Neutron Cross-Section Uncertainty and Reactivity Analysis in MOX and Metal Fuels for Sodium-Cooled Fast Reactor

Abstract

This study presents a comprehensive uncertainty and sensitivity analysis of the effective neutron multiplication factor ($k_{\mathrm{eff}}$) in a large-scale sodium-cooled fast reactor (SFR) modeled after the European Sodium Fast Reactor. Utilizing the Serpent Monte Carlo code and the ENDF/B-VII.1 cross-section library, this research investigates the impact of cross-section perturbations in key isotopes (²³⁵U, ²³⁸U, and ²³⁹Pu for both mixed oxide (MOX) and metal fuels. Particular focus is placed on the capture, fission, and inelastic scattering reactions, as well as the effects of fuel temperature on reactivity through Doppler broadening. The findings reveal that reactivity in MOX fuel is highly sensitive to the fission cross sections of fissile isotopes (²³⁹Pu and ²³⁸U, while capture and inelastic scattering reactions in fertile isotopes such as ²³⁸U play a significant role in reducing reactivity, enhancing neutron economy. Additionally, this study highlights that metal fuel configurations generally achieve a higher ($k_{\mathrm{eff}}$) compared to MOX, attributed to their higher fissile atom density and favorable thermal properties. These results underscore the importance of accurate nuclear data libraries to minimize uncertainties in criticality evaluations, and they provide a foundation for optimizing fuel compositions and refining reactor control strategies. The insights gained from this analysis can contribute to the development of safer and more efficient next-generation SFR designs, ultimately improving operational margins and reactor performance.

1. Introduction

Sodium-cooled fast reactors (SFRs) are a key Generation IV technology, offering improved fuel utilization and significant waste reduction through a fast neutron spectrum [1]. Among the available fuel options, mixed oxide (MOX) and metal fuels each have unique advantages: MOX enables efficient plutonium recycling, while metal fuel provides favorable thermal performance and higher heavy-metal loading [2]. Accurate prediction of core behavior and safety margins in these advanced systems hinges on understanding how uncertainties in neutron cross sections affect the effective neutron multiplication factor ($k_{\mathrm{eff}}$), which is strongly influenced by scattering, absorption, and fission processes [3]. Recent advances in computational methods have significantly enhanced the fidelity of reactor simulations, evolving from purely conservative safety margins to best-estimate methodologies supplemented by rigorous sensitivity and uncertainty analyses [4]. This methodological shift is particularly relevant for sodium-cooled fast reactors (SFRs), where accurate modeling of complex physical phenomena is essential for safety and performance evaluations.

In support of this transition, extensive benchmark experiments and modeling efforts have been conducted. Notably, the SNEAK-12A and 12B experimental programs have provided valuable insights into core disruption scenarios, enabling improved interpretation of reactivity feedback mechanisms and quantification of nuclear data uncertainties in liquid-metal fast breeder reactors (LMFBRs) [5,6]. In parallel, the UAM-SFR benchmark initiative has facilitated comprehensive uncertainty quantification, encompassing both simplified pin-cell models and full-core configurations [7]. Recent investigations have also examined the impact of nuclear data uncertainties on breeding ratio predictions [8] and evaluated reactivity coefficients at the end of equilibrium cycles within the European Sodium Fast Reactor (ESFR) framework [9], along with assessments of decay heat behavior under representative conditions [10]. These collective contributions have laid a robust foundation for best-estimate plus uncertainty (BEPU) methodologies in advanced fast reactor design and analysis. Building upon this foundation, the present study focuses on the energy-dependent sensitivity of key nuclear cross sections in SFR cores, aiming to refine nuclear data evaluation and enhance confidence in simulation results.

In SFR systems, the interplay of a fast neutron spectrum, elevated operating temperatures, and substantial neutron leakage increases sensitivity to nuclear data uncertainties, particularly for heavy nuclides such as ²³⁸U and ²³⁹Pu. Temperature-dependent phenomena, including Doppler broadening, further underscore the importance of accurate cross-section data for reliable reactivity predictions. Sensitivity and uncertainty evaluations play a crucial role in identifying dominant isotopic contributions to $k_{\mathrm{eff}}$, guiding nuclear data improvement efforts, and supporting the development of robust and safe reactor designs. This study focuses on a large-scale SFR at the beginning of cycle (BOC), a phase when changes to cross-section data can have a significant impact on reactor performance. Detailed simulations using the Serpent Monte Carlo code and the ENDF/B-VII.I cross-section library compare MOX-fueled and metal-fueled cores by introducing perturbations in key isotopes (²³⁵U, ²³⁸U, ²³⁹Pu, and ²⁴¹Pu). Additionally, two representative fuel temperatures (743 K and 1500 K) are considered to capture Doppler effects on reactivity in MOX fuel. Through this approach, we identify dominant reaction channels, assess temperature-dependent influences, and offer insights for optimizing fuel compositions in next-generation SFRs. Ultimately, these findings support the refinement of nuclear data libraries and improvements in uncertainty quantification, contributing to safer and more efficient reactor designs.

2. Materials and Methods

2.1. Large SFR Core Model and Atomic-Level Fuel Configurations

In this subsection, we focus on the modeling of a large sodium-cooled fast reactor (SFR) core inspired by the European Sodium-Cooled Fast Reactor (ESFR) concept [11,12]. The ESFR seeks to harness the benefits of fast-spectrum reactors, including the efficient use of fissile material, breeding potential, and reduction in long-lived waste. By adapting certain features of this design, our aim is to capture the essential physics of a large SFR core while simplifying selected elements to facilitate manageable computational analyses.

