8.4 Superparamagnetic Particles in the Mößbauer Spectrum 165
Figure 8.18 Energy levels in a
57
Fe
nucleus with and without a magnetic field.
In the excited state, the energy levels
without an external magnetic field show
electric quadrupole splitting. (a) According
to the selection rules for radiative
transitions, two emission lines are
possible. In an external magnetic field,
the quadrupole levels are split additionally
(spin-up and spin-down). In this
configuration, six transitions are
allowed. (b).
2
1
±
2
1
2
3
±
2
1
±
2
3
2
3
2
1
2
1
+
2
1
+
2
3
+
2
1
−
2
1
−
2
3
−
2
1
±
2
1
2
1
±
2
1
2
3
±
2
1
±
2
3
2
3
±
2
1
±
2
3
±
2
1
±
2
3
2
3
2
1
2
1
+
2
1
+
2
3
+
2
1
−
2
1
−
2
3
−
magn
exc
E
B
µ
∆
∝
magn
ground
E
B
µ
∆
∝
el quad
E
VQ
−
∆
=
(a)
(b)
magnetic field, these levels are split additionally, because of the two possible
orientations of the spin (spinup and spindown). The energy splitting of these
levels is proportional to the product of the magnetic moment of the nucleus in
the excited, μ exc respectively, ground state, μ ground and the magnetic field B, in
the case of crystalline materials, this is the magnetic crystal field Therefore,
one observes in the ground state two and in the excited state four energy levels.
In Figure 8.18, at the right side of each energy level, the magnetic quantum
number M (careful – in this case, as is common in the literature, M is not the
magnetic moment but the magnetic quantum number!) is indicated. The quantummechanical selection rules for radiative transitions allows transitions with
a difference ΔM of the quantum numbers
∆M ∈ −
{
}
1 0 1
, , .
(8.13)
Equation (8.13) allows two transitions without a magnetic field and six transitions in the presence of an external magnetic field. The allowed transitions
are indicated in Figure 8.18. (For example: The quantum number in the excited
state +
3
2
has an allowed transition to the ground state +
1
2
, the transition to −
1
2
is forbidden.) This selection rule allows two lines, a doublet, for nonmagnetic
specimens and six lines, a sextet, for magnetic materials. The width of the
splitting of the absorption lines allows the surrounding magnetic field,
which is, in the case of magnetic materials, the magnetic crystal field to be
calculated.
Figure 8.18 Energy levels in a
57
Fe
nucleus with and without a magnetic field.
In the excited state, the energy levels
without an external magnetic field show
electric quadrupole splitting. (a) According
to the selection rules for radiative
transitions, two emission lines are
possible. In an external magnetic field,
the quadrupole levels are split additionally
(spin-up and spin-down). In this
configuration, six transitions are
allowed. (b).
2
1
±
2
1
2
3
±
2
1
±
2
3
2
3
2
1
2
1
+
2
1
+
2
3
+
2
1
−
2
1
−
2
3
−
2
1
±
2
1
2
1
±
2
1
2
3
±
2
1
±
2
3
2
3
±
2
1
±
2
3
±
2
1
±
2
3
2
3
2
1
2
1
+
2
1
+
2
3
+
2
1
−
2
1
−
2
3
−
magn
exc
E
B
µ
∆
∝
magn
ground
E
B
µ
∆
∝
el quad
E
VQ
−
∆
=
(a)
(b)
magnetic field, these levels are split additionally, because of the two possible
orientations of the spin (spinup and spindown). The energy splitting of these
levels is proportional to the product of the magnetic moment of the nucleus in
the excited, μ exc respectively, ground state, μ ground and the magnetic field B, in
the case of crystalline materials, this is the magnetic crystal field Therefore,
one observes in the ground state two and in the excited state four energy levels.
In Figure 8.18, at the right side of each energy level, the magnetic quantum
number M (careful – in this case, as is common in the literature, M is not the
magnetic moment but the magnetic quantum number!) is indicated. The quantummechanical selection rules for radiative transitions allows transitions with
a difference ΔM of the quantum numbers
∆M ∈ −
{
}
1 0 1
, , .
(8.13)
Equation (8.13) allows two transitions without a magnetic field and six transitions in the presence of an external magnetic field. The allowed transitions
are indicated in Figure 8.18. (For example: The quantum number in the excited
state +
3
2
has an allowed transition to the ground state +
1
2
, the transition to −
1
2
is forbidden.) This selection rule allows two lines, a doublet, for nonmagnetic
specimens and six lines, a sextet, for magnetic materials. The width of the
splitting of the absorption lines allows the surrounding magnetic field,
which is, in the case of magnetic materials, the magnetic crystal field to be
calculated.