0
Chapter 16. Oxide fuels
16.1 LWR Fuel: UO2..................................................................................1
16.1.1 Fabrication of Fuel Pellets.....................................................................2
16.1.2 Microstructure of fuel.............................................................................5
16.1.3 Varieties of fuel pellets...........................................................................6
16.2 Chemistry of Nuclear Oxides............................................................8
16.2.1 Experimental methods............................................................................8
16.2.2 Nanochemistry.........................................................................................9
16.2.3 Macrochemistry.....................................................................................10
16.3 Microchemistry of nuclear oxides.................................................13
16.3.1
16.3.2
16.3.3
16.3.4
16.3.5
16.3.6
16.3.7
Characteristics of point defects in nuclear oxides............................13
Measures of concentrations in oxides................................................14
Point defects in UO2x...........................................................................16
Structural defect equilibria...................................................................18
Electronic disorder - U4+ disproportionation......................................18
Gas-phase/defect equilibrium..............................................................19
Nonstoichiometry in UO2 x fixed by p O 2 ............................................20
16.4 Mixed Oxides...................................................................................22
16.4.1 Mean valence rule..................................................................................23
16.4.2 Oxygen pressure of (U1xQ Q xQ )O 2  z ..................................................23
16.4.3 Defects in Irradiated UO2.......................................................................26
16.5 Point-defect clustering in UO2+x.....................................................28
16.6 Properties of UO2 dependent upon point defects........................32
1
16.1 LWR Fuel: UO2
There is no leeway in selecting the element that provides energy by fission; 235U is the only
naturally occurring fissile nuclide, and undergoes fission by reaction with neutrons according to:
235
U + nth  2FP + 2nf
(16.1)
where nth denotes a neutron in thermal equilibrium with the water coolant (E ~ 0.1 eV) and nf
represents a “fast”, or high-energy neutron produced by the fission process (E ~ 1 MeV). The
fluxes of the thermal and fast neutrons in an LWR are about 4x1013 n/cm2-s each. The two
fission products (FP in Eq (16.1)) are born with about 100 MeV of energy each. The energy they
lose by interaction with the electrons in the fuel material is converted to heat which is conducted
through the fuel and ultimately is deposited in the flowing coolant.
Natural uranium contains only 0.71% of the fissile isotope 235U, an amount that is insufficient for
sustaining a chain reaction in the presence of ordinary water. As a result, the uranium in LWRs is
enriched to 235U concentrations up to 5%. The other uranium isotope, 238U, does not fission
readily with thermal neutrons. However, it can be converted to 239Pu by absorbing thermal
neutrons. The sequence is:
238
U + nth  239U  239Np  239Pu
(16.2a)
The initial absorption product, 239U, decays to the neptunium isotope with a 23-minute half life,
and 239Np decays to 239Pu with a half life of 56 hours. The plutonium isotope is relatively stable
(lifetime of 24,000 years), but fissions efficiently in a thermal neutron flux:
235
Pu + nth  2FP + 2nf
(16.2b)
The thermal energy produced by the fission products from the combination of reactions (16.2a)
and (16.2b) increases with time. At the end of life of the fuel (up to 6 years), this route
contributes as much as reaction (16.1) to the reactor power. By growing in slowly, 239Pu fission
partially offsets the loss of nuclear reactivity caused by the exponential decrease in the 235U
concentration with time. Spent fuel removed from the core of a large LWR generates
approximately 1000 kg/yr of fission products and 250 kg/yr of plutonium.
The chemical form of uranium in the fuel is the oxide, UO2. The oxygen in this compound serves
no nuclear purpose, but neither is it detrimental to neutron economy. Its main purpose is to
provide a chemically inert fuel form that is also relatively resistant to radiation damage, has a
high melting point, and maintains the same cubic crystal phase throughout its entire solid range.
Compared to all of these criteria, UO2 is superior to uranium metal except for the uranium atom
density and thermal conductivity. On the negative side, the oxygen in UO2 decreases the density
of uranium in the fuel by a factor of two compared to the metallic form, with a corresponding
increase in the size of the reactor core.
Plutonium (as PuO2) can be added to UO2 during fabrication to produce a mixed oxide (MOX)
fuel. The plutonium for this purpose originates either from reprocessed spent UO2 fuel or from
decommissioned nuclear weapons.
2
The quantity that most directly controls the reactor power, the fuel temperature, and the rate of
production of fission products and neutrons is the fission density, F . This is the rate at which
reactions (16.1) and (16.2) proceed. It is proportional to the concentrations of the fissile nuclides
and the neutron concentration expressed as the thermal neutron flux th:
F = (fU235NU235 + fPu239NPu239)th
(16.3)
The rate constants are the fission cross sections for the reaction of the two fissile species with
thermal neutrons. Because the thermal neutron flux varies with position in the reactor core, so
does the fission density. A typical value of F is 1013 fissions/cm3-s.
The fission density is a measure of the rate of fission. The cumulative fissions for an irradiation
time t is called the burnup, . This quantity has three equivalent definitions.
1. The fractional burnup is the ratio of the number of fissions to the number of initial uranium
atoms (of both isotopes):
 = F t/NU
(16.4a)
where NU = 2.5x1022 atoms/cm3 is the uranium density in UO2. In LWRs, 1% burnup is
accumulated per year of full-power operation.
2. The fissions per initial metal atom, or fima, is defined by:
fima  F t /( N U  N Q )
(16.4b)
where NQ is the concentration of another oxide Q, mixed with uranium oxide.
3. The energy produced per unit mass of initial uranium This measure is the product of the
fractional burnup of Eq (16.4a) and the energy per fission:
MWd
F t kg fissioned
(16.4c)
β = 950
×
kg fissioned N U
kg U
A typical burnup of fuel discharged from LWRs is 4%, or 40 MWd/kgU.
16.1.1 Fabrication of Fuel Pellets
The standard fuel pellet is a solid cylinder of polycrystalline uranium dioxide 1 cm or less in
diameter. The two variants of the LWR, the boiling water reactor (BWR) and the pressurized
water reactor (PWR) use slightly different pellet sizes, with that for the BWR being larger in
diameter. The pellet height in both fuels is between 1 and 2 cm.
Fabrication of fuel for LWRs consists of two distinct processes. In the first, uranium
hexafluoride is converted chemically into uranium dioxide powder. In the second, high-density
pellets are produced from the starting UO2 powder.
3
A flowsheet of the conversion process is shown in Fig. 16.1. The feed material is UF6, which is
the chemical form that is used in the isotope enrichment plants. It arrives as a liquid at elevated
pressure and is vaporized by reducing the pressure and warming slightly. The resulting UF6 gas
is fed into a chemical reactor along with steam and hydrogen in nitrogen as an inert carrier gas.
The following chemical reaction produces a fine power of UO2 as reaction product:
UF6(g) + 2H2O(g) + H2(g)  UO2(s) + 6HF(g)
(16.5)
The steam serves as the source of oxygen for the oxide product. Hydrogen reduces hexavalent U
in UF6 to the tetravalent state in UO2. The hydrofluoric acid gaseous product is neutralized by
NaOH, leaving relatively benign NaF as the sole waste stream.
The remainder of the flow sheet in Fig. 16.1 consists of steps that modify the UO2 into a powder
form that is easily made into a pellet in subsequent processing. In these steps, dry nitrogen cover
gas prevents oxidation of UO2. A lubricant such as stearic acid is added to aid in pressing pellets
from the powder. A substance called a poreformer is also added to help control the quantity and
shape of the voids that remain in the finished pellet.
Fig. 16.1 Conversion of UF6 to UO2 powder
The product of the conversion process constitutes the feed to the pellet fabrication process
depicted in Fig. 16.2. the first step is pressing the powder into “green” pellets that are 50 – 60
4
percent of the theoretical density of the crystalline material (this is abbreviated as %TD). The