The ESFR core layout comprises inner and outer fuel zones, sodium coolant channels, control rods, and reflector regions, all arranged around the active core illustrated in Figure 1. These zones emulate typical radial zoning practices in large SFRs, with the outer zone generally having a slightly higher fissile content than the inner zone to help flatten the radial power distribution and reduce excessive power peaking near the core center. For this study, we focus solely on the BOC state, thereby excluding the effects of burnable poisons and control rod insertion. This approach isolates the fundamental reactivity features of fresh fuel without the added complexity of depletion or control maneuvers. Although real reactor operation covers a broad range of configurations and burnup conditions, evaluating BOC characteristics offers a well-defined baseline for gauging nuclear data sensitivity.

Two main fuel types are considered. The first type is the MOX fuel, comprising a blend of uranium and plutonium oxides, $\left((\mathrm{U},\mathrm{Pu}){\mathrm{O}}_2\right)$. Since the plutonium arises from reprocessed spent fuel, MOX presents an attractive means of managing plutonium inventories and promoting a more sustainable fuel cycle. In our model, the fuel compositions for the inner and outer core zones follow the weight ratios generally listed in

Table 1. Isotopic composition of MOX fuel in inner and outer zones.

Isotopic Composition	Weight Ratio
	Inner Zone	Outer Zone
U-234	0.000038	0.000037
U-235	0.001516	0.001474
U-238	0.756627	0.735901
Pu-238	0.002223	0.002586
Pu-239	0.069040	0.080401
Pu-240	0.029590	0.034466
Pu-241	0.016025	0.018642
Pu-242	0.006409	0.007462
Am-241	0.000740	0.000857
O-16	0.117751	0.118342

, with slight modifications to ensure criticality under nominal conditions. These compositions also capture realistic isotopic distributions, which helps approximate the reactivity and sensitivity behavior likely encountered in an operational setting.

The second fuel type is a metal alloy consisting primarily of uranium and plutonium, with a minor fraction of zirconium ($\mathrm{U}\text{-}\mathrm{Pu}\text{-}\mathrm{Zr}$). Unlike MOX, which uses oxygen to form a ceramic matrix, metal fuel omits oxygen, leading to increased heavy-metal density. This higher loading can result in greater power densities and, in certain respects, more favorable thermal properties. Nonetheless, metal fuels introduce distinct material challenges, such as swelling and fission gas release, necessitating careful design considerations for cladding and overall reactor safety. The fuel compositions for the inner and outer core zones follow the weight ratios generally listed in

Table 2. Isotopic composition of metal fuel in inner and outer zones.

Isotopic Composition	Weight Ratio
	Inner Zone	Outer Zone
U-234	0.00003793	0.00003745
U-235	0.00497	0.0049
U-238	0.6838	0.6752
Pu-238	0.00405	0.0040
Pu-239	0.1215	0.1200
Pu-240	0.0506	0.0500
Pu-241	0.0101	0.00997
Pu-242	0.00405	0.0040
Am-241	0.02025	0.0200
Zr-90	0.05212	0.05147
Zr-91	0.01137	0.01122
Zr-92	0.01737	0.01715
Zr-94	0.01759	0.01737
Zr-96	0.00283	0.00280

.

To illustrate how temperature variations can affect core behavior, we examine two temperature conditions for the MOX-fueled configuration. The first—referred to as “MOX avg.”—represents an average operating temperature around $743\mathrm{K}$. The second, “MOX hot,” approximates a significantly higher temperature of about $1500\mathrm{K}$. By contrast, the metal-fueled configuration is evaluated only at an average operational temperature near $743\mathrm{K}$, reflecting typical conditions for metal fuel. While it would be possible to examine a continuous temperature spectrum to extract detailed Doppler coefficients, we limit our scope to these two temperatures for clarity and to demonstrate a representative temperature dependence. In real applications, resonance capture and other temperature-related phenomena, especially in ²³⁸U, can substantially alter the effective neutron multiplication factor $\left(k_{\mathrm{eff}}\right)$. Observing how these effects scale between “MOX avg.” and “MOX hot” underscores the importance of accurately modeling temperature feedback mechanisms in fast reactors.

In essence, by comparing these temperature conditions and simultaneously analyzing MOX and metal fuels, we highlight how nuclear data perturbations interact with distinct fuel attributes under different thermal states. The geometry and structural setup remain consistent across both fuel types, ensuring that any observed differences in sensitivity and uncertainty outcomes arise primarily from fuel composition and temperature. This framework allows for a clearer investigation of how each fuel type responds to variations in cross sections and nuclear data, providing insight into the choice of optimal fuel configurations for fast reactors.

2.2. Uncertainty and Sensitivity Analysis

An uncertainty and sensitivity analysis in reactor physics provides deeper insight into how variations in nuclear data—particularly cross sections—affect the key performance and safety parameters of a reactor system. In this context, Serpent’s built-in sensitivity module utilizes a first-order perturbation approach based on the differential operator formalism to determine partial derivatives of $k_{\mathrm{eff}}$ with respect to relevant nuclear data in a single forward Monte Carlo simulation. Because cross-section uncertainties can significantly affect reactivity in both thermal and fast reactors, Serpent’s on-the-fly sensitivity tallies provide valuable insight into how each isotope and reaction channel influences $k_{\mathrm{eff}}$. This is particularly important for fast reactors, where certain cross sections—like inelastic scattering or high-energy capture—can introduce large uncertainties, but the method itself is broadly applicable to any reactor spectrum.

