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A novel method for active fissile mass estimation with a pulsed neutron source
Nuclear Instruments and Methods in Physics Research A 715 (2013) 62–69
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Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
A novel method for active fissile mass estimation with a pulsed neutron source C. Dubi a,n, T. Ridnik a, I. Israelashvili a, B. Pedersen b a b
Physics Department, Nuclear Research Center of the Negev, POB 9001, Beer Sheva, Israel Nuclear Security Unit, Institute for Transuranium Elements, Via E. Fermi, 2749 JRC, Ispra, Italy
art ic l e i nf o
a b s t r a c t
Article history: Received 6 September 2012 Received in revised form 1 March 2013 Accepted 1 March 2013 Available online 21 March 2013
Neutron interrogation facilities for mass evaluation of Special Nuclear Materials (SNM) samples are divided into two main categories: passive interrogation, where all neutron detections are due to spontaneous events, and active interrogation, where fissions are induced on the tested material by an external neutron source. While active methods are, in general, faster and more effective, their analysis is much harder to carry out. In the paper, we will introduce a new formalism for analyzing the detection signal generated by a pulsed source active interrogation facility. The analysis is aimed to distinct between fission neutrons from the main neutron source in the system, and the surrounding “neutron noise”. In particular, we derive analytic expressions for the first three central moments of the number of detections in a given time interval, in terms of the different neutron sources. While the method depends on exactly the same physical assumptions as known models, the simplicity of the suggested formalism allows us to take into account the variance of the external neutron source—an effect that was so far neglected. & 2013 Elsevier B.V. All rights reserved.
Keywords: Neutron detection Neutron multiplicity method Fissile mass estimation Active neutron interrogation
1. Preliminaries 1.1. Introduction Fissile mass estimation by neutron multiplicity counting may be divided into two main categories: passive methods, relying on spontaneous fissions from the tested sample, and active methods, where fissions are induced on the tested material by an external neutron source. Active methods can also be divided into two categories: fixed and pulsed source. While passive methods are simpler to handle (both in terms of the system engineering and in the mathematical analysis), they suffer from two major disadvantages: first, they are restricted to fissile materials that have spontaneous fissions, and second, they often demand relatively large measurement times, together with a large number of detectors. The multiplicity method [1–3] is traditionally used in both passive and active fissile mass estimation for distinction between the main fission source and the surrounding neutron noises. The multiplicity method was originally developed for passive interrogation, and was later extended to active interrogation facilities for both fixed source [4,5] and pulsed source [6,7]. The last two
n
Corresponding author. Tel.: þ972 52 913313. E-mail address: [email protected] (C. Dubi).
0168-9002/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nima.2013.03.004
are, to the best of our knowledge, the first (and at present the only) analytic treatments to the detection signal from an active interrogation facility with a pulsed source. Recently, the authors have introduced a new method for analyzing passive measurements, named the SVM method (see Ref. [8]), standing for Skewness, Variance and Mean. Unlike the multiplicity method, the SVM does not measure the singles, doubles and triples rate in the detection signal, but rather measures the Mean, Variance and third central moment (Skewness) of the number of detections in a given time interval. In the present work we extend the SVM method to pulsed source active measurements. In particular, we provide full analytic expressions for all the first three central moments of the number of neutron detections in a given time interval. 1.2. Motivation While active measurement facilities with a pulsed source offer some very attractive advantages, for both the nuclear safety and safeguard community, it is not fully understood how to properly analyze the detection signal in order to separate the fission source from the additional neutron sources in the system. The currently available models [6,7], while very enlightening and constructive, to the best of our knowledge, are neither properly benchmarked, nor are there any academic reports on an extensive set of laboratory tests to the model (although the theoretical results
C. Dubi et al. / Nuclear Instruments and Methods in Physics Research A 715 (2013) 62–69
were published 7 years ago). Consequently, there are many unresolved questions regarding the applicability of the current methods. For instance, applicability of the Bohnel Method [9], which is the standard neutron multiplicity counting method for handling fission chains in the tested material (and was often tested for passive interrogation), may be questioned due to the (relatively) high probability of a fast neutron that forms the tested sample to be thermalized, and then returns to the sample and creates a fission (this is true since in most active measurement facilities, unlike in passive measurement facilities, the moderator is not separated from the tested sample by a thermal absorbent, see Ref. [10]). Another effect that was not treated analytically so far is the statistical variance in the number of neutrons generated by the external neutron source. In the existing models, it is assumed that the number of neutrons generated in each pulse is fixed, while, in practice, there is an expected statistical variance in the number of neutrons generated by the neutron generator in each pulse. This gap between the theory and practice, points out to a clear justification for the construction and further development of new analytic tools describing more elaborately and in different perspectives the measurement results from active pulsed source interrogation facilities. Since the physical assumptions in the present model are the same as those assumed in Ref. [6], there is a clear resemblance to the results obtained by Hage in Ref. [6] (although a different analysis is carried out). However, using the present model, we may now give analytic treatment to at least one effect that was not fully treated in Ref. [6]—the statistical variations in the flux generated by the external neutron source.
2. Description of measurement system and defining the problem 2.1. Configuration of a pulsed source interrogation facility Pulsed source active measurement systems have, in general, the following configuration: inside the sample cavity, which is located inside a block of a neutron moderator – typically graphite or lead – a neutron source (here referred to as the external neutron source) is placed together with the tested sample. The external side of the moderator is surrounded by a thermal neutron absorber and a ring of detectors is placed around it. The neutron source releases a pulse of fast neutrons, which are then thermalized, such that at the end of the thermalization process (or “thermal buildup”), a so called “thermal pool” is created inside the sample cavity. During the thermal buildup the detectors are not operational. Once the thermal buildup is over, two procedures take place: first, the number of thermal neutrons (or equivalently, the thermal flux), decreases exponentially, due to absorption and leakage. Second, the thermal flux creates induced fissions in the tested sample. This stage is referred to as the “thermal decay”. In a typical system, the duration of the thermal buildup is about 200 μs, and the duration of the thermal decay is several milliseconds (between the two stages there is, in most cases, a transition stage, in which the detectors are not operational). The fission rate in each point of time, during the thermal decay, is proportional to both the thermal flux in the sample cavity and the mass of the tested sample. In the fission process, fast neutrons are released into the system. Some of these fast neutrons escape both the moderator block and the thermal absorbent layer surrounding it, and reach the detector ring. Since the moderator block is surrounded by a thermal absorbent, only fast neutrons that were created in the induced fissions are likely to reach the detector.
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Assuming that there are neither fast fissions due to neutrons created from the tested sample itself, nor are there any neutron sources in the system, other than the induced fission source, then the average number of detections should be proportional to the mass of the detected sample. So, after a simple calibration, we can use the active measurement system to estimate the mass of the sample. Obviously, for this to work, a single pulse of neutrons is not enough, and the entire procedure must be repeated over, with the simple law of error proportional to the reciprocal of the square root of the number of repetitions. However, in most measurement systems, the assumption that there are no additional neutron sources (or fissions coming from fast neutrons created by the tested sample) is simply not true, and the naive estimation would result in a large over-estimation of the SNM mass. Since the origin of the neutron cannot be determined by detector technology, we must use a “mathematical filter”, determining the source to noise ratio in the measurement. 2.2. Assumptions on the system and problem definition Our model assumptions are listed below. These assumptions are standard assumptions in neutron multiplicity counting (see, for instance [6]). 1. The neutrons are averaged both in energy and space: we assume a point-wise, single energy source. Effectively, this assumption allows us to refer to a single detection probability and a single die away time for all the neutrons. 2. The number of induced fissions in the sample is proportional to its mass by a known proportion: in general, there might be a correlation between the geometry of the tested sample and the number of induced fissions. However, since we assume that the thermal pool is big enough (with regard to the tested sample), this correlation is neglected. 3. The thermal flux in the thermal pool decays exponentially with a single (known) decay constant. The last two assumptions can be summarized as follows: assume that the thermal buildup is over at time T0, then for every t≥T 0 , the probability for a thermal fission in the time interval ½t,t þ dt is given by A0 exp½−λd jt−T 0 j dt
ð2:1Þ
where λd is the decay coefficient of the thermal flux, and A0 is proportional to the mass of the tested sample. The coefficient A0, which relates the fission source amplitude with the mass of the tested material, has a clear physical interpretation. The total fission rate in the tested sample is given by SðtÞ ¼ ∬ Φðt,pÞsf ρðpÞ dvp
ð2:2Þ
V
where V is the volume of the tested sample, ΦðtÞ is the thermal flux, sf is the fission microscopic cross-section and ρ is the density of the tested material. Assuming that the volume of the tested sample is small with respect to the entire sample cavity, we may assume that the flux is fixed over the volume of the tested sample, resulting in SðtÞ ¼ ΦðtÞsf ρV
ð2:3Þ
If the thermal flux in the sample cavity decays exponentially, we can write ΦðtÞ ¼ Φ0 e−λd jT 0 −tj . Since ρV is nothing more than the mass of the tested sample, comparing Eqs. (2.1) and (2.3) gives A0 ¼ Φ0 sf If the thermal flux generated by the neutron generator (after the moderation stage) is known, then A0 can be evaluated in advance. However, in most measurement systems, A0 is calibrated (together
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with other system parameters, such as the detection efficiency and the neutron die away time) by performing the measurements on a well-characterized known sample. As stated before, A0 is not truly a constant, and may vary between the different pulses. At the present stage we will assume that it is fixed, and will drop this assumption in Section 6.
2.3. The neutron sources The neutron sources affecting the measurement may be divided into three categories : Decaying sources, fixed sources and fission chains [6]. The decaying sources are the following: 1. The fission source is described in Section 2.2. 2. Leakage from the thermal pool: although the sample cavity is surrounded by a thermal absorber, it is expected that some of the “thermal pool” neutrons might eventually escape absorption and reach the detector ring. Since the number of neutrons escaping absorption should be proportional to the thermal flux in sample cavity, the amplitude of this source may be written as A1 exp½−λd jt−T 0 j:
ð2:4Þ
The sources fixed in time are the following: 1. Spontaneous fissions: this source is due to the spontaneous fissions in the even isotopes in many SNM, often present in MOX samples and other compositions containing Pu. 2. Random neutrons noise: a random neutron source with a fixed rate is expected mainly due to ðα,nÞ reaction in the sample itself, and cosmic radiation. One final neutron source is the fission chains in the tested material. Notice, this fission source, although physically very much the same as the main fission source, is not accounted for in formula 2.1. The traditional treatment to fission chains in the tested sample is usually through the well-known Bohnel method [9]. However, since in most pulsed source neutron interrogation facilities the tested sample is surrounded by a heavy reflector and is not protected by a layer of thermal absorber, the applicability of the Bohnel method is very questionable, and to the best of our knowledge, the model was not tested. One way or the other, in the present paper, as well as in Refs. [6,7], the fission chains are neglected.
