Theory of quadrupolar nuclear spin-lattice relaxation

Abstract

General theoretical expressions are derived for the nuclear spin-lattice relaxation in crystalline solids arising from the interaction of the nuclear electric quadrupole moment with the crystalline electric field. The importance of shielding, antishielding and co-valent effects for the quadrupolar spin-lattice coupling is discussed, and the complete temperature dependence of the transition probability P for a nuclear spin making a transition as a result of the thermal Raman processes is determined on the basis of some simplifying assumptions about the nature of the lattice vibrations. At low temperatures the P are developed in ascending powers of the temperature T, the first term being proportional to T7. It is found that this power series is rapidly convergent only for T smaller than about 0.02Θ, where Θ is the Debije temperature. At high temperatures, the P are developed in descending powers of T, and it is found that an expression of the form T2(a − b/T2) gives a good representation of the temperature dependence of P down to about T = ½Θ. General expressions are derived for the dependence of the probabilities P on the direction of the external magnetic field relative to the crystal axes. Finally, a simple, one-parameter model is discussed, in which the crystalline field at a nucleus is assumed to arise from a number of equal point charges placed on the nearest neighboring lattice sites. The magnitude q of these charges is a measure for the quadrupolar spin-lattice coupling, and is the adjustable parameter of the model. Detailed calculations are made with this model for crystals of the NaCl-type. The calculated relaxation times 1/P are of the same order of magnitude as the experimental ones for values of q of the order of 102 − 103e, where e is the electronic charge. The significance of this result is discussed. Finally, the dependence on the direction of the external field is calculated, and the probability for a transition Δm = ± 1 for the (111) direction is found to be about 50% larger than for the (100) direction, while the probability for a transition Δm = ± 2 for the (111) direction is about 10% smaller than for the (100) direction.