Developing diagnostic tools for low-burnup reactor samples

TL;DR

This paper evaluates and develops diagnostic tools for low-burnup reactor samples, focusing on uranium isotopic ratios and sample cooling time diagnostics, with an emphasis on their accuracy and limitations at very low fluences.

Contribution

It introduces simplified analytic formulas for uranium isotope ratios and new diagnostics for cooling times applicable to very low burnup samples.

Findings

Simple formulas agree with reactor simulations for certain fluence ranges.

$^{236}$U/$^{235}$U ratios become unreliable below $10^{19} \mathrm{n/cm^2}$.

Multiple fragment ratios are needed to identify systematic errors.

Abstract

We test common fluence diagnostics in the regime of very low burnup natural uranium reactor samples. The fluence diagnostics considered are the uranium isotopics ratios $^{235}$U/$^{238}$U and $^{236}$U/$^{235}$U, for which we find simple analytic formulas agree well with full reactor simulation predictions. Both ratios agree reasonably well with one another for fluences in the mid $10^{19}\,\mathrm{n/cm^2}$ range. However, below about $10^{19}\,\mathrm{n/cm^2}$ the concentrations of $^{236}$U are found to be sufficiently low that the measured $^{236}$U/$^{235}$U ratios become unreliable. We also derive and test diagnostics for determining sample cooling times in situations where very low burnup and very long cooling times render many standard diagnostics, such as the $^{241}$Am/$^{241}$Pu ratio, impractical. We find that using several fragment ratios are necessary to detect the presence of systematic errors, such as fractionation.

Tables

Table 1. Table 1: Theoretical errors for the fluence ( 235 U/ 238 U and 236 U/ 235 U ratios) and cooling time ( α 1 = 137 superscript 137 subscript 𝛼 1 absent \alpha_{1}=\,^{137} Cs/ 155 Eu, α 2 = 137 superscript 137 subscript 𝛼 2 absent \alpha_{2}=\,^{137} Cs/ 125 Sb, and α 3 = 155 superscript 155 subscript 𝛼 3 absent \alpha_{3}=\,^{155} Eu/ 125 Sb ratios) diagnostics given by Eq.  2 and Eq.  9 . Each cell shows the maximum expected error over a particular reactor parameter (the thermal fluence Φ t subscript Φ 𝑡 \Phi_{t} , fast flux ϕ f subscript italic-ϕ 𝑓 \phi_{f} , cooling time T c subscript 𝑇 𝑐 T_{c} , number of shutdowns N s subscript 𝑁 𝑠 N_{s} , and length of shutdowns T s subscript 𝑇 𝑠 T_{s} ) range. The overall theoretical error is the individual errors summed in quadrature, which provides a conservative maximum.

	$\Phi$ Diagnostics		$T_c$ Diagnostics
	$ϵ$	$\rho$	${\alpha }_1$	${\alpha }_2$	${\alpha }_3$
${\Phi }_t$	$\sim 0\%$	$\sim 0\%$	$3.86\%$	$0.57\%$	$-3.27\%$
${\varphi }_f$	$0.54\%$	$0.24\%$	$-0.47\%$	$-0.19\%$	$-0.14\%$
$T_c$	$0\%$	$0\%$	$-0.99\%$	$-0.12\%$	$0.89\%$
$N_s$	$0\%$	$0\%$	$0.10\%$	$0.01\%$	$-0.10\%$
$T_s$	$0\%$	$0\%$	$-0.17\%$	$-0.16\%$	$-0.14\%$
Overall	$\pm 0.54\%$	$\pm 0.24\%$	$\pm 4.02\%$	$\pm 0.63\%$	$\pm 3.40\%$