3. Notations and definitions 3.1. Generating functions In the following section we introduce the concept of the generating function, which will often appear throughout our analysis, and point out some of its features for later use. Let X be a discrete valued random variable with distribution defined by the series fan g. That is, for every n, PðX ¼ nÞ ¼ an . The generating function of X is defined by the formal power series ∞
j
g X ðxÞ ¼ ∑ x aj
ð3:5Þ
j¼0
one of the most important features of the generating function is the fact that the moments of the distribution may be written explicitly in terms of the generating function and its derivatives by
the following formulas: ∞
d g X ðxÞx ¼ 1 dx n¼1 ∞ d d x g X ðxÞ EðX 2 Þ ¼ ∑ n2 an ¼ dx dx n¼2 x¼1 ∞ d d d 3 3 x x g X ðxÞ EðX Þ ¼ ∑ n an ¼ dx dx dx n¼3 x¼1
EðXÞ ¼ ∑ nan ¼
ð3:6Þ
In some of the cases presented in the following study, the random variable of interest – as well as the corresponding distributions – will be time dependent. In such cases we will denote an ¼ an ðtÞ. Such time dependence does not affect the validity of Eqs. (3.6). In the present study we are also interested in the second and third central moments, defined by VarðXÞ ¼ EððX−EðXÞÞ2 Þ ¼ EðX 2 Þ−EðXÞ2 SkðXÞ ¼ EððX−EðXÞÞ3 Þ ¼ EðX 3 Þ−3EðXÞEðX 2 Þ þ EðX 3 Þ
ð3:7Þ
3.2. General purpose notations and definition A neutron source is defined by two attributes: the event rate (or source amplitude) and the source distribution (or neutron multiplicity). The event rate, here denoted by S and a proper sub index, describes the probability that an emission of neutron event will occur in the interval ½t,t þ dt, and the source distribution describes the probability that once an emission has occurred, exactly ν neutrons would be released. We shall denote the amplitude of the time decaying source by Sd(t) and the amplitude of fixed source by Sf. As described, the amplitude of the decaying source may be written as Sd ðtÞ ¼ ðA0 þ A1 Þexp½−λd jt−T 0 j
ð3:8Þ
(where A0 and A1 were defined in Section 2), and the fixed source amplitude may be divided into the following two components: Sf ¼ Sfission þ SRandom
ð3:9Þ
For the sake of simplicity, we here on denote A0 þ A1 ¼ AT . We denote the distribution of decaying source by qd,1 ,qd,2 ,…,qd,n and that of the fixed source by qf ,1 ,qf ,2 ,…,qf ,n . Another notation that will be often used in the present study is the Diven factors. For a given distribution fqj g∞ , the nth Diven j¼1 factor is defined as ℓ! qℓ : ðℓ−nÞ! ℓ¼n ∞
Dn ¼ ∑
The importance of the Diven factors in neutron multiplicity counting has been largely studied, see Refs. [11,12]. Notice, the Diven factors depend both on the materials and isotopic composition of the tested sample, and on the noise to source ratio. Writing down the total source (fixed and decaying) Diven factors in terms of the neutron distribution for the different materials and the noise to source ratio is a well-known (and fairly simple) algebraic procedure and can be found in many sources (see Refs. [8,1,6]).
4. The stochastic transport equation 4.1. Introduction Our next order of business is to derive and solve the stochastic transport equation for the number of detections in a given time interval. The stochastic transport model involves the dynamics through time of two functions. The first concerns the evolution through time of the detection signal generated by a single neutron, and the second is the extension of the first for a source continuous in time. Unlike the spatial transport equation, the Boltzmann
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equation, where the preservation laws are written in terms of the average number of neurons in a given volume, the preservation laws for the stochastic transport equation, are stated in terms of the event probabilities in the lifetime of a single neutron. In our context, when looking at an infitisimal time window dt, sufficiently shorter than the die away time of the neutron, a neutron coming from the sample may create exactly one of the following three events: 1. The neutron is absorbed in the detector, causing a reading in the detection signal. This event shall be referred to hereon as a detection event. 2. The neutron is lost in the system – either by leakage or by absorption – but without being detected. This event shall be referred to as a losing event. 3. The neutron does nothing. Denote by λdetection the probability per time unit that a neutron will cause a detection event, and by λlost the probability per time unit that the neutron will cause a losing event. We shall also denote λ ¼ λdetection þλlost . λ is then the probability per time unit that the neutron will create any event. In the current terminology, the average lifetime of a neutron (die away time) in the system is given by 1=λ. One should notice that both scattering and fission probabilities are not mentioned. Since the system is averaged both in space and energy, scattering of a neutron is of no importance, and may be neglected. As stated, we assume that there are no fission chains. Returning to the stochastic transport, the distribution of the detection signal is uniquely determined by three elements: The first is the behavior of a single neutron, manifested by the event probabilities λdetection and λlost . The second is the rate and the statistics of the different sources, and finally, the duration of the time interval. The objective of this section is to construct an analytical expression for the distribution of the number of counts in a time interval, using the above mentioned notations. It should be mentioned that the analysis carried out in this section is very close – often similar – to the analysis carried out in Ref. [13]. For the sake of completeness and easy reading, we sometimes repeat calculations carried out in Ref. [13]. 4.2. The detection signal generated by a single neutron Consider a neutron detector, counting the number of neutron detections on an interval of length T, starting, for convenience, at t ¼ −T and ending at t¼ 0. The interval ½−T,0 shall be referred to as the target interval (or target window). Next, consider the case where a single neutron enters the system t seconds before the target interval ends. Notice, the time of entrance, −t, may or may not be in the target interval. We denote by an(t) the probability for the neutron to generate n detector counts in the target interval. We should bear in mind that the probability must be zero for every n≥2, since we assumed that there are no fission chains. To understand the behavior of an(t), we look on an infitisimal time interval ½−ðt þ dtÞ,−t. During this time interval, exactly one of the following events may happen: 1. The neutron will be detected. The probability for such an event is λdetection dt. In such case, a detector reading in the target interval will be set if and only if −t belongs to the target interval. Notice, if −t does not belong to the target interval, the detector reading might be set (this depends on the actual setting of the system), but it will be ignored, as by definition we are only counting detections in the target interval. 2. The neutron will either leak or be absorbed without a detection reading. The probability of such an event is λlost dt. In such case,
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it is clear that no detections will be registered in the target interval. 3. The neutron will do nothing. The probability for such an event is 1−λdt. In such case, the probability for n detector readings is equal to the probability as if the neutron was released at time −t. Summing the last three options in the complete probability theorem, we may write a0 ðt þdtÞ ¼ ð1−λ dtÞa0 ðtÞ þ λlost dt þð1−H ½0,T Þλdetection dt a1 ðt þdtÞ ¼ ð1−λ dtÞa1 ðtÞ þ H ½0,T λdetection dt
ð4:10Þ
(here H ½0,T ðtÞ is the characteristic function of the interval ½0,T, satisfying H ½0,T ðtÞ ¼ 1 if 0≤t≤T and H ½0,T ðtÞ ¼ 0 otherwise). Dividing by dt and taking the limit dt-0, we obtain the differential form: a′0 ðtÞ ¼ −λa0 ðtÞ þλlost þ ð1−H ½0,T Þλdetection a′1 ðtÞ ¼ −λa1 ðtÞ þH ½0,T Þλdetection
ð4:11Þ
The initial conditions a0 ð0Þ ¼ 1,a1 ðtÞ ¼ 0 express the fact that a neutron released once the target interval has closed cannot cause a detector reading in the target interval. Eqs. (4.11) (with the proper initial conditions) may be analytically solved, resulting with a0 ðtÞ ¼ 1− a1 ðtÞ ¼
λdetection ½1−e−λt −ð1−e−λðt−TÞ ÞU 0 ðt−TÞ λ
λdetection ½1−e−λt −ð1−e−λðt−TÞ ÞU 0 ðt−TÞ λ
ð4:12Þ
Eqs. (4.12) gives us the full description of the detection signal generated by a single neutron. As can be seen, both a0 ðtÞ and a1 ðtÞ depend only on the total cross-section λ and the ratio P d ¼ λdetection =λ. Pd is known as the detection probability or the system efficiency, and describes the probability that a neutron entering the system will create a reading on the detectors. 4.3. The detection signal generated by a continues source We now turn to extend the results from the previous section to the general case of a continuous source, with a general amplitude S(t) and multiplicity distribution fqν gM ν ¼ 1 . The procedure is a wellknown technique, often referred to as the Bartlet equation, and can be found in the literature. Yet, for the sake of completeness, we will give a full description of the process. The setting is the following: As in Section 4.2, the detector is activated during the target interval ½−T,0. t seconds before the target interval is closed, the source is “activated”. We denote by bn(t) the probability that there will be exactly n detector readings in the target interval. Let us denote the following two generating functions: 1
hðx,tÞ ¼ ∑ aj ðtÞxj , j¼0
∞
gðx,tÞ ¼ ∑ bj ðtÞxj :
ð4:13Þ
j¼0
We now extend t by an infitisimal quantity dt and express bn ðt þ dtÞ. Looking at the time interval ½−ðt þ dtÞ,−t, both the fixed and decaying source may either release a neutron or not. If no neutrons were released, then the probability for n detections is now bn(t). If ν neutron were released, the probability for n detections is given by ! n
∑
j¼0
bj ðtÞ
∑
k1 þ k2 þ k3 þ ⋯ þ kν ¼ n−j
ak1 ðtÞak2 ðtÞ…akν ðtÞ
ð4:14Þ
(The last is obtained by the complete probability theorem, with the distinct events determined by the number of detection readings generated by each of the neutrons released into the system). We notice that the inner sum in formula 4.14 is exactly the ðn−jÞ th coefficient of hðx,tÞν , and thus the outer sum is exactly the
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nth coefficient of gðx,tÞhðx,tÞν . Since the probability for a source events in the interval ½−ðt þ dtÞ,t is SðtÞ dt, we may write M gðx,t þ dtÞ ¼ gðx,tÞð1−SðtÞ dtÞ þ SðtÞ dt ∑ qν gðx,tÞhðx,tÞν ν¼0
As mentioned, this is a well-known procedure and will not be repeated here. Let us write gðxÞ ¼ expðGðxÞÞ, where G(x) is well defined by Eq. (4.16). A direct computation of all first three moments results with ∞
∑ nbn ¼
or
n¼0
M gðx,t þ dtÞ−gðx,tÞ ¼ −gðx,tÞ SðtÞ−SðtÞ ∑ qν hðx,tÞν : dt ν¼0
∞
∑ n2 bn ¼
Notice, in the last we have ignored “second order” events; meaning, the probability for two source events in an infitisimal interval is ignored. Since the next step is turning to a differential form, and since second order events are proportional to dt2, these events are negligible. Taking dt-0 we then have M ∂gðx,tÞ ¼ −gðx,tÞ SðtÞ−SðtÞ ∑ qν hðx,tÞν ð4:15Þ ∂t ν¼0 Eq. (4.15) has an explicit solution, given by Z t M SðtÞ 1− ∑ qν hðx,tÞν dt : gðx,tÞ ¼ exp −
ð4:16Þ
ν¼0
0
Breaking the total source into a sum of two sources (one constant in time and one decaying in time) we have (with notations as in Eqs. (3.2)) Z t M gðx,tÞ ¼ exp AT e−λd jT−τj 1− ∑ qd,ν hðx,tÞν dτ 0
Z exp
t
0
ν¼0
M
Sf 1− ∑ qf ,ν hðx,tÞν dτ
ð4:17Þ
ν¼0
Notice that in Eq. (4.17) the generating function is presented as a product of two generating function: the first for the decaying source and the second for the fixed source. Since the fixed and decaying source are not correlated, this was expected. Eq. (4.17) depends on two time variables: the first, t, denotes the time (before the target window closes) in which the source is activated, and the second, T, is the duration of the target interval. If T0 is the time difference between two consecutive pulses, then the pulses start T0 before the target window closes . Thus, Eq. (4.17) becomes Z T 0 M gðxÞ ¼ exp AT e−λd jT 0 −τj 1− ∑ qd,ν hðx,tÞν dτ ν¼0
0
Z
∞
exp 0
M Sf 1− ∑ qf ,ν hðx,tÞν dτ
dg ¼ ðG′ðxÞhðxÞÞjx ¼ 1 ¼ G′ð1Þ dxx ¼ 1
ð4:18Þ
ν¼0
The transition from Eqs. (4.17) to (4.18) is formally tricky, as the upper bound of the integration in both integrals should be the same. To overcome this formality, we must multiply the integrated function by a step function with appropriate boundaries, this step is neglected, as the outcome is very much self explanatory. Now, the distribution depends on two time variable: T, which is embedded in hðx,tÞ, and T0.
