi;1 ni;1 Mþ ni;2 MMg g : ð24Þ þ nsf ;1 ð1þ aÞ ni;1 1
4. Multiplicity detection rates The measured quantities are the multiplicity rates. To convert the factorial moments of a single source event into detection rates of multiplicities, one has to account for the intensity of the source events and the detection efficiency. The effect of the finite measurement gate time in multiple coincidence measurements, quantified with the relative gate width factors as described in Ref. [11], will be omitted here. To find the measurable quantities such as singles doubles, etc., one needs to first find the factorial moments n~ k of the detected
Mnsf ;1 ð1 þ aÞ ¼ F en Mnsf ;1 ð1þ aÞ: ð1 þ ansf ;1 Þ
ð30Þ
Note how the scaling factor between the fission source and the total source intensity cancels out in the expression for the measurable singles. In a similar way doubles and triples can be derived as F e2 M2 M1 ð31Þ D¼ n nsf ;2 þ nsf ;1 ð1 þ aÞni;2 2 ni;1 1 T¼
F e3n M3 M1 ½3nsf ;2 ni;2 þ nsf ;1 ð1 þ aÞni;3 nsf ;3 þ 6 ni;1 1 ) M1 2 2 nsf ;1 ð1 þ aÞni;2 : þ3 ni;1 1
ð32Þ
These are the quantities one measures in multiplicity counters. It is these expressions that serve as the basis for the different approaches to find the various unknown parameters, as described in Ref. [3]. Most commonly one assumes the neutron detector efficiency en to be known, and solve the equations for fission rate (mass), F, leakage multiplication, M, and a the relative contribution from single-neutron sources. In the case of photons, the moments are considerably more complicated due to accounting for both neutrons and photons. It is still possible to derive equations for the measurable quantities of singles, doubles and triples, in a manner similar to that of neutrons. As mentioned earlier, at the level of the multiplicity rates, one has to account for the single photon generation via independent processes. To this end the ratio of the single photon source strength, Qg , to the neutrons source strength, Qs , needs to be introduced. A single calculation yields the gamma singles doubles and triples expressed as Sg ¼ ½gFð1 þ ansf ;1 Þ þ Fð1 þ ansf ;1 Þm~ 1 Dg ¼ Fð1 þ ansf ;1 Þ Tg ¼ Fð1 þ ansf ;1 Þ
m~ 2
ð34Þ
2
m~ 3 3!
ð33Þ
:
ð35Þ
Also here we have used the tilde notation for the detected moments, i.e. accounting for a detector efficiency, eg in the case of
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and
gamma photons, i.e. one has k
m~ k ¼ eg mk :
ð36Þ
For the sake of the discussion, we need to substitute Eqs. (36) and (10)–(13) into the above, to obtain Sg ¼ F eg ½gð1þ ansf ;1 Þ þ Mg fmsf ;1 þ ansf ;1 þ nsf ;1 ð1 þ aÞg 2
Dg ¼
eg F 2
msf ;2 þ 2ðmsf ;1 þ ansf ;1 Þ
ð37Þ
nsf ;1 ð1 þ aÞ Mg þ nsf ;2 M2g þ nsf ;1 ð1þ aÞg2 ð1 þ ansf ;1 Þ
e3g F n ð1 þ aÞ Mg þ 3ðmsf ;1 þ ansf ;1 Þ msf ;3 þ 3msf ;2 sf ;1 6 ð1 þ ansf ;1 Þ nsf ;2 nsf ;1 ð1 þ aÞ M2g þ g2 þ nsf ;3 M3g þ 3nsf ;2 g2 þ nsf ;1 ð1 þ aÞg3 : ð1 þ ansf ;1 Þ ð1 þ ansf ;1 Þ ð39Þ
The above shows that in these equations, there are 5 or 6 independent parameters, depending on whether the neutron leakage multiplication, M, and the gamma leakage multiplicity, Mg , are considered as one or two unknowns. Namely, these both are different, but unique functions of the first collision probability p. The other four are the fission rate F, the factors a and g and the gamma detector efficiency eg . In either case, the gamma multiplicity equations alone are insufficient for the determination of unknown sample parameters. Together with the neutron multiplicity rates, with the inclusion of the neutron detection efficiency en , there are two possibilities. If M and Mg are considered as two independent unknowns (which might enhance the accuracy of the determination of the fission rate), the only possibility to unfold algorithmically six unknowns from six equations is to consider one of the detection efficiencies known. If both are considered as known, then the equations become overdetermined. If only the neutron efficiency is considered to be known, then an analytical solution is possible. This is because then the neutron multiplicity rates decouple from the gamma multiplicities, and can be solved exactly as in the traditional case for F, M and a. In possession of these parameters, the gamma equations for the remaining parameters Mg , g and eg can be solved analytically, because it does not matter that they contain highly nonlinear versions of M and a. But this also amounts to saying that inclusion of the gamma multiplicity rates does not improve the unfolding of the most important parameter, F; in effect the gamma multiplicity rate equations can only be used to determine the gamma process parameters, which do not include the sample fission rate. If M and Mg are both expressed in terms of the first collision probability p, then the determination of this and other unknown parameters, including the detector efficiencies, is possible. However, the coupling between the neutron and gamma multiplicity equations is weak, with an ensuing large statistical uncertainty of the results as functions of measurement inaccuracies. A much stronger coupling is introduced into the equation system if also the mixed multiplicities (one double and two triples) are used. These are given as follows: Dng ¼ en eg F
ðmsf ;1 þ ansf ;1 Þnsf ;1 ð1þ aÞ M þ nsf ;2 MMg þ nsf ;1 ð1 þ aÞc1;1 ð1 þ ansf ;1 Þ ð40Þ
Tnng ¼
nsf ;2 nsf ;1 ð1 þ aÞ F ðmsf ;1 þ ansf ;1 Þ M2 þ h2 2 ð1 þ ansf ;1 Þ ð1 þ ansf ;1 Þ o þ nsf ;3 Mg M2 þ nsf ;2 ½2Mc1;1 þ Mg h2 þ nsf ;1 ð1þ aÞc2;1
en e2g F msf ;2
nsf ;2 nsf ;2 Mþ 2ðms;1 þ ansf ;1 Þ MMg 2 ð1 þ ansf ;1 Þ ð1 þ ansf ;1 Þ nsf ;1 ð1 þ aÞ þ c1;1 þ nsf ;3 MM2g þ nsf ;2 ½g2 M þ 2Mg c1;1 ð1 þ ansf ;1 Þ þ nsf ;1 ð1 þ aÞc1;2 :
ð42Þ
ð38Þ Tg ¼
Tngg ¼
e2n eg
ð41Þ
Using the neutron, gamma and mixed multiplicity rates, all unknown parameters, F, M, Mg , a, g, en and eg can be determined by assuming the neutron leakage multiplication and the gamma leakage multiplicity as two independent unknowns. The number of equations still exceeds the number of unknowns, and through the mixed multiplicity rates, a sufficient strong coupling between the equations is established. From these equations the searched parameters can be determined with a non-linear least squares non-parametric fitting to the measured values. If M and Mg are not considered as two independent unknowns, rather as functions of the first collision probability p, then the measure of redundancy is even higher. The performance of various cases of the unfolding of the unknowns will be quantitatively investigated and discussed below.
