Quadrupolar Hamiltonian coupling

If the spin-quantum number, Ik, of a spin k is larger than 1/2, we have an additional term in the Hamiltonian, the quadmpolar coupling, hPk. The quadmpolar Hamiltonian arises from the interaction between the electric-field gradient and the nuclear spin. The first-order quadrupolar Hamiltonian is given by ... 

Rapid molecular motions in solutions average to zero the dipolar and quadrupolar Hamiltonian terms. Hence, weak interactions (chemical shift and electron-coupled spin-spin couplings) are the main contributions to the Zeeman term. The chemical shift term (Hs) arises from the shielding effect of the fields produced by surrounding electrons on the nucleus ... 

There are many other terms to the Hamiltonian but for spin-1/2 nuclei in liquids they can all be ignored. The dipole-dipole (dipolar or direct coupling) Hamiltonian is important in solids and partially oriented liquids, and the quadrupolar Hamiltonian is important for spins greater than 1/2. The dipolar interaction contains a multiplier of... 

With pure quadrupole resonance experiments one observes the transitions between the energy levels of the nuclear quadrupole coupling directly. These energy levels may be obtained from the quadrupolar hamiltonian ... 

The resonance frequency of the central transition (+1/2 — —1/2) is unchanged by the first-order quadrupolar Hamiltonian. Other (satellite) transitions are, however, shifted by an amount proportional to the first-order quadrupolar coupling constant (oSb The result of this is to substantially broaden resonances from... 

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-... 

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 ... 

The dynamics of the spin system is governed by the Hamiltonian which in addition to a term reflecting external radio-frequency (rf) manipulations displays dependencies on chemical shifts, scalar (electron-mediated) J coupling, dipole-dipole coupling, and quadrupolar coupling, i.e.,... 

Based on the formulae in Eqs. (4) and (6) along with the relevant constants and operators in Table 2, it is a straightforward matter to set up all internal Hamiltonians to the first order. For the quadrupolar coupling interaction, it is often necessary to include second-order terms, as expressed by the Hamiltonian... 

The principle of sample spinning has been described in Section 3.3.3. As a result of sample spinning, the interaction Hamiltonian and thus the resonance frequency (cf. eqn (3.3.6)) becomes time dependent. For the simple case of a pair ij of dipolar coupled spins i, equivalent to a quadrupolar nucleus spin 1 like with, the time-dependent spin Hamiltonian is given by [Mehl, Schl]... 

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... 

This important equation governs, like an equation of motion, the time development of the system under a Hamiltonian H. It yields eight coupled differential equations for the coefficients c,(r). The resulting solutions for various Hamiltonians resemble rotations in an eight-dimensional space spanned by the eight nontrivial basis operators. With the convenient basis set of p, [14, 16, 68], the time evolution under Zeeman interaction can be visualized as a precession in the pi-pi, Ps-Pe Pi P planes. The axially symmetric quadrupolar interaction, on the other hand, mixes between pi and pe and between p and ps. Therefore, evolution under quadrupolar interaction does not lead to precession of magnetization within the x-y plane, but rotates it out of this plane into a not directly accessible order and back again. 

The spin Hamiltonian of the system consists of three parts The chemical shifts and/or quadrupolar interactions define the individual Hamiltonians of the spins Tj and I2 and the dipolar couplings and exchange interactions define the coupling Hamiltonian Hj j (I l > 2 Th mutual exchange of the two nuclei corresponds to a permutation of the two nuclei which exchange their individual chemical shifts and/or quadrupolar couplings with exchange rates = 21 = k. 

The options couple and weak denote a scalar coupling interaction. Couple is adequate for a strongly coupled second order spin system and weak for a coupling that can be described by an AX approximation. Dipolar describes a dipolar coupling, only the energy conserving part of the dipolar Hamiltonian are implemented. The qpolar option is reserved for the coupling interaction of a quadrupolar nucleus. Before a nucleus can appear in the interaction statement it must be defined in the nucleus statement. 

In this section, we attempt to review the solid-state Co NMR data published up to the end of 1998. The data on chemical shift tensors and quadrupole coupling constants are summarized in Tables 1-4. Data obtained by nuclear quadrupole resonance (NQR) are included wherever appropriate. Readers interested in the spin-spin coupling between spin-1 /2 and Co nuclei are referred to an article by Wasylishen and colleagues and the references cited therein. The NMR conventions employed in this review are defined in the next section. This is followed by a discussion of the techniques that have been applied to solid-state Co NMR studies with emphasis on what has been applied to Co systems. A detailed discussion on the techniques and Hamiltonians for quadrupolar nuclei is referred to the comprehensive review by Freude and Haase. 

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