Exact calculation of the response of a quadrupolar nucleus to radio frequency irradiation

The effect of a pulse in NMR is usually considered as a rotation of the frame of reference of the spin system. For spins-1/2, this concept is an important and very useful tool. The assumption behind this concept is that while the radio frequency irradiation is on, this term dominates all other interactions. Although this is usually true for spins-1/2, typical interactions for a quadrupolar nucleus can be very large and the assumption is no longer valid. The full solution is complex, but two extreme cases are already solved. If the quadrupole interaction is very small, then the assumption is valid and the pulse does act like a rotation of the frame of reference. At the other extreme, if the interaction is large and the spin, I, is half-integral, then the central transition remains relatively narrow and can be treated as a fictitious spin-1/2. The pulse then acts as a rotation, but with a scaling factor of I + 1/2. This paper treats the general case, where no approximations are made. The effects can be observed in a nutation experiment, in which the observed signal is plotted as a function of pulse width, in a simple one-pulse experiment. If the pulse acts as a rotation, then the nutation plot will be a sine wave, but otherwise it will be a sum of sinusoids. This is true even for a single-crystal sample with a single quadrupolar coupling. If the sample is a powder, then the nutation plot will be a sum of many sinusoids, since the quadrupole coupling will vary with the powder average. This paper sketches out the theory of these effects based on a full and exact description of a quadrupolar system and illustrates it with some nutation spectra of 23 Na in a powdered sample of sodium nitrate.