Chemistry Reference
In-Depth Information
where the suffixes N and e denote the nucleus and the electron. From (
1.24
), the
monopole interaction for k = 0 is given by
E
0
¼
ZZ
1
r
N
q
e
ð
r
e
Þ
q
N
ð
r
N
Þ
dr
e
dr
N
:
This interaction energy can be computed classically by considering a uniformly
charged spherical nucleus imbedded in its s-electron charge cloud and makes a
shift of the nuclear levels. Observed shift of the resonance spectrum is called as
''isomer shift'' because the shift depends on the difference in the nuclear radii of
the isomeric (ex.) and ground (gd.) states. A change in the s-electron density that
might be due to the change in valence will cause the change in isomer shift. It
implies that for
57
Fe the isomer shift values depend on the valence state of Fe
atom.
The change in c-ray energy due to the monopole interaction is therefore the
difference of two terms written for the nucleus in isomeric (ex.) and ground (gd.)
states,
DE
ex
DE
gd
¼
2p
5
Ze
2
w
ð
0
Þ
j
2
ð
R
ex
R
gd
Þ:
j
DE is the difference from the point nucleus. Observed isomer shift is the difference
between source and absorber and given by
h
i
ð
R
ex
R
gd
Þ:
2p
5
Ze
2
j
2
w
s
ð
0
Þ
j
2
Isomer shift (IS) =
j
w
a
ð
0
Þ
j
Figure
1.6
a shows the isomer shift and the expected Mössbauer absorption
spectrum.
Since the nucleus has no electronic dipole moment from the parity, the elec-
tronic dipole interaction for k = 1 does not exist. Next interaction is the electronic
quadrupole interaction for k = 2 which is given by
Z
q
N
ð
r
N
Þ
r
N
Y
2m
ð
_
N
Þ
dr
N
Z
5
p
X
2
E
2
¼
4
q
e
ð
r
e
Þ
1
r
e
Y
2m
ð
_
e
Þ
dr
e
:
m
¼
2
This is the result of the interaction of the nuclear quadrupole moment Q with
the electric field gradient EFG that is due to other charges in the solid. Nuclear
quadrupole moment Q is expressed by the left-side integral and the electric field
gradient is expressed by the right-side integral in the above equation. The nuclear
quadrupole moment reflects the deviation of the nucleus from spherical symmetry.
A flattened nucleus has negative Q while an elongated nucleus has a positive Q.
Nuclei whose spin is 0 or 1/2 are spherically symmetric and have a zero Q; thus the
ground state of
57
Fe, with j
g
= 1/2, cannot exhibit quadrupole splitting.
The electric field gradient is obtained by applying the gradient operator to the three
components of the electric field that is a vector. Consequently the electric field
gradient EFG is a 3 9 3 tensor. However, this tensor is reduced to diagonal form