Equations

$$\begin{split}\epsilon(\Phi,\epsilon_{0})&=\epsilon_{0}e^{-\Phi(\sigma^{T}_{U235}-\sigma^{T}_{U238})}\\ \rho(\Phi)&=\bigg{(}\frac{\sigma^{c}_{U235}}{\sigma^{T}_{U236}-\sigma^{T}_{U235}}\bigg{)}\bigg{(}1-e^{-\Phi(\sigma^{T}_{U236}-\sigma^{T}_{U235})}\bigg{)}.\end{split}$$

$$\begin{split}\Phi&=\frac{\ln(\epsilon_{0}/\epsilon)}{\sigma_{U235}^{T}-\sigma_{U238}^{T}}\\ \Phi&=\frac{1}{\sigma_{U235}^{T}-\sigma_{U236}^{T}}\ln\bigg{(}\frac{\sigma_{U235}^{c}-\rho(\sigma_{U236}^{T}-\sigma_{U235}^{T})}{\sigma_{U235}^{c}}\bigg{)}.\end{split}$$

$$\frac{d{N}_L}{dt}=-{\tilde{\lambda }}_L{N}_L+{\vec{Z}}_L\cdot \vec{\mathcal{F}}.$$

$$N_{L}(t)=\bigg{(}{N_{L}}_{0}-\frac{\vec{Z}_{L}\cdot\vec{\mathcal{F}}}{\tilde{\lambda}_{L}}\bigg{)}e^{-\tilde{\lambda}_{L}t}+\frac{\vec{Z}_{L}\cdot\vec{\mathcal{F}}}{\tilde{\lambda}_{L}},$$

$$N_L(t,T_c)=\Phi ({\vec{Z}}_L\cdot {\vec{\Sigma }}_{\mathrm{fiss}})e^{-{\lambda }_L{T}_c}+\mathcal{O}(({\tilde{\lambda }}_L{T}_{\mathrm{irr}}{)}^2),$$

$$\begin{array}{rl}{\alpha }_L(t,\tau ,\vec{\beta },\vec{\gamma },{N_L}_0)=& ({\lambda }_L{N_L}_0-{\vec{Z}}_L\cdot \vec{\mathcal{F}})e^{-{\lambda }_L(t+\tau )}\\ & +({\vec{Z}}_L\cdot \vec{\mathcal{F}})e^{-{\lambda }_L\tau }\times f(\vec{\beta },\vec{\gamma }),\end{array}$$

$$\begin{split}&f(\vec{\beta},\vec{\gamma})=\bigg{(}\displaystyle\prod_{i=1}^{N-1}e^{\lambda_{L}\gamma_{i}\tau}\bigg{)}+e^{-\lambda_{L}t}\times\\ &\displaystyle\sum_{i=1}^{N-1}\bigg{[}\bigg{(}\displaystyle\prod_{j=1}^{i}e^{\lambda_{L}\beta_{j}t}\bigg{)}\bigg{(}\displaystyle\prod_{k=1}^{i-1}e^{\lambda_{L}\gamma_{k}\tau}\bigg{)}-\bigg{(}\displaystyle\prod_{j=1}^{i}e^{\lambda_{L}(\beta_{j}t+\gamma_{j}\tau)}\bigg{)}\bigg{]}.\end{split}$$

$${\alpha }_{n,d}({\vec{\Sigma }}_{\mathrm{fiss}},T_c)=\frac{{\lambda }_n{\vec{Z}}_n\cdot {\vec{\Sigma }}_{\mathrm{fiss}}}{{\lambda }_d{\vec{Z}}_d\cdot {\vec{\Sigma }}_{\mathrm{fiss}}}{e}^{-({\lambda }_n-{\lambda }_d)T_c},$$

$$T_{c}=\frac{1}{\lambda_{d}-\lambda_{n}}\ln\bigg{(}\frac{\alpha_{n,d}\lambda_{d}\vec{Z}_{d}\cdot\langle\vec{\Sigma}_{\mathrm{fiss}}\rangle_{\Phi}}{\lambda_{n}\vec{Z}_{n}\cdot\langle\vec{\Sigma}_{\mathrm{fiss}}\rangle_{\Phi}}\bigg{)}.$$

