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Trapping and solution of fission Xe in UO2
Journal of Nuclear Materials North-Holland. Amsterdam
167
127 (1985) 167-169
TRAPPING AND SOLUTION OF FISSION Xe IN UO, Part 2. Solution from small overpressurized bubbles R.A. JACKSON Department
and C.R.A. CATLOW
oi_ Chemistry.
University
College London,
20 Gordon
Street, London
WCIH
OAJ,
UK
Received 7 July 1984; accepted 6 September 1984
We present results of calculations of formation energies of small aggregates, or bubbles, of Xe atoms in UO,. by a study of the energetics of resolution of gas atoms from bubbles into the UOZ lattice.
This is
followed
1. lntroductioe
3. Bubble configurations
In the previous paper [l] we reported results of calculations on the energetics of single gas atoms in UO,, and concluded that, for anion-deficient and stoichiometric UO,, neutral trivacancies were the most favourable trap sites for Xe atoms. However, the energies required to trap Xe atoms at all sites considered were high. We suggested that the trapping of a gas atom from an initially isolated state to a defect site in the lattice corresponded to solution from a large underpressurized bubble, i.e. one in which gas-atom-gas-atom interactions could be neglected, and that our calculation therefore suggested low solubility from these bubbles. In this work, we calculate the energy of formation of small aggregates of vacancies and gas atoms, with the maximum occupancy being when the number of gas atoms and cation vacancies is the same. It is then possible to calculate the energies of re-solution of gas atoms from these small bubbles into the lattice.
Calculations were carried out for bubbles containing 2,3,4 and 6 gas atoms. In each case the configuration adopted consists of the most symmetric arrangement of nearest neighbour cation vacancies, with anion vacancies chosen to maximize the cluster symmetry. For the initial configurations, the gas atoms were placed at the cation vacancy sites. The coordinates of the vacancies for each vacancy aggregate are given in table 2. Note that these coordinates are given with respect to a cation site at the origin, and anion sites at ( f 0.5, + 0.5, f 0.5). Note that vacancy aggregates containing 3 and 4 cation vacancies are not neutral; this is to maximize symmetry. In calculating the defect energies in table 3, the energies
2. Computational methods These are fully described in the previous paper, the only difference being that it is necessary to consider Xe. _. Xe interactions in the present work. A Xe.. . Xe potential calculated using the electron gas method [5,6] was used, fitted to a cubic spline. Details are given in table 1 (a) below. In addition, selected calculations were repeated using a Lennard-Jones potential calculated from gas viscosity measurements [2]. Parameters for this potential are given in table l(b).
0022-3115/85/$03.30 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
Table 1 Potential parameters for the Xe...Xe interaction gas calculation - fitted to cubic sphne potential
(a) electron
Knot position (A)
Value of function (eV)
2.0000 2.3125 2.6250 2.9375 3.2500 3.5625 3.8750 4.1875 4.5000
14.09419723 6.05730084 2.92370904 1.30349065 0.57484874 0.229894~ 0.09182871 0.03318482 0.01544404
(b) Lennard-Jones potential eV At’, C = 324 eV Ab
from
B.V.
