ti

mic

Accepted 4 November 2013Available online 25 December 2013

Keywords:Nondestructive assayNDA

e b

safeguards and burnup credit. From an analysis by basic nuclear engineering, it is shown that the physicalproperties and the isotopic content of a used fuel assembly are basically three-dimensional vector spaces.

A prac

it community is primarily interested in determining the residual

Fig. 1. First, both communities typically conduct experiments (orsimulations) to measure many fuel assemblies from a pertinentrange of values of burnup (BU), initial enrichment (IE), and cooling

quantity is correlated to the BIC set, so that any future measurement

with the csafeguardthe range

assemblies to determine their isotopic content and establcorrelation; the current practice is to conduct burnup simuto create the correlation. The burnup-credit community takes a

Tel./fax: +81 (0)29 283 4115.

1 In this paper, the term initial enrichment for mixed oxide fuel (MOX) refers tothe fraction of ssile Pu isotopes that is loaded into the fuel, assuming that depleteduranium is used for the UO2 in the MOX. Thorium-based fuels are not considered. Thecooling time is assumed to be within a practical range: greater than a few days so thatall of the very-short-lived neutron poisons (like 135Xe) have decayed away, but lessthan about one century.

Annals of Nuclear Energy 66 (2014) 3150

Contents lists available at ScienceDirect

Annals of Nuclear Energy

journal homepage: www.elsevier .com/locate /anuceneE-mail address: alan.bolind@jaea.go.jpreactivity of the assembly. Both characteristics are related to eachother, though, and both communities use nondestructive assay(NDA) to some extent, to determine these characteristics.

This practice of NDA on used fuel assemblies is illustrated in

on an unknown fuel assembly can be interpreted.Second, the values of the BIC set are correlated

teristic that is of interest to each community. Themunity in the past performed destructive assay on0306-4549/$ - see front matter 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.anucene.2013.11.010harac-s com-of fuelish thelationsTwo communities of nuclear professionals are interested incharacterizing used nuclear fuel assemblies: the safeguards com-munity and the burnup-credit community. The safeguards com-munity is primarily interested in determining the ssile orplutonium content of the fuel assembly, whereas the burnup-cred-

have been chosen because their values for a given fuel assembly areusually easy to determine from its records and because they seem tocharacterize used fuel assemblies to a large extent. The measuredquantity and the BIC variables are averages over the transversecross-section of the fuel assembly and may also be averages overthe axial length. By these experiments or simulations, the measuredSpent nuclear fuel assemblyNuclear material safeguardsBurnup creditBIC set

1. Introduction: an overview of NDof the problemBy extensive referencing of the NDA literature, the paper then shows that the BIC variables are indepen-dent variables with respect to the physical properties and the isotopes. Therefore, the knowledge of allthree BIC variables is a necessary condition for the accurate characterization of a used low- or high-enriched uranium (LEU or HEU) fuel assembly. For a plutonium mixed-oxide (MOX) fuel assembly, afourth variable for the BIC set (the curium-producing ability) is also necessary. The paper also discussesother possible variables besides the BIC set, to demonstrate that the knowledge of the BIC set is also asufcient condition in many cases. Logically, it is therefore necessary to make at least three independentNDA measurements (or four, for MOX) to achieve a unique solution (characterization) if a reliance oninformation provided by the reactor operator is to be avoided. By this fact, the common question, Whatis the accuracy of a particular NDA technique? is revealed to be a poorly posed one with regard to usedfuel assemblies. The result of the paper is a better paradigm for interpreting and improving the NDA prac-tice of both the safeguards community and the burnup-credit community.

2013 Elsevier Ltd. All rights reserved.

tice and the statement time (CT).1 (These three variables will be called collectively as theBIC set of variables, in this paper.) Historically, these BIC variablesReceived 12 May 2013Received in revised form 1 October 2013

lectively called the BIC set of variables characterize it to rst order for the purposes of nuclear-materialsThe use of the BIC set in the characterizaassemblies by nondestructive assay

Alan Michael Bolind Integrated Support Center for Nuclear Nonproliferation and Nuclear Security, Japan AtoTokai-mura, Naka-gun, Ibaraki-ken 319-1118, Japan

a r t i c l e i n f o

Article history:

a b s t r a c t

This paper explains why thon of used nuclear fuel

Energy Agency, 3-1-1 Funaishikawa Eki Higashi,

urnup, initial enrichment, and cooling time of a used fuel assembly col-

fuel assemblies are pressurized-water-reactor (PWR), low-enriched-uranium (LEU)assemblies in a conguration of 17 pins 17 pins. Copied from Lee et al. (2011).

With error bars:CT = 4 months

2.25% IE 3.30% 4 mo. CT 40 mo.

learmore circuitous route to handle the residual reactivity. First, bur-nup simulations are made for the fuel assemblies. Second, reactiv-ity calculations are made from the results of these simulations. Butrather than analyzing the reactivity of each fuel assembly speci-cally (as expressed by the innite multiplication factor, k1, oreffective multiplication factor, keff, of the assembly), the burnup-credit community instead analyzes the reactivity of the cask orstorage site in the case that it is lled with such fuel assemblies,as a function of their initial enrichment and level of burnup. (Thecooling time is always set at 5 years when assigning burnup credit,for historical reasons and because it is a bounding case. See Sec-

(a)

(b)

Fig. 1. The current logic of NDA practice for (a) safeguards and (b) burnup credit.For (b), the residual reactivity is usually expressed in terms of the reactivity of aspecic collection of similar used fuel assemblies.

32 A.M. Bolind / Annals of Nuction 4.3 of Parks et al. (2000).) From the results, a plot is made ofthe minimum burnup that is acceptable for a fuel assembly to beloaded into the cask or site. (See, for example, Fig. 1.1 in the reviewby Bevard et al. (2009).) The plot is a function of the fuel assemblysinitial enrichment. This choice of method is reasonable becausethere is currently no good way to conservatively and accuratelydetermine the reactivity of a collection of fuel assemblies (suchas in a cask or site) only from the knowledge of the reactivity ofeach individual fuel assembly. Therefore, the contribution of a fuelassembly to the reactivity of a collection of fuel assemblies is esti-mated by using its BIC-set value as a proxy, instead.

In summary, each community uses two correlations to logicallyconnect the measurement of a fuel assembly to its characteristicsof interest via its BIC-set value. One correlation is between themeasured quantity and the BIC set. The other correlation is be-tween the BIC set and the isotopic content or residual reactivity.

The problem with these measurements is illustrated in Fig. 2,for the safeguards community, and in Fig. 3, for the burnup-creditcommunity. Fig. 2 is for the differential die-away (DDA) NDA tech-nique, but it is typical of many NDA techniques (Burr et al., 2012).(See Section 2.4 below for the exceptions.) The specic problem isthat one measured value (on the vertical axis) corresponds to arange of possible values of the ssile content (on the horizontalaxis). It is therefore impossible to obtain an accurate determinationof the ssile content by using only this measured quantity in theabsence of other information. Fig. 3 shows the total neutron countrate of a set of pressurized-water-reactor (PWR) assemblies as afunction of their BU. As with Fig. 2, one value of the count rate cor-responds to a range of BU. Although the variation in BU in thisFig. 2. The differential die-away responses (count rates) of 64 used fuel assemblies,plotted against their ssile content (represented by an effective quantity of 239Pu).Each fuel assembly has a different combination of BU, IE, and CT, as indicated. The

Energy 66 (2014) 3150gure appears to be smaller than the variation in ssile contentin Fig. 2, the data in Fig. 3 are somewhat deceptive because theyare drawn from a narrow set. The fuel assemblies with the lowestBU also had the lowest IE and the longest CT; and as the BU in-creased among the set of fuel assemblies, IE increased, and CT de-creased. Therefore, the scatter of a set of fuel assemblies that trulyspanned the whole domain of the BIC set (as in Fig. 2) would bemuch greater.

Burr et al. (2012) have quantied this type of uncertainty that isexhibited in Figs. 2 and 3. They chose to analyze the Passive Neu-tron Albedo Reactivity (PNAR) NDA technique as a representativeexample. Through Monte Carlo simulations, they showed that thessile content of a used fuel assembly can be predicted to a highdegree of accuracy (accounting for 99.7% of the variance) by a para-metric equation that uses only the three BIC variables as theparameters (Eq. (1) in their paper). This equation was derived froma wide subset of the BIC domain and is therefore valid over thatwide subset (though with the caveat that the fuel assemblies beingexamined were themselves merely the product of burnup simula-tions). They furthermore made the contrasting observation that ifonly the PNAR signal is used to determine the ssile content, thenthe relative error standard deviation (RESD) in the ssile content

Fig. 3. Total neutron count rates from 36 LEU PWR assemblies. Modied fromFig. 28 in Phillips et al. (1980).

(and isotopic content) in a mostly empirical way by an extensivereferencing of the literature. It thereby explains why the BIC setcharacterizes used fuel assemblies to rst order, which has notbeen done before in the literature and which is the main resultof this paper. Section 4 approaches the question from the otherdirection by examining other factors that can cause the BIC set tofail to characterize used fuel assemblies. It thereby establishesthe limits of the applicability of the BIC set, or rather, the limitson its sufciency. Section 5 discusses the implications of these re-

BU. It is therefore more correct to recognize and include explicitly

clear Energy 66 (2014) 3150 33varies from 15% to 69% (as a function of the variation of the mea-surement RESDs from 1% to 20%, that is, of the errors in the countrates themselves). These numbers conrm the argument madeabove, that the one measured quantity the PNAR signal in thisexample cannot determine the ssile content accurately.

