Rotational resonance conditions

A very useful class of solid-state experiments exploits the rotational resonance condition, where the rotation frequency is set to be equal to a multiple... 

Fig. 2.7. Energy level diagram in the homonuelear two-spin system under near rotational resonance condition.

As mentioned earlier, magic-angle sample spinning averages out the dipolar coupling and, therefore, slows down or suppresses spin diffusion (except for the special case of the rotational-resonance condition). However, in many applications it is crucial to use MAS in order to increase the resolution of the spectrum and to obtain well-resolved lines. The homonuclear dipolar... 

Figure 19 Demonstration of the rotational resonance effect for the framework-bound ethoxy species obtained on CsX from ethyl iodide-1,2- C. The rotational resonance condition was achieved when the spinning speed was set to match the frequency difference (4646 Hz) between the signals at 68 and 17 ppm. Rotational resonance reintroduced the C- C dipolar coupling for the rigid ethoxy species, resulting in a splitting pattern that reflects the short intemuclear distance.

The APRR (adiabatic polarization transfer under rotational resonance conditions) experiment is essentially an polarization transfer experiment, with the difference that the spinning rate is not kept constant but is ramped adiabatically through the condition (Fig. 15). This leads to a more complete transfer and a more broadband resonance condition, rendering the experiment less sensitive to distributions of isotropic chemical shielding in disordered systems. 

Hou et al. have recently described a family of experiments for R2 Driven Spin Diffusion (RDSD) spectroscopy suitable for homonuclear correlation experiments under fast MAS conditions. In these RDSD experiments, since the broadened second-order rotational resonance conditions are dominated by the rf field strength and the phase shifts, as well as the size of reintroduced dipolar couplings, the different R2 sequences display unique polarization transfer behaviours and different recoupling frequency bandwidths. In the subsequent report, Hou et al. present a series of modified R2/ sequences, dubbed combined R2 -Driven (CORD), that yield broadband homonuclear dipolar recoupling and give rise to uniform distribution of cross peak intensities... 

System (A8.2)-(A8.4) defines completely the time variation of orientation and angular velocity for every path X(t). One can easily see that (A8.2)-(A8.4) describe the system with parametrical modulation, as the X(t) variation is an input noise and does not depend on behaviour of the solution of (Q(t), co(r). In other words, the back reaction of the rotator to the collective motion of the closest neighbourhood is neglected. Since the spectrum of fluctuations X(t) does not possess a carrying frequency, in principle, for the rotator the conditions of parametrical resonance and excitation (unrestricted heating of rotational degrees of freedom) are always fulfilled. In reality the thermal equilibrium is provided by dissipation of rotational energy from the rotator to the environment and... 

Polarization-transfer experiments which are based on a resonance condition, i.e. where a variable quantity in the experiment is matched to a parameter of the investigated spin system, can be carried out as a transient experiment or as an adiabatic experiment Figure 11.5 illustrates the differences between these two types of experiments. In a transient or sudden" experiment, the density operator is prepared in a state orthogonal to the effective polarization-transfer Hamiltonian (Fig. 11.5a). When the polarization-transfer Hamiltonian is switched on, the density operator starts precessing around the effective Hamiltonian, and usually maximum polarization transfer is reached after a 180° rotation. Since often the size of the effective Hamiltonian at the matching condition depends on... 

If the condition 7S1 2> 6v is not fulfilled, then the experiment is called the off-resonance spin-lock or T p experiment in a tilted rotating frame . The experiments of this type, employing off-resonance conditions to a variable extent, have actually been proposed as a tool for varying the frequencies at which the spectral densities are being sampled [49]. The advantage of this approach is that it provides a means of estimating the rotational correlation time for macromolecules in solution. 

The individual vectors p, before irradiation, are out of phase with one another and this can be represented by the vector M0 aligned in the Oz direction (Fig. 9.4). As the resonance condition is reached, all the vectors pack together and rotate in phase with B[. Hence, M0 changes direction and finally reaches an angle a with the Oz axis, which is controlled by the time and power of irradiation (Fig. 9.7). Thus M0 acquires an Mxv component in the horizontal plane that is maximum when a — 7t/2, while maintaining a component Mz in the direction of the Oz axis (except if a = tt/2). The frequency of rotation of the magnetisation vector is equal to that of the precession movement. Under these conditions, some nuclei will proceed to the second orientation allowed (in the case where I = 1/2). The system will slowly return to its original state after the irradiation is stopped. A coil is used to detect the component in the Oy direction (Fig. 9.8). 

To confirm the above conjectures we have performed a numerical simulation of equation (29) on the Brusselator model chemical reaction.46 The results are shown in Fig. 7. We start with an initial condition corresponding to a clockwise wave. Under the effect of the counterclockwise field this wave is deformed and eventually its sense of rotation is reversed. In other words, the system shows a clear-cut preference for one chirality. As a matter of fact we are witnessing an entrainment phenomenon of a new kind, whereby not only the frequency but also the sense of rotation of the system are adjusted to those of the external field. More complex situations, including chaotic behavior, are likely to arise when the resonance condition w = fl, is not satisfied, but we do not address ourselves to this problem here. 

A microwave cavity placed between SI and S2 can induce spin-flip transitions (F,Mp) = (1,1) —i (1,-1) if tuned to zvhf(H). In order to produce a positive signal, i.e. an increase in counting rate after S2 under resonance condition, S2 will be rotated by 180 degrees with respect to SI. Therefore, the (1, —1) state where Mp = —1 is defined with respect to the magnetic field direction in SI will be a (1,1) state in S2, while the (1,1) state of SI without spin flip would correspond to a (1, —1) state in S2. As a result, if the microwave frequency is off resonance, no H atoms will reach behind S2, while on resonance an increase in the number of atoms should be detected after S2. 

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