30
1 – Description of ionic crystals
For δ, the number of anionic sites is twice that of cationic sites, from
which we deduce
δ + 2 − 2
1
y = 2
or
δ = 2
1
y
Thus, (ZrO 2 ) 1−y (YO 1.5 ) y may be written in the form
Zr 1−y Y y O 2– 2
1 y V
2
1 y
Numerical evaluation of (ZrO 2 ) 1−y (YO 1.5 ) y for x = 0.08 gives
Zr
Y
O
V
.
.
.
.
0 852 0 148 1 926 0 074
Solution 1.4 – Calculation of defect concentrations
We begin by writing the reaction for doping CeO 2 by CaO:
CaO $ Ca ′′
Ce + O
#
O + V
••
O
or xCaO + (1 − x)CeO 2 $ xCa ′′
Ce + (1 − x)Ce
#
Ce + (2 − x)O
#
O + xV
••
O
Note that the concentration of oxygen vacancies is the same as that of Ca ′′
Ce .
The fluorite-type structure is shown in figure 5.
Figure 5 – Schematic diagram
of the ideal structure of ceria.
A unit cell contains four Ce sites and eight O sites. The number of sites occupied by vacancies is 4x per unit cell, which gives a vacancy concentration of
N
x
C
V
4
Av
V
m
=
#
where N Av and V m denote Avogadro constant number and the unit-cell volume,
respectively. Numerical evaluation gives
.
.
.
.
C
m ol cm
6 02 10
5 415 10
4 0 1
4 185 10
V
23
8 3
3
3
#
#
#
#
=
=
−
−
−
^
^
h
h
2
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&H