Simulation of the Measured Reactivity Distributions in the Subcritical MYRRHA Reactor

: The designed MYRRHA reactor, in its subcritical version, will be equipped with a set of detectors monitoring its condition by measuring the current value of negative reactivity, which is a crucial parameter for its safe operation. In subcritical systems, accurate and precise measurement of negative reactivity is disturbed by the so-called spatial effect, i.e., the response of detectors depends on their placement in the reactor core. This paper focuses on the Monte Carlo simulations of reactivity measurements using the area method for nat U, 238 U, 241 Am, 239 Pu, and 232 Th detectors. The simulations were performed in six positions with increasing distance from the center of the core and at three axial levels. The obtained results allow for selecting optimum locations for detectors and detector nuclides in terms of the accuracy of reactivity measurement and illustrate the dependence of the reactivity on the distance. Additionally, the possibility of using 103 Rh in self-powered neutron detectors was investigated. The influence of spatial effect in calculations using the area method was directly indicated in the MYRRHA reactor core for chosen isotopes and in-core positions. The results closest to true values were obtained for the second fuel assembly for 239 Pu, and the third fuel assembly for nat U, 238 U, 232 Th, and 241 Am; thus, these nuclides and positions should be preferred when selecting detectors for MYRRHA.


Introduction
In the future development of nuclear energy and advanced nuclear fuel cycles, the use of accelerator-driven systems (ADSs) is expected [1].These are subcritical reactors in which the proton accelerator uses a lead-bismuth eutectic (Pb-Bi) target to supply neutrons in spallation reaction, and, as a result, enabling continuous full-power operation.The unique feature of the ADS is its ability to use a wide variety of nuclear fuels with different fissionable isotopes [2].The system can effectively transmute and burn major and minor actinides from spent nuclear fuel and thus reduce its volume and radioactivity.In the ADS, the long-lived isotopes formed in the fuels of light water reactors (LWRs) due to neutron interactions can be easily transformed into short-lived fission products for the closure of the nuclear fuel cycle.Theoretical and experimental research on such systems has been ongoing for a long time [3].Currently, as part of the extensive international cooperation, the design of the first ADS of high power, called MYRRHA (Multi-purpose hYbrid Research Reactor for High-tech Applications), is advancing in Belgium [4,5].
ADSs operate as subcritical nuclear reactors with the external neutron source provided in the spallation reaction.The neutrons produced in the fission reaction in the nuclear fuel cannot hold a self-sustaining chain reaction without external neutrons, which is the main safety advantage.This approach allows for the usage of nuclear fuels containing minor actinides inducing deterioration of safety parameters, such as reactivity coefficients.Thus, one of the main scientific topics in ADS research is the development of methods allowing for a reliable estimation of reactor reactivity, which is crucial for the safe operation of the whole system.The best-known method for the reactivity estimation in subcritical systems is the Sjöstrand method, also known as the area method, and its variations [6][7][8].In this method, the responses (count rates) of the neutron detectors located in different positions in the reactor core are used to derive system reactivity.The main rationale for the usage of the area method is its simplicity, effectiveness, and high sensitivity in negative reactivity monitoring.In addition, the method is well known by the scientific community because it has been used for many years for negative reactivity estimation in subcritical systems.From a technical point of view, the area method demands fewer parameters to obtain reliable results compared to other methods [9].However, accurate and precise measurement of reactivity using the area method is disturbed by the so-called spatial effect.The responses of the neutron detectors located in different radial and axial positions in the reactor core give different values of reactivity.This effect is caused by different neutron spectra in different locations.The previous experimental studies under the European Commission FREYA (Fast Reactor Experiments for hYbrid Applications) experimental program confirmed the importance of this effect for reactivity estimation in ADS systems [10].
The main goal of this paper is to show possible differences between real (true) system reactivity and reactivity obtained by the area method using neutron detectors based on different isotopes and located in different positions in the reactor core.The final results quantify the differences between true values of the effective neutron multiplication factor and values of the effective neutron multiplication factor estimated using the response of detectors.The effective neutron multiplication factor (k) by definition is the ratio between a number of neutrons produced by fission in one generation to the number of neutrons lost in absorption and leakage in the preceding generation.The differences between both parameters directly show possible deviations in the response of neutron detectors.The developed method is based on the numerical approach, allowing for further introduction of the correction factors for real reactivity measurement during MYRRHA operation [11].
Except for the indicated scientific topic, the research on the MYRRHA reactor focuses on many scientific and technical aspects considering the construction of a full-scale demonstrator reactor.The papers mentioned below show the major mainstreams in the design of the MYRRHA system and prove that the concept of the reactor is a valuable research topic for many scientific teams.In the paper by Romojaro et al., the importance of nuclear data and data assimilation in the design of the MYRRHA system is shown [12].The authors identified the areas for improvement in nuclear data for reaching target accuracies in the integral reactor core parameters, especially in the effective neutron multiplication factor.The in-core management of the fuel assemblies in the MYRRHA core was investigated by Jaluvka et al. [13].A novel numerical tool was developed and used for loading pattern optimization with the estimation of crucial core parameters like neutron flux and k eff .The possibility of using americium-bearing fuel rods for their incineration in the MYRRHA core was investigated by Luzzi et al. [14].The authors presented the thermal behavior of fuel rods and the effects related to helium production during irradiation.The behavior of Mixed OXide fuel (MOX), which is the main fuel foreseen for MYRRHA, was presented in the paper by Magni [15].The response of the fuel pin during over-power transient irradiation conditions was investigated at the beginning and at the end of the irradiation.Magni et al. also showed the behavior of the hottest fuel pin under standard operational conditions using the TRANSURANUS code [16].The paper by Rummana et al. presents the possibility of thorium fuel usage in the MYRRHA core [17].The authors focused on the calculations of neutron fluxes and isotopic changes in nuclear fuel using two independent numerical tools-MCNPX (Monte Carlo N-Particle Transport Code) and GEANT4 (GEometry ANd Tracking).Preston et al. performed numerical simulations for the estimation of the gammaray and neutron emission rates from the MYRRHA spent fuel assemblies and compared them with the emission rates from PWR (pressurized water reactor) fuel assemblies [18].The depletion calculations help to determine techniques for further radiological verification of the spent nuclear fuel using non-destructive assay techniques.In addition, Preston et al. developed the model of a gamma spectroscopy measurement station for MYRRHA spent nuclear fuel using MCNP6.2code [19].
Section 2 presents the description of the applied numerical model of the MYRRHA reactor, the theoretical base of the applied methodology, and the developed numerical approach.Section 3 shows and analyzes the obtained results, i.e., the difference between true and simulated experimental effective neutron multiplication factors for detector isotopes and their position in the core.The discussion about the results and follow-up research is provided in Section 4. This study is concluded in Section 5.

