164
3 – Transport in ionic solids
R is the ideal gas constant, σ 1 T 1 , T 1 and σ 2 T 2 , T 2 are the coordinates
of two points 1 and 2 situated within the same linear temperature
range. We obtain the following activation energies:
E A = 2.01 eV
E B = 0.65 eV
2 The temperature range with the smallest activation energy (range B)
corresponds to pure migration. We thus deduce that the activation energy E m for migration is
E m = E B = 0.65 eV
In this range, the expression for the ionic conductivity as a function of
temperature is
T e
0
RT
E m
σ
σ
=
−
where σ 0 is a constant. The ionic conductivity is implemented by fluorine vacancies of extrinsic origin (doping with oxygen and impurities).
In range A, the expression for the activation energy is
E
E
E
2
1
A
m
f
=
+
where E f is the formation energy for the Frenkel pair corresponding to
the following reaction:
F
#
F m V
•
F + F ′
i
We thus deduce
.
E
e V
2 72
f =
In this temperature range, the expression for ionic conductivity as a
function of temperature is
T e
0
RT
E
2
E
m
f
σ
σ
=
−
+
l
c. Conductivity as a function of oxygen partial pressure in range B
At constant temperature, the expression for ionic conductivity in CaF 2 is
σ = F # u V
•
F
# [V
•
F ]
where u V
•
F
and [V
•
F ] denote the electric mobility and the fluorine vacancy
concentration, respectively.