© A. Aimetta et al., Published by EDP Sciences, 2025

1. Introduction

The TAPIRO (TAratura PIla Rapida Potenza ZerO) reactor (Fig. 1) is a fast neutrons source, in operation since April 1971 at the Casaccia research centre, and is currently gaining interest for its potential role as support facility to experimental activities of interest for Lead Fast Reactors (LFRs) technologies and Accelerator Driven Systems [1, 2]. In particular, the use of TAPIRO for the testing of materials of interest in LFRs could provide useful information for the nuclear interactions of high-energy neutrons. Potentially, these experiments could provide useful outcomes also for fusion systems, like the so-called Water Cooled Lithium Lead (WCLL) breeding blanket [3, 4] of the EU-DEMO reactor, also in consideration of previous activities carried out in TAPIRO relevant for fusion applications [5]. Similarly to LFRs, the WCLL breeding blanket will employ lead and will be characterized by high neutron energies. In fact, the potentialities of TAPIRO for next generation nuclear systems mainly lie in one of its most interesting features, i.e. its extremely compact dimension (the core volume is in the order of 800 cm³), as its composition of 93.5% enriched uranium metal, surrounded by a thick copper reflector. These characteristics lead to an almost pure fission spectrum (see Fig. 2), accessible through part of the experimental channels, which makes TAPIRO one of the most unique experimental reactors currently available worldwide, and one of the few nuclear facilities, together with VENUS-F in Belgium [6] for example, capable of supporting the fast reactor neutronic research in Europe [7]. These characteristics identify TAPIRO as an extremely valuable facility for research and development (R&D) and education and training (E&T) activities related to fast reactor noise experiments.

Fig. 1. Serpent model of TAPIRO: vertical section (left) and horizontal section (right). Zoom on the core and reflector regions. The dimensions of the depicted regions are 100 × 100 cm. The figure on the bottom show some specific regions of the core (on the left) and of the reflector (on the right) where neutron spectra have been evaluated with Serpent (Fig. 2). Each mf label is associated with a zone with a different fuel composition.

Fig. 2. Normalised neutron spectrum in the core (left) and in the reflector (right) of TAPIRO, computed with Serpent. The different colors correspond to the colors of the locations in the core and in the reflector where they have been calculated, shown in the two miniatures of the reactor.

Neutron noise analyzes the stochastic neutron fluctuations obtained from real experiments or, as in this work, computational simulations based on monte Carlo, during steady-state operation, to yield information about the kinetic parameters and reactivity of the reactor. Currently, the knowledge and experience about neutron noise experiments in fast reactors is extremely scarce, if compared to light water reactors [6, 8]. Traditional neutron noise techniques, based on point kinetics and one-speed models, have been extensively and systematically used in the past for light water reactors, but fast reactors pose some challenges, due to their inherent features:

• the shorter time scales (in TAPIRO, the generation time Λ is in the range of tens of ns, also due its composition and limited dimension);

• the ionization and the common ²³⁵U fission chambers typically adopted for experimental measurements are best suited for thermal neutrons;

• the one-group point kinetic approach adopted in noise method might need to be revisited to better grasps fast neutron physics.

In fact, it is expected that the methods of neutron noise analysis that have been developed in the past to determine the subcriticality level of thermal systems (based on point kinetics) cannot be directly applied to fast systems if accurate and reliable results are to be obtained. This is mainly due to the fact the one-group approximation could not suitable in the presence of more complicated spectra like the ones characteristic of fast neutron facilities and to the phenomenon of spatial flux redistribution, which is particularly important in highly heterogeneous reactors (like LFRs), questioning the validity of point kinetics in such systems. This supports even more the necessity to test and tailor methods for integral parameters measurements in TAPIRO, also considering that the target accuracy on these parameters is to be high, as deviations may be more impacting on the quicker dynamics of fast reactors.

The most important parameters in neutron noise theory are the reactivity ρ, the effective delayed neutron fraction β_eff and its group yields β_i and the effective prompt neutron generation time Λ_eff. In the framework of a neutron noise analysis, it is always necessary to take into account all the possible uncertainties contributing to the final result. These are mainly the statistical uncertainty associated with the fitting residuals, the uncertainty due to the arbitrary choice of the time or frequency fitting domain, the nuclear data uncertainty and the uncertainty in the calculated kinetic parameters. This work focuses on the impact of nuclear data uncertainties on the Monte Carlo calculated kinetic parameters of TAPIRO.

1.1. The TAPIRO reactor

The TAPIRO reactor is a fast neutron reactor designed to operate at low power (maximum power of ∼5 kWth, with a corresponding neutron flux in the core centre ∼4 ⋅ 10¹² cm⁻² s⁻¹) using around 25 kg of U-1.5Mo alloy as fuel, with 93.5% ²³⁵U enrichment. The core is cylindrical and is composed of seven disks with a diameter of 126 mm and a total height of 110 mm. The five lower disks can be moved vertically for emergency shutdown or to allow the installation of irradiation samples into the fuel cavity. The core is cooled by helium gas flowing on the lateral surfaces of the fuel cladding. A stainless steel pressure dome separates the inner and outer copper reflectors surrounding the core of TAPIRO. The outer reflector has an outer diameter of 820 mm and a height of 700 mm. The core and the reflector are contained into a high density borated concrete shield with a thickness of 1.75 m, which guarantees a dose rate lower than 20 μSv/h in the reactor hall when operating at 5 kWth. The control rod system (two safety rods for fast shutdown, two control rods for coarse control and reactivity removal, and one regulation rod for fine adjustments and to maintain desired power and temperature, all made of copper) can be extracted and inserted vertically moving them in the inner copper reflector. Then, experimental channels penetrate the reactor and surround the core for access to the fast neutron flux. In particular, the lower disk of the fixed part of the core has a hole connected with the diametrical channel, allowing the irradiation of samples in the neutron fission spectrum reached in the centre of the core.

TAPIRO can be used for many applications of interest for Gen-IV fast reactor designs: code validation, fast neutrons damage, nuclear data integral measurement, testing of detectors [2], and, in principle, it could provide valuable information about fast neutron noise experiments too. A more detailed description of TAPIRO can be found in [9].

