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Assessment of the impact of diversification of uranium switching on the risk of its non-energy use
Progress in Nuclear Energy 110 (2019) 75–79
Contents lists available at ScienceDirect
Progress in Nuclear Energy journal homepage: www.elsevier.com/locate/pnucene
Assessment of the impact of diversification of uranium switching on the risk of its non-energy use
T
Vasily Glebov National Research Nuclear University MEPhI, 115409, Moscow, Kashirskoe Shosse, 31, Russia
A R T I C LE I N FO
A B S T R A C T
Keywords: Diversification of switching Significant quantity Low-enriched uranium NM safe management Insider
The main objective of the IAEA verification activities is the timely detection of the switching of a significant quantity of nuclear material (NM). However, from the point of nuclear security the multiple switching of "small" quantities of nuclear material with the aim of their subsequent collection also constitutes one of the main threat to ensuring the exceptionally peaceful use of nuclear materials. In this article this threat assessments were performed under the scenario of diversification of uranium switching. It is shown that diversification of 4%uranium switching can lead to a substantial increase (up to 30 times) of the risk of uranium disuse even with carrying out proper uranium control and protection. However this effect is observed only when there are large opportunities to search for insiders (the search area is comparable to the existing set of nuclear facilities). The obtained results also prove that a reduction in the level of NM safe management at facilities could allow the collector to complete a chain of unauthorized actions with a significant probability (PSQ > 0.1).
R (M ) = P (M )⋅Y (M ) ≤ M ,
1. Introduction In the problems related with NM safe management the threat to perform a chain of the unauthorized actions (UA) may be quantitatively assessed in the following risk terms:
R = P ⋅D ,
P = Pdiv⋅Pman,
(1)
where P is a total probability to perform all UA with uranium-bearing materials; Pdiv – probability of uranium switching from nuclear site; Pman – probability to successfully complete a chain of out-site operations with uranium; D – potential damage caused by destructive application of a nuclear explosive device (NED), which can be evaluated in the term of the energy yield (Y). In the case of uranium, the energy yield depends mainly on uranium amount and uranium enrichment. From the standpoint of NM safe management, any unauthorized switching of significant quantity (IAEA Safeguards Glossary, 2002) (SQ) of nuclear materials must be completely excluded. Within the frames of the risk assessment [Masterov et al., 2016] this means that:
RNED ≤ RCED,
(2)
where indices NED and CED are used to designate nuclear and chemical explosive devices respectively. If the energy yield (Y) of NED is measured in equivalent mass of chemical explosives (TNT, for instance), then condition (2) can be rewritten in the terms of relative risk:
M ∈ [M0 (x ), M1 (x )],
(3)
where M0 and M1 are the lowest and the largest values of NM mass needed for NED manufacturing (Masterov et al., 2016). P(M) is a relative probability for NED manufacturing in the respect to probability of CED manufacturing. So, the main mission of the system of NM safe management (and, first of all, NM control and protection system) is to reduce the relative risk R(M) in such a way that its value will never exceed the acceptable level. The acceptance criteria conditions (2) were studied in (Masterov et al., 2016). If only the uranium diversion stage is considered, then the simplest relationship can be derived from general formula (1) to assess the risk of uranium switching for further NED manufacturing:
R (M , x ) = Pdiv⋅Y (Mf , x f ) = exp(−α⋅M (x ))⋅Y (Mf , x f ),
(4)
where M - mass of the switched uranium; α - the effectiveness, which characterizes the action of control measures, per unit of the switched uranium mass. As seen, for the fixed value of final uranium enrichment x f , the diversion risk is a function of uranium mass and initial uranium enrichment x . The energy yield Y (Mf , x f ) produced by final uranium product was calculated with the application of the model described in (Mark, 1993). According to the model, the energy yield from the chain fission reaction (CFR) is proportional to the cubic degree of the exponent β in time-
E-mail address: [email protected]. https://doi.org/10.1016/j.pnucene.2018.09.014 Received 27 April 2018; Received in revised form 22 August 2018; Accepted 12 September 2018 0149-1970/ © 2018 Elsevier Ltd. All rights reserved.
Progress in Nuclear Energy 110 (2019) 75–79
V. Glebov
2.2. Human factor effects on the risk of the unauthorized NM applications
dependency of neutron population growth:
Y∼
β3
,β = (kEFF − 1)/ l,
(5)
The presence of insiders at nuclear sites depends on many factors including financial well-being of the staff members, labor discipline, political, personal and so on. Also, the number of insiders depends on general level of the nuclear security culture (Mladineo and Frazar, 2013) and real mass m of nuclear materials to be diverted. It is obvious that the larger m is the lower number of potential inner adversaries d, and vice versa. The level of NM safe management may be estimated as high, if total NM amount Mdiv, which potentially could be diverted by all the insiders, would be lower the significant quantity (SQ). The following relationship
where kEFF - maximal value of effective neutron multiplication factor during CFR; l - average prompt neutron lifetime. These parameters were determined by direct modelling of the neutron multiplication process with the application of Monte Carlo code MCNP-4B (Breismeister, 2000). In the paper (Masterov et al., 2016) switching of the NM significant quantity from nuclear site was considered. This scenario meets the major technical purpose of the IAEA inspection activity in order to solve the non-proliferation problem. Nevertheless, from the standpoint of nuclear security the diversions of insignificant NM quantities by a malicious person or group for their further accumulation can also represent a serious threat to ensuring the exceptionally peaceful use of nuclear materials. That is why this problem requires studying and taking precautionary measures. The purpose of this paper is to estimate the risk of uranium non-energy use under the scenarios of diversification of its switching. The paper is organized as follows. The scenario of diversification and the model for accumulation of significant uranium quantity were considered in section 2. Section 3 introduces the metric for effectiveness of the control and protection systems. The results and discussion of modelling of multiple LEU diversions from nuclear sites are given further. Finally, the summary and conclusions are presented.