Full text

††preprint: LA-UR-16-26969

I Isotopics Introduction

Determining the reactor environment that a particular spent fuel sample experienced is critical information for non-proliferation and reactor verification. In particular, the fluence is often related to the fuel burnup and, hence, the plutonium production and grade Stepanov et al. (1980). This makes the fluence an important parameter for nonproliferation and arms reduction Wood et al. (2002). The fluence of a sample can be inferred in many ways, but is most commonly derived from isotopic ratios of actinides, such as 235U/238U or 236U/235U Boulyga and Becker (2002); Boulyga and Heumann (2006) and various plutonium ratios Kim et al. (2015). Additional methods utilize the ratios of activated isotopes in cladding and moderator material, such as the graphite isotope ratio method (GIRM) Reid et al. (1999); Gesh (2004); Gasner and Glaser (2011), or of ratios of long-lived fragments such as cesium Caruso et al. (2007); Ansari et al. (2007); Kim et al. (2015), europium Caruso et al. (2007), or neodymium Kim et al. (2015). The cooling time is often determined with ratios utilizing short-lived actinides, such as 241Pu/241Am Mayer et al. (2012), but can also be inferred by gamma spectroscopy of fragments Gauld and Francis (2010). The cooling time provides one with an estimate of the sample age, which is also pertinent for forensics and nonproliferation.

One can determine the final activities, abundances, and ratios of nuclides with detailed reactor simulations, provided a burnup history and initial fuel composition. Our goal is to invert this process, where one begins with measured isotopic abundances or ratios and then determines the reactor parameters, such as the fluence and cooling time. We focus on these two parameters as they are derived from so-called linear systems, which have simpler analytical forms, in the low burnup regime. Non-linear systems can be used to infer parameters, such as the flux and shutdown history Hayes and Jungman (2012). In our regime of interest, new cooling time diagnostics are developed and verified alongside the standard fluence diagnostics. Several cooling time diagnostics are utilized to detect the presence of systematic errors. We used low burnup archived samples, available at Los Alamos National Laboratory, to test these diagnostics. The chemical analyses to determine the abundances of the actinides and fission fragments for our low burnup samples can be found in Ref. Byerly et al. (2015) and Ref. Tandon et al. (2009).

This paper is structured as follows. The fluence diagnostics are discussed in Sec. II. Cooling time diagnostics are discussed and derived in Sec. III. The diagnostics are verified with reactor simulations and theoretical errors are generated in Sec. IV. The diagnostics are then applied to low burnup reactor samples to determine their fluence, cooling time, and sensitivity to systematic errors in Sec. V. We conclude in Sec. VI.

II Fluence Diagnostics

The fluence diagnostics considered in this work utilize the uranium isotopic ratios: 235U/238U and 236U/235U. Ratios utilizing moderator materials require a sample removal from the existing reactor, which is often not feasible or impacts future reactor design and safety. In addition, some long-lived fragments, such as 134Cs or 154Eu, are produced in extremely low concentrations for low burnup scenarios creating experimental difficulties. Finally, 239Pu cannot be used, as its accumulation is not precisely linear in fluence in low burnup scenarios, thus displaying a flux dependence. For these reasons, we focus on the uranium ratios above which are trivially related to the fluence via

[TABLE]

Here, $\epsilon$ denotes the 235U/238U ratio and $\rho$ the 236U/235U ratio. The superscripts on the cross-sections $\sigma$ are for capture ($c$) or total ($T$) reactions and $\Phi$ is the fluence  111In addition, one would group $\Phi$ and $\sigma$ by energy-groups, but we list the $1$-group values for simplicity..

We immediately note that $\epsilon$ depends on the initial ratio $\epsilon_{0}$. This implies that a measurement of $\Phi$ via the 235U/238U ratio is only valid when the initial enrichment is known. In the case of our low burnup samples, all indicated natural uranium (NU) as the initial fuel Byerly et al. (2015). On the other hand, the determination of $\Phi$ from $\rho$ is insensitive to the initial fuel, but requires a measurement of 236U, which is produced in very low quantities when the burnup is low. A final note is that a measurement of $\Phi$ using Eq. 1 will be most sensitive to the thermal fluence, as these cross-sections dominate (specifically 235U thermal fission).

