Simulations of Multiple-Pulse Experiments

As pointed out in the Introduction, the AHT and the Magnus expansion have been powerful tools for designing line-narrowing and other m.p. sequences because tractable analytical expressions can be worked out at least for the low-qrder terms f the Magnus expansion of the effective Hamiltonian F = -V- - . We follow here the notation of 

It is known that for the WAHUHA sequence all purely dipolar terms in the Magnus expansion of the effective Hamiltonian F vanish for a two-spin system in the S-pulse limit (Bowman, 1969). The lines in a WAHUHA m.p. spectrum (S-pulse limit) of a two-spin system should, therefore, be infinitely narrow, irrespective of the pulse spacing t. A two-spin system is, hence, obviously too small for out purpose. Likewise, two-spin systems are too small to meaningfully test any line-narrowing 

The computational procedure follows closely the steps of an actual m.p. experiment see Fig. 1. The spin system, which is initially in thermal equilibrium, is hit by a preparation pulse Pp. Thereafter, one component of the transverse nuclear magnetization created by Pp, say My, is measured and the measurement is repeated at intervals of the cycle time The resulting time series My(qtJ,q = 0.(2 - 1), if Fourier transformed. For simulations we accordingly first specify the initial condition of the spin system, that is, the initial value of the spin density matrix g(t) in the rotating frame. Our standard choice Pp, = P implies p(0) fy == the sum running over k = We then follow the evolution 

Aw is the offset of the spectrometer frequency from the Larmor frequency is the zz component of the chemical shift tensor a, 

The Hamiltonian [Eq. (2)] is piecewise constant in time. Therefore, we may write 

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Meakin and Jesson (48) used the Bloch equations in part of their work on the computer simulation of multiple-pulse experiments. They find that this approach is efficient for the effect upon the magnetization vector of any sequence of pulses and delays in weakly coupled spin systems. However, relaxation processes and tightly coupled spin systems cannot be dealt with satisfactorily in this way and require the use of the density matrix. 

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