following high-temperature sintering step serves four functions. First, it drives off the lubricant
added to the powder to assist pressing (care must be taken not to decompose the lubricant to
carbon, which, along with fluorine, is an undesirable impurity in the final product).
Second, the void spaces between the particles in the green pellets are nearly completely
eliminated by annealing at high temperature for several hours. This process is called sintering.
The porosity of the sintered pellets consists almost exclusively of closed spherical cavities called
closed porosity with negligible open porosity. The latter takes the form of interconnected
channels that communicate with free surfaces of the pellet. Most fuels are 95 - 96% TD.
Retention of few percent porosity is desirable. The pores serve as sinks for fission gases and
lessen their release. Swelling of the fuel due to solid fission products is also reduced by filling in
the internal voidage.
Fig. 16.2 Pellet Fabrication flow sheet
Third, the time and temperature of the sintering step also affect the size of the grains in the final
polycrystalline pellets. Simply eliminating porosity during sintering leaves grains of the same
size as the starting powder particles. More extensive annealing increases the average size of the
grains by a process termed grain growth. The usual pellet fabrication process produces
polycrystals with grain diameters about 8 m. Both sintering (porosity reduction) and grain
growth occur naturally at temperatures high enough to give reasonably rapid kinetics
(see Chap.8). However, both processes can be enhanced by additives introduced in the
preparation of the feed powder.
The fourth function accomplished during sintering is control of the oxygen-to-uranium ratio.
This characteristic of the fuel is controlled by the temperature and the ratio of H2 to H2O in the
gas fed to the sintering furnace (see Chap 7). Oxygen in excess of the 2:1 ratio of stoichiometric
UO2 is to be avoided. The excess oxygen in UO2+x (even when x  0.01) is corrosive towards the
cladding. Hyperstoichiometry also produces undesirable physical property changes in the fuel.
Excess oxygen reduces the thermal conductivity, thereby increasing the fuel temperature during
operation. Hyperstoichiometric fuel also enhances the mobility of the fission products, making
their release easier. The composition of the oxide in typical reactor fuel is UO2.005.
5
The step following sintering in Fig. 16.1b is a grinding operation designed to smooth the outer
surface of the pellet and closely control its diameter. The latter is important because, along with
the cladding inside diameter, it determines the thickness of the gap between the fuel and the
cladding in the finished fuel rod. This gap is gas-filled and constitutes a significant thermal
resistance to heat removal from the pellet during operation.
The final step in Fig. 16.1b is a low-temperature vacuum drying step intended to reduce the
concentration of adsorbed water on fuel surfaces to less than 10 ppm by weight. Improperly dried
fuel can cause cladding failure by reaction of desorbed water with Zr to produce both ZrO2 and
ZrH2. The latter embrittles the cladding and has been responsible for numerous fuel failures
during the past several decades.
16.1.2 Microstructure of fuel
Figure 16.3 shows a photomicrograph of polycrystalline UO2 after sintering. To obtain this
Fig. 16.3 Photomicrograph of
sintered UO2 showing pores and
grains. The dimension of the
grains are about 8 m
picture, the pellet is first cut with a diamond saw, then polished with fine abrasives, and finally
briefly exposed to an etching solution. Chemical etching reveals most of the grain boundaries as
the thick irregular lines enclosing individual single-crystal grains. The grain boundaries are the
intersections of the planar cut through the specimen with the faces of the polycrystalline grains
that make up the solid. (the straight lines that cross grain boundaries are polishing scratches).
The grain diameters are roughly 8 m. The junctions almost always consist of three grain
boundaries intersections called triple points. The third grain-boundary trace is sometimes missing
from a few triple points in Fig. 16.2, possibly because of inadequate etching during specimen
preparation. The porosity visible in the photomicrograph is all of the closed variety and is visible
as roughly circular black spots. The pores are less than one micron in diameter and occupy 7% of
the total specimen volume.
In UO2 destined for reactor fuel, the size of the pores is as important as the total porosity. In
operation, very fine pores (< 1 m) are eliminated by a process called radiation densification.
In essence, the pores are converted to their component vacancies, which are then removed by
sinks in the microstructure such as dislocations and grain boundaries. This process causes an
6
initial rapid reduction of the pellet diameter and concomitant collapse of the cladding on the
shrunken fuel. This potential problem is solved by removing fine pores during the sintering step
of fuel fabrication.
16.1.3 Varieties of fuel pellets
Over time, the standard cylindrical fuel pellet design has been modified in numerous ways, as
shown by the gallery in Fig. 16.3. The standard pellet is shown in (a). During operation, the
temperature distribution in the pellet and the resultant nonuniform thermal expansion causes its
shape to change to the "hourglass” shape shown in (b). The edges of the top and bottom pellet
surfaces deform the cladding with high local strains and stresses, risking perforation. Cladding
subjected to this type of deformation resembles a stalk of bamboo. To avoid this type of cladding
deformation and to minimize chipping during manufacturing and handling, the fuel pellets are
chamfered as shown in (c). Additional void space to accommodate fission product swelling and
axial thermal expansion of hot pellet centers is provided by “dishing” the top and bottom pellet
surfaces.
In a solid pellet, fission heat generation and removal creates differences in temperature between
the centerline and the pellet surface as large as 1000oC. This is undesirable for a number of
reasons, the principal one being release of fission gases from the hot center. To reduce the
centerline temperatures at the same linear power (i.e., power produced per unit fuel height), the
designs shown in (d) and (e) have been developed. Both move the heat source closer to the heat
sink (the coolant) than in the standard fuel pellet. In (d), the outer radial zone of the pellet is
more highly enriched in U-235 than the inner zone, thus reducing heat production near the center
and lowering the centerline temperature. In (e), the center is removed entirely, resulting in an
annular pellet. To produce the same power, this pellet must contain a higher enrichment than the
standard solid cylindrical pellet. However, the temperature at the inner surface of the annular
pellet is significantly lower than that of the standard cylindrical pellet for the same linear power.
Annular pellets are standard in the Russion VVER reactors.
In two designs, urania is mixed with other oxides for reasons related to the nuclear processes. In
(f), the additive is Gd2O3. Gadolinium acts as a burnable poison that allows a longer irradiation
time without excessive reactivity held in control rods at beginning-of-life.
In (g), UO2 is mixed with PuO2 to produce mixed oxide (MOX) fuel. The plutonia is in the form
of small particles in a matrix of UO2. Since Pu is the principal fissionable nuclide, these particles
are hotter than the rest of the fuel. Despite this nonuniformity of temperature, MOX fuel behaves
very similarly to standard UO2 fuel in reactor. In (h), the microstructure of the fuel is purposely
altered to increase the grain size from 8 m to as much as 40 m by using oxides such as Nb2O5
as a fuel additive. Niobium promotes grain growth during the sintering process. The objective of
the large grains is to reduce fission product release by increasing the length of the diffusion path
from the grain interior to the grain boundary.
So-called “nonfertile” or “inert matrix” fuel depicted in panel (i) is a response to the desire to
burn excess plutonium without simultaneously producing this element from a U-238 component,
as is the case with MOX fuel. Nonfertile fuels contain PuO2 dissolved in a matrix such as
(Zr,Eb)O2. Erbium also serves as a burnable poison like gadolinium.
7
high U-235
(a) classic
(c) dished and
chamfered
(b) hourglassing
at power
(e) annular
(high U-235)
low U-235
(d) zoned
enrichment
(f) Urania-gadolinia
PuO2
particles
(g) MOX
(h) large grain size
Fig.
Fig.16.4
1.3
Ceramic Oxide Pellet Designs
(i) non-fertile
8
16.2 Chemistry of Nuclear Oxides
Even without accounting for the effects of radiation, the chemistry of uranium oxide and mixed