5. Explicit formulas for the central moments After constructing and solving the stochastic transport equation, our next order of business is to calculate the first three central moments of the distribution given by the generating function in formula 4.18, in terms of the system parameters, the source amplitude and the source Diven factors. For actually solving the set of equations, one must write down the general source amplitude and Diven factors in terms of the different source amplitudes and different source distributions.
n¼0
d dg x ¼ ðG′ðxÞhðxÞ þ xðG″hðxÞ þ ðG′ðxÞÞ2 hðxÞÞÞjx ¼ 1 dx dx x ¼ 1
¼ G′ð1Þ þ G″ð1Þ þ ðG′ð1ÞÞ2
∞
∑ n3 bn ¼
n¼0
d d dg x x dx dx dx x¼1
¼ ðG′ðxÞhðxÞ þ3xðG″hðxÞ þ ðG′ðxÞÞ2 hðxÞÞÞjx ¼ 1 þ ðx2 ðG‴ðxÞhðxÞ þ 3G″ðxÞhðxÞ þ ðG′ðxÞÞ3 hðxÞÞÞjx ¼ 1 ¼ G′ð1Þ þ 3ðG′ð1Þ þ ðG′ð1ÞÞ2 Þ þ G‴ð1Þ þ 3G″ð1ÞG′ð1Þ þ ðG′ð1ÞÞ3
A straight forward computation of all three integrals in G′ð1Þ,G″ð1Þ and G‴ð1Þ gives G′ð1Þ ¼ AT P d Dd1 I d,1 þSf P d Dd1 I f ,1 G″ð1Þ ¼ AT P 2d Dd1 I d,2 þSf P d Df2 I f ,2 G‴ð1Þ ¼ AT P 3d Dd3 I 3 þ Sf P d Df3 I f ,3
ð5:19Þ
ðDfj Þ
Ddj
are the jth Diven factor of the decaying (fixed) where R∞ source, and the integral I f ,j ¼ 0 aj1 ðtÞ dt were explicitly given in R T 0 λ jT −tj j Ref. [8], and I d,j ¼ 0 e d 0 a1 ðtÞ dt is given by I d,1 ¼ Id,2 ¼
e−ðλ þ λd ÞT 0 ðλd eλd T 0 ðeλT −1Þ−λeλT 0 ðeλd T −1ÞÞ λd ðλd −λÞ
eλT−2ðλ þ λd ÞT 0 ðð2λd −λÞeλT þ 2λd T 0 ðeλT −1Þ2 þ λe2λT 0 ð2λd e2λd ðeλT −1Þ þ λeλT ðe2λd T −1ÞÞÞ 2λd ðλd −λÞð2λd −λÞ
I d,3 ¼
e−2λT−3λT 0 −3λd T 0 3λd ð3λd −λÞð2λ2 −5λλd þ 3λ2d Þ
f9λ3d e2λT þ 3λd T 0 ðeλT −1Þ3 þ 2λ3 e2λT þ 3λT 0 ð1−e3λd T Þ þ 9λλ2d ðeλT −1Þ2 ðe3λd T þ 3λT 0 −e2λT þ 3λd T 0 þ e3λd T 0 þ 3λT Þ þ λ2 λd ðeλT −1Þð9eλT þ 3λd T þ λT 0 −3e3λd þ 3λT 0 þ 2e2λT þ 3λd T 0 þ 2e4λT þ 3λd T 0 −4e3λT þ 3λd T 0 Þg
ð5:20Þ
In the simple case where we neglect all neutrons from fissions made before the target interval opens, we have T ¼ T 0 , and the last reduces to I d,1 ¼
λ−λe−λd T 0 þ λd ðe−λT 0 −1Þ ðλ−λd Þλd
I d,2 ¼ e−λd T 0
I d,3 ¼ e−λT 0
! eλd T 0 −1 eðλd −2λÞT 0 −1 2 eðλd −λÞT 0 −1 þ þ λd λd −2λ λ−λd
ð5:21Þ
e−λd T 0 −1 eðλd −λÞT 0 −1 3ðeðλd −λÞT 0 −1Þ 3ðeðλd −λÞT 0 −1Þ þ þ þ λd 3λ−λd λd −2λ λ−λd ð5:22Þ
Using the standard relations between the moments and the central moments, together with Eq. (5.19), we have EðXÞ ¼ AT P d Dd1 I d,1 þ Sf P d Dd1 I f ,1 VarðXÞ ¼ AT P 2d Dd2 I d,2 þSf P d Df2 I f ,2 þ EðXÞ SkðXÞ ¼ AT P 3d Dd3 I d,3 þ Sf P d Df3 I f ,3 þ 3VarðXÞ−2EðXÞ
ð5:23Þ
We can now observe that, if the total cross-section λ, the detection efficiency, Pd, and the length of the target interval, T, are all known, then the first three central moments can be
C. Dubi et al. / Nuclear Instruments and Methods in Physics Research A 715 (2013) 62–69
determined by the source amplitudes and the first three Diven factors.
6. The effect of source variance on the moments of the detection signal In our analysis so far, we have assumed that the number of the neutrons generated by the external source does not change between the different pulses. In practical active measurement facilities, this assumption is not true, and a statistical variation in the number of neutrons generated in each pulse is expected. Such a variation will obviously have an impact on the central moments of the detection signal. In Ref. [7], the fact that the external neutron source may have a statistical variation is mentioned. In order to apply the method developed, it is suggested that one should either take into the account the only pulses with the same number of neutrons generated, or normalize the neutron counts in each pulse with the amplitude of the pulse . This however, raises two difficulties: First, applying the first option will cause us to lose a large number of generator pulses and thus lose efficiency. Second problem, regarding both methods, is that determining the number of neutrons generated in a given pulse may turn to be a complicated task: the number of neutrons generated in a given pulse can only be tested by neutron detection, in a set of detectors often referred to as the “source monitor”. This neutron counting in the source monitor should be held during the thermalization stage (otherwise, the fission source neutrons will affect the measurement). Thus determining the number of neutrons generated should be done in a very short time interval, and the number of detections expected is not very large (in a typical system, 100–200 counts). On the other hand, during the thermalization stage, the detector ring is highly saturated, due to the extremely large number of fast neutrons coming from the source. In fact, at the very beginning of the pulse (first 100 μs), the detectors are completely saturated, and the data collection in that region is more or less useless. Therefore, on top of the expected statistical error, a dead time error is expected. What seems to be a more effective and accurate solution is to characterize the source distribution in advance, by running the pulsed source without any SNM samples in the sample cavity, and then properly account for the effect of the source variance in the total detection signal. In such a calculation, the measurements can be made in a relatively long time window (during the thermal decay), yielding a larger neutron count, and in a much smaller rate, reducing the probability of a large dead time effect. The purpose of the present section is to calculate the effect of the external source variance on the detection signal in the SVM formalism. For such purpose, we must first assume that the pulse source distribution is well characterized, meaning that the Mean, Variance and Skewness of the number of neutrons generated by the external source is known. Assume the measurement is done over N pulses, and the source amplitude on the kth pulse given is of the form Sd ðtÞ ¼ Aexp½−λd jt−T 0 j, where A is a random variable. Before we continue, some remarks are due 1. In the present section we only treat the decaying source, as the amplitude of the fixed source does not have any statistical variance. Thus, throughout this section, we assume Sf ¼ 0. Since the moments are all additive, this assumption does not pose any restrictions. 2. Looking at the first three central moments of the detection signal coming only from the decaying source, we observe that they are all linear in AT. Thus, for the sake of simplicity we may
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write EðXÞ ¼ AT B1 ,VarðXÞ ¼ AT B2 ,SkðXÞ ¼ AT B3 , where B1 ,B2 and B3 are well defined by Eq. (5.23). Next, we give the following simple proposition: Let X,E be two random variables, and assume that the conditional distribution f XjA ðxÞ is known, then the following relations hold EðXÞ ¼ EðEðXjAÞÞ VarðXÞ ¼ EðVarðXjAÞÞ−VarðEðXjAÞÞ SkðXÞ ¼ SkðEðXjAÞÞ þ EðSkðXjAÞÞ þ 3EðV ðXjAÞðEðXjAÞ−EðXÞÞÞ The first two equations may be found in many probability textbooks (see, for instance Ref. [14]), and the third equation may be derived using the same techniques. Using the upper mentioned notations, the conditioned central moment can be written as EðXjAÞ ¼ AB1 ,
VarðXjAÞ ¼ AB2 ,
SkðXjAÞ ¼ AB3
and then EðXÞ ¼ EðAB1 Þ ¼ EðAÞB1 Þ VarðXÞ ¼ EðAB2 Þ þ VarðAB1 Þ ¼ EðAÞB2 þ VarðAÞB21 SkðXÞ ¼ SkðAB1 Þ þ EðAB3 Þ þ 3EðAB2 ðAB1 −EðAÞB1 Þ ¼ SkðAÞB31 þ EðAÞB3 þ 3B1 B2 EðA2 −EðAÞAÞ ¼ SkðAÞB31 þ EðAÞB3 þ 3B1 B2 VarðAÞ
ð6:24Þ
Notice, if A is constant (that is, VarðAÞ ¼ SkðAÞ ¼ 0,EðAÞ ¼ AT ) then the last three equations are reduced to the moments of the decaying source in Section 5.
