The origin of spin–spin coupling
المؤلف:
Peter Atkins، Julio de Paula
المصدر:
ATKINS PHYSICAL CHEMISTRY
الجزء والصفحة:
ص528-530
2025-12-09
23
The origin of spin–spin coupling
Spin–spin coupling is a very subtle phenomenon, and it is better to treat J as an empirical parameter than to use calculated values. However, we can get some insight into its origins, if not its precise magnitude—or always reliably its sign—by considering the magnetic interactions within molecules. A nucleus with spin projection mI gives rise to a magnetic field with z-component Bnuc at a distance R, where, to a good approximation,
The angle θ is defined in (6). The magnitude of this field is about 0.1 mT when R = 0.3 nm, corresponding to a splitting of resonance signal of about 104 Hz, and is of the order of magnitude of the splitting observed in solid samples (see Section 15.3a). In a liquid, the angle θ sweeps over all values as the molecule tumbles, and 1–3 cos2θ averages to zero. Hence the direct dipolar interaction between spins cannot account for the fine structure of the spectra of rapidly tumbling molecules. The direct inter action does make an important contribution to the spectra of solid samples and is a very useful indirect source of structure information through its involvement in spin relaxation (Section 15.11).
Spin–spin coupling in molecules in solution can be explained in terms of the polarization mechanism, in which the interaction is transmitted through the bonds. The simplest case to consider is that of 1JXY where X and Y are spin-
nuclei joined by an electron-pair bond (Fig. 15.20). The coupling mechanism depends on the fact that in some atoms it is favourable for the nucleus and a nearby electron spin to be parallel (both α or both β), but in others it is favourable for them to be antiparallel (one α and the other β). The electron–nucleus coupling is magnetic in origin, and may be either a dipolar interaction between the magnetic moments of the electron and nuclear spins or a Fermi contact interaction. A pictorial description of the Fermi contact interaction is as follows. First, we regard the magnetic moment of the nucleus as arising from the circulation of a current in a tiny loop with a radius similar to that of the nucleus (Fig. 15.21). Far from the nucleus the field generated by this loop is indistinguishable from the field generated by a point magnetic dipole. Close to the loop, however, the field differs from that of a point dipole. The magnetic interaction between this non-dipolar field and the electron’s magnetic moment is the contact
Fig. 15.20 The polarization mechanism for spin–spin coupling (1JHH). The two arrangements have slightly different energies. In this case, J is positive, corresponding to a lower energy when the nuclear spins are antiparallel.
Fig. 15.21 The origin of the Fermi contact interaction. From far away, the magnetic f ield pattern arising from a ring of current (representing the rotating charge of the nucleus, the pale green sphere) is that of a point dipole. However, if an electron can sample the field close to the region indicated by the sphere, the field distribution differs significantly from that of a point dipole. For example, if the electron can penetrate the sphere, then the spherical average of the field it experiences is not zero.
interaction. The lines of force depicted in Fig. 15.21 correspond to those for a proton with α spin. The lower energy state of an electron spin in such a field is the β state. In conclusion, the contact interaction depends on the very close approach of an electron to the nucleus and hence can occur only if the electron occupies an s orbital (which is the reason why 1JCH depends on the hybridization ratio). We shall suppose that it is energetically favourable for an electron spin and a nuclear spin to be antiparallel (as is the case for a proton and an electron in a hydrogen atom). If the X nucleus is α, a β electron of the bonding pair will tend to be found nearby (because that is energetically favourable for it). The second electron in the bond, which must have αspin if the other is β, will be found mainly at the far end of the bond (because electrons tend to stay apart to reduce their mutual repulsion). Because it is energetically favourable for the spin of Y to be antiparallel to an electron spin, a Y nucleus with β spin has a lower energy, and hence a lower Larmor frequency, than a Y nucleus with α spin. The opposite is true when X is β, for now the α spin of Y has the lower energy. In other words, the antiparallel arrangement of nuclear spins lies lower in energy than the parallel arrangement as a result of their magnetic coupling with the bond electrons. That is, 1JHH is positive.
To account for the value of 2JXY, as in H-C-H, we need a mechanism that can transmit the spin alignments through the central C atom (which may be 12C, with no nuclear spin of its own). In this case (Fig. 15.22), an X nucleus with α spin polarizes the electrons in its bond, and the α electron is likely to be found closer to the C nucleus. The more favourable arrangement of two electrons on the same atom is with their spins parallel (Hund’s rule, Section 10.4d), so the more favourable arrangement is for the α electron of the neighbouring bond to be close to the C nucleus. Consequently, the βelectron of that bond is more likely to be found close to the Y nucleus, and therefore that nucleus will have a lower energy if it is α. Hence, according to this mechanism, the lower Larmor frequency of Y will be obtained if its spin is parallel to that of X. That is, 2JHH is negative.
The coupling of nuclear spin to electron spin by the Fermi contact interaction is most important for proton spins, but it is not necessarily the most important mechanism for other nuclei. These nuclei may also interact by a dipolar mechanism with the electron magnetic moments and with their orbital motion, and there is no simple way of specifying whether J will be positive or negative.
Fig. 15.22 The polarization mechanism for 2JHH spin–spin coupling. The spin information is transmitted from one bond to the next by a version of the mechanism that accounts for the lower energy of electrons with parallel spins in different atomic orbitals (Hund’s rule of maximum multiplicity). In this case, J < 0, corresponding to a lower energy when the nuclear spins are parallel.
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