oxides has been theoretically and experimentally investigated for over half of a century. The aim
of these efforts is to understand how the partial pressure of oxygen in equilibrium with the solid,
p O 2 , depends on temperature and composition, as defined by the oxygen-to-metal ratio O/M and
the cation fraction of the other metal ion, y. The symbol for such a mixed oxide is (U1-yQ y)O2  x,
where Q is the second cation. The latter is typically an actinide such as Pu or Th, one or more
fission products that are soluble in the oxide (La, Y, Eu, Sr) or a neutron absorber such as Gd.
The deviation from stoichiometry is quantified by the quantity x; the subscript 2-x signifies a
hypostochiometric oxide and 2+x means hyperstoichiometry..
The zirconium oxide corrosion scale that grows on Zircaloy cladding as a result of exposure to
high-temperature water is an example of a non-fissile oxide whose behavior is critical to the
lifetime of LWR fuel elements. This section is primarily devoted to the theoretical aspects of
oxide chemistry, but a brief description of the experimental techniques employed is given below.
16.2.1 Experimental methods
The wide array of techniques employed to determine the chemical behavior of nuclear oxides
include the following
1. Thermogravimetry [Ref 1, Sect. 11.4.1, Refs 2 - 10]
A sample is suspended inside a quartz or alumina tube in a furnace. The wire holding the
specimen is attached to a microbalance. A gas of known oxygen partial pressure flows through
the tube past the specimen. Examples are:
- O2 in an inert gas
- a gas mixture such as CO2/CO, which fixes p O 2 by the equilibrium: CO(g) + 1 2 O2 = CO2(g)
The microbalance measures weight changes as temperature or oxygen pressure is changed. The
specimen mass is then converted to the O/M ratio, or x = O/M - 2.
2. Galvanic Cell [Ref 1, Sect. 11.4.2, Refs. 11 and 12]
An all-solid assembly consisting of a metal/metal oxide mixture (e.g., Ni/NiO) that establishes
the oxygen pressure by the equilibrium Ni(s) + 1 2 O2 = NiO(s). Oxygen ions are transferred to or
from the test oxide via a piece of cubic zirconia until the oxygen potential (RTln p O 2 ) is the
same at both electrodes
3. Electrical Conductivity [Refs 6, 11, 12]
Standard conductivity techniques are used, usually in conjunction with thermogravimetry.
Conductivity measurements provide additional insight into the defect structure of the oxides.
4, Neutron scattering [Refs 13,14]
These studies were the first (and only ones) that revealed the existence of the now wellestablished cluster of point defects in high-temperature, high-hyperstoichiometry uranium
dioxide.
9
16.2.2 Nanochemistry
By the term nanochemistry, we mean theoretical investigation of equilibrium states of the oxides
at the atomic level. In the ab-initio method [15], the Schrodinger wave equation is solved for the
many-electron/atomic nucleus structure of the interacting ions. The potential energy of the
system is calculated for a number of configurations, such as along the path of a diffusing ion.
To reduce the computational demands of the ab-initio method, the interactions between entire
ions are instead represented by empirical potentials. Only two-body interactions are considered;
many-body effects are neglected.
A method considerably less detailed than the ab-initio technique goes by the generic name
molecular dynamics. The basic information needed for the simulation by this method is the
potential between pairs of ions, (r). The interionic force is the derivative of  with respect to r,
the distance between them. Common to all potential functions is the long-range Coulomb
potential,
ziz j 1
(rij ) 
(16.6)
4o rij
rij is the distance separating ions i and j, zi and zj are the charges of the ions (e.g., 4+ for a U4+
cation) and o is the The other components of the interionic potential are determined either from
ab-initio computations or by fitting to crystal properties. The short-range potential contains a
repulsive component expressed by the Born-Meyer potential and an attractive term representing
the Lennard-Jones potential:
r / 
(rij )  Ae ij  B / 6rij
(16.7)
where A,  and B are empirical constants, obtained by using the model to compute known crystal
properties such as the lattice parameter, the elastic constants and the specific heat. Sophisticated
molecular dynamic methods also include the energy contained in covalent bonds between atoms
or ions.
There are various techniques for minimizing computer demands by dividing the system into
regions in which appropriate simplifications are made. For example, in the rigid ion model, the
ions interact with one another via long-range electrostatic (i.e., Coulomb) forces. The shell model
adds the polarization of the ions due to these forces. Another simplification restricts the
individual ion-ion potential calculation to a certain radius around the defect. Beyond this sphere,
the medium is treated as a continuum, that is, a charge distribution without discrete charges. The
number of particles in the system ranges from a few thousand to millions, depending on the level
of detail with which the chosen method calculates energies. Whatever mathematical
simplification is employed, the equilibrium state of the crystal with a particular point defect is
the one for which the sum of the pair potentials is a minimum.
In addition to treating thermal systems characterized by the temperature, molecular dynamics is
the standard method for treating collision cascades generated by collisions of neutrons or
energetic ions such as fission fragments with the stationary ions of the crystal (Chap. 19). In this
application, both potential and kinetic energies are involved.
10
The outcomes of these calculations include:
- The intrinsic point defects that minimize the system energy (e.g., anion Frenkel defect in UO2)
- The charge states of the defects in nonstoichiometric oxides (O/M  2)
- The extent of electronic disorder (e.g., transfer of an electron from one U4+ ion to another to
produce a U3+ ion and a U5+ ion)
- relaxation of lattice ions surrounding a defect.
- The energies and entropies of formation of defects
- migration energies - the activation energies of diffusion of O2- and U4+ ions
Table 16.1 shows the results of applying the molecular dynamic method to UO2. It is clear that
from both experiment and modeling that the anion Frenkel pair is the lowest energy defect. The
Schottky defect, although much less stable than the anion Frenkel, must be considered because it
provides the cation vacancies by which uranium diffusion occurs. However, the low migration
energy for anion vacancies means that O2- is much more mobile in UO2 than U4+.
Table 16.1 Energies of defect processes in UO2 from Table 17 of Ref. 16
Energy, eV*
Process
Experiment
Calculation
Anion Frenkel
4.8
3.70.5
Cation Frenkel
9.0
19.4
Schottky
7.0
7.3
Anion vacancy migration
0.5
0.5
Anion interstitial migration
1.0
0.6
Cation vacancy migration
2.4
6.0
Cation interstitial migration
2.0
8.8
* 1 eV = 96.5 kJ/mole
16.2.3 Macrochemistry
By macrochemistry is meant treatment of chemical equilibria without delving into the behavior
of the atomic species involved, as was presented in Sect. 2.8. As an example, the equilibrium
between metallic uranium, oxygen and the hypostoichiometric oxide UO2-x' is described by the
reaction:
U(s) + (1-x'/2)O2(g) = UO2-x'(s)
(16.8)
for which the law of mass action is:
[1]
K
(16.9)
[1] p (O12 x '/ 2 )
The activities of the two solids are unity because both are pure phases that coexist at opposite
boundaries of the two-phase region of the U-O phase diagram (Fig. 16. ).
The macrochemical approach to reaction (16.8) merely says that if K were known, the O2
pressure in equilibrium with the oxide at the lower boundary of this phase (O/M = 2-x') could be
determined. However, this approach provides no means for knowing K, nor does it provide a
method for determining p O 2 in equilibrium with UO2-x where x < x' (i.e., in the region of Fig. 16.
11
labeled UO2  x). For this purpose, a more sophisticated approach, such as species representation
proposed by Lindemer and Besmann [17], is needed. This method represents nonstoichiometric
urania (or other oxide) by a solution of two species of different O/U ratios. The two components
are in equilibrium with gaseous oxygen according to the "reaction":
UO2 + O2(g) = UaOb
(16.10)
The two hypothetical species forming the solid are the "solvent" UO2 and the "solute" UaOb. In
fitting the data to the model, the subscripts a and b are varied and the oxygen pressure computed
as a function of the O/U ratio for several temperatures. The a,b combination hat produces oxygen
pressures that best agrees with experimental results is chosen.
The first step is to determine the coefficients  and  in reaction (16.10). Balancing U on the two
sides of the reaction gives  = a. The analogous O balance is 2 + 2 = b. These two equations
are solved for  and  in terms of a and b:
2a
2

