Potential Model for Actinide Oxides (v1.0)
Many Bodied Actinide Oxide Potentials
We have developed a new potential model describing a range of actinide oxides which includes manybody effects to improve the description of their thermomechanical properties.
 Enquiries:
This work is published in: M.W.D. Cooper, M.J.D Rushton, R.W. Grimes, J. Phys. Condens. Matter 26 (2014) 105401.
We would be grateful if you could cite this paper if you make use, or derive, any work from the materials contained here.
This page provides a brief description of the model and gives examples of its ability to reproduce thermal expansion, elastic properties and specific heat for a wide range of actinide oxide systems:
 CeO_{2}
 ThO_{2}
 UO_{2}
 NpO_{2}
 PuO_{2}
 AmO_{2}
 CmO_{2}
Importantly the oxygenoxygen interactions are fixed across the actinide oxide series keeping open the ability to model mixed oxides. Currently this page hosts descriptions of the model suitable for the GULP and LAMMPS codes, we will be providing tabulation files for DL_POLY in the near future. Furthermore, we will be releasing the EAM tabulation python library used during the development of the model in the next few weeks. If you haven any questions please get in touch.
##Overview
Downloads
File  Description 

GULP_Example_UO2.zip  Example GULP UO_{2} energy minimisation. See below for more. 
GULP_AmO2.lib  GULP AmO_{2} potential library file. 
GULP_CeO2.lib  GULP CeO_{2} potential library file. 
GULP_CmO2.lib  GULP CmO_{2} potential library file. 
GULP_NpO2.lib  GULP NpO_{2} potential library file. 
GULP_PuO2.lib  GULP PuO_{2} potential library file. 
GULP_ThO2.lib  GULP ThO_{2} potential library file. 
GULP_UO2.lib  GULP UO_{2} potential library file. 
LAMMPS_example.zip  LAMMPS energy minimisation and equilibration run for UO_{2}. Described in more detail below 
LAMMPS_AmO2.fs  LAMMPS pair_style eam/fs potential tabulation for AmO_{2}

LAMMPS_CeO2.fs  LAMMPS pair_style eam/fs potential tabulation for CeO_{2}

LAMMPS_CmO2.fs  LAMMPS pair_style eam/fs potential tabulation for CmO_{2}

LAMMPS_NpO2.fs  LAMMPS pair_style eam/fs potential tabulation for NpO_{2}

LAMMPS_PuO2.fs  LAMMPS pair_style eam/fs potential tabulation for PuO_{2}

LAMMPS_ThO2.fs  LAMMPS pair_style eam/fs potential tabulation for ThO_{2}

LAMMPS_UO2.fs  LAMMPS pair_style eam/fs potential tabulation for UO_{2}

##Using the Model in LAMMPS [id_lammps_model]
Example
In order to show how the potential model can be used in LAMMPS a working example has been provided. The LAMMPS_example.zip provides an example where a 4×4×4 UO_{2} supercell is energy minimised then equilibrated under NPT conditions at a temperature of 300K for 50ps. This should give a good idea of how the other LAMMPS potential tabulations can be used.
Running the Example
 Unzip the LAMMPS_example.zip archive.
 In your terminal enter the example directory.
 Run LAMMPS:
lammps in EAM_equil.lmpin log EAM_equil.lmpout
Details
Both the EAM and pairwise interactions of the actinide oxide potential
described below are tabulated for use in LAMMPS
using the eam/fs
pair_style, as described in the LAMMPS
manual. The pair_coeff
command is used to assign the elements within tabulated EAM file to
LAMMPS' species IDs. The following example shows how the LAMMPS_UO2.fs
table file is used in LAMMPS.
pair_style hybrid/overlay coul/long ${SR_CUTOFF} eam/fs
pair_coeff * * coul/long
pair_coeff * * eam/fs embed_UO2.fs O U
Notes:

Oxygen is assumed to have ID = 1 and uranium ID = 2

NB: if the IDs were reversed, such that U = 1 and O = 2,
then then the species would be reversed in the
pair_coeff
command as follows:pair_coeff * * eam/fs embed_UO2.fs U O