The obvious solution to this problem is to use additional infor-mation to obtain an accurate determination. Such information canbe added implicitly or explicitly. It can be added implicitly byapplying the NDA only to a limited set of fuel assemblies, such asfuel assemblies from the same spent-fuel pool. In this way, mostof the variables are implicitly kept xed or within a tight dynamicrange (Phillips et al., 1980), and so the relationships illustrated inFigs. 2 and 3 each collapse more or less to a one-to-one relation-ship (bijection). Such NDA results are inherently relative, applyingonly to fuel assemblies within the limited set, and so they are mostuseful just for checking for outlying fuel assemblies (Bevard et al.(2009), Sections 3.4 and 7.1). Explicitly adding more informationto the NDA measurement enables more general correlations to bemade. If the BU, IE, or CT of the fuel assemblies is known a priorifrom their records or burnup simulations, then this informationcan be used to select a more accurate correlation between the mea-sured NDA quantity and the ssile content or residual reactivity.For example, Burr et al. (2012) have shown that including accuratevalues of BU and CT (or of IE and BU) along with the NDA signalwhen calculating the ssile content produces a more accurate re-sult than using just the NDA signal alone.

Burrs results also reveal that the uncertainty in Figs. 2 and 3can be classied neither as random error (since knowledge ofone or more BIC variables improves the result) nor as short-termsystematic error (since the discrepancies are not all in one generaldirection), according to the commonly used denitions in the bookof International Target Values (Zhao et al., 2010). Instead, theuncertainty is not inherent to each fuel assembly itself but is ratherthe uncertainty of choosing the fuel assembly to be measured i.e.,where its unknown BIC-set value happens to be in relation to thecalibration curve through the BIC-set domain (e.g., the curve inFig. 3). It is this choosing that is random. The explicit addition ofinformation about the values of one or more BIC variables for a gi-ven fuel assembly therefore reduces the uncertainty by assistingthe NDA practitioner to adjust the calibration curve to be moreappropriate for interpreting the NDA measurement result for thatgiven fuel assembly.

The purpose of this paper is to explain the physical reasons be-hind this phenomenon that has been merely observed by Burret al.: why this uncertainty exists and why the knowledge of theBIC set, in particular, reduces this uncertainty. Thus, this paperhas two corresponding thrusts. First, it explains why the BIC setcharacterizes used fuel assemblies that are immersed in water, atleast to rst order. Second, it shows how this fact denes a frame-work for conducting and interpreting NDA measurements of usedfuel assemblies.

Various parts of these explanations are well-known in the liter-ature, and this paper draws extensively from them. The originalityof this paper is the collection of these parts into a cohesive whole a paradigm that provides both a clearer interpretation of past re-sults and the ability to predict ways to improve NDA practice. Thepapers priority is thus to derive an understanding from physicalprinciples rather than to address specic pragmatic concerns. Suchan understanding will nonetheless be useful for directing other re-search to address such concerns.

This paper has four main sections, Sections 2-5. Section 2 reex-amines and claries the logic of NDA practice (Fig. 1), which mustrst be done if any systematic treatment is to be performed. It

A.M. Bolind / Annals of Nuthereby establishes the framework for connecting the BIC variablesto the physics of the used fuel assembly. Section 3 proceeds tomake this connection between the BIC set and the physicsthe physical-properties vector space in the logic of NDA practice,and this addition is an essential part of the arguments of the restof this paper.

Measured QuantitiesMQ

Physical Properties

(of the single fuel assembly, at the time of the NDA

measurement)

(oatim

m

l BUIECT

(CP)

BIEC

(C

Isotopic Content Iso t

Physical Property(the fuel assemblys contribution to the

reactivity of the entire collection

of fuel assemblies)

Ph(thc

o

ys

For Safeguards

For Burnup Creditsults. It thereby demonstrates the importance of adopting this par-adigm for the interpretation and improvement of NDA practice onused fuel assemblies.

2. Theory: the nature of the vector spaces in NDA practice

2.1. The existence of the vector space of physical properties

The rst step to constructing a better paradigm for NDA practiceis to recognize explicitly that a vector space of the physical proper-ties of the used fuel assembly at the time of the NDA measure-ments should be included in the overall logic of Fig. 1. The resultis shown in Fig. 4. The physical properties are physical in the senseof being related to the physics of the generation and transport ofneutrons, photons, heat, or other particles or energy throughoutthe fuel assembly. The physical properties may be able to be de-tected directly (e.g., the gamma-ray activity), or they may needto be deduced (e.g., the multiplication of neutrons).

The reason that the physical-properties vector space must berecognized and included is that the physical properties are thecurrent characteristics of a fuel assembly and can be repeatedlydetermined or veried, whereas the BIC set describes only the pasthistory of the assembly and therefore cannot be repeated for thatparticular assembly. This is to say that the NDA measurementsare actually measuring the physical properties, not the BICvariables, in contradiction to the common parlance in both thesafeguards and burnup-credit communities. For example, the onlyway to actually measure BU is to measure it while the fuel assem-bly is being burned in the reactor, such as by putting a neutron-uxmonitor next to the fuel assembly or by detecting the antineutrinosassociated with ssion events (Bernstein et al., 2008). This contra-diction may have originated with total (energy-independent)gamma-ray measurements, since the gamma-ray activity of a usedfuel assembly is almost completely comprised of gamma-raysemitted from the radioactive decay of ssion products. Thegamma-ray activity is thus an indication of the number of ssionsthat have taken place that is, an indication of the burnup. Mostphysical properties, especially the neutronic properties, cannotbe so closely identied with a particular BIC variable, though;and even the gamma-ray activity depends on CT as well as onFig. 4. The corrected logic for proper NDA practice on used fuel assemblies (CP inthe BIC vector space is for MOX fuel and is discussed in Section 3).

term, this term either is zero, for a steady-state NDA measurement,or is a known output (measured quantity), for a time-dependentNDA measurement such as neutron coincidence counting or a dif-ferential die-away (DDA) measurement. That is, the term either is

lear Energy 66 (2014) 31502.2. The dominance of the neutronic physics

The next step is to recognize that the neutronic physics is themost important physics of a used fuel assembly and actually gov-erns all the other relevant physics. This recognition is supportedby three obvious facts. First, the burning of nuclear fuel in a nuclearreactor is a neutronic process: neutron-induced ssion. Second,uranium and plutonium are required to be determined and safe-guarded precisely because of their ssile isotopes. Third, the resid-ual reactivity of a used fuel assembly which is important forensuring criticality safety is also a neutronic property. Therefore,the processes by which a fuel assembly comes to be in the status ofused and also the reasons for caring about the characteristics ofthe used fuel assembly are all derived from the neutronic physicsof the fuel assembly.

2.3. The tri-dimensionality of the physical-properties and isotopic-content vector spaces

The previous statements may have been obvious, but they arethe necessary justication for this next step. Since the neutronicphysics governs the relevant physics and isotopes of the used fuelassembly, it is valid to examine an equation that describes the neu-tronic physics of the fuel assembly that is, the neutron diffusionequation to determine the dimensionality of the physical-proper-ties vector space and of the isotopic-content vector space. Theemphasis is on the state of the used fuel assembly at the time ofthe NDA measurement, rather than during the burning of the fuelin the reactor. It is also assumed that the used fuel assembly is im-mersed in cooling water unless otherwise stated.

The one-speed neutron-diffusion equation for the used fuelassembly is as follows:

S 1keff

mRf/ Ra/ Dr2/ 1v

@/@t

1

The symbols have their usual nuclear-engineering meanings(Duderstadt and Hamilton, 1976; Lamarsh and Baratta, 2001). Notethat the 1/keff coefcient should be included only when criticality isbeing examined, by setting S and o//ot to zero. Also, delayed neu-trons are not included separately in this equation, for simplicityof argument. The rst two terms on the left-hand side of Eq. (1) rep-resent the production of neutrons: the rst one by primary-neutronsources and the second one by induced ssion. The last two termson the left-hand side represent the loss of neutrons: the rst oneby absorption and the second one by diffusion (i.e., leakage). Theterm on the right-hand side represents the change in the neutronux during a transient. Of course, each term changes with positioninside the fuel assembly.

The fact that the physical-properties vector space is basicallythree dimensional can be seen by recognizing that although thereare ve terms in this equation, two of them are constant or known.Relative to the magnitude of the neutron ux, the diffusion termstays almost constant with the burning of the fuel, because boththe geometry of the fuel assembly and the diffusion coefcient ateach location within the fuel assembly stay constant regardless ofthe burning of the fuel (Nauchi et al., 2011). (Nauchi et al. alsodetermined that the neutron leakage probability is constant, too;see Section 2.5.) Nauchi et al. explained this constancy of the diffu-sion coefcient by recognizing that the neutron diffusion is domi-nated by elastic scattering in water, and the water in and aroundthe fuel assembly obviously does not change with burnup. An addi-tional argument, notmentioned byNauchi et al., is that the inelasticscattering in the fuel also does not change greatly with burnup, be-

34 A.M. Bolind / Annals of Nuccause the quantity of the most prevalent heavy isotope in the fuel(238U for LEU and MOX; 235U for very enriched HEU) remains nearlyconstant over typical ranges of burnup. As for the time-dependentzero or is represented by s, a characteristic die-away time. This rec-ognition is also supported by the fact that the term contains nomaterial-related variables. It is seen then that only three terms inthe neutron-diffusion equation vary with the burning of the fuel,and since the neutronic physics governs the physical properties ofthe used fuel assembly, the vector space of the physical propertiesmust therefore be basically three dimensional.

The dimensionality of the isotopic-content vector space canlikewise be determined by examining this equation. Since Ra, bydenition, includes both absorption that leads to ssion andabsorption that just captures the neutron, it is helpful to separaterst these two kinds of absorption and to rearrange the equationaccordingly:

S 1keff

m 1

Rf/ Ra;capture/ Dr2/ 1v

@/@t

2

The second term now is the net neutron production from ssion. Inthis equation, the rst three terms correspond to the three kinds ofisotopes that affect the neutron physics of the fuel assembly: theprimary neutron sources (corresponding to S), the ssile isotopes(corresponding to Rf), and the neutron-capturing isotopes (corre-sponding to Ra,capture). Again, the last two terms are constant orknown. Therefore, the isotopic-content vector space is also three-dimensional, consisting of three groups of isotopes.