7. Numerical results and concluding remarks 7.1. Initial numerical testing As a first step of validation of the method, we have tested our results on a computer based simulator. Before introducing the results, it is important to state that the results do not indicate by any means the physical validity of the model, as all the physical assumptions of the model (point mode, single die away time and no fission chains) were assumed by the simulation itself. Therefore, the results should be viewed as a “sanity check” for the results. The method was tested over 10 simulations. In each simulation both the induced fission rate and the noise rate were altered in the following manner: the induced fission rate (at the beginning of the detection window) was valued 5:4 104 U and the noise rate was valued 5:4 104 ð1−UÞ, with U changing from 1 to 0 in jumps of 0.1. In each simulation, 119,000 histories were taken, which is equivalent to 119,000 pulses from the neutron source. The length of the detection window was T ¼ 10−3 , the thermal pulse decay coefficient was λd ¼ 104 , the average lifetime of a neutron was 1=λ ¼ 5 10−5 and the efficiency was 0.35 (since all the rates are accidental, the measurement – as well as the time scales – is all in arbitrary units). The source did not have any statistical variance, thus Eqs. (6.24) were not relevant. The results are presented in Table 1. As can be seen, there is a very good correspondence between the simulation results and the expected values given by the model. 7.2. Numerical comparison with existing models Once the initial validation of the formulas has been done, we turn to more elaborated examples, comparing the performance of the SVM and the earlier methods, demonstrating the possible advantages of using the SVM formulas. When comparing the current SVM model with the multiplicity model, three comparisons are due: the first is comparing the
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Table 1 Numerical simulation results. Sample Mean \ Var \ Sk columns introduce the simulated results, while Analytic Mean \ Var \ Sk introduce the predicted results. U
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
Sample Mean
Analytic Mean
Sample Var
Analytic Var
Sample Sk
Analytic Sk
40.69 38.24 36.73 34.1 31.91 29.76 27.60 25.38 23.22 21.06 18.86
40.4 38.45 36.08 33.91 31.74 29.59 27.42 25.26 23.09 20.93 18.77
66.22 61.99 56.67 51.51 47.62 42.08 37.63 37.63 28.21 23.49 18.91
65.41 60.75 56.08 51.42 46.75 42.35 37.42 37.42 28.09 23.43 18.77
127.54 120.81 101.65 93.04 85.13 74.24 66.84 48.85 40.40 29.18 18.83
135.23 116.67 105.79 94.91 84.03 73.15 62.28 51.34 40.52 29.64 18.77
inaccuracies due to the model assumptions, the second comparison regards the convergence rate (or, equivalently, the statistical error), and the third comparison concerns the use of the formulas for the source variation, rather than using the source monitor for normalizing the detection signal. For the first comparison, since the model assumptions are exactly the same, there is a full agreement between the results. For instance, the expression for the first moment in Ref. [6] can be explicitly obtained by using the approximation eλT ¼ 1 (which is assumed in Ref. [6]) in the first formula in (5.21). Notice, this does not mean that the two methods will give the same approximation, since the two methods sample a different random variable, and a statistical error is always expected—but rather the two methods will converge to the same estimation of the fissile mass. Turning to the second comparison, we recall that the two methods sample different random variables: while in the SVM method we sample the first three central moments, in the multiplicity method described in Ref. [6] we sample the Singles, Doubles and Triples rate, which are, eventually, proportional to the first three factorial moments (an exception here is the first moment, which is exactly the same in both methods), see Eqs. (3.43), (3.46) and (3.52) in Ref. [6]. On the other hand, we notice that both the central and the factorial moments may be written as simple algebraic expression of the moments (of the same distribution), with the same dominant part. That is, both the nth central and factorial moment may be written in the form EðX n Þ þ ðalgebraic combination of lower termsÞ. For instance, the second central moment – the variance – may be written as VarðXÞ ¼ EðX 2 Þ−E2 ðXÞ, and the second factorial moment may be written as EðX 2 Þ−EðXÞ. Since the dominant error comes from the highest moment sampled, this means that both methods converge in a similar rate. This fact was demonstrated experimentally for passive measurements in Ref. [8]. Once again, this does not mean that the two methods will give the same result, but rather for a sufficiently large amount of statistics, both methods will yield the same error-bar. To demonstrate the advantage in applying formulas 6.24, we carried out three numerical simulations with the following specifications: we have simulated a Pu decaying source, with an average initial source amplitude of EðAÞ ¼ 105 , the source standard deviations were 5%, 10% and 15%. The thermal die away time 1=λd ¼ 895 μs, the fast die away time was 1=λ ¼ 40 μs, the detection efficiency was P d ¼ 0:15, the source to noise ratio was U¼ 0.3, the target interval was T ¼ 1500 μs, and we have opened the target interval 500 μs after the pulse has started, thus T 0 ¼ 2000 μs. The simulation also registered the exact source amplitude in each pulse, which will serve as our source monitor. The results for the SVM method are given in Table 2. Once again, there is a very good correspondence between the simulation results and the expected values given by the model. As expected,
Table 2 Numerical simulation results with source variation. Sample Mean \ Var \ Sk columns introduce the simulated results, while Analytic Mean \ Var \ Sk introduce the predicted results. Source variance (%)
Sample Mean
Analytic Mean
Sample Var
Analytic Var
Sample Sk
Analytic Sk
5 10 15
16.451 16.451 16.451
16.458 16.458 16.458
22.568 24.544 27.899
22.627 24.658 28.044
37.153 45.440 58.585
37.358 45.486 59.034
Table 3 Numerical simulation with source monitor normalization. Sample Mean \ Var \ Sk columns introduce the simulated results, while Analytic Mean \ Var \ Sk introduce the predicted results. Source monitor uncertainty (%)
Analytic Mean
Sample Mean
Analytic Var
Sample Var
Analytic Sk
Sample Sk
0 1 2 3 4 5 8
16.458 16.458 16.458 16.458 16.458 16.458 16.458
16.449 16.450 16.456 16.467 16.477 16.494 16.561
21.949 21.949 21.949 21.949 21.949 21.949 21.949
22.005 22.037 22.138 22.324 22.575 22.925 24.358
34.648 34.648 34.648 34.648 34.648 34.648 34.648
35.358 35.510 36.270 37.611 39.643 42.260 54.03
the largest error is in the third moment, with a maximal error of less than 1%. In the present numerical simulation we observe a better correspondence between the simulation results and the numerical prediction, since the total count is smaller, thus the second and third central moments are less sensitive (Table 3). Next, we used the source monitor readings to normalize the number of detections in each pulse. This was done first with exact source amplitude, and then redone with a 1%, 2%, 3%, 4%, 5% and 8% uncertainty in the source monitor. Results are given in the table below. As can be observed, the uncertainty on the source amplitude has no effect on the mean (this was fully expected), little effect on the variance and a very large effect on the skewness. As can be seen, for an uncertainty of 5%, we see a 20% error in the skewness. When looking at an 8% uncertainty – which is expected in a typical source monitor, counting an average of 200 neutrons – we see a 10% error in the variance and a 58% error in the skewness. Even a 5% uncertainty should require about 400 readings in the source monitor detectors, which is much more than is expected. To remain in the same error bar as the SVM method, an uncertainty of 3% is needed – or a total of about 1300 counts in source monitor detector – an extremely high number.
8. Conclusions In the paper, we have introduced a new method for analyzing the detection signal arriving from an active interrogation facility. This method offers one main potential advantage over the existing model: the effect of the source variation is fully modeled. Looking at the numerical results, we see that using the analytic model presented in the paper for the source variance, rather than normalizing the pulses with the source detection readings, may reduce the error in the measured values for the second and third moments dramatically. While the method was tested numerically, the method was not yet tested in any actual measurement facility. Therefore, before the method can be claimed to be operational, a proper benchmarking procedure must be done. In particular, the physical assumption
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made (single energy group, point model and no fission chains) must be carefully checked.
References [1] M.S. Krick, Thermal Neutron Multiplicity Counting of Samples with Very Low Fission Rate, LA-UR-97-2649, 1997. [2] P. Baeten, M. Bruggeman, R. Carchon, Nuclear Instruments and Methods in Physics 428 (2–3) (1999) 466. [3] N. Ensslin,W.C. Harker, M.S. Krick, D.G. Langner, M.M. Pickvell, J.E. Stewart, Application Guide to Neutron Multiplicity Counting, Los Alamos Manual, LA13422-M. [4] N. Ensslin, W.H. Geist, M.S. Krick, M.M. Pickrell, Active Neutron Multiplicity Counting, LA-UR-07-1403. [5] M.M. Pickrell, An Alternative Method for Analyzing Active Multiplicity Data, LA-UR-05-4278.