NB: if the IDs were reversed, such that U = 1 and O = 2,
then then the species would be reversed in the

pair_style hybrid/overlay
is used to combinecoul/long
with the EAM model (eam/fs
), this tells LAMMPS to calculate the electrostatic interactions between ions (NB: akspace_style
must also be defined). 
Ion charges of 2.2208 for uranium and 1.1104 for oxygen should be specified. This can be performed either in a LAMMPS file read in using
read_data
or using theset charge
directive. The variable
${SR_CUTOFF}
is used to define the cutoff parameter forcoul/long
.
Using the Model in GULP
GULP library files are provided for each actinide oxide. These can be downloaded using the links in the downloads table.
Example:
An example showing how to energy minimise a single fluorite UO_{2}
unitcell using the GULP_UO2.lib
file is provided in the
GULP_Example_UO2.zip.
Running the Example
 Unzip GULP_Example_UO2.zip.
 From the terminal enter the example directory
 Run the provided GULP file:
gulp < GULP_UO2_fluorite_example.gin
Note: The other GULP libraries provided within the
downloads table can be used with the
GULP_UO2_fluorite_example.gin
by changing the library
directive to
refer to another model file.
Details
In GULP the actinide oxide potentials are defined using the species
, buck
, morse
, manybody
, eam_functional
and eam_density
keywords without the need for tabulation. Ionic charges for the description of Coulombic interactions are defined using the species
keyword.
species 2
U core 2.220800
O core 1.110400
Short range Buckingham and Morse parameters are defined using the buck
and morse
keywords respectively.
buck
O core U core 448.778865087 0.387757677453 0.0 0.00 11.00
O core O core 830.283447557 0.352856254215 3.88437209048 0.00 11.00
U core U core 18600.0 0.27468 0.0 0.00 11.00
morse
O core U core 0.6607973133 2.05815247834 2.38051231271 0.000 11.000
The short range and long range cutoffs for the Embedded Atom Method (EAM) are defined using the manybody
keyword.
manybody
O core U core 1.500 11.000
U core U core 1.500 11.000
O core O core 1.500 11.000
The square root form of the embedding function and associated \(G_\alpha\) parameters are defined using the eam_functional
keyword.
eam_functional square_root
U core 1.8061282781
O core 0.690017059438
The \(\sigma_\beta(r_{ij})\) functions and associated
\(n_\beta\) parameters are defined using the eam_density
keyword.
eam_density power 8
U core 3450.99492623
O core 106.855913747
##Model Description and Parameters [id_model_description]
Our approach used the embedded atom method to introduce manybody effects to the more conventional Buckingham + Morse pairwise description. This approach is advantageous as it allows the inclusion of manybody interactions in MD simulations without the tricky massless entities required by the shell model.
The actinide oxide potential set has two distinct components: pairwise and manybody. The energy, \(E_i\), of atom \(i\) with respect to all other atoms can be written as follows:
\[
E_{\textit{i}} =
\color{orange} \frac{1}{2}\sum_\textit{j}\phi_{\alpha\beta}(r_{ij})
\color{magenta} G_\alpha \sqrt{\sum_\textit{j}\sigma_{\beta}(r_{\textit{ij}})}
\]
Pairwise Component
The first, pairwise term describes short range cationoxygen pair interactions. This uses the conventional BuckinghamMorse potential with long range Coulombic interactions also included:
\[ \definecolor{buck}{RGB}{255,0,0} \definecolor{coulomb}{RGB}{0,0,255} \definecolor{morse}{RGB}{0,128,0} \phi_{\alpha\beta}(r_{\textit{ij}}) =
\color{coulomb} \frac{q_{\alpha}q_{\beta}}{4\pi \epsilon_0 r_{\textit{ij}}} \color{black} +
\color{buck} A_{\alpha\beta}\exp \left(\frac{r_{\textit{ij}}}{\rho_{\alpha\beta}}\right)\frac{C_{\alpha\beta}}{r_{\textit{ij}}^{6}} \color{black} +
\color{morse} D_{\alpha\beta}[\exp(2\gamma_{\alpha\beta}(r_{\textit{ij}}r_0))2\exp(\gamma_{\alpha\beta}(r_{\textit{ij}}r_0))] \]
For Coulombic interactions the charges of the ions are nonformal such that \(q_\alpha=Z^{eff}_\alpha e \), however, they are proportional to their formal charges ensuring the system is charge neutral (for tetravalent cations \(Z^{eff}_\alpha\) = 2.2208 and for oxygen \(Z^{eff}_\alpha\) = 1.1104 ). \(A_{\alpha\beta}\), \(\rho_{\alpha\beta}\), \(C_{\alpha\beta}\), \(D_{\alpha\beta}\), \(\gamma_{\alpha\beta}\) and \(r_0\) are empirical parameters that describe the Buckingham and Morse potentials. For cationcation and oxygenoxygen interactions the covalent Morse term is not required (i.e. D=0). The pairwise parameters are now summarised:
Interaction  \(\phi_B (r_{ij})\)  \(\phi_M (r_{ij})\)  