At the heart of this sensitivity methodology is the idea that small changes in a cross section $\sigma \left(E\right)$ (or its group-wise representation in discrete energy bins) lead to a proportional change in $k_{\mathrm{eff}}$. To quantify this relationship, Serpent computes sensitivity coefficients $S_i\left(E\right)$. For each isotope i in a specific energy bin E, $S_i\left(E\right)$ captures how much $k_{\mathrm{eff}}$ changes (as a fraction of its original value) when ${\sigma }_i\left(E\right)$ undergoes a small fractional change. Mathematically, if $\mathrm{\Delta }{\sigma }_i\left(E\right)$ is a small perturbation in a cross section, then the change in $k_{\mathrm{eff}}$ can be approximated as follows:

$$\mathrm{\Delta }{k}_{\mathrm{eff}}\approx S_i\left(E\right){\displaystyle \frac{\mathrm{\Delta }{\sigma }_i\left(E\right)}{{\sigma }_i\left(E\right)}}{k}_{\mathrm{eff}}.$$

(1)

The final quantity $S_i\left(E\right)$ is dimensionless and thereby readily comparable across different isotopes, reactions, or energy ranges. By examining the energy dependence, one can immediately see which energy group contributions are most critical. If the magnitude of $S_i\left(E\right)$ is large for a particular isotope–reaction pair at certain energies, it indicates that $k_{\mathrm{eff}}$ is highly sensitive to uncertainties or changes in that cross section at those energies.

A key reason sensitivity analysis is so vital in fast reactor applications is the higher neutron energies involved. Fast reactors rely primarily on fission events induced by higher-energy neutrons, which changes the relative importance of specific isotopes and reaction channels compared to thermal reactors. For example, in many fast reactor designs, ²³⁸U captures and inelastic scattering reactions play a major role in the overall reactivity balance. Whereas thermal reactors rely heavily on moderating neutrons to lower energies, fast reactors are more sensitive to the resonance behaviors of ²³⁸U and the availability of fertile-to-fissile conversion routes (e.g., breeding ²³⁹Pu) [13].

In this study, we focus on capturing how perturbations in a subset of reaction channels—²³⁸U capture, ²³⁸U inelastic scattering, and fission reactions of ²³⁹Pu and ²³⁵U—affect $k_{\mathrm{eff}}$. Historically, these channels are among the largest contributors to uncertainty in fast reactor cores because of their strong influence on the neutron economy. For instance, ²³⁸U inelastic scattering can moderate neutron energies slightly within the fast spectrum, thus affecting subsequent fission probabilities. Meanwhile, the capture reaction in ²³⁸U competes with fission events, potentially driving the breeding of ²³⁹Pu but also removing neutrons from the fission chain. Understanding how these channels respond to perturbations is therefore paramount for any robust uncertainty quantification or reactor optimization effort.

To add further complexity, the analysis of MOX fuels versus metal fuels highlights distinct sensitivity profiles. MOX fuels contain significant amounts of ²³⁹Pu (and possibly other transuranics), creating a fission spectrum that is somewhat different from that of traditional metal fuels, which might consist primarily of uranium and plutonium in metallic form. In MOX-fueled systems, the fraction of fissions occurring in ²³⁹Pu can be higher depending on burnup and initial fuel composition, thus altering how $k_{\mathrm{eff}}$ responds to changes in cross sections for capture and fission. By contrast, metal fuels may leverage the higher density and improved thermal conductivity of metallic uranium and plutonium, leading to different neutronic characteristics—especially regarding resonance self-shielding and local power distributions.

When $\sigma$ values or reaction channels differ between MOX and metal fuels, the resulting energy spectra of neutrons, fission product yields, and breeding rates also diverge. The direct perturbation approach in Serpent allows us to trace these differences back to exact isotopic and energy-bin dependencies. Thus, for core designers or analysts, reviewing the sensitivity coefficients for these two fuel types side by side provides both a gauge of where nuclear data improvements would have the highest impact and a clearer understanding of the core’s operational margin under various uncertainties. This knowledge is invaluable when setting research priorities (e.g., new differential measurements for certain cross sections) or validating and refining nuclear data libraries to better match the needs of fast reactor simulations.

2.3. A First-Order Estimation of the Average Doppler Coefficient

Although we do not present a full reactivity feedback study in this work, it is instructive to consider an approximate measure of how fuel temperature affects the reactor’s neutronic behavior. In particular, changes in $k_{\mathrm{eff}}$ as the fuel temperature varies are closely tied to the Doppler effect, which is one of the most important feedback mechanisms in a nuclear reactor. The Doppler effect, in the context of reactor physics, primarily arises because the resonance absorption cross sections of certain isotopes broaden with increasing temperature [14]. As the temperature of the fuel increases, these resonance peaks become wider, leading to an increased probability of neutron capture by the fuel. This phenomenon provides a negative feedback to the fission reaction, helping to stabilize the reactor against rapid power excursions.

