Solutions to exercises
25
4. Expressions for associated defect pairs written in Kröger-Vink notation and
with values given for Q e , Q r , and Q n :
Structure element
Q r
Q n
Q e Notation
a. Gd
3+
ion in substitution position
in CeO 2 and an oxygen vacancy
+ 3
+ 4 − 2
= + 2
+ 1 (Gd Ce V O )
•
b. Chlorine and sodium vacancies
in NaCl
0
− 1 + 1
= 0
0 (V Cl V Na )
#
Solution 1.2 – Notation for doping reactions
A – MgO doped with Al 2 O 3
1. For MgO, the equilibrium associated with Schottky disorder is expressed as:
0 $ V ′′
Mg + V
••
O
The reaction for doping MgO by Al 2 O 3 is written as follows:
a. with Al in an interstitial position:
Al 2 O 3 $ 2Al
3•
i + 3O
#
O + 3V ′′
Mg
or
Al 2 O 3 + 3V
••
O $ 2Al
3•
i + 3O
#
O
The electroneutrality equation is
2[V
••
O ] + 3[Al
3•
i ] + p = 2[V ′′
Mg ] + n
where n and p respectively denote the concentrations of electrons and
holes.
b. with Al in the substitution position of magnesium:
Al 2 O 3 $ 2Al
•
Mg + 3O
#
O + V ′′
Mg
or
Al 2 O 3 + V
••
O $ 2Al
•
Mg + 3O
#
O
In this case, the electroneutrality equation is
2[V
••
O ] + [Al
•
Mg ] + p = 2[V ′′
Mg ] + n
2. The cationic sub-lattice in MgO is face-centered-cubic (fcc), where the
octahedral sites are occupied by oxygen, leading to a compact structure.
Because the cationic radius of magnesium is close to that of aluminum,
substitution is favored.