MYRRHA Numerical Model
The MYRRHA reactor consists of a tank filled with liquid Pb-Bi eutectic coolant, fuel assemblies containing nuclear fuel, and a surrounding neutron reflector; see Figure 1.The tank houses elements of the reactivity control system for normal operation and the protection system for emergency shutdown.Parts of both systems are neutron detectors located in the core and the reflector.
help to determine techniques for further radiological verification of the spent nuclear fuel using non-destructive assay techniques.In addition, Preston et al. developed the model of a gamma spectroscopy measurement station for MYRRHA spent nuclear fuel using MCNP6.2code [19].
Section 2 presents the description of the applied numerical model of the MYRRHA reactor, the theoretical base of the applied methodology, and the developed numerical approach.Section 3 shows and analyzes the obtained results, i.e., the difference between true and simulated experimental effective neutron multiplication factors for detector isotopes and their position in the core.The discussion about the results and follow-up research is provided in Section 4. This study is concluded in Section 5.

MYRRHA Numerical Model
The MYRRHA reactor consists of a tank filled with liquid Pb-Bi eutectic coolant, fuel assemblies containing nuclear fuel, and a surrounding neutron reflector; see Figure 1.The tank houses elements of the reactivity control system for normal operation and the protection system for emergency shutdown.Parts of both systems are neutron detectors located in the core and the reflector.The homogeneous geometry numerical model MYRRHA FASTEF (FAst Spectrum Transmutation Experimental Facility) applied for the Monte Carlo neutron transport simulations was developed in line with the design of the MYRRHA system [20][21][22].The indicated model was used because it is well described in the scientific literature, which allowed for its adaptation to the presented research.The reactor core consists of 151 positions for assemblies arranged in a hexagonal pattern; see Figure 2. The center of the core contains a spallation source, six oin-pile sections (IPS) for irradiations of material samples, six B4C control rods, and three B4C shutdown rods.The B4C used for neutronabsorbing material contains 90% of 10 B. The elements are placed in the core, between 69 fuel assemblies.The central part of the core is surrounded by 24 Pb-Bi dummies for reduction of neutron leakage and 42 assemblies similar to the fuel assemblies but filled with yttrium zirconium oxide (YZrO) pellets for the protection of structural elements against fast neutrons.For the reason of numerical simulation, the non-homogeneous structure of fuel assemblies containing detailed geometry of fuel rods was implemented [23].Each fuel assembly contains 127 MOX fuel rods with an active high of 65 cm and a pellet diameter of 0.54 cm.The fuel pins are fixed inside a hexagonal ferritic martensitic steel T91 wrapper with a thickness of 2 mm.The gap between fuel assemblies is 3 mm and was designed for fuel handling purposes.The pitch of the fuel rod equals 8.4 mm.The The homogeneous geometry numerical model MYRRHA FASTEF (FAst Spectrum Transmutation Experimental Facility) applied for the Monte Carlo neutron transport simulations was developed in line with the design of the MYRRHA system [20][21][22].The indicated model was used because it is well described in the scientific literature, which allowed for its adaptation to the presented research.The reactor core consists of 151 positions for assemblies arranged in a hexagonal pattern; see Figure 2. The center of the core contains a spallation source, six oin-pile sections (IPS) for irradiations of material samples, six B 4 C control rods, and three B 4 C shutdown rods.The B 4 C used for neutron-absorbing material contains 90% of 10 B. The elements are placed in the core, between 69 fuel assemblies.The central part of the core is surrounded by 24 Pb-Bi dummies for reduction of neutron leakage and 42 assemblies similar to the fuel assemblies but filled with yttrium zirconium oxide (YZrO) pellets for the protection of structural elements against fast neutrons.For the reason of numerical simulation, the non-homogeneous structure of fuel assemblies containing detailed geometry of fuel rods was implemented [23].Each fuel assembly contains 127 MOX fuel rods with an active high of 65 cm and a pellet diameter of 0.54 cm.The fuel pins are fixed inside a hexagonal ferritic martensitic steel T91 wrapper with a thickness of 2 mm.The gap between fuel assemblies is 3 mm and was designed for fuel handling purposes.The pitch of the fuel rod equals 8.