Mdiv = SQ
defines the exact upper value for the number of insiders (d = k) at high level of the NM safe management. If the following inequality is correct, Mdiv > SQ,
(8)
then, the level of NM safe management at nuclear sites should be estimated as low. 2.2.1. In the case of insignificant quantity of insiders in nuclear sites The probability to collect significant NM quantity can be obviously increased if the number of insiders d increases too. Therefore the upper value for the number of insiders (d = k) should be used to determine the upper value of the diversion risk from nuclear sites with high level of the NM safe management. In general, the illegal uranium collector could investigate not all N nuclear sites but only n of them because of the limited capability for searching for insiders. In this case, the first question of accumulation of significant quantity calls for the following second question: how large is a probability for the illegal uranium collector to contact and agree with k insiders, if the search is carried out on a part of nuclear sites (n of N, n ≥ k)? The question was answered in (Balakrishnan and Navzorov, 2003). The probability for the illegal uranium collector to contact with k insiders Pcol (k ) is defined by hyper-geometrical distribution and, at high level of the NM safe management, is equal to:
2. The model of diversification of uranium switching from nuclear site 2.1. Scenario for accumulation of significant uranium quantity by means of multiple diversions Let N be a characteristic for total scale of nuclear activity in a country or region, i.e. N is the total number of nuclear sites and nuclear facilities where uranium is used under their own systems of uranium control and protection. A collector of illegal uranium investigates nuclear sites and tries to establish new contacts or use the already available contacts with the nuclear sites staff for further uranium diversion. Maybe, some staff members of nuclear sites, insiders, are able to commit UA with NM for the certain remuneration. The collector is not informed about the insiders, and their possible contacts with the collector should be considered as random events. The following question arises whether the probability for the illegal uranium collector to contact with insiders, have a deal with them on uranium diversion and, finally, accumulate significant uranium quantity is really great. Let us assume that the illegal uranium collector is the owner of sufficiently large material resources, and, therefore, (s)he is able for the agreement of financial remuneration for uranium diversion with the insider. It is generally known that the major barrier created by nuclear security is an inaccessibility of nuclear materials in the respect to any UA. That is why administration of a nuclear site must pay much attention to the proper maintenance of high-efficiency NM control and protection system. This is a powerful barrier and deterrent for insiders in the respect of any UA with nuclear materials. So, it may be certainly assumed that d ≪ N,
(7)
Pcol (k ) = CNn −−kk / CNn ,
(9)
Сba
where - binomial coefficients. If uranium switching is diversified, then formula (4) for the diversion risk of uranium for future NED manufacturing should be re-written in the following form:
R (k , α, n) = PSQ (k , α, n)⋅Y (Mf , x f ) = Pdiv (k, α )⋅Pcol (k, n)⋅Y (Mf , x f ), (10) where Pdiv (k , α ) – the probability of uranium diversion, when effectiveness of uranium control and protection system is equal to α; k – multiplicity for splitting of significant uranium quantity on several insignificant quantities; Pcol (k , n) – the probability to collect significant uranium quantity from n nuclear sites; Y (Mf , x f ) – energy yield from final uranium state. 2.2.2. In the case of significant quantity of insiders in nuclear sites If the level of NM safe management is low, then the following inequality takes place: d > k. The probability for the illegal uranium collector to contact with l insiders (from their total number d) is defined by general-type hyper-geometrical distribution:
(6)
where d – the total number of insiders at all nuclear sites. Let k be a multiplicity for diversification of significant NM quantity. To analyze the impact of diversification on risk of unauthorized uranium application, we assume k = 2, 3, …, 10. A successful switching of significant NM quantity requires k ≤ d . In accordance with limitation (6), the following inequality can be obtained: N ≫ k. For the range of accepted multiplicity values k, the latter inequality may be re-written as N > 102.
Pcol (l, d, n) = Cdl⋅CNn −−ld/ CNn ,
Сba
(11)
where - binomial coefficients. Then, the probability of uranium diversion and accumulation of significant uranium quantity can be calculated by using the following 76
Progress in Nuclear Energy 110 (2019) 75–79
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formula: d
PSQ (k , d, n) = Pdiv (k )⋅Pcol (k , d , n) = Pdiv (k )⋅( ∑ Pcol (l, d, n))=Pdiv (k ) l=k d
⋅( ∑ (Cdl⋅CNn −−ld/ CNn )).
(12)
l=k
3. Risk assessment for low-enriched uranium 3.1. Particularity of risk assessment for diversion of low-enriched uranium from nuclear site The urgent necessity to perform the risk assessment for diversions of low-enriched uranium (LEU, XLEU ∈ [0.71%; 20%]) is primarily defined by large-scale energy utilization of LEU at nuclear power plants and by potential possibility of LEU conversion from the reactor-grade state into the weapon-grade state. Supposing limited technological capabilities of potential adversaries, manufacturing of a primitive gun-type uraniumbased NED may be considered as the most feasible threat. Maximal acceptable risk for uranium of given enrichment x is defined by condition (3). This risk is limited from above by maximal uranium mass M1 to be used for NED manufacturing (Masterov et al., 2016). The mass M1 (x ) depends on uranium enrichment and becomes maximal for natural uranium M1 (x = 0.71%) . Therefore, maximal acLEU for low-enriched uranium, in general, is limited from ceptable risk Rtol above by M1 (x = 0.71%) , i.e.