Inverting Eq. 1 produces the fluence diagnostics we will apply to the low burnup samples

[TABLE]

Measurement of the values of $\epsilon$ or $\rho$ are typically accomplished by chemical separation Natsume et al. (1972); Abernathey et al. (1972); Marsh, S.F. and Ortiz, M.R. and Abernathey, R.M. and Rein, J.E. (1974), followed by gamma spectroscopy Kim et al. (2015), thermal ionization mass spectrometry (TIMS) Byerly et al. (2015), or inductively coupled plasma mass spectrometry (ICP-MS) Boulyga and Becker (2002); Boulyga and Heumann (2006). Specifically, the measurement of 236U is made difficult as isobaric interferences arise when small quantities of 236U exist amidst large 235U and 238U quantities. Additionally, the $\alpha$-decay peak of 235U can interfere in a 236U/238U measurement done via $\alpha$-spectrometry Sánchez et al. (1992); Iturbe (1992).

III Cooling Time Diagnostics

Cooling time diagnostics must be selected specifically for the context of low burnup samples. For example, the 241Pu/241Am ratio cannot be used as neither 241Pu nor 241Am are produced in appreciable amounts at low burnups. Similar issues preclude the use of other unstable actinides from NU fuel or 134Cs and 154Eu, both of which are non-linear nuclides Huber and Jaffke (2016) that rely on neutron capture as their primary production channel. This indicates that common cooling time diagnostics that utilize same-species ratios to avoid fractionation Freiling and Kay (1966), such as the 134Cs/137Cs ratio Navarro et al. (2011), are invalid. In addition, the special case of extremely long cooling times 222We define the cooling time as the sum of all pure decay periods, including shutdowns. ($\sim 20\,\mathrm{yr}$) invalidate the use of some major decay heat tags, such as 106Ru and 144Ce Bergelson et al. (2013). Thus, the cooling time diagnostic requires nuclides that are appreciably produced in low burnup scenarios, have long half-lives, and are easy to chemically separate and analyze. These requirements naturally lead one to the so-called ‘linear’ fission fragments, described by:

The linear fragment $N_{L}$ has a halflife such that its decay constant $\lambda_{L}$ satisfies $\lambda_{L}T_{\mathrm{irr}}\ll 1$. 2. 2.

The fission product cumulative yields for $N_{L}$ are large. 3. 3.

The beta-parents of $N_{L}$ have halflives such that they are in equilibrium during $T_{\mathrm{irr}}$.

These fragments are dubbed ‘linear’ as their production is linear in the fluence $\Phi=\phi T_{\mathrm{irr}}$. The first criteria ensures that the fragment is long- lived relative to the irradiation period of the reactor. The second criteria demands that the fragment is appreciably produced in fission. The third criteria allows one to derive a simple analytical expression for $N_{L}$, independent of its $\beta$- parents. For our low burnup purposes, 85Kr, 125Sb, 137Cs, and 155Eu are linear fragments. Next, we proceed to derive the simple expressions for these and verify that they satisfy the criteria above with detailed reactor simulations.

All nuclides in a reactor environment are governed by depletion equations, which form the basis for constructing an interaction matrix between the various nuclides. This is the structure utilized by many reactor simulation codes Oak Ridge National Laboratory (2011); Meplan (2009), which often solve these massive ($\sim 2000$ species) systems as an eigenvalue problem Pusa and Leppänen (2010). In our case, we can utilize linear fragments to construct a simple isolated system, which resembles a Bateman equation Bateman (1910),

[TABLE]

The positive (negative) terms denote production (depletion) channels and we use an effective decay constant $\tilde{\lambda}=\lambda+\phi\sigma^{T}$. We note that the full depletion equation, which resembles Eq. 1 of Ref. Isotalo and Aarnio (2011), reduces to Eq. 3 after applying $\lambda_{i}\gg b_{j,i}\lambda_{j}$ (criteria $3$), noting that $\sigma_{j,i}\phi\ll b_{j,i}\lambda_{j}$ is satisfied for most fragments 333An exception to this is the 135Xe$(n,\gamma)^{136}$Xe cross-section., and adding an explicit fission term. Thus, Eq. 3 states that a linear fragment $N_{L}$ is produced in fission with a fission rate vector $\vec{\mathcal{F}}$ and a cumulative yield $\vec{Z}_{L}$, both of which span the major fissiles, and is depleted through its decay and neutron-capture.

