7 Application of Mössbauer Spectroscopy to Li-Ion and Na-Ion …
329
electronic densities at the nucleus, ρ s (0) (source) and ρ a (0) (absorber), and therefore,
different energy transitions: E a and E s . The isomer shift is given by δ = E a −
E s , which corresponds to the energy shift of the resonance line relative to the
source. The isomer shift is a relative quantity and the origin can be chosen arbitrary.
A reference material is often considered instead of the source and ρ s (0) is replaced
by ρ ref (0). This defines the zero-velocity of the isomer shift scale. In this chapter, the
commonly used references α-Fe for
57 Fe and BaSnO 3 for
119 Sn are considered, but
others can be found in the literature, which should be taken into account to compare
the isomer shifts originating from different works. The expression of the isomer shift
obtained from Eq. (7.2) is
δ =
Ze
2
6ε 0
r
2
n
ρ(0) − ρ re f (0)
(7.3)
Since r
2
n is constant for a given isotope, the following simplified expression
is often used for chemical applications:
δ = α
ρ(0) − ρ re f (0)
(7.4)
where the constant α is positive for
119 Sn and negative for
57 Fe, leading to opposite
variations of δ versus ρ(0) for these two isotopes. The value of α can be evaluated
from the correlation between the experimental values of δ and the theoretical values
of ρ(0) for a series of reference materials [44–49]. For example, the values of ρ(0)
were evaluated for a series of tin compounds with density functional theory (DFT),
using the linearized augmented plane wave (LAPW) method [50]. A good δ−ρ(0)
linear correlation is observed (Fig. 7.3) as expected from Eq. (7.4). The calibration
constant α obtained by linear regression for this series of compounds can then be used
with ρ(0) calculated by the LAPW method and Eq. (7.4) to evaluate the theoretical
values of δ for other tin based phases as illustrated in Sects. 7.5 and 7.6.
The isomer shift provides information about the local environment of a Mössbauer
atom through the electron density at the nucleus. Many studies have been devoted
to the interpretation of the isomer shift from the evaluation of ρ(0) or from some
more qualitative but very fruitful approaches [51–53]. For instance, it is possible to
relate the variations of the isomer shift to the valence electron populations of the
Mössbauer atoms. First, it is important to note that only the s electrons and some
relativistic p electrons are inside the nucleus, and the main contributions to ρ(0)
come, in descending order, from 1s, 2s, 3s, etc. electrons. However, changes in ptype and d-type wavefunctions influence the s-type wavefunctions (shielding effect)
even at the nucleus. Since most of the modifications in the local environment of the
Mössbauer atoms affect the chemical bonds and the valence electron density, one
can consider, as a first approximation, that the contribution of core electrons ρ c (0)
is constant and ρ(0) mainly changes with the valence electron contribution ρ v (0).
As shown by atomic calculations, the latter term strongly increases with the number
of s-type valence electrons and decreases, but to a lesser extent, with the number of
329
electronic densities at the nucleus, ρ s (0) (source) and ρ a (0) (absorber), and therefore,
different energy transitions: E a and E s . The isomer shift is given by δ = E a −
E s , which corresponds to the energy shift of the resonance line relative to the
source. The isomer shift is a relative quantity and the origin can be chosen arbitrary.
A reference material is often considered instead of the source and ρ s (0) is replaced
by ρ ref (0). This defines the zero-velocity of the isomer shift scale. In this chapter, the
commonly used references α-Fe for
57 Fe and BaSnO 3 for
119 Sn are considered, but
others can be found in the literature, which should be taken into account to compare
the isomer shifts originating from different works. The expression of the isomer shift
obtained from Eq. (7.2) is
δ =
Ze
2
6ε 0
r
2
n
ρ(0) − ρ re f (0)
(7.3)
Since r
2
n is constant for a given isotope, the following simplified expression
is often used for chemical applications:
δ = α
ρ(0) − ρ re f (0)
(7.4)
where the constant α is positive for
119 Sn and negative for
57 Fe, leading to opposite
variations of δ versus ρ(0) for these two isotopes. The value of α can be evaluated
from the correlation between the experimental values of δ and the theoretical values
of ρ(0) for a series of reference materials [44–49]. For example, the values of ρ(0)
were evaluated for a series of tin compounds with density functional theory (DFT),
using the linearized augmented plane wave (LAPW) method [50]. A good δ−ρ(0)
linear correlation is observed (Fig. 7.3) as expected from Eq. (7.4). The calibration
constant α obtained by linear regression for this series of compounds can then be used
with ρ(0) calculated by the LAPW method and Eq. (7.4) to evaluate the theoretical
values of δ for other tin based phases as illustrated in Sects. 7.5 and 7.6.
The isomer shift provides information about the local environment of a Mössbauer
atom through the electron density at the nucleus. Many studies have been devoted
to the interpretation of the isomer shift from the evaluation of ρ(0) or from some
more qualitative but very fruitful approaches [51–53]. For instance, it is possible to
relate the variations of the isomer shift to the valence electron populations of the
Mössbauer atoms. First, it is important to note that only the s electrons and some
relativistic p electrons are inside the nucleus, and the main contributions to ρ(0)
come, in descending order, from 1s, 2s, 3s, etc. electrons. However, changes in ptype and d-type wavefunctions influence the s-type wavefunctions (shielding effect)
even at the nucleus. Since most of the modifications in the local environment of the
Mössbauer atoms affect the chemical bonds and the valence electron density, one
can consider, as a first approximation, that the contribution of core electrons ρ c (0)
is constant and ρ(0) mainly changes with the valence electron contribution ρ v (0).
As shown by atomic calculations, the latter term strongly increases with the number
of s-type valence electrons and decreases, but to a lesser extent, with the number of