A criticism of these conclusions might be that Eqs. (1) and (2)are only one-speed and so do not account for the physics of theneutron energy spectrum. Neutron energy should therefore be con-sidered as another dimension, perhaps. However, the neutron en-ergy spectrum is determined by neutron scattering, whichremains constant with the burning of the fuel, as has already beenexplained. (It must be reiterated that the energy spectrum at thetime of the NDA measurement is what is in view here.) Therefore,neutron energy is indeed a secondary property, and its inuencecan be treated as a minor uncertainty in the values of the other,more important properties.

The fact that some NDA techniques specically analyze the neu-tron energy spectrum may seem to contradict this simplication.Such techniques include neutron resonance transmission analysis(NRTA) (Sterbentz and Chichester, 2010, 2011), neutron resonancecapture analysis (NRCA) (Postma and Schillebeeckx, 2000), leadslowing-down spectroscopy (LSDS) (Sawan and Conn, 1974; War-ren et al., 2011), and self-interrogation neutron resonance densi-tometry (SINRD) (Hu et al., 2012b; LaFleur et al., 2012). Thesetechniques operate by analyzing the resonances of specic iso-topes, though, rather than the energy spectrum of the fuel assem-bly as a whole. (In its most practical conguration that uses only235U ssion chambers, SINRD analyzes the 0.3 eV resonances ofall three ssile isotopes together as a group.) In other words, theseNDA measurements are more or less direct measurements of indi-vidual isotopes, rather than measurements of a physical propertyof the fuel assembly. Therefore, the quantity of the isotope of inter-est, not neutron scattering and moderation, governs the measure-ment. The neutron-energy characteristics of the fuel assembly as awhole merely limit the applicability of such NDA techniques.2

2 One might say that the very difcult problem of self-shielding in the LSDStechnique is a symptom of a partial failure of these arguments for that technique.Although LSDS does try to analyze a unique, resonance-based signal from each ssileisotope, the phenomenon of self-shielding is actually a property of the fuel assembly

as a whole. LSDS requires the absence of water in the fuel assembly; and withoutwater in between the pins, the neutron-energy characteristics of the fuel assemblybecome much more signicant.

clearThough the argument has already been made that the neutronicphysics governs the physical properties and isotopes, it is never-theless appropriate to show explicitly how the gamma-ray physicsand associated isotopes do follow the neutronics, in fact. The de-layed gamma (DG) NDA technique follows the neutronics becausethe delayed gamma-rays come from induced ssion, primarily inthe ssile isotopes (according to the instruments design) (Mozinet al., 2012). Prompt-gamma activation analysis (PGAA) is basedon neutron capture in the neutron absorbers (Falley et al., 1979).Passive gamma-ray measurements, whether energy-independent(total, TG) or energy-dependent (PG), rely on gamma-rays emittedby the radioactive decay of ssion products (Galloway et al., 2012;Reilly et al., 1991). Admittedly, there is not a direct connection be-tween the isotopes that produce these passive gamma-rays and theneutronic physics of the fuel assembly at the time of the NDA mea-surement, since these isotopes do not play a signicant role in s-sion, neutron capture, or neutron scattering. An indirect link doesexist, though, because these isotopes were created through the in-duced ssion that occurred in the nuclear reactor. The induced-s-sion term of Eqs. (1) and (2) at the time of the NDA measurementmust obviously be connected with the induced-ssion term of Eqs.(1) and (2) during the burning of the fuel assembly in the reactor.From another perspective, the passive gamma-rays are clearlylinked to BU; and since BU is linked, in turn, to the neutronic phys-ical properties and isotopes at the time of the NDA measurement(as will be proven in Section 3), the passive gamma-rays must like-wise be linked to them as well. Therefore, the gamma-ray physicsis tied to the neutronic physics, in general (though see alsoSection 3.3).

2.4. The NDA techniques that are less dependent upon the BIC set

It was mentioned in the previous subsection that the NRTA,NRCA, LSDS, and SINRD NDA techniques analyze the characteristicneutron-energy resonances of specic isotopes. Similarly, the nu-clear resonance uorescence (NRF) technique analyzes the charac-teristic gamma-ray resonances of specic isotopes (Hayakawaet al., 2010), and the X-ray uorescence (XRF) technique analyzesthe characteristic X-rays of specic elements (Freeman et al.,2010; Reilly et al., 1991). By the arguments made above, suchNDA techniques should not depend upon the BIC set signicantlyor at all. On the other hand, all these techniques except NRF aremoderately to severely limited by the attenuation of the character-istic signal through the intervening material between the point ofgeneration of the signal and the location of the detector. Suchattenuation prevents the NDA techniques from detecting a signi-cant signal from the inner fuel pins of a used fuel assembly. Thislack of information is unacceptable for safeguards, since inner pinscould be removed (i.e., a partial defect) without the loss being de-tected. For this reason, it is presumed in this paper that such NDAtechniques must be used in a complementary way with NDA tech-niques that are able to detect the inner pins but are dependentupon the BIC set.

2.5. The composition of the physical-properties and isotopic-contentvector spaces

The discussion of Eqs. (1) and (2) has already indicated some ofthe various physical properties and groups of isotopes that can beincluded in the respective vector spaces. Nevertheless, it is neces-sary to list both the main properties and isotopes that can be in-cluded and those that cannot, and to provide justication forthese listings. In the various extant and proposed NDA techniques,

A.M. Bolind / Annals of Numany physical properties are examined, and many different iso-topes are considered. These facts do not contradict the tri-dimen-sionality of each space, because the many physical properties arenot all independent of each other and because the many isotopesmust be grouped appropriately.

The rst term, S, in Eqs. (1) and (2) corresponds to the quantityor production rate of primary neutrons in the used fuel assembly.This quantity (or rate) will henceforth be designated by the symbolNPRI. Primary neutrons are created chiey by the spontaneous s-sion of 242Cm, 244Cm, and 240Pu but also by (a,n) reactions andby photossion, to a much lesser extent (Reilly et al., 1991). Be-cause the spontaneous ssion of these three isotopes dominatesthe production of primary neutrons, these isotopes can be consid-ered collectively as one of the variables in the isotopic-content vec-tor space.

The second and third terms in Eq. (1) correspond to neutronmultiplication. There are many ways to represent neutron multi-plication, but this paper will represent it by the leakage multiplica-tion, ML. It is to be understood that ML can be translated into theother representations for both active and passive measurements,at least in theory (Schulze et al., 1980; Suzaki, 1991; Tsuji et al.,2003; Ueda et al., 1992, 1993; Wrz et al., 1990).ML is the numberof neutrons that leak out of the fuel assembly per initial neutron(Reilly et al., 1991). For passive neutron measurements, ML has asimple yet powerful relationship with both NPRI and the measuredneutron count rate outside the fuel assembly (Eq. (14.1) of Reillyet al. (1991), and Eq. (17) of Henzl et al. (2013)):

Total Neutron Count Rate eMLNPRI 3(Here, e is the efciency with which the neutron detector can detectneutrons that are outside the fuel assembly, in contrast to someother denitions.) It must be emphasized that although the primaryneutrons multiply differently than do neutrons introduced into thefuel assembly from an external source, the various representationsof such multiplication are all connected and are not independent.

The second term in Eq. (2) corresponds to the ssile isotopes.The safeguards community denes an effective 239Pu content(239Pueff) that is a weighted sum of all three ssile isotopes 235U, 239Pu, and 241Pu (Tobin et al., 2012). These are the only threessile isotopes that are considered in this paper, although somehigher ssile isotopes, such as 245Cm, might be important for usedMOX fuel (Roque et al., 2003). The weighting of the isotopes de-pends on the particular NDA technique being considered, but thisminor distinction will be neglected in this paper except inSection 5.1.

The third term in Eq. (2) corresponds to the neutron absorbers.Only those isotopes that change signicantly with burning are rel-evant for NDA; neutron capture in 238U in used LEU fuel assembliesis irrelevant for this reason, for instance. Therefore, the neutronabsorbers can be divided into transuranic isotopes and ssionproducts (which include those isotopes made by neutron capturein the direct ssion products). All of the transuranic isotopes onthe 242Cm and 244Cm nucleosynthesis pathways (Fig. 5, Section 3.1)can be considered as neutron absorbers, because all of them(including 239Pu and 241Pu) have thermal and epithermal neu-tron-capture cross sections that are larger than those of 238U(Brown et al., 2013; Chadwick et al., 2011). As for the ssion prod-ucts, the burnup-credit community has recognized that a mere f-teen isotopes are responsible for about 80% of the neutroncapturing in ssion products (Roque et al., 2003; Toubon et al.,2003). These fteen isotopes are 95Mo, 99Tc, 101Ru, 103Rh, 109Ag,133Cs, 143Nd, 145Nd, 147Sm, 149Sm, 150Sm, 151Sm, 152Sm, 153Eu, and155Gd. Parks et al. (2000) add 151Eu to this list. The same set of s-sion products applies to both uranium and MOX fuels. For safe-guards, 155Eu should also be included in the list because it decaysto 155Gd. (It is not included in the burnup-credit list because the

Energy 66 (2014) 3150 35burnup-credit community always assumes a 5-year cooling time.)Thus, the ssion-product neutron absorbers can be represented bythis set of 16 + 1 isotopes. In short, the neutron-absorbing isotopes

learare one group, consisting of two subgroups of transuranic and s-sion-product isotopes.

The third term in Eq. (2) can sometimes also be identiedexplicitly with the physical property of neutron capture, such asthrough the NDA technique of prompt gamma-ray activation anal-ysis (PGAA) (Falley et al., 1979). Usually, though, the intense pas-sive gamma-ray activity of the used fuel assembly makes itimpossible to distinguish such neutron-capture gamma-rays.Therefore, neutron capture is usually incorporated into the physi-cal property of neutron multiplication, instead.