2. Neutronic analysis of TAPIRO

The nuclear data uncertainty propagation analysis has been performed employing a high-fidelity Serpent (version 2.2.1) [10] Monte Carlo model of TAPIRO, provided by Argonne National Lab (ANL) and ENEA. The geometry model of TAPIRO used for the Monte Carlo neutronics assessment of the facility is shown in Figure 1. The results obtained with Serpent (see Tab. 1) have been compared with the ones obtained by ANL and ENEA using the MCNP code (version 6.2). All the simulations have been run using the ENDF-B/VIII.0 nuclear data library evaluated at 296.3 K and the results have shown a good agreement between the two codes, both in terms of k_eff, average number of neutrons produced per fission $\overline{\nu }$, β_eff, Λ_eff and prompt decay constant α_p. The formal definitions of the three kinetic parameters are the following [11]:

$$\begin{array}{cc}\hfill {\beta }_{\mathrm{eff}}& =\frac{⟨{\varphi }^{†},{\mathbf{F}}^{\mathrm{d}}\varphi ⟩}{⟨{\varphi }^{†},\mathbf{F}\varphi ⟩},\hfill \end{array}$$(1)

$$\begin{array}{cc}\hfill {\mathrm{\Lambda }}_{\mathrm{eff}}& =\frac{⟨{\varphi }^{†},\frac{1}{v}\varphi ⟩}{⟨{\varphi }^{†},\mathbf{F}\varphi ⟩},\hfill \end{array}$$(2)

$$\begin{array}{cc}\hfill {\alpha }_{\mathrm{p}}& =\frac{\rho -{\beta }_{\mathrm{eff}}}{{\mathrm{\Lambda }}_{\mathrm{eff}}},\hfill \end{array}$$(3)

where ϕ and ϕ^† are the direct and adjoint fluxes, F is the fission operator, and F^d the delayed neutron emission operator and v the neutron velocity.

The Serpent simulations for the neutronic assessment of TAPIRO have been run with 10⁶ neutrons per generation, 2050 active cycles and 50 inactive cycles (tests with a larger number of inactive cycles have shown that the fission source has already converged after 50 inactive cycles). Equations (1) and (2) show that the kinetic parameters require the knowledge of the adjoint flux to be computed. However, in Monte Carlo codes, the adjoint flux can not be estimated in a straightforward and direct way like the forward flux. Therefore, Serpent can calculate the kinetic parameters with several methods specifically developed for the evaluation of a surrogate of the adjoint flux, like the Meulekamp [12] and the Nauchi [13] methods, based on the concept of next fission probability (NFP). However, in this work, the technique based on the Iterated Fission Probability (IFP) [14], which considers fission chains traced several generations in the future for the evaluation of adjoint-weighted quantities, has been chosen. Formally, the IFP corresponds to the the asymptotic population size, that is the expected number of neutrons in the system after a sufficient amount of time has passed since the initiating event (i.e., the insertion of one neutron in a phase space point of a critical system). In [14], it has been proven that the asymptotic population size is proportional to the value of the adjoint flux in the same phase space point where the source neutron has been introduced. Thus, estimating the asymptotic population size with Monte Carlo represents the basis for the evaluation of kinetic quantities like β_eff and Λ_eff in Serpent. The procedure used in Serpent 2 to implement the IFP is not the only possible one and it is different, for example, to the one implemented in MCNP: the approach in Serpent has been developed in order to simplify the implementation of the method, at the cost of a slight increase in memory requirement [15].

It is worth mentioning that the prompt neutron generation time of TAPIRO, which can be interpreted as the importance-weighted average time taken for a single prompt neutron to reproduce itself by fission, is in the order of nanoseconds, much smaller than the one for light water reactors and more similar to results obtained in neutronic calculations of fast reactors employing liquid metals (e.g., ALFRED and ASTRID), as shown in Table 2. This small value in TAPIRO can be explained starting from the mathematical definition of Λ_eff in equation (2). The small size and high enrichment of the TAPIRO core increases the probability of fission per neutron collision (i.e., the denominator of Eq. (2) increases) and induces a higher neutron leakage from the core (i.e., the numerator of Eq. (2) decreases). The presence of the copper reflector, which is less effective than other materials at reflecting fast neutrons, further intensifies this effect, leading to an extremely small value of Λ_eff.

Finally, observing the neutron flux sampled over the XMAS-172 energy gird in different regions of TAPIRO (Fig. 2), it can be noticed that almost pure fission spectrum obtained in all the seven disks of the TAPIRO core. For what concerns the neutron flux in the copper reflector, it is still a fast neutron flux, making it possible to exploit the removable reflector sector for studies of interest for Gen-IV reactors, simply replacing one of the six sectors of the outer copper reflector (in violet in Fig. 2) with the aimed mockups.

Once the integral parameters have been calculated with Serpent, it is possible to perform the nuclear data uncertainty propagation on such kinetic parameters. Generalised Perturbation Theory (GPT) calculations have been run with Serpent using the energy grids ECCO-33 [21] and XMAS-172 [21], to assess the impact of using different energy grids on the nuclear data uncertainties. Moreover, also the impact of using different nuclear data libraries have been accounted, performing computations with both ENDF/B-VIII.0 and the JEFF-3.3 libraries. Finally, to check the consistency of GPT results, also the Unscented Transform (UT) technique has been applied. In the following sections additional details on the chosen methodologies are provided.

3. Uncertainty Propagation techniques

Several possibilities to perform the propagation of nuclear data uncertainties exist. Some of them, like the GPT, described in the following Section 3.1, have been specifically developed for nuclear systems and require the modification of the employed neutronic code to be implemented; therefore, these are known as intrusive techniques. A possible alternative is to employ non-intrusive techniques, like the so-called Unscented Transform (see Sect. 3.2), which in principle can be applied to any field of engineering and that, in this particular case, are based on the generation of a large number of perturbed nuclear data libraries and on the use of each of these libraries in a different neutronic simulation. Then, information about the nuclear data uncertainty can be retrieved with the post-processing of the neutronic results.