Theoretical Approach
The main objective of the performed research was to check the accuracy of reactivity measurements using the area method proposed by Sjöstrand and improved by Talamo [24][25][26][27].The method is based on the measurement of signal from the neutron detectors (count rates) after a short proton beam pulse, which leads to the emission of neutrons from the Pb-Bi target.Immediately after the pulse neutron flux increases, it reaches the maximum and undergoes exponential decay.The flux decays to the value representing the equilibrium generation of delayed neutrons in the fuel.Figure 3 shows the visualization of the typical neutron count distribution after pulse.The ratio of the prompt (AP) to delay (AD) neutron area is proportional to the ratio of reactivity (ρ) to delay neutron fraction (βeff); see Equation (1) [7,28].The effective neutron multiplication factor is then obtained by modifying Equation (1) to Equation ( 2

Theoretical Approach
The main objective of the performed research was to check the accuracy of reactivity measurements using the area method proposed by Sjöstrand and improved by Talamo [24][25][26][27].The method is based on the measurement of signal from the neutron detectors (count rates) after a short proton beam pulse, which leads to the emission of neutrons from the Pb-Bi target.Immediately after the pulse neutron flux increases, it reaches the maximum and undergoes exponential decay.The flux decays to the value representing the equilibrium generation of delayed neutrons in the fuel.Figure 3 shows the visualization of the typical neutron count distribution after pulse.The ratio of the prompt (A P ) to delay (A D ) neutron area is proportional to the ratio of reactivity (ρ) to delay neutron fraction (β eff ); see Equation (1) [7,28].The effective neutron multiplication factor is then obtained by modifying Equation (1) to Equation (2) using the standard reactivity definition ρ = (k eff − 1)/k eff .The effective delayed neutron fraction β eff is obtained from Monte Carlo numerical simulations.
A The areas for prompt and delayed neutrons are obtained in numerical simulations fission or radiative capture reaction rates for particular detectors' isotopes in the tim function by MCNP6.2 software on the basis of the following theoretical approach: T total area underneath the curve (A = AP + AD) after neutron pulse in period T proportional to the chosen reaction rate (r) at delayed neutron equilibrium for particu detectors and finally to the neutron detector count rates, which is shown in Equation ( The first integral in the Equation ( 3) is equal to the integral on reaction rates when t delayed neutrons are not at equilibrium (r1).This integral finally can be split into tw integrals representing prompt and delayed neutron areas.However, after the neutr pulse, some delayed neutrons correspond to area AP and prompt to area AD-hence correction should be introduced.

𝐴 𝐴 𝐴 𝑟 𝑡 𝑑𝑡 𝑟 𝑡 𝑑𝑡 𝑟 𝑡 𝑑𝑡 𝑟 𝑡 𝑑𝑡
The numerical setup for calculations of both areas should take into account t reaction rates at the delayed neutron equilibrium state (r).Therefore, the new approa defined in Equation ( 4) was derived by Talamo and used for calculations presented in th paper [27].The reaction rates of delayed neutrons not at equilibrium (r1) are normaliz to the period T and added to the total reaction rate for a single pulse (r1(t)), which giv the reaction rate at delayed neutron equilibrium.This approach is a fast and efficient w to calculate the so-called pulse superimposition methodology [29].Practically, t approach follows the introduction of the corrections based on the period T to the AP a AD, which add contribution from the prompt to the delayed neutron areas a contribution of delayed neutrons to the prompt neutron area and thus correspondi reaction rates.