5. Application As it was discussed in Section 4, unlike for the case of using only the neutron multiplicities, the complexity of the expressions for the gamma photons prevents the possibility of using analytical inversion of the multiplicity rate expressions. Hence we propose the use of artificial neural network (ANN) techniques for the unfolding of sample parameters from the measured multiplicity rates. As a first application, the use of ANNs can be tested on the known case of determining three unknown sample parameters from the three neutron multiplicities, in which case the unfolding can be performed also analytically. However, the use of ANNs offers some advantages already for this relatively simple case. Namely, the analytical inversion is only possible as long as only three unknowns are attempted to be retrieved from the three multiplicity rates. This has the effect that normally the neutron efficiency needs to be known in advance. With ANN techniques, there is a larger flexibility, since ANNs can utilize the rich information in the non-linearity of the expressions to unfold more parameters than the number of expressions. Hence there is a chance that in addition to the usual three parameters, also the detector efficiency can be retrieved. First we will discuss the test of the traditional case of determining three unknowns from three neutron factorial moments. Results are obtained after the initial training of the network was completed, the training data were generated for parameters corresponding to samples ranging from a few tens of grams to the kilogram range. The trained network was tested by further sample vectors generated the same way as the training set (approximately 20% of all the input data were used for testing and validation for the different networks). The network used for this task is depicted in Fig. 2, it had two hidden layers of 25 and 15 nodes respectively. Results show that the parameters F, a and p can be evaluated with a relative error smaller than about 0.002% (Table 1), whereas the mean errors and standard deviations are even smaller. The parameters were unfolded from the three neutron rates only, which is the recent routine analytical procedure, and which serves as a good indicator that the ANNs are consistent with methods in the case of using neutron assay only.
ARTICLE IN PRESS A. Enqvist et al. / Nuclear Instruments and Methods in Physics Research A 615 (2010) 62–69
Tansig hidden layer
Input multiplicity rates (3)
Tansig hidden layer
Linear output layer
67
Output parameters (3)
(15 nodes)
(25 nodes)
Fig. 2. Schematic outline of the artificial neural network for the case of using the neutron rates only and determining three output parameters.
10
Table 1 Training results of ANN, using the neutron equations to simulating large plutonium samples in the kg range.
Probability density estimates
2
a
p
0.0001 1.98e 9 7.56e 6 1.28e 6 9.34e 6
0.0021 4.14e 7 2.11e 4 7.15e 6 1.42e 4
0.0001 4.96e 9 3.24e 5 2.56e 6 3.29e 5
[S, D]
6
4
2
x 104 fission rate alpha ratio p
1.5
Observations
Max. abs. rel. error (%) Mean error of training (%) Standard deviation of training (%) Mean error of test data (%) Standard deviation of test data (%)
[S, D/S] 8
Fission rate (F)
0 0
0.2
0.4
0.6
0.8
1
Correlation coefficients Fig. 4. Histograms of the correlation coefficients by using bootstrap procedure.
1
0.5
0 0
20
40 60 100 equally spaced points
80
100
Fig. 3. Probability density estimates of the relative errors of the unfolded parameters for the case with three neutron rates.
In Fig. 3 one can see the probability density estimate for the relative errors of the unfolded parameters. The estimates are based on using a kernel smoothing function and an associated bandwidth to estimate the density. Since the parameters have different physical units and domains, the x-axis was re-scaled for all three parameters in terms of sampling bins such that the estimated distributions are directly comparable for their widths. The figure gives a visual illustration that the fission rate can be unfolded with the smallest standard deviation (largest accuracy), compared to the factor a and probability p from the three neutron multiplicity rates. This behaviour is beneficial since the fission rate is the most important parameter to decide accurately, being the basis for determining the sample mass. To further improve the accuracy of the ANN, one can choose how to organize the output parameters. A quick investigation of the dependence in the formulae, reveals that the dependence of the accuracy of the results on M and ð1aÞ is more straightforward than that on p and a. The final calculation needed to get p from M, can
then be done manually with very little effort, while decreasing the time and computing power needed to train the ANN. Various strategies can also be used for the input parameters. Instead of using S, D, T for the neutron rates, one can choose the parameters: S, D=S, T=S, which proves to be more independent. In order to investigate if the correlation coefficients are statistically significant, we have applied a bootstrap procedure for two first inputs in combinations [S,D] and [S, D/S]. A histogram of the results obtained after resampling both input vectors 100 times and computing the correlation between the inputs on each sample is given in Fig. 4. This shows a strong evidence that we can reduce the value of the correlation coefficient between inputs such as S and D from approximately 0.9 to 0.2 when we use the second configurations of the inputs. This behaviour is evident also for the gamma and mixed rates where the correlation between parameters is reduced significantly when using the following input parameters: S, D/S, T/S, Sg , Dg =Sg , Tg =Sg , Dng , Tnng =Dng , Tngg =Dng . When using also the gamma and mixed rates, the results indicate that more parameters such as the gamma ratio, g, and the gamma detector efficiency, eg , can be unfolded in addition to the parameters that were unfolded from the neutron rates only. Due to the complicated form of the dependence of the detection rates on the sample parameters, the accuracy is reduced when using all nine inputs and unfolding more parameters. This is valid even though the ANN was made more complex in the case of using all input rates and then consisted of two hidden layers with 35 and 15 nodes respectively. This layout was used to obtain the results in Table 2 and Fig. 5. To make a deeper analysis of the sensitivity of the ANN when trying to apply it to real measured data, we perturbed the analytically obtained ‘‘clean’’ inputs with a random noise at a
ARTICLE IN PRESS 68
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Table 2 Accuracy of ANNs when unfolding sample parameters from an over determined system. Maximal variations (%)
Input þ noise
Without noise S þ 5% noise D þ 5% noise T þ 5% noise Sg þ 5% noise Dg þ 5% noise Tg þ 5% noise Dng þ 5% noise Tnng þ 5% noise Tngg þ 5% noise
a
p
F
g
eg
3:080e þ 00 2:978e þ 00 5:825e þ 00 6:341e þ 00 9:553e þ 00 3:765e þ 01 1:808e þ 01 2:718e þ 01 1:136e þ 01 3:883e þ 00 1:6766e þ 02
1.056e 01 2.514e 01 7.804e 01 8.593e 01 4.023e 01 1:536e þ 00 1:357e þ 00 8.461e 01 6.211e 01 1.063e 01 4:494e þ 00
4.704e 02 1.255e 01 1.066e 01 1.867e 01 1.442e 01 3.623e 01 2.693e 01 1.318e 01 1.658e 01 6.883e 02 1:851e þ 00
3.100e 01 1:435e þ 00 1:792e þ 00 3:669e þ 00 2:439e þ 00 1:596e þ 01 1:460e þ 01 3:291e þ 00 3:601e þ 00 3.807e 01 2:786e þ 01
2.200e 01 5.924e 01 9.448e 01 1:072e þ 00 6.930e 01 1:217e þ 01 7:758e þ 00 1:867e þ 00 1:205e þ 00 2.564e 01 4:012e þ 01
3.2457e 003
1.9557e 002
1.8702e 001
2.7251e 001
2.3318e 001
2:3705e þ 000
2:9004e þ 000
Mean error of test data (%) 1:6581e þ 000
All þ 5% noise
Standard deviation of test data (%) 1:5577e þ 001
5.2392e 001
frequency
3000 2000 1000 0 −20
−15
−10
−5
0
5
10
15
20
25
2
2.5
relative errors (%) for gamma the efficiency, εγ frequency
2000 1000 0 −2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
Although the maximum errors might seem large, one should observe that the average error is much smaller, which is demonstrated in the histograms of a few of the unfolded parameters (Fig. 5). It should be pointed out that the formulae used for the detection rates are all based on the point-model and therefore are not exact. However in the field of multiplicity measurements they have proved to be accurate enough, and thus we have based the extensions to gamma and mixed rates on the same formalism. A more detailed quantitative and qualitative analysis of the performance of the ANN based unfolding procedure is given in a companion paper [12].