viscosity
data
A = 1183480
168
R.A. Jackson,
C. R.A. Catlow / Trapptng und solemn
Table 2 Coordinates of vacancies for vacancy aggregate containing 2.3,4 and 6 cation vacancies. No. of cation Vacancies 2
3
Cation vacancy coordinates 0.0, 0.0, 0.0 0.0, 1.0, 1.0
1.0, 1.0,o.o l.O,O.O, - 1.0 0.0, 1.0, - 1.0
1 .o,1 .o,0.0 1.0, 0.0, - 1.0 0.0, l,O, - 1.0 0.0,&O, 0.0
1.0, 1.0,o.o - 1.0, 0.0,o.o 0.0, 2.0, 0.0 0.0, 0.0, 0.0 0.0, 1.0, 1.0 0.0, 1.0, - 1.0
Anion vacancy coordinates +0.5,
0.5, - 0.5,
0.5, 0.5 0.5, -0.5 0.5, 1.5
0.5, 0.5, -0.5 1.5, 0.5, -0.5 - 0.5, 0.5, -0.5 0.5, 1.5, -0.5 0.5, -0.5, -0.5 0.5. 0.5, 0.5 0.5, 0.5, - 1.5 0.5, 0.5, -0.5 1.5, 0.5, -0.5 - 0.5, OS, -0.5 0.5, 1.5, -0.5 0.5, -0.5, -0.5 0.5, 0.5, 0.5 0.5, 0.5, -1.5 -0.5, 0.5, -0.5, 0.5, -0.5, 0.5, - 0.5, 0.5, -0.5, 0.5, 1.5, - 1.5,
1.5, 1.5, 0.5, 0.5, 1.5, 1.5, 0.5, 0.5, 1.5, 0.5, 1.5, OS,
0.5 0.5 0.5 0.5 -0.5 -0.5 -0.5 -0.5 1.5 -1.5 0.5 -0.5
of Xe in ilO_, - 2
Table 3
Defect
energies for bubble
Bubble
Defect enerav CeV)
l/l 2/-J 2/Q
114.72 220.01 225.16 (224.49) 333.66 428.82 434.78 (434.57) 445.16 (442.42) 659.08
3/3 4/l 412 4/4 b/6
Table 4 Energetics of resolution processes Process
energy (eV per dissociated
2/2-+2(1/l) 3/3 -+ 2/2 + l/l 4/4 -* 3/3 + l/l 6/6-+4/4+2(1/l)
2.14 (2.48) 6.22 3.22 7.76
4. Results of calculations 4.1. Energerics
of bubble formation
Calculations were carried out for the formation energy of the bubbles listed in table 3. The notation adopted is that an “u/b” bubble is one in which there are “n” cation vacancies and “b” gas atoms. Energies quoted in the first column were calculated using the spline Xe.. .Xe potential; energies in brackets were calculated using the Lennard-Jones potential.
gas atom)
Table 5 Energetics of bubble dissociation Process
Energy (eV per gas atom)
2/2 3/3 4/4 6,‘6
2.14 (2.48) 3.50 3.43 (4.07) 4.88
-+ 2(1/l) -+ 3(1/l) -+ 4(1/l) -+6(1/l)
4.2. Solution energies
of a sufficient number of anion vacancies was added so that the results all refer to a neutral assembly of cation and anion vacancies.
formation
of gas
atoms
These results can be used to calculate the energy required to re-solve one or more gas atoms, each with an accompanying neutral trivacancy, into the lattice. The solution processes and associated energies are reported in table 4. In addition, the energy required for each bubble to dissociate completely into gas atoms and neutral trivacancies can be calculated, and these energies are given in table 5.
5. Discussion In the previous paper [l) results were presented for the process corresponding to the re-solution of Xe atoms from large underpressurized bubbles to defect sites in
R.A.
Jacksurr.
C. R.A.
Cutlow
/ Trapprng
the lattice. The lower energies for this process were around 9 eV. In this paper we show that the energies involved in trapping Xe atoms from small underpressurized bubbles is considerably less, typically in the range 2-8 eV. This energy difference could be relevant to the question of whether fission gas diffuses through the lattice by thermal re-solution from bubbles [3]. or by biased bubble migration [4]. A further point of interest is the fact that we have shown the two-atom bubble to be bound. The difference between the energies calculated using the spline and Lennard-Jones Xe...Xe potentials gives a useful indication of the uncertainties due to choice of potential in these calculations. Finally, the methods used have been applied to bubbles containing up to 6 gas atoms, although calculations on larger bubbles would clearly be possible.
Acknowledgements
We thank the UKAEA
the Safety and Reliability Directorate of for financial support, and acknowledge
cmd so&ton
of Xe m 50,
169
- 2
useful discussion with Drs. I.R. Breariey, D.A. MacInnes and P.W. Winter of SRD, Culcheth, and with Drs. J.R. Matthews and M.H. Wood of AERE, Harwell.
References
[l] R.A. Jackson (1985) 161. [2] A.A. Clifford,
and
C.R.A.
CatIow,
J. Nucl.
Mater.
127
P. Gray and N. Platts, J. Chem. Sot. Faraday
Trans. I 73 (1977) 381. f3] D.A. MacInnes and I.R. Brearley,
J. Nucl. Mater. 107 (1982) 123. [4] M.H. Wood and J.R. Matthews, J, Nucl. Mater. 102 (1981)
223. [S] R.G. Gordon and Y.S. Kim, J. Chem. Phys. 56 (1972) 3122. [6I P.T. Wedepohl, Pm-e. Phys. See. 92 (1967) 79.