Solving Eq. 3 yields

[TABLE]

with an initial nuclide abundance ${N_{L}}_{0}$. We note that most linear fragments satisfy $\tilde{\lambda}_{L}\approx\lambda_{L}$, which can be verified by solving for the critical flux when decay and neutron channels have equal rates. Standard reactor fluxes are far below the critical fluxes of most fragments, ensuring that decay channels dominate. An exception to this is 155Eu and, in high thermal flux reactors Ilas (2012), 85Kr. We include the effective decay constant in our derivations for generality. One can easily verify that our selected fragments are linear in nature using reactor simulations. We use a finite-difference methods solver for the interaction matrix, where the included nuclear data can be varied. A sample irradiation history is given by $4$ cycles of $[5,19]\mathrm{hr}$ ON/OFF periods and a thermal flux of $\phi_{t}=8.5\times 10^{13}\,\mathrm{n/cm^{2}/sec}$. The resulting relative abundances for our linear fragments and, for comparison, two non-linear fragments (152,154Eu) are shown in Fig.1.

The minimum layer of nuclear data considered just our fragment of interest (FOI). This physically represents the case when each FOI is given by Eq. 4. Layer $1$ added the $\beta$-parents. Layer $2$ added the $(n,\gamma)$ parent. Layer $4$ included the primary, secondary, and (in some cases) tertiary $(n,\gamma)$ channels as well as all of their $\beta$-parents with halflives greater than $30\,\mathrm{sec}$. We also included a simulation of all nuclides with available data ($\sim 700$). From Fig. 1, one can verify that 85Kr, 125Sb, 137Cs, and 155Eu are linear as they have very little dependence on the layer of nuclear data and, thus, are accurately given by Eq. 4. None of the fragments studied varied significantly between the major fission yields libraries Chadwick et al. (2011); Kellett et al. (2009); Shibata et al. (2011).

To derive the cooling time diagnostic, we first expand Eq. 4 with $\tilde{\lambda}T_{\mathrm{irr}}\ll 1$ (criteria $1$) and arrive at

[TABLE]

once we have set ${N_{L}}_{0}=0$, accounted for the decay after a cooling time $T_{c}$, and separated the fission rate vector into a weighted fission cross-section and the flux through the relation $\vec{\mathcal{F}}=\vec{\Sigma}_{\mathrm{fiss}}\phi$. The expansion to arrive at Eq. 5 is easily valid for all fragments used here, except 155Eu which deviates from it by $\sim 1-3\%$ due to its large cross-section. As $\vec{\Sigma}_{\mathrm{fiss}}$ is the fission cross-sections weighted by the fissile abundances, one can determine $\vec{\Sigma}_{\mathrm{fiss}}$ with similar chemical analyses as those used for the fragments Byerly et al. (2015).

Universally setting ${N_{L}}_{0}=0$ appears to exclude cases with multiple irradiation cycles. Suppose we have a distribution of irradiation and cooling times described in Fig. 2, where $t$ and $\tau$ are the total irradiation and cooling times across all cycles.