6. Conclusions
relative errors (%) for the p parameter frequency
2000 1000 0 −1
−0.8 −0.6 −0.4 −0.2
0
0.2
0.4
0.6
0.8
1
relative errors (%) for fission rate, F Fig. 5. Histogram showing the distribution of the errors of the predicted parameters: F, p and eg , when 5% random noise have been added to the input data.
level of 5% in magnitude to each rate individually as well as to all rates at once. This is to simulate the fact that measured detection rates will always have some statistical uncertainties, and it is important that the neural networks can handle inputs that are not perfectly acquired. Table 2 presents the maximal variations of the unfolded parameters. Some of the unfolded parameters show much larger variation compared to others. However, the main parameter of interest is the fission rate, F, which relates to the sample mass, and it shows very little variation also when the input data contain noise. This fact is naturally very promising. The table also lists the mean error and standard deviation in the last case of 5% noise on all data to give additional insight to how significant the errors are expected to be when using realistic data.
The present paper shows that by taking all possible auto- and cross-factorial moments of the neutron and gamma counts into account, one has nine expressions which are functions of six independent parameters. The expressions for higher order factorial moments become increasingly long, and show a complicated dependence on some of the parameters. The above factorial moments were converted into detection rates, which was hitherto done for neutrons only. It has also been demonstrated that just as in the case of single neutron emission through (a, n) reactions, one can also phenomenologically account for single photon emission, both in connection to the (a, n) reactions, but also for photons emitted in processes not related to neutron generations, with the help of a gamma parameter, g, defined as the ratio of the rate of single gamma emissions to the rate of spontaneous fission. It was shown that taking only the neutron and gamma detection rates, the complexity of the unfolding does not increase, but the accuracy and effectiveness of the unfolding does not get improved compared to using neutron multiplicities only. Accounting for also the mixed detection rates, especially when all nine multiplicity rates are used, theoretically increases the performance of the unfolding (number of parameters unfolded, achievable accuracy, etc.) but at the expense of the increased complexity of the unfolding problem.
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Due to the highly non-linear character of the formulae when also the mixed moments are used, artificial neural networks were suggested for handling the task of unfolding the sample parameters, and especially the fission rate which is closely linked to the sample mass. By using the equations of the multiplicity rates to generate training data for the networks, we were able to prove the applicability of the approach. The results shown are very promising and of good accuracy. The training and performance of ANNs using all moments for both neutrons and photons is computationally more demanding, but still within manageable range. It has also been demonstrated that the ANN approach can be used also as a replacement for pure neutron assay without loss off accuracy compared to the conventional unfolding of three sample parameters. Since real measurements data will always contain statistical uncertainties the network was tested with data that had random noise added to it, to simulate more realistic input data. The ANN still showed very good accuracy when predicting the sample parameters, where the largest uncertainties were found not in the fission rate, but rather in other parameters such as the alpha ratio.
69
Acknowledgement This work was supported by the Swedish Radiation Safety Authority (SSM). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
[12]
¨ K. Bohnel, Nucl. Sci. Eng. 90 (1985) 75. W. Hage, D.M. Cifarelli, Nucl. Instr. and Meth. A 236 (1985) 165. D.M. Cifarelli, W. Hage, Nucl. Instr. and Meth. A 251 (1986) 550. I. Pa´zsit, S.A. Pozzi, Nucl. Instr. and Meth. A 555 (1–2) (2005) 340. A. Enqvist, I. Pa´zsit, S.A. Pozzi, Nucl. Instr. and Meth. A 566 (2006) 598. I. Pa´zsit, L. Pa´l, Neutron Fluctuations—A Treatise on the Physics of Branching Processes, Elsevier Science Ltd., London, New York, Tokyo, 2008. A. Enqvist, M. Flaska, S.A. Pozzi, Nucl. Instr. and Meth. A 595 (2008) 426. S.D. Clarke, M. Flaska, S.A. Pozzi, P. Peerani, Nucl. Instr. and Meth. A 599 (2009) 221. I. Pa´zsit, A. Enqvist, L. Pa´l, Nucl. Instr. and Meth. A 603 (2009) 541. R. Sanchez, CEA, Personal communication, 2008. N. Ensslin, W.C. Harker, M.S. Krick, D.G. Langner, M.M. Pickrell, J.E. Stewart, Application Guide to Neutron Multiplicity Counting, Los Alamos Report LA13422-M, 1998. S. Avdic, A. Enqvist, I. Pa´zsit, ESARDA Bull. 43 (2009) 21.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Sample characterization using both neutron and gamma multiplicities Andreas Enqvist a,, Imre Pa´zsit a, Senada Avdic a,b a b
Department of Nuclear Engineering, Chalmers University of Technology, SE-412 96 G¨ oteborg, Sweden Faculty of Science, Department of Physics, University of Tuzla, 75000 Tuzla, Bosnia and Herzegovina
a r t i c l e in f o
a b s t r a c t
Article history: Received 8 December 2009 Received in revised form 28 December 2009 Accepted 8 January 2010 Available online 18 January 2010
Formulas for multiplicity counting rates (singles, doubles, etc.), used for the unfolding of parameters of an unknown sample, can be derived from those for the corresponding factorial moments. So far such rates were derived only for neutrons. The novelty of this paper is related to the derivation of the individual gamma and mixed neutron–gamma detection rates as well as to investigation of the possibilities and actual algorithms for the sample parameter unfolding. Taking the individual gamma and mixed neutron–gamma moments up to third order, together with the neutron moments, there will be nine auto- and cross-factorial moments and corresponding multiplicity rates, as well as a larger number of unknowns than for pure neutron detection. The total number of measurable multiplicities exceeds the number of unknowns, but on the other hand the structure of the additional equations is substantially more complicated than that of the neutron moments. Since an analytical inversion of the highly non-linear system of over-determined equations is not possible, the use of artificial neural networks (ANNs) is suggested, which can handle both the nonlinearity and the redundance in the measured quantities in an effective and accurate way. The use of ANNs is demonstrated with good results on the unfolding of various combination of multiplicities for certain combinations of the unknown parameters, including the sample fission rate, the leakage multiplication and the a and g ratios. & 2010 Elsevier B.V. All rights reserved.