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[6] W. Hage, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 551 (11 October (2–3)) (2005) 396. [7] W. Hage, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 614 (2010) 72. [8] C. Dubi, T. Ridnik, I. Israelshvili, J. Bagi, J. Huszti, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 673 (1 May) (2012) 111. [9] K. Bohnel, Nuclear Science and Engineering 90 (1985) 75. [10] A. Ocherashvili, E. Roesgen, A. Beck, E.N. Caspi, M. Mosconi, J-M. Chrochemore, B. Pedersen, Journal of Instrumentation 7 (2012). [11] B.C. Diven, H.C. Martin, R.F. Taschek, J. Terrell, Physical Review 101 (1956) 1012. [12] S. Okajima, Y. Yamane, Y. Takemoto, T.i. Sakurai, Journal of Nuclear Science and Technology 37 (August (8)) (2000) 720. [13] D.M. Cifarelli, W. Hage, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 251 (1 November (3)) (1986) 550. [14] S. Ross, A First Course in Probability, 6th edition, Pearson Prentice Hall.
Contents lists available at SciVerse ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
A novel method for active fissile mass estimation with a pulsed neutron source C. Dubi a,n, T. Ridnik a, I. Israelashvili a, B. Pedersen b a b
Physics Department, Nuclear Research Center of the Negev, POB 9001, Beer Sheva, Israel Nuclear Security Unit, Institute for Transuranium Elements, Via E. Fermi, 2749 JRC, Ispra, Italy
art ic l e i nf o
a b s t r a c t
Article history: Received 6 September 2012 Received in revised form 1 March 2013 Accepted 1 March 2013 Available online 21 March 2013
Neutron interrogation facilities for mass evaluation of Special Nuclear Materials (SNM) samples are divided into two main categories: passive interrogation, where all neutron detections are due to spontaneous events, and active interrogation, where fissions are induced on the tested material by an external neutron source. While active methods are, in general, faster and more effective, their analysis is much harder to carry out. In the paper, we will introduce a new formalism for analyzing the detection signal generated by a pulsed source active interrogation facility. The analysis is aimed to distinct between fission neutrons from the main neutron source in the system, and the surrounding “neutron noise”. In particular, we derive analytic expressions for the first three central moments of the number of detections in a given time interval, in terms of the different neutron sources. While the method depends on exactly the same physical assumptions as known models, the simplicity of the suggested formalism allows us to take into account the variance of the external neutron source—an effect that was so far neglected. & 2013 Elsevier B.V. All rights reserved.
Keywords: Neutron detection Neutron multiplicity method Fissile mass estimation Active neutron interrogation
1. Preliminaries 1.1. Introduction Fissile mass estimation by neutron multiplicity counting may be divided into two main categories: passive methods, relying on spontaneous fissions from the tested sample, and active methods, where fissions are induced on the tested material by an external neutron source. Active methods can also be divided into two categories: fixed and pulsed source. While passive methods are simpler to handle (both in terms of the system engineering and in the mathematical analysis), they suffer from two major disadvantages: first, they are restricted to fissile materials that have spontaneous fissions, and second, they often demand relatively large measurement times, together with a large number of detectors. The multiplicity method [1–3] is traditionally used in both passive and active fissile mass estimation for distinction between the main fission source and the surrounding neutron noises. The multiplicity method was originally developed for passive interrogation, and was later extended to active interrogation facilities for both fixed source [4,5] and pulsed source [6,7]. The last two
n
Corresponding author. Tel.: þ972 52 913313. E-mail address: [email protected] (C. Dubi).
0168-9002/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nima.2013.03.004
are, to the best of our knowledge, the first (and at present the only) analytic treatments to the detection signal from an active interrogation facility with a pulsed source. Recently, the authors have introduced a new method for analyzing passive measurements, named the SVM method (see Ref. [8]), standing for Skewness, Variance and Mean. Unlike the multiplicity method, the SVM does not measure the singles, doubles and triples rate in the detection signal, but rather measures the Mean, Variance and third central moment (Skewness) of the number of detections in a given time interval. In the present work we extend the SVM method to pulsed source active measurements. In particular, we provide full analytic expressions for all the first three central moments of the number of neutron detections in a given time interval. 1.2. Motivation While active measurement facilities with a pulsed source offer some very attractive advantages, for both the nuclear safety and safeguard community, it is not fully understood how to properly analyze the detection signal in order to separate the fission source from the additional neutron sources in the system. The currently available models [6,7], while very enlightening and constructive, to the best of our knowledge, are neither properly benchmarked, nor are there any academic reports on an extensive set of laboratory tests to the model (although the theoretical results
C. Dubi et al. / Nuclear Instruments and Methods in Physics Research A 715 (2013) 62–69
were published 7 years ago). Consequently, there are many unresolved questions regarding the applicability of the current methods. For instance, applicability of the Bohnel Method [9], which is the standard neutron multiplicity counting method for handling fission chains in the tested material (and was often tested for passive interrogation), may be questioned due to the (relatively) high probability of a fast neutron that forms the tested sample to be thermalized, and then returns to the sample and creates a fission (this is true since in most active measurement facilities, unlike in passive measurement facilities, the moderator is not separated from the tested sample by a thermal absorbent, see Ref. [10]). Another effect that was not treated analytically so far is the statistical variance in the number of neutrons generated by the external neutron source. In the existing models, it is assumed that the number of neutrons generated in each pulse is fixed, while, in practice, there is an expected statistical variance in the number of neutrons generated by the neutron generator in each pulse. This gap between the theory and practice, points out to a clear justification for the construction and further development of new analytic tools describing more elaborately and in different perspectives the measurement results from active pulsed source interrogation facilities. Since the physical assumptions in the present model are the same as those assumed in Ref. [6], there is a clear resemblance to the results obtained by Hage in Ref. [6] (although a different analysis is carried out). However, using the present model, we may now give analytic treatment to at least one effect that was not fully treated in Ref. [6]—the statistical variations in the flux generated by the external neutron source.
2. Description of measurement system and defining the problem 2.1. Configuration of a pulsed source interrogation facility Pulsed source active measurement systems have, in general, the following configuration: inside the sample cavity, which is located inside a block of a neutron moderator – typically graphite or lead – a neutron source (here referred to as the external neutron source) is placed together with the tested sample. The external side of the moderator is surrounded by a thermal neutron absorber and a ring of detectors is placed around it. The neutron source releases a pulse of fast neutrons, which are then thermalized, such that at the end of the thermalization process (or “thermal buildup”), a so called “thermal pool” is created inside the sample cavity. During the thermal buildup the detectors are not operational. Once the thermal buildup is over, two procedures take place: first, the number of thermal neutrons (or equivalently, the thermal flux), decreases exponentially, due to absorption and leakage. Second, the thermal flux creates induced fissions in the tested sample. This stage is referred to as the “thermal decay”. In a typical system, the duration of the thermal buildup is about 200 μs, and the duration of the thermal decay is several milliseconds (between the two stages there is, in most cases, a transition stage, in which the detectors are not operational). The fission rate in each point of time, during the thermal decay, is proportional to both the thermal flux in the sample cavity and the mass of the tested sample. In the fission process, fast neutrons are released into the system. Some of these fast neutrons escape both the moderator block and the thermal absorbent layer surrounding it, and reach the detector ring. Since the moderator block is surrounded by a thermal absorbent, only fast neutrons that were created in the induced fissions are likely to reach the detector.
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Assuming that there are neither fast fissions due to neutrons created from the tested sample itself, nor are there any neutron sources in the system, other than the induced fission source, then the average number of detections should be proportional to the mass of the detected sample. So, after a simple calibration, we can use the active measurement system to estimate the mass of the sample. Obviously, for this to work, a single pulse of neutrons is not enough, and the entire procedure must be repeated over, with the simple law of error proportional to the reciprocal of the square root of the number of repetitions. However, in most measurement systems, the assumption that there are no additional neutron sources (or fissions coming from fast neutrons created by the tested sample) is simply not true, and the naive estimation would result in a large over-estimation of the SNM mass. Since the origin of the neutron cannot be determined by detector technology, we must use a “mathematical filter”, determining the source to noise ratio in the measurement. 2.2. Assumptions on the system and problem definition Our model assumptions are listed below. These assumptions are standard assumptions in neutron multiplicity counting (see, for instance [6]). 1. The neutrons are averaged both in energy and space: we assume a point-wise, single energy source. Effectively, this assumption allows us to refer to a single detection probability and a single die away time for all the neutrons. 2. The number of induced fissions in the sample is proportional to its mass by a known proportion: in general, there might be a correlation between the geometry of the tested sample and the number of induced fissions. However, since we assume that the thermal pool is big enough (with regard to the tested sample), this correlation is neglected. 3. The thermal flux in the thermal pool decays exponentially with a single (known) decay constant. The last two assumptions can be summarized as follows: assume that the thermal buildup is over at time T0, then for every t≥T 0 , the probability for a thermal fission in the time interval ½t,t þ dt is given by A0 exp½−λd jt−T 0 j dt
ð2:1Þ
where λd is the decay coefficient of the thermal flux, and A0 is proportional to the mass of the tested sample. The coefficient A0, which relates the fission source amplitude with the mass of the tested material, has a clear physical interpretation. The total fission rate in the tested sample is given by SðtÞ ¼ ∬ Φðt,pÞsf ρðpÞ dvp
ð2:2Þ
V
where V is the volume of the tested sample, ΦðtÞ is the thermal flux, sf is the fission microscopic cross-section and ρ is the density of the tested material. Assuming that the volume of the tested sample is small with respect to the entire sample cavity, we may assume that the flux is fixed over the volume of the tested sample, resulting in SðtÞ ¼ ΦðtÞsf ρV
ð2:3Þ
If the thermal flux in the sample cavity decays exponentially, we can write ΦðtÞ ¼ Φ0 e−λd jT 0 −tj . Since ρV is nothing more than the mass of the tested sample, comparing Eqs. (2.1) and (2.3) gives A0 ¼ Φ0 sf If the thermal flux generated by the neutron generator (after the moderation stage) is known, then A0 can be evaluated in advance. However, in most measurement systems, A0 is calibrated (together
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with other system parameters, such as the detection efficiency and the neutron die away time) by performing the measurements on a well-characterized known sample. As stated before, A0 is not truly a constant, and may vary between the different pulses. At the present stage we will assume that it is fixed, and will drop this assumption in Section 6.
