Corresponding author: Daniil M. Arkhangelsky (

Academic editor: Georgy Tikhomirov

The paper considers a computational study of the importance function effect on the accuracy of calculating the effective fraction of delayed neutrons, β_{eff}, and generation time of instantaneous neutrons using the MCU Monte Carlo code based on the example of three criticality experiments from the ICSBEP handbook. In the MCU code, the importance function has a piecewise constant form: the computational model is broken down into a finite number of registration objects, and the neutron importance is calculated in each. The obtained importance values are used then to calculate the kinetic functionals due to which the calculation accuracy for the latter depends on the resolution. Three types of the importance function spatial partition (axial, radial, combined) have been studied. The numerical simulation results have shown that the axial component of the neutron importance function in all experiments has practically no effect on the calculation accuracy for β_{eff}_{eff} estimate. Using combined partition, as compared with radial partition, improves the calculation accuracy insignificantly (< 1%).

Arkhangelsky DM, Daichenkova YuS, Kalugin MA, Oleynik DS, Shkarovsky DA (2023) The effect of the importance function resolution on the accuracy of calculating the functionals of the neutron kinetics in water critical assemblies by Monte Carlo method. Nuclear Energy and Technology 9(3): 171–175.

Presently, the Monte Carlo method has been increasingly often used for solving transport problems in nuclear reactor design (

Critical in this respect is to improve the accuracy of estimating the neutron kinetics functionals: effective fraction of delayed neutrons, β_{eff}_{eff}

This paper looks into the effect of the importance function resolution on the accuracy of β_{eff} and Λ estimations.

As part of the study, experiments were considered involving low-enriched uranium in composite systems with a thermal spectrum conducted on light-water critical assemblies, ZR-6 (LCT-015) and Stend P (LCT-053 and LCT-085) from the ICSBEP Handbook (

The paper presents a brief description of the algorithms and methods offered by the MCU code, a description of computational models, and an analysis of the results obtained.

The MCU code calculates Λ and β_{eff}

Λ = (^{+}^{+}

β_{eff} = (^{+}^{+}

where ^{+} is the generation density adjoint function which has the meaning of neutron importance (

However, it is difficult to obtain both these functions in a continuous form using the Monte Carlo method. The computational model is therefore broken down into a finite number of registration objects, that is, a set of points in a phase space in which the required functions are registered. Thus, a conditionally critical problem can be presented in a matrix form

_{eff}

where ^{th} component of which fits the neutron generation density in the respective object, and _{ij}

The neutron importance function can be obtained by solving the equation adjoint with (3)

_{eff}^{+} = ^{T}ψ^{+}, (4)

where the transposed matrix ^{T} has the meaning of an adjoint operator.

Therefore, the matrix of fissions, ^{+}, a neutron importance function partitioned by registration objects.

The obtained values are used then to obtain estimates (1) and (2) that depend on the importance function resolution.

The evaluation of the new methodology undertaken in (

This paper deals with light-water critical facilities, ZR-6 (LCT-015) and Stend P (LCT-053 and LCT-085) (see the ICSBEP Handbook).

The Stend P experimental facility represents a triangular lattice of fuel rods submerged in a water-filled tank of stainless steel. Light water is used as the moderator. The facility’s criticality is controlled through the selection of the moderator level. The fuel rod spacing in the triangular lattice is 1.27 cm. There are bottom and top support grids to retain the rods in their required positions. Uranium dioxide, UO_{2}, is used as the fuel composition. The fuel enrichment in the LCT-053 and LCT-085 facility versions under consideration is 4.4% and 6.5% respectively.

The Stend P computational model’s vertical and horizontal cross-sections are shown in Figs

Vertical cross-section of the Stend P computational model.

Horizontal cross-section of the Stend P computational model.

Similarly to the Stend P, the ZR-6 zero-power experimental facility consists of an assembly of fuel rods submerged in a stainless-steel tank. Light water with a boric acid addition (5.8 g/l) is used as the moderator. The triangular fuel lattice spacing is 1.27 cm, and the fuel is UO2 with an enrichment of 4.4 wt. %.

The ZR-6 computational model’s vertical and horizontal cross-sections are shown in Figs

Vertical cross-section of the ZR-6 computational model.

Horizontal cross-section of the ZR-6 computational model.

As part of the study, different methods were used to partition the fuel columns into registration objects (

Let us consider the results of calculating β_{eff} and Λ with taking into account and without taking into account the importance function with different resolutions of the neutron importance function.