Keywords: Nuclear safeguards Neutron and gamma multiplicities Joint moments Materials control and accounting Neural networks
1. Introduction Multiplicity detection rates, based on higher order factorial moments of the neutron counts from an unknown sample, can be used to determine sample parameters [1–3]. The factorial moments here refer to those of the total number of neutrons generated in the sample by one initial source event (spontaneous fission or ða; nÞ reaction). Due to internal multiplication through induced fission, the probability distribution of the total number of generated neutrons will deviate from that by the initial source event, the deviation being a function of the sample mass via the first collision probability of the initial neutrons. From here the detection rates can be calculated by accounting for the initial source event rates, which further corroborates the dependence of the multiplicity rates on the sample mass since source event rate (sample fission rate) depends linearly on the sample mass. This gives a possibility to determine the sample mass. Multiplicity rate measurements of the first three multiples enable the recovery of three unknowns, which are usually taken as the sample leakage multiplication (related to the first collision
Corresponding author. Tel.: + 46 31 772 30 86; fax: + 46 31 772 30 79.
E-mail addresses: [email protected] (A. Enqvist), [email protected] (I. Pa´zsit), [email protected] (S. Avdic). 0168-9002/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2010.01.022
probability, itself being a function of sample mass), the ratio a of the intensity of neutron production via (a; nÞ reactions to spontaneous fission, and the spontaneous fission rate, the latter being the most important parameter. This approach is forced to leave the detector efficiency undetermined which needs to be predetermined experimentally. Alternative approaches, such as assuming the sample multiplication to be known, could allow for the detector efficiency to be unfolded. Recently it was suggested that in addition to neutron multiplicity counting, gamma multiplicities be also taken into account [4–6]. The motivation for using gamma counting is manifold: higher gamma multiplicity per fission, larger penetration through most of the strong neutron absorbers, and the relatively easy detection by organic scintillation detectors, whose potential use in nuclear safeguards have been recently investigated [7,8]. The goal is still the same, i.e. to determine the above factors, plus the further unknowns introduced, such as the gamma leakage multiplicity, the ratio g of single gamma to fission gamma intensity and gamma detector efficiency. These can though be handled since three neutron and three gamma multiplicities can be measured automatically, so one has still as many unknowns as measured quantities. In addition, there exists the further possibility of using the joint moments of the neutron and gamma counts, which supply further independent measured data to determine still the same
ARTICLE IN PRESS A. Enqvist et al. / Nuclear Instruments and Methods in Physics Research A 615 (2010) 62–69
number of unknowns. Accounting also for the joint moments up to third order, there are altogether nine factorial moments. Hence the problem becomes overdetermined. The multiplicity expressions depend highly non-linearly on most of the searched parameters and hence constitute a system of non-linear equations of order up to 9. The consequences of this fact on the possibility of unfolding will be discussed in some detail in the paper. To make maximal use of the redundant information from the measurement when also the joint moments are used, the unfolding of the parameters has to be performed by least-square type unfolding methods. Actually, there is a conceptually simple non-parametric unfolding method for such a purpose, the artificial neural networks (ANNs), whose use will be demonstrated here. In this paper we give the definitions of the quantities used and list all nine factorial moments. To give insight into the information contained in the joint moments, the dependence of the lowest order joint neutron–gamma moment on the non-leakage probability will be given quantitatively. In addition to the factorial moments, the multiplicity detection rates for the gamma photons and for mixed neutron–gamma detections will also be given. These have not been given previously. The derivation of converting the factorial moments into multiplicity detection rates follows the method described in a recent clarifying note on the relationship between factorial moments and the single, double and triple coincidence rates [9]. This will clearly show the relation between the measurable multiplicity rates and the factorial moments, when accounting for both multiple emission (spontaneous fission) and single emission ða; nÞ source events. In the last section a test and an analysis of the unfolding procedure is given for various combinations of the multiplicity rates and the parameters taken as unknown.
2. Definitions The following definitions and conventions will be used. Random variables and their moments referring to neutrons will be denoted by n, and those for gamma photons by m. Variables referring to spontaneous fission will have a subscript sf, and those referring to induced fission a subscript i. For the factorial moments, there will always be a second index, giving the order of the moment. Hence, nsf ;2 will stand for /nðn1ÞS in case of spontaneous fission. In addition, we will distinguish between two sets of variables for both neutrons and photons, depending on whether they belong to a source event (all processes, i.e. spontaneous fission and ða; nÞ processes included), or to the total number of generated neutrons, which accounts for the internal multiplication (superfission). The parameters belonging to the first set will be written with a subscript indication, such as ns;2 ; ni;3 (the subscript s alluding to ‘‘source’’), whereas those of the second set will be denoted with just a numerical subscript indicating their order such as n1 ; m3 . The factorial moments corresponding to the distribution of neutrons or gammas emitted in fission, whether induced or spontaneous, are nuclear constants and are known in advance. However, as is usual in such work, it is practical to include the (unknown) contribution from generation of single neutrons and photons, such as by ða; nÞ processes for the neutrons, into the moments related to spontaneous fission, thus creating a source moment with a clear dependence on the different types of source events. Inclusion of single neutron generating processes in the calculations has long been applied. However there is a need for introducing a similar correction for gamma photons, since they also can be produced either in bunches (in the spontaneous fission process) or as singular gamma photons, for example in the same
63
ða; nÞ-reactions which lead to the emission of single neutrons, in radiative capture of neutrons, and finally from independent ‘‘background’’ sources such as beta-decay. The need for accounting for such single photon generating processes was suggested recently by Sanchez [10]. Further, there is also the possibility of producing individual gamma photons through inelastic scattering of the neutrons. Accounting for this possibility as well as for radiative capture is though deferred to later work. To account for the presence of single neutron producing events, one introduces the statistics of the total source events as a weighted average of the two processes of spontaneous fission and ða; nÞ reactions [1,6]. As mentioned earlier, quantities belonging to such a generalized source event will be denoted by a subscript s. Hence, we will use
ns;n ¼
nsf ;n ð1þ ad1;n Þ 1 þ ansf ;1
ð1Þ
as source moments for neutrons. The factor a is defined as
a¼
Qa : Qf nsf ;1
Here, Qf and Qa are the intensities of spontaneous fission and ða; nÞ processes, respectively, and hence a stands for the ratios of the mean neutron production intensity of ða; nÞ and spontaneous fission processes. For gamma photons produced also in connection to ða; nÞ-reactions, the source distribution changes and leads to the following modified source moments:
ms;n ¼
msf ;n þ d1;n ansf ;1 : 1þ ansf ;1
ð2Þ
Here, nsf ;n and msf ;n are the true moments of spontaneous fission (i.e. nuclear constants), whereas ns;n and ms;n are the ones corrected for the inclusion of production of neutrons and gammas by reactions other than fission. The moments relating to induced fission remains unchanged for neutrons and photons (ni;n and mi;n respectively). The second set of moments and random variables concerns the distribution of the total number of generated or detected neutrons and gammas due to one initial source event, with internal ¨ multiplication included (‘‘superfission’’ in Bohnels terminology [1]). These will not be denoted by any lettered subscript, only with a single number, expressing the order of the moment. The purpose of the calculations is to express these latter type of variables with the ones given by the nuclear constants, based on the distributions from spontaneous and induced fission, and the parameter a, describing the (unknown) relative intensity of production of single neutrons. To obtain this relationship we need also the first collision probability p. For simplicity of the description, absorption will be neglected; however, detection efficiency will later be taken into account by the factors en and eg for the neutrons and gamma photons, respectively. Absorption for the gamma photons can actually be included into the efficiency factor eg in an exact way, due to the fact that gamma photons do not take part in the internal multiplication. This fact has the significance that regarding a fissile sample itself (any possible shielding disregarded), the internal absorption in the sample is only significant for the gamma photons but not for the neutrons.