We recursively insert Eq. 4 into itself as an initial condition for the following irradiation and cooling period to verify that distributing the total irradiation and cooling time in a generalized way is a negligible effect on our linear fragments. We find that the final activity ($\alpha_{L}=\lambda_{L}N_{L}$) of a purely linear fragment (i.e. $\tilde{\lambda}_{L}=\lambda_{L}$) with a generic distribution of $t$ and $\tau$ over $N$ cycles is given by

[TABLE]

with a pre-irradiation initial abundance ${N_{L}}_{0}$ and the function $f(\vec{\beta},\vec{\gamma})$ is given as a sum and product of exponentials over the additional $N-1$ cycles

[TABLE]

This complex function for $N$ cycles reduces to unity when $N=1$. One can show that criteria $1$, and the fact that the individual elements of $\vec{\beta}$ and $\vec{\gamma}$ are limited by unitarity, restricts Eq. 7 to very small deviations from $1$. We analyzed generic values for $\vec{\beta}$ and $\vec{\gamma}$ within our expected $t$ and $\tau$ ranges and found that Eq. 7 is well-constrained to $\lesssim 1\%$ deviations from unity. An exception to this is 125Sb, which showed larger deviations when the decay time is concentrated towards earlier cycles (i.e. when $\gamma_{1}\gg\gamma_{k>1}$), but this is disfavored for our samples. As no fragments are expected in pre-irradiated fuel, we determine that ${N_{L}}_{0}=0$ is a valid assumption at the start of irradiation and any subsequent cooling time diagnostic will now include intermediate shutdowns.

With ${N_{L}}_{0}=0$, the final abundance of a linear fragment can be expressed as in Eq. 5. A ratio of the activities of two linear fragments removes the explicit dependence on $\Phi$ and creates the expression

[TABLE]

which is a direct measure of the total cooling time. One can correct Eq. 8 with higher order expansion terms to account for linear fragments with large neutron-capture components, but this will create a dependence on $T_{\mathrm{irr}}$. For large fast fluences, $\Phi$ must remain in Eq. 8 so as to account for fast fissions: $\Phi_{g}\times(\vec{Z}^{g}_{n}\cdot\vec{\Sigma}^{g}_{\mathrm{fiss}})$, with an implied sum over the neutron energy groups $g$.

As mentioned previously, the final value of $\vec{\Sigma}_{\mathrm{fiss}}$ is known from a measurement of fissile isotopics. However, $\vec{\Sigma}_{\mathrm{fiss}}$ varies over the irradiation period. Therefore, one must average the weighted fission cross-sections so as not to bias Eq. 8 towards U or Pu fissions. The averaging is conducted linearly over the fluence $\Phi$ because $T_{\mathrm{irr}}$ is unknown. One can use the thermal fluence derived from Eq. 2 as the fluence endpoint and the initial value of $\vec{\Sigma}_{\mathrm{fiss}}$ reflected natural uranium for our samples Byerly et al. (2015). This fluence-averaged value $\langle\vec{\Sigma}_{\mathrm{fiss}}\rangle_{\Phi}$ becomes a critical factor when predicting fragments that have cumulative yields with large plutonium components.

Inverting Eq. 8 reveals the cooling time diagnostic

[TABLE]

Due to the pole in Eq. 9, two linear fragments with similar decay constants $\lambda_{n}\simeq\lambda_{d}$, such as a ratio of 90Sr and 137Cs, can produce large errors in the cooling time, but there are theoretical methods for removing these Cetnar (2006). For fragments with large cross-sections, one can expand Eq. 4 to $\mathcal{O}((\tilde{\lambda}T_{\mathrm{irr}})^{2})$, but this introduces an unverifiable value for $T_{\mathrm{irr}}$ and only corrects the cooling time by a few percent.

IV Verification

In Sec. II and Sec. III, we listed diagnostics for the thermal fluence and cooling time. These diagnostics were verified with the use of the reactor simulation described in Sec. III. Over $70$ sample cases were evaluated with layer $4$ nuclear data to determine the validity of the analytical calculations. The cases spanned a range of reasonable values for the thermal flux $\phi_{t}$, cooling time $T_{c}$, fast flux $\phi_{f}$, irradiation time $T_{\mathrm{irr}}$, number of shutdowns $N_{s}$, and shutdown length $T_{s}$. The derived values for $\Phi$ and $T_{c}$, using Eq. 2 and Eq. 9, were compared with those used as input to the simulation. We found that the only parameter that affected the fluence diagnostic is the introduction of a fast flux $\phi_{f}$ as it slightly increases the final $\rho$ and $\epsilon$ values, which could be mistaken for a larger thermal fluence. Using the maximum expected fast flux, the diagnostic of Eq. 2 returned the input fluence within $\sim 0.5\%$ for both the 235U/238U and 236U/235U ratios. The situation for the cooling time diagnostic was much more complicated.