3.1. Generalised Perturbation Theory

The Generalised Perturbation Theory (GPT) is a perturbation technique mainly used for sensitivity analyses in fission reactors [22, 23]. Thanks to the estimation of the solution of both the direct and adjoint transport equation, the GPT allows, in a single simulation, evaluating the sensitivity S_X^R of a set of responses R (e.g. k_eff, β_eff, …) to a set of arbitrary input parameters (cross sections, $\overline{\nu }$, χ, …). A sensitivity coefficient is therefore defined as the relative variation of a R due to a relative variation of an input parameter X:

$$\begin{array}{c}\hfill S_X^R=\frac{\partial R/R}{\partial X/X}\end{array}$$(4)

Traditionally, the estimation of the sensitivity of k_eff lies within the scope of Standard Perturbation Theory (SPT), however in this work we will always refer to GPT since it can be seen as an extended version of the SPT, allowing to calculate the sensitivity of a generic response of interest, even k_eff.

Besides its capability of evaluating kinetic paramaters of fission systems, the Serpent code has been selected in this work also because the GPT is implemented in the code: GPT has been implemented in Serpent based on a collision history approach, fully described in [24]. In the context of sensitivity calculations with Serpent, the choice of a proper energy grid is particularly important. In fact, even being Serpent a continuous energy MC code, a discrete energy grid for the scoring of the sensitivities needs to be selected, in order to avoid too large statistical uncertainties when the energy group structure is too thin. In this case it is necessary not only to choose a not too refined energy grid, but also to define it with appropriate energy boundaries positions, according to the nuclear system under study.

Sensitivity profiles obtained thanks to the GPT are extremely valuable per se to gain insights about the reactor physics under analysis. Additionally, they can also be used to propagate the uncertainties of a generic nuclear data X to a response of interest R according to the so called first-order sandwich rule,

$$\begin{array}{c}\hfill {\sigma }_X^R=\sqrt{{\overrightarrow{\mathrm{S}}}_X^{{}^R}\mathrm{rcov}[X]{\overrightarrow{\mathrm{S}}}_X^{{}^{R^T}}},\end{array}$$(5)

where σ_X^R is the response uncertainty due to the uncertainty in the input parameter X, ${\overrightarrow{\mathrm{S}}}_X^{{}^R}$ is the relative sensitivity of response R to the input X, and rcov[X] is the relative covariance matrix of X. The relative covariance matrix must be obtained employing the same energy grid used to perform the GPT calculation (and upon which the sensitivity profiles are obtained). This procedure can be performed exploiting the ERRORR module of the nuclear data processing code NJOY [25, 26]. Thanks to the sandwich rule, it is possible to rank the nuclides and the reaction channels according to their contribution to the total uncertainty too.

Since the evaluation of sensitivity profiles depends on the choice of the energy grid, one can expect that it can impact also the value of the response uncertainty. However, previous works such as [27] show that the final result should not be so sensitive to the choice of the energy grid, in particular if only a rough estimate of the uncertainty is required. In the present work, calculations with the energy grids ECCO-33 and XMAS-172 [21] have been performed to assess their influence on the nuclear data uncertainties. However, it is important to remind that using finer grids, which should be the best choice in principle, dramatically increases the RAM memory required by the GPT simulation and represents also an issue in terms of statistical convergence.

3.2. Unscented Transform

The GPT is an intrusive technique, since its implementation in Serpent required the modification of the code. There are alternative uncertainty techniques which are non-intrusive, in the sense that they do not require any modification of the code, but simply to perform many model evaluations, each time with a different set of input parameters (cross sections, $\overline{\nu }$, χ, …). In the field of nuclear data, the reference non-intrusive method is the so called Total Monte Carlo (TMC) [28], which requires the generation of perturbed inputs. This step is carried out in this work thanks to the open-source Python package SANDY [29]. SANDY has been developed to perturb cross section data from ENDF-6 files to the ACE (A Compact ENDF) format required by Serpent, exploiting the information on the best estimate data and the covariance of the nuclear data. The complete probability distribution of a generic response due to the uncertainties of a generic input parameter (e.g. the nuclear data uncertainty of a specific nuclide) can be then obtained thanks to the TMC, providing a much larger amount of information with respect to the GPT, which only gives the standard deviation of the distribution. However, the main drawback of the TMC is its slow convergence ($\propto 1/\sqrt{\mathrm{N}}$ where N is the number of simulations), requiring to perform a non-negligible amount of simulations to obtain reliable and converged results. Considering the decorrelation between the statistical and epistemic uncertainties, the resulting uncertainty of the response R is a sum of two terms:

$$\begin{array}{c}\hfill {\sigma }_R=\sqrt{{\sigma }_{\mathrm{al}}^2+{\sigma }_{\mathrm{ep}}^2},\end{array}$$(6)

where σ_al² is the aleatory variance, associated with the statistical noise of each Monte Carlo simulation, while σ_ep² is the epistemic uncertainty due to the nuclear data. According to equation (6), the epistemic uncertainty can be calculated starting from σ_R, which is the results provided by TMC, and subtracting the statistical uncertainty. Thus, if one is interested in knowing the impact of nuclear data uncertainties, a requirement of TMC is that the aleatory variance is much smaller than the epistemic one, which translates into simulating a number of neutron histories sufficient to guarantee that the statistical uncertainty is negligible. This requirement further increases the amount of computational time required by TMC.

Faster alternatives to TMC have been developed, like the fast Total Monte Carlo (fTMC) and the fast GRS methods [30] and the Embedded Monte Carlo (EMC) [31], and have been applied in previous works [32, 33]. Another technique that allows to perform the uncertainty propagation in a smaller amount of time with respect to the TMC, at the cost of limiting the amount of information that can be obtained to the mean value and standard deviation, is the Unscented Transform (UT) [34, 35].