3. Factorial moments Here we only list the various single and mixed factorial moments without any details of the underlying derivation. The principles of derivation through master equations can be found in Refs. [1,4–6].
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being the gamma leakage multiplicity per one initial neutron.1
3.1. Neutrons
m2 ¼
The factorial moments of neutrons are given below
n1 ¼
1p ns;1 ¼ Mns;1 1pni;1
pni;1 o 1
(
) M1 M1 2 ð3ns;2 ni;2 þ ns;1 ni;3 Þ þ 3 ns;1 n2i;2 : ni;1 1 ni;1 1
ð12Þ
m3 ¼ ð4Þ
ð6Þ
In these formulae the factorial moments of the combined source events are used. In order to be able to unfold the sample parameters, including the unknown factor a, one has to re-write these formulae in terms of the factorial moments of spontaneous fission and a. This is easily achieved by using Eq. (1), the results being:
n1 ¼
M ð1 þ ansf ;1 Þ
nsf ;1 ð1 þ aÞ
M2 M1 n2 ¼ nsf ;2 þ nsf ;1 ð1 þ aÞni;2 ð1 þ ansf ;1 Þ ni;1 1 M3 M1 ½3nsf ;2 ni;2 þ nsf ;1 ð1 þ aÞni;3 n3 ¼ nsf ;3 þ ð1 þ ansf ;1 Þ ni;1 1 # 2 M1 nsf ;1 ð1þ aÞn2i;2 : þ3 ni;1 1
ðmsf ;1 þ ansf ;1 Þ nsf ;1 ð1 þ aÞ nsf ;2 nsf ;1 ð1 þ aÞ Mg þ M2 þ g2 ð1 þ ansf ;1 Þ g ð1 þ ansf ;1 Þ ð1 þ ansf ;1 Þ ð1 þ ansf ;1 Þ
and
is called the leakage multiplication and p is the first collision probability. The higher order moments are p M1 ð5Þ n2 ¼ M2 ns;2 þ ns;1 ni;2 ¼ M2 ns;2 þ ns;1 ni;2 ni;1 1 1pni;1
n3 ¼ M3 ns;3 þ
þ2
ð3Þ
where 1p ; M 1pni;1
msf ;2 1þ ansf ;1
ð7Þ
n ð1 þ aÞ Mg þ 3ðmsf ;1 þ ansf ;1 Þ m þ 3msf ;2 sf ;1 ð1þ ansf ;1 Þ sf ;3 ð1 þ ansf ;1 Þ nsf ;2 nsf ;1 ð1þ aÞ M2 þ g2 þ nsf ;3 M3g þ 3nsf ;2 g2 ð1 þ ansf ;1 Þ g ð1 þ ansf ;1 Þ þ nsf ;1 ð1 þ aÞg3 : 1
ð13Þ
In Eqs. (12) and (13) gn are the factorial moments of the photon distribution initiated by a single neutron, which are defined as [6] dn gðzÞ : ð14Þ gn ¼ n dz z ¼ 1 This leads for g2 and g3 to the following expressions: g2 ¼
M1
ni;1 1
fmi;2 þ 2mi;1 ni;1 Mg þ ni;2 M2g g
ð15Þ
and g3 ¼
M1
fm þ 3mi;2 ni;1 Mg þ 3mi;1 ½ni;2 M2g þ ni;1 g2 ni;1 1 i;3 þ ni;3 M3g þ 3ni;2 Mg g2 g:
ð16Þ
ð8Þ
ð9Þ
As discussed in Ref. [9], these formulae are incorrectly given in Ref. [11] and in several other publications. The multiplicity rates found in the literature including Ref. [11] are, however, correct.
Since these do not contain the ns;n and ms;n , these expressions do not change for a compound source compared to a pure spontaneous fission source. From Eqs. (10) and (13) one can note that the ða; nÞ processes affect also the photon equations, in a way analogous to the neutron moment equations, due to single photon emission processes accompanying the ða; nÞ reactions. The appearance of the factor a becomes also highly non-linear, due to its occurring in multiple ways in the process. 3.3. Mixed moments
3.2. Photons The gamma processes accounted for in this paper will be as follows: (a) spontaneous fission, (b) single gamma production in ða; nÞ processes, and (c) independent generation of single gammas from radioactive processes other than fission. Out of these, the first two are needed to be considered already at the level of the factorial moments; the third one, the ‘‘background’’ gamma generation, can be conveniently considered at the level of the multiplicity rates. As mentioned, radiative capture and gamma generation in inelastic scattering will not be considered in the present work. Although these can expected to occur with a low relative frequency, they have the significance that they influence the neutron–gamma correlations. The factorial moments of the photons (which are more complicated due to the simple fact that photons do not selfmultiply, rather they depend on the multiplication of neutrons), are given below. The detailed derivation can be found in Ref. [6]. Again, the moments are re-written in terms of the true fission neutron and gamma photon factorial moments and the factor a by using Eqs. (1) and (2). This leads to the following expressions:
m1 ¼
msf ;1 þ ansf ;1 nsf ;1 ð1 þ aÞ þ Mg ð1 þ ansf ;1 Þ ð1 þ ansf ;1 Þ
ð10Þ
pmi;1 1pni;1
ð11Þ
For the mixed moments the following expressions can be derived: /mlS ¼
ðmsf ;1 þ ansf ;1 Þnsf ;1 ð1 þ aÞ M þ nsf ;2 MMg þ nsf ;1 ð1 þ aÞc1;1 ð1 þ ansf ;1 Þ ð1 þ ansf ;1 Þ 1
ð17Þ /m ðm 1ÞlS ¼
1 ð1þ ansf ;1 Þ
ðmsf ;1 þ ansf ;1 Þ
nsf ;2 nsf ;1 ð1 þ aÞ M2 þ h2 ð1 þ ansf ;1 Þ ð1 þ ansf ;1 Þ
þ nsf ;3 Mg M2 þ nsf ;2 ½2Mc1;1 þ Mg h2 þ nsf ;1 ð1 þ aÞc2;1
ð18Þ and nsf ;2 1 Mþ 2ðms;1 þ ansf ;1 Þ msf ;2 /mlðl1ÞS ¼ ð1 þ ansf ;1 Þ ð1 þ ansf ;1 Þ nsf ;2 nsf ;1 ð1 þ aÞ MMg þ c1;1 þ nsf ;3 MM2g ð1 þ ansf ;1 Þ ð1 þ ansf ;1 Þ þ nsf ;2 ½g2 Mþ 2Mg c1;1 þ nsf ;1 ð1þ aÞc1;2 :
ð19Þ
with Mg
1 In the previous literature this quantity was termed ‘‘gamma leakage multiplication’’. However since gamma photons do not multiply themselves, the term ‘‘multiplicity’’ feels much more justified.