The calculation results from the LCT-085, LCT-053 and LCT-015 experiments, as well as the value of the estimate correction using the neutron importance are presented in Tables

LCT-085 experiment calculation results, case 13

Partition type | β_{eff} |
Λ | ||
---|---|---|---|---|

Value, pcm | Δ, % | Value, 10^{-5} s |
Δ, % | |

Importance not taken into account | 815.6 | – | 3.1 | – |

Axial | 816.2 | -0.1 | 3.1 | 0.1 |

Radial | 822.1 | -0.8 | 2.7 | 15.7 |

Combined | 822.4 | -0.9 | 2.7 | 15.9 |

Radial components of neutron generation density and neutron importance: a) – generation density in LCT-085; b) – importance in LCT-085; c) – generation density in LCT-053; d) – importance in LCT-053; e) – generation density in LCT-015; f) – importance in LCT-015.

Axial components of neutron generation density and neutron importance: a) – generation density in LCT-085; b) – importance in LCT-085; c) – generation density in LCT-053; d) – importance in LCT-053; e) – generation density in LCT-015; f) – importance in LCT-015.

As shown by the results, the following regularities can be identified in all experiments: the contribution from the radial partition of the importance function is larger than that from the axial partition. In particular, when Λ is calculated, taking into account the importance function has little effect on the β_{eff} computation accuracy. To explain the obtained results, one needs to consider the radial and axial dependence diagrams for the generation density and the neutron importance.

LCT-053 experiment calculation results, case 11

Partition type | β_{eff} |
Λ | ||
---|---|---|---|---|

Value, pcm | Δ, % | Value, 10^{-5} s |
Δ, % | |

Importance not taken into account | 804.3 | – | 3.3 | – |

Axial | 806.6 | -0.3 | 3.3 | 0.6 |

Radial | 810.4 | -0.8 | 2.9 | 14.5 |

Combined | 812.7 | -1.0 | 2.8 | 15.3 |

All radial distributions show an abrupt jump in the neutron generation density on the assembly periphery which is the result of the water-uranium ratio growth on the reflector boundary. The neutrons born in the high importance region can get into the reflector and, after they return, will be highly likely absorbed in the low importance region in the external fuel rod row. To obtain the reliable value of Λ, the long lifetime of such neutrons needs to be included in the final estimate with a weight (importance function) since the secondary neutrons they produce have a small importance due to a high probability of leakage. The increase in the ratio between the neutron generation density on the periphery and in the assembly center leads to an increased effect from taking into account the importance (

LCT-015 experiment calculation results, 163/161

Partition type | β_{eff} |
Λ | ||
---|---|---|---|---|

Value, pcm | Δ, % | Value, 10^{-5} s |
Δ, % | |

Importance not taken into account | 778 | – | 2.72 | – |

Axial | 783 | -0.6 | 2.60 | 2.0 |

Radial | 783 | -0.6 | 2.46 | 10.3 |

Combined | 788 | -1.2 | 2.44 | 11.5 |

The monotonies of the generation density and neutron importance axial distributions coincide in the entire interval due to which taking into account the importance function does not make a major contribution to the Λ estimate adjustment (up to 2%).

As compared with radial partition, combined partition leads to a slightly improved accuracy of estimates (≈ 1%) as the result of the minor radial partition effect.

Noteworthy, taking into account the importance function has little effect on the accuracy of calculating β_{eff} (≈ 1%) as the result of rather a soft spectrum of the system. The difference in the energy of prompt and delayed neutrons levels off rapidly due to good moderating properties of the medium.

The paper describes in brief the algorithms offered in the MCU code for taking into account the importance function when calculating the effective fraction of delayed neutrons and the prompt neutron generation time.

The effect of the importance function resolution on the accuracy of computing the functionals under consideration was investigated using three partition types (axial, radial, combined).

An analysis of the calculation results has shown that radial partition contributes decisively to the improved accuracy of estimating the prompt neutron generation time (up to 16%) as the result of the neutron generation density jump on the assembly periphery. Peripheral neutrons of minor importance are born largely by the neutrons that have returned from the reflector and have a long lifetime. To obtain correct estimates, the lifetime of these needs to be included in the final estimate with a weight in the form of the importance function at the absorption point.

Since the functions under consideration have similar monotonies, axial partition does not contribute greatly to the improved accuracy of estimates, due to which combined distribution corrects the estimates, this correction being nearly identical to that for radial partition.

Taking into account the importance function has little effect on the accuracy of calculating the effective fraction of delayed neutrons (≈ 1%) β_{eff} in all options due to the good moderating properties of the system: the difference in energy between prompt and delayed neutrons levels off rapidly.

It is therefore required to increase the neutron importance resolution in the peripheral regions in which the neutron spectrum changes abruptly.

The results presented in the paper have been obtained using the computational resources from the Joint Computing Cluster of NRC Kurchatov Institute,

Russian text published: Izvestiya vuzov. Yadernaya Energetika (ISSN 0204-3327), 2023, n. 2, pp. 5–14.