The fth term in Eq. (2) the o//ot term corresponds to acharacteristic die-away time, s, for a pulse of neutrons in the fuelassembly to dissipate, as was previously mentioned. The die-awaytime is not actually a separate and independent physical property,though. Instead, it represents the net effect of all the other physicalproperties on the neutron ux. Henzl et al. (2012, 2013) haveshown that s changes proportionally with changes in ML for exter-nal-source neutrons. Therefore, it can be said that s and ML repre-sent the same physical property.

Besides the four physical properties discussed so far, there is an-other way to represent the physics in Eq. (2): namely, through theneutron fate probabilities (Croft et al., 2012; Reilly et al., 1991). Thelast three terms on the left-hand side of Eq. (2) correspond respec-tively to the only three ways that neutrons can be removed fromthe fuel assembly: by being absorbed and inducing ssion, bybeing captured, and by permanently escaping out of the fuel(leakage). The three probabilities that correspond to these threefates (designated by pf, pc, and pl, respectively) are thus fundamen-tal neutronic properties of every fuel assembly. Since the fates aremutually exclusive, the fate probabilities sum to one(pf + pc + pl = 1).

Nevertheless, only the ssion and capture probabilities are rel-evant for characterizing an LEU or HEU fuel assembly, since theleakage probability corresponds to the diffusion term, which stayspractically constant. Nauchi et al. (2011) have demonstrated thatthe leakage probability for a given type of LEU fuel assembly(e.g., PWR vs. boiling-water-reactor (BWR)) in water does notchange with BU. There are two ways to view this fact: (1) the con-stancy of the neutron diffusion (pl = constant) and (2) the compen-sating changes in pf and pc (1 pl = pf + pc = constant). The rstviewpoint has already been justied; the second viewpoint is jus-tied next.

Nauchi et al. (2011) found that the decline in the macroscopicssion cross section of the 235U in the fuel assembly because ofburnup is compensated by increases in the macroscopic absorptioncross sections of the ssion products and the transuranic isotopesthat are produced. Even the burnout of Gd burnable poison has acompensating mechanism, through changes in the neutron energyspectrum. The Gd hardens the spectrum in the fresh fuel, which de-creases the effective microscopic absorption cross of the 235U butalso increases the production of plutonium isotopes through cap-ture in 238U. The result is that the decrease in the macroscopiccross section of the Gd during burning is compensated by increasesin the 235U effective microscopic cross section and in the transura-nic isotopes macroscopic cross sections.

Furthermore, although they did not specically study or men-tion it, their results also indicate that pl should be practically con-stant with IE and CT, too. This conclusion comes from the fact thatthey did compare fuel assemblies with and without Gd burnablepoison (at constant 4.8% IE), and they found that both had the sameleakage probability. The microscopic ssion cross section of 235U atthermal energies is more than seven orders of magnitude greaterthan the capture cross section of 238U, yet the capture cross section

235

36 A.M. Bolind / Annals of Nucof natural Gd is two orders of magnitude greater than that of U(Brown et al., 2013; Chadwick et al., 2011). Since Gd poisons have anegligible effect on pl, it is reasonable to conclude that changes inIE also should not affect it, excepting perhaps a gross change suchas from LEU to HEU. For slight changes in IE in HEU, the physicsshould be the same. As for CT, BU more dramatically reduces thessile content and increases the absorber content than does CT.Since BU does not signicantly affect pl, it is reasonable to concludethat CT also should not affect it. Therefore, the leakage probabilitycannot be relevant to characterization, except in the gross sense ofdistinguishing one type of fuel assembly from another (e.g., PWRfrom BWR) (Nauchi et al., 2011).

Since Nauchi et al. did not perform simulations for MOX fuelassemblies, it is not yet clear whether or not pl is constant for them,too. The fact that the Pu isotopes all have larger cross sections (forssion and/or capture) than 238U may have enough signicance tochange pl with one or more of the BIC variables. Nevertheless, evenif pl is not constant in MOX fuel, it probably does not vary dramat-ically, based on the above considerations for uranium fuelassemblies.

As for gamma-rays (including X-rays), they are unlike neutronsand neither perpetuate themselves in a chain reaction nor multi-ply, in general. There are therefore only two fates for gamma-rays:capture (pc) and leakage (pl). The capture of gamma-rays is betterknown as attenuation, particularly in three-dimensional geometryin which scattering is not considered a fate (in contrast with one-dimensional, beam attenuation). Gamma-rays are attenuated byinteractions with both the nuclei and the electrons of the sur-rounding material (Lamarsh and Baratta, 2001). The material com-position of a fuel assembly stays roughly constant during burningand cooling, though. Since the attenuation does not change withchanges in the physical properties and isotopes, it cannot be rele-vant to the characterization of the fuel assembly, except to limitthe ability to measure the gamma-rays. The only gamma-ray phys-ical property that does change with burning, then, is the produc-tion of gamma-rays, either passively (such as through radioactivedecay) or actively (such as through uorescence or ssion); andthis activity is, of course, a direct function of the quantities of thegamma-ray emitting isotopes in the fuel assembly.

2.6. A comparison between the NDA of used fuel assemblies and theNDA of puried materials, scrap, or waste

Many readers may have experience with the NDA of puriednuclear materials, scrap, or contaminated waste and may wonderwhy some physical properties that are associated with such NDAhave not been included in the list of physical properties for NDAof used fuel assemblies. Furthermore, such readers may not under-stand why NDA of used fuel assemblies is apparently so muchmore difcult than NDA of other materials or samples. This sectionprovides a clear answer in light of the foregoing subsections.

The NDA of puried materials, scrap, or waste (which willhenceforth be abbreviated as traditional NDA) differs from theNDA of used fuel assemblies in three critical ways: (1) a smallergeometry, (2) a fast neutron spectrum, and (3) a direct associationof the count rate(s) with the particular isotope(s) of interest. Afourth difference which is not a true difference is that manytimes the accuracy of traditional NDA is not required to be good,because the NDA is serving only as a rough double-checking ofotherwise well-known materials. (See, for example, Section 15.5of Reilly et al., 1991.) Since the presumed goal of this paper is toreduce the uncertainty of used-fuel-assembly NDA below the 15%RESD threshold found by Burr et al. (2012) (see Section 1, above),only the rst three differences will be considered.

(1) The smaller geometry in some traditional NDA applications

Energy 66 (2014) 3150reduces the attenuation of the signal. For instance, a smalltube containing dissolved nuclear material can be analyzedthoroughly with gamma-ray spectroscopy, but gamma-ray

spectroscopy can examine only the surface of a (much lar-

at the time of the NDA measurements. The goal is to determine the

A.M. Bolind / Annals of Nuclear Energy 66 (2014) 3150 37ger) used fuel assembly. This point was discussed in Sec-tion 2.4. Another, related physical property is that smallsamples necessarily have less self-generated radiation thanlarger samples of the same material. For this reason, signalsthat can be detected from small samples of fuel may be over-whelmed by the intense gamma-ray and X-ray backgroundof an entire used fuel assembly.

(2) In traditional neutron NDA techniques, such as neutron coin-cidence (or multiplicity) counting (Ensslin et al., 1998; Reillyet al., 1991), the neutron energy spectrum in the samplebeing measured is kept as fast as possible. The neutronspeed is maintained by keeping the sample in air rather thanin water, by avoiding including other moderating material inthe sample and sample chamber, by putting a cadmium lineraround the sample chamber, and by limiting the size or den-sity of the sample (particularly in the case of containers ofcontaminated waste). When the neutron spectrum is fast,the induced-ssion term and the neutron-capture term inEqs. (1) and (2) become small relative to the primary-neu-tron term, since the neutron capture and ssion cross sec-tions of most isotopes decrease rapidly with increasingneutron energy. Therefore, neutron capture can be ignoredor subsumed into the detection efciency (see page 487 ofReilly et al. (1991)), and the coincidence from induced s-sion from (a,n) neutrons can be treated as an error to be cor-rected.3 By these simplications, the detected neutrons can beassociated more or less directly with the primary-neutronsource term and isotopes. These considerations apply evenfor NDA of larger volumes of contaminated waste whichoften contains some moderating material such as paper orplastic because the attenuation and self-shielding of thewaste are minimized by keeping the density of the wastesmall or by limiting the amount of the nuclear material inthe waste.

(3) The third feature of traditional NDA a direct association ofthe count rate(s) with the particular isotope(s) of interest is self-evident for those NDA techniques that measure char-acteristic signals (such as resonances). This difference alsoexists, though, for neutron coincidence (multiplicity) count-ing; that is, the primary-neutron-emitting isotopes are theisotopes of interest or are closely related to the isotopes ofinterest. For instance, the dominant source isotope in typicalMOX powder is 240Pu, because the 244Cm in the used-fuelfeedstock is removed during reprocessing. By measuring NPRI(the primary neutrons), the quantity of 240Pu can be deter-mined; and by combining this knowledge with an a prioriknowledge (or gamma-ray measurement) of the plutoniumisotopic vector of the MOX, the total quantity of plutoniumcan be determined. Neutron multiplicity counting can some-times even be used to distinguish among two or more spon-taneously ssioning isotopes, because different isotopeshave different probability distributions of the multiplicityof the neutrons emitted from each ssion event. (This multi-plicity distribution is denoted by P(m); its mean is m.) Thus inan ideal case, the isotopes can be measured by the effects oftheir multiplicity distributions on the higher moments of thetime distributions of the detected neutrons (Ensslin et al.,1998).