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neutrons per one initial event. This requires the introduction of the detector efficiency and for the first few moments these factorial moments are obtained as
Covariance of neutrons and photons 106 105 Covariance, Cov (ν,μ)
65
n~ 1 ¼ en n1
ð25Þ
104
n~ 2 ¼ e2n n2
ð26Þ
103
n~ 3 ¼ e3n n3 :
ð27Þ
102
We also need to account for the fact that a measured doublet could be the result of detecting a certain number of particles from a higher order multiplet, and that the detection rates Ck , k ¼ 1; 2; 3 are also related to the total source intensity. Using Ck as the notation for the k-th order multiplet (such that C1 ¼ S, C2 ¼ D, etc.) and the total neutron source rate Qs Qf þQa it follows that
X n~ n! n~ Ck ¼ Qs ð28Þ ¼ Qs ekn PðnÞ ¼ Qs k : k! k!ðnkÞ! k n
101 100 10−1 10−2 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Reaction probability (p)
Qs ¼ F þQa ¼ Fð1 þ ansf ;1 Þ:
Fig. 1. Covariance between neutrons and photons.
In the above (as found in Ref. [6]) one has m ni;1 Mþ ni;2 MMg c1;1 ¼ p i;1 1pni;1 c2;1 ¼
S ¼ F en ð1þ ansf ;1 Þ ð20Þ
p fm ½n M2 þ ni;1 h2 þ ni;3 M2 Mg þ ni;2 ½h2 Mg þ 2Mc1;1 g 1pni;1 r;1 i;2 p fm n Mþ 2mi;1 ½ni;2 MMg þ ni;1 c1;1 þ ni;3 MM2g 1pni;1 i;2 i;1 þ ni;2 ½2Mg c1;1 þ Mg2 g:
ð22Þ
Here h2 is the second factorial moment of the total number of neutrons generated in the sample, internal multiplication included, by one single neutron. This and other moments can be found in Ref. [4] M1 n M2 : h2 ¼ ni;1 1 i;2
ð29Þ
In the case of singles for neutrons the following expression is derived as
ð21Þ c1;2 ¼
Using the a-factor, the source factor can be expressed as
ð23Þ
One reason for finding and using also the joint moments can be illustrated by the covariance between neutrons and photons. As shown in Fig. 1 there is a very strong relationship between the two particle types, for increasing values of the probability to induce fission p. ðnsf ;1 ð1þ aÞÞ 1 ðmsf ;1 þ ansf ;1 Þ Mþ nsf ;2 MMg Covfm ; lg ¼ ð1 þ ansf ;1 Þ ð1 þ ansf ;1 Þ M1 fmi;1 ni;1 Mþ ni;2 MMg g : ð24Þ þ nsf ;1 ð1þ aÞ ni;1 1
4. Multiplicity detection rates The measured quantities are the multiplicity rates. To convert the factorial moments of a single source event into detection rates of multiplicities, one has to account for the intensity of the source events and the detection efficiency. The effect of the finite measurement gate time in multiple coincidence measurements, quantified with the relative gate width factors as described in Ref. [11], will be omitted here. To find the measurable quantities such as singles doubles, etc., one needs to first find the factorial moments n~ k of the detected
Mnsf ;1 ð1 þ aÞ ¼ F en Mnsf ;1 ð1þ aÞ: ð1 þ ansf ;1 Þ
ð30Þ
Note how the scaling factor between the fission source and the total source intensity cancels out in the expression for the measurable singles. In a similar way doubles and triples can be derived as F e2 M2 M1 ð31Þ D¼ n nsf ;2 þ nsf ;1 ð1 þ aÞni;2 2 ni;1 1 T¼
F e3n M3 M1 ½3nsf ;2 ni;2 þ nsf ;1 ð1 þ aÞni;3 nsf ;3 þ 6 ni;1 1 ) M1 2 2 nsf ;1 ð1 þ aÞni;2 : þ3 ni;1 1
ð32Þ
These are the quantities one measures in multiplicity counters. It is these expressions that serve as the basis for the different approaches to find the various unknown parameters, as described in Ref. [3]. Most commonly one assumes the neutron detector efficiency en to be known, and solve the equations for fission rate (mass), F, leakage multiplication, M, and a the relative contribution from single-neutron sources. In the case of photons, the moments are considerably more complicated due to accounting for both neutrons and photons. It is still possible to derive equations for the measurable quantities of singles, doubles and triples, in a manner similar to that of neutrons. As mentioned earlier, at the level of the multiplicity rates, one has to account for the single photon generation via independent processes. To this end the ratio of the single photon source strength, Qg , to the neutrons source strength, Qs , needs to be introduced. A single calculation yields the gamma singles doubles and triples expressed as Sg ¼ ½gFð1 þ ansf ;1 Þ þ Fð1 þ ansf ;1 Þm~ 1 Dg ¼ Fð1 þ ansf ;1 Þ Tg ¼ Fð1 þ ansf ;1 Þ
m~ 2
ð34Þ
2
m~ 3 3!
ð33Þ
:
ð35Þ
Also here we have used the tilde notation for the detected moments, i.e. accounting for a detector efficiency, eg in the case of
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and
gamma photons, i.e. one has k
m~ k ¼ eg mk :
ð36Þ
For the sake of the discussion, we need to substitute Eqs. (36) and (10)–(13) into the above, to obtain Sg ¼ F eg ½gð1þ ansf ;1 Þ þ Mg fmsf ;1 þ ansf ;1 þ nsf ;1 ð1 þ aÞg 2
Dg ¼
eg F 2
msf ;2 þ 2ðmsf ;1 þ ansf ;1 Þ
ð37Þ
nsf ;1 ð1 þ aÞ Mg þ nsf ;2 M2g þ nsf ;1 ð1þ aÞg2 ð1 þ ansf ;1 Þ
e3g F n ð1 þ aÞ Mg þ 3ðmsf ;1 þ ansf ;1 Þ msf ;3 þ 3msf ;2 sf ;1 6 ð1 þ ansf ;1 Þ nsf ;2 nsf ;1 ð1 þ aÞ M2g þ g2 þ nsf ;3 M3g þ 3nsf ;2 g2 þ nsf ;1 ð1 þ aÞg3 : ð1 þ ansf ;1 Þ ð1 þ ansf ;1 Þ ð39Þ
The above shows that in these equations, there are 5 or 6 independent parameters, depending on whether the neutron leakage multiplication, M, and the gamma leakage multiplicity, Mg , are considered as one or two unknowns. Namely, these both are different, but unique functions of the first collision probability p. The other four are the fission rate F, the factors a and g and the gamma detector efficiency eg . In either case, the gamma multiplicity equations alone are insufficient for the determination of unknown sample parameters. Together with the neutron multiplicity rates, with the inclusion of the neutron detection efficiency en , there are two possibilities. If M and Mg are considered as two independent unknowns (which might enhance the accuracy of the determination of the fission rate), the only possibility to unfold algorithmically six unknowns from six equations is to consider one of the detection efficiencies known. If both are considered as known, then the equations become overdetermined. If only the neutron efficiency is considered to be known, then an analytical solution is possible. This is because then the neutron multiplicity rates decouple from the gamma multiplicities, and can be solved exactly as in the traditional case for F, M and a. In possession of these parameters, the gamma equations for the remaining parameters Mg , g and eg can be solved analytically, because it does not matter that they contain highly nonlinear versions of M and a. But this also amounts to saying that inclusion of the gamma multiplicity rates does not improve the unfolding of the most important parameter, F; in effect the gamma multiplicity rate equations can only be used to determine the gamma process parameters, which do not include the sample fission rate. If M and Mg are both expressed in terms of the first collision probability p, then the determination of this and other unknown parameters, including the detector efficiencies, is possible. However, the coupling between the neutron and gamma multiplicity equations is weak, with an ensuing large statistical uncertainty of the results as functions of measurement inaccuracies. A much stronger coupling is introduced into the equation system if also the mixed multiplicities (one double and two triples) are used. These are given as follows: Dng ¼ en eg F
ðmsf ;1 þ ansf ;1 Þnsf ;1 ð1þ aÞ M þ nsf ;2 MMg þ nsf ;1 ð1 þ aÞc1;1 ð1 þ ansf ;1 Þ ð40Þ
Tnng ¼
nsf ;2 nsf ;1 ð1 þ aÞ F ðmsf ;1 þ ansf ;1 Þ M2 þ h2 2 ð1 þ ansf ;1 Þ ð1 þ ansf ;1 Þ o þ nsf ;3 Mg M2 þ nsf ;2 ½2Mc1;1 þ Mg h2 þ nsf ;1 ð1þ aÞc2;1
en e2g F msf ;2
nsf ;2 nsf ;2 Mþ 2ðms;1 þ ansf ;1 Þ MMg 2 ð1 þ ansf ;1 Þ ð1 þ ansf ;1 Þ nsf ;1 ð1 þ aÞ þ c1;1 þ nsf ;3 MM2g þ nsf ;2 ½g2 M þ 2Mg c1;1 ð1 þ ansf ;1 Þ þ nsf ;1 ð1 þ aÞc1;2 :
ð42Þ
ð38Þ Tg ¼
Tngg ¼
e2n eg
ð41Þ
Using the neutron, gamma and mixed multiplicity rates, all unknown parameters, F, M, Mg , a, g, en and eg can be determined by assuming the neutron leakage multiplication and the gamma leakage multiplicity as two independent unknowns. The number of equations still exceeds the number of unknowns, and through the mixed multiplicity rates, a sufficient strong coupling between the equations is established. From these equations the searched parameters can be determined with a non-linear least squares non-parametric fitting to the measured values. If M and Mg are not considered as two independent unknowns, rather as functions of the first collision probability p, then the measure of redundancy is even higher. The performance of various cases of the unfolding of the unknowns will be quantitatively investigated and discussed below.