Calculations
The Monte Carlo neutron transport simulations were performed using the Mon Carlo N-Particle Transport Code (MCNP6.2) with the MYRRHA core loaded with fre fuel assemblies.In order to obtain sufficient precision of the results, source neutron puls with a width of 1 µs were used, producing a total of 2 × 10 7 fission neutrons.T experimental results, carried out using the area method, were simulated by determini the neutron flux density time distributions in the positions selected for neutron detecto Then, using the JEFF3.3(Joint Evaluated Fission and Fusion) cross-section libraries The areas for prompt and delayed neutrons are obtained in numerical simulations of fission or radiative capture reaction rates for particular detectors' isotopes in the time function by MCNP6.2 software on the basis of the following theoretical approach: The total area underneath the curve (A = A P + A D ) after neutron pulse in period T is proportional to the chosen reaction rate (r) at delayed neutron equilibrium for particular detectors and finally to the neutron detector count rates, which is shown in Equation (3).The first integral in the Equation ( 3) is equal to the integral on reaction rates when the delayed neutrons are not at equilibrium (r 1 ).This integral finally can be split into two integrals representing prompt and delayed neutron areas.However, after the neutron pulse, some delayed neutrons correspond to area A P and prompt to area A D -hence, a correction should be introduced.
The numerical setup for calculations of both areas should take into account the reaction rates at the delayed neutron equilibrium state (r).Therefore, the new approach defined in Equation ( 4) was derived by Talamo and used for calculations presented in this paper [27].The reaction rates of delayed neutrons not at equilibrium (r 1 ) are normalized to the period T and added to the total reaction rate for a single pulse (r 1 (t)), which gives the reaction rate at delayed neutron equilibrium.This approach is a fast and efficient way to calculate the so-called pulse superimposition methodology [29].Practically, the approach follows the introduction of the corrections based on the period T to the A P and A D , which add contribution from the prompt to the delayed neutron areas and contribution of delayed neutrons to the prompt neutron area and thus corresponding reaction rates.

