Most nuclei in the periodic table that possess spin are quadrupolar (i.e. spin ℓ > 1/2). It is well-known that, in the solution-state, a J-coupling between the I (ℓ = 1/2) and S (ℓ >1/2) spins “vanishes” (or more precisely induces a scalar relaxation effect) in the spectrum of the I spin(s) when the following condition is satisfied:
That is to say, J-couplings to quadrupolar spins are not directly observable when the rate of relaxation (1/T1) of the S spins significantly exceeds the J-coupling. Sometimes this is referred to as “self-decoupling”, a term coined by Spiess, Haeberlen, and Zimmermann.
But there are exceptions. The relaxation of the S spins is usually overwhelmingly dominated by the quadrupolar mechanism, whose contribution (in the extreme narrowing limit) is given by:
Here, the norm of the quadrupolar coupling tensor is related to the quadrupolar coupling constant (QCC):
And the quadrupolar coupling constant itself is:
Here, it is worth explaining why I have written the equation in an unconventional way. The nuclear quadrupole moment Q (an intrinsic nuclear property which is non-zero for ℓ > 1/2) is proportional to the eccentricity of the spheroid charge distribution unique to each nuclear species, and has units of area. On the other hand, Vzz (in units voltage/area) represents the principal component of the electric field gradient (EFG) tensor, and EFGs may technically be present even at spin-0 or spin-1/2 nuclei. It is clear that Q and Vzz couple to produce a voltage at the nucleus. Finally, half the Josephson constant (½KJ, in units of frequency/voltage) is the fundamental quantum of voltage-driven oscillation. This basic physical picture is rarely appreciated; a run-of-the-mill QCC of ~150 kHz corresponds to a ~100 picovolt potential difference at the nucleus.
Now it is simple to figure out the exceptions when quadrupolar relaxation may be slow enough to observe J-couplings:
- Quadrupolar spins with an intrinsically small nuclear quadrupole moment Q. Examples include 2H and 17O.
- Molecules where the quadrupolar spin experiences a small electric field gradient V. Examples are molecules with intrinsically high symmetry, such as 14NH4+ , but it is worth noting that this criterion may not be sufficient for spins where Q is massive.
- A larger heteronuclear J-coupling, which is increasingly possible when the I and/or S spin(s) has a larger atomic number and gyromagnetic ratio. But again, the J-coupling must be “large” relative to the quadrupolar relaxation rate.
Now, suppose you actually observe, in the I spectrum, a well-resolved coupling to a single quadrupolar spin S. It is well-known that one would observe 2S+1 Lorentzian peaks with equal areas/integrals. It is less well-known that the Lorentzian linewidths of these peaks would not be equal:
This remarkable effect was described by the Nobel laureate John Pople in 1958, as well as Masuo Suzuki & Ryogo Kubo in 1963, but it appears that the name “Pople-Suzuki-Kubo effect” never really caught on.
Prominent examples where the “PSK effect” have been observed include the hexafluorides, e.g. of niobium (93Nb, I=9/2), antimony (121/123Sb with I=5/2 and 7/2 respectively), or bismuth (209Bi, I=9/2). Insanely enough, there are even examples involving the unpleasant nucleus 235U (I=7/2).
However, my favorite example (and I must admit some personal bias) is certainly the fine paper by Stuart J. Elliott et al. from the Levitt group. The paper not only shows a beautiful spectrum:
But impressively, the paper also provides a hidden gem in the supporting information: a general expression for the linewidths that was derived with the help of the commutator relations of spherical tensor operators. The expression may be written:
And I have provided a triangle of the linewidth coefficients k(m) [note that the intensities would be provided by 1/k(m)] here:
