Quadrupole coupling constants, defining equations

The formulation of energy and matrix element expressions depends on the coupling scheme which is adopted for nuclear spins / and rotational angular momentum 7. The resulting expressions can always be written in terms of quadmpole coupling parameters eQqjj- which are defined as 

Letting Lbe the potential at the nucleus in question due to all other charges, matrix elements taken of the electric field gradient tensor U are given by matrix elements of the second derivative of V directed 

The rotational dependence of can be calculated by transformation to the principal inertial axes system (g = a, b,c) yielding 

The quadrupole-couphng tensor is symmetric, zL s diagonal elements obey Laplace s 

The first-order quadrupole coupling energy for one quadmpolar nucleus in the molecule can be written [55Tow, 64Tha] 

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Figure 8.11. Relationship between the nuclear quadrupole coupling constant xq for a number of magnesium compounds and minerals and A. the octahedral distortion index DI defined in Equation (8.1). Regression line fitted only to the most reliable data points (full circles), less reliable data points (full squares) are included for interest only. B. the octahedral shear strain il defined in Equation (8.2). Regression line fitted to all data points. From MacKenzie and Meinhold (1994a) by permission of the copyright owner.

Figure 8.31. A. Relationship between the " i nuclear quadrupole coupling constants of ATiOs-type titanates and their structural shear strain i(( defined in equation (8.3). B. Relationship between the Ti isotropic chemical shifts (ppm) and the mean Ti-O bond lengths (A) for a series of ATi03 compounds with the perovskite structure. Note that compounds with the ilmenite stmcture do not fit this relationship and are not included here. From Padro et al. (2000), by permission of the copyright owner.

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