Homonuclear two-spin system

Vega, and coworkers as a sequence of well-placed ideal (i.e., infinitely strong) 7r-pulses serving to reintroduce the zero-quantum (ZQ) part of the homonuclear dipole-dipole coupling operator in a homonuclear two-spin system. The coherent averaging due to MAS is in the toggling frame of the n-pulses partially disrupted by a differential chemical shift term and thereby recoupling takes place. 

By choosing C = a>r/4, the first-order effective Hamiltonian in a homonuclear two-spin system looks as follows ... 

The raising operator I+, acting as an operator, raises the a state of a single spin to the p state. All of these operators can be represented as matrices. In the case of the homonuclear two-spin system (Ha and Hb), these are 4 x 4 matrices. For example, the raising operator 1+ can be represented by the following matrix, which acts on the vector that describes the aa state to give a new vector that describes the Pa state ... 

For the heteronuclear two-spin system we found that we could create an average of the two X spin states by decoupling. Clearly such heteronuclear decoupling is inapplicable to the homonuclear two-spin system, but examination of the Hamiltonian of Eq. 7.6 suggests another approach—manipulate the spin system to make the two spin terms equal on the average so that pulse sequences. Prior to our discussion of pulse sequences in Chapters 9—11, we are not prepared to treat this rather complex process in detail, but a simple pictorial presentation gives the essential features. 

In an isolated homonuclear two-spin system in the extreme narrowing limit relaxing exclusively via the dipole-dipole mechanism, the steady-state NOE is predicted to be -1-50% and independent of intemuclear separation, rjs. However, the initial rate at which the NOE grows is proportional to rj . 

Figure 8.23. Schematic illustration of the dependence of the ROE and transient NOE for an isolated homonuclear two-spin system as a function of molecular tumbling rates.

Figure Bl.13.7. Simulated NOESY peak intensities in a homonuclear two-spin system as a flinetion of the mixing time for two different motional regimes. (Reproduced by permission of Wiley from Neuhaus D 1996 Encyclopedia of Nuclear Magnetic Resonance ed D M Grant and R K Harris (Chichester Wiley) pp 3290-... 

The interaction of interest in RR is the dipolar interaction, an orientationally dependent through-space spin-spin coupling, which leads to a perturbation of the CS-perturbed Zeeman energy levels. For a homonuclear two-spin system, the truncated dipolar Hamiltonian operator is given by the following ... 

The terms pc and py correspond to 1/Tic and 1/Tih, respectively, and CTCH is the cross-relaxation rate. It should be stressed that the simplicity of the above equation is a consequence of the rareness of the I spins and of the dominant strength of the dipolar interaction between directly bonded nuclei. The situation for homonuclear proton spin systems is often more complicated, since the protons usually constitute a much larger spin system, and a separation into distinct two-spin systems may be not valid in this case. The broadband irradiation of the protons yields, in a steady state, Mhz = 0 and M z = Mj (1 rj). The factor 1 + 77 is called, as introduced above, the nuclear Overhauser enhancement factor. The NOE factor is related in a simple way to the equilibrium magnetizations of the I- and S-spins (which are proportional to the magnetogyric ratios 71 and 7s), the cross-relaxation rate and the relaxation rate of the I-spin ... 

In the homonuclear AX case, e.g. a proton-proton two-spin system, the 180" pulse affects both nuclei. The doublet vectors are reflected at the x z plane. Inverting the precession states of all other coupling nuclei, the 180" pulse also inverts rotation of both AX components (Fig. 2.38(c)). At time 2z, the vectors will be aligned antiparallel along + x (Fig. 2.38(c)) the resultant is zero and no signal will be detected. 

For abundant nuclei with spin V2, the spectrum is often dominated by heteronuclear or homonuclear dipolar interactions, i.e., the interactions between the magnetic moments of two neighbouring spins. In this case there is no isotropic contribution and q is zero, so that Equation 14.1 simplifies correspondingly. For a two-spin system one obtains a spin Hamiltonian of the form ... 

Let us start by considering a molecule with two coupled nuclei (A and B) of the same isotope (e.g., H). There are three independent variables that describe the system completely the chemical shifts (8 or 5v) of A and B and their homonuclear coupling constant 7. The exact appearance of the NMR spectrum for this system, that is, the position and intensity of each line, can be calculated from the values of these three variables (and the operating frequency of the instrument if 8 values are used). The general solution for the two-spin system is a four-line spectrum, with each line having the position and intensity listed below ... 

FIGURE 7.1 Energy levels for a two-spin system, (a) Homonuclear (b) heteronuclear. 

FIGURE 7.6 Powder patterns for homonuclear dipole coupling. Dashed line represents a two-spin system. Solid line shows broadening from other nearby nuclei in a multispin system. 

For a heteronuclear two-spin system, we focus on terms in the spectral density at (t)h which do not induce transitions of the S spins, so the second term can be ignored to give an overall rate constant of (2 Wj + Wt + W2). On the other hand, for a homonuclear system, I = S, so the two terms can be combined and simplified to give a rate constant (2W1 + 2W2). Note that W0 does not appear, a consequence of the fact that for a homonuclear system — cos 0. 

Consider now the two-spin system, in which chemical shifts and scalar coupling come into play. In Chapter 6 we discussed the two-spin system in detail, both the weakly coupled AX system and the general case, AB. To illustrate the application of the density matrix, we concentrate first on the AX system and then indicate briefly how the results would be altered for AB. To simplify the notation, we call the nuclei I and S, rather than A and X, and use the common notation in which the spin operators and their components are designated, for example, Ix and Sx, rather than the more cumbersome 4(A) or /. Although the I-S notation is usually applied to heteronuclear spin systems, we use it here to include homonuclear systems (e.g., H-H) as well. 

It should be clear from the foregoing that the two-dimensional J-spectrum of a homonuclear AX spin system subject to a non-selective 180° refocussing pulse should consist of four lines arranged in a 2 x 4 grid. Their coordinates (o)j,o)2) are J/2, 6 + J/2 —Jjl, 6, — J/2 J/2, + /2 and — J/2, — J/2. For systems with second-order... 

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