Fig. 1. Dependencies of the probabilities Pdiv and Pcol on the SQ splitting multiplicity k.
LEU R (M , x ) ≤ M1 (x = 0.71%) ≡ Rtol .
3.2. The impact of the main parameters of diversification (k, α , d, n) on the risk of uranium disuse
NM management; - Possibility for a collector of searching the insiders. In the model we used here the first factor is characterized by the multiplicity for splitting k of the significant uranium quantity, the second factor is characterized by effectiveness α of NM control and protection system, the third factor is characterized by the number d of insiders who are able, under certain circumstances, to commit UA with nuclear materials and the last factor – by the searching scale n.
(13)
In accordance to the main mission of NM control and protection systems, its effectiveness is satisfactory only if the system provides the acceptable risk in the respect of unauthorized NM applications. Inequality (13) allows us to find the minimal satisfactory effectiveness αtol , at which the diversion risk is still within the acceptable LEU . The value αtol can be determined from (13) with aprange R ≤ Rtol plication of rather simple iterative algorithm. In fact, due to the monotonic dependence (4), the inequality (13) defines the metrics for effectiveness of uranium control and protection system as applied for the chosen value of uranium enrichment x . Namely, uranium control and protection system can be estimated as satisfactory, if α ≥ αtol , because the diversion risk is within the accepLEU . Vice versa, uranium control and protection table range R ≤ Rtol system can be estimated as unsatisfactory, if α < αtol , because the diversion risk is outside of the acceptable range. Minimal satisfactory values for effectiveness of uranium control and protection system are presented in Table 1 for some typical uranium enrichments. Since the dependence M (x ) is determined by the enrichment process, that is M (x ) ∼ 1/(x − xW ) ( xW - waste enrichment), it follows from (13) and (4) that minimal satisfactory effectiveness (αtol ) increases with increasing enrichment of uranium. For LEU with fixed enrichment the possibility to divert and collect significant NM quantity is defined at least by the following four important factors:
3.2.1. The impacts produced by multiplicity for splitting of significant uranium quantity and by the searching scale for insiders on the risk Here the variants of multiple LEU (4% 235U) diversions for further accumulation of significant uranium quantity under the conditions of high level of NM safe management at sites (d = k; α = αtol ) were studied. Dependencies of the probabilities Pdiv and Pcol on the SQ splitting multiplicity k and the number of nuclear sites n are presented in Fig. 1. The contrary tendencies are seen in variations of the probabilities Pdiv and Pcol, when the SQ splitting multiplicity increases. Besides, the probability for collection of significant uranium quantity Pcol demonstrates monotonous growth, if the number of nuclear sites n where the illegal uranium collector searches for insiders increases too. These circumstances allowed us to predict a possible appearance of maximal point in the total risk at multiple splitting of significant uranium quantity on the smaller quantities to be diverted. Let Rrel designate the relative diversification risk, i.e. ratio of the total risk at the SQ splitting multiplicity k to the single diversion risk, namely Rrel (k) = R(k)/R(1). Dependencies of the relative diversification risk Rrel on the SQ splitting multiplicity k are shown in Fig. 2. As seen the diversification gain is either absent at all or insignificant in the cases when the illegal uranium collector deals with small number of nuclear sites, i.e. when n is less than 10% from the total number of nuclear sites N. Maximal diversification effects are as follows: max max Rrel = 8.2 for n = 300 and Rrel = 29.5 for n = 600. These effects can be achieved only for small values of the SQ splitting multiplicity (k ∼ 2÷4) and for large number of nuclear sites (n ∼ N). The latter fact means that the illegal uranium collector has to search for insiders in substantial part of available nuclear sites, if he wants to achieve a
- The mass of LEU to be diverted; - Availability and effectiveness of NM control and protection systems at nuclear sites; - Impact of human factor at nuclear sites. It means that the staff members must be adherent to the basic principles of safe and secure
Table 1 Minimal satisfactory effectiveness of uranium control and protection system for some typical values of uranium enrichment. Uranium enrichment, %
0.71
2
4
9.645
20
Minimal satisfactory effectiveness, 1/kg
αnat = 0.00136
α2 = 0.005
α 4 = 0.01
αLEU = 0.0272
max αLEU = 0.053
77
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If effectiveness of NM control and protection system is satisfactory (α > α4 ), then the diversification demonstrates the pronounced effect with sharp splash of the relative diversification risk (especially for exmax cess effectiveness - αLEU ). At the same time, the expressed slump tendency for the risk probability PSQ with an increase α is observed. As a result, at excess effectiveness α the risk probability appears to be so negligibly small (PSQ ∼ 10−8÷10−6 at n = 100) that practically excludes the realization of the winnings from diversification. Slight elevation of the risk probability PSQ at n → N is compensated by decreasing the secrecy of the searching process and by possibility of its interruption because of too wide involvement of many participants into UA chains. Therefore, despite such a large positive diversification effect at high effectiveness of NM control and protection system (α > α4 ), the real implementation of the diversification process seems unfeasible.
