Chemistry Reference
In-Depth Information
Fig. 2.4 In the case of a non-zero electric field gradient (EFG), electric quadrupole interaction as
visualized by the precession of the quadrupole moment vector about the field gradient axis sets in
and splits the degenerate I = 3/2 level into two substates with magnetic spin quantum numbers
m
I
=±3/2 and ±1/2. This gives rise to two transition lines with equal probability (intensity).
The energy difference between the two sub states DE
Q
is observed in the spectrum as the
separation between the two resonance lines
split excited state and the unsplit ground state. The ground state with I = 1/2 has
no quadrupole moment and remains therefore unsplit, but still twofold degenerate.
This degeneracy can be lifted by magnetic dipole interaction (Zeeman effect, see
below). The same holds for the two substates of the excited I = 3/2 level, which
are still twofold degenerate after electric quadrupole interaction. This becomes
apparent by looking at the expression for the quadrupolar interaction energies E
Q
derived from perturbation theory:
3m
I
I
ð
I
þ
1
Þ
eQV
zz
4I
ð
2I
1
Þ
E
Q
ð
I
;
m
I
Þ¼
ð
Þ:
for axial symmetry
For
57
Fe Mössbauer spectroscopy, electric quadrupole interaction in the absence
of magnetic dipole interaction leads to a doublet, the separation of the two resonance
lines giving the quadrupole interaction energy DE
Q
which is proportional to the
quadrupole moment eQ and the electric field gradient. The electric field E at the
nucleus is the negative gradient of the potential,-
r
V, and the electric field gradient
EFG is given by the nine components V
ij
¼
o
2
V
=
oioj
ð
i
;
j
¼
x
;
y
;
z
Þ
of the 3 9 3s
rank EFG tensor. Only five of these components are independent because of the
symmetric form of the tensor, i.e. V
ij
= V
ji
and because of Laplace's equation which
requires that the tensor be traceless:
r
V
ii
= 0. In the principal axes system the off-
diagonal elements vanish, and for axial symmetry (fourfold or threefold axis of
symmetry passing through the Mössbauer nucleus yielding V
xx
= V
yy
) the EFG is
then solely given by the tensor component V
zz
. For non-axial symmetry the asym-
metry parameter g = (V
xx
- V
yy
)/V
zz
is required in addition. When choosing the
principal axes ordering such that V
zz
C V
xx
C V
yy
, the asymmetry parameter range
becomes 0 B g B 1.
In principle, there are two sources which can contribute to the total EFG: (1)
charges (or dipoles) on distant ions surrounding the Mössbauer atom in non-cubic
symmetry, usually termed lattice contribution to the EFG; (2) anisotropic (non-
cubic) electron distribution in the valence shell of the Mössbauer atom, usually
