Effective coupling tensor

Whereas the term 0( > 3) can often be neglected in practice, the effective fields and the effective coupling tensors are decisive for the transfer of magnetization in Hartmann-Hahn experiments. If the basis sequence of duration is repeated n times, the propagator U nTff) is simply given by the nth power of... 

In general, the full isotropic coupling tensor between two homonuclear spins i and j can only be preserved by a multiple-pulse sequence if Ui (t) Uj U) [i.e., if fl j (t) = ap(t)] for 0 < / < Tj,. For multiple-pulse sequences with constant rf amplitude this is only possible for spin pairs with small offset differences p - Vj v, that is, for small angles 0-j. In general, the average coupling tensors are nonisotropic. Even if isotropic effective coupling tensors (with c J = c Jy = c 0 and c Jp = 0 for a = p)... 

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... 

Fig. 6. Classification schemes for Hartmann-Hahn experiments based on (A) the aggregation state of the sample, (B) nuclear species of the spins between which magnetization is transferred, (C) dynamics of magnetization transfer and its reach within a spin system, (D) isotropic or nonisotropic magnetization transfer, (E) magnitude of effective fields, (F) type of effective coupling tensors, (G) active bandwidth of the sequence, (H) type of multiple-pulse sequence, and (I) suppression of cross-relaxation.

Even in the absence of relaxation, Hartmann-Hahn transfer depends on a large number of parameters pulse sequence parameters (multiple-pulse sequence, irradiation frequency, average rf power, etc.) and spin system parameters (size of the spin system, chemical shifts, /-coupling constants). For most multiple-pulse sequences, these parameters may be destilled into effective coupling tensors, which completely determine the transfer of polarization and coherence in the spin system. This provides a general classification scheme for homo- and heteronuclear Hartmann-Hahn experiments and allows one to characterize the transfer properties of related... 

For example, suppose the effective fields and Bf of two spins i and j are oriented in the x-z plane of the rotating frame. If the effective coupling tensor [see Eq. (67)] is isotropic... 

If all zero-quantum phases can be adjusted to zero, the effective coupling tensors have the simple form... 

Isotropic effective /-coupling tensors (/) with a scaling factor s 1 are characteristic for ideal homonuclear Hartmann-Hahn experiments and, in particular, for homonuclear isotropic mixing experiments (see Section X). Isotropic effective /-coupling tensors can also be created between heteronuclear spins i and m (see Section XI) however, this results in a reduced effective coupling constant with a scaling factor i 1/3 [see Eq. (115)]. 

Fig. 7. Schematic representation of characteristic effective coupling topologies for three coupled spins. Thick, thin, dashed, and dotted lines stand for isotropic iC = /), planar (C - = P longitudinal (C > = L), and vanishing (C > = O) effective coupling tensors, respectively. The effective coupling topologies III, IPP, and PPP shown in (A), (B), and...

Spin Systems with Planar Effective Coupling Tensors... 

In the case of an isotropic effective coupling tensor, the effective coupling between two spins with offsets v,- and Vj is given by... 

In heteronuclear triple-resonance experiments (TCP Majumdar and Zuiderweg, 1995), the spin system corresponds to an effective PPP coupling topology if planar effective coupling tensors are created. In an ISQ system, triple-resonance Hartmann-Hahn transfer from a spin / to a spin Q is only efficient if I//5I J,q (Glaser, 1993c Majumdar and... 

The MLEV-16 sequence, which contains rf pulses with orthogonal phases, has poor heteronuclear transfer characteristics. The effective coupling tensors are neither planar nor isotropic. For two spins on-resonance, the average coupling tensor has the form... 

Hartmann-Hahn mixing sequences that create nonisotropic effective coupling tensors of the form -t- I S ) are only able to transfer x... 

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