Calculations
The Monte Carlo neutron transport simulations were performed using the Monte Carlo N-Particle Transport Code (MCNP6.2) with the MYRRHA core loaded with fresh fuel assemblies.In order to obtain sufficient precision of the results, source neutron pulses with a width of 1 µs were used, producing a total of 2 × 10 7 fission neutrons.The experimental results, carried out using the area method, were simulated by determining the neutron flux density time distributions in the positions selected for neutron detectors.Then, using the JEFF3.3(Joint Evaluated Fission and Fusion) cross-section libraries of neutron fission and radiative capture reactions, the rates of these reactions were determined for six selected detector materials ( nat U, 238 U, 239 Pu, 241 Am, 232 Th, and 103 Rh) and for three time intervals (0-0.8 s, 0.8-1.0s, and 1.0-1.0× 10 5 s).The choice of detector materials for quantitative analysis was based mainly on the evaluations of the fission cross-sections.The three types of nuclides with high fission cross-sections in the specified energy ranges were proposed: 239 Pu for thermal neutrons, nat U and 241 Am for resonance neutrons, and 238 U and 232 Th for fast neutrons, where the fission cross-section is determined by threshold energy.The foreseen measurement method is a real-time electronic detector based on a fission chamber (ionization caused by fission products following the fission reaction).In this case, the fission reaction rates are calculated for nat U, 238 U, 239 Pu, 241 Am, and 232 Th isotopes.In addition, for the measurements in the very high neutron fluxes, not suitable for the fission chamber, the 103 Rh detector was proposed.In this case, the ionizing radiation emitted after neutron radiative capture is used for electric signal generation, so the radiative capture reaction rates were calculated [30].The discretization of the reaction rates for three time intervals is necessary for the application of the improved area method.In order to increase the statistics of calculations, the detectors were placed concentrically in groups of six, around the vertical axis and at increasing distances from the center of the reactor core in sets of assemblies I-VI; see Figure 2. The fragments of fuel assemblies at three levels, measured from the plane of vertical symmetry of the system (level zero), are considered as detector volumes; see Figure 4. Therefore, one set of simulations contained 108 individual runs (6 × 6 × 3).Finally, based on the obtained reaction rates, the effective neutron multiplication factor was calculated using the area method (Equation ( 2)).It is worth noting that the mass of the fissionable isotope in the detector affects the signal from the detector.However, the value of the measured reactor reactivity is obtained from the ratio of two quantities (areas) from the same detector, and as a relative one, it does not depend on this mass.The uncertainty of the determined reactivity is lower at a higher signal from the detector.
neutron fission and radiative capture reactions, the rates of these reactions we determined for six selected detector materials ( nat U, 238 U, 239 Pu, 241 Am, 232 Th, and 103 Rh) a for three time intervals (0-0.8 s, 0.8-1.0s, and 1.0-1.0× 10 5 s).The choice of detect materials for quantitative analysis was based mainly on the evaluations of the fissi cross-sections.The three types of nuclides with high fission cross-sections in the specifi energy ranges were proposed: 239 Pu for thermal neutrons, nat U and 241 Am for resonan neutrons, and 238 U and 232 Th for fast neutrons, where the fission cross-section determined by threshold energy.The foreseen measurement method is a real-tim electronic detector based on a fission chamber (ionization caused by fission produ following the fission reaction).In this case, the fission reaction rates are calculated for nat 238 U, 239 Pu, 241 Am, and 232 Th isotopes.In addition, for the measurements in the very hi neutron fluxes, not suitable for the fission chamber, the 103 Rh detector was proposed.this case, the ionizing radiation emitted after neutron radiative capture is used for elect signal generation, so the radiative capture reaction rates were calculated [30].T discretization of the reaction rates for three time intervals is necessary for the applicati of the improved area method.In order to increase the statistics of calculations, t detectors were placed concentrically in groups of six, around the vertical axis and increasing distances from the center of the reactor core in sets of assemblies I-VI; s Figure 2. The fragments of fuel assemblies at three levels, measured from the plane vertical symmetry of the system (level zero), are considered as detector volumes; s Figure 4. Therefore, one set of simulations contained 108 individual runs (6 × 6× 3).Final based on the obtained reaction rates, the effective neutron multiplication factor w calculated using the area method (Equation ( 2)).It is worth noting that the mass of t fissionable isotope in the detector affects the signal from the detector.However, the val of the measured reactor reactivity is obtained from the ratio of two quantities (areas) fro the same detector, and as a relative one, it does not depend on this mass.The uncertain of the determined reactivity is lower at a higher signal from the detector.The measure of the accuracy of the reactivity measurement is defined as t difference between the experimentally measured neutron multiplication factor for a giv detector position (kexp) and its true value (kkcode).The effective neutron multiplication fact The measure of the accuracy of the reactivity measurement is defined as the difference between the experimentally measured neutron multiplication factor for a given detector position (k exp ) and its true value (k kcode ).The effective neutron multiplication factor obtained in calculations using the KCODE mode of the MCNP6.2code (k kcode ) is considered as a value close to the true value [25].For the adopted numerical model of the MYRRHA system, the reference values of k kcode and β eff obtained in MCNP6.2 modeling are 0.95636 ± 0.00002 and 335.0 ± 2.0 pcm (percent mille, one-thousandth of a percent), respectively.These values Energies 2024, 17, 2565 7 of 15 are quite close to those published by other authors, especially by Romojaro et al. [31]; see Table 1.The obtained k kcode and β eff values were fixed for further evaluation of differences (∆ = k exp − k kcode ), which are considered to be the final result.The difference straightforwardly quantifies the magnitude of the spatial effect for each position and detector.Figure 5 presents the flowchart of the developed method.-320 [33] 0.95944 ± 0.00018 - [34] system, the reference values of kkcode and βeff obtained in MCNP6.2 modeling are 0.956 0.00002 and 335.0 ± 2.0 pcm (percent mille, one-thousandth of a percent), respectiv These values are quite close to those published by other authors, especially by Romo et al. [31]; see Table 1.The obtained kkcode and βeff values were fixed for further evalua of differences (Δ = kexp − kkcode), which are considered to be the final result.The differe straightforwardly quantifies the magnitude of the spatial effect for each position detector.Figure 5 presents the flowchart of the developed method.

Spatial Effect
Table 2 shows the results obtained for five isotopes chosen for the fission cham neutron detectors.For a given isotope, statistically equal results were obtained for symmetrical horizontal positions, and average values were adopted.The analysis focu on the dependence of the differences in the isotope, radial, and axial positions in the c

