Zeeman-quadrupolar Hamiltonian

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

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 ... 

When the quadrupolar Hamiltonian is considered as a perturbation on the Zeeman Hamiltonian, there is no general analytical solution for the eigenvalues of in the (very rare) case when and are of comparable magnitude. When the split-... 

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

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... 

The Zeeman effect must be mentioned in the case of nitrogen it behaves normally when 77 = 0 but, in the general case of a non-zero asymmetry, the Zee-man part of the hamiltonian no longer commutes with the quadrupolar part and there appears to be no first-order Zeeman effect, The second-order treatment of the perturbation yields the following values for the transition frequen cies 81 ... 

Except for some quadrupolar effects, all the interactions mentioned are small compared with the Zeeman interaction between the nuclear spin and the applied magnetic field, which was discussed in detail in Chapter 2. Under these circumstances, the interaction may be treated as a perturbation, and the first-order modifications to energy levels then arise only from terms in the Hamiltonian that commute with the Zeeman Hamiltonian. This portion of the interaction Hamiltonian is often called the secular part of the Hamiltonian, and the Hamiltonian is said to be truncated when nonsecular terms are dropped. This secular approximation often simplifies calculations and is an excellent approximation except for large quadrupolar interactions, where second-order terms become important. 

The isotropic g and a values are now replaced by two 3x3 matrices representing the g and A tensors and which arise from the anisotropic electron Zeeman and hyperfine interaction. Other energy terms may also be included in the spin Hamiltonian, including the anisotropic fine term D, for electron-electron interactions, and the anisotropic nuclear quadrupolar interaction Q, depending on the nucleus. Usually the quadrupolar interachons are very small, compared to A and D, are generally less than the inherent linewidth of the EPR signal and are therefore invisible by EPR. They are readily detected in hyperfine techniques such as ENDOR and HYSCORE. All these terms (g. A, D) are anisotropic in the solid state, and must therefore be defined in terms of a tensor, which will be explained in this section. 

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. 

Urf being the selected radio frequency and H the homogeneous field applied. This original setup was then widely used for the determination of the magnetic and quadrupolar hyperfine structure (hfs) constants A and B. Hereby one has to consider that the additional magnetic field further splits the atomic energy levels now characterized by F into (2F + 1) sublevels and mixes states of the same Mp but different F values. A Zeeman term has therefore to be added to the hyperfine Hamiltonian according to... 

NMR spectra cannot be measured in solids in the same way in which they are routinely obtained in solutions because NMR lines from solids are too broad. In solution all interactions apart from chemical shift and indirect coupling are averaged to zero by thermal motions of molecules. Magnetic interactions in the solid state are described by a Hamiltonian H [1], which is a sum of several contributions Zeeman interaction (the same as in solution), direct dipole-dipole interaction, magnetic shielding (giving chemical shifts), scalar spin-spin coupling to other nucleus, and for nuclei with / >1/2 also quadrupolar interactions. 

The High-Field Approximation In most NMR experiments the nuclear Zeeman interaction with the static external magnetic field is much stronger than all other interactions of the nuclear spins. As a result of these differences in the size, it is usually possible to treat these interactions in first order perturbation theory, i.e. use only those terms which commute with the Zeeman Hamiltonian, the so called secular terms. This approximation is called the high field approximation. While the single particle interactions like CSA or quadrupolar interaction have a unique form, for bilinear interactions, one has to distinguish between a homonuclear and a hetero-nuclear case. The secular parts of Hamiltonians discussed in the previous section are collected in Table 1. 

For an ensemble of nuclei with spin / > 1/2 experiencing no quadrupolar interaction, such as in an isotropic liquid or a crystal with cubic symmetry, the equations describing the time evolution of the density matrix under action of static and RF magnetic fields are a natural extension of the / = 1/2 case. The Hamiltonian contains only the Zeeman and RF terms the effects of RF pulses are described by rotations of the spin operators around the transverse axes in the rotating frame, whereas free evolution corresponds to rotations around the z-axis. 

As already shown, the total NMR Hamiltonian, from which the spin energy levels are obtained, is the sum of terms representing the Zeeman (Hq), chemical shift (Hcs)< dipole-dipole coupling (Hdd), and quadrupolar (Hq) interactions for nuclei with spins greater than V . 

Remembering that we are restricting our discussion to the cases when the Zeeman term determines the axis of quantization of the nuclear spins, then the quadrupolar interaction contributes a term, in hertz, to the spin Hamiltonian of... 

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