Two-spin system

Figure Bl.13.7. Simulated NOESY peak intensities in a homoniielear two-spin system as a fiinetion of the mixing time for two different motional regimes. (Reprodiieed by pennission of Wiley from Neiihaiis D 1996 Encyclopedia of Nuclear Magnetic Resonance ed D M Grant and R K Harris (Chiehester Wiley) pp 3290-301.)...

In an isolated two-spin system, the NOE (or, more accurately, the slope of its buildup) depends simply on where d is the distance between two protons. The difficulties in the interpretation of the NOE originate in deviations from this simple distance dependence of the NOE buildup (due to spin diffusion caused by other nearby protons, and internal dynamics) and from possible ambiguities in its assignment to a specific proton pair. Mofec-ufar modeling methods to deaf with these difficulties are discussed further below. 

Figure 1.4. Two-spin system of type AX with a chemical shift difference which is large compared with the coupling constants (schematic)...

In contrast, in a two-spin system the two nuclei coupled with each other by the coupling constant, J, will have four energy levels available for transitions (Fig. 5.56). Such a system not only has single-quantum coher-... 

Figure 5.56 Energy-level diagram representing two nuclei in a two-spin system coupled with each other.

In Equation (5), we can first notice (i) the factor 1/r6 which makes the spectral density very sensitive to the interatomic distance, and (ii) the dynamical part which is the Fourier transform of a correlation function involving the Legendre polynomial. We shall denote this Fourier transform by (co) (we shall dub this quantity "normalized spectral density"). For calculating the relevant longitudinal relaxation rate, one has to take into account the transition probabilities in the energy diagram of a two-spin system. In the expression below, the first term corresponds to the double quantum (DQ) transition, the second term to single quantum (IQ) transitions and the third term to the zero quantum (ZQ) transition. 

In spite of the apparent simplicity of the method, its drawback comes from the fact that a two-spin system has been assumed. It provides merely global information spanning all protons prone two interact by dipolar coupling with the considered carbon. Selective information requires pulsed experiments stemming from the general solution of Equation (14) given below. 

In a heteronuclear two-spin system the build-up of AP magnetization for coherence transfer during a delay A is determined by the amplitude function... 

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 concept of cross-polarization as applied to solid state NMR was implemented by Pines et al. [20]. A basic description of the technique is the enhancement of the magnetization of the rare spin system by transfer of magnetization from the abundant spin system. Typically, the rare spin system is classified as 13C nuclei and the abundant system as H spins. This is especially the case for pharmaceutical solids and the remaining discussion of cross-polarization focuses on these two spin systems only. 

Figure 12.3 outlines the essential features of the PASADENA/PHIP concept for a two-spin system. If the symmetry of the p-H2 protons is broken, the reaction product exhibits a PHIP spectrum (Fig. 12.3, lower). If the reaction is carried out within the high magnetic field of the NMR spectrometer, the PHIP spectrum of the product consists of an alternating sequence of enhanced absorption and emission lines of equal intensity. This is also true for an AB spin system due to a compensating balance between the individual transition probabilities and the population rates of the corresponding energy levels under PHIP conditions. The NMR spectrum after the product has achieved thermal equilibrium exhibits intensities much lower than that of the intermediate PHIP spectrum. 

Cross-relaxation and auto-relaxation for a two-spin system is described by the Solomon [6] equation ... 

In general, a relatively direct and straightforward means of analysis may be performed in the case of slow exchange on the chemical shift time-scale by combining the relaxation matrices of the free and bound state with the kinetic matrix to describe the effect of exchange [12]. For the two spin systems described above the expanded relaxation matrix R can be written as ... 

Since the tensor D is strictly proportional to the inverse cube of the distance between the unpaired electrons for two spin systems and is regarded as a measure of the inverse volume of the delocalized spins in higher-spin systems, the absolute Z)-values become smaller as S increases. Since the epr spectral width is approximated by 2D, it is customarily the case that the g = 2 region becomes crowded with spectral peaks with increasing S. 

In an alternative formulation of the Redfield theory, one expresses the density operator by expansion in a suitable operator basis set and formulates the equation of motion directly in terms of the expectation values of the operators (18,20,50). Consider a system of two nuclear spins with the spin quantum number of 1/2,1, and N, interacting with each other through the scalar J-coupling and dipolar interaction. In an isotropic liquid, the former interaction gives rise to J-split doublets, while the dipolar interaction acts as a relaxation mechanism. For the discussion of such a system, the appropriate sixteen-dimensional basis set can for example consist of the unit operator, E, the operators corresponding to the Cartesian components of the two spins, Ix, ly, Iz, Nx, Ny, Nz and the products of the components of I and the components of N (49). These sixteen operators span the Liouville space for our two-spin system. If we concentrate on the longitudinal relaxation (the relaxation connected to the distribution of populations), the Redfield theory predicts the relaxation to follow a set of three coupled differential equations ... 

A spin system comprising just two protons i.e. an AX or an AB system) is always exceptionally easy to analyse because, independent of the value of the ratio of Av/J, the spectrum always consists of just four lines with each pair of lines separated by the coupling constant J. The only distortion from the first-order pattern consists of the gradual reduction of intensities of the outer lines in favour of the inner lines, a characteristic "sloping" or "tenting" towards the coupling partner. A series of simulated spectra of two-spin systems are shown in Figure 5.7. 

Fig. 2. Trajectories of some Cartesian product operators of a two-spin system under a 270°...

Using the procedure described above, signals of two spin systems are accumulated during the entire experiment and the individual subspectra are obtained after editing of the original spectra. This provides a two-fold reduction in the total experimental time when compared to 1D experiments which employ two selective pulses for the selection of the magnetization transfer pathway. 

Figure 1 shows the pulse sequence of the C HSQC experiment supplemented by a spin-lock pulse to suppress the signals from C-bound protons. The experiment is readily described in terms of Cartesian product operators [9]. For a two spin system consisting of a proton spin H coupled to a C spin C, the relevant coherence transfer pathway is... 

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 ... 

Figure 4 shows the results of a simulation of CT-VPP-REDOR evolution curves calculated for the two different S-I4. five-spin systems and the S-I two-spin system from Figure 2. Again the distances were chosen so that the resulting second moments as calculated employing Equation (12) are identical for the five-spin and the two-spin systems. As expected, the AS/So values at fpp/TR=0.5 match the data points of the REDOR curves (cf. Figures 2 and 4).

---