The One-Spin System

To provide a concrete application of the tools we have developed, let us now apply the general density matrix approach to an ensemble of one-spin systems (with I = V. To set up the density matrix according to Eqs. 11.1 and 11.5, we need basis functions j) and (f)2, which are simply a and (3. Because these are eigenfunctions of the wave functions are simple i (1) = a and t f2) = /3, giving the four coefficients needed for Eq. 11.5 the values 

We can then write the matrix elements of p (which at equilibrium is the same in the rotating and laboratory frames) as 

35 the first term is just a constant times the unit matrix l.This term is of little interest, as it does not vary with time and it does not contribute to quantities that we wish to measure. It is customary, therefore, to simplify the 

In Section 2.3 we described the effect of the operators I ly, Iz, I+ and I on the functions a and /3. Here we summarize these relations in terms of matrices with a and fd as the basis functions. Thus, 

As we shall see, it is very helpful to be able to compute the behavior of the spin system from the sort of matrix multiplications that we have already carried out. On the other hand, it is often possible to simplify the algebraic expressions by using the corresponding spin operators. In fact, this is the concept of the product operator formalism that we discuss later. Note that from Eqs. 11.35 and 11.36, the (redefined) density matrix at equilibrium can be written in operator form as 

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We now look at the evolution of the density matrix for the one spin system as the magnetization precesses in the rotating frame. Once more, we apply Eq. 11.16 in the same manner we did in Section 11.2 to take into account the effect of the rotating frame. In this case we obtain... 

By calculating (Mxy), as we did for the one-spin system, we could demonstrate the two precessing components of magnetization that beat with each other as they go in and out of phase. 

These complications require some carefiil analysis of the spin systems, but fiindamentally the coupled spin systems are treated in the same way as uncoupled ones. Measuring the z magnetizations from the spectra is more complicated, but the analysis of how they relax is essentially the same. 

It may be concluded that during the contact time in the competing process for the energy in the various spin systems, the carbon atoms are trying to reach thermal equilibrium with the proton polarization, which is in itself decreasing with a time constant, (Tig, H). In fact the protons undergo spin diffusion and can be treated together, whereas the carbon atoms behave individually. Therefore one implication is that we can also expect to obtain a C-13 spin polarization proportional to the proton polarization. 

Taking into account spin diffusion among the proton spin system (multiple proton-proton cross-relaxation rates), one obtain order... 

Fig. 11 ROCSA spectra for Val-18 of APn-25 fibrils. The C spectrum was measured at 9.39 T under a MAS frequency of 20 kHz. The other spectra were measured at 14.09 T under the frequency of 11 kHz. Upper traces are experimental spectra. Lower traces are best-fit simulations for one-spin system. The rf carrier frequency is at 0 Hz. (Figure and caption adapted from [158]. Copyright [2003], American Institute of Physics)...

Fig. 12 C -detected C CSA patterns of the SHPrP109 i22 fibril sample. The upper and lower traces correspond to the experimental and simulated spectra, respectively. Simulations correspond to the evolution of a one-spin system under the ROCSA sequence. The only variables are the chemical shift anisotropy and the asymmetry parameter. A Gaussian window function of 400 Hz was applied to the simulated spectmm before the Fourier transformation. (Figure and caption adapted from [164], Copyright (2007), with permission from Elsevier)...

Recapitulating, the SBM theory is based on two fundamental assumptions. The first one is that the electron relaxation (which is a motion in the electron spin space) is uncorrelated with molecular reorientation (which is a spatial motion infiuencing the dipole coupling). The second assumption is that the electron spin system is dominated hy the electronic Zeeman interaction. Other interactions lead to relaxation, which can be described in terms of the longitudinal and transverse relaxation times Tie and T g. This point will be elaborated on later. In this sense, one can call the modified Solomon Bloembergen equations a Zeeman-limit theory. The validity of both the above assumptions is questionable in many cases of practical importance. 

The TOCSY spectrum has a diagonal set of peaks (open circles) as well as peaks which are off the diagonal (filled circles). The off-diagonal peaks occur at positions where a proton on the Fi axis is in the same spin system as one on the F2 axis. In the schematic spectrum on the right, there are two superimposed isolated 3-spin systems (Hm, Ha2, Has) and (Hxi. Hx2, Hxs) and the cross peaks clearly indicate which resonances belong to each spin system. 

By applying the NOESY step first, this experiment allows us to jump from one spin system to another or to overcome a bottleneck in TOCSY transfer caused by an occurrence of a small coupling constant in the chain of J-connectivities. Both these features are illustrated in ID NOESY-TOCSY spectra of the type VI group B Streptococcus capsular polysaccharide (1). 

Although Vj does not appear explicitly in Eqs. (3.32) and (3.34), the interaction influences the driving potential and the vector potential via in Eq. (3.29). In general, there is no one-body potential and vector potential that satisfies Eqs. (3.32) and (3.34), hence the driving potential in Eq. (3.32) is unrealistic. However, it turns out that the driving potential and the vector potential for rotation of the orientation and translation of the wave function distribution of one-component gas of charged particles without spin are the same as those for the one-particle system in Section 3.5.3. 

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