332
P.-E. Lippens
where Q is an average value obtained from ˆ
Q and usually defined as the nuclear
quadrupole moment, which depends on the nuclear spin value I, V ZZ is the EFG
component of highest absolute value in the principal axis system, ˆ
I X , ˆ
I Y and ˆ
I Z are
the nuclear spin operators and η is the asymmetry parameter defined by
η =
V X X − V Y Y
V Z Z
(7.9)
where V XX and V YY are the two other components of the EFG tensor in the principal
axis system such that |V X X | ≤ |V Y Y | ≤ |V Z Z |. It should be noted that 0 ≤ η ≤ 1.
The nuclear energy levels are obtained by the diagonalization of the H QI matrix
in the basis of the nuclear spin states |I, m I >, where m I is the nuclear magnetic
quantum number.
For the first excited nuclear state of
57 Fe and
119 Sn (I = 3/2), H QI can be diagonalized analytically, giving two twofold degenerated energy levels of opposite signs for
the states |3/2, ± 1/2> and |3/2, ± 3/2>, respectively, that are separated by the energy
=
1
2
eQV Z Z
1 +
η
2
3
1/2
(7.10)
called the quadrupole splitting. Thus, there are two possible nuclear transitions
between the ground state (I = 1/2) and the first excited state (I = 3/2). The corresponding Mössbauer spectrum is formed by a doublet of two resonant lines whose
separation is given by . Equation (7.10) indicates that the quadrupole splitting is
proportional to the nuclear quadrupole moment of the excited state (I = 3/2). Its value
can be determined experimentally or from the linear correlation between the experimental values of and the theoretical values of EFG for a series of crystalline phases
[49, 56–58]. As an example, the experimental values of the quadrupole splitting are
reported against the DFT-LAPW values of the electronic term | V Z Z |
1 +
η
2
3
1/2
for a series of tin compounds (Fig. 7.4). The slope of the regression line gives Q =
10.5 fm
2 in agreement with the experimental value Q = 10.9(8) fm
2 [59].
Both V ZZ and η provide information about the anisotropic charge distribution
around the nucleus. For the 1/2–3/2 nuclear transition, these two terms cannot be
obtained independently from the experimental value of the quadrupole splitting as
shown by Eq. (7.10). This is nevertheless possible with a conventional experimental
setup and the application of an external magnetic field.
For other nuclear transitions, the quadrupole interactions are more complex,
requiring the numerical diagonalization of H QI given by Eq. (7.8) in the basis of
the nuclear spin states. This is, for example, the case of the 5/2–7/2 nuclear transition of
121 Sb. The 5/2 nuclear ground state and the 7/2 nuclear excited state are split
into three and four substates, respectively, leading to eight allowed transitions. In
that case, the values of V ZZ and η can be determined from the Mössbauer spectra,
but there are often large uncertainties on the values of η [60].
P.-E. Lippens
where Q is an average value obtained from ˆ
Q and usually defined as the nuclear
quadrupole moment, which depends on the nuclear spin value I, V ZZ is the EFG
component of highest absolute value in the principal axis system, ˆ
I X , ˆ
I Y and ˆ
I Z are
the nuclear spin operators and η is the asymmetry parameter defined by
η =
V X X − V Y Y
V Z Z
(7.9)
where V XX and V YY are the two other components of the EFG tensor in the principal
axis system such that |V X X | ≤ |V Y Y | ≤ |V Z Z |. It should be noted that 0 ≤ η ≤ 1.
The nuclear energy levels are obtained by the diagonalization of the H QI matrix
in the basis of the nuclear spin states |I, m I >, where m I is the nuclear magnetic
quantum number.
For the first excited nuclear state of
57 Fe and
119 Sn (I = 3/2), H QI can be diagonalized analytically, giving two twofold degenerated energy levels of opposite signs for
the states |3/2, ± 1/2> and |3/2, ± 3/2>, respectively, that are separated by the energy
=
1
2
eQV Z Z
1 +
η
2
3
1/2
(7.10)
called the quadrupole splitting. Thus, there are two possible nuclear transitions
between the ground state (I = 1/2) and the first excited state (I = 3/2). The corresponding Mössbauer spectrum is formed by a doublet of two resonant lines whose
separation is given by . Equation (7.10) indicates that the quadrupole splitting is
proportional to the nuclear quadrupole moment of the excited state (I = 3/2). Its value
can be determined experimentally or from the linear correlation between the experimental values of and the theoretical values of EFG for a series of crystalline phases
[49, 56–58]. As an example, the experimental values of the quadrupole splitting are
reported against the DFT-LAPW values of the electronic term | V Z Z |
1 +
η
2
3
1/2
for a series of tin compounds (Fig. 7.4). The slope of the regression line gives Q =
10.5 fm
2 in agreement with the experimental value Q = 10.9(8) fm
2 [59].
Both V ZZ and η provide information about the anisotropic charge distribution
around the nucleus. For the 1/2–3/2 nuclear transition, these two terms cannot be
obtained independently from the experimental value of the quadrupole splitting as
shown by Eq. (7.10). This is nevertheless possible with a conventional experimental
setup and the application of an external magnetic field.
For other nuclear transitions, the quadrupole interactions are more complex,
requiring the numerical diagonalization of H QI given by Eq. (7.8) in the basis of
the nuclear spin states. This is, for example, the case of the 5/2–7/2 nuclear transition of
121 Sb. The 5/2 nuclear ground state and the 7/2 nuclear excited state are split
into three and four substates, respectively, leading to eight allowed transitions. In
that case, the values of V ZZ and η can be determined from the Mössbauer spectra,
but there are often large uncertainties on the values of η [60].