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1 – Description of ionic crystals
Exercise 1.9 – Crystallographic and thermodynamic study
of thorium dioxide ThO 2
Thorium dioxide ThO 2 crystallizes as a cubic system. The symmetry of thorium
dioxide is described by the space group Fm3m. The structure is composed of
a face-centered-cubic lattice of Th
4+
ions, with the tetrahedral sites occupied
by O
2–
ions. Anionic Frenkel-type disorder dominates in ThO 2 .
A – Crystallographic study
1. Show the structure of ThO 2 in perspective and in standard [001] projection.
Is the stacking compact for the face-centered-cubic (fcc) Th
4+
ions?
2. Calculate the lattice parameter of thorium dioxide.
Data
r(Th
4+
) = 1.02 A c r(O
2−
) = 1.40 A c
3. Study the sites occupied by interstitial oxygen atoms.
B – Equilibrium with gas phase
1. Pure ThO 2
a. Write the various equilibria and the relationships between the concentrations
of structure defects, the equilibrium constants, and, where appropriate,
the oxygen partial pressure.
Remarks:
2 write only the independent equilibria,
2 use the following notation:
• K e for the electronic equilibrium constant,
• K g for the equilibrium constant for the gas phase,
• K AF for the equilibrium constant for the formation of intrinsic
defects.
2 K g and K AF are defined for oxide ions in their normal position and
with an activity equal to one.
b. Establish the laws of variation in concentration of each type of defect
as a function of oxygen partial pressure. Assimilate the activity to the
concentration of each species. Consider successively the low-pressure,
intermediate-pressure, and high-pressure domains by making the
1 – Description of ionic crystals
Exercise 1.9 – Crystallographic and thermodynamic study
of thorium dioxide ThO 2
Thorium dioxide ThO 2 crystallizes as a cubic system. The symmetry of thorium
dioxide is described by the space group Fm3m. The structure is composed of
a face-centered-cubic lattice of Th
4+
ions, with the tetrahedral sites occupied
by O
2–
ions. Anionic Frenkel-type disorder dominates in ThO 2 .
A – Crystallographic study
1. Show the structure of ThO 2 in perspective and in standard [001] projection.
Is the stacking compact for the face-centered-cubic (fcc) Th
4+
ions?
2. Calculate the lattice parameter of thorium dioxide.
Data
r(Th
4+
) = 1.02 A c r(O
2−
) = 1.40 A c
3. Study the sites occupied by interstitial oxygen atoms.
B – Equilibrium with gas phase
1. Pure ThO 2
a. Write the various equilibria and the relationships between the concentrations
of structure defects, the equilibrium constants, and, where appropriate,
the oxygen partial pressure.
Remarks:
2 write only the independent equilibria,
2 use the following notation:
• K e for the electronic equilibrium constant,
• K g for the equilibrium constant for the gas phase,
• K AF for the equilibrium constant for the formation of intrinsic
defects.
2 K g and K AF are defined for oxide ions in their normal position and
with an activity equal to one.
b. Establish the laws of variation in concentration of each type of defect
as a function of oxygen partial pressure. Assimilate the activity to the
concentration of each species. Consider successively the low-pressure,
intermediate-pressure, and high-pressure domains by making the