may not actually involve spin-spin interactions, characterizes the return of the
ensemble to thermal equilibrium at some “spin temperature.” The spin-lattice
1
return of the ensemble of nuclei to thermal equilibrium with the surrounding
crystal, usually called the “lattice.” The processes which determine T
1
involve
exchange of energy between the nuclei and their surroundings––that is
nuclear transitions are induced by the (random) fluctuations of the
surroundings. In contrast, the interactions that are usually most effective for
T
2
relaxation are those where there is no (net) gain or loss of energy from the
nuclear ensemble. In general several mechanisms will contribute to the
relaxation.
Typically only one of several possible transitions is excited and detected
for NQR and so the phrase “the return to thermal equilibrium” is usually
applied in the effective spin-½ sense. That is, the thermal ensemble includes
belonging to the lattice. This can give rise to relaxation characterized by
multiple exponentials. A nice derivation of the effect for Nb (I = 9/2) is given
by Chen and Slichter [29].
2.2.5.1 Spin-Lattice Relaxation and Temperature-Dependent Frequency Shifts
The coupling between the nuclei and the surrounding crystal will be due to
time-dependent magnetic and/or electric quadrupole interactions. Magnetic
interactions include those due to paramagnetic impurities and, in metals, the
conduction electrons. A much weaker magnetic interaction can occur via a
time-dependent spin-spin interaction. For materials with two readily available
isotopes (e.g., Cl, Cu, Ga, Br, Rb, Sb, etc.), whether the relaxation is
dominated by magnetic or electric quadrupole interactions can usually be
determined by comparing the ratio of the relaxation times to the ratios of the
magnetic dipole and electric quadrupole moments, respectively, for the two
isotopes.
The contribution to the relaxation by the conduction electrons in metals is
due to the so-called Korringa result, just as in NMR [30]. Hence, this
contribution is very sensitive to changes in the electron density of states at the
Fermi level, such as what one expects near superconducting phase transitions
[31]. Korringa relaxation is also evident in some semiconductors [32].
Paramagnetic impurities generally contribute a constant to the rate, as they do
for NMR.
In many NQR measurements of non-metals, the spin-lattice relaxation time
and the temperature coefficient of the NQR frequency are both a result of
lattice dynamics. This can be understood using the simple model proposed by
Bayer [33]. Consider an electric quadrupole Hamiltonian that has been rotated
about the y-axis by a small angle E. The new Hamiltonian (in the old principal
axis coordinates) is given by
80 2. Nuclear Quadrupole Resonance Spectroscopy
relaxation time, T , which is the inverse of the relaxation rate, characterizes the
just the two nuclear energy levels used, with the remaining levels now