UT is based on the fact that it is easier to approximate an input distribution than an output distribution, and this yields better results especially when the model is non-linear. The input distribution is approximated in a smarter way with respect to the random sampling typical of TMC, thanks to the generation of a set of sigma points $\overrightarrow{\chi }$ that can be used to reconstruct the input distribution. Generally, 2k+1 sigma points are enough to obtain a good approximation of the input, where k represents the dimension of the input perturbed data. In this specific case, for a given nuclide, a given number of reaction channels n_MT and a given number of energy groups n_G over which the covariance matrix is collapsed, k is given by:

$$\begin{array}{c}\hfill k=n_{\mathit{MT}}·n_G.\end{array}$$(7)

This number can be extremely high if the number of reaction channels and energy groups used to collapse the covariance matrix is high. However, one strength of the UT method is that it is possible to reduce k (and, as a consequence, the number of simulations required by the UQ analysis) applying a suitable reduction technique based on the application of Singular Value Decomposition (SVD) [36].

The set of sigma points $\overrightarrow{\chi }$ can be computed according to the following expressions which provide the value of the i-th sigma points:

$$\begin{array}{cc}\hfill {\chi }^{[0]}& =\mu \hfill \end{array}$$(8)

$$\begin{array}{cc}\hfill {\chi }^{[i]}& =\mu +{\left(\sqrt{(k+\lambda )\widehat{C}}\right)}_i\mathrm{for}i=1,\dots ,k\hfill \\ \hfill {\chi }^{[i]}& =\mu -{\left(\sqrt{(k+\lambda )\widehat{C}}\right)}_{i-n}\mathrm{for}i=k+1,\dots ,2k,\hfill \end{array}$$

where μ is the mean vector of the input (in this case, the nominal nuclear data values from the nuclear data library), λ is an arbitrary spreading parameter that has a non-negligible impact on the results of UT, and $\widehat{C}$ is the corresponding nuclear data covariance matrix. UT assumes that the distribution of the input parameters is Gaussian and generates sigma points which take into account the correlation between data. As a consequence, the sigma points are generated symmetrically with respect to the mean value of the distribution (i.e., the nominal value of the cross section in this case). In this work, a value of λ equal to 0.5 has been chosen. This value has proven to be the best one when Monte Carlo codes and, as a consequence, statistical uncertainties are involved [32, 35]. Each sigma point is then associated with a weight ω:

$$\begin{array}{cc}\hfill {\omega }^{[0]}& =\frac{\lambda }{k+\lambda }\hfill \end{array}$$(9)

$$\begin{array}{cc}\hfill {\omega }^{[i]}& =\frac{1}{2(k+\lambda )}\mathrm{for}i=1,\dots ,2k.\hfill \end{array}$$

Equations (8) and (9) guarantee that the mean μ and the covariance $\widehat{C}$ of the input parameters can be retrieved as:

$$\begin{array}{cc}\hfill \mu & =\sum _{i=0}^{2k}{\omega }^{[i]}{\chi }^{[i]},\hfill \end{array}$$(10)

$$\begin{array}{cc}\hfill \widehat{C}& =\sum _{i=0}^{2k}{\omega }^{[i]}({\chi }^{[i]}-\mu )({\chi }^{[i]}-\mu {)}^T.\hfill \end{array}$$(11)

Each of the sigma points calculated with equation (8) is then used as a nuclear data input for different independent Serpent simulations (here, Serpent is represented as a non-linear function ℳ′), similarly to what happens in TMC, and the weighted mean and weighted covariance of the transformed distribution are calculated as:

$$\begin{array}{cc}\hfill {\mu }^{\prime }& =\sum _{i=0}^{2k}{\omega }^{[i]}{\mathcal{M}}^{\prime }({\chi }^{[i]}),\hfill \end{array}$$(12)

$$\begin{array}{cc}\hfill {\widehat{C}}^{\prime }& =\sum _{i=0}^{2k}{\omega }^{[i]}({\mathcal{M}}^{\prime }({\chi }^{[i]})-{\mu }^{\prime })({\mathcal{M}}^{\prime }({\chi }^{[i]})-{\mu }^{\prime }{)}^T.\hfill \end{array}$$(13)

The diagonal of the covariance matrix $\widehat{C}\prime$ represents the uncertainty on the responses of interest due to the nuclear data uncertainties, propagated through the neutronic code itself.

UT presents some drawbacks that can be tackled in different ways. The first issue is represented by the computation of the square root of $\widehat{C}$ in equation (8). Generally, computing the square root of a matrix is non-trivial. However, in the case of covariance matrices which, by definition, are symmetric and positive semi-definite matrices, this problem can be handled with the Singular Value Decomposition (SVD). According to the SVD, it is possible to write:

$$\begin{array}{c}\hfill \widehat{C}=V\mathrm{\Sigma }{V}^T,\end{array}$$(14)

where the matrix Σ is a diagonal matrix composed by the singular values of $\widehat{C}$. When $\widehat{C}$ is positive semi-definite, its square root can be computed as:

$$\begin{array}{c}\hfill \sqrt{\widehat{C}}=V{\mathrm{\Sigma }}^{1/2}.\end{array}$$(15)

Unfortunately, it is quite likely that, due to inconsistencies in the covariance matrices contained in the ENDF files and to round-off errors occurring in the extraction and collapsing processes of the multi-group covariances from the nuclear data libraries, $\widehat{C}$ is not positive semi-definite. Thus, before applying the UT, the statsmodels Python package [37] is used to estimate the closest semi-positive definite matrix to the original one, according to [38]. In this way, it is possible to apply equations (14) and (15) to nuclear covariance matrices too.

A byproduct of SVD that in this case can be extremely useful is represented by the singular values matrix Σ. In fact, if one orders from the highest to the lowest the singular values of strongly correlated covariance matrices, like nuclear data covariance matrices, it is possible to observe that, except for the first few singular values, the others are negligible. This means that, exploiting the inherent strong correlation of nuclear data covariance matrices, it is possible to maintain a satisfactory accuracy in the computation of $\widehat{C}$ by reducing the number of active singular values t from the initial r non-zero singular values. The t-th singular value at which the k-dimensional covariance matrix is truncated can be selected according to the so called energy of the singular values:

$$\begin{array}{c}\hfill E_t=\frac{{\displaystyle \sum _{i=1}^t{\sigma }_i^2}}{{\displaystyle \sum _{i=1}^r{\sigma }_i^2}},\end{array}$$(16)

where σ_l² is the l-th singular value. This truncation allows to decrease the number of sample points and of Serpent simulations from 2k+1 to 2t+1. As a consequence, UT can be computationally cheaper than GPT and TMC. In fact, the number of simulations required by TMC does not depend on the dimension of the covariance matrix, but only on the statistical convergence of the moments of the response distributions.