and

(16.11)
b  2a
b  2a
In analyzing UO2+x, Lindemer and Besmann tested the following "solutes": UO3; U2O5; U3O7;
U10/3O23/3 and U2O9. Note that none of these "compounds" (including the UO2 "solvent") are real
chemical entities. They are constructs intended to mimic the actual chemistry of the oxide.
Different solute species are required for different ranges of hyperstoichiometry.
2.01 < O/U < 2.2
In this approximate nonstoichiometry range, the best fit was achieved with the "solute" U3O7.
With  and  determined from Eq (16.11) for a = 3, b = 7, the equilibrium "reaction" is
6UO2 + O2(g) = 2U3O7
(16.10a)
According to Eq (2.57), the equilibrium condition is given by the following relation between the
chemical potentials:
2 U3O7  6 UO2   O 2
(16.12)
Assuming the solution to be ideal, the chemical potentials of the two uranium-oxygen
components are given by Eq (2.36) with the activity of each replaced by the mole fraction. The
chemical potential of O2 is given by Eq (2.44). Substituting these into Eq (16.12) yields:
(16.13)
g o  6RT ln x UO2  RT ln p O2  2RT ln x U3O7
go is the standard-state free energy change of reaction (16.10a), R = 8.314 J/mole-K and T is
the temperature in Kelvins. The sum of the mole fractions of the "solution" components is:
x UO2  x U 3O7  1
(16.14a)
In 1 mole of the "solution", there are:
n U  x UO2  3x U 3O7 moles U and n O  2x UO2  7 x U3O7 moles O
so the O/U ratio of the oxide is:
12
2x UO2  7 x U3O7
nO
 2x 
nU
x UO2  3x U3O7
Solving for the two mole fractions from Eqs (16.14a) and (16.14b) produces:
1  3x
x
x UO2 
and
x U 3O 7 
1  2x
1  2x
(16.14b)
(16.14c)
Substituting these results into Eq (16.13) yields:
 x (1  2x ) 2 
h o s o
(16.15)
ln p O2 

 2 ln 
3 
RT
R
 (1  3x ) 
where the standard free energy change of the reaction has been expressed in terms of the
corresponding enthalpy and entropy changes according to Eq (2.62). Equation (16.15) is the
 x (1  2x ) 2 
2 ln 
3 
 (1  3x ) 
Figure 16.5 Fit of Eq (16.15) to a compendium of data at 1500 K (Ref. 17)
13
desired result. In addition to choosing the chemical "components" that result in the stoichiometry
dependence in the square brackets of Eq (16.15), fitting the model to the data involves choosing
the "equilibrium" parameters ho and so. Figure 16.5 shows the fit of Eq (16.15) to a
compendium of UO2+x data for x > 0.01. The thermodynamic parameters are: ho = - 316
kJ/mole and so = 126 J/mole-K.
There is a distinct change in slope at x just a bit below 0.01. This required choosing a solute
species different from U3O7, for which U2O4.5 provided the best agreement with ln p O 2 vs x data.
The analog of Eq (16.15) is:
 2x (1  2x ) 
360 214
ln p O2 

 4 ln 
2 
RT
R
 (1  4x ) 
(16.16)
Although not noted by the authors of Ref. 17, the data in the range 0.01 < x < 0.1 lie along a line
with a slope that is less than that deduced for the U3O7 case discussed above. It will be shown in
Sect. 16.4 that such slope changes has been recognized by later analyses of oxygen pressure vs
stoichiometry data.
PuO2-x and mixed oxides
The species representation model has been applied to PuO2-x, (U,Pu)O2-x[18,19] and
(U,RE)2x[20] (RE includes the rare earths La, Eu, Dy,...Y). The significant advantage of the
model is that for the two-cation systems, the proper solutes and the reaction parameters (ho and
so) of the single-cation oxides (e.g., the results for UO2-x and PuO2-x are directly applicable to
(U,Pu)O2-x). Even for multi-cation oxides such as (U,Pu,Am)O2-x [21], the "solutes" AmO2,
UO2, PuO2 and Am5/4O2 sufficed. The reactions involved in the modeling of this system were:
4Am5/4O2 + O2 = 5AmO2
3Pu4/3O2 + O2 = 4PuO2
1 Am
5/4O2 + 5 8 UO2 + O2 = 1516 Am1/3U2/3O4
4
2 Am
5/4O2 + 5 6 PuO2 + O2 = 5 3 Am1/2Pu1/2O3
3
In addition, in the ternary (U,Q)O2x or quarternary (U,Q,W)O2x oxides, the assumption of ideal
behavior of the solid "solution" is usually not sufficient to achieve a good fit to the p O 2 vs x data.
Simple nonideality formulations such as regular-solution theory (Eq (2.35)) are often used for
this purpose. Of course, this introduces another adjustable parameter into the analysis.
16.3 Microchemistry of nuclear oxides
Between the nanochemical approach of Sect. 16.2.2 and the macrochemical treatment such as
that described in Sect. 16.2.3 is a methodology that analyzes the chemical aspects of nuclear
oxides at an intermediate scale. This "microchemical" method combines the atomic picture of
point defects with equilibrium thermodynamics for characterizing their formation and the effect
of the environment on them. The output of this method are quantitative relations between oxygen
pressure, temperature and oxide stoichiometry.
16.3.1 Characteristics of point defects in nuclear oxides(See also Sects. 4.3 and 4.4)
Several criteria distinguish point defects:
14
1. They are either randomly distributed in the crystal as isolated entities or the individual defects
agglomerate into clusters.
2. Their concentrations are determined by thermodynamics (intrinsic) or by the presence of
impurity cations (extrinsic).
3. They are present on the anion (oxygen) sublattice and/or the cation (metal) sublattice
4. Structural defects
(a) a structural point defect is an ion missing from a sublattice (vacancy) or an ion in a
non-regular location (interstitial).
(b) the Frenkel defect is a vacancy and interstitial of the same ionic species;
(c) the Schottky defect is a vacancy on both the anion and cation sublattices
5. Electrical Defects
Electrical defects are cations of valences different from the normal valence in the stoichiometric
compound. For example, in oxides, oxygen is always O2-and uranium is usually in the 4+
oxidation state. However, under appropriate external environments, other oxidation states of
uranium appear. If the oxide is in air, U5+ and U6+ form; the product of mining of uranium ore is
U3O8, in which the uranium cations consist of two U5+ for every U6+ (to balance the negative
charges on O2-, viz, 25 + 16 = 82). The U3+ state is not as easily stabilized; exposing UO2 to
high-purity hydrogen gas at high temperature produces UO2-x, but U2O3 does not exist.
The ease with which nonstoichiometric oxides are stabilized is reflected in the metal-oxygen
phase diagram. The U-O phase diagram shown in Fig. 16.6 shows a wide range of single-phase
UO2+x ( to the right of the vertical dashed line at O/U = 2) that starts to open at ~ 500 K.
Hypostoichiometric urania, UO2-x does not appear until ~ 1500 K, an indication of the less stable
U3+ ion.
The corresponding phase diagram for the zirconium-oxygen system is shown in Fig. 28.16.
There is no hyperstoichiometric region because stable valence states greater than Zr4+ do not
exist. The very thin hypostoichiometric region shown hatched at the extreme right of the diagram
indicates that Zr3+ is difficult to produce.
Some metal ions have only one stable state. The only stable ion of aluminum is Al3+ ; barium and
strontium possess only divalent ions, and the alkali metals exist in compounds only as Na+, K+
and Cs With the exception of cerium, the rare earths (La, etc) exist only in the 3+ oxidation state.
The growth of rare-earth fission products in UO2 nuclear fuel is an important example of 3+
valence cations replacing a tetravalent metal ion in an oxide (Sect. 16.4.3). +.
16.3.2 Measures of concentrations in oxides
So far, five measures of defect concentrations in nonstoichiometric binary ionic solids have been
introduced. With O2- as the anion and U4+ as the cation, they are:
- oxygen-to-metal ratio, O/U, which appears as the abcissa in Fig 16.6
- deviation from stoichiometry For oxides such as UO2x these are related to O/U by:
O/U = 2-x (hypo), 2+x (hyper)
- atom fraction (or percent) of oxygen, as on the Zr-O phase diagram of Fig. 28.16:
(16.17a)
15
atom fraction O =
2x
2x
(hypo ),
(hyper )
3 x
3 x
(16.17b)
Fig. 16.6 The uranium-oxygen
phase diagram
- Kroger-Vink Notation (Ref. 18, pp 6 - 8)
This system applies to volumetric concentrations (moles or molecules per unit volume) and is
fully explained in Sect. 4.3.2. For UO2, this system is written as:
VO = doubly-positively charged anion vacancy
VU"" = vacancy on the cation (uranium) sublattice.
U I  = cation on a cation interstitial site.
O"I = oxygen ion on an anion interstitial site
U' U
= trivalent uranium ion (i.e., U3+) on a normal cation sublattice site
U U = U5+ on a cation sublattice site
UU = normal uranium ion (U4+) on a cation sublattice site
OO = oxygen ion on an anion sublattice site
Fixed-valence impurity ions in UO2 are designated in a similar manner:
Q M = pentavalent cation (Q5+) on a regular cation site.
Q'M = trivalent cation (Q3+) on a regular cation site.
16
In the Kroger-Vink system, electrical charges are deviations from the normal valences on the two
sublattices. Superscript dots are relative positive charges and superscript apostrophes indicate
relative negative charges. The subscript letter indicates the sublattice on which the defect is
located: sub O for the anion sublattice and sub U for the cation sublattice.
The concentrations are denoted by the symbol enclosed in brackets. The concentrations of
regular anion and cation sites are [OO]* and [UU]*, respectively, whereas the concentrations of
O and U ions on these sites are [OO] and [UU], respectively.
In addition to the regular sublattices, there are sublattices for anion and cation interstitials. These
would usually be designated by sub IO and sub IU, respectively. However, the fluorite unit cell
shown in Fig. 3.12a consists of 8 oxygen simple cubes only half of which are occupied by
uranium (Fig. 3.12b shows a single empty anion-cornered cube). The empty cubes are sites for
uranium interstitials, which are stabilized by the nearest-neighbor oxygen ions.
The sites for O2- interstitials are probably the same as those for the U4+ ions, namely the empty
cubes with corners occupied by oxygen anions. Thus the interstitial site density is denoted by
[I]* for both anions and cations. The relations between the site densities are:
[I]* = [UU]* = 1 2 [OO]*
(16.18)
- Site fraction is the fraction of the lattice sites available for a particular defect or ion that are
filled with that defect. Site fractions are related to Kroger-Vink concentrations as explained in
Sect. 16.3.2. The site fractions for the point defects in UO2x are:
- anion vacancy:
xVO = [ VO ]/[OO]*
(16.19a)
"
- anion interstitial:
xIO = [ O I ]/[I]*
- cation vacancy:
xVU = [ VU ]/[UU]*
- cation interstitial:
xIU = [ U I
- aliovalent cation
xU' = [U'U]/[UU]* or xU. = [ U U ]/[UU]*
(16.19b)
""

