186 5.4  NMR and Other Radio Frequency and Microwave Resonance Spectroscopies

in biophysical investigations than solid-​state NMR, molecular reorientation averages out this

anisotropic effect.

5.4.3  OTHER NMR ENERGY COUPLING PROCESSES

The overall NMR Hamiltonian function includes the sum of several independent Hamiltonian

functions for not only the Zeeman interaction and chemical shift coupling, but also terms

relating to other energy coupling factors. There are spin–​spin coupling, which includes both

dipolar coupling (also known as magnetic dipole–​dipole interactions) and J-​coupling (also

known as scalar coupling or indirect dipole–​dipole coupling). And there is also a nuclear E-​

field coupling called “quadrupolar coupling.”

In dipolar coupling, the energy state of a nuclear magnetic dipole is affected by the mag­

netic field generated by the spin of other nearby magnetic atomic nuclei, since over short

distances comparable to typical covalent bond lengths (but dropping off rapidly with distance

r between nuclei with a 1/​r³ dependence), nuclei experience the B-​field generated from each

other’s spin in addition to the external (and in general shielded) magnetic field. This coupling

is proportional to the product of the two associated magnetogyric ratios (whether from the

same or different atoms) and can result in additional splitting of the chemical shift values

depending on the nearby presence of other nuclei.

Several magnetic atomic nuclei used in NMR are not spin-​1/​2 nuclei, and in these cases,

the charge distribution in each nucleus may be nonuniform, which results in an electrical

quadrupole moment, though these have a limited application in biophysics. An electrical

quadrupole moment may experience the E-​field of another nearby electrical quadrupole

moment, resulting in quadrupolar coupling. In liquid-​state NMR, however, since molecular

motions are relatively unconstrained, molecular reorientation averages out any fixed shift on

resonance frequency due to dipolar or quadrupolar coupling but can result in broadening of

the chemical shift peaks.

However, in solid-​state NMR, and also NMR performed in solution but on liquid crystals,

molecular reorientation cannot occur. Although liquid-​state/​solution NMR has the most

utility in biophysics, solid-​state NMR is useful for studying biomineral composites (e.g.,

bone, teeth, shells) and a variety of large membrane protein complexes (e.g., transmembrane

chemoreceptors and various membrane-​associated enzymes) and disease-​related aggregates

of proteins (e.g., amyloid fibrils that form in the brains of many patients suffering with

various forms of dementia), which are inaccessible either with solution NMR or with x-​ray

diffraction methods. Solid-​state NMR results in peak broadening and shifting of mean energy

levels in an anisotropic manner, equivalent to ~10 ppm for dipolar coupling, but as high as

~10⁴ ppm in the case of quadrupolar coupling. There are also significant anisotropic effects

to the chemical shift. To a certain extent, anisotropic coupling interactions can be suppressed

by inducing rotation of the solid sample around an axis of angle ~54.7°, known as the “magic

angle” relative to the external B-​field, in a process known as “magic-​angle spinning” requiring

a specialized rotating sample stage, which satisfies the conditions of zero angular dependence

since (3cos²θ − 1) =​ 0.

In liquid-​state NMR, the most significant coupling interaction in addition to the Zeeman

effect and the chemical shift is J-​coupling. J-​coupling is mediated through the covalent bond

linking the atoms associated with two magnetic nuclei, arising from hyperfine interactions

between the nuclei and the bonding electrons. This results in hyperfine structure of the NMR

spectrum splitting a single chemical shift peak into multiple peaks separated by a typical

amount of ~0.1 ppm given by the J-​coupling constant. The multiplicity of splitting of a chem­

ical shift peak is given by the number of equivalent magnetic nuclei in neighboring atoms n

plus one, that is, the n +​ 1 rule.

The example of this rule often quoted is that of the ¹H NMR spectrum of ethanol

(Figure 5.4a), which illustrates several useful features of NMR spectra. Carbon atom 1 (C1),

part of a methyl group, is covalently bound to C2, which in turn is bound to two ¹H atoms, and

the nucleus (a proton) of each has one of two possible orientations (parallel, p, or antiparallel,