2.3. The neutron sources The neutron sources affecting the measurement may be divided into three categories : Decaying sources, fixed sources and fission chains [6]. The decaying sources are the following: 1. The fission source is described in Section 2.2. 2. Leakage from the thermal pool: although the sample cavity is surrounded by a thermal absorber, it is expected that some of the “thermal pool” neutrons might eventually escape absorption and reach the detector ring. Since the number of neutrons escaping absorption should be proportional to the thermal flux in sample cavity, the amplitude of this source may be written as A1 exp½−λd jt−T 0 j:
ð2:4Þ
The sources fixed in time are the following: 1. Spontaneous fissions: this source is due to the spontaneous fissions in the even isotopes in many SNM, often present in MOX samples and other compositions containing Pu. 2. Random neutrons noise: a random neutron source with a fixed rate is expected mainly due to ðα,nÞ reaction in the sample itself, and cosmic radiation. One final neutron source is the fission chains in the tested material. Notice, this fission source, although physically very much the same as the main fission source, is not accounted for in formula 2.1. The traditional treatment to fission chains in the tested sample is usually through the well-known Bohnel method [9]. However, since in most pulsed source neutron interrogation facilities the tested sample is surrounded by a heavy reflector and is not protected by a layer of thermal absorber, the applicability of the Bohnel method is very questionable, and to the best of our knowledge, the model was not tested. One way or the other, in the present paper, as well as in Refs. [6,7], the fission chains are neglected.
3. Notations and definitions 3.1. Generating functions In the following section we introduce the concept of the generating function, which will often appear throughout our analysis, and point out some of its features for later use. Let X be a discrete valued random variable with distribution defined by the series fan g. That is, for every n, PðX ¼ nÞ ¼ an . The generating function of X is defined by the formal power series ∞
j
g X ðxÞ ¼ ∑ x aj
ð3:5Þ
j¼0
one of the most important features of the generating function is the fact that the moments of the distribution may be written explicitly in terms of the generating function and its derivatives by
the following formulas: ∞
d g X ðxÞx ¼ 1 dx n¼1 ∞ d d x g X ðxÞ EðX 2 Þ ¼ ∑ n2 an ¼ dx dx n¼2 x¼1 ∞ d d d 3 3 x x g X ðxÞ EðX Þ ¼ ∑ n an ¼ dx dx dx n¼3 x¼1
EðXÞ ¼ ∑ nan ¼
ð3:6Þ
In some of the cases presented in the following study, the random variable of interest – as well as the corresponding distributions – will be time dependent. In such cases we will denote an ¼ an ðtÞ. Such time dependence does not affect the validity of Eqs. (3.6). In the present study we are also interested in the second and third central moments, defined by VarðXÞ ¼ EððX−EðXÞÞ2 Þ ¼ EðX 2 Þ−EðXÞ2 SkðXÞ ¼ EððX−EðXÞÞ3 Þ ¼ EðX 3 Þ−3EðXÞEðX 2 Þ þ EðX 3 Þ
ð3:7Þ
3.2. General purpose notations and definition A neutron source is defined by two attributes: the event rate (or source amplitude) and the source distribution (or neutron multiplicity). The event rate, here denoted by S and a proper sub index, describes the probability that an emission of neutron event will occur in the interval ½t,t þ dt, and the source distribution describes the probability that once an emission has occurred, exactly ν neutrons would be released. We shall denote the amplitude of the time decaying source by Sd(t) and the amplitude of fixed source by Sf. As described, the amplitude of the decaying source may be written as Sd ðtÞ ¼ ðA0 þ A1 Þexp½−λd jt−T 0 j
ð3:8Þ
(where A0 and A1 were defined in Section 2), and the fixed source amplitude may be divided into the following two components: Sf ¼ Sfission þ SRandom
ð3:9Þ
For the sake of simplicity, we here on denote A0 þ A1 ¼ AT . We denote the distribution of decaying source by qd,1 ,qd,2 ,…,qd,n and that of the fixed source by qf ,1 ,qf ,2 ,…,qf ,n . Another notation that will be often used in the present study is the Diven factors. For a given distribution fqj g∞ , the nth Diven j¼1 factor is defined as ℓ! qℓ : ðℓ−nÞ! ℓ¼n ∞
Dn ¼ ∑
The importance of the Diven factors in neutron multiplicity counting has been largely studied, see Refs. [11,12]. Notice, the Diven factors depend both on the materials and isotopic composition of the tested sample, and on the noise to source ratio. Writing down the total source (fixed and decaying) Diven factors in terms of the neutron distribution for the different materials and the noise to source ratio is a well-known (and fairly simple) algebraic procedure and can be found in many sources (see Refs. [8,1,6]).
4. The stochastic transport equation 4.1. Introduction Our next order of business is to derive and solve the stochastic transport equation for the number of detections in a given time interval. The stochastic transport model involves the dynamics through time of two functions. The first concerns the evolution through time of the detection signal generated by a single neutron, and the second is the extension of the first for a source continuous in time. Unlike the spatial transport equation, the Boltzmann
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equation, where the preservation laws are written in terms of the average number of neurons in a given volume, the preservation laws for the stochastic transport equation, are stated in terms of the event probabilities in the lifetime of a single neutron. In our context, when looking at an infitisimal time window dt, sufficiently shorter than the die away time of the neutron, a neutron coming from the sample may create exactly one of the following three events: 1. The neutron is absorbed in the detector, causing a reading in the detection signal. This event shall be referred to hereon as a detection event. 2. The neutron is lost in the system – either by leakage or by absorption – but without being detected. This event shall be referred to as a losing event. 3. The neutron does nothing. Denote by λdetection the probability per time unit that a neutron will cause a detection event, and by λlost the probability per time unit that the neutron will cause a losing event. We shall also denote λ ¼ λdetection þλlost . λ is then the probability per time unit that the neutron will create any event. In the current terminology, the average lifetime of a neutron (die away time) in the system is given by 1=λ. One should notice that both scattering and fission probabilities are not mentioned. Since the system is averaged both in space and energy, scattering of a neutron is of no importance, and may be neglected. As stated, we assume that there are no fission chains. Returning to the stochastic transport, the distribution of the detection signal is uniquely determined by three elements: The first is the behavior of a single neutron, manifested by the event probabilities λdetection and λlost . The second is the rate and the statistics of the different sources, and finally, the duration of the time interval. The objective of this section is to construct an analytical expression for the distribution of the number of counts in a time interval, using the above mentioned notations. It should be mentioned that the analysis carried out in this section is very close – often similar – to the analysis carried out in Ref. [13]. For the sake of completeness and easy reading, we sometimes repeat calculations carried out in Ref. [13]. 4.2. The detection signal generated by a single neutron Consider a neutron detector, counting the number of neutron detections on an interval of length T, starting, for convenience, at t ¼ −T and ending at t¼ 0. The interval ½−T,0 shall be referred to as the target interval (or target window). Next, consider the case where a single neutron enters the system t seconds before the target interval ends. Notice, the time of entrance, −t, may or may not be in the target interval. We denote by an(t) the probability for the neutron to generate n detector counts in the target interval. We should bear in mind that the probability must be zero for every n≥2, since we assumed that there are no fission chains. To understand the behavior of an(t), we look on an infitisimal time interval ½−ðt þ dtÞ,−t. During this time interval, exactly one of the following events may happen: 1. The neutron will be detected. The probability for such an event is λdetection dt. In such case, a detector reading in the target interval will be set if and only if −t belongs to the target interval. Notice, if −t does not belong to the target interval, the detector reading might be set (this depends on the actual setting of the system), but it will be ignored, as by definition we are only counting detections in the target interval. 2. The neutron will either leak or be absorbed without a detection reading. The probability of such an event is λlost dt. In such case,
