Analytical description of theI= 5/2 quadrupole nutation experiment

Semi-analytical description of the S=9/2 quadrupole nutation NMR experiment: multinuclear application to (113) In and (115) In in indium phosphide

The density matrix of a spin S=9/2 excited by a radiofrequency pulse is calculated. The interaction involved during the excitation of the spin system is first-order quadrupolar. Consequently, the results are valid for any ratio of the quadrupolar coupling ωQ to the pulse amplitude ω1 . The behavior of the central transition intensities versus the pulse length is discussed. The (115) In and (113) In nuclei in a powdered sample of indium phosphide (InP) are used to illustrate the results. It is found that the ratio of the quadrupolar coupling constants determined in this work is in excellent agreement with the ratio of the quadrupole moments of the two nuclei.

Complete description of the interactions of a quadrupolar nucleus with a radiofrequency field. Implications for data fitting

We present a theory, with experimental tests, that treats exactly the effect of radiofrequency (RF) fields on quadrupolar nuclei, yet retains the symbolic expressions as much as possible. This provides a mathematical model of these interactions that can be easily connected to state-of-the-art optimization methods, so that chemically-important parameters can be extracted from fits to experimental data. Nuclei with spins >1/2 typically experience a Zeeman interaction with the (possibly anisotropic) local static field, a quadrupole interaction and are manipulated with RF fields. Since RF fields are limited by hardware, they seldom dominate the other interactions of these nuclei and so the spectra show unusual dependence on the pulse width used. The theory is tested with (23)Na NMR nutation spectra of a single crystal of sodium nitrate, in which the RF is comparable with the quadrupole coupling and is not necessarily on resonance with any of the transitions. Both the intensity and phase of all three transitions are followed as a function of flip angle. This provides a more rigorous trial than a powder sample where many of the details are averaged out. The formalism is based on a symbolic approach which encompasses all the published results, yet is easily implemented numerically, since no explicit spin operators or their commutators are needed. The classic perturbation results are also easily derived. There are no restrictions or assumptions on the spin of the nucleus or the relative sizes of the interactions, so the results are completely general, going beyond the standard first-order treatments in the literature.

Investigation of the central line of 11B in hexagonal boron nitride by a one-dimensional single pulse nutation NMR experiment

Spin-3/2 central line intensities are calculated for any ratio of the quadrupolar coupling constant to the radiofrequency field for the one-pulse NMR experiment. The sequence is then used to obtain the quadrupolar coupling constant on a powder sample of hexagonal boron nitride. This technique is found to be particularly useful when the NMR spectrum is featureless.

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 23Na in a powdered sample of sodium nitrate.

Spin-7/2 nutation and Hahn-echo amplitudes in model compounds and application to the tetrahedral cluster Co4(CO)12

Spin-7/2 central line intensities are calculated for any ratio of the quadrupolar coupling constant to the radiofrequency field for both one-pulse and two-pulse Hahn-echo experiments and tested by 59Co NMR on the model compounds, Na3[Co(NO2)6] and trans-[Co(en)2(NO2)2]NO3. The Hahn-echo sequence is then used to obtain the quadrupole coupling constant and asymmetry parameter on a powder sample of Co4(CO)12. This technique is found to be particularly useful when the NMR spectrum is broad and featureless.

Spin-5/2 Hahn echoes in solids

The Hahn echoes for a spin I = 5/2 system were derived by considering the first-order quadrupole interaction throughout the experiment. Consequently, our results are valid for any ratio of the quadrupole coupling, omega Q, to the amplitude of the RF pulse, omega RF. The use of simplified density operators reduced the computation time dramatically. Furthermore, this approach allows a better understanding of the origin of the Hahn echoes in half-integer quadrupole spins. The tau 4 = tau 2 Hahn echoes were illustrated with the 27Al nucleus in a polycrystalline sample of KAl(SO4)2.12H2O. The quadrupole coupling constant and the asymmetry parameter of 27Al were determined by two methods: from the lineshape of the powder pattern and from the variation of the central line integrated intensity as a function of the second pulse duration.

Application of one-dimensional nutation nuclear magnetic resonance to 51V in ferroelastic BiVO4

The central line intensity of a spin I = 7/2, excited by a radio frequency (rf) pulse, is calculated by taking into account the first-order quadrupolar interaction during excitation. Thus, the result is valid for any ratio of quadrupolar coupling to pulse amplitude. The quadrupolar coupling of the nuclei vanadium 51V in a single crystal of ferroelastic BiVO4 is determined using this one-dimensional (1D) mutation method.

Determination of spin-5/2 quadrupolar coupling with two-pulse sequences

The density matrix of a spin-5/2 system excited by two in-phase pulses separated by a delay tau 2 is calculated from the equilibrium state to the end of the second pulse. The interaction involved throughout the computation is the first-order quadrupolar interaction. Consequently, the results are valid for any ratio of the quadrupolar coupling to the amplitude of the radio-frequency pulse. It is shown that single- and multi-quantum coherences developed during the first pulse are detected at the end of the second pulse through single-quantum coherences. Other two-pulse sequences with various phase cyclings as well as the rotary echo sequence are also discussed and illustrated with the nuclei 27Al in a single crystal of corundum Al2O3.