\(\alpha  \beta\)  \(A_{\alpha\beta}\) / eV  \(\rho_{\alpha\beta}\) / Å  \(D_{\alpha\beta}\) / eV  \(\gamma_{\alpha\beta}\) / Å^{–1}  \(r_0\) / Å  
CeCe  18600  0.2664  0.0       
ThTh  18600  0.2884  0.0       
UU  18600  0.2747  0.0       
NpNp  18600  0.2692  0.0       
PuPu  18600  0.2637  0.0       
AmAm  18600  0.2609  0.0       
CmCm  18600  0.2609  0.0       
CeO  351.341  0.3805  0.0  0.7193  1.869  2.356 
ThO  315.544  0.3959  0.0  0.6261  1.860  2.498 
UO  448.779  0.3878  0.0  0.6608  2.058  2.381 
NpO  360.436  0.3830  0.0  0.7638  1.840  2.368 
PuO  377.395  0.3793  0.0  0.7019  1.980  2.346 
AmO  364.546  0.3774  0.0  0.7467  1.955  2.325 
CmO  356.083  0.3775  0.0  0.7613  2.081  2.311 
OO  830.283  0.3529  3.8843       
Embedded Atom Component
The second term uses the Embedded Atom Method to introduce a subtle manybody perturbation to the conventional pairwise description. This manybody dependency is achieved by summing a set of pairwise functions, \(\mathrm{\sigma_\beta(r_{ij})}\), between atom \(i\) and its surrounding atoms. \(\mathrm{\sigma_\beta(r_{ij})}\) is proportional to the 8th power of the interionic separation with \(n_\beta\) being the constant of proportionality and is defined by the species of the surrounding atom \(j\):
\[\sigma_{\beta}(r_{\textit{ij}}) =\frac{n_{\beta}}{r_{\textit{ij}}^8}\]
\(\mathrm{\Sigma\sigma_\beta(r_{ij})}\) is then passed through a nonlinear embedding function. The embedding function used means that the manybody energy perturbation is proportional to the square root of \(\mathrm{\Sigma\sigma_\beta(r_{ij})}\) with \(G_\alpha\) being the constant of proportionality and is defined by the species of atom \(i\). The EAM parameters are given in the following table:
Species  \(G_\alpha\) / eVÅ^{1.5}  \(n_\beta\) / Å^{5} 

Ce  0.308  1556.803 
Th  1.185  1742.622 
U  1.806  3450.995 
Np  0.343  1796.945 
Pu  1.231  1456.773 
Am  0.333  1631.091 
Cm  0.494  1503.704 
O  0.690  106.856 
##Comparison with Experimental Data [id_model_validation]
The potential was successfully fitted for all actinide oxides to experimental data of thermal expansion via MD and elastic constants via static calculations. The agreement obtained between model predictions and experimental data is now given.
###Thermal Expansion: [id_graph_thermal_expansion]
Comparison of experimental data (solid lines) with the data points from MD simulations obtained using the potential model for CeO_{2}, ThO_{2}, UO_{2}, NpO_{2}, PuO_{2}, AmO_{2} and CmO_{2}.
###Bulk Modulus: [id_graph_bulk_modulus]
Comparison of experimental data for UO_{2} bulk modulus (open circles) with the data points from MD simulations (red halfcircles) using the new potential.
###Specific Heat Capacity: [id_graph_specific_heat_capacity]
Change in enthalpy relative to 300K as a function of temperature calculated using the new potential. Polynomials have been fitted to the data for PuO_{2} ThO_{2} and UO_{2}, shown in green, purple and red respectively. The polynomial form is based on that used for the experimental data, also shown in black. The derivative of the enthalpy gives the specific heat and is shown alongside the enthalpy data.