7 Application of Mössbauer Spectroscopy to Li-Ion and Na-Ion …
335
The effective magnetic field at the nucleus comes from external and internal
(in the Mössbauer atom) magnetic fields. In the latter case, the magnetic field at
the nucleus originates from the spin imbalance at the nucleus (Fermi contact),
the interaction between nuclear and electronic orbital moments, and the dipolar
interaction between nuclear and electron spin moments. For example, the hyperfine magnetic field of
57 Fe in α-Fe is mainly due to the Fermi contact term arising
from the difference between spin-up and spin-down electron densities at the nucleus
ρ ↑ (0) − ρ ↓ (0). The spin densities at the nucleus ρ ↑ (0) and ρ ↓ (0) are due to s-type
electrons that are affected by the shielding effects of Fe 3d ↑ and Fe 3d ↓ valence electrons, respectively. The intensity of the hyperfine magnetic field can be obtained from
the energy difference between the two external lines of the Mössbauer spectrum corresponding to the two nuclear transitions between states |I, m I > : |1/2,−1/2 > − |3/2,
−3/2 > and |1/2,1/2 > − |3/2,3/2 > . According to Eq. (7.15), the observed value of
10.6 mm s
−1 gives a hyperfine magnetic field of 33 T.
Although, the magnetic properties of electrode materials are not of direct interest
for applications in batteries, they provide additional information that can help in the
characterization of materials (crystallinity, particle size) or reaction mechanisms.
Finally, it should be noted that the electric quadrupole and hyperfine magnetic interactions are not only described by a number, as the isomer shift, but they both depend
on direction (EFG axes, magnetic field direction). If these two interactions occur
simultaneously, the Hamiltonian resulting from Eqs. (7.8) and (7.13) must be generally solved numerically. In that case, the shape of the Mössbauer spectrum is more
complex than line splitting. An example of such Mössbauer spectrum is given for
antiferromagnetic FeSn 2 in Sect. 7.6.1.
7.3.5 Recoil-Free Fraction
The recoil-free fraction, f , is the probability of recoilless emission (source) or
absorption (absorber) of γ-rays and is given by
f = ex p
−k
2
x
2
(7.16)
where k is the γ-ray wavenumber and
x
2
is the mean-square amplitude of the nucleus
vibrations in the direction of γ-rays. A significant Mössbauer effect requires a large
recoil-free fraction, which is obtained with a low value of k, i.e. of the γ-ray energy,
typically lower than 150 keV, and a small value of
x
2
. The nucleus displacements
depend on the phonon spectrum of the solid that usually shows complex dispersion
and must be evaluated by computational methods. However, analytical expressions
can be derived by considering the Debye model. In the Debye model, the density of
phonon states is assumed to increase as ω
2 up to the Debye frequency ω D = K B θ D /,
where θ D is the Debye temperature, K B is the Boltzmann constant and è is the reduced
Planck constant. Simple analytical expressions can be obtained for T θ D
335
The effective magnetic field at the nucleus comes from external and internal
(in the Mössbauer atom) magnetic fields. In the latter case, the magnetic field at
the nucleus originates from the spin imbalance at the nucleus (Fermi contact),
the interaction between nuclear and electronic orbital moments, and the dipolar
interaction between nuclear and electron spin moments. For example, the hyperfine magnetic field of
57 Fe in α-Fe is mainly due to the Fermi contact term arising
from the difference between spin-up and spin-down electron densities at the nucleus
ρ ↑ (0) − ρ ↓ (0). The spin densities at the nucleus ρ ↑ (0) and ρ ↓ (0) are due to s-type
electrons that are affected by the shielding effects of Fe 3d ↑ and Fe 3d ↓ valence electrons, respectively. The intensity of the hyperfine magnetic field can be obtained from
the energy difference between the two external lines of the Mössbauer spectrum corresponding to the two nuclear transitions between states |I, m I > : |1/2,−1/2 > − |3/2,
−3/2 > and |1/2,1/2 > − |3/2,3/2 > . According to Eq. (7.15), the observed value of
10.6 mm s
−1 gives a hyperfine magnetic field of 33 T.
Although, the magnetic properties of electrode materials are not of direct interest
for applications in batteries, they provide additional information that can help in the
characterization of materials (crystallinity, particle size) or reaction mechanisms.
Finally, it should be noted that the electric quadrupole and hyperfine magnetic interactions are not only described by a number, as the isomer shift, but they both depend
on direction (EFG axes, magnetic field direction). If these two interactions occur
simultaneously, the Hamiltonian resulting from Eqs. (7.8) and (7.13) must be generally solved numerically. In that case, the shape of the Mössbauer spectrum is more
complex than line splitting. An example of such Mössbauer spectrum is given for
antiferromagnetic FeSn 2 in Sect. 7.6.1.
7.3.5 Recoil-Free Fraction
The recoil-free fraction, f , is the probability of recoilless emission (source) or
absorption (absorber) of γ-rays and is given by
f = ex p
−k
2
x
2
(7.16)
where k is the γ-ray wavenumber and
x
2
is the mean-square amplitude of the nucleus
vibrations in the direction of γ-rays. A significant Mössbauer effect requires a large
recoil-free fraction, which is obtained with a low value of k, i.e. of the γ-ray energy,
typically lower than 150 keV, and a small value of
x
2
. The nucleus displacements
depend on the phonon spectrum of the solid that usually shows complex dispersion
and must be evaluated by computational methods. However, analytical expressions
can be derived by considering the Debye model. In the Debye model, the density of
phonon states is assumed to increase as ω
2 up to the Debye frequency ω D = K B θ D /,
where θ D is the Debye temperature, K B is the Boltzmann constant and è is the reduced
Planck constant. Simple analytical expressions can be obtained for T θ D