We used the following ratios for the cooling time diagnostic: 137Cs/155Eu ($\alpha_{1}$), 137Cs/125Sb ($\alpha_{2}$), and 155Eu/125Sb ($\alpha_{3}$) 444Diagnostics using 85Kr were removed as it may have experienced volatile leakage.. The derived cooling time was found to vary with all major reactor parameters listed above. As the total $\Phi_{t}$ increased, the errors on Eq. 9 increased linearly, but this was shown to be mediated somewhat by the averaging of $\vec{\Sigma}_{\mathrm{fiss}}$. The increase of $\phi_{f}$ created an underestimation of $T_{c}$ proportional to the additional fast cumulative yields of the fragments used in Eq. 9. Increasing the cooling time served to decrease the errors on all $T_{c}$ diagnostics as the deviation from end-of-cycle activity ratios becomes more severe for longer $T_{c}$. Finally, the shutdown history is shown to have a very small impact, in agreement with the derivation in Sec. III. The maximum theoretical errors in percent for the expected reactor parameters and the largest overall theoretical error are provided in Tab. 1.

Overall, from Tab. 1, one can see that the diagnostics derived in Eq. 2 for the fluence have extremely small theoretical errors and one can expect the correct fluence within $\sim 0.5\%$. For the cooling time diagnostic, the theoretical errors are more substantial as the fragment systems are more complex. Overall, one can expect the correct cooling time within $\sim 4\%$, $\sim 0.6\%$, and $\sim 3.4\%$ for the 137Cs/155Eu, 137Cs/125Sb, and 155Eu/125Sb ratios, respectively. The linear-averaging in Sec. III returned the lowest errors, but ignores the quadratic behavior of 239Pu at low burnup. We verified that removing 239Pu fissions and calculating Eq. 9 to $\mathcal{O}((\tilde{\lambda}T_{\mathrm{irr}})^{2})$ can effectively eliminate these errors. We note that these errors are strictly theoretical and contain no systematic errors, such as fractionation or experimental uncertainties. We have also calculated the expected 239Pu abundance using a similar analytical method with errors of $\sim 0.25\%$, but this requires knowledge of many reactor parameters, so we have excluded it from our analysis. The theory errors of Tab. 1 are lower than the experimental measurement errors. With these notes in mind, we use these diagnostics to determine the thermal fluence and extract information about systematic errors from three cooling time diagnostics.

V Experimental Application

Ten archived samples were analyzed for their U and Pu isotopics, as well as the activities of several fission fragments. The actinides were separated and measured as described in Ref. Byerly et al. (2015). In short, U metal or UO3 samples are dissolved in HNO3, then loaded and separated on anion exchange columns to achieve separation of Pu from U. Isotope ratios and isotope dilution measurements were determined by TIMS. Fission fragments were measured by gamma spectrometry Tandon et al. (2009). Samples H and K were in UO3 form, while the remainder were uranium metal.

Both fluence diagnostic methods were attempted, but discrepancies were observed between the 236/235U and 235/238U ratios in very-low burnup cases as shown in Fig. 3.

