_{6}↔ UF

_{5}+ F and UF

_{6}↔ UF

_{4}+ F

_{2}

Corresponding author: Konstantin Yu. Khromov (Khromov_KY@nrcki.ru)

Academic editor: Yury Korovin

Quantum-mechanical methods are used to assess the energy barriers to dissociation and recombination reactions of UF_{6} ↔ UF_{5} + F and UF_{6} ↔ UF_{4} + F_{2}. The energy characteristics of these reactions are found to be strongly asymmetric: the dissociation reaction barriers exceed the recombination reactions barriers by more than 4 eV. The equilibrium atomic configurations of F_{2}, UF_{4}, UF_{5} and UF_{6} have been determined using precision quantum mechanical calculations. The U-F bond lengths obtained as a result of the calculations are in good agreement with experimental data. It was found that the decay reaction UF_{6} → UF_{5} + F is either barrier-free, or the energy barrier for such a reaction is less than the resolving power of the method (~ 0.1 eV). For the decay of UF_{6} → UF_{4} + F_{2}, there is an energy barrier with a height of about 0.3 eV. An initial approximation was proposed for the arrangement of UF_{6} atoms in order to find the saddle points of the UF_{6} dissociation reactions. In this initial configuration, all 7 atoms of the UF_{6} molecule are located in the same plane. The F atoms are located at the vertices of a regular hexagon, and the U atom is at the center of such a hexagon. The results of this work can be used to determine the constants of thermal reactions of dissociation and recombination UF_{6} ↔ UF_{5} + F и UF_{6} ↔ UF_{4} + F_{2}. These constants are necessary for modeling the physicochemical processes occurring during the enrichment of spent nuclear fuel (

Closing the fuel cycle requires new approaches to enrichment of reprocessed uranium. Until now, uranium recycle has been limited mainly to recycle of once reprocessed ^{236}U isotope content, which, in turn, leads to an increase in the ^{232}U content (^{232}U due to non-observance of the limitation on the content of this isotope during the enrichment of regenerated uranium. The problem can be solved by using double cascades, which allow for the purification of enriched regenerated uranium. An example of such a cascade is shown in Fig. ^{232}U isotope. This uranium fraction (selection of _{2} in Fig. ^{232}U concentration in it, depending on the recycle number and enrichment in the second cascade, exceeds by several orders of magnitude the admissible limit on the ^{232}U content in low enriched uranium (

Scheme of a double cascade for recycling regenerated uranium: _{1} – flow of regenerated uranium; _{1} – flow of selection of the first stage, power supply of the second stage; _{2} – flow of selection of the second cascade (waste of purification from the ^{232}U isotope); _{1} – flow of the dump of the first stage; _{2} – flow of the heavy fraction of the second cascade; _{P}_{0} – final product (commercial

In terms of further treatment, this material is, in principle, problematic. The only available method at present is dilution with waste uranium (^{235}U isotope. An attempt to prevent these losses is the development of a double cascade with the return of the extraction flow, in which the fraction containing the ^{232}U isotope with a high ^{235}U content is diluted with a new batch of regenerate and again fed to the inlet of the cascade (

Dissociation reactions are one of the main reactions that occur with UF_{6} under the influence of an internal radiation source and thermal influence:

UF_{6} → UF_{5} + e– + F → UF_{5} + F and UF_{6} ↔ UF_{4} + F_{2}.

To determine the rates of both thermal and radiation-induced dissociation, energy barriers are of great importance, which must overcome individual atoms of the starting materials in order to achieve a local minimum of potential energy corresponding to the reaction products. Such processes, which underlie chemical kinetics, are currently described using the transition-state theory (

The experimental determination of such barriers and the atomic geometry corresponding to the transition state causes enormous difficulties associated with the short residence time of atoms near the transition state. Only recently, after the development of the femtochemical infrared spectroscopy method (_{6} dissociation by the presence of a non-thermal radiation-induced contribution.

Thus, theoretical computational methods for determination are extremely important. In quantum chemistry, reliable methods have been developed for calculating the energy characteristics of atoms, molecules, and crystals (

In this work, quantum-chemical methods are used to study energy barriers and transition states of reactions.