167
127 (1985) 167-169
TRAPPING AND SOLUTION OF FISSION Xe IN UO, Part 2. Solution from small overpressurized bubbles R.A. JACKSON Department
and C.R.A. CATLOW
oi_ Chemistry.
University
College London,
20 Gordon
Street, London
WCIH
OAJ,
UK
Received 7 July 1984; accepted 6 September 1984
We present results of calculations of formation energies of small aggregates, or bubbles, of Xe atoms in UO,. by a study of the energetics of resolution of gas atoms from bubbles into the UOZ lattice.
This is
followed
1. lntroductioe
3. Bubble configurations
In the previous paper [l] we reported results of calculations on the energetics of single gas atoms in UO,, and concluded that, for anion-deficient and stoichiometric UO,, neutral trivacancies were the most favourable trap sites for Xe atoms. However, the energies required to trap Xe atoms at all sites considered were high. We suggested that the trapping of a gas atom from an initially isolated state to a defect site in the lattice corresponded to solution from a large underpressurized bubble, i.e. one in which gas-atom-gas-atom interactions could be neglected, and that our calculation therefore suggested low solubility from these bubbles. In this work, we calculate the energy of formation of small aggregates of vacancies and gas atoms, with the maximum occupancy being when the number of gas atoms and cation vacancies is the same. It is then possible to calculate the energies of re-solution of gas atoms from these small bubbles into the lattice.
Calculations were carried out for bubbles containing 2,3,4 and 6 gas atoms. In each case the configuration adopted consists of the most symmetric arrangement of nearest neighbour cation vacancies, with anion vacancies chosen to maximize the cluster symmetry. For the initial configurations, the gas atoms were placed at the cation vacancy sites. The coordinates of the vacancies for each vacancy aggregate are given in table 2. Note that these coordinates are given with respect to a cation site at the origin, and anion sites at ( f 0.5, + 0.5, f 0.5). Note that vacancy aggregates containing 3 and 4 cation vacancies are not neutral; this is to maximize symmetry. In calculating the defect energies in table 3, the energies
2. Computational methods These are fully described in the previous paper, the only difference being that it is necessary to consider Xe. _. Xe interactions in the present work. A Xe.. . Xe potential calculated using the electron gas method [5,6] was used, fitted to a cubic spline. Details are given in table 1 (a) below. In addition, selected calculations were repeated using a Lennard-Jones potential calculated from gas viscosity measurements [2]. Parameters for this potential are given in table l(b).
0022-3115/85/$03.30 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
Table 1 Potential parameters for the Xe...Xe interaction gas calculation - fitted to cubic sphne potential
(a) electron
Knot position (A)
Value of function (eV)
2.0000 2.3125 2.6250 2.9375 3.2500 3.5625 3.8750 4.1875 4.5000
14.09419723 6.05730084 2.92370904 1.30349065 0.57484874 0.229894~ 0.09182871 0.03318482 0.01544404
(b) Lennard-Jones potential eV At’, C = 324 eV Ab
from
B.V.
