AX spin system

The transitions between energy levels in an AX spin system are shown in Fig. 1.44. There are four single-quantum transitions (these are the normal transitions A, A, Xi, and X2 in which changes in quantum number of 1 occur), one double-quantum transition 1% between the aa and j8 8 states involving a change in quantum number of 2, and a zero-quantum transition 1% between the a)3 and fia states in which no change in quantum number occurs. The double-quantum and zero-quantum transitions are not allowed as excitation processes under the quantum mechanical selection rules, but their involvement may be considered in relaxation processes. 

Figure 1.44 Transitions between various energy levels of an AX spin system. A, and Aj represent the single-quantum relaxations of nucleus A, while Xi and Xj represent the single-quantum relaxations of nucleus X. W2 and are double- and zero-quantum transitions, respectively.

Figure 2.3 Spin-echo experiment. The behavior of nucleus X in an AX spin system is shown. (A) Application of the second 180° pulse to nucleus X in the AX hetero-nuclear system results in a spin-flip of the two X vectors across the x -axis. But the direction of rotation of the two X vectors does not change, and the two vectors therefore refocus along the —y axis. The spin-echo at the end of the t period along the -y axis results in a negative signal. (B) When the 180° pulse is applied to nucleus A in the AX heteronuclear system, the spin-flip of the X vectors...

Spin-Edio Production in a Heteronudear AX Spin System... 

In a heteronuclear AX spin system in which nucleus X is being observed, three different cases can be considered, depending on whether the 180° pulse is applied to nucleus A only, nucleus X only, or simultaneously to both nuclei A and X (Fig. 2.3). 

M and are magnetization components of the X vector of a heteronuclear AX spin system, shown here at a certain delay after the application of a 90° pulse. Draw the vector positions and their direction of rotation after each of the following radiofrequency pulses ... 

Figure 2.7 Pulse sequence for a heteronuclear AX spin system representing polarization transfer from H to nuclei.

Figure 5.25 Energy levels for an AX spin system representing only singlequantum transitions.

A 90° Gaussian pulse is employed as an excitation pulse. In the case of a simple AX spin system, the delay t between the first, soft 90° excitation pulse and the final, hard 90° detection pulse is adjusted to correspond to the coupling constant JJ x (Fig- 7.2). If the excitation frequency corresponds to the chemical shift frequency of nucleus A, then the doublet of nucleus A will disappear and the total transfer of magnetization to nucleus X will produce an antiphase doublet (Fig. 7.3). The antiphase structure of the multiplets can be removed by employing a refocused ID COSY experiment (Hore, 1983). 

In the pulse sequence for a homonuclear COSY experiment, the first 90° pulse flips the z magnetization into the x direction and into the xy plane. Considering an AX spin system (one in which the nuclei have very different chemical shifts) having two doublets due to spin-spin coupling, the magnetization will include four components processing at different frequencies. During a... 

Figure 9.7 The NMR spectrum of an AX spin system (shown schematically). Left NMR transitions. Right Resulting NMR spectrum.

Fig. l.ll. Energy levels of an AX spin system (a) and interaction of the nuclear spins A and X with I = 1/2, involving the bonding electrons (ethane molecular orbital model). 

Figure A1.7 Nuclear spin states and normal absorption spectrum for the AX spin system.

In their 31P 1H nuclear magnetic resonance (NMR) spectra, the 1,4,2-diphosphastibolide salts display an AX spin system (<5 = 278 ppm, Ox = 309 ppm,. /ax = 55 Hz) with the low field signal considerably broadened, which suggests that it originates from the phosphorus atom adjacent to the quadrupolar antimony center < 1997JOM291, 2001JOM61>. 

Figure 7 Exchange of spectrum lines in a mutually exchanging AX spin system with coupling between the nuclei.

Use the dipolar Hamiltonian in Eq. 7.8 to compute the energy levels and spectra for the A part of an AX spin system. Ignore chemical shifts and indirect coupling. Use wave functions and other pertinent results from Chapter 6. 

The rationale for INEPT can best be understood by looking first at a somewhat simpler continuous wave experiment for transferring polarization, selective population transfer (SPT). SPT can be understood simply in terms of the populations of energy levels, whereas INEPT requires consideration of coherent precessing magnetization. Figure 9.9 shows the energy levels and populations of an AX spin system, which we take to be H and 13C, respectively, in this example. At equilibrium the populations conform to a Boltzmann distribution. Because the H... 

An AX spin system is esqiected to show a pair of doublets (assiuning that both A and X were nuclei for which 7=1/2). If a second radiofi equency is then applied at ftie resonance fiequency of nucleus A, the signal corresponding to X will collapse to a singlet This process is called spin decoupling Mid in... 

Many workers have in fact used density matrix methods for the calculation of line shapes and intensities in multiple resonance experiments, and two excellent reviews of the background theory are available. (49, 50) In addition there is also a simple guide (51) to the actual use of the method which is capable of predicting the results of quite elaborate experiments. Major applications have included the calculation of the complete double resonance spectrum from an AX spin system which gives 12 transitions in all (52) an extremely detailed study of the relaxation behaviour of the AX2 systems provided by 1,1,2-trichloroethane and 2,2-dichloroethanol (53) the effects of gating and of selective and non-selective pulses on AB and AX spin systems and the importance of the time evolution of the off-diagonal elements of the density matrix in repetitively pulsed FT NMR and spin-echo work (54) the use of double resonance to sort out relaxation mechanisms and transient responses (55) the calculation of general multiple resonance spectra (56) and triple resonance studies of relaxation in AB and AX spin systems. (57)... 

FIG. 8. View of the transverse magnetization projected on to the xy-plane in the rotating frame for a heteronuclear AX spin system. The pulse at time f = 0 is applied along the x-direction to tip the magnetization from the z- to the y-axis, and the pulses at time t = t are applied along the y-direction. (a) Formation of unmodulated spin-echo, (b) Formation of modulated spin-echo. Adapted from ref. 145. 

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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