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. 

Ignoring relaxation, the spin evolution is governed by the Liouville-von-Neumann equation... 

In the following section, we explain the basic protocols used for removing the second-order quadrupolar broadening based on the refocusing of the second-order quadrupolar interaction. These protocols rely on mechanical reorientation of the rotor axis (DAS) or use a combination of sample spinning and rf manipulation of the spins evolution (MQMAS and STMAS). Experimental aspects of these methods, as well as methods for data processing and analysis, are described in Sects. 5.3 and 5.4. 

In this section we analyze the dynamics of a spin subject to a classical random field and derive the equation of motion for the spin dynamics (the spin-evolution operator), averaged over the fluctuations. Following the discussion of the case with quantum fluctuations, we first analyze the dynamics in a stationary field B and a random field exactly as in the quantum case one can reduce the analysis of the dissipative corrections to the Berry phase accumulated over a conic loop to the problem with a stationary field by going over to a rotating frame. 

The time Tev depends on the mechanism of spin evolution. For the dipole-dipole mechanism tcv = / gpD, where D is the dipole-dipole energy of interaction, p is the Bohr magneton. The values Tev=2.7T0"11 s, and TPaii=3.1 10 9 s were calculated for TBPDA(C6o)2-... 

A different approach to simplifying RP spin evolution for calculation is the semiclassical approach developed by Schulten and coworkers.In this approximation. 

FIGURE 9.2 Vector models (projections) illustrating pair substitution. The labels 1 and 2 denote the electron spin of the first and the second radical of the pairs. The observed proton is contained in the first radical. Its spin state, la) or IP), is displayed at the respective leftmost projection. The radical pairs are bom in the triplet state, and the product is formed from the singlet state c gives the singlet character. First radical pair RPi, positive g-value difference, zero hyperfine coupling constant second radical pair RP2, equal g values, positive hyperfine coupling constant. For the situations without pair substitution, the spin evolutions under the influence of the Zeeman and the hyperfine interaction have been separated for clarity. Further explanation, see text. 

This procedure could be followed, when possible, analytically, or by truncating the Fioquet Hamiltonian and using computer calculations. However, because this numerical approach can become cumbersome, it is almost always more practical to perform exact time ordering for actual numerical spin evolution calculations. The Fioquet approach is then only used when physical insight is required and when perturbation theory can be utilised to estimate He// and the D fc s. 

The distance dependence of the exchange integral / couples spin evolution and diffusive motion of the radical pair. However, / is such a strongly varying function that during a diffusive excursion there are almost instantaneous transitions... 

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. 

Although photo-QDNP spectroscopy has come of age by now, new applications still arise. As has emerged, the CIDNP effect connects diffusion, chemical reactivity, and spin evolution in a unique way it also combines the analytical potential of NMR spectroscopy with a sensitivity to species as short-lived as a nanosecond or even less. Hence, photo-ClDNP spectroscopy provides very diverse and deep insight into both chemical and physical processes, and yields information that is often inaccessible by other techniques. A method as powerful and versatile as this certainly deserves to be more widely known, and more frequently applied. 

With R being either diag(l - X, 1,1) or diag(l — X, 1,0), spin evolution and chemical reactivity can be separated [21] by introducing [30] a quantity F that gives the amount of geminate product formed for X = 1 from radical pairs with a triplet precursor without initial phase correlation, and a quantity A, the spin independent reaction probability,... 

In a radiation track, the geminate radical ion pairs originate in singlet state in which the partner spins are antiparallel and the total electron spin S is zero. Description of the pair spin evolution in an arbitrary magnetic field is a fairly complex problem. Common, however, is the fact that the expression for the... 

There are several approaches [4, 25] that try to determine the fraction of spin-correlated radical ion pairs in radiolysis. The transient emission and absorption [25] suffer, however, from the lack of exact data on the extinction coefficients and luminescence quantum yields of intermediate products. The magnetic field effect technique [4] is more straightforward. However, it requires a detailed knowledge of spin evolution in zero field which is a problem in most cases. 

When applying the quantum beats technique, it is essential that only the singlet-correlated pairs contribute to the oscillating component of spin evolution. Therefore, for systems with a single oscillation frequency, the singlet state population Ps t) is determined by... 

Two narrow peaks are clearly observed on all experimental curves, and some systems showed also the third and fourth peaks. The position of the strongest second peak corresponds to the published hfi constants. The form of the curves indicates that spin relaxation has substantial effect on spin evolution. The curves have a smoothly rising background and the amplitudes of peaks decay with time. 

Obviously, maximum time of spin evolution cannot be infinite (which would be pointless anyway, because of relaxation). This limit can be represented by multiplying signal by step function n(ti, t2..., t ) ... 

Evolution involves imparting phase character to the spins in the sample. Mixing involves having the phase-encoded spins pass their phase information to other spins. Evolution usually occurs prior to mixing and is termed tj (not to be confused with Tj the relaxation time ), but in some 2-D NMR pulse sequences the distinction is blurred, for example in the correlation spectroscopy (COSY) experiment. Evolution often starts with a pulse to put some magnetization... 

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