5. Application As it was discussed in Section 4, unlike for the case of using only the neutron multiplicities, the complexity of the expressions for the gamma photons prevents the possibility of using analytical inversion of the multiplicity rate expressions. Hence we propose the use of artificial neural network (ANN) techniques for the unfolding of sample parameters from the measured multiplicity rates. As a first application, the use of ANNs can be tested on the known case of determining three unknown sample parameters from the three neutron multiplicities, in which case the unfolding can be performed also analytically. However, the use of ANNs offers some advantages already for this relatively simple case. Namely, the analytical inversion is only possible as long as only three unknowns are attempted to be retrieved from the three multiplicity rates. This has the effect that normally the neutron efficiency needs to be known in advance. With ANN techniques, there is a larger flexibility, since ANNs can utilize the rich information in the non-linearity of the expressions to unfold more parameters than the number of expressions. Hence there is a chance that in addition to the usual three parameters, also the detector efficiency can be retrieved. First we will discuss the test of the traditional case of determining three unknowns from three neutron factorial moments. Results are obtained after the initial training of the network was completed, the training data were generated for parameters corresponding to samples ranging from a few tens of grams to the kilogram range. The trained network was tested by further sample vectors generated the same way as the training set (approximately 20% of all the input data were used for testing and validation for the different networks). The network used for this task is depicted in Fig. 2, it had two hidden layers of 25 and 15 nodes respectively. Results show that the parameters F, a and p can be evaluated with a relative error smaller than about 0.002% (Table 1), whereas the mean errors and standard deviations are even smaller. The parameters were unfolded from the three neutron rates only, which is the recent routine analytical procedure, and which serves as a good indicator that the ANNs are consistent with methods in the case of using neutron assay only.
ARTICLE IN PRESS A. Enqvist et al. / Nuclear Instruments and Methods in Physics Research A 615 (2010) 62–69
Tansig hidden layer
Input multiplicity rates (3)
Tansig hidden layer
Linear output layer
67
Output parameters (3)
(15 nodes)
(25 nodes)
Fig. 2. Schematic outline of the artificial neural network for the case of using the neutron rates only and determining three output parameters.
10
Table 1 Training results of ANN, using the neutron equations to simulating large plutonium samples in the kg range.
Probability density estimates
2
a
p
0.0001 1.98e 9 7.56e 6 1.28e 6 9.34e 6
0.0021 4.14e 7 2.11e 4 7.15e 6 1.42e 4
0.0001 4.96e 9 3.24e 5 2.56e 6 3.29e 5
[S, D]
6
4
2
x 104 fission rate alpha ratio p
1.5
Observations
Max. abs. rel. error (%) Mean error of training (%) Standard deviation of training (%) Mean error of test data (%) Standard deviation of test data (%)
[S, D/S] 8
Fission rate (F)
0 0
0.2
0.4
0.6
0.8
1
Correlation coefficients Fig. 4. Histograms of the correlation coefficients by using bootstrap procedure.
1
0.5
0 0
20
40 60 100 equally spaced points
80
100
Fig. 3. Probability density estimates of the relative errors of the unfolded parameters for the case with three neutron rates.
