Chemistry Reference
In-Depth Information
Consequently, there remain only two independent components, usually chosen
as V
zz
and g which is the asymmetry parameter defined by
g
¼
V
xx
V
yy
V
zz
:
and then 0
g
1.
The excited state of
57
Fe has spin 3/2 and has a positive Q value showing a
quadrupole splitting. From the quadrupole interaction Hamiltonian, the eigen-
values are
Usually the components are chosen as
j
V
zz
j
[ V
xx
j
j
V
yy
h
i
ð
1
þ
g
2
=
3
Þ
2
;
m
j
¼
j
;
j
1
;;;;
j
:
V
zz
Q
4j
ð
2j
1
Þ
3m
j
j
ð
j
þ
1
Þ
E
Q
¼
Figure
1.6
b shows the quadrupole splitting of
57
Fe excited state j = 3/2 and the
resultant Mössbauer spectrum which is a doublet.
As described in the first part of this section, when the nucleus has a nuclear
magnetic dipole moment, the magnetic hyperfine interaction with magnetic state of
its own electrons lifts all of their (2j ? 1)-fold degeneracy of the nuclear levels as
the nuclear Zeeman effect. For
57
Fe, both of excited and ground states have nuclear
magnetic dipole moments and interact with the magnetic fields being created by
the electronic states like electron spin-polarization, electron orbital current and
dipole field by other electron spin or external magnetic field. Nuclear levels of
excited and ground states are divided into six different levels. Excited state
(j
e
= 3/2) splits into four sublevels and ground state (j
g
= 1/2) splits into two
sublevels and eight c transitions between excited and ground states can be
expected. However, the radiation field is M1 and its selection rule excludes the
transition of Dm
¼
2 and resultant Mössbauer absorption spectrum consists of 6
lines as shown in Fig.
1.6
c. The hyperfine magnetic field from a single electron is
þ
:
3r
ð
S
r
Þ
r
2
S
r
5
8p
3
L
r
3
j
2
þ
H
¼
2l
B
S w
ð
0
Þ
j
The first term is the Fermi contact interaction and is only operable for s elec-
trons. The second term is due to the orbital current. The third term represents the
dipole field due to the electron spin. These two latter terms are generally smaller
than the contact term and vanish for s-state ions. For
57
Fe in Fe
3+
(S = 5/2,
L = 0), the contact interaction gives about -60 T. For
57
Fe in Fe
2+
(S = 2,
L = 2), the field is somewhat smaller because of smaller spin and also appreciable
positive orbital contribution. At room temperature the hyperfine magnetic field at
57
Fe in metallic iron is -33 T and this is the reference value to determined the
hyperfine magnetic field in magnetic materials using
57
Fe Mössbauer spectros-
copy. Nuclear levels of
57
Fe under magnetic field and the expected Mössbauer
spectrum are shown in Fig.
1.6
c.
Figure
1.6
c shows the
57
Fe spectrum obtained from magnetically ordered
materials like ferromagnet or antiferromagnet that have a cubic symmetry to
