) N bIS ~ IS (AS + to - \\ Zt Ui3 < F >) (5.154) where b is the amplitude of the oscillating magnetic field. The resonance condition is clearly UJ = -(AS-1-J2 Uij < Iz (5.155) dh dt Chapter 5. The general Suhl-Nakamura interaction 86 In other words, the frequency pulling effect predicted classically is the same as the effect predicted quantum mechanically in equation 5.144 except for the factor of 1/2 in the second term of equationclassicalpulling. Classical simulations were performed with an array of 15 by 15 nuclear spins. The spins were first thermalized without including the interaction terms. The interactions and the oscillating magnetic field were then added, and the system was evolved for a short period of time, At. The change in P was measured for each spin, and the absorbed power is then A ~ P - ^ = A * ) - / ; ( * = o) 3 oc < / ; - / / > (5.156) where the average <> is taken over all the spins in the lattice, If = Pit — At) and If = P(t = 0). At was chosen in order to remain in the limit of C W NMR. The centre frequency of the resonance was obtained by plotting the P as a function of frequency. The result of several simulations at the same temperature (10 mK) with longer and longer range interactions is shown in figure 5.39. The results of several simulations at different temperatures is shown in figure 5.40. The agreement with the theory is reasonable, even with a short range interaction. The discrepancy may be partly due to errors associated with the strength of the oscillating magnetic field. In order to perform the simulations in a reasonable time, the oscillating field was stronger by five orders of magnitude than it is experimentally. The discrepancy between the theory and the simulations was reduced when the oscillating field was reduced. A simulation involving an array of 12 by 12 triplets of spins was also performed, and the results can be seen in figure 5.41. Only interactions within a triplet of spins were included in the calculation. The quantity plotted is < If — If > where the average is over the Mnl spins or one of the Mn2 spins. It appears there are only two resonances. Chapter 5. The general Suhl-Nakamura interaction 87 14 12 10 £ 8 3 6 0 y = 2.046x , ' , ' V y = 1 -8656x - 0.0084 0 1 2 3 4 5 6 7 (1/2)U(k=0) (MHz) Figure 5.39: The frequency pulling for a variety of interactions at T = 10 mK. Each data point is calculated by increasing the range of Uij. The first point corresponds to only a self interaction of 2 MHz. A nearest neighbour interaction of 1 MHz was added, then a second nearest neighbour interaction of 0.5 MHz and finally a third nearest neighbour interaction of 0.2 MHz. The solid line is the best fit to the data, and the dashed line is the prediction of equation 5.155 because at T = 10 mK, < T >= 2.046. Chapter 5. The general Suhl-Nakamura interaction 88 7 0 -| , , , , 1 0 0.5 1 1.5 2 2.5