Biomedical Engineering Reference
In-Depth Information
Table 17.4
Representative nuclear
spin quantum numbers
Isotopic species
I
1
H
1/2
2
D
1
12
C
0
13
C
1
/
2
17
O
5/2
1 also possess a nuclear quadrupole
Q
n
. It is usually defined in
terms of the nuclear charge distribution ρ
n
(
r
)as
Many nuclei with
I
≥
⎛
⎝
ρ
n
3
x
2
r
2
dτ
3
ρ
n
xy
dτ
3
ρ
n
xz
dτ
⎞
⎠
−
3
ρ
n
yx
dτ
ρ
n
3
y
2
r
2
dτ
3
ρ
n
yz
dτ
1
e
Q
n
=
−
(17.29)
3
ρ
n
zx
dτ
3
ρ
n
zy
dτ
ρ
n
3
z
2
r
2
dτ
−
This definition gives a traceless tensor. Here the integration is over the nuclear charge
distribution. Nuclear wavefunctions are pretty hard to come by, and we normally have to
determine the components of
Q
n
by experiment.
In a molecule, a given nucleus will generally experience an electric field gradient due
to the surrounding electrons. Electric fields are vector quantities and so an electric field
gradient is a tensor quantity. The electric field gradient at nucleus n is usually written
q
n
.
The energy of interaction
U
between the nuclear quadrupole and the electric field gradient is
6
(
Q
n
)
ij
(
q
n
)
ij
e
U
=−
(17.30)
the largest of the
diagonal components of
q
n
. The quantity
eQ
n
q
n
/
h
is referred to as the
quadrupole coupling
constant
. According to Townes and Dailey (1949), since filled shells and s orbitals have
spherical symmetry, and since d and f orbitals do not penetrate near the nucleus, the quadru-
pole coupling constant should be largely due to any p electrons present in the valence shell.
Molecular quadrupole coupling constants are usually determined from the hyperfine
structure of pure rotational spectra or from electric beam and magnetic beam reson-
ance spectroscopies. Nuclear magnetic resonance, electron spin resonance and Mossbäuer
spectroscopies are also routes to this property.
I can use the well-studied series HCN, FCN and ClCN to illustrate a couple of points.
These molecules, including many isotopic species, have been exhaustively studied by spec-
troscopic techniques. An interesting feature of the experimental geometries is that the CN
bond lengths are almost identical in length yet the
14
N quadrupole coupling constant (QCC)
is quite different (Table 17.5).
Table 17.6 shows results calculated at the HF/6-311G** level of theory, including
geometry optimization.
In principal axes, the interaction is determined by
Q
n
=
(
Q
n
)
zz
and
q
n
=