NDA for used fuel assemblies stands in contrast. The intensegamma-ray and X-ray radiation that is generated by a used fuel3 This effect of (a,n) neutrons is handled by a correction factor in coincidencecounting (Ensslin et al., 1979; Krick, 1980; Reilly et al., 1991) and by the solution ofthree simultaneous equations in multiplicity counting (Ensslin et al., 1998).direction (increase or decrease) and relative magnitude of the ef-fects, and the results are summarized at the end of this section,in Table 1. The next section (Section 4) discusses the ways in whichtime, energy, and geometry can cause the BIC set to fail to charac-terize a fuel assembly. In short, this section discusses how the BICset does characterize the physical properties and isotopic content,whereas the next section discusses how the BIC set does not char-acterize them in certain situations. Readers who are not interestedin the detailed arguments for these assertions are advised to readonly the summaries in Sections 3.4 and 4.5 and then to proceed di-rectly to the implications in Section 5.

3.1. The effect of the BIC set on NPRI and the primary-neutron-emittingisotopes4

3.1.1. The effect on LEU fuelFor LEU fuel, the number of internal primary neutrons (NPRI) is

increased by BU, decreased by IE, and decreased by CT. BU in-creases NPRI by producing 240Pu, 242Cm, and 244Cm through neutroncapture in 238U and the successively higher actinides (Fig. 5). As BUis increased, NPRI is dominated rst by 240Pu, then by 242Cm, and -nally by 244Cm (Bosler et al., 1982).

Note that the nucleosynthesis pathways for 242Cm and 244Cmdiverge at 241Pu. If a 241Pu atom radioactively decays, then it be-comes 241Am; and neutron capture in 241Am produces 242Cm. Ifthe 241Pu atom captures a neutron, instead, then it becomesassembly, plus the attenuation due to the size of the fuel assembly,limit or even prohibit the application of traditional photon-spec-troscopic NDA techniques. As for neutron-coincidence NDA tech-niques (Croft et al., 2011a), the spontaneous ssion from thecurium isotopes dominates NPRI for all but the least burnups. Thetraditional logic would thus try to correlate the coincidence signalwith the amount of 244Cm in the fuel assembly. But unlike 240Pu,the amount of 244Cm is not interesting on its own; its chief utilityis as an indication of BU. Similarly, since the (a,n) source is alsodependent on BU, there is no longer a direct advantage of separat-ing it out. Furthermore, the neutron multiplication and captureterms in Eqs. (1) and (2) cannot be neglected, because the waterin the fuel assembly thermalizes the neutron spectrum. Therefore,the point model becomes invalid. The neutron multiplication andcapture ruin the ability of neutron multiplicity counting to distin-guish isotopes, in particular.

Therefore, both the logic and the practicality of distinguishingthe (a,n) sources from the spontaneous-ssion sources fail forNDA of used fuel assemblies. Also, isotopes cannot be distin-guished by their neutron multiplicities, P(m). Lastly, it is clear thatthe non-negligible contributions of the induced-ssion and neu-tron-capture terms in Eqs. (1) and (2) are largely responsible forcausing the NDA of used fuel assemblies to be more complicatedand historically less accurate than the traditional NDA of othersamples.

3. Results (1): the independence of the BIC variables withrespect to the physical properties and the isotopic content

The previous section (Section 2) argued that the physical-prop-erties and isotopic-content vector spaces exist, are three-dimen-sional, and are ruled by the neutronic physics. It also speciedwhat physical properties and groups of isotopes are included inthe respective vector spaces. This section now proves the indepen-dence of the BIC variables with respect to the physical propertiesand the isotopes, by examining how the BIC variables affect them4 This section is organized by fuel type.

The problem is that the concentration of 238U by which 240Pu,242Cm, and 244Cm are produced can no longer be taken forgranted as the concentration of 235U is increased beyond several

lear Energy 66 (2014) 3150242Pu; and successive capture in 242Pu produces 244Cm. This differ-ence in the nucleosynthesis pathways is important because theradioactive decay of 241Pu causes the path for 242Cm to dependon the irradiation history of the fuel assembly, whereas the pathfor 244Cm does not signicantly do so (Tiitta and Hautamki,2001). (See Section 4.1.)

IE decreases NPRI in LEU fuel, for a given level of BU. (SeeFig. 66 in Bosler et al. (1982), Fig. 6 in Schulze et al. (1980),Fig. 8 in Tiitta and Hautamki (2001), and Fig. 16 in Tiitta et al.(2001).) This net effect of IE is complicated, though, because thereare actually two competing processes. One process is related tothe denition of BU, and the other is related to changes to theneutron energy spectrum.

BU is dened by the total number of ssion events per unitvolume or mass of fuel, not by the number of ssion events perneutron that is within that volume or mass (see Chapter 18 ofReilly et al. (1991)). More enriched fuel, then, undergoes moression than less enriched fuel does for the same neutron ux.Therefore, the less enriched fuel needs more neutron uence toreach the same level of BU. The quantity of 238U is practically thesame in both kinds of fuel, though. Since 239Pu and the othertransuranic isotopes are created through neutron capture in 238U,the higher uence of less enriched fuel necessarily produces more240 242 244

Fig. 5. Main nucleosynthesis pathways for 242Cm and 244Cm (Hsue et al., 1979;Phillips et al., 1980; Sasahara et al., 2004). Broad arrows represent neutron capture;narrow arrows represent b-decay.

38 A.M. Bolind / Annals of NucPu, Cm, and Cm. The process will be called the neutron-uence effect in the remainder of this paper. By this rst process,therefore, IE decreases NPRI.

The other process is caused by the fact that more burnable poi-son, chemical shim, or control-rod insertion is needed at startup touse more highly enriched fuel. These poisons preferentially absorbthermal neutrons and thereby slightly harden the neutron energyspectrum, which increases the rate of production of transuranicisotopes for the same amount of neutron ux. (See Parks et al.(2000) and Section 4.2.) By this second, albeit indirect, process, IEincreases the amounts of 240Pu, 242Cm, and 244Cm, and increasesNPRI.

The rst process the neutron-uence effect is more signi-cant than the second process, so that the net effect of more IE isless NPRI for a given level of BU. In other words, increased IE gener-ates 240Pu, 242Cm, and 244Cm at a faster rate but for much less time(where time is measured in terms of neutron uence).

CT decreases NPRI in all fuels via the radioactive decay of 242Cm(half-life = 163 days) and 244Cm (half-life = 18.1 years).

3.1.2. The effect on HEU fuelFor HEU fuel, NPRI is affected by BU and CT in the same way as

for LEU fuel, but the process by which IE affects the fuel is changed.

tially present in theMOX fuel. For the sake of argument, let it be de-

nedas an effectivequantity of 243Am, since 243Am is the last isotopeon the nucleosynthesis pathway to 244Cm if the reasonable assump-tion is made that all of the 244Am b-decays into 244Cm. The nucleo-synthesis pathway to 242Cm is ignored because neutron NDAmeasurements must be made after the 242Cm has decayed away,to achieve the best accuracy; see Section 4.1. All of the uraniumand transuranic isotopes on the nucleosynthesis pathway to 244Cm(Fig. 5) contribute to CP. Notably, the ssile isotopesmust also be in-cluded in CP, because their capture cross sections, rather than theirssion cross sections, are important in this case. The contribution ofeach isotope to CP should be a function of its quantity in the fuel andof its probability for producing 244Cm.

5 Of course, 238U also produces 240Pu, which contributes to NPRI on its own. Butsince 240Pu is on the nucleosynthesis pathway for both 242Cm and 244Cm and since240Pu is a minor contributor to NPRI in comparison to these curium isotopes for BUpercent. As IE increases, it begins to dramatically decrease NPRIby removing 238U from the fuel assembly. This effect is in additionto the other effects that apply to LEU fuel. For high enrichmentsand reasonably low burnups, though, the effect of hardening theneutron spectrum by means of neutron poisons saturates. One ofthe reasons is that as enrichment increases dramatically, nuclearengineers simply decrease the size of the reactor instead ofincreasing the quantities of neutron absorbers. Therefore, the dom-inant effects of IE are the neutron-uence effect and the removal of238U. This situation was observed in the neutron NDA measure-ments by Phillips et al. (1980) of spent MTR fuel elements (93%enrichment).

3.1.3. The effect on MOX fuelFor MOX fuel, BU and CT affect NPRI in the same way as they af-

fect LEU and HEU fuel. IE must take a new meaning, though, sincethe ssile material is not 235U but 239Pu and 241Pu. For this paper,the term initial enrichment will be retained for MOX fuel but willrefer to the fraction (or percentage) of ssile isotopes relative tothe total content heavy metal (i.e., U and Pu). In other words, IE al-ways equals the initial ssile content of the fuel assembly. Thisconvention is the same as the one used in the OECD/NEA BurnupCredit Criticality Benchmark Study (OConnor and Liem, 2003),and it allows the continued use of the abbreviation IE with MOXfuel.

The fact that non-ssile isotopes of plutonium are also includedin the initial condition of a MOX fuel assembly cannot be ignored,though. (Other transuranic isotopes could also be included but ide-ally are not.) They increase the rate at which 240Pu, 242Cm, and244Cm are produced in the fuel, as compared to LEU fuel. (Of course,initially including more 240Pu in the fresh fuel also causes more240Pu to be present in the used fuel at discharge.) As mentionedin Section 2, this inuence cannot be taken into account by BUand IE only, since IE represents the ssile isotopes only and sinceBU is a process rather than an initial condition of the fuel assembly.Therefore, it is necessary to include a fourth variable in the BIC set.This fourth variable will be called the curium-production ability,with the abbreviation CP, in this paper.5 With the inclusion of CP,the BIC set expands to four dimensions and becomes the BICC set,for MOX fuel. The fact that the BICC set is four-dimensional butthe physical characteristics and isotopic content are three-dimen-sional will be discussed later in Section 5.2.

CP is an effective quantity of the curium-producing isotopes ini-greater than about 15 GWd/tU (Bosler et al., 1982), it is not a misnomer to call thisfourth variable as simply the curium-production ability. Henzl et al. (2013) make asomewhat similar observation.

