Electric-field-gradient tensor quadrupolar interactions

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 ... 

In this equation e is the charge on the electron, / is the spin quantum number of the nucleus, and Av is the separation between adjacent lines of the 2/-component multiplet. This spectrum arises because the quadrupolar interaction pushes some energy levels up and some down the transitions between adjacent energy levels (recall that the only allowed transitions are between adjacent levels) are shifted by amounts proportional to mz. The center of gravity of the spectrum is not affected to first order. Equation (15.6) applies only to axially symmetric electric field gradient tensors, such as those experienced by 2H in a typical C-D bond. The... 

Figure 3 Energy level splittings and transitions for half-integer quadrupolar nuclei, as illustrated for the example of a spin-3/2 nucleus, (Oy is the NMR transition frequency in the absence of the quadrupolar interaction, Wy is the quadrupolar frequency, and 0 is the angle between the magnetic field and the principal axis of the electric field gradient tensor.

Classically, the electric field gradient at a nucleus is produced by the arrangement of charges (i.e., other nuclei and electrons) about that nucleus 66). If the nucleus is quad-rupolar, as in the case of 27A1, then the interaction of its nuclear quadrupole moment, eQ, with the largest component of the EFG tensor, V33, is defined as the quadrupolar coupling constant, CQ ... 

In a rotating molecule containing one quadrupolar nucleus there is an interaction between the angular momentum J of the molecule and the nuclear spin momentum I. The operator of this interaction can be written as a scalar product of two irreducible tensor operators of second rank. The first tensor operator describes the nuclear quadrupole moment and the second describes the electrical field gradient at the position of the nucleus under investigation. 

Nuclei with spin quantum number I > can possess electric quadrupole moments. The nuclear quadrupole is directly coupled to the nuclear spin and the electrostatic interaction dominates the nuclear spin relaxation. The quadrupolar interaction is simply the first non-vanishing term in the Taylor expansion of the electrostatic interaction between the charge distribution of the nucleus and that of its surrounding. It can be expressed as a direct product between the nuclear quadrupole tensor and the electric field gradient (EFG) at the nucleus [6]. 

While eq N) is the largest principal component of the electric field gradient (EFG) tensor at the site of the quadrupolar nucleus. All the information about the structure of the molecule and interactions is in the eq N) term. In the gas-phase (at low symmetry positions) the EFG is entirely of intermolecular origin. In the condensed phases (liquids and solids), intermolecular interactions may have a substantial influence on the observed EFG. 

It must be stressed that there are fundamental differences in the natures of the different nuclei discussed above. Si and P are spin 1/2 nuclei, and MAS yields particularly simple spectra with complete averaging of the chemical shift tensor components. The average isotropic shift values are field independent and correspond to the solution chemical shifts. AI and O, however, are quadrupolar nuclei with nonintegral spins greater than 1, and their solid-state spectra are often much more complex. Since the quadrupolar interaction depends on local electric field gradients at the nucleus studied, solid-state NMR of quadrupolar nuclei can yield further information about the local environment. More complicated spinning techniques have been introduced to average the anisotropies in the spectra of quadrupolar systems, and these new methods have increased interest in the use of such nuclei as probes of local microstructure. 

For Zn SSNMR spectroscopy, the observed spectrum is usually dominated by the nuclear electric quadrupolar interaction unless the zinc sits at a site with spherical symmetry (such as tetrahedral, octahedral and cubic sites). This anisotropic interaction originates from the coupfing between the quadrupole moment (eQ) of the nucleus and the electric field gradient (EFG) at the nuclear site. The EFG is described by a second-rank tensor that can be diagonalized in a principal axis system (PAS). In the PAS, the EFG is described by three components, which are ordered in such that I Fxx I kVy I ud they satisfy FxxT TyyT Tzz fi-The quadrupolar interaction is characterized by two parameters one is quadrupolar coupling constant ... 

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