Fig. 2. Dependencies of the relative diversification risk on the SQ splitting multiplicity at different scales of searching for insiders.
3.3. Impact of human factor (d) on the risk and probability for diversion and collection of significant uranium quantity
significant gain from the diversification. Even if the collector has investigated many nuclear sites, the growth of the SQ splitting multiplicity (k) more than 4÷5 does't increase the relative diversification risk.
As in the previous cases, the problem on application of the diversification process for the collection of significant quantity is considered further. It is presumed that nuclear sites are equipped with high-effective uranium control and protection systems (α = αtol ). Our task is to evaluate the diversification risk and the probability PSQ for diversion and collection of significant uranium quantity at different levels of the NM safe management (d) at nuclear sites. The following relationship may be written:
3.2.2. The impact produced by effectiveness of NM control and protection system on the diversification risk of NED manufacturing In this section, we evaluated the impact produced by effectiveness α of LEU control and protection system on the relative risk when diversification of switching (d = k). Maximal values of the relative diversification risk and the risk probabilities are presented in Table 2 for different effectiveness of LEU control and protection system and for different numbers of nuclear sites involved in the diversification process. If effectiveness of NM control and protection system is unsatisfactory (α < α4 ), then the relative diversification risk does not increase with the growth of the SQ splitting multiplicity k (except of slight elevation for the cases of α2 and n = 600). Thus the diversification makes no sense in this case.
PSQ = Pdiv (α, k )⋅Pcol (d ),
(14) d
where Pdiv (α, k ) = exp(−α⋅SQ/ k ) , Pcol (d ) = ∑l = k (Cdl⋅CNn −−ld/ CNn ) ,k – the SQ splitting multiplicity; d – the total number of insiders at all nuclear sites. Dependencies of relative risk Rrel (k , d ) on the splitting SQ multiplicity and the number of insiders are shown in Fig. 3. The dependencies demonstrate obvious growth of the relative diversification risk when d increases, i.e. when the level of NM safe management degrades. The tendency of the risk growth substantially strengthens when the SQ splitting multiplicity k increases because of the larger probability for SQ collection. Dependencies of the risk probability PSQ (formula (12)) on the number d of insiders and the number n of nuclear sites are shown in Fig. 4. Similarly to the relative diversification risk, the risk probability also demonstrates obvious monotonous growth (two orders of magnitude at 100 nuclear sites) when the level of NM safe management degrades. In comparison with high level of the NM safe management (d = k),
Table 2 Maximal relative diversification risk and the risk probability PSQ for different effectiveness α of LEU control and protection system and for different numbers of nuclear sites n.
Fig. 3. Dependencies of the relative LEU (4% number of insiders. 78
235
U) diversification risk on the
Progress in Nuclear Energy 110 (2019) 75–79
V. Glebov
of the main diversification parameters on the risk of uranium disuse were carried out: - In order to assess an impact, the metric of effectiveness (α ) for uranium control and protection systems was determined; - For high level of NM safe management (α = αtol ,d = k ) at sites, multiple LEU (4% 235U) diversions can increase the risk up to max max = 8.2 at n = 300 and Rrel = 29.5 at n = 600. This values: Rrel diversification effect could be achieved only in the case when illegal uranium collector might search for insiders at many nuclear sites (n ∼ N ); - Despite the large positive diversification effect, observed with increasing of the efficiency of the NM control and protection systems (α > αtol ), real implementation of diversification process seems unfeasible because of small values of the probabilities; - Degradation of the level of NM safe management (d > k) in combination with the strong capabilities of the illegal uranium collector (n ∼ N) can lead to an increase of the risk and the probability of successful completion of UA chain PSQ > 0.1, even if NM control and protection system are maintained with satisfactory effectiveness (α = αtol ).
Fig. 4. Dependencies of successful completion of UA chain on the total number of insiders at nuclear sites.
maximal points of the risk probability PSQ and the relative diversification risk Rrel shifted towards the larger values of the SQ splitting multiplicity. It follows from the presented dependencies that even if satisfactoryeffective NM control and protection systems are in operation at nuclear sites, degradation of the level of NM safe management (d > k) in combination with strong capabilities of the illegal uranium collector (n ∼ N) can lead to a successful completion of UA chain with substantial value of the risk probability (PSQ > 0.1).
Acknowledgments The work was supported by the Ministry of Education and Science of the Russian Federation under the Project 13.9748.2017/8.9. References Balakrishnan, N., Navzorov, V.B., 2003. A Primer on Statistical Distributions. John Wiley & Sons, Inc., pp. 83–88. Briesmeister, J.F., 2000. MCNP – a General Monte Carlo N-particle Transport Code. Version 4C. Los Alamos National Laboratory Report LA-13709-M (April 2000). IAEA Safeguards Glossary, 2002. International Nuclear Verification Series No. 3. Mark, C.J., 1993. Explosive properties of reactor-grade plutonium. Sci. Global Secur. 4, 111–128. Masterov, S.V., Kalugin, N.K., Glebov, V.B., 2016. Assessment of risks and control conditions for some scenarios on unauthorized usage of nuclear materials. Int. J. Risk Assess. Manag. 19, 346–361. Mladineo, S.V., Frazar, S.L., 2013. The importance of Safeguards culture. Nonproliferation Rev. 20, 509–523.