This probability of an isotope to produce 244Cm is, in turn, afunction of its capture cross section6 and its position along the244Cm nucleosynthesis pathway. It is also a function of the neutronuence. A simplied method to determine these probabilities wouldbe to assume constant neutron ux in the reactor and to use the gen-eralized Bateman equations, treating the nucleosynthesis pathwaylike an inverted radioactive-decay chain and replacing decay con-

the additional IE on NPRI is a competition between (1) the increasein CP from the replacement of the 238U by the 239Pu and (2) theneutron-uence effect on the 240Pu and higher isotopes that werealready in the fuel. The outcome of this competition can be roughlyestimated by comparing the CP of the pure 239Pu against the CP ofthe reactor-grade MOX powder.8 If the CP of 239Pu is the lesser, thenthe neutron-uence effect must dominate, and IE must decreaseNPRI; this case is like the case of LEU fuel. If the CP of 239Pu is signif-icantly greater, then the quantities of the higher Pu isotopes in theso-called reactor-grade fuel are negligible, and IE clearly must in-crease NPRI; this case is like the rst example, above. If the CP of239Pu is greater but close in value, though, the outcome is unclearand depends on the burnup, since BU increases the neutron uence

A.M. Bolind / Annals of Nuclear Energy 66 (2014) 3150 39stants with capture cross sections (Cetnar, 2006). The probabilityfor an isotope would then be the ratio of the quantity of 244Cm pro-duced per initial quantity of that isotope, as a function of the neutronuence. In the extreme case, the method would become a burnupsimulation, somewhat similar to that described by Fig. 12 of the pa-per by Sasahara et al. (2004).7 Determining a proper method for cal-culating the probability for each isotope may be difcult, though,and it is unnecessary for the scope of this paper. It is sufcient hereto recognize that the probabilities of the isotopes that are lower onthe nucleosynthesis pathway must be much less than the probabil-ities of the higher isotopes, since the transmutation of the lower iso-topes must pass through the higher isotopes on the way to 244Cm.

Note that CP could be dened for LEU and HEU fuel, too. Itwould basically be the amount of 238U in the fuel, since the produc-tion of 244Cm from neutron capture in 235U must be extremelysmall in comparison with the production from 238U, due to themuch lower position of 235U in the nucleosynthesis pathway. SinceIE and CP are mutually exclusive for uranium fuel, they are notindependent variables for this fuel, though. Therefore, the four-dimensional BICC set collapses to the three-dimensional BIC setfor LEU and HEU fuel. Due to the prevalence of LEU and HEU fuelover MOX fuel worldwide, this paper will continue to use theshorter acronym, BIC, for general cases, unless specically referringto MOX fuel.

With this distinction between IE and CP having been made, it isnow possible to consider the effects of these two variables on NPRIfor MOX fuel. The effect of CP is obvious; by denition, CP increasesNPRI. The effect of IE is not so clear, though, because the effect is afunction of CP, too, since 239Pu and 241Pu are included in CP.

This effect of IE is easiest to see conceptually by means of twoexamples. In the rst example, more plutonium of a given isotopicvector is added to the fresh MOX fuel. This additional Pu must nec-essarily increase NPRI, since all of the Pu isotopes even the ssileisotopes are farther along the 244Cm nucleosynthesis pathwaythan is the 238U that is replaced by the Pu. In other words, the addi-tional Pu has a greater CP than the 238U that it replaces. Of course,the additional 239Pu and 241Pu also increase the reactivity (i.e., ML,see the next section) and reduce the amount of neutron uence re-quired to achieve a given BU. There is little to no neutron-uenceeffect in this example, though, because the non-ssile isotopes ofPu are being added equally with the ssile isotopes (i.e., the isoto-pic vector of the plutonium in the MOX fuel is remaining constant)and because the production of 244Cm by the Pu isotopes likelydominates the production by 238U for all but the least IE.

In the second example, pure 239Pu oxide is added to a baseamount of fresh, reactor-grade MOX fuel. For conceptual simplic-ity, assume no dilution, but instead assume that an equivalentamount of only 238U is removed from the MOX fuel to keep the to-tal heavy-metal content constant. In this example, the net effect of

6 Note that capture cross sections have units of area (barns or cm2) and so are notprobabilities. It is therefore not valid to multiply them together to achieve a jointcross section, in the way that a joint probability is the product of the individualprobabilities.

7 The probabilities for each isotope in this gure by Sasahara et al. are with respectto all of the nuclear-reaction modes of only that same isotope; i.e., the probabilitiesare conditional upon a reaction in that isotope. They are therefore not on an absolute

scale that encompasses all of the isotopes, by which the probabilities of one isotopecan be combined with those of another to form joint probabilities, which is necessaryhere.and thereby increases the probabilities of the lower isotopes to form244Cm. In other words, there is a competition, in this case, betweenthe neutron-uence-effects reduction of the production of 244Cmfrom the small amounts of 240Pu and higher isotopes initially inthe fuel, on the one hand, and the additional 239Pus increase ofthe production of 244Cm from the 238U that it replaces, on the otherhand.9

MOX fuel is typically created according to the rst example i.e., the isotopic vector of the Pu that is added to the MOX is con-stant and therefore the second example is largely irrelevant fromthe standpoint of creating the MOX fuel. The safeguards commu-nity must work backwards and solve for the unknown initial com-position of the fuel, though. From this standpoint, the secondexample is important because it illuminates the interconnected-ness of IE and CP (and even BU).

Note that this issue of the interconnectedness of IE and CP can-not arise in LEU and HEU fuel because the CP of 235U is less than theCP of 238U (not to mention the mutual exclusivity of the 235U and238U concentrations). The same principle, though, would apply inthe case of adding 235U to MOX fuel.

An extreme case that illustrates the effect of CP on NPRI is thesimulation study by Gross (1965). In this study, two kinds of plu-tonium fuel were irradiated. One kind was weapons-grade pluto-nium. The other kind was plutonium that had been created byirradiating depleted uranium; and it contained much more of thenon-ssile Pu isotopes than the weapons-grade plutonium con-tained; that is, it had a greater CP. Of course, the addition of thesenon-ssile isotopes correspondingly diluted the concentrations ofthe ssile isotopes, so that the IE of this second kind was less thanthe IE of the rst kind. From the combination of Fig. 1 (BU) andFig. 22 (244Cm) in Grosss paper, it is clear that the combined effectof IE and CP in this second case was to increase NPRI. This case is anextreme case, though, because the plutonium fuel was pure pluto-nium, not MOX; it did not contain any UO2. Therefore, the fuel wasthe Pu version of HEU, in which IE and CP were not independentvariables. Since IE was so large, this extreme case is best viewedas illustrating the effect of CP.

8 Technically, the comparison should be made with the reactor-grade MOX powderafter the 238U has been removed, but this detail is unimportant since the argument isregarding differential quantities.

9 To the best of the authors knowledge, the literature does not yet contain enoughexperimental or simulation data to conrm this conceptual approach. The properexperiment or simulation would have to burn several MOX fuel assemblies containingdifferent amounts of 239Pu and 241Pu (i.e., varying IE) but having the same amounts ofother Pu isotopes (i.e., almost constant CP). (The amount of 238U would have tochange slightly, of course, to accommodate the changes in 239Pu and 241Pu.)Furthermore, the assemblies would have to be burned to the same BU. In theexperiments and simulations described in the literature thus far, not all of these

conditions have been satised simultaneously. Thus, the effect of IE has not beenisolated from the effects of CP and BU.

40 A.M. Bolind / Annals of Nuclear3.2. The effect of the BIC set on ML, the ssile and capturing isotopes,and the fate probabilities10

3.2.1. The effect of BUBy denition, BU consumes ssile material, by ssion and neu-

tron capture, though it also generates 239Pu and 241Pu through neu-tron capture in 238U (Fig. 5). In enriched uranium fuel, theproduction of 239Pu and 241Pu does not offset the loss of 235U andthe production of neutron absorbers, so that pf and ML always de-crease with BU. In natural uranium fuel (e.g., CANDU fuel) and de-pleted uranium fuel, a so-called plutonium peak occurs duringthe initial period of irradiation (Canadian Nuclear Safety Commis-sion, 1993). In this case, the larger ssion cross section of 239Puactually does offset the loss of 235U and the production of ssionproducts, causing pf and ML to increase temporarily with BU. Withfurther burning, the net production rate of 239Pu and 241Pu de-creases because of increased destruction of these isotopes throughssion, and the reactivity then declines with BU, as usual. For thesake of a simple argument, the rest of this paper will exclude theconsideration of natural and depleted uranium fuel, for this reason.Of course, future work must extend the discussion to cover suchcases, especially for safeguards purposes.

Unlike in uranium fuel, BU always decreases the amount of239Pu in MOX fuel (excepting the extremely unusual case thatthe MOX is made with mostly 241Pu). Also, BU may increase or de-crease the amount of 241Pu in MOX fuel, depending on its initialconcentration and on the balance between its production from238U, 239Pu, and 240Pu and its destruction by ssion, neutron cap-ture, and radioactive decay.

BU creates neutron absorbers and increases pc and thereby fur-ther decreases ML. BU produces the neutron-capturing transuranicisotopes and ssion products by capture and ssion, respectively.These two subgroups of isotopes have been described inSection 2.5.

The different ssile isotopes produce different proportions ofthe individual isotopes in the set of 16 important ssion products.Furthermore, the capturing ability (or reactivity worth) of the en-tire set also changes according to the ssile isotope. An initial at-tempt to estimate the relative reactivity worth of the ssionproducts from the different ssile isotopes might compare thecumulative ssion-product yields. For each ssile isotope, thecumulative ssion-product yields (Brown et al., 2013; Chadwicket al., 2011) of the 16 important ssion products can be multipliedby their thermal-neutron radiative-capture microscopic cross sec-tions (Chang, 2000). The result can be summed for each ssile iso-tope to obtain an estimate of the macroscopic capture cross sectionof its ssion products. The conclusion is that the ssion-productmacroscopic capture cross section for 235U is about three fourthsof the cross section for 239Pu; and in turn, the cross section for239Pu is about four fths of the cross section for 241Pu. Also, the s-sions of 239Pu and 241Pu produce roughly ve to seven times more155Eu than the ssion of 235U (Brown et al., 2013; Chadwick et al.,2011). It might be hypothesized, therefore, that preferential burn-ing of the heavier ssile isotopes should result in more neutron-capturing ssion products in the used fuel assembly, and that thereshould be more neutron capture in ssion products in used MOXfuel than in used uranium fuel.