viscosity
data
A = 1183480
168
R.A. Jackson,
C. R.A. Catlow / Trapptng und solemn
Table 2 Coordinates of vacancies for vacancy aggregate containing 2.3,4 and 6 cation vacancies. No. of cation Vacancies 2
3
Cation vacancy coordinates 0.0, 0.0, 0.0 0.0, 1.0, 1.0
1.0, 1.0,o.o l.O,O.O, - 1.0 0.0, 1.0, - 1.0
1 .o,1 .o,0.0 1.0, 0.0, - 1.0 0.0, l,O, - 1.0 0.0,&O, 0.0
1.0, 1.0,o.o - 1.0, 0.0,o.o 0.0, 2.0, 0.0 0.0, 0.0, 0.0 0.0, 1.0, 1.0 0.0, 1.0, - 1.0
Anion vacancy coordinates +0.5,
0.5, - 0.5,
0.5, 0.5 0.5, -0.5 0.5, 1.5
0.5, 0.5, -0.5 1.5, 0.5, -0.5 - 0.5, 0.5, -0.5 0.5, 1.5, -0.5 0.5, -0.5, -0.5 0.5. 0.5, 0.5 0.5, 0.5, - 1.5 0.5, 0.5, -0.5 1.5, 0.5, -0.5 - 0.5, OS, -0.5 0.5, 1.5, -0.5 0.5, -0.5, -0.5 0.5, 0.5, 0.5 0.5, 0.5, -1.5 -0.5, 0.5, -0.5, 0.5, -0.5, 0.5, - 0.5, 0.5, -0.5, 0.5, 1.5, - 1.5,
1.5, 1.5, 0.5, 0.5, 1.5, 1.5, 0.5, 0.5, 1.5, 0.5, 1.5, OS,
0.5 0.5 0.5 0.5 -0.5 -0.5 -0.5 -0.5 1.5 -1.5 0.5 -0.5
of Xe in ilO_, - 2
Table 3
Defect
energies for bubble
Bubble
Defect enerav CeV)
l/l 2/-J 2/Q
114.72 220.01 225.16 (224.49) 333.66 428.82 434.78 (434.57) 445.16 (442.42) 659.08
3/3 4/l 412 4/4 b/6
Table 4 Energetics of resolution processes Process
energy (eV per dissociated
2/2-+2(1/l) 3/3 -+ 2/2 + l/l 4/4 -* 3/3 + l/l 6/6-+4/4+2(1/l)
2.14 (2.48) 6.22 3.22 7.76
4. Results of calculations 4.1. Energerics
of bubble formation
Calculations were carried out for the formation energy of the bubbles listed in table 3. The notation adopted is that an “u/b” bubble is one in which there are “n” cation vacancies and “b” gas atoms. Energies quoted in the first column were calculated using the spline Xe.. .Xe potential; energies in brackets were calculated using the Lennard-Jones potential.
gas atom)
Table 5 Energetics of bubble dissociation Process
Energy (eV per gas atom)
2/2 3/3 4/4 6,‘6
2.14 (2.48) 3.50 3.43 (4.07) 4.88
-+ 2(1/l) -+ 3(1/l) -+ 4(1/l) -+6(1/l)
4.2. Solution energies
of a sufficient number of anion vacancies was added so that the results all refer to a neutral assembly of cation and anion vacancies.
formation
of gas
atoms
These results can be used to calculate the energy required to re-solve one or more gas atoms, each with an accompanying neutral trivacancy, into the lattice. The solution processes and associated energies are reported in table 4. In addition, the energy required for each bubble to dissociate completely into gas atoms and neutral trivacancies can be calculated, and these energies are given in table 5.
5. Discussion In the previous paper [l) results were presented for the process corresponding to the re-solution of Xe atoms from large underpressurized bubbles to defect sites in
R.A.
Jacksurr.
C. R.A.
Cutlow
/ Trapprng
the lattice. The lower energies for this process were around 9 eV. In this paper we show that the energies involved in trapping Xe atoms from small underpressurized bubbles is considerably less, typically in the range 2-8 eV. This energy difference could be relevant to the question of whether fission gas diffuses through the lattice by thermal re-solution from bubbles [3]. or by biased bubble migration [4]. A further point of interest is the fact that we have shown the two-atom bubble to be bound. The difference between the energies calculated using the spline and Lennard-Jones Xe...Xe potentials gives a useful indication of the uncertainties due to choice of potential in these calculations. Finally, the methods used have been applied to bubbles containing up to 6 gas atoms, although calculations on larger bubbles would clearly be possible.
Acknowledgements
We thank the UKAEA
the Safety and Reliability Directorate of for financial support, and acknowledge
cmd so&ton
of Xe m 50,
169
- 2
useful discussion with Drs. I.R. Breariey, D.A. MacInnes and P.W. Winter of SRD, Culcheth, and with Drs. J.R. Matthews and M.H. Wood of AERE, Harwell.
References
[l] R.A. Jackson (1985) 161. [2] A.A. Clifford,
and
C.R.A.
CatIow,
J. Nucl.
Mater.
127
P. Gray and N. Platts, J. Chem. Sot. Faraday
Trans. I 73 (1977) 381. f3] D.A. MacInnes and I.R. Brearley,
J. Nucl. Mater. 107 (1982) 123. [4] M.H. Wood and J.R. Matthews, J, Nucl. Mater. 102 (1981)
223. [S] R.G. Gordon and Y.S. Kim, J. Chem. Phys. 56 (1972) 3122. [6I P.T. Wedepohl, Pm-e. Phys. See. 92 (1967) 79.