4. Conclusions Analysis of the diversification process aimed at collecting significant uranium quantity from nuclear sites allowed us to make the following conclusions. 1. A model for risk assessment in conditions of multiple uranium diversions is proposed. 2. Within the framework of the model numerical studies of an impact
79
Contents lists available at ScienceDirect
Progress in Nuclear Energy journal homepage: www.elsevier.com/locate/pnucene
Assessment of the impact of diversification of uranium switching on the risk of its non-energy use
T
Vasily Glebov National Research Nuclear University MEPhI, 115409, Moscow, Kashirskoe Shosse, 31, Russia
A R T I C LE I N FO
A B S T R A C T
Keywords: Diversification of switching Significant quantity Low-enriched uranium NM safe management Insider
The main objective of the IAEA verification activities is the timely detection of the switching of a significant quantity of nuclear material (NM). However, from the point of nuclear security the multiple switching of "small" quantities of nuclear material with the aim of their subsequent collection also constitutes one of the main threat to ensuring the exceptionally peaceful use of nuclear materials. In this article this threat assessments were performed under the scenario of diversification of uranium switching. It is shown that diversification of 4%uranium switching can lead to a substantial increase (up to 30 times) of the risk of uranium disuse even with carrying out proper uranium control and protection. However this effect is observed only when there are large opportunities to search for insiders (the search area is comparable to the existing set of nuclear facilities). The obtained results also prove that a reduction in the level of NM safe management at facilities could allow the collector to complete a chain of unauthorized actions with a significant probability (PSQ > 0.1).
R (M ) = P (M )⋅Y (M ) ≤ M ,
1. Introduction In the problems related with NM safe management the threat to perform a chain of the unauthorized actions (UA) may be quantitatively assessed in the following risk terms:
R = P ⋅D ,
P = Pdiv⋅Pman,
(1)
where P is a total probability to perform all UA with uranium-bearing materials; Pdiv – probability of uranium switching from nuclear site; Pman – probability to successfully complete a chain of out-site operations with uranium; D – potential damage caused by destructive application of a nuclear explosive device (NED), which can be evaluated in the term of the energy yield (Y). In the case of uranium, the energy yield depends mainly on uranium amount and uranium enrichment. From the standpoint of NM safe management, any unauthorized switching of significant quantity (IAEA Safeguards Glossary, 2002) (SQ) of nuclear materials must be completely excluded. Within the frames of the risk assessment [Masterov et al., 2016] this means that:
RNED ≤ RCED,
(2)
where indices NED and CED are used to designate nuclear and chemical explosive devices respectively. If the energy yield (Y) of NED is measured in equivalent mass of chemical explosives (TNT, for instance), then condition (2) can be rewritten in the terms of relative risk:
M ∈ [M0 (x ), M1 (x )],
(3)
where M0 and M1 are the lowest and the largest values of NM mass needed for NED manufacturing (Masterov et al., 2016). P(M) is a relative probability for NED manufacturing in the respect to probability of CED manufacturing. So, the main mission of the system of NM safe management (and, first of all, NM control and protection system) is to reduce the relative risk R(M) in such a way that its value will never exceed the acceptable level. The acceptance criteria conditions (2) were studied in (Masterov et al., 2016). If only the uranium diversion stage is considered, then the simplest relationship can be derived from general formula (1) to assess the risk of uranium switching for further NED manufacturing:
R (M , x ) = Pdiv⋅Y (Mf , x f ) = exp(−α⋅M (x ))⋅Y (Mf , x f ),
(4)
where M - mass of the switched uranium; α - the effectiveness, which characterizes the action of control measures, per unit of the switched uranium mass. As seen, for the fixed value of final uranium enrichment x f , the diversion risk is a function of uranium mass and initial uranium enrichment x . The energy yield Y (Mf , x f ) produced by final uranium product was calculated with the application of the model described in (Mark, 1993). According to the model, the energy yield from the chain fission reaction (CFR) is proportional to the cubic degree of the exponent β in time-
E-mail address: [email protected]. https://doi.org/10.1016/j.pnucene.2018.09.014 Received 27 April 2018; Received in revised form 22 August 2018; Accepted 12 September 2018 0149-1970/ © 2018 Elsevier Ltd. All rights reserved.
Progress in Nuclear Energy 110 (2019) 75–79
V. Glebov
2.2. Human factor effects on the risk of the unauthorized NM applications
dependency of neutron population growth:
Y∼
β3
,β = (kEFF − 1)/ l,
(5)
The presence of insiders at nuclear sites depends on many factors including financial well-being of the staff members, labor discipline, political, personal and so on. Also, the number of insiders depends on general level of the nuclear security culture (Mladineo and Frazar, 2013) and real mass m of nuclear materials to be diverted. It is obvious that the larger m is the lower number of potential inner adversaries d, and vice versa. The level of NM safe management may be estimated as high, if total NM amount Mdiv, which potentially could be diverted by all the insiders, would be lower the significant quantity (SQ). The following relationship
where kEFF - maximal value of effective neutron multiplication factor during CFR; l - average prompt neutron lifetime. These parameters were determined by direct modelling of the neutron multiplication process with the application of Monte Carlo code MCNP-4B (Breismeister, 2000). In the paper (Masterov et al., 2016) switching of the NM significant quantity from nuclear site was considered. This scenario meets the major technical purpose of the IAEA inspection activity in order to solve the non-proliferation problem. Nevertheless, from the standpoint of nuclear security the diversions of insignificant NM quantities by a malicious person or group for their further accumulation can also represent a serious threat to ensuring the exceptionally peaceful use of nuclear materials. That is why this problem requires studying and taking precautionary measures. The purpose of this paper is to estimate the risk of uranium non-energy use under the scenarios of diversification of its switching. The paper is organized as follows. The scenario of diversification and the model for accumulation of significant uranium quantity were considered in section 2. Section 3 introduces the metric for effectiveness of the control and protection systems. The results and discussion of modelling of multiple LEU diversions from nuclear sites are given further. Finally, the summary and conclusions are presented.