(16.19c)
]/[I]*
(16.19d)


- aliovalent impurity xQ' = [ Q' U ]/[UU]* or xQ. = [ Q U ]/[UU]*
(1619e)
(16.19f)
The Kroger-Vink concentration unit is required for writing the condition of electrical neutrality
in a defected crystal. The equilibria relating the partners in a defect are usually expressed in
terms of site fractions.
16.3.3 Point defects in UO2x
site-filling:
[OO]* = [OO] + [ VO ]
[UU]* = [UU] + [U'U] + [ U U ]
electrical neutrality:
"
2[ O I ] + [U'U] = 2[ VO ] + [ U U ]
(16.20a)
(16.20b)
(16.21)
Dividing by the cation site density [UU]* and with the aid of Eq (16.18) and the appropriate sitefraction definitions of Eqs (16.19), Eq (16.21) equation becomes:
17
2 xIO + xU' = 4xVO + xU.
(16.21a)
O [O O ] * [V ]  [O ]
16.22a)

 2x
U
[U U ] *
using Eqs (16.19a) and (16.19b), this becomes::
(16.22b)
x IO  2x VO  x
Figure 16.7 shows the three principal point defects in uranium oxide. The top cube contains an
anion Frenkel defect, the middle cube a Schottky defect and the lower cube represents the main
electrical defect
composition/stoichiometry:
..
O
"
I
18
1
Fig.Structural
16.7 Electrical
and structural
point defects in uranium dioxide
16.3.4
defect equilibria
Anion-Frenkel defects in UO2 are produced by moving an anion from a regular site on the anion
sublattice to a site on the interstitial sublattice:
"
OO = VO + O I
(16.23)
for which the law of mass action is:
K FO  x VO x IO  e s FO / R e   FO / RT
(16.24)
where the subscript FO denotes Frenkel defects on the anion (oxygen) sublattice. R = 8.314
J/mole-K is the gas constant and T is in Kelvins.
Substituting Eq (16.22b) into Eq (16.24) gives:


x VO  1 4 x 1  8K FO / x 2  1
and


x IO  1 2 x 1  8K FO / x 2  1
(16.25)
In the limit as x 0, these concentrations reduce to:
x VO  K FO / 2
x IO  2K FO
(16.25a)
Example
Estimates of the thermodynamic parameters for KFO in UO2 are sFO ~ 4 J/mole-K and
FO ~ 360 kJ/mole K. What is KFO at 1500oC and what are the site fractions of the anion point defects?
Eq (16.24) becomes:
KFO =1.7exp(-43000/T)
(16.24a)
at 1500oC (1773 K), Eq (16.24a) gives: KFO =510-11. If the oxide is exactly stoichiometric, the site
fractions of anion defects from Eq (16.25a) are: xVO = 510-6, xIO = 1.110-5.
Schottky defects:
UU + 2OO = 2 VO + VU'''' + (UU + 2OO)surface
(16.26)
are minor species in UO2, where anion Frenkel defects predominate. Because of the dominance
of the latter, their analyses above remain valid. All that is needed to determine the cation vacancy
fraction is to substitute the appropriate expression for xVO into the Schottky equilibrium constant:
K S  x 2VO x VU
(16.26a)
and solve for xVU.
UO2 - xVO is given by Eq (16.25a)
x VU  2 K S / K FO
(16.26b)
UO2-x - xVO  x/2 from Eq (16.22b)
xVU = 4KS/x2
(16.26c)
1
Cation Frenkel defects are neglected; the formation energy is too large (Table 16.1)
19
UO2+x - xIO  x from Eq (16.22b)
xVU = x2 KS/ K 2FO
(16.26d)
16.3.5 Electronic disorder - U4+ disproportionation
U4+ ions spontaneously donate an electron to a nearby cation, thereby changing the valence
states of the donor and the recipient. The equilibrium constant for the reaction:
2UU = U'U + U U
(16.27)
is:
K dis  x U ' x U.
(16.27a)
where x U' and x U. are the fractions of U 3 and U 5 on the cation sublattice .
Kdis is poorly known but the best estimate is:
Kdis = 0.04exp(-17000/T)
(16.27b)
Example: What are the fractions of the aliovalent cations in stoichiometric UO2 at 1500 oC?
For x = 0, Eq (16.22b) gives xIO = 2xVO, which, when substituted into Eq (16.21a), yields xU' = xU.
From Eq (16.27b), Kdis = 310-6, so Eq (16.27a) gives:
x U'  x U.  1.7 10 3
16.3.6 Gas-phase/defect equilibrium
In order to arrive at a connection between stoichiometry and oxygen potential, a final relation
between the point-defect concentrations and the environment is needed. This is supplied by the
equilibrium between the solid and the oxygen partial pressure in the gas phase, for which the
reaction is:
OO + 2UU = 1 2 O2(g) + 2U'U + VO
(16.28)
In the reverse of this reaction, an oxygen atom ( 1 2 O2) enters a vacant anion sublattice site ( VO )
at the same time extracting an electron from each of two U3+ cations (2U'U). The result is an O2ion on a previously-vacant anion sublattice site (OO) and two U4+ cations on regular cation sites
(2UU). The equilibrium condition for the reaction is expressed by:
K RE 
[Vo.. ][ U ' U ] 2 p O 2
[O O ][ U U ] 2
the subscript RE means that the equilibrium constant refers to a "REDOX", or
oxidation/reduction reaction. KRE is given by:
KRE = 2106exp(-1105/T)
(16.29)
(16.29a)
The numerical values correspond to an entropy of reaction sRE = 120 J/mole-K and a reaction
enthalpy RE = 830 kJ/mole. Both of these large positive values correspond to the high-enthalpy,
disordered (especially O2(g)) right-hand side of reaction (16.28) compared to the stable, tightlybound normal lattice ions on the left-hand side. At 1500oC. KRE = 710-19.
p O 2 in Eq (16.29) is fixed by one of the methods described in Sect. 16.2.1.
20
Because the point defect site fractions are < 0.002 at 1500oC (see examples above), Eqs (16.20a)
and (16.20b) can be approximated by [OO]* = [OO] and [UU]* = [UU] and Eq (16.29) reduces to:
K RE  x 2U ' x VO p O 2
(16.29b)
16.3.7 Nonstoichiometry in UO2 x fixed by p O 2
"
The principal point defects in hyperstoichiometric urania are anion interstitials O I and
pentavalent uranium, UU.. In Eq (16.29b), xVO is eliminated using Eq (16.24) and xU' is
expressed in terms of x U. via Eq (16.27a), yielding:
x IO x U .  B p O 2
(16.30)
where:
2
B  K FO K dis
/ K RE
(16.31)
In order to determine the relation between the nonstoichiometry parameter x in UO2+x and the
oxygen partial pressure, a relation between xU. and xIO is needed to accompany Eq (16.30). This
relation is supplied by Eq (16.21a) in which the same substitutions yield:
2xIO + Kdis/xU. = 4KFO/xIO + xU.
(16.21b)
Eliminating xIO between Eqs (16.21b) and (16.30) yields.
2B p O 2
x
2
U.

