Second-order quadrupole

The technique can be applied to both spinning and static samples, taking advantage of the fact that chemical shifts are proportional to the Larmor frequency while second-order quadrupole couplings are inversely proportional to the Larmor frequency (9). Thus, the spectrum of the CT can be seen as the projection along a specific angle a of a 2D chemical shift/second-order quadrupolar correlation spectrum. The angle a. for a spectrum acquired at u>o is defined as... 

The ZB and WZ forms of ZnS could be distinguished on the basis of their isotropic 67Zn and 33S static NMR chemical shifts [63] however, estimates of small CSAs obtained from static 33S NMR of the WZ forms of ZnS and CdS were obtained by ignoring any small second-order quadrupole broadening [65], which may account for discrepancies when NQCCs are assumed to be responsible for the broadenings [65]. 

In this section analytical expressions for ENDOR transition frequencies and intensities will be given, which allow an adequate description of ENDOR spectra of transition metal complexes. The formalism is based on operator transforms of the spin Hamiltonian under the most general symmetry conditions. The transparent first and second order formulae are expressed as compact quadratic and bilinear forms of simple equations. Second order contributions, and in particular cross-terms between hf interactions of different nuclei, will be discussed for spin systems possessing different symmetries. Finally, methods to determine relative and absolute signs of hf and quadrupole coupling constants will be summarized. 

The hfs (or quadrupole) tensors of geometrically (chemically) equivalent nuclei can be transformed into each other by symmetry operations of the point group of the paramagnetic metal complex. For an arbitrary orientation of B0 these nuclei may be considered as nonequivalent and the ENDOR spectra are described by the simple expressions in (B 4). If B0 is oriented in such a way that the corresponding symmetry group of the spin Hamiltonian is not the trivial one (Q symmetry), symmetry adapted base functions have to be used in the second order treatment for an accurate description of ENDOR spectra. We discuss the C2v and D4h covering symmetry in more detail. 

If the hfs of the nucleus under consideration is not resolved in the EPR spectrum, all nuclear spin states are simultaneously saturated and a sign determination using ENDOR line intensities is not possible. In this case the relative signs may sometimes be determined from second order hf contributions. This method has been applied by DuVarney and Spaeth74) to determine the sign of the 41K electric quadrupole moment using F centres in KC1. 

The line widths and shapes to be expected for cubic crystals containing point defects have been derived by Cohen and Reif for both first and second order quadrupole interaction 97). In particular, for point defect concentrations greater than about 0.1 (in terms of probability /, of a lattice site being occupied by a defect) distributed in a random fashion over various possible lattice sites, the second order interaction gives rise to a lopsided central component whose shape is given by (97)... 

The quadrupole perturbed NMR spectriun of Rb (I = 312) consists of the central hne and two satelUte Unes. In D-RADP-x the nuclear quadrupole couphng constant decreases linearly with x from 9.6 MHz (x = 0) to 6.6 MHz (x = 0.95) [10], whereas the Larmor frequency amounts to 98.163 MHz in a Bo field of 7 T. Due to the substitutional disorder, and the corresponding loss of the translational invariance, the satelHte transitions are inhomogeneously broadened that much, that only the central transition can be directly observed. The position of the corresponding resonance line can be calculated using standard NMR-NQR theory [16], where the second order shift is given by ... 

Before closing this chapter, it is important to emphasize the context in which the transition rate expressions obtained here are most commonly used. The perturbative approach used in the above development gives rise to various contributions to the overall rate coefficient for transitions from an initial state i to a final state i these contributions include the electric dipole, magnetic dipole, and electric quadrupole first order tenns as well contributions arising from second (and higher) order terms in the perturbation solution. 

As if can be seen in fhese simulations, all MAS spectra are shifted towards low frequencies wifh respecf fo fhe true isotropic chemical shift—an effect known as the second-order quadrupole-induced shift. The centre of gravity ( g) of fhese lineshapes is given by ° ... 

However, commonly due to a spread of quadrupole parameters caused by structural variation, with the extreme case being glassy or amorphous materials, the distinct features of fhe lineshape can be lost. This is particularly true if Cq is small. Then an approach which records spectra at several magnetic fields will allow bofh fhe second-order quadrupole effect parameter Pq, defined as... 

In the case of quadrupole nuclei with spin Z> 1/2, the frequency at the center of gravity, v g, of the CT is shifted by the second-order quadrupole shift ... 

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