# Potential Model for Fission Gases in Actinide Oxides

NEW v1.0

Buckingham pair potentials have been developed for Xe and Kr in CeO2, ThO2, UO2 and PuO2, including any solid solution of these oxides. The new parameters are fully compatible with v1.2 of the CRG potential.

The following page describes our new potentials for Kr and Xe that were derived for use with our existing many bodied actinide oxide potentials (namely CeO2, ThO2, UO2, PuO2).

## Version History

Versions Description Relevant Papers
v1.0 - 23/08/2016 Buckingham pair potentials for the fission gases Xe and Kr have been developed [1], [2],[3],[4],

## Publications

• Details of the derivation and validation of the potentials can be found in the following paper:
• If you make use of this potential model in your own work, we would be grateful if you could cite this paper and also the following papers that describe the MO2 model with which Kr and Xe potentials are used:
• Finally, the gas-gas interactions that we employed were derived by K.T. Tang and J.P. Toennies this should be recognised by citing:

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Files Description
CeThUNpPuAmCmO.eam.alloy LAMMPS tabulation for all actinide oxides for use with pair_style eam/alloy
CRG_gas_potentials.lmptab LAMMPS tabulation for Xe-Xe, Xe-Kr and Kr-Kr Tang-Toennies potentials and the newly developed Xe-Ce, Kr-Ce, Xe-Th, Kr-Th, Xe-U, Kr-U, Xe-Pu, Kr-Pu and Kr-Xe potentials for use with pair_style table.
Example_1-LAMMPS_mixed_bubble.zip LAMMPS files for example 1 equilibration of mixed Xe-Kr fission gas bubble in UPuO2 mixed oxide.
Example_2-LAMMPS_mixed_bubble.zip LAMMPS files for example 2 equilibration of mixed Xe-Kr fission gas bubble in UPuO2 mixed oxide using analytical Buckingham potentials.

## Fission Gas Xe and Kr Interactions

Due to inert nature of the fission gases Xe and Kr it was not necessary to use Morse, EAM or Coulomb contributions in the description of their interactions. Rather a simple Buckingham description of Xe and Kr with O, Ce, Th, U and Pu is used. These parameters were developed by fitting to DFT forces calculated for MD configurations at 300 K, 1500 K, 3000 K and 5000 K. For gas-gas interactions the potential of Tang-Toennies was used.

The fission product gas, MO2 interactions are described using the Buckingham potential form:

$V(r_{ij}) = A_{\alpha\beta}\exp \left(\frac{-r_{\textit{ij}}}{\rho_{\alpha\beta}}\right)-\frac{C_{\alpha\beta}}{r_{\textit{ij}}^6}$

the $A_{\alpha\beta}$, $\rho_{\alpha\beta}$ and $C_{\alpha\beta}$ potential parameters for these interactions are given in the following table.

$\alpha - \beta$ $A_{\alpha\beta}$ / eV $\rho_{\alpha\beta}$ / Å $C_{\alpha\beta}$ / eVÅ^6
Xe-O 1877.5990 0.3401910 46.478139
Xe-Ce 6308.6510 0.2891647 18.013067
Xe-Th 6238.0739 0.3130134 20.013067
Xe-U 6606.3980 0.2981080 19.013067
Xe-Pu 6591.3310 0.2861837 18.013067
Kr-O 1862.8460 0.3231814 39.908723
Kr-Ce 7885.8137 0.2747064 17.112414
Kr-Th 7797.5924 0.2973627 19.012431
Kr-U 8257.9975 0.2832030 18.062414
Kr-Pu 8239.1637 0.2718745 17.112414

### Examples

The files contained in the downloads table can be used in several ways to define an MO2 actinide system containing Xe or Kr. The following examples demonstrate these different methods.

In each case, an 8×8×8 (U0.5,Pu0.5)O2 supercell, containing a mixed Xe-Kr fission gas bubble, is equilibrated under NPT conditions at 300K for a period of 10ps. For each example, a slightly different way of describing the potential model is demonstrated, however each should give the same result.

#### Example 1: LAMMPS table combining lattice-lattice and lattice-gas interactions. Separate table for gas-gas interactions

In order to show how the Xe and Kr fission gas potential can be used with the MOX potential model in LAMMPS a working example has been provided. The Example_1-LAMMPS_mixed_bubble.zip file contains an NPT equilibration. The potential model is defined in the following lines:

variable O equal 1
variable U equal 2
variable Xe equal 3
variable Pu equal 4
variable Kr equal 5

set type $O charge -1.1104 set type${U} charge 2.2208
set type ${Xe} charge 0 set type${Pu} charge 2.2208
set type ${Kr} charge 0 kspace_style pppm 1.0e-5 variable SR_CUTOFF equal 11.0 pair_style hybrid/overlay coul/long${SR_CUTOFF} eam/alloy table linear 1000 pppm
pair_coeff   *    *    coul/long
pair_coeff   *    *    eam/alloy CeThUNpPuAmCmO.eam.alloy O U NULL Pu NULL
pair_coeff   ${Xe}${Xe} table CRG_gas_potentials.lmptab Xe-Xe 11
pair_coeff   ${Kr}${Kr} table CRG_gas_potentials.lmptab Kr-Kr 11
pair_coeff   ${Xe}${Kr} table CRG_gas_potentials.lmptab Kr-Xe 11

pair_coeff   ${O}${Xe} table CRG_gas_potentials.lmptab Xe-O 11
pair_coeff   ${U}${Xe} table CRG_gas_potentials.lmptab Xe-U 11
pair_coeff   ${O}${Kr} table CRG_gas_potentials.lmptab Kr-O 11
pair_coeff   ${U}${Kr} table CRG_gas_potentials.lmptab Kr-U 11
pair_coeff   ${Xe}${Pu} table CRG_gas_potentials.lmptab Xe-Pu 11
pair_coeff   ${Pu}${Kr} table CRG_gas_potentials.lmptab Kr-Pu 11


