9 Mössbauer Spectroscopic Studies on Atomic Diffusion in Materials
471
=
2
τ
⎛
⎝ 1 −
1
Z
j
exp
−i k · R j
⎞
⎠
(9.20)
For poly-crystals, we average the Eq. (9.20) overall possible jump directions for
|k| · |R n | 1,
=
2
τ
f M .
(9.21)
Here f M is called “Mössbauer correlation factor,” Wolf calculated to be 0.64 [30]
in the case of bcc lattice, which is related to a probability of the Mössbauer probe
jumping to the vacancies. On the other hand, in the tracer diffusivity, we have to
consider a tracer correlation factor, f T = 0.73, for self-diffusion in bcc lattice.
D =
R
2
6τ
f T
(9.22)
Combining both equations of (9.21) and (9.22), we finally obtain a relation
between the tracer diffusivity, D, and the line broadening , as in equation of
(9.23).
D =
R
2
12
f T
f M
(9.23)
Assuming a vacancy mechanism with a jump distance of R for the self-diffusion in
γ- and δ-phases of pure Fe, the self-diffusivity was estimated from the line broadening
using the above equation. We plot the results as a function of 1/T in Fig. 9.25, in
comparison with the tracer diffusivities for γ- and δ-phases [22]. The diffusivities in
both phases are in good agreement, indicating that the diffusion theory in Mössbauer
spectroscopy can be applied even for studying an unknown diffusion mechanism such
as fast impurity diffusion in metals. This result was the first experimental proof that
Mössbauer spectroscopy can provide the atomistic information on the long-range
self-diffusion based on the Singwi and Sjölander theory [26].
We further applied the high-temperature technique for studying the Fe fast diffusion in β-Zr-Fe [43], β-ZrNb-Fe [44], α-Zr and α-Ti [45], β-Ti-Fe [46] and also the
short-range ordering and the clustering properties in Au-Fe and Cu-Fe [47–50], and
the Fe diffusion and segregations in Si single crystals and multi-crystals for Si solar
cells [51–54].
471
=
2
τ
⎛
⎝ 1 −
1
Z
j
exp
−i k · R j
⎞
⎠
(9.20)
For poly-crystals, we average the Eq. (9.20) overall possible jump directions for
|k| · |R n | 1,
=
2
τ
f M .
(9.21)
Here f M is called “Mössbauer correlation factor,” Wolf calculated to be 0.64 [30]
in the case of bcc lattice, which is related to a probability of the Mössbauer probe
jumping to the vacancies. On the other hand, in the tracer diffusivity, we have to
consider a tracer correlation factor, f T = 0.73, for self-diffusion in bcc lattice.
D =
R
2
6τ
f T
(9.22)
Combining both equations of (9.21) and (9.22), we finally obtain a relation
between the tracer diffusivity, D, and the line broadening , as in equation of
(9.23).
D =
R
2
12
f T
f M
(9.23)
Assuming a vacancy mechanism with a jump distance of R for the self-diffusion in
γ- and δ-phases of pure Fe, the self-diffusivity was estimated from the line broadening
using the above equation. We plot the results as a function of 1/T in Fig. 9.25, in
comparison with the tracer diffusivities for γ- and δ-phases [22]. The diffusivities in
both phases are in good agreement, indicating that the diffusion theory in Mössbauer
spectroscopy can be applied even for studying an unknown diffusion mechanism such
as fast impurity diffusion in metals. This result was the first experimental proof that
Mössbauer spectroscopy can provide the atomistic information on the long-range
self-diffusion based on the Singwi and Sjölander theory [26].
We further applied the high-temperature technique for studying the Fe fast diffusion in β-Zr-Fe [43], β-ZrNb-Fe [44], α-Zr and α-Ti [45], β-Ti-Fe [46] and also the
short-range ordering and the clustering properties in Au-Fe and Cu-Fe [47–50], and
the Fe diffusion and segregations in Si single crystals and multi-crystals for Si solar
cells [51–54].