Mdiv = SQ
defines the exact upper value for the number of insiders (d = k) at high level of the NM safe management. If the following inequality is correct, Mdiv > SQ,
(8)
then, the level of NM safe management at nuclear sites should be estimated as low. 2.2.1. In the case of insignificant quantity of insiders in nuclear sites The probability to collect significant NM quantity can be obviously increased if the number of insiders d increases too. Therefore the upper value for the number of insiders (d = k) should be used to determine the upper value of the diversion risk from nuclear sites with high level of the NM safe management. In general, the illegal uranium collector could investigate not all N nuclear sites but only n of them because of the limited capability for searching for insiders. In this case, the first question of accumulation of significant quantity calls for the following second question: how large is a probability for the illegal uranium collector to contact and agree with k insiders, if the search is carried out on a part of nuclear sites (n of N, n ≥ k)? The question was answered in (Balakrishnan and Navzorov, 2003). The probability for the illegal uranium collector to contact with k insiders Pcol (k ) is defined by hyper-geometrical distribution and, at high level of the NM safe management, is equal to:
2. The model of diversification of uranium switching from nuclear site 2.1. Scenario for accumulation of significant uranium quantity by means of multiple diversions Let N be a characteristic for total scale of nuclear activity in a country or region, i.e. N is the total number of nuclear sites and nuclear facilities where uranium is used under their own systems of uranium control and protection. A collector of illegal uranium investigates nuclear sites and tries to establish new contacts or use the already available contacts with the nuclear sites staff for further uranium diversion. Maybe, some staff members of nuclear sites, insiders, are able to commit UA with NM for the certain remuneration. The collector is not informed about the insiders, and their possible contacts with the collector should be considered as random events. The following question arises whether the probability for the illegal uranium collector to contact with insiders, have a deal with them on uranium diversion and, finally, accumulate significant uranium quantity is really great. Let us assume that the illegal uranium collector is the owner of sufficiently large material resources, and, therefore, (s)he is able for the agreement of financial remuneration for uranium diversion with the insider. It is generally known that the major barrier created by nuclear security is an inaccessibility of nuclear materials in the respect to any UA. That is why administration of a nuclear site must pay much attention to the proper maintenance of high-efficiency NM control and protection system. This is a powerful barrier and deterrent for insiders in the respect of any UA with nuclear materials. So, it may be certainly assumed that d ≪ N,
(7)
Pcol (k ) = CNn −−kk / CNn ,
(9)
Сba
where - binomial coefficients. If uranium switching is diversified, then formula (4) for the diversion risk of uranium for future NED manufacturing should be re-written in the following form:
R (k , α, n) = PSQ (k , α, n)⋅Y (Mf , x f ) = Pdiv (k, α )⋅Pcol (k, n)⋅Y (Mf , x f ), (10) where Pdiv (k , α ) – the probability of uranium diversion, when effectiveness of uranium control and protection system is equal to α; k – multiplicity for splitting of significant uranium quantity on several insignificant quantities; Pcol (k , n) – the probability to collect significant uranium quantity from n nuclear sites; Y (Mf , x f ) – energy yield from final uranium state. 2.2.2. In the case of significant quantity of insiders in nuclear sites If the level of NM safe management is low, then the following inequality takes place: d > k. The probability for the illegal uranium collector to contact with l insiders (from their total number d) is defined by general-type hyper-geometrical distribution:
(6)
where d – the total number of insiders at all nuclear sites. Let k be a multiplicity for diversification of significant NM quantity. To analyze the impact of diversification on risk of unauthorized uranium application, we assume k = 2, 3, …, 10. A successful switching of significant NM quantity requires k ≤ d . In accordance with limitation (6), the following inequality can be obtained: N ≫ k. For the range of accepted multiplicity values k, the latter inequality may be re-written as N > 102.
Pcol (l, d, n) = Cdl⋅CNn −−ld/ CNn ,
Сba
(11)
where - binomial coefficients. Then, the probability of uranium diversion and accumulation of significant uranium quantity can be calculated by using the following 76
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formula: d
PSQ (k , d, n) = Pdiv (k )⋅Pcol (k , d , n) = Pdiv (k )⋅( ∑ Pcol (l, d, n))=Pdiv (k ) l=k d
⋅( ∑ (Cdl⋅CNn −−ld/ CNn )).
(12)
l=k
3. Risk assessment for low-enriched uranium 3.1. Particularity of risk assessment for diversion of low-enriched uranium from nuclear site The urgent necessity to perform the risk assessment for diversions of low-enriched uranium (LEU, XLEU ∈ [0.71%; 20%]) is primarily defined by large-scale energy utilization of LEU at nuclear power plants and by potential possibility of LEU conversion from the reactor-grade state into the weapon-grade state. Supposing limited technological capabilities of potential adversaries, manufacturing of a primitive gun-type uraniumbased NED may be considered as the most feasible threat. Maximal acceptable risk for uranium of given enrichment x is defined by condition (3). This risk is limited from above by maximal uranium mass M1 to be used for NED manufacturing (Masterov et al., 2016). The mass M1 (x ) depends on uranium enrichment and becomes maximal for natural uranium M1 (x = 0.71%) . Therefore, maximal acLEU for low-enriched uranium, in general, is limited from ceptable risk Rtol above by M1 (x = 0.71%) , i.e.