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it is clear that no detections will be registered in the target interval. 3. The neutron will do nothing. The probability for such an event is 1−λdt. In such case, the probability for n detector readings is equal to the probability as if the neutron was released at time −t. Summing the last three options in the complete probability theorem, we may write a0 ðt þdtÞ ¼ ð1−λ dtÞa0 ðtÞ þ λlost dt þð1−H ½0,T Þλdetection dt a1 ðt þdtÞ ¼ ð1−λ dtÞa1 ðtÞ þ H ½0,T λdetection dt
ð4:10Þ
(here H ½0,T ðtÞ is the characteristic function of the interval ½0,T, satisfying H ½0,T ðtÞ ¼ 1 if 0≤t≤T and H ½0,T ðtÞ ¼ 0 otherwise). Dividing by dt and taking the limit dt-0, we obtain the differential form: a′0 ðtÞ ¼ −λa0 ðtÞ þλlost þ ð1−H ½0,T Þλdetection a′1 ðtÞ ¼ −λa1 ðtÞ þH ½0,T Þλdetection
ð4:11Þ
The initial conditions a0 ð0Þ ¼ 1,a1 ðtÞ ¼ 0 express the fact that a neutron released once the target interval has closed cannot cause a detector reading in the target interval. Eqs. (4.11) (with the proper initial conditions) may be analytically solved, resulting with a0 ðtÞ ¼ 1− a1 ðtÞ ¼
λdetection ½1−e−λt −ð1−e−λðt−TÞ ÞU 0 ðt−TÞ λ
λdetection ½1−e−λt −ð1−e−λðt−TÞ ÞU 0 ðt−TÞ λ
ð4:12Þ
Eqs. (4.12) gives us the full description of the detection signal generated by a single neutron. As can be seen, both a0 ðtÞ and a1 ðtÞ depend only on the total cross-section λ and the ratio P d ¼ λdetection =λ. Pd is known as the detection probability or the system efficiency, and describes the probability that a neutron entering the system will create a reading on the detectors. 4.3. The detection signal generated by a continues source We now turn to extend the results from the previous section to the general case of a continuous source, with a general amplitude S(t) and multiplicity distribution fqν gM ν ¼ 1 . The procedure is a wellknown technique, often referred to as the Bartlet equation, and can be found in the literature. Yet, for the sake of completeness, we will give a full description of the process. The setting is the following: As in Section 4.2, the detector is activated during the target interval ½−T,0. t seconds before the target interval is closed, the source is “activated”. We denote by bn(t) the probability that there will be exactly n detector readings in the target interval. Let us denote the following two generating functions: 1
hðx,tÞ ¼ ∑ aj ðtÞxj , j¼0
∞
gðx,tÞ ¼ ∑ bj ðtÞxj :
ð4:13Þ
j¼0
We now extend t by an infitisimal quantity dt and express bn ðt þ dtÞ. Looking at the time interval ½−ðt þ dtÞ,−t, both the fixed and decaying source may either release a neutron or not. If no neutrons were released, then the probability for n detections is now bn(t). If ν neutron were released, the probability for n detections is given by ! n
∑
j¼0
bj ðtÞ
∑
k1 þ k2 þ k3 þ ⋯ þ kν ¼ n−j
ak1 ðtÞak2 ðtÞ…akν ðtÞ
ð4:14Þ
(The last is obtained by the complete probability theorem, with the distinct events determined by the number of detection readings generated by each of the neutrons released into the system). We notice that the inner sum in formula 4.14 is exactly the ðn−jÞ th coefficient of hðx,tÞν , and thus the outer sum is exactly the
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nth coefficient of gðx,tÞhðx,tÞν . Since the probability for a source events in the interval ½−ðt þ dtÞ,t is SðtÞ dt, we may write M gðx,t þ dtÞ ¼ gðx,tÞð1−SðtÞ dtÞ þ SðtÞ dt ∑ qν gðx,tÞhðx,tÞν ν¼0
As mentioned, this is a well-known procedure and will not be repeated here. Let us write gðxÞ ¼ expðGðxÞÞ, where G(x) is well defined by Eq. (4.16). A direct computation of all first three moments results with ∞
∑ nbn ¼
or
n¼0
M gðx,t þ dtÞ−gðx,tÞ ¼ −gðx,tÞ SðtÞ−SðtÞ ∑ qν hðx,tÞν : dt ν¼0
∞
∑ n2 bn ¼
Notice, in the last we have ignored “second order” events; meaning, the probability for two source events in an infitisimal interval is ignored. Since the next step is turning to a differential form, and since second order events are proportional to dt2, these events are negligible. Taking dt-0 we then have M ∂gðx,tÞ ¼ −gðx,tÞ SðtÞ−SðtÞ ∑ qν hðx,tÞν ð4:15Þ ∂t ν¼0 Eq. (4.15) has an explicit solution, given by Z t M SðtÞ 1− ∑ qν hðx,tÞν dt : gðx,tÞ ¼ exp −
ð4:16Þ
ν¼0
0
Breaking the total source into a sum of two sources (one constant in time and one decaying in time) we have (with notations as in Eqs. (3.2)) Z t M gðx,tÞ ¼ exp AT e−λd jT−τj 1− ∑ qd,ν hðx,tÞν dτ 0
Z exp
t
0
ν¼0
M
Sf 1− ∑ qf ,ν hðx,tÞν dτ
ð4:17Þ
ν¼0
Notice that in Eq. (4.17) the generating function is presented as a product of two generating function: the first for the decaying source and the second for the fixed source. Since the fixed and decaying source are not correlated, this was expected. Eq. (4.17) depends on two time variables: the first, t, denotes the time (before the target window closes) in which the source is activated, and the second, T, is the duration of the target interval. If T0 is the time difference between two consecutive pulses, then the pulses start T0 before the target window closes . Thus, Eq. (4.17) becomes Z T 0 M gðxÞ ¼ exp AT e−λd jT 0 −τj 1− ∑ qd,ν hðx,tÞν dτ ν¼0
0
Z
∞
exp 0
M Sf 1− ∑ qf ,ν hðx,tÞν dτ
dg ¼ ðG′ðxÞhðxÞÞjx ¼ 1 ¼ G′ð1Þ dxx ¼ 1
ð4:18Þ
ν¼0
The transition from Eqs. (4.17) to (4.18) is formally tricky, as the upper bound of the integration in both integrals should be the same. To overcome this formality, we must multiply the integrated function by a step function with appropriate boundaries, this step is neglected, as the outcome is very much self explanatory. Now, the distribution depends on two time variable: T, which is embedded in hðx,tÞ, and T0.
5. Explicit formulas for the central moments After constructing and solving the stochastic transport equation, our next order of business is to calculate the first three central moments of the distribution given by the generating function in formula 4.18, in terms of the system parameters, the source amplitude and the source Diven factors. For actually solving the set of equations, one must write down the general source amplitude and Diven factors in terms of the different source amplitudes and different source distributions.
n¼0
d dg x ¼ ðG′ðxÞhðxÞ þ xðG″hðxÞ þ ðG′ðxÞÞ2 hðxÞÞÞjx ¼ 1 dx dx x ¼ 1
¼ G′ð1Þ þ G″ð1Þ þ ðG′ð1ÞÞ2
∞
∑ n3 bn ¼
n¼0
d d dg x x dx dx dx x¼1
¼ ðG′ðxÞhðxÞ þ3xðG″hðxÞ þ ðG′ðxÞÞ2 hðxÞÞÞjx ¼ 1 þ ðx2 ðG‴ðxÞhðxÞ þ 3G″ðxÞhðxÞ þ ðG′ðxÞÞ3 hðxÞÞÞjx ¼ 1 ¼ G′ð1Þ þ 3ðG′ð1Þ þ ðG′ð1ÞÞ2 Þ þ G‴ð1Þ þ 3G″ð1ÞG′ð1Þ þ ðG′ð1ÞÞ3
A straight forward computation of all three integrals in G′ð1Þ,G″ð1Þ and G‴ð1Þ gives G′ð1Þ ¼ AT P d Dd1 I d,1 þSf P d Dd1 I f ,1 G″ð1Þ ¼ AT P 2d Dd1 I d,2 þSf P d Df2 I f ,2 G‴ð1Þ ¼ AT P 3d Dd3 I 3 þ Sf P d Df3 I f ,3
ð5:19Þ
ðDfj Þ
Ddj
are the jth Diven factor of the decaying (fixed) where R∞ source, and the integral I f ,j ¼ 0 aj1 ðtÞ dt were explicitly given in R T 0 λ jT −tj j Ref. [8], and I d,j ¼ 0 e d 0 a1 ðtÞ dt is given by I d,1 ¼ Id,2 ¼
e−ðλ þ λd ÞT 0 ðλd eλd T 0 ðeλT −1Þ−λeλT 0 ðeλd T −1ÞÞ λd ðλd −λÞ
eλT−2ðλ þ λd ÞT 0 ðð2λd −λÞeλT þ 2λd T 0 ðeλT −1Þ2 þ λe2λT 0 ð2λd e2λd ðeλT −1Þ þ λeλT ðe2λd T −1ÞÞÞ 2λd ðλd −λÞð2λd −λÞ
I d,3 ¼
e−2λT−3λT 0 −3λd T 0 3λd ð3λd −λÞð2λ2 −5λλd þ 3λ2d Þ
f9λ3d e2λT þ 3λd T 0 ðeλT −1Þ3 þ 2λ3 e2λT þ 3λT 0 ð1−e3λd T Þ þ 9λλ2d ðeλT −1Þ2 ðe3λd T þ 3λT 0 −e2λT þ 3λd T 0 þ e3λd T 0 þ 3λT Þ þ λ2 λd ðeλT −1Þð9eλT þ 3λd T þ λT 0 −3e3λd þ 3λT 0 þ 2e2λT þ 3λd T 0 þ 2e4λT þ 3λd T 0 −4e3λT þ 3λd T 0 Þg
ð5:20Þ
In the simple case where we neglect all neutrons from fissions made before the target interval opens, we have T ¼ T 0 , and the last reduces to I d,1 ¼
λ−λe−λd T 0 þ λd ðe−λT 0 −1Þ ðλ−λd Þλd
I d,2 ¼ e−λd T 0
I d,3 ¼ e−λT 0
! eλd T 0 −1 eðλd −2λÞT 0 −1 2 eðλd −λÞT 0 −1 þ þ λd λd −2λ λ−λd
ð5:21Þ
e−λd T 0 −1 eðλd −λÞT 0 −1 3ðeðλd −λÞT 0 −1Þ 3ðeðλd −λÞT 0 −1Þ þ þ þ λd 3λ−λd λd −2λ λ−λd ð5:22Þ
Using the standard relations between the moments and the central moments, together with Eq. (5.19), we have EðXÞ ¼ AT P d Dd1 I d,1 þ Sf P d Dd1 I f ,1 VarðXÞ ¼ AT P 2d Dd2 I d,2 þSf P d Df2 I f ,2 þ EðXÞ SkðXÞ ¼ AT P 3d Dd3 I d,3 þ Sf P d Df3 I f ,3 þ 3VarðXÞ−2EðXÞ
ð5:23Þ
We can now observe that, if the total cross-section λ, the detection efficiency, Pd, and the length of the target interval, T, are all known, then the first three central moments can be
C. Dubi et al. / Nuclear Instruments and Methods in Physics Research A 715 (2013) 62–69
determined by the source amplitudes and the first three Diven factors.
