Effective Hamiltonian Magnus expansion

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... 

With the help of the Magnus expansion, the effective Hamiltonian that is created by a time-dependent Hamiltonian during the time Tj, can be divided into contributions of different orders (Haeberlen, 1976 Ernst et at., 1987) ... 

This effective Hamiltonian is again not unique but can be chosen such that its eigenvalue differences are smaller than l/2o t. Maricq [100-102] and others [14, 103] have demonstrated that the Magnus expansion of the effective Hamiltonian in AHT and the van Vleck transformation approach of the Floquet Hamiltonian are equivalent. At the time points krt the Floquet solution for the propagator in Eq. 24 has the form... 

This expression is identical to the zero and first AHT terms obtained when the first-order Magnus expansion terms are calculated using the integral expressions and the Fourier expansion of Hint t)- Thus the effective Hamiltonian to first-order in AHT differs from the first-order van Vleck expansion, Eq. 61b. This difference has been discussed by Goldman [98], Mehring [14] and others [103, 104] and it was shown that the additional term in < H > should be discarded... 

The power of ultrafast MAS can easily be understood with AHT, as is explained in the seminal paper by Maricq and Waugh [52]. The solution to the periodic Hamiltonian problem H t) is obtained with a Magnus expansion that provides an effective Hamiltonian Hgg- acting on the spin system during a rotor period. This is relevant in the case of stroboscopic observation, that is, when a spectral window of or a sampling dwell time equal to Tr= 1/Vr is used HefF governs the shapes of the sidebands in the MAS spectrum and, indirectly, the resolution that can be achieved. On the other hand, the decay of the rotational echo is responsible for the shape of the spinning sideband pattern [52]. 

Mananga presents the possibility of applying the Floquet-Magnus expansion to the most useful interaetions known in solid-state NMR using the magic-echo scheme. The results of the effective Hamiltonians of these theories and average Hamiltonian theory are presented. ... 

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