As will be seen in more detail below, in order to use NQR spectroscopy
one must have available an isotope with a nuclear spin I > ½, which has a
reasonably high isotopic abundance, and which is at a site in a solid that has
symmetry lower than tetragonal. The most common NMR isotopes,
1
H,
13
C,
and
15
N cannot be used since they have a nuclear spin ½. Of course,
12
C and
16
O cannot be used either as they have nuclear spin 0. Table 2.1 shows a
selection of potential nuclei including those most commonly used for NQR, as
well as a few others of possible interest.
2.2 BASIC THEORY
2.2.1 The Nuclear Electric Quadrupole Interaction
Since a nuclear wavefunction has a definite state of parity, a multipole
expansion of the fields due to the nucleus yields electric 2
n
-poles, where n is
even (monopole, quadrupole, etc.) and magnetic 2
n
-poles, where n is odd
(dipoles, octupoles, etc.). In general these multipole moments become weaker
very rapidly with increasing n. In a molecule or in a solid, the nucleus will be
at an equilibrium position where the electric field is zero, and so in the
absence of a magnetic field the first non-zero interaction is with the electric
quadrupole moment of the nucleus. Higher moments, if they exist, are
generally much too weak to affect NQR measurements [7–9].
A non-zero electric quadrupole moment arises for nuclei that are
classically described as prolate (“stretched”) or oblate (“squashed”)
spheroids. The nuclear charge distribution has axial symmetry and the axis of
symmetry coincides with the direction of the nuclear angular momentum and
the nuclear magnetic dipole moment. In general, an electric quadrupole
moment is described by a 3 u 3 symmetric, traceless tensor Q. For a nucleus
such a tensor can be determined using a single value that describes how
prolate or oblate the nucleus is, plus a description of the orientation of the
nucleus. Since the charge distribution for a nucleus with spin 0 or ½ is
spherical, such nuclei will have no electric quadrupole moment.
If the charge distribution within the nucleus is known, the amount by
which the sphere is prolate or oblate is determined by the (scalar) nuclear
quadrupole moment Q, which can be calculated using
WU
³
drzeQ )3(
22
(2.1)
where the z-axis is along the direction of axial symmetry, e is the magnitude
of the charge on an electron, and U is the nuclear charge density as a function
of position. While such computations may be done by a nuclear physicist to
check a new model for the nucleus, the NQR spectroscopist uses values
determined experimentally. Values of Q are conveniently expressed in units
of 10
–24
cm
2
= 1 barn.
66 2. Nuclear Quadrupole Resonance Spectroscopy