The sigma points can be generated, again, using the SANDY Python package, simply applying the following vector p_SP as the perturbation factor to the data contained in the ENDF files:

$$\begin{array}{cc}\hfill p_{\mathit{SP}}^{[0]}& =1\hfill \end{array}$$(17)

$$\begin{array}{cc}\hfill p_{\mathit{SP}}^{[i]}& =1+\sqrt{(t+\lambda )\widehat{C}}\mathrm{for}i=1,\dots ,k\hfill \\ \hfill p_{\mathit{SP}}^{[i]}& =1+\sqrt{(t+\lambda )\widehat{C}}\mathrm{for}i=k+1,\dots ,2k,\hfill \end{array}$$

It is possible that some perturbation factors are smaller than 0 or larger than 2: in these cases, SANDY truncates the perturbation factors to 0 and 2, respectively, distorting the results of the UT. In this respect, a possible solution is to employ a variant of the UT, called the General Constrained Sigma Point (GCSP) method [35, 39], which can be used to force all the perturbation coefficients to stay in the range [0,2].

Finally, as a general comment, it is important to remember that the nuclear data uncertainty (NDU) values strongly depend on the selected covariance matrices and, as a consequence, on the nuclear data library. On the other hand, the sensitivity profiles are almost independent of the nuclear data library [40], being relative variations, thus, with GPT, one can choose different covariance matrices to compute the nuclear data uncertainties due to different libraries after having performed the sensitivity calculation once. On the other hand, non-intrusive techniques require to select the covariance matrix at the moment the perturbed nuclear data are generated. This means that, in order to assess the impact of the nuclear data uncertainties of different libraries with non-intrusive techniques, one has to perform a full set of simulations for each library. In this work, the ENDF/B-VIII.0 [41] and the JEFF-3.3 [42] libraries have been employed to retrieve the nuclear data covariance matrices.

4. Results

This section presents the results in terms of sensitivity obtained with GPT and the nuclear data uncertainties calculated with two different techniques: GPT and UT. For the reader’s convenience, the main parameters used to run the Serpent simulations described in the following sections are summed up in Table 3. The number of perturbed parameters is also provided, since, at least for GPT, the memory consumption and the computational time scale with the amount of perturbed parameters.

4.1. Sensitivity results

A first GPT Serpent simulation has been performed employing the ECCO-33 energy grid and perturbing all the nuclides in TAPIRO and all the reaction channels available. The ECCO-33 grid is quite coarse, but it can be used to obtain preliminary results in order to perform a ranking of the most important nuclides and reaction channels. Once the ranking has been prepared, it is possible to perform a new GPT simulation with a finer grid or with a larger neutron population in order to obtain better results. In this way, the amount of RAM and the computational time required by GPT simulation is kept under a reasonable level in both cases. The preliminary ECCO-33 GPT simulation has been run considering as responses of interest the k_eff, the β_eff and the Λ_eff, with 10⁵ neutrons per cycles (ten times smaller than the number of neutrons employed to obtain the results in Table 1, due to computational memory issues), 2050 active cycles and 50 inactive cycles for the fission source convergence. The simulation, run on 30 CPUs, required 78 GB of RAM and 130 hours to be completed. The most important nuclides and reaction channels in terms of Integral Sensitivity Coefficient (ISC), obtained integrating the sensitivity profiles over the energy domain, are shown in Table 4 and in Figure 3. The most impacting nuclides are ²³⁵U, ⁶³Cu, ⁶⁵Cu, ²³⁸U (structural materials, Mo and the concrete have a negligible impact), while, concerning reaction channels, they are prompt and delayed neutron yields (${\overline{\nu }}_p$ and ${\overline{\nu }}_d$, respectively, elastic scattering (n, n), fission (n, f) and radiative capture (n, γ) cross sections.

Fig. 3. Relative ISC values for the keff, βeff and Λeff, computed with Serpent. The errorbars represent the 1σ stochastic uncertainty.

Observing the results in Figure 3, it is clear that the strongest contribution to k_eff comes from ²³⁵U (as expected, due to the high uranium enrichment of TAPIRO), similarly to β_eff, which is particularly sensitive to the average number of neutrons produced per fission; it is also interesting to observe that the generation time is quite sensitive to copper: this is in line with what has been previously said about the role of the copper reflector in leading to a Λ_eff in the order of few nanoseconds. Moreover, all the results in Figure 3 are consistent with the physics of TAPIRO: for example, in the case of Λ_eff, as already discussed, having a higher scattering in the reflector would lead to higher values of Λ_eff, while having a larger fission cross section in ²³⁵U would further decrease the generation time, reducing the time between two fission events.

For what concerns β_eff, it is possible to see that the contribution of ν_d and ν_p of ²³⁵U are almost the same, but with opposite sign. This result can be explained considering that β_eff can be approximately estimated as the ratio between ν_dΣ_f and ν_pΣ_f (where Σ_f is the macroscopic fission cross section), since ν_p ∼ ν_t. In particular, considering that the core of TAPIRO is almost completely composed of ²³⁵U, it is possible to state that β_eff ∼ (ν_dΣ_f)_U235/(ν_pΣ_f)_U235. According to these considerations, it results that the sensitivities are almost equal to 1: doubling ν_d of ²³⁵U will approximately double β_eff, while doubling ν_p will have the opposite effect. On the opposite, the contribution of ²³⁸U is extremely small (mainly due to its extremely low concentration, and, thus, to the small number of fissions occurring in ²³⁸U). The only contribution is given by ν_p, with a positive sign. The positive sign seems to be in contradiction with what has been stated for ²³⁵U. However, it is possible that this comes from the fact that, in the case of ²³⁸U the indirect effect of ν_p (i.e., the increase of β_eff due to the increase of the neutron flux caused by the increase of ν_p) is larger than the direct one (which, by definition, decreases β_eff). Additionally, the results obtained for β_eff in TAPIRO confirms that this quantity is mostly affected by the composition of the nuclear fuel.