The fluences determined in samples D through K were all nearly self-consistent. Sample C reported fluences that deviate more strongly. Samples A and B were contaminated with 236U memory effects, so their values were not used. The chemical analyses of the remaining samples were conducted at a later time, correcting the 236U issue. Overall, it appears that our method of extracting the thermal fluence via Eq. 2 is accurate and self-consistent for the majority of samples with $\Phi\geq 10^{19}\,\mathrm{n/cm^{2}}$. Below this limit, the low concentrations of 236U created experimental difficulties in acquiring the fluence with multiple methods. Thus, one can determine the thermal fluence with two independent diagnostics in samples with appreciable amounts of 236U, but must rely solely on the 235/238U ratio in extremely low-burnup samples with trace levels of 236U. The $\epsilon$ diagnostic is only valid when $\epsilon_{0}$ is known, so the $\rho$ diagnostic should be used if enough 236U is present. The average error between the two diagnostics was $19.9\%$.

In determining the total cooling time, we used the ratios identified in Sec. IV. Figure 4 illustrates the agreement and tension between the different diagnostics. A few samples performed relatively well, but most demonstrated disagreement between the three cooling time diagnostics.

In particular, the 155Eu-based determinations of $T_{c}$ showed disagreement with the 137Cs/125Sb ratio as the inferred fluence rises. Leakage of volatile fission fragments, such as 85Kr, can occur at the $\gtrsim 13\%$ level in PWR fuels Metz. V et. al. (2013) so these ratios were removed. A portion of the bias from 155Eu-based diagnostics can be explained by the over-estimation of the 239Pu-component when linearly averaging $\vec{\Sigma}_{\mathrm{fiss}}$ and the need to compute $T_{c}$ to second order, but these errors will only approach those in Tab. 1. The 137Cs/125Sb diagnostic seemed to match the average reported sample age of $20\,\mathrm{yr}$. The 125Sb abundance was not measured in sample A. The average diagnostic discrepancy was found to be $\sim 37\%$ between the 155Eu-based diagnostics and the 137Cs/125Sb ratio. The use of multiple $T_{c}$ diagnostics allows one to detect the presence of systematic errors, such as fractionation, when diagnostics do not agree and a single consistent cooling time when they do. This technique must be used in the very low burnup regime as traditional same-species ratios are impractical.

VI Conclusion

The work conducted here demonstrates that the thermal fluence can be determined in low burnup samples using the 235U/238U and 236U/235U ratios. These ratios are common fluence diagnostics, which were verified with detailed reactor simulations and then experimentally demonstrated to be accurate and self- consistent when enough 236U is produced above the detection threshold. The average discrepancy between the two fluence diagnostics in our low burnup samples was $19.9\%$ for $\Phi>10^{19}\,\mathrm{n/cm^{2}/sec}$.

The low burnup of our reactor samples required new cooling time diagnostics to be derived, as the concentrations of standard diagnostic tags are below detection thresholds. The new cooling time diagnostics utilized simple linear fission fragments with long half-lives and considerable fission yields. Four such fragments were identified and the derived diagnostics were verified in low burnup scenarios. The experimentally determined cooling times were shown to be consistent in some samples, but varied by $\sim 37\%$ on average. In addition, leakage of volatile gases invalidated the diagnostics using 85Kr. Overall, the 137Cs/125Sb ratio seemed to agree with the average sample age across all samples. Differing results for the cooling time, as measured by several diagnostics, proved to be indicative of systematic errors, such as fractionation, even in the very low burnup regime.

The fluence and cooling time derivation should be conducted in tandem, where the $\Phi$ determination would be used to derive $\langle\vec{\Sigma}_{\mathrm{fiss}}\rangle_{\Phi}$ and verify that the sample has a burnup low enough to validate the simple analytical expressions for $T_{c}$. This work provides verification of fluence diagnostics and new cooling time diagnostic techniques to determine the presence of systematic errors in low burnup samples, both of which have applications in non-proliferation and verification.

Acknowledgements.

We would like to thank the analytical chemistry team: L. Colletti, E. Lujan, K. Garduno, T. Hahn, L. Walker, A. Lesiak, P. Martinez, F. Stanley, R. Keller, M. Thomas, K. Spencer, L. Townsend, D. Klundt, D. Decker, and D. Martinez. Los Alamos National Laboratory supported this work through LDRD funding. This publication is LA-UR-16-26969.