K dis
4K FO 2

x .  x U.
x U.
B p O2 U
(16.32)
or:
B p O2 
1
4


32K FO

x 2U . x U .  K dis / x U .  1 

1
2




x
.

K
/
x
.
U
dis
U


(16.32a)
The + sign applies when xU. > Kdis/xU., or in the hyperstoichiometric region where U5+
electrically balances the anion interstitials that represent the excess oxygen.
Combining Eqs (16.21a) and (16.22b) and replacing xU' using Eq (16.27) produces:
x=
and solving for
xU.
yields:
1
2
(xU. - Kdis/xU.)

x U .  x 1  1  K dis / x 2
(16.33)

(16.33a)
Specifying the temperature fixes the three equilibrium constants. Two applications of these
equations are often encountered:
Case I x is specified. What is p O 2 ?
xU. is
calculated from Eq (16.33a) then substituted into Eq (16.32a) for the oxygen pressure.
21
Case II p O 2 is fixed. What is x?
Eq (16.32) is solved for xU. (numerically) which is then substituted into Eq (16.33) to give x.
The results of the computations for these two cases are plotted as ln p O 2 vs log(x) in Figure
16.8. Of equal interest as the curve proper are the three regions of the variable x in which the
curve can more-or-less be represented by straight lines. The literature on this topic is invariably
expressed in these terms.
Slope 0 - Below x ~ 10-5, the oxide is effectively stoichiometric. From Eq (16.22b) at x = 0,
xIO = 2xVO, from which Eq (16.21a) gives xU' = xU. . Equation (16.25a) is x IO  2K FO and
from Eq (16.27a), xU. = Kdis. Substituting these into Eq (16.30) yields:
2K 2RE
2 (7  10 19 ) 2
(16.34)
p O 2 x 0 

 2.2  10 15 atm
2
11
6 2
K FO K dis (5  10 )(3  10 )
which corresponds to the horizontal dashed line in the lower left-hand corner of Fig. 16.8.
 
Slope 1/2 - Following the slope 0 portion of the curve is a region 10-5  x  ~610-4 in which the
curve can be roughly represented by a line of slope 1/2. This line originates from the values of
the terms in the electrical neutrality equation in the general solution for this zone. Here, xVO and
xIO are much smaller than xU. and xU', so Eq (16.21a) reduces to xU. ~ xU' = Kdis. Equation
(16.30) becomes:
K K
(16.34b)
x  x IO  FO dis p O2  260 p O2
K RE
Slope 1/6 - in the range ~610-4  x  ~ 410-3, the oxide is sufficiently hyperstoichiometric that
"
the only intrinsic defects of significance are the anion interstitial O I and the pentavalent cation
U . . Substituting what remains of the charge balance (Eq (16.21a)), xU. = 2xIO and xIO = x
(Eq(16.22b)) into Eq (16.30) yields the approximation:
 K K2 
x   FO dis 
 4K RE 
1/ 3
p1O/26  0.054 p1O/26
(16.34c)
which corresponds to the dashed line in this range.
The ln p O 2 vs stoichiometry x plots of Figs 16.5 and 16.8 are distinctly different in shape,
although the data they represent are for temperatures that differ by ~ 300o. The entire curve in
Fig. 16.8 corresponds to the data to the left of the dividing line in Fig. 16.5, which roughly fits:
x p1O/24 . This dependence of the oxygen pressure on stoichiometry is intermediate between
those shown in Fig. 16.8. However, neither plot shows the very steep decrease in p O as exact
2
stoichiometry is approached (Fig. 16.9a). The reason for this failure is the lack of modeling the
chemistry (macro- or micro-) of the hypostoichiometric oxide.
22
-10
-15
-20
lnpO
2
slope 1/6
-25
-30
slope 1/2
-35
slope 0
-40
-6
-5
-4
-3
-2
log(x)
Fig. 16.8 O2 pressure in equilibrium with hyperstoichiometric urania at 1500oC
Kdis = 310-6; KFO = 510-11; KRE = 710-19
16.4 Mixed Oxides
It has long been known that the oxygen potential of mixed oxides such as (U,Th)O2+x consisting
of cations of the same valence is very little different from that of UO2+x. On the other hand, the
oxygen pressure over (U,La)O2+x is very different from that of UO2+x of the same x. Thorium is a
very stable 4+ cation, and so does not affect the uranium valence in the mixed oxide. However,
trivalent lanthanum on the cation sublattice is effectively negative (in Kroger-Vink notation) and
so must be balanced by an effective positive charge. This is supplied during fabrication of the
mixed oxide by conversion of some U4+ to U5+ (UU.) or by creation of anion vacancies,
VO , which produces a large increase in the equilibrium oxygen pressure.
The oxygen potentials of mixed oxides wherein one cation is fixed-valence and the other
possesses a variable valence are functions of the latter's average valence. Moreover, the oxygen
potential is independent of the fixed-valence element. Examples are:
- (U,Ce)O2-x - uranium has a fixed valence of 4+ (because of the difficulty of stabilizing U3+
except at high temperature) and the cerium valence lies between 3+ and 4+[20].
- (U,Gd)O2+x - Gd has a fixed valence of 3+ and the uranium valence varies between
4+ and 5+[21].
- (U,Th)O2+x - Th valence is 4+ and U valence lies between 4+ and 5+[22].
-(U,Y)O2+x - Y3+ is mixed with either U4+ and U5+ or U4+ and anion vacancies [23].
23
16.4.1 Mean valence rule
In the cases where the positive charge consists of a mixture of U3+, U4+ and U5+, the oxygen
potentials are uniquely a function of the average valence of the uranium ions. This
approximation is known as the mean valence rule [24], or valence-control rule [25,26].
Moreover, this rule applies to the pure oxide, UO2+x as well, as to mixed oxides, and herein lies
its utility. Because so much effort has been expended in understanding the thermochemistry of
the pure oxide, the ability to apply this knowledge base to mixed oxides is a great advantage.
16.4.2 Oxygen pressure of (U1xQ Q xQ )O 2  z
In this general case, Q is a fixed-valence ion (either Q3+ or Q5+) mixed in the oxide at a cation
site fraction of xQ. In the following development, xVO is eliminated in terms of xIO using
Eq (16.24) and xU. replaces xU' via Eq (16.27a)
The cation site-filling condition that replaces Eq (16.20b) is:
1 = xQ + xU + xU. + Kdis/xU.
(16.20c)
where xU is the cation site fraction of U4+.
The equation of electrical neutrality that replaces Eq (16.21b) is:
Kdis/xU. + 2xIO = 4KFO/xIO + xU. + nxQ
(16.21c)
where n = +1 if the impurity ion is pentavalent and n = -1 if it is trivalent.
Equation (16.22b) is
x IO  2K FO / x IO   z
(16.22c)
Given the temperature (which fixes the equilibrium constants), the cation site fraction xQ, and
the oxygen-to-metal ratio of the mixed oxide (z), the procedure for determining p O 2 is as
follows:
1. Solve Eq (16.22c) for xIO


x IO  1 2 z 1  8K FO / z 2  1
(16.22d)
The + sign is used for (U1xQ Q xQ )O2  z and the - sign for (U1xQ Q xQ )O2  z
2. Solve Eq (16.21c) for xU.


