334
P.-E. Lippens
i = X, Y, Z. Equations (7.11) and (7.12) show that the sign of V ZZ reflects the
accumulation (V ZZ < 0) or the depletion (V ZZ > 0) of the Sn 5p electrons along the Z
axis relative to the average number of the remaining Sn 5p electrons in the XY plane
perpendicular to Z.
For the 1/2–3/2 nuclear transition, the sign of V ZZ cannot be determined experimentally except for combined electric quadrupole and magnetic interactions.
However, the magnitude of is of particular interest since it provides information about the anisotropy of the charge density around the Mössbauer atom, which
depends on the Sn local structure and chemical bonds. For battery applications,
this parameter can be used for the analysis of structural relaxations induced by Li
or Na insertion or for the characterization of electrochemically formed small and
poorly crystallized particles in two-phase, alloying and conversion reactions. The
quadrupole splitting can also help to understand differences between metastable and
stable crystalline phases as described in Sect. 7.6.1 for Li 7 Sn 2 .
7.3.4 Magnetic Splitting
The interaction between the magnetic moment of the nucleus μ and the magnetic
field at the nucleus B is described by the Hamiltonian
H = −μ.B
(7.13)
with
μ = g I μ n ˆ
I
(7.14)
where g I is the nuclear g-factor that depends on the nuclear spin number I, μ n is the
nuclear magneton and ˆ
I is the nuclear spin operator. The nuclear energy levels are
E = −g I μ n Bm I
(7.15)
where the nuclear magnetic quantum number m I takes the 2I + 1 values −I, −I +
1,…, I–1, I, leading to the splitting of the nuclear states into 2I + 1 substates (nuclear
Zeeman splitting). For
57 Fe and
119 Sn isotopes, the ground (I = 1/2) and excited (I
= 3/2) states are split into two (m I=1/2 = ± 1/2) and four (m I=3/2 = ± 1/2, ± 3/2)
substates, respectively, leading to six allowed dipole transitions by considering the
selection rules m I = 0, ± 1. In that case, the Mössbauer spectrum is a sextet formed
by six resonant absorption lines with intensities depending on the magnetization and
γ-ray directions. For polycrystalline samples, the relative intensities are in the ratio
3:2:1:1:2:3.
P.-E. Lippens
i = X, Y, Z. Equations (7.11) and (7.12) show that the sign of V ZZ reflects the
accumulation (V ZZ < 0) or the depletion (V ZZ > 0) of the Sn 5p electrons along the Z
axis relative to the average number of the remaining Sn 5p electrons in the XY plane
perpendicular to Z.
For the 1/2–3/2 nuclear transition, the sign of V ZZ cannot be determined experimentally except for combined electric quadrupole and magnetic interactions.
However, the magnitude of is of particular interest since it provides information about the anisotropy of the charge density around the Mössbauer atom, which
depends on the Sn local structure and chemical bonds. For battery applications,
this parameter can be used for the analysis of structural relaxations induced by Li
or Na insertion or for the characterization of electrochemically formed small and
poorly crystallized particles in two-phase, alloying and conversion reactions. The
quadrupole splitting can also help to understand differences between metastable and
stable crystalline phases as described in Sect. 7.6.1 for Li 7 Sn 2 .
7.3.4 Magnetic Splitting
The interaction between the magnetic moment of the nucleus μ and the magnetic
field at the nucleus B is described by the Hamiltonian
H = −μ.B
(7.13)
with
μ = g I μ n ˆ
I
(7.14)
where g I is the nuclear g-factor that depends on the nuclear spin number I, μ n is the
nuclear magneton and ˆ
I is the nuclear spin operator. The nuclear energy levels are
E = −g I μ n Bm I
(7.15)
where the nuclear magnetic quantum number m I takes the 2I + 1 values −I, −I +
1,…, I–1, I, leading to the splitting of the nuclear states into 2I + 1 substates (nuclear
Zeeman splitting). For
57 Fe and
119 Sn isotopes, the ground (I = 1/2) and excited (I
= 3/2) states are split into two (m I=1/2 = ± 1/2) and four (m I=3/2 = ± 1/2, ± 3/2)
substates, respectively, leading to six allowed dipole transitions by considering the
selection rules m I = 0, ± 1. In that case, the Mössbauer spectrum is a sextet formed
by six resonant absorption lines with intensities depending on the magnetization and
γ-ray directions. For polycrystalline samples, the relative intensities are in the ratio
3:2:1:1:2:3.