Quadrupolar spin theory

A general treatment of spin coupled with quadrupolar spins was given using density matrix theory.27 This formulation enables one to calculate, on the same theoretical basis, the lineshapes of the systems with different ratios of the quadrupolar interaction to the Zeeman interaction. Additionally, it includes the spinning sidebands very naturally. 

As an example, quadrupole nutation NMR of nuclei with half-integer quadrupolar spin in zeolitic materials can distinguish between nuclei of the same chemical element subjected to different quadrupole interactions, the signals of which overlap in conventional spectra. The situation is favourable for half-integer quadrupolar spins since the m=l/2 <-> m= -1/2 transition for these nuclei is broadened by the quadrupole interaction only in second-order perturbation theory. The technique can be usefully applied for the determination of the local environment of A1 in zeolitic catalysts (28). It allows discrimination between species of similar chemical shift but different quadrupolar coupling constants (see Figure 5). The main difficulty in the interpretation is the complex spectmm that results from a nutation experiment since it can consist of many overlapping powder patterns (29). 

Theory of Quadrupolar Spins Line Narrowing Techniques... 

THEORY OF QUADRUPOLAR SPINS LINE NARROWING TECHNIQUES... 

The authors acknowledge fruitful discussions with Shimon Vega on various aspects of the theory of quadrupolar spins and MQMAS. We also acknowledge Shimon Vega and Thomas Brauniger for a very careful reading of the manuscript and their critical comments. 

The theory of satellite transition MAS NMR of half-integer quadrupolar spins has been presented and a 2D experiment that uses satellite transitions to obtain isotropic NMR spectra has been described. Pulse sequences of the 2D satellite transition experiment have been described. SateUite transition excitation and coherence transfer have been studied using numerical simulations. Experimental results with model samples show high efficiencies of the experiment for acquisition of high-resolution isotropic NMR of half-integer quadrupolar nuclei. 

Computing Electric Field Gradient Tensors to Study Cl, and Quadrupolar Spins Recent advances in theory and computing power have made the computation of the electric field gradient (EFG) tensor parameters possible by first-principles methods and made quadrupolar spins a more valuable source of chemical information in solids. This section focuses on recent advances on the computational and experimental studies of APIs exploiting Cl, and quadrupolar nuclei. 

In quadrupolar nuclei, the situation differs notably the quadrupolar interaction only affects spins with I>% and is created by electric field gradient resulting from the asymmetry of charge distribution around the nucleus of interest. The quadrupolar interaction is characterized by the nuclear quadrupolar coupling constant Cq (from 0 in symmetrical environments to tens or hundreds of MHz) and an asymmetry parameter T]q. NMR spectra are usually recorded when Cq Vl the Larmor frequency of the quadrupolar spin. In such a case, the NMR spectrum can easily be simulated First, the first-order quadrupolar Hamiltonian, which is the quadrupolar interaction Hamiltonian truncated by the Larmor frequency, has to be taken into account. The first-order quadrupolar interaction (or the zeroth-order term in perturbation theory) is an inhomogeneous interaction and is modulated by MAS and does not affect symmetrical transition —m +m. Therefore, in half-integer spins, the single-quantum central transition (CT, i.e., —1/2 +1/2) is not affected by the first-order quadrupolar inter-... 

Measurements of spin-lattice relaxation rate as a function of the magnetic field/resonance frequency are commonly referred to as nuclear magnetic relaxation dispersion, NMRD. Measurements of this kind, when performed over a broad range, are an invaluable source of information on frequency-resolved molecular motions in complex biological or colloidal systems. Several years ago, Halle proposed a theory for relaxation of quadrupolar spins in dynamically heterogeneous systems... 

The most important examples of 2S states to be described in this book are CO+, where there is no nuclear hyperfine coupling in the main isotopomer, CN, which has 14N hyperfine interaction, and the Hj ion. A number of different 3E states are described, with and without hyperfine coupling. A particularly important and interesting example is N2 in its A 3ZU excited state, studied by De Santis, Lurio, Miller and Freund [19] using molecular beam magnetic resonance. The details are described in chapter 8 the only aspect to be mentioned here is that in a homonuclear molecule like N2, the individual nuclear spins (1 = 1 for 14N) are coupled to form a total spin, It, which in this case takes the values 2, 1 and 0. The hyperfine Hamiltonian terms are then written in terms of the appropriate value of h As we have already mentioned, the presence of one or more quadrupolar nuclei will give rise to electric quadrupole hyperfine interaction the theory is essentially the same as that already presented for1 + states. 