In Fig. 3 one can see the probability density estimate for the relative errors of the unfolded parameters. The estimates are based on using a kernel smoothing function and an associated bandwidth to estimate the density. Since the parameters have different physical units and domains, the x-axis was re-scaled for all three parameters in terms of sampling bins such that the estimated distributions are directly comparable for their widths. The figure gives a visual illustration that the fission rate can be unfolded with the smallest standard deviation (largest accuracy), compared to the factor a and probability p from the three neutron multiplicity rates. This behaviour is beneficial since the fission rate is the most important parameter to decide accurately, being the basis for determining the sample mass. To further improve the accuracy of the ANN, one can choose how to organize the output parameters. A quick investigation of the dependence in the formulae, reveals that the dependence of the accuracy of the results on M and ð1aÞ is more straightforward than that on p and a. The final calculation needed to get p from M, can
then be done manually with very little effort, while decreasing the time and computing power needed to train the ANN. Various strategies can also be used for the input parameters. Instead of using S, D, T for the neutron rates, one can choose the parameters: S, D=S, T=S, which proves to be more independent. In order to investigate if the correlation coefficients are statistically significant, we have applied a bootstrap procedure for two first inputs in combinations [S,D] and [S, D/S]. A histogram of the results obtained after resampling both input vectors 100 times and computing the correlation between the inputs on each sample is given in Fig. 4. This shows a strong evidence that we can reduce the value of the correlation coefficient between inputs such as S and D from approximately 0.9 to 0.2 when we use the second configurations of the inputs. This behaviour is evident also for the gamma and mixed rates where the correlation between parameters is reduced significantly when using the following input parameters: S, D/S, T/S, Sg , Dg =Sg , Tg =Sg , Dng , Tnng =Dng , Tngg =Dng . When using also the gamma and mixed rates, the results indicate that more parameters such as the gamma ratio, g, and the gamma detector efficiency, eg , can be unfolded in addition to the parameters that were unfolded from the neutron rates only. Due to the complicated form of the dependence of the detection rates on the sample parameters, the accuracy is reduced when using all nine inputs and unfolding more parameters. This is valid even though the ANN was made more complex in the case of using all input rates and then consisted of two hidden layers with 35 and 15 nodes respectively. This layout was used to obtain the results in Table 2 and Fig. 5. To make a deeper analysis of the sensitivity of the ANN when trying to apply it to real measured data, we perturbed the analytically obtained ‘‘clean’’ inputs with a random noise at a
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Table 2 Accuracy of ANNs when unfolding sample parameters from an over determined system. Maximal variations (%)
Input þ noise
Without noise S þ 5% noise D þ 5% noise T þ 5% noise Sg þ 5% noise Dg þ 5% noise Tg þ 5% noise Dng þ 5% noise Tnng þ 5% noise Tngg þ 5% noise
a
p
F
g
eg
3:080e þ 00 2:978e þ 00 5:825e þ 00 6:341e þ 00 9:553e þ 00 3:765e þ 01 1:808e þ 01 2:718e þ 01 1:136e þ 01 3:883e þ 00 1:6766e þ 02
1.056e 01 2.514e 01 7.804e 01 8.593e 01 4.023e 01 1:536e þ 00 1:357e þ 00 8.461e 01 6.211e 01 1.063e 01 4:494e þ 00
4.704e 02 1.255e 01 1.066e 01 1.867e 01 1.442e 01 3.623e 01 2.693e 01 1.318e 01 1.658e 01 6.883e 02 1:851e þ 00
3.100e 01 1:435e þ 00 1:792e þ 00 3:669e þ 00 2:439e þ 00 1:596e þ 01 1:460e þ 01 3:291e þ 00 3:601e þ 00 3.807e 01 2:786e þ 01
2.200e 01 5.924e 01 9.448e 01 1:072e þ 00 6.930e 01 1:217e þ 01 7:758e þ 00 1:867e þ 00 1:205e þ 00 2.564e 01 4:012e þ 01
3.2457e 003
1.9557e 002
1.8702e 001
2.7251e 001
2.3318e 001
2:3705e þ 000
2:9004e þ 000
Mean error of test data (%) 1:6581e þ 000
All þ 5% noise
Standard deviation of test data (%) 1:5577e þ 001
5.2392e 001
frequency
3000 2000 1000 0 −20
−15
−10
−5
0
5
10
15
20
25
2
2.5
relative errors (%) for gamma the efficiency, εγ frequency
2000 1000 0 −2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
Although the maximum errors might seem large, one should observe that the average error is much smaller, which is demonstrated in the histograms of a few of the unfolded parameters (Fig. 5). It should be pointed out that the formulae used for the detection rates are all based on the point-model and therefore are not exact. However in the field of multiplicity measurements they have proved to be accurate enough, and thus we have based the extensions to gamma and mixed rates on the same formalism. A more detailed quantitative and qualitative analysis of the performance of the ANN based unfolding procedure is given in a companion paper [12].
6. Conclusions
relative errors (%) for the p parameter frequency
2000 1000 0 −1
−0.8 −0.6 −0.4 −0.2
0
0.2
0.4
0.6
0.8
1
relative errors (%) for fission rate, F Fig. 5. Histogram showing the distribution of the errors of the predicted parameters: F, p and eg , when 5% random noise have been added to the input data.
level of 5% in magnitude to each rate individually as well as to all rates at once. This is to simulate the fact that measured detection rates will always have some statistical uncertainties, and it is important that the neural networks can handle inputs that are not perfectly acquired. Table 2 presents the maximal variations of the unfolded parameters. Some of the unfolded parameters show much larger variation compared to others. However, the main parameter of interest is the fission rate, F, which relates to the sample mass, and it shows very little variation also when the input data contain noise. This fact is naturally very promising. The table also lists the mean error and standard deviation in the last case of 5% noise on all data to give additional insight to how significant the errors are expected to be when using realistic data.
The present paper shows that by taking all possible auto- and cross-factorial moments of the neutron and gamma counts into account, one has nine expressions which are functions of six independent parameters. The expressions for higher order factorial moments become increasingly long, and show a complicated dependence on some of the parameters. The above factorial moments were converted into detection rates, which was hitherto done for neutrons only. It has also been demonstrated that just as in the case of single neutron emission through (a, n) reactions, one can also phenomenologically account for single photon emission, both in connection to the (a, n) reactions, but also for photons emitted in processes not related to neutron generations, with the help of a gamma parameter, g, defined as the ratio of the rate of single gamma emissions to the rate of spontaneous fission. It was shown that taking only the neutron and gamma detection rates, the complexity of the unfolding does not increase, but the accuracy and effectiveness of the unfolding does not get improved compared to using neutron multiplicities only. Accounting for also the mixed detection rates, especially when all nine multiplicity rates are used, theoretically increases the performance of the unfolding (number of parameters unfolded, achievable accuracy, etc.) but at the expense of the increased complexity of the unfolding problem.
ARTICLE IN PRESS A. Enqvist et al. / Nuclear Instruments and Methods in Physics Research A 615 (2010) 62–69
Due to the highly non-linear character of the formulae when also the mixed moments are used, artificial neural networks were suggested for handling the task of unfolding the sample parameters, and especially the fission rate which is closely linked to the sample mass. By using the equations of the multiplicity rates to generate training data for the networks, we were able to prove the applicability of the approach. The results shown are very promising and of good accuracy. The training and performance of ANNs using all moments for both neutrons and photons is computationally more demanding, but still within manageable range. It has also been demonstrated that the ANN approach can be used also as a replacement for pure neutron assay without loss off accuracy compared to the conventional unfolding of three sample parameters. Since real measurements data will always contain statistical uncertainties the network was tested with data that had random noise added to it, to simulate more realistic input data. The ANN still showed very good accuracy when predicting the sample parameters, where the largest uncertainties were found not in the fission rate, but rather in other parameters such as the alpha ratio.
69
Acknowledgement This work was supported by the Swedish Radiation Safety Authority (SSM). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
[12]
¨ K. Bohnel, Nucl. Sci. Eng. 90 (1985) 75. W. Hage, D.M. Cifarelli, Nucl. Instr. and Meth. A 236 (1985) 165. D.M. Cifarelli, W. Hage, Nucl. Instr. and Meth. A 251 (1986) 550. I. Pa´zsit, S.A. Pozzi, Nucl. Instr. and Meth. A 555 (1–2) (2005) 340. A. Enqvist, I. Pa´zsit, S.A. Pozzi, Nucl. Instr. and Meth. A 566 (2006) 598. I. Pa´zsit, L. Pa´l, Neutron Fluctuations—A Treatise on the Physics of Branching Processes, Elsevier Science Ltd., London, New York, Tokyo, 2008. A. Enqvist, M. Flaska, S.A. Pozzi, Nucl. Instr. and Meth. A 595 (2008) 426. S.D. Clarke, M. Flaska, S.A. Pozzi, P. Peerani, Nucl. Instr. and Meth. A 599 (2009) 221. I. Pa´zsit, A. Enqvist, L. Pa´l, Nucl. Instr. and Meth. A 603 (2009) 541. R. Sanchez, CEA, Personal communication, 2008. N. Ensslin, W.C. Harker, M.S. Krick, D.G. Langner, M.M. Pickrell, J.E. Stewart, Application Guide to Neutron Multiplicity Counting, Los Alamos Report LA13422-M, 1998. S. Avdic, A. Enqvist, I. Pa´zsit, ESARDA Bull. 43 (2009) 21.