Corresponding author: Konstantin Yu. Khromov ( khromov_ky@nrcki.ru ) Academic editor: Yury Korovin
© 2021 Konstantin Yu. Khromov, Andrey V. Orlov, Ivan A. Belov, Vladimir A. Nevinitsa.
This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Citation:
Khromov KYu, Orlov AV, Belov IA, Nevinitsa VA (2021) Determination of the energy characteristics of the reactions UF_{6} ↔ UF_{5} + F and UF_{6} ↔ UF_{4} + F_{2}. Nuclear Energy and Technology 7(3): 239244. https://doi.org/10.3897/nucet.7.74152

Quantummechanical 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 UF 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 barrierfree, 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 (SNF).
Double cascade, radiolysis, uranium hexafluoride, regenerated uranium, closed fuel cycle
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 SNF irradiated to relatively low burnups. However, the trends in the development of the fuel cycle are such that the stocks of SNF with low burnup are almost exhausted, but instead of them new ones have been accumulated, already with a higher burnup in significantly increased volumes. The need to reduce the mass of disposed waste leads to an understanding of the need for multiple recycle of regenerated nuclear materials and, first of all, uranium, which makes up more than 90% of spent nuclear fuel. The physics of multiple recycling of uranium is such that in the sequence of recycles, from the previous one to the next, there is a continuous increase in the ^{236}U isotope content, which, in turn, leads to an increase in the ^{232}U content (
Scheme of a double cascade for recycling regenerated uranium: E_{1} – flow of regenerated uranium; P_{1} – flow of selection of the first stage, power supply of the second stage; P_{2} – flow of selection of the second cascade (waste of purification from the ^{232}U isotope); W_{1} – flow of the dump of the first stage; W_{2} – flow of the heavy fraction of the second cascade; F_{P} – LEU diluent flow; P_{0} – final product (commercial LEU).
In terms of further treatment, this material is, in principle, problematic. The only available method at present is dilution with waste uranium (
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 radiationinduced 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 transitionstate 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 (
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, quantumchemical 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  UF bond length (FF for F_{2}) calculation, pm  UF bond length (FF 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 d_{min} = 1 nm to d_{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 d = 1.6 nm, the change in the energy of the UF_{4} molecule did not exceed 0.01 eV. Subsequently, all the calculations were performed at d = 1.6 nm.
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.
To determine the accuracy of quantumchemical 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
Various methods are used to determine the geometry and energy of transition states, for example, the Nudged Elastic Band (NEB) (
The NEB method is fairly easy to use: it is enough to create several atomic configurations obtained by linear interpolation of atomic configurations corresponding to two minima of potential energy. It is rather difficult to use the NEB method for finding the transition states of chemical reactions, since formally one of the potential energy minima corresponds to the reaction products at infinity, which causes problems with the convergence of numerical algorithms.
However, for a qualitative assessment of the barrier to a chemical reaction, one can use the ideas embodied in the NEB method. For this purpose, several atomic configurations have been constructed by interpolating the atomic positions in the UF_{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.
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.
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 78 and 910, 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 2N directions in the 3N configuration space, where N is the number of atoms, local minima will be observed. The local minima will correspond to positive squares of natural frequencies and, therefore, real natural frequencies.
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.
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, exchangecorrelation 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, http://computing.nrcki.ru/.