Magnetically equivalent nuclei

Cross-relaxation The mutual intermolecular or intramolecular relaxation of magnetically equivalent nuclei, e.g., through dipolar relaxation. This forms the basis of nOe experiments. 

When several magnetically equivalent nuclei are present in a radical, some of the multiplet lines appear at exactly the same field position, i.e., are degenerate , resulting in variations in component intensity. Equivalent spin-1/2 nuclei such as 1H, 19F, or 31P result in multiplets with intensities given by binomial coefficients (1 1 for one nucleus, 1 2 1 for two, 1 3 3 1 for three, 1 4 6 4 1 for four, etc.). One of the first aromatic organic radical anions studied by ESR spectroscopy was the naphthalene anion radical,1 the spectrum of which is shown in Figure 2.2. The spectrum consists of 25 lines, a quintet of quintets as expected for hyperfine coupling to two sets of four equivalent protons. 

First order ENDOR frequencies of nonequivalent nuclei or of pairs of magnetically equivalent nuclei are given by Eq. (3.3) which is derived from the direct product spin base. To obtain correct second order shifts and splittings, however, adequate base functions have to be used. We start the discussion of second order contributions with the most simple case of a single nucleus and will then proceed to more complex nuclear spin systems. 

Analogous formulae are valid for magnetically equivalent nuclei. 

RULE 2 Interaction of nucleus A with a group of n magnetically equivalent nuclei X (of spin IX), produces a multiplet of (2nx, lx+ 1) peaks,... 

A way out of this dilemma is to perform an NMR-type experiment on the paramagnetic centre. NMR provides an inherently higher resolution than EPR, since only one line pair at frequences v is obtained for each nucleus (or set of magnetically equivalent nuclei). The lines appear, to first order, symmetrically spaced around the Larmor frequency v of the respective nucleus for nuclei with I = i and a hfc A the resonance condition is... 

Nuclei may be chemically equivalent but magnetically non-equivalent. To be magnetically equivalent nuclei must couple in exactly the same way to all other nuclei in the system. Thus the two protons in 1,1-difluoroethylene are magnetic-... 

The H- ll9Sn HDMR experiments may often be applied to the analysis of type A X spin systems with degenerate transitions and only two groups of magnetically equivalent nuclei. For example, the energy level diagram for an A3X system is shown in Fig. 3. 

The pairs of fluorines in all of these molecules, except those in 1,1-difluoroethene, would also be magnetically equivalent. In order to be magnetically equivalent, nuclei that are chemically equivalent must have identical coupling constants to any other particular nucleus in the molecule, and it can be seen that the two protons in 1,1-difluoroethene do not have the same spatial relationship with respect to a given fluorine substituent. For example, the Fa substituent has a cis relationship to Ha, but a trans relationship to Hb (Scheme 2.30). A spin system such as this one is represented as an AA XX system, which contrasts with the A2X2, A2X, and A2XY systems in Scheme 2.29 wherein both fluorines in each of these systems have identical 2/HF coupling constants. 

We describe this situation by saying that the two Hb s are chemically equivalent (i.e., they are symmetry equivalent and occur at the same chemical shift), but they are not magnetically equivalent. For two (or more) nuclei to be magnetically equivalent, they must not only be chemically equivalent but also be equally coupled to any other nucleus. Thus, all magnetically equivalent nuclei are also chemically equivalent, but not all chemically equivalent nuclei are magnetically equivalent. We will use the same letter with a prime to indicate when one nucleus is chemically but not magnetically equivalent to another. So, we can relabel structure 9-1 as follows ... 

Spin systems (collections of interacting nuclei) are often labeled (Pople notation) by assigning a letter from the alphabet to each set of magnetically equivalent nuclei. Nuclei that are close (but not identical) in chemical shift are given letters that are close in the alphabet (e.g., AB). Two nuclei that are chemically equivalent but not magnetically equivalent are assigned the same letter but one letter is primed (e.g., AA ). 

AA BB Pople designation for a coupled four-spin system consisting of two chemically but not magnetically equivalent nuclei in each of two sets... 

For sets of magnetically equivalent nuclei, first-order analysis is usually considered applicable when (vA — r,u)//AB — 7. [When 7 < (vK — Pb)/Jab < 20, there is some distortion of intensities from the pattern given in Table 6.2, but the multiplet is still recognizable.] The deviation in intensities always occurs in the direction of making the lines near the center of the overall spectrum more intense and those toward the edges less intense. 

For a pair of magnetically equivalent nuclei (e.g., two protons), symmetrized wave functions must be used, as in Eqs. 6.42 and 6.44, and the resulting calculation gives eigenvalues and transitions as described in Section 6.11, except that the adjacent symmetric energy levels are separated by... 

One caveat that must be made is that ENDOR signal intensities, in contrast to those in diamagnetic NMR, are not reliable indicators of the relative numbers of magnetically equivalent nuclei. The process by which an NMR transition affects the EPR signal intensity that is being monitored to yield a given ENDOR signal is complicated. " " Many relaxation factors are involved, both nuclear and electronic (see Sections 4.5 4.6 in in addition to simply numbers of nuclei. 

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