The statistical precision is quite satisfactory for all the results shown in Table 4, except for two terms in the β_eff column, where the relative uncertainty is around 100%. These results are not reliable, however this should not be an issue since their large uncertainty is mainly due to the small value of the corresponding sensitivities.

Figures 4 and 5 shows the sensitivity profiles of the responses of interest for ²³⁵U and ⁶³Cu respectively. The plots show that the statistical uncertainty is negligible also in the sensitivity profiles and not only in the integral values of the sensitivity. Another interesting feature arising from these plots is that below 10 eV the sensitivity is almost null for all the responses. Thus, it could be possible to define a new energy grid specific for TAPIRO with finer bins for energies larger than 10 eV and one single energy bin below this value, without the risk of losing important information about the physics of the system.

Fig. 4. Sensitivity profiles of 235U using the ECCO-33 energy grid.

Fig. 5. Sensitivity profiles of 63Cu using the ECCO-33 energy grid.

Additional results with GPT adopting the XMAS-172G energy grid have been obtained. Here, only ²³⁵U and ⁶³Cu have been considered, to limit the memory consumption, perturbing only the elastic (n, n), fission (n, f) and capture cross sections (n, γ), together with fission neutron yield $\overline{\nu }$. Since it can be expected that larger statistical uncertainties will be obtained due to the finer energy grid employed, 3 × 10⁵ neutron histories per cycles have been simulated. The simulation required 46 GB of RAM (i.e., around 25 MB/energy group/nuclide/reaction channel) and 21 hours of simulations on 30 CPUs. Figure 6 shows that, thanks to the reduction of the number of nuclides and perturbations of interest, it has been possible to obtain results with small statistical uncertainty even with a finer energy grid. The results are extremely similar to the ones obtained with the ECCO-33 energy grid, without adding any new relevant information to the results.

Fig. 6. Relative sensitivity profiles of 235U using the XMAS-172 energy grid.

4.2. Nuclear data uncertainty results

Having calculated the sensitivity profiles from the GPT Serpent simulation, the nuclear data uncertainties can be calculated according to the sandwich rule (Eq. (5)). Table 5 shows the uncertainties obtained using covariance matrices from the JEFF-3.3 library, while Table 6 has been obtained with the ENDF/B-VIII.0 library. The statistical uncertainties in Tables 5 and 6 have been computed without considering the energy correlation between the sensitivity bins. These Tables contain interesting macroscopic results, like the fact that the ENDF/B-VIII.0 misses covariance matrices for copper, while the JEFF-3.3 misses information about the covariance of delayed neutron yield ${\overline{\nu }}_d$, which indeed in the case of ENDF/B-VIII.0 results to be one of the most important contributors to the total uncertainty. Another quantity that is missing in JEFF-3.3 is the covariance matrix of the prompt neutron yield ${\overline{\nu }}_p$ for ²³⁸U, even if in the case of TAPIRO this does not represent an issue, since its contribution to the NDU of TAPIRO is negligible (see Tab. 6). It is also interesting to observe that a higher sensitivity does not always coincide with a higher uncertainty, either due to the fact that the corresponding covariance in the nuclear data library is small or that the sensitivity profile is not “in phase” with the nuclear data variance. This is the case, for example, of the impact of ²³⁵U ${\overline{\nu }}_p$ on the uncertainty of k_eff. Figure 7 shows that the sensitivity to ${\overline{\nu }}_p$ is higher where the data variance is generally smaller, leading to a final NDU smaller than expected. Moreover, the uncertainty of ²³⁵U ${\overline{\nu }}_p$ in the ENDF/B-VIII.0 is higher than the one in the JEFF-3.3, except for the region where the sensitivity is high (i.e., between 10⁵ and 10⁷ eV). As a result, the NDU arising from the JEFF-3.3 is higher than the one from the ENDF/B-VIII.0. This feature is further demonstrated by Figure 8, where the nuclear data uncertainty profiles of ²³⁵U obtained with the ENDF/B-VIII.0 library are shown.

Tables 7 and 8 show the total uncertainty, obtained summing the uncertainties due to all the main reaction channels and to the fission neutron yield $\overline{\nu }$, due to the main nuclides in TAPIRO and their impact on the responses of interest, using JEFF-3.3 and ENDF/B-VIII.0 libraries. With both libraries, the total uncertainty on the k_eff is larger than 900 pcm (in line with other nuclear data uncertainty studies performed on fast systems [35, 43–46]), which is non-negligible and questions the accuracy of the result obtained with the unperturbed and nominal input parameters. In particular, the general trend is that fast systems employing uranium show higher uncertainties than fast systems with plutonium. Moreover, it can be noticed that the statistical uncertainties on the nuclear data uncertainties of β_eff are quite high, mainly due to the small amount of delayed neutrons sampled during criticality simulations. This is a well-known issue when GPT calculations are performed in MC codes; however, recently, in the framework of the TRIPOLI-4 code, a technique for the reduction of the statistical uncertainty of the sensitivities of β_eff has been proposed [47], based on the artificial increase of the number of sampled delayed neutrons after each fission event. The implementation of a similar strategy in Serpent should be quite straightforward, since it just requires to substitute the delayed neutron yield with the total neutron yield and to adjust the weight of the neutrons to avoid biased results.

Fig. 7. Relative sensitivity per unit lethargy profile versus variance of ${\overline{\nu }}_p$ from the ENDF/B-VIII.0 (left figure) and JEFF-3.3 (right figure) of keff.

Fig. 8. Relative nuclear data uncertainty profiles of 235U obtained employing the ENDF/B-VIII.0 library.

