Inhomogenous interaction

When only inhomogeneous interactions are involved and the dynamic effects are neglected, the Hamiltonian becomes exactly solvable and frequently in analytical forms. For example, the lineshapes of the chemical shift, heteronuclear dipolar and quadrupolar interactions can be calculated exactly and analytically, either for static or for rotating samples. ... 

In 1964, Pierre Hohenberg and Walter Kohn published a theory of an inhomogeneous interacting electron gas. These authors put forward an idea that the electronic density plays a central role in properties of this gas. The authors in fact are the founders of the density functional theory (DFT). 

The quadrupolar-echo experiment represents the most widely used experiment for the observation of quadrupolar nuclei.For half-integer nuelei, it may be tuned to observe only the central transition (1/2-1/2), which is perturbed by the quadrupolar interaction only to second order, thus allowing the observation of less dominant anisotropic interactions. A significant improvement in sensitivity can be obtained by ncorporating a spin-echo method such as the Carr-Purcell-Meiboom-Gill sequence into the detection period. " The powder line-shape splits into a manifold of sidebands, from which information on the homogeneous and inhomogeneous interactions can be extracted from the line-shape of the sidebands and their envelope, respectively. 

For CSA alone (see Fig. 3.1A), the commutators in the first-, second-, and higher-order terms vanish, and the stroboscopic observation of the spin system is well described by the zeroth-order term alone. This is described as an inhomogeneous interaction in the description of Portis [54]. The same situation is observed in the case of the heteronuclear dipolar coupHng of a pair of atoms, or for a pair of like spins in the absence of CSA, or if their CSA tensors have the same orientations (Fig. 3.1C) [52,55]. 

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-... 

Inhomogeneous interaction (Gaussian line shape function)... 

For a polydisperse system that contains inter-particle (inhomogeneity) interferences, Eq. (6.34) cannot be used by simple multiplication by the structured factor, as in Eq. (6.33). Although this has been used for significantly narrow distributions [37, 38, 59], there are better ways to properly account for structure in a polydisperse system. One such method, for relatively narrow size distributions, is by using the so-called local monodisperse approximation [60]. This approximation ignores the interference between different sized inhomogeneities and assumes that only like-sized inhomogeneities interact. 

A number of factors complicate the experimental measurements of CSA. These include a distribution in the values of the CSA nomandom orientation of the chains in the sample inhomogeneous interactions, such as residual line broadening due to incomplete proton decoupling and carbon-carbon dipolar coupling and molecular motion [30]. 

---