Spatial Effect
Table 2 shows the results obtained for five isotopes chosen for the fission chamber neutron detectors.For a given isotope, statistically equal results were obtained for six symmetrical horizontal positions, and average values were adopted.The analysis focuses on the dependence of the differences in the isotope, radial, and axial positions in the core.The lowest difference ∆ was obtained for the 239 Pu detector located in the second assembly at the top or central axial level, where k exp is underestimated at about 40 pcm.The largest difference was obtained for 238 U and 232 Th detectors located in the first assembly at the bottom level.In this case, k exp is underestimated at about 1110 pcm.The foregoing results for 239 Pu, 238 U, and 232 Th detectors also state the lowest and highest underestimation of k exp .On the contrary, the lowest overestimation of about 76.5 pcm corresponds to the 238 U and 232 Th detectors located in the third assembly at the central level, and the highest of about 269 pcm corresponds to the 239 Pu located in the sixth fuel assembly at the central level.In general, the maximal underestimation is about four times larger than the maximal overestimation.
The general trend shows that in the case of bottom detectors, both negative (assemblies I-II) and positive differences (assemblies III-VI) are lower than in the case of central and top detectors.Despite the fact that the absolute value of the differences depends on the isotope and radial position, it was directly observed that the lowest differences for all axial levels were obtained for all isotopes in the third assembly.The difference in this assembly does not exceed 5 pcm and corresponds to the simulation uncertainty.The highest absolute differences were obtained for all isotopes in the first assembly.
Figures 6-10 show the variability of ∆ with the distance of the detector from the center for all isotopes and axial levels.The monotonic increase from negative values in the center (lower than k kcode ) to positive (higher than k kcode ) was observed.This tendency does not depend on the detector isotope.The final positive value of the difference in the most peripheral sixth fuel assembly equals on average about 250 pcm.However, the initial negative values in most inner fuel assemblies can be classified into three groups.The underestimation for 239 Pu is 385 pcm, 915 pcm for nat U and 241 Am on average, and 1057 pcm for 238 U and 232 Th on average.show the variability of Δ with the distance of the detector from the center for all isotopes and axial levels.The monotonic increase from negative values in the center (lower than kkcode) to positive (higher than kkcode) was observed.This tendency does not depend on the detector isotope.The final positive value of the difference in the most peripheral sixth fuel assembly equals on average about 250 pcm.However, the initial negative values in most inner fuel assemblies can be classified into three groups.The underestimation for 239 Pu is 385 pcm, 915 pcm for nat U and 241 Am on average, and 1057 pcm for 238 U and 232 Th on average.the variability of Δ with the distance of the detector from the center for all isotopes and axial levels.The monotonic increase from negative values in the center (lower than kkcode) to positive (higher than kkcode) was observed.This tendency does not depend on the detector isotope.The final positive value of the difference in the most peripheral sixth fuel assembly equals on average about 250 pcm.However, the initial negative values in most inner fuel assemblies can be classified into three groups.The underestimation for 239 Pu is 385 pcm, 915 pcm for nat U and 241 Am on average, and 1057 pcm for 238 U and 232 Th on average.

Self-Powered 103 Rh Detectors
Online monitoring of the subcritical system using self-powered neutron detectors (SPND) with immediate operation was postulated by Gandini [35].Taking into account the expected strong neutron fluxes near the center of the MYHRRA reactor core-too strong for the fission chambers-the preliminary simulations of the response of 103 Rh detectors, arranged in the same way as the fission chambers discussed in the previous paragraphs, was performed.The detector responses were approximated as the product of

Self-Powered 103 Rh Detectors
Online monitoring of the subcritical system using self-powered neutron detectors (SPND) with immediate operation was postulated by Gandini [35].Taking into account the expected strong neutron fluxes near the center of the MYHRRA reactor core-too strong for the fission chambers-the preliminary simulations of the response of 103 Rh detectors, arranged in the same way as the fission chambers discussed in the previous paragraphs, was performed.The detector responses were approximated as the product of the neutron flux density and the radiative capture reaction cross-section (n, γ).Such an approach does not take into account further phenomena occurring in the detector, such as the emitted particle type, intensity and spectrum, and the effect related to the decay of isotopes produced in the detector materials [36].Nevertheless, the (n, γ) reaction rates and their uncertainties were calculated and, as a result, the assessment of the ∆ = k exp − k kcode values and their uncertainties was performed.After statistical processing of the raw results from all detector positions, an estimate of the distribution of simulated responses using the area method was obtained and presented in Table 3 and Figure 11.It can be seen that the results for the assemblies furthest from the center (VI) present high uncertainty of about 120 pcm and are less reliable.However, towards the center of the system for the assemblies I-V, the uncertainties are lower, and the results approach the distributions foreseen for the fission chambers.It can be seen that the shapes of the distributions in both cases are quite similar.The maximum negative differences of about 150 pcm were obtained for the first assembly and the maximum positive differences of about 200-260 pcm were obtained for the assemblies IV-V.Similarly to the fission chamber, the dependence of the difference on the axial level was observed, but its scale and tendency are difficult to assess due to quite high uncertainties.Generally, the results closest to the true value of the effective neutron multiplication factor were obtained for the second assembly.

Discussion
The values of kexp obtained using the area method depend on the reaction rates calculated in numerical Monte Carlo simulations.The reaction rates in turn depend on the neutron fluxes for the given position in the core, calculated by tallying procedures.However, the absolute value of neutron flux and its spectra depends on the surroundings