Fig. 1. Dependencies of the probabilities Pdiv and Pcol on the SQ splitting multiplicity k.
LEU R (M , x ) ≤ M1 (x = 0.71%) ≡ Rtol .
3.2. The impact of the main parameters of diversification (k, α , d, n) on the risk of uranium disuse
NM management; - Possibility for a collector of searching the insiders. In the model we used here the first factor is characterized by the multiplicity for splitting k of the significant uranium quantity, the second factor is characterized by effectiveness α of NM control and protection system, the third factor is characterized by the number d of insiders who are able, under certain circumstances, to commit UA with nuclear materials and the last factor – by the searching scale n.
(13)
In accordance to the main mission of NM control and protection systems, its effectiveness is satisfactory only if the system provides the acceptable risk in the respect of unauthorized NM applications. Inequality (13) allows us to find the minimal satisfactory effectiveness αtol , at which the diversion risk is still within the acceptable LEU . The value αtol can be determined from (13) with aprange R ≤ Rtol plication of rather simple iterative algorithm. In fact, due to the monotonic dependence (4), the inequality (13) defines the metrics for effectiveness of uranium control and protection system as applied for the chosen value of uranium enrichment x . Namely, uranium control and protection system can be estimated as satisfactory, if α ≥ αtol , because the diversion risk is within the accepLEU . Vice versa, uranium control and protection table range R ≤ Rtol system can be estimated as unsatisfactory, if α < αtol , because the diversion risk is outside of the acceptable range. Minimal satisfactory values for effectiveness of uranium control and protection system are presented in Table 1 for some typical uranium enrichments. Since the dependence M (x ) is determined by the enrichment process, that is M (x ) ∼ 1/(x − xW ) ( xW - waste enrichment), it follows from (13) and (4) that minimal satisfactory effectiveness (αtol ) increases with increasing enrichment of uranium. For LEU with fixed enrichment the possibility to divert and collect significant NM quantity is defined at least by the following four important factors:
3.2.1. The impacts produced by multiplicity for splitting of significant uranium quantity and by the searching scale for insiders on the risk Here the variants of multiple LEU (4% 235U) diversions for further accumulation of significant uranium quantity under the conditions of high level of NM safe management at sites (d = k; α = αtol ) were studied. Dependencies of the probabilities Pdiv and Pcol on the SQ splitting multiplicity k and the number of nuclear sites n are presented in Fig. 1. The contrary tendencies are seen in variations of the probabilities Pdiv and Pcol, when the SQ splitting multiplicity increases. Besides, the probability for collection of significant uranium quantity Pcol demonstrates monotonous growth, if the number of nuclear sites n where the illegal uranium collector searches for insiders increases too. These circumstances allowed us to predict a possible appearance of maximal point in the total risk at multiple splitting of significant uranium quantity on the smaller quantities to be diverted. Let Rrel designate the relative diversification risk, i.e. ratio of the total risk at the SQ splitting multiplicity k to the single diversion risk, namely Rrel (k) = R(k)/R(1). Dependencies of the relative diversification risk Rrel on the SQ splitting multiplicity k are shown in Fig. 2. As seen the diversification gain is either absent at all or insignificant in the cases when the illegal uranium collector deals with small number of nuclear sites, i.e. when n is less than 10% from the total number of nuclear sites N. Maximal diversification effects are as follows: max max Rrel = 8.2 for n = 300 and Rrel = 29.5 for n = 600. These effects can be achieved only for small values of the SQ splitting multiplicity (k ∼ 2÷4) and for large number of nuclear sites (n ∼ N). The latter fact means that the illegal uranium collector has to search for insiders in substantial part of available nuclear sites, if he wants to achieve a
- The mass of LEU to be diverted; - Availability and effectiveness of NM control and protection systems at nuclear sites; - Impact of human factor at nuclear sites. It means that the staff members must be adherent to the basic principles of safe and secure
Table 1 Minimal satisfactory effectiveness of uranium control and protection system for some typical values of uranium enrichment. Uranium enrichment, %
0.71
2
4
9.645
20
Minimal satisfactory effectiveness, 1/kg
αnat = 0.00136
α2 = 0.005
α 4 = 0.01
αLEU = 0.0272
max αLEU = 0.053
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If effectiveness of NM control and protection system is satisfactory (α > α4 ), then the diversification demonstrates the pronounced effect with sharp splash of the relative diversification risk (especially for exmax cess effectiveness - αLEU ). At the same time, the expressed slump tendency for the risk probability PSQ with an increase α is observed. As a result, at excess effectiveness α the risk probability appears to be so negligibly small (PSQ ∼ 10−8÷10−6 at n = 100) that practically excludes the realization of the winnings from diversification. Slight elevation of the risk probability PSQ at n → N is compensated by decreasing the secrecy of the searching process and by possibility of its interruption because of too wide involvement of many participants into UA chains. Therefore, despite such a large positive diversification effect at high effectiveness of NM control and protection system (α > α4 ), the real implementation of the diversification process seems unfeasible.