The theory of the magnetic hyperfine interactions in NCI is essentially the same as that already described for the PF radical in the previous section, except that the nuclear spins / are 1 for 14N and 3/2 for 35C1. The form of the effective Hamiltonian for the quadrupole interaction and its matrix elements for two different quadrupolar nuclei was described in some detail in chapter 8 when we discussed the electric resonance spectra of CsF and LiBr. We now use the same case (b) hyperfine-coupled basis set as was used for PF. The quadrupole Hamiltonian for the two nuclei can be written as the sum of two independent terms as follows ... 

Contents 1. Introduction 104 2. Theory 106 2.1. Specifically for spin-1 nuclei 112 2.2. Two-axis jump processes 114 2.3. 2-by-2-site jump 114 2.4. 2-by-3-site jump 115 3. Numerical Simulations 117 3.1. Spin-1 nuclei 118 3.2. Half-integer quadrupolar nuclei 118 4. Results and Discussion 119 4.1. Spin-1 nuclei 119 4.2. Dynamic effects in 14N MAS spectra by SQ or DQ coherences 123 4.3. Multi-axis jump processes 124 4.4. Half-integer quadrupolar nuclei 129 5. Conclusions 134 Acknowledgements 135 References 135... 

We now calculate the perturbation to the Zeeman field due to the quadrupolar interaction by means of average Hamiltonian theory.This is accomplished by transforming TYq to the Zeeman interaction frame and then applying the spherical tensor rotation properties to the spin elements 72,The resulting quadrupolar Hamiltonian TTq in the rotating frame is given by ... 

Because H linewidths in many elastomers are of the order of 3 kHz or less, spin-echo and gradient-echo imaging techniques can be applied. Contrast is introduced by suitable filters like Ti and double-quantum filters or by use of the spectroscopic dimension. Parameter images of T2, the double-quantum signal intensity, or the quadrupolar coupling strength are evaluated and rescaled according to theory or experimental calibration data (cf. Section 7.1.6). 

The application of temperature-dependent line shapes and the measurements of second moments in more complex organic solids like polymers followed soon after. Even nowadays, this simple method still has its place in the characterization of materials like solid polymer electrolytes where the line widths and Ti relaxation of the charge carriers provide information about their mobility that can be correlated with the electrical conductivity of the material. More detailed information can be obtained from cases in which the interaction is well defined, i.e., when an anisotropic single-spin interaction dominates the spectrum. Typical cases are the chemical shielding anisotropy (CSA) and quadrupolar interaction for which the theory is well developed. 

MQCs are not excited uniformly and the efficiency with which the various orders of MQC are excited depends specifically on the parameters of the spin system (dipolar couplings, scalar couplings, quadrupolar couplings, chemical shifts) in the spin system and the choice of the preparation time t. Many researchers have co-added spectra acquired with different preparation times to ensure that all transitions are observed with reasonable intensity. A number of broadband excitation techniques have been developed,13-15 where the value of t in the preparation sequence has been varied either in a pseudo-random or systematic fashion to achieve a more uniform excitation in the multiple quantum domain. An experimental search method has been used to optimise the delays in the preparation period of the MQ excitation sequence16 and Wimperis17 used average Hamiltonian theory to propose... 

We now consider relaxation of quadrupolar ions in greater detail. The theory presented applies to any spin- quadrupolar nucleus, though the experimental examples discussed are of 7Rb+ in agarose gels. 

Probably of more relevance to the practising organic chemist is the influence quadrupolar nuclei have on the spectra of spin- /2 nuclei, by virtue of their mutual scalar coupling. Coupling to a quadrupolar nucleus of spin 1 produces, in theory, 21 -f- 1 lines so, for example, the carbon resonance of CDCI3 appears as a 1 1 1 triplet ( H has 1 = 1) by virtue of the 32 Hz coupling. 

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