In order to quantify this effect, we examine the difference in $k_{\mathrm{eff}}$ between two temperatures relevant to the operation of MOX fuel: $743\mathrm{K}$ (which might represent an overall average operating temperature) and $1500\mathrm{K}$ (a higher temperature that could be reached at the center of the MOX fuel rod). To form a rough estimate of the average Doppler coefficient between two big different temperatures, we define the following:

$${\alpha }_{\mathrm{Doppler}}\approx {\displaystyle \frac{{\rho }_{1500}-{\rho }_{743}}{T_{1500}-T_{743}}}={\displaystyle \frac{\mathrm{\Delta }\rho }{\mathrm{\Delta }T}},$$

(2)

where $\rho$ denotes reactivity in pcm (percent mille) and is given by

$$\rho ={\displaystyle \frac{k_{\mathrm{eff}}-1}{k_{\mathrm{eff}}}}.$$

In Equation (2), ${\rho }_{1500}$ and ${\rho }_{743}$ are the reactivities at $1500\mathrm{K}$ and $743\mathrm{K}$, respectively, while $T_{1500}$ and $T_{743}$ represent those respective temperatures in Kelvin.

The average Doppler coefficient evaluated between 743 K and 1500 K offers a convenient integral measure of how reactivity evolves with temperature over a wide range. Because the interval is large, the result cannot be treated as the true differential (instantaneous) Doppler coefficient; instead, it represents a temperature-averaged value obtained from the exact definition applied in finite-difference form. It is important to emphasize that this quantity does not strictly isolate the canonical fuel-temperature (Doppler) effect. Any physics that changes between the two states—such as shifts in scattering cross sections or small density variations—can also contribute to the observed difference in $k_{\mathrm{eff}}$. Consequently, the value reported here should be interpreted as an aggregate reactivity change dominated by, but not exclusively due to, Doppler broadening. In a more detailed treatment, one would separately account for these effects or use a methodology designed to focus purely on the resonance capture phenomenon within the fuel. Nevertheless, the quantity calculated via Equation (2) still provides a valuable first-order indicator of the negative feedback strength associated with temperature increases in MOX fuel.

Assuming that the uncertainties in reactivity at each temperature are independent and random, the uncertainty in the Doppler coefficient can be estimated using standard error propagation as follows:

$$\delta {\alpha }_{\mathrm{Doppler}}={\displaystyle \frac{1}{\mathrm{\Delta }T}}\sqrt{{\left(\delta \rho \left(T_1\right)\right)}^2+{\left(\delta \rho \left(T_2\right)\right)}^2},$$

(3)

where $\delta \rho \left(T_1\right)$ is the uncertainty in reactivity at temperature $T_1$, and $\delta \rho \left(T_2\right)$ is the uncertainty at temperature $T_2$.

The sign of ${\alpha }_{\mathrm{Doppler}}$ is expected to be negative, reflecting that an increase in temperature leads to a net decrease in reactivity. This inherently stabilizing effect is particularly important in fast-reacting scenarios, such as unprotected transients, where the fuel temperature can rise rapidly. A sufficiently negative Doppler coefficient ensures that as the fuel warms, the resonance absorption of neutrons becomes more pronounced, thus lowering the neutron multiplication factor and mitigating any sudden power excursion. From a reactor design perspective, the magnitude of the Doppler coefficient can influence numerous operational and safety parameters. For instance, reactors with fuels or isotopic compositions that exhibit a strong negative Doppler feedback tend to be more forgiving under transient conditions. Conversely, if this feedback is weak, the reactor operator or automatic control systems must rely more heavily on other reactivity control mechanisms (e.g., control rods, soluble boron, or burnable poisons) to maintain safety margins.

3. Results and Discussion

3.1. Core Reactivity and Doppler Effect

As summarized in

Table 3. $k_{\mathrm{eff}}$ and corresponding statistical uncertainty at the beginning of cycle.

Fuel	Metal (743 K)	MOX (743 K)	MOX (1500 K)
$k_{\mathrm{eff}}$	1.30788	1.09474	1.08573
Standard Deviation ($\sigma$)	0.00019	0.00021	0.00021

, metal fuel typically achieves a higher effective neutron multiplication factor, underscoring its strong suitability for sustaining reactor criticality. One principal reason is the higher density of metal fuel, which permits a larger number of fissile atoms in a given volume and thereby enhances the likelihood of neutron-induced fission events. Additionally, metal fuel generally possesses superior thermal conductivity compared to oxide-based fuels. This characteristic improves heat removal from the core, helping to maintain relatively uniform operating temperatures and reduce risks related to excessive heat buildup [15]. On the other hand, mixed oxide (MOX) fuels, commonly composed of a blend of plutonium oxide (${\mathrm{PuO}}_2$) and uranium oxide (${\mathrm{UO}}_2$), exhibit a comparatively lower $k_{\mathrm{eff}}$, particularly at higher temperatures. A primary factor contributing to this behavior is Doppler broadening, where the resonant absorption peaks broaden due to increased atomic vibrations within the heated fuel. As these peaks widen, more neutrons are captured by non-fissile isotopes rather than inducing fission, thus decreasing the overall neutron economy [16].

While metal fuel demonstrates a notable advantage in maintaining criticality—largely due to its higher $k_{\mathrm{eff}}$ and favorable thermophysical properties—MOX fuel faces a reduction in $k_{\mathrm{eff}}$ at elevated temperatures as a consequence of Doppler broadening. Specifically, Equations (2) and (3) yield an average Doppler coefficient of $-1.0\mathrm{pcm}/\mathrm{K}$ and an associated uncertainty of $0.033\mathrm{pcm}/\mathrm{K}$ for the MOX core. This value falls within the typical range of Doppler feedback observed in fast reactor fuels [17]. Although Doppler broadening is a major contributor to negative temperature reactivity in oxide fuel, we note that changes in scattering cross sections also influence reactivity in fast systems. A more thorough breakdown of spectral shift, leakage, and conduction feedback would require advanced core-coupled simulations, but here, we restrict our focus to $k_{\mathrm{eff}}$-based sensitivity findings.