The neutron-absorbing ssion products are also being burnedout while they are being created in the reactor, though, which actsto minimize the differences. Furthermore, the data in the burnup-credit literature, such as presented by Roque et al. (2003), indicatethat the ssion products actually have less of an effect on the reac-tivity of used MOX fuel than on the reactivity of used LEU fuel. Thesource of the discrepancy is the fact mentioned before, that the10 This section is organized by BIC variable.transuranic isotopes have larger epithermal cross sections than238U, including signicant resonances in this energy region. Thus,the additional quantities of transuranic isotopes in MOX fuel tendto absorb the neutrons before they can slow down to thermal ener-gies, where the ssion product cross sections are largest (exceptingthe 0.09 eV resonance of 149Sm (Brown et al., 2013; Chadwick et al.,2011). This process might be likened to a decrease in the resonanceescape probability, p, of the four-factor formula for keff (Duderstadtand Hamilton, 1976). The transuranic isotopes harden the neutronenergy spectrum, although the total number of neutrons staysbasically the same on a per-ssion basis. By intercepting theneutrons before the ssion products have a chance to capturethem, the transuranic isotopes decrease the relative magnitude ofthe reactivity worth of the ssion products, despite their slightlyincreased macroscopic thermal cross section in used MOX fuel.This effect is what has been reported in the burnup-creditliterature (Roque et al., 1999, 2003).

From an NDA perspective, the burnup-credit publications prob-ably overstate this effect, though. The burnup-credit results comefrom models of innite lattices of fuel assemblies, which is areasonable approach for that community since burnup credit isconcerned with collections of used fuel assemblies. Typical NDAmeasurements are of a lone fuel assembly in cooling water, though.The lone fuel assembly has a more thermalized neutron spectrumthan the collection of fuel assemblies, because the extra watersurrounding the lone fuel assembly moderates and reects theneutrons. The more-thermalized spectrum helps to preserve thereactivity worth of the ssion-product neutron absorbers. Further-more, pl is much greater for a lone fuel assembly than for acollection of fuel assemblies. Correspondingly, the impact ofchanges in pc on ML is less for the lone fuel assembly. Lastly, theconceptual distinction between capture in transuranic isotopesand capture in ssion products is largely an articial constructwith regard to NDA. To the majority of NDA techniques, a capturedneutron is just a captured neutron regardless of what isotope cap-tures it, since they do not analyze the neutron-energy spectrum(Sections 2.3 and 2.4).

For NDA, the distinction between thermal-neutron capture inssion products and resonance-neutron capture in transuranic iso-topes is relevant mostly to the comparison between uranium fueland MOX fuel. For a given type of fuel, the buildup of transuranicisotopes by BU and their effect on the ssion products and ssileisotopes in the used fuel assembly after discharge are an inherentpart of burnup and occur more or less consistently from one fuelassembly to another. In other words, the domains of the other vec-tor spaces illustrated in Fig. 4 and their correlations with the BICset (or BICC set) will be different for each type of fuel, but this factdoes not substantially alter the logic of the correlations. The mainpurpose of this discussion has been to show clearly the connectionbetween BU and neutron capture in used fuel assemblies, whileaddressing the supercially conicting result in the burnup-creditliterature.

3.2.2. The effect of IEIE always increases ML, pf, and the total amount of ssile mate-

rial remaining in the fuel assembly after discharge from the reac-tor. Its effect on the individual ssile isotopes depends on thetype of fuel, though.

For LEU and HEU fuel, IE always increases the amount of 235Uremaining in the used fuel. The effect of IE on 239Pu and 241Pu ismultifaceted like its effect on NPRI. The lessened neutron uenceand the hardening of the spectrum both have the same effects on239Pu and 241Pu as they do on NPRI, and again as with NPRI, the neu-

Energy 66 (2014) 3150tron-uence effect is stronger and causes IE to decrease the quan-tities of 239Pu and 241Pu. A third effect is manifested at higherburnups with these isotopes, though, because they are ssile. At

clearsome level of BU at about 12 GWd/tU for 239Pu and at about38 GWd/tU for 241Pu for the PWR assemblies studied by Bosleret al. (1982) enough of these two ssile isotopes has been createdthat BU then proceeds signicantly by their ssion. After thesepoints, IE actually increases the quantities of 239Pu and 241Pu inthe used fuel assembly, by the simple fact that if more 235U is pres-ent, then more of it and less of the 239Pu and 241Pu must burn inorder to increase BU (Fig. 19 and 21 in Bosler et al. (1982)). In sum-mary, IE initially decreases the amounts of 239Pu and 241Pu in theused fuel assembly, but after a certain level of BU, IE then increasesthe amounts of 239Pu and 241Pu.

MOX fuel is similar, although the neutron energy spectrum af-fects only the relative burning of 239Pu and 241Pu since there isnot a signicant amount of 235U in the fuel. More initial 239Puand 241Pu results in more of these isotopes remaining after burn-ing; but just as with the effects on NPRI, the IE and CP variablesare interconnected and lead to various results when only one orthe other is varied. Specically, adding only more 239Pu to the ini-tial fuel while keeping the initial quantity of 241Pu constant may in-crease or decrease the amount 241Pu remaining in the used fuel.The outcome depends on the initial amount of 241Pu in the fueland the CP of the fuel; and like the effect of IE on NPRI for MOX fuel,it is a competition between the neutron-uence effect on 240Pu andthe capture of neutrons in 239Pu.

IE also affects the neutron-capturing ssion products, the trans-uranic isotopes, and pc. In LEU and HEU fuel, IE increases the pro-portion of 235U that is burned relative to 239Pu and 241Pu andthereby decreases the combined macroscopic capture cross sectionof all the ssion products. By the neutron-uence effect, IE also de-creases the production of the transuranic isotopes, which all havelarger capture cross sections than 238U does. Therefore, IE de-creases pc in used uranium fuel assemblies.

In MOX fuel, IE and CP together increase the quantities of neu-tron-capturing transuranic isotopes, by displacing 238U in the fuel.Similarly, IE and CP together increase the proportion of 241Puburned relative to 239Pu, thereby slightly increasing the quantitiesof the important thermal-neutron-capturing ssion products.However, the result of the neutron-uence effect is not clear. Thecapture cross sections of the transuranic isotopes do not monoton-ically increase (or decrease) along the 244Cm nucleosynthesis path-way, which is unlike their probabilities for creating 244Cm andincreasing NPRI. Since pl may not be constant for MOX fuel (Sec-tion 2.5), it can be tentatively predicted that IE and CP togetherslightly increase pc in used MOX fuel assemblies.

3.2.3. The effect of CTCT affects the ssile isotopes by consuming 241Pu by radioactive

decay (half-life = 14.325 years) (Hu et al., 2012a). CT produces cap-turing isotopes by radioactive decay, mainly 155Gd (stable) from155Eu (half-life = 4.753 years), and 241Am (half-life = 432.6 years)from 241Pu (Hu et al., 2012a; Sonzogni, 2013). (Again, it is assumedthat CT is long enough that all of the very-short-lived neutron poi-sons (like 135Xe) have decayed away.) Apart from 155Eu and 155Gd,the other 15 most important ssion products are either stable orhave very long half-lives, so they are not affected by CT. The decayof 155Eu into 155Gd thus causes the set of 16 ssion products which includes 155Gd to increase with CT. Therefore, CT decreasespf, increases pc, and decreases ML.

3.3. The effect of the BIC set on gamma-ray physical properties andisotopes

Regarding the production of gamma-rays, BU produces radioac-

A.M. Bolind / Annals of Nutive ssion products and increases the passive gamma-ray activity,and CT eliminates the radioactive ssion products and reduces thegamma-ray activity. For example, Rinard (1983) gives twopower-law equations for the relationship of the total gamma-rayactivity (denoted here by Cc) to BU and CT:

Cc aBUCTb for CT > 12 months 4

Cc cBU1:4150:00234CT for CT > 36 months 5In these equations, a, b, and c are positive tting constants. Notethat the total gamma-ray activity does not follow such a powerlaw at cooling times less than 1 year (Fig. 1 by Rinard (1983)), forreasons that will be discussed below in Section 4.1. After the restof the gamma-ray-emitting ssion products have sufciently de-cayed away, the activity is dominated by 137Cs, because of its rela-tively long half-life (30.1 years) and large yield from ssion (Phillipset al., 1980; Rinard, 1983). After this point, the power-law equationreduces to a single exponential (Willman et al., 2006):

Cc aBUekCT 6In all three relations, BU increases Cc, but CT decreases Cc.

Perhaps surprisingly, IE does not have a practical effect oneither the total gamma-ray activity or on the activity of 137Cs.The chief reason is that BU, not IE, represents the total number ofssions and therefore the amount of ssion products and theirgamma-rays. This logic is obvious for MOX fuel, but for LEU andHEU fuel, it is more subtle and has two parts. Firstly, IE changesjust the proportion of ssions that occur in 235U versus in 239Puand 241Pu; again, it does not change the total number of ssions.Secondly, the ssions of 235U and 239Pu emit practically the sameamounts of 137Cs and almost the same amounts of the sum ofthe other gamma-ray-emitting ssion products (though the exactisotopic composition of these other ssion products does vary be-tween the ssions of 235U and 239Pu). For the same reason, bothLEU and MOX used fuels produce about the same total gamma-ray activity and activity from 137Cs, for a given BU (Murphy, 1996).

