A2 spin system

Figure 9.6 Molecular hydrogen and NMR spectroscopy, (a) Energy levels of an A2 spin system depending on the strength of the magnetic field (Zeeman effect), (b) NMR transitions and resulting NMR spectrum (shown schematically).

The A2 spin system is just one illustration of the lack of splitting due to spin coupling in spectra where certain types of equivalence exist among the nuclei. We shall state without proof an extremely important NMR theorem and its corollary ... 

When the molecule in question contains more than one carbon atom, the 13C satellites often become more complex. Consider, for example, the molecule CHC12CHC12, the proton resonance of which is shown in Fig. 6.14b. The ordinary spectrum is a single line because of the magnetic equivalence of the two protons. On the other hand, the approximately 2.2% of the molecules that contain one 13C and one 12C have protons that are not magnetically equivalent. The proton resonance spectrum of these molecules is an ABX spectrum in which vA v[t. A simulation of the ABX spectrum is also shown in Fig. 6.14. The value of 3Jhh (i.e., /Aij in the ABX spectrum) can readily be observed, even though it is not obtainable from the spectrum of the fully 12C molecule, which constitutes an A2 spin system. 

For the para-H2 state, the Hamiltonian is given as a pure A2 spin system. 

Fast spinning MAS spectra, even in the case of J-resolved 2D experiments do not allow to distinguish between the presences of one or two chemically equivalent nuclei since under fast spinning MAS conditions the situations is comparable to liquid state NMR i.e. chemical shift anisotropies are averaged out. However, under static conditions, the chemical shift tensors of two chemically equivalent nuclei do not coincide and both nuclei have in general dilferent chemical shifts at certain orientations of the molecules with respect to the magnetic field axis, i.e. they are magnetically inequivalent. Thus under static conditions, the spin system may be described as an AB or AX system, instead of an A2 spin system. 

In 1972 Wegner [25] derived a power-series expansion for the free energy of a spin system represented by a Flamiltonian roughly equivalent to the scaled equation (A2.5.28). and from this he obtained power-series expansions of various themiodynamic quantities around the critical point. For example the compressibility... 

From the pair of Ay and A2 states, the latter is expected to be of lower energy. If Hund s rule is applicable to this system, then the higher spin system is preferred. Also, AM 1 calculations prefer the triplet state at both the HF level and various levels of Cl treatment. 

Any such collection of sets insulated from all other sets, is a spin system. The following are examples of first-order spin system AX (two doublets), A2X (doublet, triplet), A2X2 (two triplets), A3X2 (triplet, quartet). The reasoning is obvious In A2X for example, the A2 set is split into a doublet by the n + 1 neighboring protons in this case there is one proton in the X set the two protons in the A2 set account for the triplet. 

If two protons in the same set (i.e., chemical-shift-equivalent protons in the same multiplet) couple equally to every other proton in the spin system, they are also magnetically equivalent, and the usual Pople notations apply A2, B2, X2 etc. However, if two protons in a set are not magnetically equivalent, the following notations apply AA, BB, XX, etc. To rephrase Two chemical-shift-equivalent protons are magnetically equivalent if they are symmetrically disposed with respect to each proton in the spin system. Obviously magnetic equivalence presupposes chemical-shift equivalence. In other words, do not test for magnetic equivalence unless the two protons in question are chemical-shift equivalent. 

FIGURE 6.9 (a) Schematic representation of an AMX spectrum, showing the spin states associated with each spectral line. The situation depicted is that for all three fs positive. The signs of m are indicated. (b) Spectrum of 2,3-dichloropyridine at 60 MHz, an example of an AMX spin system. (c-e) Results of selective decoupling in 2,3-dichloropyridine, with decoupling frequencies centered between the following lines (c) Mi and M2, (d) M3 and M4, (e) and A2. 

In the weak coupling limit the density operator of the spin system does not evolve, that is, the polarization of the first spin is invariant aU) = A2 (see Fig. lA). In the intermediate case of strong coupling (Fig. IB), the density operator evolves in an oscillatory fashion. Starting from cr(0) = /j, terms Ujyl2x hxhy and U xhx + are created periodically, as... 

Figure A3-4 Evaluation of the transition probabilities in a two-spin system with identical chemical shifts (A2).

A further refinement involves the concept of magnetic-equivalent nuclei, also termed spin-coupling equivalent nuclei . If nuclei in the same set (i.e., chemical-shift equivalent nuclei) couple equally to any nucleus (probe nucleus) in the same spin system, they are magnetic equivalent, and the designations A2, X2, and so on apply. However, if the nuclei are not magnetic equivalent, the designations A A XX are used. The designation AA BB is used if AiVJ < 8. 

Schroder et have presented the phase modulation for the A2 isochronous spin system, which allowed them to measure dipolar couplings between these nuclei in H NMR in vivo. The method has been demonstrated for phosphocrea-tine methylene protons in a PRESS pulse sequence. The same authors have used dipolar couplings in their study of molecular dynamics of camosine in vivo. 

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