3.2. Energy-Integrated Sensitivity Coefficients for Major Nuclides

The energy-integrated sensitivity coefficients of $k_{\mathrm{eff}}$ to various nuclear reaction cross sections provide a quantitative measure of how perturbations in nuclear data influence the neutron multiplication factor across different fuel configurations. Specifically, these coefficients quantify how perturbations in reaction cross sections influence $k_{\mathrm{eff}}$, helping to identify key isotopes and reaction channels that significantly affect reactor performance.

Table 4. Energy-integrated average sensitivity of different cross sections for important isotopes in metal, MOX, and MOXHot fuels.

XS Type	Isotope	Metal	MOXAvg	MOXHot
Inelastic	Total	$-0.12181000$	$-0.10684600$	$-0.10941800$
	U-235	$-0.00033801$	$-0.00016476$	$-0.00008074$
	U-238	$-0.06330520$	$-0.06678080$	$-0.06875840$
	PU-238	$-0.00012152$	$-0.00006842$	$-0.00009895$
	PU-239	$-0.00682883$	$-0.00344073$	$-0.00340254$
	PU-240	$-0.00333061$	$-0.00178913$	$-0.00179746$
	PU-241	$-0.00116242$	$-0.00122586$	$-0.00129699$
	PU-242	$-0.00027888$	$-0.00047779$	$-0.00038794$
	Am-241	$-0.00205158$	$-0.00006882$	$-0.00003439$
Capture	Total	$-0.27133900$	$-0.36869300$	$-0.36266400$
	U-235	$-0.00214106$	$-0.00116189$	$-0.00114546$
	U-238	$-0.14685900$	$-0.24744800$	$-0.24668400$
	PU-238	$-0.00202289$	$-0.00211734$	$-0.00203558$
	PU-239	$-0.03806030$	$-0.05088130$	$-0.04787780$
	PU-240	$-0.01864860$	$-0.02065230$	$-0.02024120$
	PU-241	$-0.00339389$	$-0.01022360$	$-0.00968387$
	PU-242	$-0.00123934$	$-0.00384590$	$-0.00374746$
	Am-241	$-0.02941990$	$-0.00191747$	$-0.00187506$
Fission	Total	$+0.63324400$	$+0.68070900$	$+0.68633300$
	U-235	$+0.01372460$	$+0.00698240$	$+0.00693773$
	U-238	$+0.04833730$	$+0.07453830$	$+0.07584650$
	PU-238	$+0.01008220$	$+0.00812210$	$+0.00820255$
	PU-239	$+0.45260100$	$+0.41319200$	$+0.41732500$
	PU-240	$+0.04267300$	$+0.03482300$	$+0.03487790$
	PU-241	$+0.04871790$	$+0.13696800$	$+0.13687800$
	PU-242	$+0.00254044$	$+0.00531447$	$+0.00545129$
	Am-241	$+0.01453560$	$+0.00074055$	$+0.00077990$

presents the sensitivity coefficients for major reaction types—inelastic scattering, capture, and fission—for selected isotopes in a metal-fueled core, a MOX-fueled core at an average temperature, and a MOX-fueled core under hot conditions.

The fission cross-section sensitivity results reaffirm the dominant role of ²³⁹Pu in fast reactor systems. In all three cases, ²³⁹Pu fission exhibits the highest positive sensitivity, meaning that uncertainties in its cross-section data could lead to substantial changes in $k_{\mathrm{eff}}$. Among the three reactor configurations, the metal-fueled core shows the highest sensitivity to ²³⁹Pu fission (+0.4526), emphasizing its strong contribution to reactivity. The MOX-fueled core exhibits slightly lower sensitivities at both average (+0.4132) and hot temperatures (+0.4173), though the difference remains within a reasonable margin. Additionally, ²⁴¹Pu fission contributes positively, with a notably higher sensitivity in the MOX systems (+0.1369) compared to the metal-fueled core (+0.0487), reflecting the role of higher plutonium isotopes in MOX-fueled fast reactors.

The sensitivity of ²³⁸U capture is negative across all core configurations, consistent with its neutron-absorbing effect, which reduces $k_{\mathrm{eff}}$. This impact is most pronounced in the MOX cores (−0.2474 for average and −0.2467 for hot conditions), compared to the metal-fueled system (−0.1469). The greater magnitude of negative sensitivity in the MOX cores can be attributed to a higher fraction of ²³⁸U and the influence of a harder neutron spectrum. The slight difference between MOX (avg.) and MOX (hot) suggests that the Doppler effect and spectral variations modestly influence capture sensitivity. Similar trends are observed for ²³⁹Pu capture, which remains negative, though with a smaller magnitude relative to ²³⁸U capture.

The inelastic scattering sensitivity coefficients are generally negative, as inelastic scattering removes neutrons from the thermalization process and reduces neutron availability for fission. Among the isotopes, ²³⁸U exhibits the most substantial negative sensitivity, particularly in the MOX-fueled cases (−0.0668 for average and −0.0688 for hot), compared to −0.0633 in the metal core. The trend suggests that inelastic scattering contributes to spectral hardening, and its effects are more pronounced in MOX systems due to their plutonium content and neutron energy distribution.