Fig. 2. Dependencies of the relative diversification risk on the SQ splitting multiplicity at different scales of searching for insiders.
3.3. Impact of human factor (d) on the risk and probability for diversion and collection of significant uranium quantity
significant gain from the diversification. Even if the collector has investigated many nuclear sites, the growth of the SQ splitting multiplicity (k) more than 4÷5 does't increase the relative diversification risk.
As in the previous cases, the problem on application of the diversification process for the collection of significant quantity is considered further. It is presumed that nuclear sites are equipped with high-effective uranium control and protection systems (α = αtol ). Our task is to evaluate the diversification risk and the probability PSQ for diversion and collection of significant uranium quantity at different levels of the NM safe management (d) at nuclear sites. The following relationship may be written:
3.2.2. The impact produced by effectiveness of NM control and protection system on the diversification risk of NED manufacturing In this section, we evaluated the impact produced by effectiveness α of LEU control and protection system on the relative risk when diversification of switching (d = k). Maximal values of the relative diversification risk and the risk probabilities are presented in Table 2 for different effectiveness of LEU control and protection system and for different numbers of nuclear sites involved in the diversification process. If effectiveness of NM control and protection system is unsatisfactory (α < α4 ), then the relative diversification risk does not increase with the growth of the SQ splitting multiplicity k (except of slight elevation for the cases of α2 and n = 600). Thus the diversification makes no sense in this case.
PSQ = Pdiv (α, k )⋅Pcol (d ),
(14) d
where Pdiv (α, k ) = exp(−α⋅SQ/ k ) , Pcol (d ) = ∑l = k (Cdl⋅CNn −−ld/ CNn ) ,k – the SQ splitting multiplicity; d – the total number of insiders at all nuclear sites. Dependencies of relative risk Rrel (k , d ) on the splitting SQ multiplicity and the number of insiders are shown in Fig. 3. The dependencies demonstrate obvious growth of the relative diversification risk when d increases, i.e. when the level of NM safe management degrades. The tendency of the risk growth substantially strengthens when the SQ splitting multiplicity k increases because of the larger probability for SQ collection. Dependencies of the risk probability PSQ (formula (12)) on the number d of insiders and the number n of nuclear sites are shown in Fig. 4. Similarly to the relative diversification risk, the risk probability also demonstrates obvious monotonous growth (two orders of magnitude at 100 nuclear sites) when the level of NM safe management degrades. In comparison with high level of the NM safe management (d = k),
Table 2 Maximal relative diversification risk and the risk probability PSQ for different effectiveness α of LEU control and protection system and for different numbers of nuclear sites n.
Fig. 3. Dependencies of the relative LEU (4% number of insiders. 78
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U) diversification risk on the
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of the main diversification parameters on the risk of uranium disuse were carried out: - In order to assess an impact, the metric of effectiveness (α ) for uranium control and protection systems was determined; - For high level of NM safe management (α = αtol ,d = k ) at sites, multiple LEU (4% 235U) diversions can increase the risk up to max max = 8.2 at n = 300 and Rrel = 29.5 at n = 600. This values: Rrel diversification effect could be achieved only in the case when illegal uranium collector might search for insiders at many nuclear sites (n ∼ N ); - Despite the large positive diversification effect, observed with increasing of the efficiency of the NM control and protection systems (α > αtol ), real implementation of diversification process seems unfeasible because of small values of the probabilities; - Degradation of the level of NM safe management (d > k) in combination with the strong capabilities of the illegal uranium collector (n ∼ N) can lead to an increase of the risk and the probability of successful completion of UA chain PSQ > 0.1, even if NM control and protection system are maintained with satisfactory effectiveness (α = αtol ).
Fig. 4. Dependencies of successful completion of UA chain on the total number of insiders at nuclear sites.
maximal points of the risk probability PSQ and the relative diversification risk Rrel shifted towards the larger values of the SQ splitting multiplicity. It follows from the presented dependencies that even if satisfactoryeffective NM control and protection systems are in operation at nuclear sites, degradation of the level of NM safe management (d > k) in combination with strong capabilities of the illegal uranium collector (n ∼ N) can lead to a successful completion of UA chain with substantial value of the risk probability (PSQ > 0.1).
Acknowledgments The work was supported by the Ministry of Education and Science of the Russian Federation under the Project 13.9748.2017/8.9. References Balakrishnan, N., Navzorov, V.B., 2003. A Primer on Statistical Distributions. John Wiley & Sons, Inc., pp. 83–88. Briesmeister, J.F., 2000. MCNP – a General Monte Carlo N-particle Transport Code. Version 4C. Los Alamos National Laboratory Report LA-13709-M (April 2000). IAEA Safeguards Glossary, 2002. International Nuclear Verification Series No. 3. Mark, C.J., 1993. Explosive properties of reactor-grade plutonium. Sci. Global Secur. 4, 111–128. Masterov, S.V., Kalugin, N.K., Glebov, V.B., 2016. Assessment of risks and control conditions for some scenarios on unauthorized usage of nuclear materials. Int. J. Risk Assess. Manag. 19, 346–361. Mladineo, S.V., Frazar, S.L., 2013. The importance of Safeguards culture. Nonproliferation Rev. 20, 509–523.
4. Conclusions Analysis of the diversification process aimed at collecting significant uranium quantity from nuclear sites allowed us to make the following conclusions. 1. A model for risk assessment in conditions of multiple uranium diversions is proposed. 2. Within the framework of the model numerical studies of an impact
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