PHYSICA L RE VIEW B VOLUME 11, NUM BEE, 5 1 MARCH 1975 Spin-lattice relaxation in the dipolar and rotating frames due to correlated ultraslow motions: Order-disorder-type crystals V' S. Zumer University of Ljubjlana, Ljubjlana, Yugoslavia (Received 4 December 1974) In the ultraslow-motion case an expression for the dipolar spin-lattice relaxation time is derived, which is convenient for both correlated and uncorrelated motions. T,D is related to the first derivative of the dipolar Hamiltonian autocorrelation function. The Slichter-Ailion expression for TiD is obtained as a special case. Results are applied to the order-disorder-type crystals which can be described by an Ising model, where the necessary correlation functions have simple forms in the random-phase approximation. The relaxation in the rotating frame is briefly discussed. A summary is given of critical effects on the nuclear-spin-lattice relaxation for both ultraslow motion and faster motion. I. INTRODUCTION Spin-lattice relaxation is a useful tool for ele- mentary motion studies. Each of the relaxation methods covers a characteristic frequency region. Motions which we shall study are ultraslow, which means that their correlation frequencies 7 ' are small compared to the I. armor frequency wo of the observed nuclei in the local dipolar field. Under these conditions, measurements of T» and weak radio-frequency- (rf) field T„measurements are advantageous. Such ultraslow motions strongly relax magnetiza- tion of the dipolar- and weak-rf-fieM energy reser- voir, while in the case of strong rf field the cor- responding energy reservoir is weakly coupled to the lattice. Therefore expressions for the relaxa- tion rates obtained with the perturbation approxima- tion are valid' only for an rf field strong compared to the local dipolar field. Ultraslow motions are in fact fast but infrequent jumps, so that the sudden-quantum approximation can be used. Using this approximation Slichter and Ailion' (SA) have derived expressions for T» due to several types of uncorrelated ultraslow motions. We shall derive general expressions for T» and T„, which can be applied equally well to correlat- ed and uncorrelated ultraslow motions. The result relates T» to the first derivative of the ultraslow- motion correlation function. The SA expression for uncorrelated molecular reorientations is ob- tained as a special case. Qur results are applied to ultraslow motions in order-disorder-type crystals, where dynamics can be well described by a, kinetic Ising model in the interesting frequency region. The correlation functions are expressed in terms of the known Ising- model random-phase approximation (HPA) expres- sion ' (see also Appendix) for y(q, &u). In Sec. V conclusions are made about the possi- bility of the nuclear-magnetic-relaxation observa- tions of critical phenomena near order-disorder- type phase transitions. II. DIPOLAR SPIN-LATTICE RELAXATION I,et us consider a system of & identical nuclei with spins I coupled only by the magnetic dipolar interactions. The Hamiltonian of such a system in a strong external magnetic field is K = Kz-+KD where K~ is the Zeeman part and KD the dipolar part. The dipolar energy reservoir, whose spin- lattice relaxation rate is T, D, is represented only by the secular part of KD: where cIO' = ~'e'[s„l„. — , '(f„, f„+f„z„)—I, . . &0) 1 3 cos Hz& p t2 'Vfj z, , is the distance between the i th and j th nucleus, and 0;, . is the angle between r, , and magnetic field B. The spin-lattice relaxation is caused by motional modulations of the dipolar interactions. These mo- tions can be molecular flips, reorientations, or single-ion jumps. In the case of ultraslow motions the characteristic correlation time v is longer than the spin-spin relaxation time T2, which is of order Therefore it can be assumed that immediately after each jump or reorientation, which is nearly instantaneous compared to v, the spin temperature of the dipolar energy reservoir is established. ~ The spin order in the dipolar reservoir can be well described by a density matrix in the high-tempera- ture approximation, which, following Goldman, reads p(t) = l — X'(t)P(t), 1830