The slight variations between MOX (avg.) and MOX (hot) cases indicate that temperature-dependent spectral shifts alter sensitivity coefficients, albeit marginally. The Doppler broadening effect leads to minor changes in capture and inelastic scattering coefficients, with ²³⁸U capture displaying a small decrease in magnitude at higher temperatures. However, the overall trends remain consistent, underscoring the primary role of ²³⁹Pu fission and ²³⁸U capture in determining the reactivity of these fast reactor systems.

These results highlight the most critical isotopes and reactions in the studied core configurations, reaffirming that uncertainties in ²³⁹Pu fission and ²³⁸U capture cross sections could have the greatest impact on $k_{\mathrm{eff}}$ predictions. Understanding these sensitivities is essential for improving nuclear data libraries and refining reactor physics calculations.

3.3. Energy-Dependent Sensitivity Analysis of $k_{\mathrm{eff}}$ to ²³⁹Pu Fission Cross Sections

The energy-dependent sensitivity of $k_{\mathrm{eff}}$ to Pu-239 fission cross sections provides crucial insights into how variations in fission probabilities affect neutron multiplication in different fuel configurations. Figure 2 shows three plotted distributions, each representing the sensitivity profiles for a metal-fueled core, a MOX-fueled core at an average temperature, and a MOX-fueled core under hot conditions. Each configuration exhibits distinct patterns in response to perturbations in the Pu-239 fission cross section, revealing the underlying spectral effects and temperature dependencies in fast reactor systems.

In the metal-fueled core, the sensitivity profile shows a consistently strong positive correlation, with peak values occurring in the fast neutron energy region, particularly above ${10}^4$ eV. This trend indicates that changes in the Pu-239 fission cross section significantly impact neutron multiplication, reinforcing the dominant role of Pu-239 as the primary fissile material in the system. The sensitivity remains elevated across a broad energy spectrum, gradually decreasing at higher neutron energies. The absence of thermalized neutrons in the system ensures that the sensitivity remains largely constrained to fast energy ranges, with no significant influence from lower-energy interactions.

For the MOX-fueled core at an average temperature, the sensitivity distribution retains a similar overall shape but exhibits a slight reduction in magnitude compared to the metal-fueled configuration. The spectral hardening effect in MOX fuel, due to the presence of a mixed plutonium-uranium composition, leads to a redistribution of neutrons into higher energy regions, affecting the reactivity response. Although the sensitivity remains positive throughout the fast energy range, the slight decrease suggests that additional interactions, including neutron capture and spectral moderation effects, influence the overall reactivity balance.

In the MOXHot configuration, the sensitivity profile undergoes a more pronounced transformation. The overall magnitude of sensitivity reduces significantly, with some regions exhibiting near-zero or slightly negative values. The reduction in sensitivity can be attributed to Doppler broadening effects, which increase the likelihood of resonance absorption and alter the neutron flux distribution. The spectral shift at elevated temperatures causes a redistribution of neutron interactions, reducing the dominance of ²³⁹Pu fission in determining $k_{\mathrm{eff}}$. This trend highlights the impact of temperature-dependent resonance broadening on the system’s reactivity response, demonstrating how increased thermal motion affects neutron-induced fission probabilities.

Comparing all three configurations, the energy-dependent sensitivity analysis underscores the fundamental role of ²³⁹Pu fission in sustaining reactivity in fast-spectrum systems. The transition from metal to MOX fuels introduces spectral modifications that slightly reduce sensitivity, while temperature elevation in MOXHot leads to a more significant reduction due to enhanced Doppler broadening. These observations emphasize the need for precise modeling of neutron interactions and temperature effects to ensure accurate predictions of reactivity behavior in fast reactor cores.

3.4. Energy-Dependent Sensitivity Analysis of $k_{\mathrm{eff}}$ to ²³⁸U Capture Cross Sections

The sensitivity of $k_{\mathrm{eff}}$ to the ²³⁸U capture cross section reveals the extent to which neutron absorption in ²³⁸U affects the overall neutron multiplication factor in different core configurations. The negative sensitivity values across all energy regions indicate that an increase in the ²³⁸U capture cross section leads to a decrease in $k_{\mathrm{eff}}$, as more neutrons are removed from the fission chain reaction without contributing to new fissions. Figure 3 compares three distributions, illustrating this effect in a metal-fueled core, a MOX-fueled core at an average temperature, and a MOX-fueled core under hot conditions.

In the metal-fueled core, the sensitivity profile remains negative throughout the energy spectrum, with the strongest effect occurring in the intermediate energy region, between ${10}^3$ and ${10}^5$ eV. This pattern suggests that neutron capture in ²³⁸U primarily influences reactivity through interactions occurring in the epithermal range. The gradual decrease in sensitivity magnitude at higher neutron energies reflects the reduced probability of capture reactions in the fast spectrum, where fission dominates neutron interactions. The overall sensitivity magnitude is moderate, indicating that, while ²³⁸U capture plays a role in reactivity suppression, its impact is less pronounced compared to primary fissile isotopes, such as ²³⁹Pu.