Discussion
The values of k exp obtained using the area method depend on the reaction rates calculated in numerical Monte Carlo simulations.The reaction rates in turn depend on the neutron fluxes for the given position in the core, calculated by tallying procedures.However, the absolute value of neutron flux and its spectra depends on the surroundings of the detector.Therefore, the next stage of the numerical analysis will focus on the calculations considering the changing environment around neutron detectors.In particular, the influence of the strong neutron-absorbing materials will be investigated for the assemblies III and IV, which are located near B 4 C rods.Additionally, the insertion level of the control rods affects the response of the detectors located at different axial levels.The influence of the less absorbing elements like Pb-Bi, YZrO dummies, and IPS will also be checked.The of such an analysis will follow the creation of the response matrix defining the detector's responses in the function of geometrical configuration around the detector.Part of the calculations will take into consideration fuel burnup and associated changes in isotopic fuel composition and thus in neutron flux and spectrum as well as in other core parameters [37,38].This approach will help to estimate reactivity correction factors for a given core configuration and irradiation time and may be part of an online reactivity control and monitoring system.The preliminary calculations were already performed.The configuration of the MYRRHA core was changed in such a way that the detectors in each row were surrounded only by fuel assemblies.Such a core configuration is not foreseen for a real reactor but allows for the investigation of the indicated effect.The preliminary results show the maximal reduction of differences of about 10-15% for 232 Th detectors located in the fifth assembly compared with the difference obtained in reference calculations for the original core configuration, which proves the existence and importance of the effect.
The additional analysis will consider the investigation of new isotopes suitable for fission chamber detectors and self-power neutron detectors (e.g., 27 Al).The type of detector determines the type of nuclear reaction used for response estimation, i.e., either radiative capture or fission.The intensity of the particular reaction depends on its cross-section for the given neutron energy.Thus, the strongest response of the detector is expected for the location where the spectrum fits the highest cross-section ranges and finally gives the highest reaction rates.Hence, a detailed analysis of the neutron spectra for given positions in the reactor core is foreseen.The neutron spectra will be calculated for energy group structure with many intervals (e.g., 100), which will additionally indicate the best nuclide for detector material.This approach, however, requires more complex calculations with larger discretization of the MYRRHA fuel assemblies in individual cells for neutron detectors.Moreover, the detailed geometry and material composition of the detectors should be proposed and included in the numerical model.This will allow for the estimation of the more precise detector's response for the area method and estimation of k exp .One of the interesting scientific topics may also be a feasibility study on applying monocrystalline diamond neutron detectors for negative reactivity measurements using the area method [39].Such calculations are planned for further research.
Finally, the quantified differences in reactivity should be correlated to the difference between the real reactivity measurements in the MYRRHA reactor prototype.However, the MYRRHA reactor is under development, and construction has not started yet; thus, the validation of the reactivity monitoring system at the current stage is not possible, but of course is planned in the future.

Conclusions
Based on the results obtained using the developed methodology for the estimation of the difference between the simulated experimental and the true value of effective neutron multiplication factors, the following conclusions can be made: (a) The k kcode as well as β eff obtained in MCNP6.2 modeling are quite close to the values reported by other teams working on MYRRHA subcritical reactors, which proves the reliability of the developed numerical model.
(b) The influence of the spatial effect in calculations of k exp using the area method was directly indicated in the MYRRHA reactor core for the chosen isotopes and in-core positions.(c) The magnitude of the spatial effect depends on the detector's isotope as well as its radial position in the core.The influence on the axial position in the fuel is less significant.(d) The distributions of ∆ in the MYRRHA reactor model range from negative values of about 1000-1100 pcm for 238 U and 232 Th and 400 pcm for 239 Pu, close to the center of the core, to positive values of about 200-300 pcm for all nuclides, on the core periphery.(e) The results (differences) closest to zero were obtained for assembly II, about 40-50 pcm for 239 Pu, and assembly III, about 50-100 pcm for nat U, 238 U, 232 Th, and 241 Am; thus, these nuclides and positions should be preferred when selecting detectors for MYRRHA.(f) Pilot results show that it should be possible to use self-powered neutron detectors based on 103 Rh for real negative reactivity measurements.

Figure 1 .
Figure 1.Cross-sections of the MYRRHA reactor tank, reactor core, and fuel assembly.

Figure 1 .
Figure 1.Cross-sections of the MYRRHA reactor tank, reactor core, and fuel assembly.
4 mm.The fuel pin cladding is made of DIN (Deutsche Institut für Normung) 1.4970-austenitic stainless steel stabilized by 15-15Ti with a thickness of Energies 2024, 17, 2565 4 of 15 0.45 mm.The gap between the fuel pellet and cladding is filled with helium.The MOX fuel contains 30% of plutonium dioxide PuO 2 and 70% of natural uranium dioxide UO 2 , by weight.The reactor core is placed in the reactor vessel filled with the Pb-Bi eutectic.The fission neutron source was numerically placed on the side (cylindrical) and frontal (front and rear) surfaces of the spallation target.Energies 2024, 17, x FOR PEER REVIEW 4 of 16 fuel pin cladding is made of DIN (Deutsche Institut für Normung) 1.4970-austenitic stainless steel stabilized by 15-15Ti with a thickness of 0.45 mm.The gap between the fuel pellet and cladding is filled with helium.The MOX fuel contains 30% of plutonium dioxide PuO2 and 70% of natural uranium dioxide UO2, by weight.The reactor core is placed in the reactor vessel filled with the Pb-Bi eutectic.The fission neutron source was numerically placed on the side (cylindrical) and frontal (front and rear) surfaces of the spallation target.

