Quadrupolar Hamiltonian

In accordance with the works we assume that for analysing the effect of RF pulses on the quadrupolar spin system, it is sufficient to consider only three t)q)es of interactions quadrupolar interaction, homonuclear dipole-dipole interaction and the nuclear-spin interaction with the magnetic component of the RF field. Before the initiation of the multi-pulse sequence the quadrupolar system is described by two Hamiltonians quadrupolar Hq and the part of the homonuclear dipole-dipole Hamiltonian Hj secular in relation to the quadrupole Hamiltonian, the sum of which can be regarded as the effective Hamiltonian of the spin system independent of the time factor. 

Chemical shift Hamiltonian Dipolar interaction Hamiltonian Nuclear-nuclear interaction Hamiltonian Quadrupolar interaction Hamiltonian... 

The low-temperature Mossbauer spectra of the spinel type oxides, NiCr204 [14,18] (Fig. 7.6b) and NiFe204 [3, 18], have been found to exhibit combined magnetic dipole and electric quadrupole interaction (Fig. 7.7). For the evaluation of these spectra, the authors have assumed a small quadrupolar perturbation and a large magnetic interaction, as depicted in Fig. 7.3 and represented by the Hamiltonian [3]... 

The Hamiltonian describing the quadrupolar interaction in the laboratory frame (L), in the units of radians/s, can be written using the spherical tensor formalism as [1,6, 24]... 

When the Zeeman interaction is much larger than the quadrupolar frequency (coo coq), the averaged quadmpolar Hamiltonian over a Larmor period,... 

When the sample is spun in a rotor with a spinning frequency coR about an axis at angle [>RI with respect to Z> 0, the first-order quadrupolar Hamiltonian can be rewritten as... 

Fig. 2 The values of second-order P2 and fourth-order P4 Legendre polynomials as functions of [iRL. The two polynomials do not have a common root, i.e., the anisotropic terms in the second-order quadrupolar Hamiltonian cannot both be canceled at once, regardless of the orientation of the rotor axis...

However, in the presence of a strong quadrupolar interaction, the second-order cross-terms between the quadrupolar interaction and the CSA [27, 28] or the heteronuclear dipolar interaction [29, 30] also have to be taken into account. For instance, upon including the CSA in the expression of the time-dependent Hamiltonian in (9), the second-order term becomes... 

Let us calculate the frequencies of transitions between Zeeman eigenstates s) and r), assuming that the nuclei are only subjected to an isotropic chemical shift and the first- and second-order quadrupolar interaction. As seen in Sect. 2.1, the Hamiltonian that governs the spin system in the frame of the Zeeman interaction (the rotating frame) is... 

During an rf-field irradiation, the Hamiltonian governing a quadrupolar spin system can be written in the rotating frame as... 

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 ... 

When the electric field gradient at the nucleus exerted by the electrons is nonzero, the nuclear levels will be split. The eigenvalues of the quadrupolar interaction Hamiltonian are given by... 

As previously observed, the dominant terms in the Hamiltonian which describe a spin system in the solid state are the dipolar and quadrupolar terms. In the case of nuclei with 1 = 1/2 (such as H, 13C, 19F and 29Si) the quadrupolar interaction is zero. The dipolar Hamiltonian HD (for a homonuclear spin system) has the general form ... 

Proton and fluorine are the most frequently studied nuclei in the solid state NMR of polymers. However, useful information can be obtained from deuterium resonance, (2H I = 1). In the presence of nuclei with spin I > 1/2, which possess a quadrupolar moment, the most dominant interaction occurs between the quadrupolar moment and an electric field gradient tensor V, generated by the C—D bonding electrons. Consequently, the only dominant interaction, besides Hz (Eq. (1)), will be described by the quadrupolar Hamiltonian, which for a single spin has the form ... 

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 ... 

In the principal axis system of Vy, in which the tensor is diagonal, the quadrupolar Hamiltonian for a single spin is... 

Quadrupolar interactions can offer direct information on the dynamics of organics within zeolite crystals. Eckman and Vega (304) studied the 2H quadrupolar echo decay in perdeuterated p-xylene adsorbed on zeolite ZSM-5. The deuterium quadrupolar interaction usually dominates the spin Hamiltonian, so that the powder pattern can be used as a test for models of molecular motion. At -75°C and 25°C typical rigid-lattice spectra were obtained. At 100°C however, the resonance arising from the aromatic deuterons was motionally narrowed, while the methyl resonance was not, The authors conclude that p-xylene molecules reorient about an axis which passes through the C3 axes of the methyl groups. 

The energy levels for the Zeeman -Quadrupolar Hamiltonian are given by... 

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