x U.  1 2 b 1  4K dis / b 2  1
(16.21d)
where
b = nxQ +4KFO/xIO - 2xIO
(16.35)
24
The + sign in Eq (16.21d) applies for b < 0 and the - sign for b > 0.
3. The mean uranium valence is:
VU 
3x U '.  4x U  5x U .
x U '.  x U  x U .
(16.36)
Eliminating xU between this equation and Eq (16.20c) and expressing xU' in terms of xU. using
Eq (16.27) yields:
x .  K dis / x U .
VU  4  U
(16.36a)
1 xQ
4. The mean-valence rule is invoked and applied to UO2x , where xU. is determined from VU by
first solving Eq (16.36a) (with xQ removed):


4K dis
(16.37)
x U .  1 2 VU  4  1 
 1
2
(VU  4)


The + sign applies when VU > 4 and the - sign when VU < 4.
5. then determining p O 2 from Eq (16.32a).
Example At 1500oC, what is the equilibrium oxygen pressure for (U0.99Q0.01)O2 z? The impurity ion is
Q3+ . The equilibrium constants are given in the caption of Fig. 16.8.
Following the above steps produces the plots in Figs.16.9a- 16.9c.
Fig. 16.9a Oxygen pressure in
equilibrium with (U0.99Q0.01)O2z
at 1500oC; O/M = 2+z
The extremely rapid decrease in
p O 2 as O/M approaches 2.00 is characteristic of all mixed oxides
involving uranium as the major constituent. The drop-off in the hypostoichiometric region is slower than
in UO2 x because of the presence of the trivalent impurity ion.
25
Fig. 16.9b Point defect
site fractions in
(U0.99Q0.01)O2z at 1500oC
O/M = 2+z
The structural defects (anion vacancy and interstitial) are not affected by the trivalent impurity and the
curves for xIO and xVO cross at O/M = 2, as they would in pure UO2  x. The charge defects (U3+ and U5+),
however, are sensitive to the presence of Q3+, which causes the U5+ site fraction to begin increasing well
before exact stoichiometry. These two point-defect curves cross at ~ 1.995.
Fig. 16.9c
Mean uranium valence in
(U0.99Q0.01)O2z at 1500oC
as a function of O/M = 2+z
The U valence decreases as the oxide becomes hypostoichiometric. Because of the trivalent impurity
cation, VU remains > 4.00 well below O/M < 2.00.
26
16.4.3 Defects in Irradiated UO2
Irradiation of nuclear fuel (UO2) produces a vast array of fission products that interact with the
remaining fuel in a very complicated way. Because of its importance to the performance of fuel
elements as the burnup increases, the chemistry of irradiated fuel has been extensively studied
[27 - 30]. For the purposes of this book, this complex system is simplified in order to illustrate
how burnup changes the oxygen potential of the fuel. The following are neglected:
1. Molybdenum partitioning between an oxide phase (e.g. BaMoO4 - Mo is not soluble in UO2)
and the noble-metal phase (with Rh, Ru, Pd) [Ref. 1, Sect. 12.4.2]
2. Oxygen tied up in ternary oxide phases, zirconates, molybdates, uranates.
3. Clusters formed by binding of oxygen vacancies to soluble fission products.
4. Blockage of lattice sites by insoluble fission products (e.g., Xe)
5. Schottky defects - consequently, the cation sublattice is structurally (but not electrically)
perfect
6. Uranium Frenkel defect formation (only oxygen Frenkel defects are treated)
7. Details of the retention of fission products in solution - all soluble fps (especially the rareearths and Zr) are lumped into a single pseudo-species with fission yield Yfp (<1) and valence
Vfp (<4). The remaining fission products are insoluble in UO2 and are rejected from the solid.
8. The effect of irradiation on point-defect concentrations. This is a very significant
simplification, but it permits thermodynamics to be applied.
9. Decrease of the free volume in the fuel rod due to gap closure caused by fuel swelling.
10. Reaction of oxygen released from the fuel with the cladding inner surface
- The fresh fuel consists of [UU]* ions of U4+ per unit volume filling all cation lattice sites and
[OO]* oxygen ions in all anion lattice sites.
- F t fissions per unit volume2 replace U4+ on cation lattice sites with Yfp F t fission-product ions,
leaving (1 - Yfp) F t empty cation sites.
- the remaining cations collapse to
[UU]o = [UU]* - (1 - Yfp) F t = [UU]*[1 - (1-Yfp)]
structurally-perfect sublattice sites per unit volume.  is the burnup expressed as:
 = F t /[ U U ] *
which is the fissions per initial metal atom, or fima.
(16.38)
(16.39)
The cation site fraction of soluble fission products is:
x fp 
Yfp F t
[ U U ] * (1  Yfp )F t

Yfp 
1  (1  Yfp )
(16.40)
cation site-filling
The cation sublattice is structurally perfect, but contains soluble fission product ions and uranium
ions of valences 3+, 4+ and 5+:
1 = xfp + xU' + xU + xU.
(16.41)
2
F is fissions/s per unit volume and t is time in seconds
27
anion site-filling
The crystal structure of the solid is still fluorite, so the concentration of anion sublattice sites in
the collapsed structure is:
[OO]o = 2[UU]o
(16.42)
where [UU]o is given by Eq (16.38).
Filling of the anion sublattice sites gives:
[OO]o = [OO] + [ VO ]
(16.43)
Oxygen conservation
The collapse of the cation sublattice to eliminate the empty sites results in rejection of oxygen
from the solid. This oxygen reappears in the gas phase of the free volume in the fuel element:
"
[OO]* = [OO] + [ O I ] + [Og]
(16.44)
where Og represents gas-phase oxygen (actually as O2). Its concentration is negligibly small and
Og can henceforth be omitted from the oxygen balance.
Eliminating [OO] by Eq (16.43), [OO]o by Eq (16.42) and [UU]o by Eq (16.38) yields:
"
2 = 2[1 - (1-Yfp)] - [ VO ]/[UU]* + [ O I ]/[UU]*
The site fraction of vacancies on the anion sublattice is
[VO.. ]
[VO.. ]
[VO.. ] /[ U U ] *
x VO 

 12
[O O ]o 2[ U U ]o
1  (1  Yfp )  
and the site fraction of anion interstitials on the interstitial sublattice is:
[O 'I' ] /[ U U ] *
[O 'I' ]
x IO 