In this example the Buckingham potential gas-lattice interactions and Tang-Toennies gas-gas interactions have been tabulated in CRG_gas_potentials.lmptab.

Note, in this example the ID of each species in the mixed_bubble_MOX.lmpstruct structure file have been stored in variables, allowing variable placeholders such as ${Xe} and ${Pu} to be used. This allows the input file to be more easily understood.

Note: due to a limitation of LAMMPS, the species labels in the pair_coeff lines must appear in strict numerical order - the order of the variable placeholders does matter.

Here the hybrid/overlay, pair_style allows the different components of the potential model to be combined:

• coul/long: specifies that electrostatic interactions should be calculated using the kspace_style defined as pppm.
• eam/alloy: reads in the CRG actinide model from the CeThUNpPuAmCmO.eam.alloy table file.
• The pair_coeff line for the eam/alloy table specifies the mapping between LAMMPS species IDs and the species labels used in the table file:
• pair_coeff * * eam/alloy CeThUNpPuAmCmO.eam.alloy O U NULL Pu NULL
• The O U NULL Pu NULL at the end of the line states that ID 1 = O, 2 = U and 4 = Pu.
• The 3rd and 5th entries are NULL indicating that species 3 and and 5 (Xe and Kr) are not contained in the EAM file.
• table: the pair_coeff table lines specify that the gas interactions are read from the table file CRG_gas_potentials.lmptab file.

The example can be run by invoking LAMMPS as follows:

lammps -in equil.lmpin


Output will be written to a file named log.lammps.

#### Example 2: LAMMPS using tabulated description of Xe-Xe, Kr-Kr and Xe-Kr interactions. Analytical pair_style buck description of gas-lattice interactions. Tabulated EAM potentials for lattice interactions

The gas-lattice interactions given here use the Buckingham potential form - in the previous example these were tabulated along with the gas-gas interactions. A table file was used because the Tang-Toennies potential form is quite complex and is not available as one of LAMMPS' standard pair_styles. By comparison, the Buckingham potential form used for the lattice-gas interactions, (see above) is available through the pair_style buck command in LAMMPS. The following example shows how the analytical Buckingham potentials can be combined with tabulated gas-gas interactions and a separate EAM table file for the description of the lattice.

The input files for this example can be found in the Example_2-LAMMPS_mixed_bubble.zip file which is fundamentally the same as Example 1 albeit with a modified description of the potential model. This is now shown:

variable O equal 1
variable U equal 2
variable Xe equal 3
variable Pu equal 4
variable Kr equal 5

set type $O charge -1.1104 set type${U} charge 2.2208
set type ${Xe} charge 0 set type${Pu} charge 2.2208
set type ${Kr} charge 0 kspace_style pppm 1.0e-5 variable SR_CUTOFF equal 11.0 pair_style hybrid/overlay coul/long${SR_CUTOFF} eam/alloy table linear 1000 pppm buck ${SR_CUTOFF} pair_coeff * * coul/long pair_coeff * * eam/alloy CeThUNpPuAmCmO.eam.alloy O U NULL Pu NULL pair_coeff${Xe} ${Xe} table CRG_gas_potentials.lmptab Xe-Xe 11 pair_coeff${Kr} ${Kr} table CRG_gas_potentials.lmptab Kr-Kr 11 pair_coeff${Xe} ${Kr} table CRG_gas_potentials.lmptab Kr-Xe 11 pair_coeff${O}  ${Xe} buck 1877.5990 0.3401910 46.478139 pair_coeff${U}  ${Xe} buck 6606.3980 0.2981080 19.013067 pair_coeff${Xe} ${Pu} buck 6591.3310 0.2861837 18.013067 pair_coeff${O}  ${Kr} buck 1862.8460 0.3231814 39.908723 pair_coeff${U}  ${Kr} buck 8257.9975 0.2832030 18.062414 pair_coeff${Pu} ${Kr} buck 8239.1637 0.2718745 17.112414 #pair_coeff${Xe} ${Ce} buck 6308.6510 0.2891647 18.013067 #pair_coeff${Xe} ${Th} buck 6238.0739 0.3130134 20.013067 #pair_coeff${Kr} ${Ce} buck 7885.8137 0.2747064 17.112414 #pair_coeff${Kr} \${Th} buck 7797.5924 0.2973627 19.012431


In this example the pair_style buck is overlaid with the EAM and table pair styles and the gas-lattice potential parameters are specified. For convenience the remaining interactions between gases and Ce and Th are provided although they are commented out in the example above.