Even though the GPT in Serpent does not provide directly the sensitivity of the decay constant α_p, its uncertainty can be calculated with the classic formula for the uncertainty propagation which states that, for a function α_p = (ρ − β)/Λ, the uncertainty σ_α on α_p is given by:

$$\begin{array}{c}\hfill {\sigma }_{\alpha }\simeq \sqrt{\frac{{\sigma }_{\rho }^2+{\sigma }_{\beta }^2-2{\theta }_{\rho \beta }{\sigma }_{\rho }{\sigma }_{\beta }}{{\mathrm{\Lambda }}^2}+\frac{{(\rho -\beta )}^2{\sigma }_{\mathrm{\Lambda }}^2}{{\mathrm{\Lambda }}^4}-\frac{2(\rho -\beta ){\theta }_{(\rho -\beta )\mathrm{\Lambda }}{\sigma }_{(\rho -\beta )}{\sigma }_{\mathrm{\Lambda }}}{{\mathrm{\Lambda }}^3}},\end{array}$$(18)

where θ represents the correlation coefficient between the different parameters, which is induced by the nuclear data [35, 48]. In equation (18), all the quantities are known, except for the correlation coefficients, which could be estimated performing neutronic evaluations of the system after the random sampling of the nuclear data. However, one can imagine three representative examples: no correlations between the parameters (θ = 0), unitary and positive correlation (θ = 1) and unitary and negative correlation (θ = −1). The uncertainties of α_p are shown in Tables 9 and 10: it can be seen that the uncertainties due to nuclear data on the decay constant are huge, and they are weakly sensitive to the level of correlation. This is mainly due to the fact the nuclear data uncertainty propagated from the multiplication factor to the reactivity, according to:

$$\begin{array}{c}\hfill {\sigma }_{\rho }\simeq \sqrt{\frac{1}{{\mathrm{k}}_{\mathrm{eff}}^2}{\left(\frac{{\sigma }_{{\mathrm{k}}_{\mathrm{eff}}}}{{\mathrm{k}}_{\mathrm{eff}}}\right)}^2}\end{array}$$(19)

is extremely high when the system is near criticality, as in this case. In fact, according to equation (19), the standard deviation on ρ is similar to the relative uncertainty on k_eff, which means that if the reactivity of a generic fission system is smaller than the relative uncertainty of k_eff (which, due to nuclear data uncertainties, generally ranges between 0.40% and 1.70% in fission reactors [49, 50]), the relative uncertainty on ρ will be larger than 100%. It can be shown that, using equations (18) and (19), in order to have reasonable values of uncertainties on α_p, around 5–10%, the total uncertainty on k_eff (i.e., statistical plus nuclear data uncertainties) should be of ∼10 pcm in this TAPIRO configuration, which is non-feasible. Thus, other methods for evaluating the reactivity and its uncertainty should be adopted in order to obtain reliable values of α_p, when the system configuration is close to criticality.

Table 11 shows the GPT uncertainties obtained with XMAS-172 and JEFF-3.3 covariance matrices and their difference with respect to ECCO-33 results according to the following expression:

$$e_r=\frac{s_{\mathrm{XMAS}}-s_{\mathrm{ECCO}}}{s_{\mathrm{XMAS}}}$$(20)

The results are similar to the ones obtained with the ECCO-33 energy grid (Tab. 8), suggesting that the ECCO-33 energy grid can be a suitable grid for the sensitivity and uncertainty quantification analysis of TAPIRO, avoiding to excessively load the RAM with a fine energy grid like the XMAS-172. Figure 9 sums up the total uncertainties computed in this work. Since the sensitivity profiles obtained with the two grids (Figs. 4 and 6) are extremely similar, the differences in the results are mainly due to small differences in the covariance matrices obtained with the two grids, as it can be seen in Figure 10.

Fig. 9. Total relative uncertainties due to 235U, 63Cu, 238U and 65Cu using different energy grids and different nuclear data libraries. The errorbars represent the 2σ uncertainty.

Fig. 10. ECCO-33 and XMAS-172 covariance matrices of 63Cu capture cross sections. The dashed lines represent the energy boundaries of the ECCO-33 energy grid.

Then, as an additional example, the unscented transform has been applied to evaluate the uncertainty due to ⁶³Cu in order to test this alternative technique and to check the consistency of the GPT results. In principle, the UT requires to perform 2k+1 simulations, where k is the dimension of the input data covariance matrix i.e., the number of singular values. However, applying the singular value decomposition (SVD) of the matrix it is possible to reduce the number of simulations required, thanks to the fact that the original matrix can be reconstructed keeping only the largest singular values, at the cost of a small approximation. Only the elastic scattering (n, n), the radiative capture (n, γ) and the inelastic scattering (n, n′) to the first three excited states have been considered to generate the 33 groups covariance matrices, since from the GPT they result to be the main contributors in the case of ⁶³Cu. Thanks to the SVD, it has been possible to pass from 331 sigma points (i.e., number of Serpent simulations) to only 95 sigma points (i.e., only 47 singular values out of 165 have been kept), with the original covariance matrix not being significantly altered after this truncation. SANDY has been used to extract the covariance matrix and to generate the 95 perturbed ACE files according to equation (17). Each simulation has been performed with 4 × 10⁵ neutrons per cycle, 2050 active cycles and 50 inactive cycles, requiring 1 hour of computation per simulation with 30 CPUs and only 5 GB of RAM, with the memory requirement independent of the number of perturbed parameters. Since the criticality calculations used in the context of the UT are not affected by issues in terms of computational cost, it is possible to increase the amount of neutron histories with respect to GPT to have a better statistics. Considering that each simulation is statistically independent from the others and, in principle, can be run in parallel, the UT could be much faster than the GPT and also less memory consuming. The results (see Tab. 12) are similar to the GPT ones, which is in line with other results found in the literature [35], except for the β_eff, for which more neutron histories would have been needed in order to obtain reliable results. In fact, if the statistical uncertainty is larger than the NDU, the application of the UT can lead to negative values of uncertainties, which obviously have to be discarded. The differences with respect to GPT results can be due to the fact that in the UT less reaction channels have been considered, to the truncation of the covariance matrix and also to the fact that the UT can be used to propagate uncertainties in non-linear systems too, while the GPT is by definition a linear technique. The nuclear data uncertainties obtained so far can be propagated in the neutron noise framework to assess their impact on the results of the analysis. An alternative is to use SANDY to generate a limited set of perturbed ACE files (e.g., two ACE files per nuclide at ±1σ from the nominal value) to be used in the neutron noise analysis and to generate detector signals with these perturbed ACE files. The latter solution requires to perform at least two neutron noise simulations, which are extremely time consuming. Thus, from this point of view, exploiting the uncertainties calculated in this work should be the faster way. However, both solutions will be tested in future works in order to assess possible discrepancies between the two approaches.