[ U U ]o 1  (1  Yfp )  
Using the above two equations and Eq (16.24) in Eq (16.45) yields:
(16.45)
1 = (1 - KFO/xIO +
(16.46)
1
2
xIO)[1 - (1-Yfp)]
Electrical neutrality
In Kroger-Vink concentration notation, electrical neutrality in the irradiated fuel is:
"
2[ VO ] + [UU.] = (4-Vfp) [fp] + [UU'] + 2[ O I ]
or, with Eqs (16.24) and (16.27a):
4KFO/xIO + xU. = (4-Vfp)xfp + Kdis/xU. + 2xIO
(16.47)
(16.48)
xVO and xU' are expressed by Eqs (16.24) and (16.27a), respectively.
The solution is straightforward. Equation (16.46) is solved for x IO as a function of burnup  and
the result used in Eq (16.48) to calculate xU.. The oxygen pressure is given by Eq (16.30).
Figure 16.10 shows these results plotted as functions of burnup (bu), expressed in the MWd/kgU,
units more common than fima for LWRs.  and bu are related by: = 1.2210-3bu.
28
.
xU
xU'
Fig. 16.10 Point defect site fractions and oxygen pressure as functions of burnup.
Equilibrium constants from the caption of Fig. 16.8
It is clear from this graph that fission is an oxidizing process, although because of neglecting O2
reaction with the cladding inside surface, the buildup of p O 2 on the graph is exaggerated. The
relative negative charges introduced by the fission products (xfp) and the anion interstitials (xIO)
together electrically neutralize the positively-charged pentavalent uranium (xU.).
16.5 Point-defect clustering in UO2+x
In 1964, Willis' neutron-diffraction measurements [13] on single-crystal UO2.12 at 800oC
definitively showed that anion vacancies and interstitials existed in distinct complexes, or
clusters, rather than as individual point defects scattered randomly in the fluorite lattice, as has
been assumed in the preceding sections. Subsequently, the structure of the cluster was shown
[14] to be that depicted in Fig. 16.11.
This cluster contains two oxygen interstitial ions, which are labeled "2" in Fig. 16.11. The two
ions are located along a <110> direction that passes through a side of the inner cube in the
diagram. Because of the close packing of anions in this region, the two closest lattice anions
("1") move away from the interstitials along <111> directions. Simultaneously, four nearby U4+
ions are oxidized to U5+ to balance the negative charges introduced by the two oxygen
interstitials. The cluster consists of 4U. +2 VO +2 OI1 +2 O I 2 point defects. This cumbersome
combination is more conveniently denoted by (4:2:2:2).
An intrinsic equilibrium that relates the cluster concentration to those of the other point defects
is:
.
"
4UU + 2OO +2 O I = (4:2:2:2)
(16.49)
The four U5+ ions supply this component of the cluster; the two normal anions (OO) become the
No. 2 interstitials and the two anion interstitials enter the cluster without change.
29
Fig. 16.11 Willis (4:2:2:2) cluster
The next equilibrium involving the cluster involves extraction of an electron from a neighboring
U4+ ion:
.
UU + (4:2:2:2) = (4:2:2:2)' + UU
(16.50)
Unlike the structure of the cluster deduced by Willis, the electrical characteristic contained in the
above equilibrium has not been determined by experiment; rather, as shown below, it is needed
to produce the observed dependence of the equilibrium oxygen pressure on stoichiometry.
Adding the preceding two reactions gives:
.
"
UU + 3UU + 2OO +2 O I = (4:2:2:2)'
for which the law of mass action (in terms of site fractions) is:
KW 
x W'
x 3U. x 2IO
(16.51)
(16.52)
30
where xW is defined as the concentration of defects divided by the cation site density:
x W' 
[( 4 : 2 : 2 : 2)' ]
[U U ] *
(16.53)
Expressing the equilibrium that determines the cluster concentration by Eq (16.51) does not imply that
this reaction is the actual mechanism of cluster formation. For example, an alternative formation reaction
is [31- 34]:
.
2OO + O2 + 5UU = (4:2:2:2)' + UU
(16.54)
In this formulation of the equilibrium, the No. 2 oxygen interstitials in the cluster are obtained directly
from gaseous oxygen. The law of mass action for this reaction is given by:
KD 
x W' x U .
p O2
(16.55)
The connection between the equilibrium constants KW and KD is obtained by adding to reaction (16.51):
twice reaction (16.23):
"
2OO = 2 O I + 2 VO
2(16.23)
and 4 times reaction (16.27):
.
8UU = 4UU' + 4UU
4(16.23)
and subtracting twice reaction (16.28):
2 VO + 4UU + O2 = 2OO + 4UU
-2(16.28)
This combination produces reaction (16.54). The corresponding relation between the equilibrium
constants is:
KD 
4
K W K 2FO K dis
K 2RE
or
KW 
K 2RE K D
4
K 2FO K dis
(16.56)
Using the values of KRE, KFO and Kdis that were employed in the preceding analysis of point defects in the
absence of clustering and an approximate value of KD ~ 105 at x = 0.1 and 1500oC [Ref. 31, Fig.3] gives:
KW ~
(7  10 19 ) 2 (10 5 )
 2  1011
11 2
6 4
(5  10 ) (3  10 )
(16.56a)
In the following, conversion of concentrations [i] to site fractions xi employs the identifications
[OO]  [OO]* and [UU]  [UU]*. These approximations are justified because the concentrations
of the point defects are small compared to those of the normal ions.
The anion site-filling condition that modifies Eq (16.20a) is:
[OO]* = [OO] + [ VO ] +2 [(4:2:2:2)']
The last term on the right accounts for the two vacant anion sites in the cluster.
The total concentration of oxygen ions in the solid is:
(16.57)
31
"
[O]tot = [OO] + [ O I ] +4[(4:2:2:2)']
(16.58)
The last term on the right accounts for the four interstitials in the cluster.
The cation sublattice is perfect (structurally). In the presence of clusters, hyperstoichiometry is
expressed by: 2+x = [O]tot/[UU]* or, eliminating [OO] from the above two equations,
Eq (16.22b) becomes:
x = xIO - 2KFO/xIO + 2xW'
(16.59)
The condition of electrical neutrality that replaces Eq (16.21a) (neglecting the U3+ contribution)
is:
2KFO/xIO + xU. = 2xIO + xW'' + Kdis/xU.
(16.60)
in which xVO has been eliminated by Eq (16.24) and xU' by Eq (16.27a).
Simultaneous solution of Eqs (16.52), (16.59) and (16.60) yield xW', xIO and xU. as functions of
the nonstoichiometry variable x. These functions are displayed in Fig. 16.12.
Fig. 16.12 Point defects in hypostoichiometric UO2 at 1500oC
This plot reveals how far towards stoichiometric UO2 the cluster survives before decomposing
into individual anion vacancies. The concentration of the [4:2:2:2] cluster drops rapidly with
decreasing x while the anion vacancy site fraction xIO slowly grows. The two curves cross at
x ~ 10-3, thereby attesting to the tenacity of the cluster. At about the same concentration, x IO
reaches a maximum and begins to decrease as well. This is due to the reduced need to provide
negative charges to balance the U5+ concentration, which as shown by the dashed curve in the
figure, is also decreasing.
32
16.6 Properties of UO2 dependent upon point defects
Anion Frenkel defects greatly influence the diffusivity of oxygen ions in UO2. The vacancies on
the cation sublattice produced by the Schottky process are responsible for the diffusion
coefficient of uranium ions in UO2.
Another property affected by the creation of point defects in ionic crystals is the heat capacity.
At relatively low temperatures (but not approaching absolute zero), the heat capacity is nearly
temperature-independent with a value of 3R per gram atom (R is the gas constant). Per mole of
UO2, this classical value of the heat capacity is (CV)lattice = 9R. This contribution to the heat
capacity arises from the vibrations of the atoms in the lattice.
The creation of structural point defects by the anion Frenkel process (Eq (16.23)) provides an
additional component to the heat capacity, as does the electrical defect arising from
disproportionation of U4+ (Eq (16.27)):
CV = (CV)lattice + (CV)struct+ (CV)elect.
(16.63)
The extra energy in the solid due to creation of the anion Frenkel defects is the concentration
times the energy FO (J/mole of defects ). The excess energy due to the point defects is:
eFO = xIOFO = FO 2K FO
J /mole UO2
The excess heat capacity arising from the point defects is the derivative of eFO with respect to
temperature, or, using Eq (16.24) for KFO:
C V struct 1 de FO 1   FO  2
(16.64)



 K FO
R
R dT
2  RT 
with FO = 360 kJ/mole and KFO given by Eq (16.24a).
The analogous contribution from U4+ disproportionation is:
C V elect. 1 de dis   dis  2


 K dis
R
R dT  RT 
(16.65)
where dis = 140 kJ/mole and Kdis is given by Eq (16.27b).
The two defect contributions and the total heat capacity are shown in Fig. 16.13. The electrical
defect contribution is larger than the structural component, but with the equilibrium constants
used here, together they add very little to the classical heat capacity. At 3000 K, the combined
effect is only 7% of the total.
33
Fig. 16.13 Point defect components of the specific heat of UO2