On the other hand, IE does affect the quantities of 134Cs and154Eu. These isotopes are produced by neutron capture in other, s-sion-product isotopes (Phillips et al., 1980; Willman et al., 2006).By the neutron-uence effect, IE decreases the quantities of theseisotopes in the used fuel assembly (Willman et al., 2006). ForMOX fuel, it is unclear if CP affects the production of 134Cs and154Eu.

At extremely short cooling times, the different yields of short-lived, gamma-ray-emitting ssion products from the ssions of235U, 239Pu, and 241Pu may be used to differentiate among theseisotopes. These different ssion products emit gamma-rays at dif-ferent energies, which can be distinguished by an energy-sensitivegamma-ray detector. These facts are the basis of the delayed gam-ma (DG) technique (Mozin et al., 2012). At reasonably long coolingtimes, though, these short-lived isotopes are gone; and corre-spondingly the reason to make energy-sensitive measurements,rather than just total-activity measurements, changes. The goal isno longer to distinguish among the ssile isotopes but rather totry to infer BU by accounting for CT by considering ratios of gam-ma-ray emitting ssion products (Phillips et al., 1980; Willmanet al., 2006). A solution for both BU and CT is possible if the two iso-topes in the ratio have either different production rates from BU ordifferent half-lives or have both differences. Then the two equa-tions for the two isotopes are independent and can be solvedsimultaneously. This algorithm is the basis of the energy-depen-dent (spectroscopic) passive-gamma (PG) technique. In these spe-cic ways that they form ratios of characteristic gamma-rays todetermine relative quantities, the DG and PG techniques may beconsidered to be slightly independent from the BIC set (see Sec-

Energy 66 (2014) 3150 41tion 2.3). However, both techniques are correspondingly limitedby attenuation and high background, so they fall under the samerestrictions as the other NDA techniques discussed in Section 2.4.

thees anshi

T

ownownown

ownp

OWOWOWOW

rres

learTable 1The dependence of relevant physical properties and isotopes, in used fuel assemblies, onincreases (up) or decreases (down). Each table entry assumes that the other BIC variablgreatly or predominantly affect the quantity. Question marks indicate that the relatio

Quantity For LEU and HEU fuela

BU IE C

Neutronic propertiesNPRI UP DOWN DML DOWN UP Ds DOWN UP Dpl pf DOWN UP Dpc UP DOWN U

Gamma-ray propertiesCc, total UP DCc, Cs-137 UP DCc, Cs-133 UP Down DCc, Eu-154 UP Down D

R Co

42 A.M. Bolind / Annals of Nuc3.4. (Main Result) The proof of the independence of the BIC variableswith respect to the physical properties and isotopic content

Table 1 summarizes the results of this section and shows howthe BIC set (or BICC set) affects the relevant physical propertiesand isotopes. From this table, it is clear that the different propertiesvary in different ways with the BIC set: not just in magnitude buteven in direction. If the logic is inverted (Fig. 4), it is also clear thatthe three BIC variables must be independent functions of thephysical properties and of the isotopic content (see Fig. 6 for anexample). This deduction plus the earlier recognition of the tri-dimensionality of the vector spaces of the physical propertiesand isotopic content (Section 2) are the main results of this paper.They explain why the BIC set characterizes used fuel assemblies aswell as it does, as quantied by Burr et al. (2012) (Section 1).

The independence of the BIC variables with respect to the phys-ical properties of the used fuel assembly by itself also implies anindependence with respect to the physical properties of the assem-bly in a collection of fuel assemblies. The same physics that is rep-resented by Eq. (2) applies to the fuel assembly when it is in the

resonance

Specic isotopes244Cm & 240Pu UP DOWN Down235U DOWN UP 239Pu UP First down then up 241Pu UP First down then up Down155Eu UP Down DOW16 Fission Productsb UP Down Up

a Natural and depleted uranium fuels are excluded from this analysis. See Section 3.2b These are the set of the 16 most important ssion products for neutron capture. Seec Additional ssile isotopes will increase ML, s, and pf; additional capturing isotopes wd These relationships assume CP is increasing because 239Pu and 241Pu are being added to the MOX fuel.

Fig. 6. Illustration of how the BIC set can be a basis for an example vector ofphysical properties (NPRI,ML, Cc) (left schematic) and vice versa (right schematic).The non-orthogonal vectors in each schematic are pointing in the correspondingdirections of maximum increase with respect to the orthogonal vectors, inaccordance with Table 1.collection, albeit that the geometry and neutron energy spectrumchange. Therefore, the BIC variables are also independent with re-spect to the variables that are of concern to the burnup-credit com-munity, including the reactivity of the collection of used fuelBIC variables. As BU, IE, CT, or CP increases, the table indicates whether each quantityre being held constant. The bold-faced, capitalized entries represent the variables thatp is unclear.

For MOX fuel

BU IE CT CP

UP Up? Down UPDOWN UP Down ?c

DOWN UP Down ?c

Down?DOWN UP Down ?c

UP Up? Up Up?

N UP DOWN N UP DOWN N UP Down DOWN ?N UP Down DOWN ?

ponds to the individual isotopes.

UP Up? Down UPNot applicable with depleted UO2 in MOXDOWN UP Upd

? Up? Down Upd

N UP ? DOWN ?UP ? Up ?

Sections 2.5 and 3.2.ill decrease them.

Energy 66 (2014) 3150assemblies.

4. Results (2): the possible failure modes in the ability of the BICset to characterize used fuel assemblies

In Section 2.3, it was argued that the physical-properties andisotopic-content vector spaces are basically three-dimensional.The basis of this argument was the observation that only threeterms in the neutron diffusion equation (Eq. (2)) change signi-cantly with the BIC set. The geometry and neutron-energy spec-trum of the used fuel assembly were specically dismissed ashaving negligible inuence at the time of the NDA measurement.

In this section, it is recognized that the geometry and neutronenergy spectrum of the fuel assembly during burning in the reactorare signicant and that their effects may not be fully representedby the BU variable. Furthermore, irradiation history that is, therate at which the fuel assembly is burned is not taken intoaccount by BU or CT. Also, IE does not directly account for burnablepoisons. These and similar additional factors may cause the BIC setto fail to adequately characterize used fuel assemblies.

The meaning of such failure can be explained by using the CPvariable as an example. In Section 3.1, it was argued that neitherIE nor BU accounts for the presence of non-ssile transuranicisotopes in fresh MOX fuel, but such isotopes must be taken intoaccount because they signicantly increase the resulting quantityof 244Cm in the used fuel assembly. Therefore, the CP variablewas introduced as a fourth variable in the BICC set, therebyexpanding it to four dimensions. Like the other BICC variables,though, CP represents a past aspect of the fuel assembly. The vectorspaces of the physical properties and isotopic content are stillpresent characteristics of the fuel assembly and remain

three-dimensional. Therefore, there is a mismatch between thedimension of the BICC vector space and the dimensions of the othervector spaces, so that the correlations from the BICC set to theother vector spaces are surjective but not injective (Fig. 4).

be considered to be the problem of cooling (i.e., radioactive decay)while the fuel is being burned, whereas the CT parameter repre-

A.M. Bolind / Annals of Nuclearsents cooling after the fuel has been burned. Apart from this choiceof the starting time, the chief distinction between the two conceptsis that precursor and daughter isotopes can readily undergo neu-tronic reactions in the high neutron ux in the reactor while thefuel is being burned, but they cannot substantially do so after thefuel has been permanently discharged from the reactor.

If only the passive gamma-ray isotopes are considered, theproblem of irradiation history largely becomes its own solution(Rinard, 1983). The short-lived isotopes are the ones that are mostdependent on irradiation history, but they also are the ones thatdecay away the fastest after the fuel assembly has been dischargedfrom the reactor. After 13 years of CT, the total gamma-ray activ-ity is dominated by the longer-lived isotopes (134Cs, 137Cs, and154Eu) and follows Eqs. (4)(6) (Cobb et al., 1982; Phillips et al.,1980; Rinard, 1983; Willman et al., 2006). That is, the passive gam-ma-ray properties and isotopes are mostly governed by the BIC setfor CT greater than 1 year, and they are certainly governed by theBIC set after 3 years (Fig. 14 in Cobb et al. (1982)). This fact is trueeven though 134Cs and 154Eu are produced through neutron cap-ture, because the neutron-capture events in their nucleosynthesispathways are in isotopes that either are stable or have long half-lives in comparison with fuel-assembly irradiation periods(Figs. A-4A-7 in Phillips et al. (1980)).11

11 Some readers may object to considering a waiting period of three years to be asolution to the problem of irradiation history, since after this time period, suchindicators as 95Nb, 103Ru, 95Zr, 144Ce + 144Pr, and 106Ru + 106Rh are no longer usablebecause they have decayed away (Phillips et al., 1980). Nevertheless, the point beingWhereas it is possible to proceed logically from the BICC set todetermine the physical properties (and groups of isotopes), it isnot possible to reverse the logic to determine the BICC set fromthe physical properties. More importantly to the current argument,if CP is ignored for MOX fuel, then the correlations from the result-ing three-dimensional BIC set will fail even to be surjective.

The various other factors that inuence the fuel assembly dur-ing burning in the reactor affect the NDA logic in a similar fashion.If it can be shown that the BIC set or BICC set should actually haveeven more dimensions, then the correlations to the other vectorspaces will fail to be surjective; and in such a case, it can be saidthat the BIC set or BICC set fails to characterize used fuel assem-blies. The purpose of this section is to show, from the literature,that these other factors usually are inherently secondary or canbe made to be secondary in comparison to the BIC or BICC vari-ables. (The question of whether or not an NDA practitioner wantsto make such efforts to mitigate these other factors is outside thepurview of this paper.)

This section is subdivided into the failure modes associatedwith time dependence, those associated with neutron energydependence, those associated with geometric dependence, and -nally those associated with other factors. The section is summa-rized in Sectio