In the MOX-fueled core at an average temperature, the sensitivity profile exhibits a deeper negative trend, suggesting that ²³⁸U capture has a more substantial effect on reactivity in MOX fuel than in the metal core. The increased negative sensitivity is attributed to the spectral hardening effect in MOX fuel, where the higher proportion of plutonium leads to a greater presence of high-energy neutrons that contribute to capture interactions. The distribution retains a similar shape to the metal case but with a more pronounced depression in the intermediate energy range, emphasizing the stronger influence of ²³⁸U capture in suppressing reactivity in MOX-fueled systems.

The MOXHot configuration exhibits further variations, with an even deeper negative sensitivity in the intermediate energy region compared to the MOXAvg case. The influence of Doppler broadening at elevated temperatures leads to increased neutron capture probabilities due to the widening of resonance absorption peaks. This effect enhances the suppression of $k_{\mathrm{eff}}$ by ²³⁸U capture as evidenced by the increased magnitude of negative sensitivity. The spectral shift caused by higher temperatures results in a broader and slightly shifted sensitivity profile compared to the MOXAvg case, reinforcing the importance of temperature effects on neutron capture dynamics.

Comparing all three configurations, the energy-dependent sensitivity analysis highlights the increasing role of ²³⁸U capture in suppressing reactivity as the core transitions from metal to MOX fuel and experiences temperature-induced spectral shifts. While ²³⁸U capture is inherently a neutron-absorbing process that reduces $k_{\mathrm{eff}}$, its effect becomes more significant in MOX cores due to changes in the neutron energy distribution. The enhanced impact in MOXHot underscores the need for precise modeling of Doppler broadening effects, as resonance absorption plays a crucial role in shaping reactor behavior under varying temperature conditions.

3.5. Energy-Dependent Sensitivity Analysis of $k_{\mathrm{eff}}$ to ²³⁸U Inelastic Scattering Cross Sections

Examining the sensitivity of $k_{\mathrm{eff}}$ to the ²³⁸U inelastic scattering cross section clarifies how neutron moderation influences reactivity across different core configurations. Figure 4 shows the sensitivity coefficients for three cases—a metal-fueled core, a MOX-fueled core at average temperature, and a MOX-fueled core under hot conditions—indicating how $k_{\mathrm{eff}}$ responds in each. The results underscore the role of inelastic scattering in redistributing neutron energy and reveal its varying impact among different reactor environments.

In the metal-fueled core, the sensitivity remains close to zero over a large portion of the neutron energy spectrum, with a slight negative trend appearing in the higher energy range above ${10}^5$ eV. This indicates that inelastic scattering in ²³⁸U does not significantly alter the neutron population in the metal-fueled system but has a minor negative effect at fast neutron energies. The negative sensitivity suggests that an increase in the ²³⁸U inelastic scattering cross section leads to a slight decrease in $k_{\mathrm{eff}}$, likely due to the redistribution of fast neutrons into lower energy ranges where their contribution to fission is less effective. However, the relatively small magnitude of sensitivity values implies that the influence of inelastic scattering in this configuration is secondary compared to capture and fission reactions.

In the MOX-fueled core at an average temperature, the sensitivity distribution follows a similar pattern but exhibits a slightly more pronounced negative region. The increased spectral hardening in MOX fuel leads to a higher probability of inelastic interactions, which results in a more noticeable redistribution of neutron energy. The sensitivity remains small in magnitude but extends across a broader range of energies, particularly in the fast spectrum region. This observation suggests that inelastic scattering plays a more active role in shaping the neutron distribution in MOX fuel compared to the metal-fueled core, but its overall impact on reactivity remains limited.

In the MOXHot configuration, the sensitivity profile deepens further, showing a more distinct negative region at high neutron energies. The effect of Doppler broadening at elevated temperatures increases the probability of inelastic interactions, leading to a more significant shift in neutron energy distribution. This shift further suppresses $k_{\mathrm{eff}}$ as neutrons are scattered into less reactive energy regions. The increasing negative sensitivity at higher energies suggests that temperature-induced spectral modifications enhance the role of inelastic scattering in neutron redistribution. The broader and slightly deeper sensitivity profile compared to the MOXAvg case indicates that temperature effects amplify the influence of inelastic scattering in fast reactor systems.

Comparing all three configurations, the sensitivity analysis of ²³⁸U inelastic scattering demonstrates that while its impact on $k_{\mathrm{eff}}$ is generally modest, it becomes more relevant in MOX fuels and at higher temperatures. The increasing magnitude of negative sensitivity from metal to MOX to MOXHot highlights the importance of accurately modeling inelastic scattering effects in fast reactor simulations, particularly under varying temperature conditions. While less influential than fission and capture reactions, inelastic scattering remains a key mechanism in neutron spectral redistribution and contributes to the overall neutron energy balance in fast-spectrum reactor cores.

3.6. Reactor Design and Nuclear Data Improvement

The sensitivity analysis provides valuable insights for enhancing reactor design by identifying key nuclear data that impact core performance and safety margins. Sensitivity studies emphasize that the accuracy of ²³⁹Pu fission cross sections directly influences the precision of $k_{\mathrm{eff}}$ estimates. A reduction in uncertainty for these cross sections helps reactor designers fine-tune fuel compositions and loading patterns. As a result, designers can more effectively achieve target breeding ratios or specific reactivity levels. Significant uncertainties in ²³⁸U capture and inelastic scattering reactions often compel designers to adopt conservative design assumptions to ensure safety. By improving cross-section data for these reactions, designers can reduce unnecessary safety margins and optimize the reactor core geometry. This improvement also helps minimize fluctuations in reactivity throughout the reactor’s operational life.

