Toggling frame

In cases where ideal (i.e., instantaneously short) rf pulses are used, it is useful instead to use a so-called toggling frame approach where the Hamiltonian by each pulse is toggled to another frame. For the case where N ideal rf events have occurred, the Hamiltonian for the free precession period following the Mh pulse is described by... 

Vega, and coworkers as a sequence of well-placed ideal (i.e., infinitely strong) 7r-pulses serving to reintroduce the zero-quantum (ZQ) part of the homonuclear dipole-dipole coupling operator in a homonuclear two-spin system. The coherent averaging due to MAS is in the toggling frame of the n-pulses partially disrupted by a differential chemical shift term and thereby recoupling takes place. 

The basic RFDR element consists of two 7i-pulses each centered in the middle of each rotor period in two consecutive rotor periods leading to a cyclic rf field. In the toggling frame of ideal (i.e., infinitely strong) 7i-pulses, the Hamiltonian of the chemical shift terms and the homonuclear dipolar coupling interaction is split up into three commuting parts ... 

If the (doubly) rotating frame and the toggling frame coincide at the beginning and at the end of the basis sequence [i.e., if the basis sequence is cyclic with = 1], then U(t ) = Uo°Krh) and Provided... 

Hamiltonian ft is well approximated by the time-averaged Hamiltonian in the toggling frame ... 

In the toggling frame that is defined by alone, isotropic homonuclear coupling terms are invariant, because = 0, that =... 

Bazzo and Boyd, 1987 Fujiwara and Nagayama, 1989 Abramovich et al., 1995). In this alternative toggling frame, the only remaining term is the coupling term. j°F(t), which is defined as... 

In the toggling frame, the time evolution is effected by the propagator... 

Hence, each spin i has its own transformation operator t (r), which transforms the spin operators (with a = x, y, or z) in the toggling frame ... 

Homonuclear isotropic coupling terms are not invariant in this toggling frame and can be transformed into nonisotropic bilinear terms ... 

Using Eqs. (91)-(93) and (96), it can be shown that the matrix elements of the time-dependent homonuclear coupling tensor in the toggling frame are constrained by the conditions... 

For the matrix elements of the average coupling tensor in the toggling frame. 

In heteronuclear Hartmann-Hahn experiments, where two (or more) nuclear species are irradiated simultaneously with the same multiple-pulse sequence, the heteronuclear coupling terms have the following form in the toggling frame ... 

For the average heteronuclear coupling tensor J , in the toggling frame, the matrix elements fulfill the conditions... 

In this case, the average coupling tensor between two spins i and j in the toggling frame yields a good estimate of the effective coupling tensors in the rotating (or doubly rotating) frame ... 

The effective coupling tensor between two coupled spins in the toggling frame is only a good approximation of the effective coupling tensor in the (doubly) rotating frame if the higher order contributions in the Baker-Campbell-Hausdorff expansion [see Eq. (119)] can be neglected. This is the case if the term... 

Consider the simplest of these sequences, WHH4, where the cycle has four pulses and five time-intervals as shown in Figure 2.20. If the dipolar coupling is dominant then, in the toggling frame, the magnetisation is rotated successively to different directions, the average dipolar interaction being... 

The last equation defines the average Hamilton operator in the toggling frame, which is given in zeroth order by the average (H -f + H )/3. For the Hamilton operators of the chemical shielding (A. = o- in Section 3.1.1) and of the dipole-dipole interaction (A = D) are inserted. They are given in the rotating frame by... 

After transformation to a toggling frame defined by the sequence of tt pulses and in the limit of -shaped tt pulses, the Hamiltonian assumes the simple form... 

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