UF_{6} ↔ UF_{5} + F and UF_{6} ↔ UF_{4} + F_{2}.

Calculated and experimental values of interatomic distances in F_{2}, UF_{4}, UF_{5} и UF_{6} molecules

Molecule | U-F bond length (F-F for F_{2}) calculation, pm |
U-F bond length (F-F for F_{2}) experiment, pm |
Experiment source |
---|---|---|---|

F_{2} |
142.4 | 144 | (Chomaker |

UF_{4} |
207.4 | 205.6 ± 0.1 | ( |

UF_{5} |
203.3 | 200 | (Jones Llewellyn and Ekberg Scott 1977) |

UF_{6} |
202.4 | 199.8 | ( |

For quantum mechanical calculations, we used the Quantum Espresso program (

Periodic boundary conditions were used in the calculations. In order to neglect the mutual influence of molecules in neighboring periodic patterns, we selected the size of the unit cell. We used cubic unit cells with an edge length from _{min} = 1 nm to _{max} = 2 nm. We selected the required cell size using the example of the UF_{4} molecule. By means of numerical experiments, we determined that with an increase in the unit cell size above _{4} molecule did not exceed 0.01 eV. Subsequently, all the calculations were performed at

To determine the equilibrium geometric configurations, the positions of the atoms were optimized by searching for local minima of the potential energy. The atomic geometries of the F_{2}, UF_{4}, UF_{5}, and UF_{6} molecules optimized in this way are shown in Fig.

Optimized atomic geometries of the F_{2}, UF_{4}, UF_{5}, and UF_{6} molecules.

To determine the accuracy of quantum-chemical methods for calculating the structure of molecules, the bond lengths were compared with the experimental values. The comparison is shown in Table

As a first step to assess the energy barrier of the dissociation reactions UF_{6} → UF_{5} + F and UF_{6} → UF_{4} + F_{2}, we compared the energy of the UF_{6} molecule with the sum of the energies of the UF_{5} molecule and the F atom and the sum of the energies of the UF_{4} and F_{2} molecules, respectively. The energies are given in Table

Energies of the UF_{6} molecule and the UF_{5} + F and UF_{4} + F_{2} complexes

Configuration | Energy, eV | Energy difference with UF_{6}, eV |
---|---|---|

UF_{6} |
–45.732 | 0 |

UF_{5} + F |
–41.227 | 4.505 |

UF_{4} + F_{2} |
–38.428 | 7.304 |

Based on the data in Table _{6} → UF_{5} + F and UF_{6} → UF_{4} + F_{2}) and recombination reactions (UF_{5} + F → UF_{6} and UF_{4} + F_{2} → UF_{6}). The energy differences between the configurations are very high: 4.505 and 7.304 eV, respectively. At the same time, the energy of the UF_{6} molecule is lower. Thus, the rates of the recombination reactions will be much higher than the rates of the dissociation reactions. To quantitatively determine the rates of forward and reverse reactions, it is necessary to determine the energy characteristics of the saddle point between the configurations, the corresponding reactants and reaction products.

Various methods are used to determine the geometry and energy of transition states, for example, the Nudged Elastic Band (

The

However, for a qualitative assessment of the barrier to a chemical reaction, one can use the ideas embodied in the _{6} molecule and at the extreme positions of the UF_{5} + F and UF_{4} + F_{2} complexes. In these extreme positions, the fluorine atom and molecule are at the maximum distances possible, taking into account the chosen size of the simulation volume and periodic boundary conditions, from the atoms included in UF_{5} and UF_{4}, respectively. The extreme atomic configurations corresponding to the maximum distance F from UF_{5} and F_{2} from UF_{4} are shown in Fig.

Atomic configurations corresponding to the maximum distance of F from UF_{5} (left) and F_{2} from UF_{4} (right), used to determine the barriers to dissociation and recombination reactions.