5. Conclusions

At first, a neutronic analysis of TAPIRO has been performed employing the Serpent Monte Carlo code, in order to calculate the main integral parameters of interest to be used in the neutron noise analysis of the facility: the multiplication factor, the effective delayed neutron fraction, the effective prompt neutron generation time and the prompt neutron decay constant (k_eff, β_eff, Λ_eff and α_p, respectively). The results are comparable with the ones obtained with MCNP by ENEA and ANL. Then, the GPT Serpent simulation of the TAPIRO reactor employing the ECCO-33 energy grid has allowed to point out that the integral parameters required by the noise analysis of the system (i.e., ${\mathrm{k}}_{\mathrm{eff}}$, β_eff and Λ_eff) are mostly sensitive to the nuclear data of four nuclides: ²³⁵U, ²³⁸U, ⁶³Cu and ⁶⁵Cu. The sensitivity profiles calculated with the GPT have been used to compute the nuclear data uncertainty thanks to the sandwich rule employing both the ENDF/B-VIII.0 and the JEFF-3.3 nuclear data libraries. This analysis indicates that there are some non-negligible discrepancies between the two libraries, mainly due to the fact that some covariances are missing, i.e. for copper in ENDF/B-VIII.0 and for ${\overline{\nu }}_p$ of ²³⁸U and ${\overline{\nu }}_d$ in JEFF-3.3. These gaps limit the reliability of uncertainty quantification and should guide future nuclear data evaluation efforts. An additional GPT simulation has been performed using the XMAS-172 energy grid showing that it is not necessary to employ a too fine grid for the sensitivity and uncertainty analysis of TAPIRO, suggesting that the ECCO-33 could be an enough refined grid. Finally, an alternative uncertainty propagation technique called unscented transform (UT) has been applied to propagate the nuclear data uncertainty of ⁶³Cu. The results of the UT method are similar to GPT, making the UT an interesting alternative technique, in particular when neutronic codes which can not perform sensitivity calculations are used, since its requirements in terms of RAM and computational time are less expensive. However, the UT should be further validated in the future since here only two nuclides have been tested and the comparison has been made only with a linear technique (i.e, GPT).

This study shows that the impact of the nuclear data uncertainties on the responses of interest is non-negligible, reaching up to 1000 pcm in the case of k_eff. In particular, it has been shown that the impact of nuclear data uncertainties on the prompt decay constant α_p are huge. Thus, strategies for mitigating this issue, like using subcritical configurations or experimental determination should be considered. The results of this study are planned to be exploited to further propagate the nuclear data uncertainties in the framework of the noise analysis of the TAPIRO reactor, foreseen in future works.

Acknowledgments

The authors are grateful to Argonne National Laboratory and ENEA for the fruitful discussions about TAPIRO and for producing a converged model of TAPIRO useful for Monte Carlo evaluations. A particular thanks to Dr. Alberto Talamo (ANL) for having provided a model of TAPIRO converted for its use with the Serpent code. Computational resources were provided by HPC@POLITO, a project of Academic Computing within the Department of Control and Computer Engineering at Politecnico di Torino (http://www.hpc.polito.it).

Funding

The present work was carried out within the Contract No. 3F-60145, supported by Argonne National Laboratory.

Conflicts of interest

The authors have nothing to disclose.

Data availability statement

Data associated with this article cannot be disclosed.

Author contribution statement

Conceptualization, A. Aimetta and S. Dulla; Methodology, A. Aimetta and S. Dulla; Investigation, A. Aimetta and S. Dulla; Resources, S. Dulla; Data Curation, A. Aimetta; Writing – Original Draft Preparation, A. Aimetta; Writing – Review & Editing, A. Aimetta, S. Dulla; Visualization, A. Aimetta; Supervision, S. Dulla; Project Administration, S. Dulla; Funding Acquisition, S. Dulla.

References

All Tables

All Figures

Fig. 1. Serpent model of TAPIRO: vertical section (left) and horizontal section (right). Zoom on the core and reflector regions. The dimensions of the depicted regions are 100 × 100 cm. The figure on the bottom show some specific regions of the core (on the left) and of the reflector (on the right) where neutron spectra have been evaluated with Serpent (Fig. 2). Each mf label is associated with a zone with a different fuel composition.

In the text

	Fig. 2. Normalised neutron spectrum in the core (left) and in the reflector (right) of TAPIRO, computed with Serpent. The different colors correspond to the colors of the locations in the core and in the reflector where they have been calculated, shown in the two miniatures of the reactor.
In the text

	Fig. 3. Relative ISC values for the keff, βeff and Λeff, computed with Serpent. The errorbars represent the 1σ stochastic uncertainty.
In the text

	Fig. 4. Sensitivity profiles of 235U using the ECCO-33 energy grid.
In the text

	Fig. 5. Sensitivity profiles of 63Cu using the ECCO-33 energy grid.
In the text

	Fig. 6. Relative sensitivity profiles of 235U using the XMAS-172 energy grid.
In the text

	Fig. 7. Relative sensitivity per unit lethargy profile versus variance of ${\overline{\nu }}_p$ from the ENDF/B-VIII.0 (left figure) and JEFF-3.3 (right figure) of keff.
In the text

	Fig. 8. Relative nuclear data uncertainty profiles of 235U obtained employing the ENDF/B-VIII.0 library.
In the text

	Fig. 9. Total relative uncertainties due to 235U, 63Cu, 238U and 65Cu using different energy grids and different nuclear data libraries. The errorbars represent the 2σ uncertainty.
In the text

	Fig. 10. ECCO-33 and XMAS-172 covariance matrices of 63Cu capture cross sections. The dashed lines represent the energy boundaries of the ECCO-33 energy grid.
In the text