Spin-lattice relaxation of heavy spin-1/2 nuclei in diamagnetic solids: A Raman process mediated by spin-rotation interaction

We present a theory for the nuclear spin-lattice relaxation of heavy spin-1/2 nuclei in solids, which explains within an order of magnitude the unexpectedly effective lead and thallium nuclear spin-lattice relaxation rates observed in the ionic solids lead molybdate, lead chloride, lead nitrate, thallium nitrate, thallium nitrite, and thallium perchlorate. The observed rates are proportional to the square of the temperature and are independent of magnetic field. This rules out all known mechanisms usually employed to model nuclear spin relaxation in lighter spin-1/2 nuclei. The relaxation is caused by a Raman process involving the interactions between nuclear spins and lattice vibrations via a fluctuating spin-rotation magnetic field. The model places an emphasis on the time dependence of the angular velocity of pairs of adjacent atoms rather than on their angular momentum. Thus the spin-rotation interaction is characterized not in the traditional manner by a spin-rotation constant but by a related physical parameter, the magnetorotation constant, which relates the local magnetic field generated by spin rotation to an angular velocity. Our semiclassical relaxation model involves a frequency-mode description of the spectral density that can directly be related to the mean-square amplitudes and mode densities of lattice vibrations in the Debye model.

Figure 1

Relative positions of two neighboring atoms participating in an acoustical lattice vibration mode. Closed and open circles are atoms in equilibrium and displaced positions, respectively. Four cases of relative orientations of the propagation vector $\mathbf{k}$, the displacement vector $\mathbf{\xi }$, and the interatomic vector $\mathbf{a}$ are illustrated. Left and right columns: Planar wave propagation perpendicular and parallel to $\mathbf{a}$, respectively. Top and bottom rows: Longitudinal and transverse modes, respectively.

Figure 2

Two-dimensional frequency space showing the locus of points whose sum or difference equals the NMR frequency. These lines represent the trajectory of integration for evaluation of the Raman contributions to relaxation. Because ${\omega }_0\ll {\omega }_D$, the contribution of the sum section is negligible, while the difference sections nearly collapse to two lines coinciding with the diagonal.

Figure 3

The absolute values of magnetorotation constants $\mathit{\Gamma }$ of nuclear isotopes vs their atomic weights.