6. The effect of source variance on the moments of the detection signal In our analysis so far, we have assumed that the number of the neutrons generated by the external source does not change between the different pulses. In practical active measurement facilities, this assumption is not true, and a statistical variation in the number of neutrons generated in each pulse is expected. Such a variation will obviously have an impact on the central moments of the detection signal. In Ref. [7], the fact that the external neutron source may have a statistical variation is mentioned. In order to apply the method developed, it is suggested that one should either take into the account the only pulses with the same number of neutrons generated, or normalize the neutron counts in each pulse with the amplitude of the pulse . This however, raises two difficulties: First, applying the first option will cause us to lose a large number of generator pulses and thus lose efficiency. Second problem, regarding both methods, is that determining the number of neutrons generated in a given pulse may turn to be a complicated task: the number of neutrons generated in a given pulse can only be tested by neutron detection, in a set of detectors often referred to as the “source monitor”. This neutron counting in the source monitor should be held during the thermalization stage (otherwise, the fission source neutrons will affect the measurement). Thus determining the number of neutrons generated should be done in a very short time interval, and the number of detections expected is not very large (in a typical system, 100–200 counts). On the other hand, during the thermalization stage, the detector ring is highly saturated, due to the extremely large number of fast neutrons coming from the source. In fact, at the very beginning of the pulse (first 100 μs), the detectors are completely saturated, and the data collection in that region is more or less useless. Therefore, on top of the expected statistical error, a dead time error is expected. What seems to be a more effective and accurate solution is to characterize the source distribution in advance, by running the pulsed source without any SNM samples in the sample cavity, and then properly account for the effect of the source variance in the total detection signal. In such a calculation, the measurements can be made in a relatively long time window (during the thermal decay), yielding a larger neutron count, and in a much smaller rate, reducing the probability of a large dead time effect. The purpose of the present section is to calculate the effect of the external source variance on the detection signal in the SVM formalism. For such purpose, we must first assume that the pulse source distribution is well characterized, meaning that the Mean, Variance and Skewness of the number of neutrons generated by the external source is known. Assume the measurement is done over N pulses, and the source amplitude on the kth pulse given is of the form Sd ðtÞ ¼ Aexp½−λd jt−T 0 j, where A is a random variable. Before we continue, some remarks are due 1. In the present section we only treat the decaying source, as the amplitude of the fixed source does not have any statistical variance. Thus, throughout this section, we assume Sf ¼ 0. Since the moments are all additive, this assumption does not pose any restrictions. 2. Looking at the first three central moments of the detection signal coming only from the decaying source, we observe that they are all linear in AT. Thus, for the sake of simplicity we may
67
write EðXÞ ¼ AT B1 ,VarðXÞ ¼ AT B2 ,SkðXÞ ¼ AT B3 , where B1 ,B2 and B3 are well defined by Eq. (5.23). Next, we give the following simple proposition: Let X,E be two random variables, and assume that the conditional distribution f XjA ðxÞ is known, then the following relations hold EðXÞ ¼ EðEðXjAÞÞ VarðXÞ ¼ EðVarðXjAÞÞ−VarðEðXjAÞÞ SkðXÞ ¼ SkðEðXjAÞÞ þ EðSkðXjAÞÞ þ 3EðV ðXjAÞðEðXjAÞ−EðXÞÞÞ The first two equations may be found in many probability textbooks (see, for instance Ref. [14]), and the third equation may be derived using the same techniques. Using the upper mentioned notations, the conditioned central moment can be written as EðXjAÞ ¼ AB1 ,
VarðXjAÞ ¼ AB2 ,
SkðXjAÞ ¼ AB3
and then EðXÞ ¼ EðAB1 Þ ¼ EðAÞB1 Þ VarðXÞ ¼ EðAB2 Þ þ VarðAB1 Þ ¼ EðAÞB2 þ VarðAÞB21 SkðXÞ ¼ SkðAB1 Þ þ EðAB3 Þ þ 3EðAB2 ðAB1 −EðAÞB1 Þ ¼ SkðAÞB31 þ EðAÞB3 þ 3B1 B2 EðA2 −EðAÞAÞ ¼ SkðAÞB31 þ EðAÞB3 þ 3B1 B2 VarðAÞ
ð6:24Þ
Notice, if A is constant (that is, VarðAÞ ¼ SkðAÞ ¼ 0,EðAÞ ¼ AT ) then the last three equations are reduced to the moments of the decaying source in Section 5.
7. Numerical results and concluding remarks 7.1. Initial numerical testing As a first step of validation of the method, we have tested our results on a computer based simulator. Before introducing the results, it is important to state that the results do not indicate by any means the physical validity of the model, as all the physical assumptions of the model (point mode, single die away time and no fission chains) were assumed by the simulation itself. Therefore, the results should be viewed as a “sanity check” for the results. The method was tested over 10 simulations. In each simulation both the induced fission rate and the noise rate were altered in the following manner: the induced fission rate (at the beginning of the detection window) was valued 5:4 104 U and the noise rate was valued 5:4 104 ð1−UÞ, with U changing from 1 to 0 in jumps of 0.1. In each simulation, 119,000 histories were taken, which is equivalent to 119,000 pulses from the neutron source. The length of the detection window was T ¼ 10−3 , the thermal pulse decay coefficient was λd ¼ 104 , the average lifetime of a neutron was 1=λ ¼ 5 10−5 and the efficiency was 0.35 (since all the rates are accidental, the measurement – as well as the time scales – is all in arbitrary units). The source did not have any statistical variance, thus Eqs. (6.24) were not relevant. The results are presented in Table 1. As can be seen, there is a very good correspondence between the simulation results and the expected values given by the model. 7.2. Numerical comparison with existing models Once the initial validation of the formulas has been done, we turn to more elaborated examples, comparing the performance of the SVM and the earlier methods, demonstrating the possible advantages of using the SVM formulas. When comparing the current SVM model with the multiplicity model, three comparisons are due: the first is comparing the
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Table 1 Numerical simulation results. Sample Mean \ Var \ Sk columns introduce the simulated results, while Analytic Mean \ Var \ Sk introduce the predicted results. U
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
Sample Mean
Analytic Mean
Sample Var
Analytic Var
Sample Sk
Analytic Sk
40.69 38.24 36.73 34.1 31.91 29.76 27.60 25.38 23.22 21.06 18.86
40.4 38.45 36.08 33.91 31.74 29.59 27.42 25.26 23.09 20.93 18.77
66.22 61.99 56.67 51.51 47.62 42.08 37.63 37.63 28.21 23.49 18.91
65.41 60.75 56.08 51.42 46.75 42.35 37.42 37.42 28.09 23.43 18.77
127.54 120.81 101.65 93.04 85.13 74.24 66.84 48.85 40.40 29.18 18.83
135.23 116.67 105.79 94.91 84.03 73.15 62.28 51.34 40.52 29.64 18.77
inaccuracies due to the model assumptions, the second comparison regards the convergence rate (or, equivalently, the statistical error), and the third comparison concerns the use of the formulas for the source variation, rather than using the source monitor for normalizing the detection signal. For the first comparison, since the model assumptions are exactly the same, there is a full agreement between the results. For instance, the expression for the first moment in Ref. [6] can be explicitly obtained by using the approximation eλT ¼ 1 (which is assumed in Ref. [6]) in the first formula in (5.21). Notice, this does not mean that the two methods will give the same approximation, since the two methods sample a different random variable, and a statistical error is always expected—but rather the two methods will converge to the same estimation of the fissile mass. Turning to the second comparison, we recall that the two methods sample different random variables: while in the SVM method we sample the first three central moments, in the multiplicity method described in Ref. [6] we sample the Singles, Doubles and Triples rate, which are, eventually, proportional to the first three factorial moments (an exception here is the first moment, which is exactly the same in both methods), see Eqs. (3.43), (3.46) and (3.52) in Ref. [6]. On the other hand, we notice that both the central and the factorial moments may be written as simple algebraic expression of the moments (of the same distribution), with the same dominant part. That is, both the nth central and factorial moment may be written in the form EðX n Þ þ ðalgebraic combination of lower termsÞ. For instance, the second central moment – the variance – may be written as VarðXÞ ¼ EðX 2 Þ−E2 ðXÞ, and the second factorial moment may be written as EðX 2 Þ−EðXÞ. Since the dominant error comes from the highest moment sampled, this means that both methods converge in a similar rate. This fact was demonstrated experimentally for passive measurements in Ref. [8]. Once again, this does not mean that the two methods will give the same result, but rather for a sufficiently large amount of statistics, both methods will yield the same error-bar. To demonstrate the advantage in applying formulas 6.24, we carried out three numerical simulations with the following specifications: we have simulated a Pu decaying source, with an average initial source amplitude of EðAÞ ¼ 105 , the source standard deviations were 5%, 10% and 15%. The thermal die away time 1=λd ¼ 895 μs, the fast die away time was 1=λ ¼ 40 μs, the detection efficiency was P d ¼ 0:15, the source to noise ratio was U¼ 0.3, the target interval was T ¼ 1500 μs, and we have opened the target interval 500 μs after the pulse has started, thus T 0 ¼ 2000 μs. The simulation also registered the exact source amplitude in each pulse, which will serve as our source monitor. The results for the SVM method are given in Table 2. Once again, there is a very good correspondence between the simulation results and the expected values given by the model. As expected,
Table 2 Numerical simulation results with source variation. Sample Mean \ Var \ Sk columns introduce the simulated results, while Analytic Mean \ Var \ Sk introduce the predicted results. Source variance (%)
Sample Mean
Analytic Mean
Sample Var
Analytic Var
Sample Sk
Analytic Sk
5 10 15
16.451 16.451 16.451
16.458 16.458 16.458
22.568 24.544 27.899
22.627 24.658 28.044
37.153 45.440 58.585
37.358 45.486 59.034
Table 3 Numerical simulation with source monitor normalization. Sample Mean \ Var \ Sk columns introduce the simulated results, while Analytic Mean \ Var \ Sk introduce the predicted results. Source monitor uncertainty (%)
Analytic Mean
Sample Mean
Analytic Var
Sample Var
Analytic Sk
Sample Sk
0 1 2 3 4 5 8
16.458 16.458 16.458 16.458 16.458 16.458 16.458
16.449 16.450 16.456 16.467 16.477 16.494 16.561
21.949 21.949 21.949 21.949 21.949 21.949 21.949
22.005 22.037 22.138 22.324 22.575 22.925 24.358
34.648 34.648 34.648 34.648 34.648 34.648 34.648
35.358 35.510 36.270 37.611 39.643 42.260 54.03
the largest error is in the third moment, with a maximal error of less than 1%. In the present numerical simulation we observe a better correspondence between the simulation results and the numerical prediction, since the total count is smaller, thus the second and third central moments are less sensitive (Table 3). Next, we used the source monitor readings to normalize the number of detections in each pulse. This was done first with exact source amplitude, and then redone with a 1%, 2%, 3%, 4%, 5% and 8% uncertainty in the source monitor. Results are given in the table below. As can be observed, the uncertainty on the source amplitude has no effect on the mean (this was fully expected), little effect on the variance and a very large effect on the skewness. As can be seen, for an uncertainty of 5%, we see a 20% error in the skewness. When looking at an 8% uncertainty – which is expected in a typical source monitor, counting an average of 200 neutrons – we see a 10% error in the variance and a 58% error in the skewness. Even a 5% uncertainty should require about 400 readings in the source monitor detectors, which is much more than is expected. To remain in the same error bar as the SVM method, an uncertainty of 3% is needed – or a total of about 1300 counts in source monitor detector – an extremely high number.
8. Conclusions In the paper, we have introduced a new method for analyzing the detection signal arriving from an active interrogation facility. This method offers one main potential advantage over the existing model: the effect of the source variation is fully modeled. Looking at the numerical results, we see that using the analytic model presented in the paper for the source variance, rather than normalizing the pulses with the source detection readings, may reduce the error in the measured values for the second and third moments dramatically. While the method was tested numerically, the method was not yet tested in any actual measurement facility. Therefore, before the method can be claimed to be operational, a proper benchmarking procedure must be done. In particular, the physical assumption
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made (single energy group, point model and no fission chains) must be carefully checked.
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