Magnus expansion

If the pulse strength of the simultaneous PIPs is further increased to 2/i = 44.48 kHz and 2nAfx=x (m= 1), (fi/Af) in Eq. (89) is no longer a small value and the Magnus-expansion in the coherent averaging theory may not converge. Consequently, the compensation in the null region vanishes almost completely. 

Although the AHT is powerful in predicting which terms in the Magnus expansion are zero, it is rather weak in estimating what residual linewidths... 

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... 

Note that in both the MREV and BR-24 graphs of Fig. 6 the squares and the triangles fall on straight lines. The slope of the tp = 0 line for BR-24 is s = 3, whereas the slope of the other three lines is 5 = 2. The slopes of the => 0 lines for the MREV and the BR-24 sequences thus confirm the prediction of the AHT that says that second-order terms in the Magnus expansion should dominate the residual width for the MREV sequence, that is, whereas the BR-24 sequence is designed in such a way that the purely dipolar second-order terms are cancelled in the limit tp => 0, that is, 8v, ... 

The AHT combined with the Magnus expansion predicts how the resolution in m.p. pulse spectra should depend on the pulse spacing t. Previously, attention was mainly (and unconsciously) focused on the limit-... 

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) ... 

Independent of the experimental implementation, idealized Hartmann-Hahn transfer functions can be calculated for characteristic zero-quantum coupling topologies (see Section V.B). Except for simple two-spin systems, transfer functions are markedly different for different zero-quantum coupling tensor types (see Fig. 10). This difference results from different commutator sequences that occur in the Magnus expansion of the density operator zero-quantum coupling topologies are shown schematically in Fig. 11. 

Evans, W. A. B. (1968). On some application of the Magnus expansion in nuclear magnetic resonance. Ann. Phys. 48, 72-93. 

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... 

Averaging over the Larmor period (Uo/27r is performed by taking the zero- and first-order terms in the Magnus expansion.They are notated here as and TtCq in order to stress their equivalence to the first- and second-order terms of standard perturbation theory 2... 

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]. 

The simple form of the spin propagator might have inspired Haeberlen and Waugh [29] to make use of the Magnus expansion to gain insight into... 

The operator t/, in its time-ordered exponential form, readily follows from the Magnus expansion [16]... 

Given the separation of slow rotational from fast vibrational motions outlined in section 4.2, unperturbed rotations are described by the Hamiltonian Hr, and the translational-rotational coupling is V R.n = ViR. u).d). Using again the Magnus expansion one finds that the wave operator sr - where [16]... 

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. ... 

Floquet-Magnus expansion is also used to study the effect of chemical shift anisotropy in solid-state NMR of rotating solids. The chemical shift interaction is irradiated with two types of rf pulse sequences BABA and C7. The criteria for the chemical shift anisotropy to be averaged out in each rotor period are obtained. ... 

From the nature of the Magnus expansion in terms of commutators one sees immediately that an analytical one step solution is obtained when W and V commute, if W is a constant diagonal matrix. 

As mentioned in Section 2.5, optical excitation in the practically somewhat trivial case of the degenerate many-level problem with a constant diagonal matrix W can be treated by exact integration, because the Magnus expansion terminates after the first term. Closed expressions for the two-level (also the three-level) problem can thus be obtained, and can be compared, for instance, with the quasiresonant approximation, identical with the RWA for the two-level model. Since the conditions for the QRA are obviously not satisfied (x = co in this case and not x tu, as required by the QRA), there are then sizeable differences between the exact and the approximate results. ... 

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