Figure 2 .
Figure 2. Horizontal cross-section of the MYRRHA reactor core with concentric positions foreseen for neutron detectors (I-VI).The blue color around the core indicates the Pb-Bi eutectic.The numbers in brackets show the number of elements.
) using the standard reactivity definition ρ = (keff − 1)/keff.The effective delayed neutron fraction βeff is obtained from Monte Carlo numerical simulations.

Figure 2 .
Figure 2. Horizontal cross-section of the MYRRHA reactor core with concentric positions foreseen for neutron detectors (I-VI).The blue color around the core indicates the Pb-Bi eutectic.The numbers in brackets show the number of elements.

Energies 2024 ,Figure 3 .
Figure 3. Visualization of the typical d-normalized neutron count distribution after pulse.

Figure 3 .
Figure 3. Visualization of the typical d-normalized neutron count distribution after pulse.

Figure 4 .
Figure 4. Vertical cross-section of the central channel and two adjacent fuel assemblies.T boundaries of the detector areas (black horizontal lines) and their sizes are marked on the right.

Figure 4 .
Figure 4. Vertical cross-section of the central channel and two adjacent fuel assemblies.The boundaries of the detector areas (black horizontal lines) and their sizes are marked on the right.

Figure 5 .
Figure 5. Graphical representation of the applied method.

Figure 5 .
Figure 5. Graphical representation of the applied method.

Figure 6 .
Figure 6.Variation in Δ values for nat U detectors placed in fuel assemblies at six distances from the center on three vertical levels.

Figure 7 .
Figure 7. Variation in Δ values for 238 U detectors placed in fuel assemblies at six distances from the center on three vertical levels.

Figure 6 .
Figure 6.Variation in ∆ values for nat U detectors placed in fuel assemblies at six distances from the center on three vertical levels.

Figures 6- 10
show the variability of Δ with the distance of the detector from the center for all isotopes and axial levels.The monotonic increase from negative values in the center (lower than kkcode) to positive (higher than kkcode) was observed.This tendency does not depend on the detector isotope.The final positive value of the difference in the most peripheral sixth fuel assembly equals on average about 250 pcm.However, the initial negative values in most inner fuel assemblies can be classified into three groups.The underestimation for 239 Pu is 385 pcm, 915 pcm for nat U and 241 Am on average, and 1057 pcm for 238 U and 232 Th on average.
show the variability of Δ with the distance of the detector from the center for all isotopes and axial levels.The monotonic increase from negative values in the center (lower than kkcode) to positive (higher than kkcode) was observed.This tendency does not depend on the detector isotope.The final positive value of the difference in the most peripheral sixth fuel assembly equals on average about 250 pcm.However, the initial negative values in most inner fuel assemblies can be classified into three groups.The underestimation for 239 Pu is 385 pcm, 915 pcm for nat U and 241 Am on average, and 1057 pcm for 238 U and 232 Th on average.

Figure 6 .
Figure 6.Variation in Δ values for nat U detectors placed in fuel assemblies at six distances from the center on three vertical levels.

Figure 7 .
Figure 7. Variation in Δ values for 238 U detectors placed in fuel assemblies at six distances from the center on three vertical levels.

Figure 7 .
Figure 7. Variation in ∆ values for 238 U detectors placed in fuel assemblies at six distances from the center on three vertical levels.

Figure 8 .
Figure 8. Variation in Δ values for 241 Am detectors placed in fuel assemblies at six distances from the center on three vertical levels.

Figure 9 .
Figure 9. Variation in Δ values for 239 Pu detectors placed in fuel assemblies at six distances from the center on three vertical levels.

Figure 8 . 16 Figure 8 .
Figure 8. Variation in ∆ values for 241 Am detectors placed in fuel assemblies at six distances from the center on three vertical levels.

Figure 9 .
Figure 9. Variation in Δ values for 239 Pu detectors placed in fuel assemblies at six distances from the center on three vertical levels.

Figure 9 . 16 Figure 10 .
Figure 9. Variation in ∆ values for 239 Pu detectors placed in fuel assemblies at six distances from the center on three vertical levels.Energies 2024, 17, x FOR PEER REVIEW 11 of 16

Figure 10 .
Figure 10.Variation in ∆ values for 232 Th detectors placed in fuel assemblies at six distances from the center on three vertical levels.

Figure 11 .
Figure 11.Variation in Δ values for 103 Rh SPND detectors placed in fuel assemblies at six distances from the center on three vertical levels.

Figure 11 .
Figure 11.Variation in ∆ values for 103Rh SPND detectors placed in fuel assemblies at six distances from the center on three vertical levels.

Table 1 .
The literature values of k kcode and β eff .

Table 1 .
The literature values of kkcode and βeff.

Table 2 .
Average values of Δ obtained in numerical simulations.

Table 2 .
Average values of ∆ obtained in numerical simulations.

Table 3 .
The average values of ∆ for 103 Rh detectors.