The binomial distribution is given by:
P p k; n; p
ð
Þ¼
n
k
p
k 1 À p
ð
Þ
nÀk
ð9:8Þ
where P p is probability to getting k successes in n trials and p is the probability of
success.
In ABO 3 crystal structure, every iron cation has six other cations in the
B-sublattice in his coordination. The probability of finding iron in the coordination
is 0.75 and niobium 0.25 based on the stoichiometry of the considered material.
Thus, based on the binomial distribution, (Eq. 9.9) can be the calculated probability
of different Fe and Nb configurations in the iron coordination. From all the possible
configurations, we selected only those which probability exceeds about 5%. The
most probable configurations are (5, 1), (4, 2), (6, 0), (3, 3) where the first number
in the bracket represents the number of Fe and the second Nb ions in the coordination. The sum of the selected probabilities is 0.963, and this value is used to
renormalize them. It is assumed that the particular neighborhoods of iron atoms
produce the individual subspectra that contribute to the overall Mössbauer effect
pattern and areas of the subspectra should follow the probabilities. Therefore, the
obtained spectrum can be fitted using four Zeeman sextets of the Lorentzian line
shape. The calculated probabilities and the Bi 0.5 Pb 0.5 (Fe 0.75 Nb 0.25 )O 3 hyperfine
interaction parameters are presented in Table 9.2.
We can easily find out that the probabilities of different configurations of the
measured iron neighbors in B-site follow very well area of the subspectral components. This clearly confirms the random distribution of Fe
3+ and Nb
5+ cations in
B-sites. This kind of analysis gives the possibility to assign the specific configuration set of the hyperfine interaction parameters. Thus, deeper conclusions concerning local iron environment and properties can be drawn, and this will be
discussed further.
Magnetic Ordering Temperature
Mössbauer spectroscopy can be used to obtain magnetic ordering temperature. To
do this, at first, one should somehow estimate the range of searching the temperature. In case of the Bi 0.5 Pb 0.5 (Fe 0.75 Nb 0.25 )O 3, we can assume that the substitution
of Fe
3+ by Nb
5+ dilutes the magnetic sublattice and the magnetic ordering
Table 9.2 Hyperfine interaction parameters of the Bi 0.5 Pb 0.5 (Fe 0.75 Nb 0.25 )O 3 at 77 K [12]
No
(Fe, Nb)
P p
Area (%)
IS (mm/s)
B hf (T)
QS (mm/s)
1.
5, 1
0.370
36.1(5)
0.453(5)
53.30(4)
–0.013(6)
2.
4, 2
0.308
30.2(5)
0.517(8)
51.59(5)
–0.010(7)
3.
6, 0
0.185
19.4(5)
0.631(10)
53.34(7)
–0.029(10)
4.
3, 3
0.137
14.3(5)
0.532(10)
49.79(10)
0.030(14)
9 Mössbauer Spectroscopy of Magnetoelectric Perovskite Oxides
293
P p k; n; p
ð
Þ¼
n
k
p
k 1 À p
ð
Þ
nÀk
ð9:8Þ
where P p is probability to getting k successes in n trials and p is the probability of
success.
In ABO 3 crystal structure, every iron cation has six other cations in the
B-sublattice in his coordination. The probability of finding iron in the coordination
is 0.75 and niobium 0.25 based on the stoichiometry of the considered material.
Thus, based on the binomial distribution, (Eq. 9.9) can be the calculated probability
of different Fe and Nb configurations in the iron coordination. From all the possible
configurations, we selected only those which probability exceeds about 5%. The
most probable configurations are (5, 1), (4, 2), (6, 0), (3, 3) where the first number
in the bracket represents the number of Fe and the second Nb ions in the coordination. The sum of the selected probabilities is 0.963, and this value is used to
renormalize them. It is assumed that the particular neighborhoods of iron atoms
produce the individual subspectra that contribute to the overall Mössbauer effect
pattern and areas of the subspectra should follow the probabilities. Therefore, the
obtained spectrum can be fitted using four Zeeman sextets of the Lorentzian line
shape. The calculated probabilities and the Bi 0.5 Pb 0.5 (Fe 0.75 Nb 0.25 )O 3 hyperfine
interaction parameters are presented in Table 9.2.
We can easily find out that the probabilities of different configurations of the
measured iron neighbors in B-site follow very well area of the subspectral components. This clearly confirms the random distribution of Fe
3+ and Nb
5+ cations in
B-sites. This kind of analysis gives the possibility to assign the specific configuration set of the hyperfine interaction parameters. Thus, deeper conclusions concerning local iron environment and properties can be drawn, and this will be
discussed further.
Magnetic Ordering Temperature
Mössbauer spectroscopy can be used to obtain magnetic ordering temperature. To
do this, at first, one should somehow estimate the range of searching the temperature. In case of the Bi 0.5 Pb 0.5 (Fe 0.75 Nb 0.25 )O 3, we can assume that the substitution
of Fe
3+ by Nb
5+ dilutes the magnetic sublattice and the magnetic ordering
Table 9.2 Hyperfine interaction parameters of the Bi 0.5 Pb 0.5 (Fe 0.75 Nb 0.25 )O 3 at 77 K [12]
No
(Fe, Nb)
P p
Area (%)
IS (mm/s)
B hf (T)
QS (mm/s)
1.
5, 1
0.370
36.1(5)
0.453(5)
53.30(4)
–0.013(6)
2.
4, 2
0.308
30.2(5)
0.517(8)
51.59(5)
–0.010(7)
3.
6, 0
0.185
19.4(5)
0.631(10)
53.34(7)
–0.029(10)
4.
3, 3
0.137
14.3(5)
0.532(10)
49.79(10)
0.030(14)
9 Mössbauer Spectroscopy of Magnetoelectric Perovskite Oxides
293