Zero-quantum dephasing

The most recent, successful and widely applicable approach to suppressing zero-quantum contributions employs a pulsed field gradient method for the net dephasing of these coherences in a single scan with no requirement for experiment repetition [88]. Since PFGs alone cannot manipulate ZQCs, the gradients must be applied in a rather ingenious manner to be effective, tbe operation of wbicb may be understood as follows. 

To begin we again consider tbe approach described above for generating differential evolution of ZQCs by variation of tbe total duration of tbe z-filter delay, An alternative approach to this would be to maintain r f at a fixed duration and to place within this a 180° pulse that moves throughout the delay on each repetition of the experiment [89] (see Fig. 10.38b). This pulse introduces a spin-echo to the z-filter in which the ZQ evolution is refocused such that the net evolution is restricted only to the residual period r f. Relocation of the 180° pulse for each experiment thus leads to differential evolution between them. 

The degree of attenuation A qc depends upon the zero-quantum frequency Gzqc (here in rad/s) and the duration of the frequency sweep according to the following equation [88]  

This approximates to IQzqcT when the sine term oscillations are ignored and allows an estimate of the sweep durations that need to be employed. Thus, in the case of a spectrum recorded at 600 MHz with /-coupled spins separated by 0.5 ppm (that is. 

The application of the zero-quantum dephasing element has been illustrated in earlier chapters within sequences such as TOCSY (Section 5.7.4) and NOESY (Section 8.7.2), so here we consider only the steps necessary for the calibration of the element itself. For the dephasing to perform correctly, one needs to match the frequency profile of the swept pulse with the frequency spread imposed on the spectrum by the simultaneous field gradient, and the calibration sequence of Fig. 10.40 is designed to achieve this. 

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Even more efficient single-scan zero-quantum dephasing is possible if adiabatically switched gradients are used in combination with on-resonance spin-locking. Experimental and theoretical details of this technique can be found in the paper by Davis et al. (1993). 

Figure 5.72. Zero-quantum interference in selective ID TOCSY spectra, (a) Partial spectmm and the corresponding regions of ID TOCSY recorded with 80-ms DIPSI-2 (b) without zero-quantum suppression and c) with the zero-quantum dephasing scheme shown in Fig. 5.74 employing adiabatic smoothed CHIRP pulses with 40 kHz sweep widths and durations of IS and 10 ms.

Figure 5.74. The zero-quantum dephasing scheme applied to TOCSY. The boxed regions contain the dephasing elements in which the gradients are applied during the swept inversion pulse the two elements are of different durations to avoid accidental refocusing. Gi represents a purge gradient.

Figure 10.38. Approaches to suppressing zero-quantum coherences (a) the basic r-filter, (b) the spin-echo z-filter and (c) the zero-quantum dephasing element employing an rf frequency sweep/ gradient pulse combination see text for details.

Figure 10.39. Schematic representation of events during the zero-quantum dephasing element. The effective timing of the 180° refocusing pulse and hence the net zero-quantum evolution time becomes dependent upon spatial location along the z axis, meaning the zero-quantum coherences evolve to differing extents according to the position of the participating spins in the sample.

Figure 10.40. The zero-quantum dephasing calibration sequence.

Figure 10.41. Execution of the zero-quantum dephasing calibration routine see text for discussions.

This modification reflects the fact that a p-quantum coherence dephases at a rate proportional to p. Thus, double-quantum coherences dephase twice as fast as single-quantum coherences yet zero-quantum coherences are insensitive to field gradients. When the eoherence involves different nuclear species, allowance must be made for the magnetogyric ratio and coherence order for each, such that ... 

A number of methods have been developed to suppress contributions to the spectrum from zero-quantum coherence. Most of these utilise the property that zero-quantum coherence evolves in time, whereas z-magnetization does not. Thus, if several experiments in which the zero-quantum has been allowed to evolve for different times are co-added, cancellation of zero-quantum contributions to the spectrum will occur. Like phase cycling, such a method is time consuming and relies on a difference procedure. However, it has been shown that if a field gradient is combined with a period of spin-locking the coherences which give rise to these zero-quantum coherences can be dephased. Such a process is conveniently considered as a modified purging pulse. 

This frequency can, under certain circumstances, become spatially dependent and thus the zero-quantum coherence in the tilted frame will dephase. This is in contrast to the case of zero-quantum coherence in the laboratory frame which is not dephased by a gradient pulse. 

Reff = observed dipolar coupling constant t = time T20 = spin term in the spherical tensor representation of the dipolar Hamiltonian = zero-quantum relaxation time constant U = propagator = magne-togyric ratio of spin / A/ = anisotropy of the indirect spin-spin interaction 0 = angle between the applied field and the internuclear vector A = dephasing parameter /Uq = permeability of free space Vj. = rotor frequency in Hz 1/, = isotropic resonant frequen-... 

The short-time spike in the decay, which can be attributed to the dephasing of many quantum beat terms (all with + 1 phases), represents the irreversible flow of vibrational energy out of the zero-order state prepared by the laser. The long-time component, although weakly modulated, represents an equilibration in the distribution of vibrational energy subsequent to the initial energy flow process. 

One dynamical example, two-level quantum beats including decay (but not including collision induced depopulation and dephasing), illustrates the power of this complex H formalism. (See Section 6.5.3 for a detailed discussion of quantum beats in the strong coupling limit.) Consider two zero-order states, Ei = ei — iTi/2 and E2 = ei — iT2/2, where state 1 is bright and narrow and state 2 is dark and broad (T2 >> Ti). 

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