For each interpolation configuration between the UF_{6} molecule and the extreme positions of the atoms in the complex, the total energy was calculated without relaxation of the atomic positions. The energy dependences of a system of atoms along such interpolation paths are shown in Fig.

Values for different interpolation configurations of atomic positions of the complex UF_{4} + F_{2} (left) and UF_{5} + F (right). Configuration No. 0 corresponds to the UF_{6} molecule, Configuration No. 11 corresponds to the most distant atomic positions shown in Fig.

For the recombination reaction UF_{4} + F_{2} → UF_{6}, the energy barrier was estimated using this algorithm. This barrier is approximately equal to 0.34 eV. The barrier for the dissociation reaction is 7.64 eV. For the reactions UF_{6} ↔ UF_{5} + F, it was not possible to estimate the height of the energy barrier due to its low accuracy. However, for pairs of configurations 7-8 and 9-10, the energies are very close. Apparently, the energy barrier can be realized for atomic geometries close to these configurations.

The ultimate goal is not only to determine energy reactions, but also to calculate the reaction constants, which will make it possible to calculate the kinetics of the reactions. For this purpose, it is necessary to know not only the height of the energy barrier, but also the vibrational spectra of the initial molecule and the atomic configuration corresponding to the transition state.

The dimer method is used to determine the transition state (

By displacing the atoms along the eigenvector corresponding to the imaginary natural frequency, in the dimer method, it is possible to adjust the position of the saddle point. As a rule, transition states in chemical reactions correspond to saddle points of the first kind, when only one of the natural frequencies is imaginary. However, there are reports in the literature on the possible existence of saddle points of the second kind, when two natural frequencies are imaginary (

A good approximation for starting the search for a saddle point is, apparently, a configuration when all atoms are located in the same plane. Indeed, if we minimize the potential energy of the system under the additional condition that atoms cannot be displaced in one of the directions, then, at least for 2

To search for an initial approximation to the saddle point of the reactions UF_{6} ↔ UF_{5} + F and UF_{6} ↔ UF_{4} + F_{2}, the positions of atoms in one plane were optimized. The resulting atomic configuration is shown in Fig.

Atomic configuration of the U atom and six F atoms, obtained by minimizing the potential energy when the atoms are located in the same plane. The U atom is in the center of the regular hexagon, and the F atoms are in the vertices of the hexagon.

Note also that the reaction constant for the decay of UF_{6} → UF_{5} + F was estimated experimentally in (

With the help of fundamental quantum mechanical methods, a study of the energy barriers to the reactions UF_{6} ↔ UF_{5} + F and UF_{6} ↔ UF_{4} + F_{2} has begun. The calculated energy differences between the UF_{6} molecule and the UF_{5} + F and UF_{4} + F_{2} complexes exceed 5 eV, which indicates strongly different rates of dissociation and recombination reactions. Moreover, the energy of the UF_{6} molecule is lower than that of the complexes; therefore, the rate of recombination reactions is expected to be higher than the rate of dissociation reactions. For a more accurate determination of the energy characteristics of these reactions, an initial configuration has been prepared for the search for atomic configurations corresponding to the saddle point. For the recombination reaction UF_{4} + F_{2} → UF_{6}, an energy barrier of 0.34 eV was obtained. For the recombination reaction UF_{5} + F → UF_{6} within the framework of the relatively simple approach used, no energy barrier was found. Further research is required. Such studies should include, first of all, calculations of the energy characteristics of the indicated reactions using more accurate, in comparison with those used in this work, exchange-correlation functionals for the electronic subsystem. These calculations will require significant computing resources but can be performed using modern computing clusters.

The results of this work can be used for determining the constants of the considered chemical reactions and, therefore, for the correct parametrization of physicochemical models describing the kinetics of the processes occurring during the decay of UF_{6}.

This work was supported by the National Research Center Kurchatov Institute (Order No. 1879 dated August 22, 2019).

The results of the work were obtained using the computing resources of the Joint Compute Cluster of the NRC Kurchatov Institute,

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* Russian text published: Izvestiya vuzov. Yadernaya Energetika (ISSN 0204-3327), 2021, n. 2, pp. 59–70.