Although achieving a stable $k_{\mathrm{eff}}$ is essential, other performance goals such as extending fuel burnup, achieving power distribution uniformity, and managing transient responses are equally important. Enhanced nuclear data enable more reliable simulation results, allowing designers to anticipate the reactor’s behavior under different operating conditions. Consequently, this facilitates better-informed decisions for control strategies and fuel management techniques. Fast-spectrum reactors rely heavily on intrinsic feedback mechanisms, such as Doppler broadening and coolant density changes, to maintain operational stability. A sensitivity and uncertainty analysis identifies which isotopes’ cross-section data should be refined to improve simulations of these feedback effects. The accurate modeling of these mechanisms is critical for ensuring the reactor’s safe response to power fluctuations and varying operational conditions.

Sensitivity analysis also plays a pivotal role in guiding improvements to nuclear data libraries. It highlights where experimental efforts and data evaluations should be focused to enhance reactor simulations. Large sensitivity coefficients for ²³⁹Pu fission and ²³⁸U capture reactions indicate that experimental measurements in these energy regions will have the greatest impact on reducing overall uncertainty in reactor performance predictions. Such targeted experiments are essential for refining cross-section data that directly influence SFR behavior. Benchmark experiments play a crucial role in validating nuclear data. For fast reactors, integral experiments should be designed to replicate SFR spectra and operating conditions as closely as possible. Accurate benchmarks can help identify gaps or errors in current nuclear data libraries, such as ENDF or JEFF, and provide a clear direction for improvement.

To maintain the accuracy of nuclear data libraries, it is essential to integrate new experimental results into libraries, like ENDF and JEFF. Feedback from sensitivity and uncertainty studies helps evaluators systematically compare simulations with measurements, thereby identifying discrepancies and guiding updates to reaction models and covariance data. Reducing uncertainties in nuclear data translates directly into more reliable reactor performance predictions and safety margin assessments. Lower uncertainties strengthen the technical basis for safety evaluations and facilitate the regulatory approval process for advanced reactor designs. As a result, both designers and regulators gain confidence in the accuracy and robustness of simulation-based safety cases.

4. Conclusions

A Serpent-based sensitivity analysis of $k_{\mathrm{eff}}$ in a sodium-cooled fast reactor (SFR) core, fueled with both MOX and metal fuels, provided insights into the dominant contributors to reactor reactivity. By applying Serpent’s built-in sensitivity and uncertainty module in conjunction with ENDF/B-VII.I cross-section data, it became evident that ²³⁹Pu fission plays a substantial positive role in determining $k_{\mathrm{eff}}$ for both metal and MOX cores, while ²³⁸U capture exerts a significant negative contribution under fast-spectrum conditions. An increase in fuel temperature from 743 K to 1500 K in MOX assemblies led to a drop of about 758 pcm in $k_{\mathrm{eff}}$, which corresponds to a Doppler coefficient of approximately $-1$ pcm/K. This temperature effect demonstrates the importance of resonant absorption broadening and spectral shifts in fast-reactor designs that rely on minimal moderator content.

These findings have several important implications for future SFR design and safety evaluations. The identification of ²³⁹Pu fission and ²³⁸U capture as dominant contributors to $k_{\mathrm{eff}}$ suggests that improving the accuracy of cross-section data for these isotopes should be a priority for nuclear data libraries. Reducing uncertainties in these key reactions can lead to more reliable predictions of core behavior, improved fuel composition strategies, and more accurate safety margin assessments.

Furthermore, the insights gained from this study can guide core design optimization efforts aimed at achieving better reactivity control and enhanced operational stability. For example, the higher $k_{\mathrm{eff}}$ observed in metal fuels highlights their potential advantages for achieving greater neutron economy and higher power densities. However, the greater sensitivity of MOX fuels to temperature-induced reactivity changes underscores the need for careful thermal management in MOX-fueled SFR designs.

From a safety perspective, the quantified Doppler coefficient provides a valuable indicator of negative reactivity feedback, a crucial mechanism for mitigating power excursions and ensuring safe reactor operation under transient conditions. This knowledge can inform the development of more effective reactor control systems and emergency response strategies.

Planned future work includes extending the current sensitivity analysis across a range of burnup states to investigate how isotopic evolution influences reactivity responses over time. To enhance the accuracy of Doppler feedback characterization, the Doppler coefficient will be computed using small incremental temperature changes, enabling the capture of potential non-linearities that may be obscured by broader temperature intervals. This approach will support a more precise quantification of temperature-dependent reactivity effects, which are critical to safety assessments in fast reactors. Further investigations will also focus on the systematic computation of the full covariance matrix and its decomposition into correlation matrices, enabling a more transparent understanding of how inter-reaction and inter-isotope cross-correlations propagate into core physics uncertainties. Explicit treatment of off-diagonal covariance elements will help assess their relative importance and provide improved confidence in the robustness of reactor performance predictions.

Complementary studies will examine discrete reactivity feedback mechanisms, including coolant voiding and axial expansion effects, to establish a more comprehensive view of dynamic safety behavior in sodium-cooled fast reactor (SFR) cores. Moreover, the use of alternative nuclear data libraries, such as JEFF and JENDL, will be explored to evaluate how library-specific differences influence the resulting sensitivity profiles and associated uncertainties. By integrating these expanded investigations into the reactor analysis and design workflow, this research aims to support the development of more accurate, resilient, and safety-optimized next-generation SFR concepts while systematically reducing data-driven performance uncertainties.