Coherent Spin Evolution

In comparison with other spectroscopic techniques, NMR is blessed with short-range interactions that render it possible to characterize and influence the spin evolution over relatively long periods of time without excessive loss from dissipative processes. This implies that the spin evolution to a large extent (but certainly not exclusively) may be described as unitary evolution of coherence/polarization potentially supplemented with corrections due to relaxation. 

In our introduction to the physics of NMR in Chapter 2, we noted that there are several levels of theory that can be used to explain the phenomena. Thus far we have relied on (1) a quantum mechanical treatment that is restricted to transitions between stationary states, hence cannot deal with the coherent time evolution of a spin system, and (2) a picture of moving magnetization vectors that is rooted in quantum mechanics but cannot deal with many of the subder aspects of quantum behavior. Now we take up the more powerful formalisms of the density matrix and product operators (as described very briefly in Section 2.2), which can readily account for coherent time-dependent aspects of NMR without sacrificing the quantum features. 

Rhim et al. have shown that application of a suitable sequence of strong RF pulses can reverse the time evolution of the coherent spin state caused by the homonuclear dipolar interaction [28]. Consequently, the influence of the dipo-... 

This again has no parallel in liquid-phase CIDNP. It is based on a coherent process in correlated radical pairs, without the involvement of electron-spin polarizations but again with the anisotropy of the hyperfine interaction playing a major role different lifetimes of the singlet and triplet pairs break the symmetry of the spin evolution. 

Before moving forward, the chemical biology reader should appreciate that Mz(f) polarisation and recovery towards steady state bulk magnetisation Mby spin-lattice relaxation are primarily indicative of changes in the levels of spin state occupancy. By contrast, My(f) coherence and evolution by spin-spin relaxation are both primarily indicative of transitions between spin state energy levels. In this context, the nuclear Overhauser effect (NOE) is a very important through-space effect in NMR spectroscopy that results directly from dipolar coupling-mediated spin-lattice relaxation. 

In this experiment, the phase acquired by the observer spin coherence during the first interpulse delay of length Ti is exactly compensated by the phase acquired during the second interpulse delay of length Xi after the n inversion pulse. At time 2xi, the coherence thus has zero phase. This time can thus be identified with dipolar evolution time t = 0 and corresponds to the time immediately after the nl2 pulse in the basic PELDOR experiment Spin evolution for t > 0 is analogous to basic PELDOR with time X2 replacing time x. For Xi > 200 ns, interference between pump and observer pulses can be safely neglected at t = 0. 

To conclude this section on 2D NMR of liquid crystals, some studies of more exotic liquid crystalline systems are pointed out. Polymer dispersed nematic liquid crystals have attracted much attention because of their applications as optical display panels. Deuteron 2D quadrupole echo experiments have been reported [9.28] in the isotropic and nematic phases of / -deuterated 5CB dispersed in polymers. A similar technique was used [9.29] to study two model bilayer membranes. Both studies allow determination of the lineshape F(u ) due to quadrupolar interactions and the homogeneous linewidth L(u ) of the individual lines [9.28]. The 2D quadrupole echo experiment has also been used [9.30] to separate chemical shift and quadrupolar splitting information of a perdeuterated solute dissolved in a lyotropic liquid crystal. The method was compared with the multiple-quantum spectroscopy that is based on the observation of double-quantum coherence whose evolution depends on the chemical shift but not on the quadrupolar splitting. The multiple-quantum method was found to give a substantial chemical shift resolution. The pulse sequences for these methods and their treatment using density matrix formalism were summarized [9.30] for a spin 1=1 system with non-zero chemical shift. Finally, 2D deuteron exchange NMR was used [9.31] to study ring inversion of solutes in liquid crystalline solvents. 

In the previous section, we learned how the recombination probability of a radical pair depended on the coherent mixing of the spin states caused by the S-To mixing. On the other hand, it also depends on the incoherent mixing caused by transversal and longitudinal relaxation. In this section, we will show that one can obtain a qualitative understanding of these effects by considering the spin evolution described by the Bloch equations. From textbooks of Quantum Mechanics, any Hermitian operator F has the following property ... 

The delay is generally kept at Vi x> The coupling constant Jcc for direcdy attached carbons is usually between 30 and 70 Hz. The first two pulses and delays (90J -t-180 2-t) create a spin echo, which is subjected to a second 90J pulse (i.e., the second pulse in the pulse sequence), which then creates a double-quantum coherence for all directly attached C nuclei. Following this is an incremented evolution period tu during which the double quantum-coherence evolves. The double-quantum coherence is then converted to detectable magnetization by a third pulse 0,, 2, and the resulting FID is collected. The most efficient conversion of double-quantum coherence can... 

If j Rf is exactly the magic angle and infinite spinning speed is assumed, the first-order anisotropic terms are zero for both single and DQ coherence (33). This does not hold true for finite spinning speed, but a complete averaging of the first-order effect occurs at the exact rotor cycles. Therefore, the x evolution time has to match exactly a multiple of the rotor period. The second-order anisotropy refocusing occurs for... 

For spin 3/2 and thus q = 2 corresponding to the two symmetric ST and DQ transitions (Fig. 14), it follows that k = 8, meaning that the time evolution on the DQ coherences has to be set to 8t/9. Several protocols for the acquisition and processing of STARTMAS data have been described [202]. The simplest method is to use only the data points acquired stroboscopically at times t = nx, which can be... 

Several basic 2D schemes have been applied in these experiments, based on (1) the use of the SQ evolutions during t and f2, in analogy to the NOESY experiment, (2) the MQ-SQ experiment, in which the evolutions of MQ coherences during t are correlated with the CTs of recoupled spins to enhance the resolution in analogy to MQMAS, and (3) the DQCT-SQ protocol similar to that used in DQMAS NMR of spin-1/2 nuclei. 

The first case has already been considered section 2.0 the second case leads to a strong classical spin-orbit coupling, which is reflected in a Hamiltonian nature of the classical combined dynamics. In both situations the procedure is to find a suitable approximate Hamiltonian Hq( ) that propagates coherent states exactly along appropriate classical spin-orbit trajectories (x(l,),p(t),n(l,)). (For problems with only translational degrees of freedom this has been suggested in (Heller, 1975) and proven in (Combescure and Robert, 1997).) Then one treats the full Hamiltonian as a perturbation of the approximate one and calculates the full time evolution in quantum mechanical perturbation theory (via the Dyson series), i.e., one iterates the Duhamel formula... 

Under such circumstances, the evolution of a spin system has to be calculated in different rotating frames defined by the corresponding PIPs and a special case may arise, where a spin experiences an on-resonance excitation but off-resonance evolution in the conventional rotating frame. Unpredictable results may occur if the phase coherence in PIPs fails. Unfortunately, to date, no... 

---