Quadrupole interaction energy tensor

Quadrupole interaction energy tensor X Residual resistivity (solid state) Pr... 

The nuclear quadrupole interaction energy tensor / is usually quoted in MHz, corresponding to the value of eQq/h, although the h is usually omitted. 

For all charges external to the nucleus, the potential V(r) satisfies Laplace s equation, 2 =i V = 0. For this source of V(r), the second term in (18.38) vanishes. The first term, which does not vanish, is the electric quadrupole interaction energy, Eq, with the (classical) quadrupole moment tensor of the nucleus given by Q = / p(r)(3x - r ) dr. Electrons having s character penetrate the nucleus, however, since for such electrons the wave function does not vanish at the origin [as with the Fermi contact interaction, eq. (18.10)]. For these electrons, 2 =.i = Aire Q)f, so that the second term in (18.38) becomes... 

The quadrupole interaction will perturb all energy levels to first order (oc3cos 0 — 1, where 6 is the angle between the external magnetic field and the Z-axis of the PAS of the EFG tensor), apart from the If Cq is... 

Laplace s equation, V V = 0, means that the number of unique elements needed to evaluate an interaction energy can be reduced. For the second moment this amounts to a transformation into a traceless tensor form, a form usually referred to as the quadrupole moment [5]. Transformations for higher moments can be accomplished with the conditions that develop from further differentiation of Laplace s equation. With modern computation machinery, such reduction tends to be of less benefit, and on vector machines, it may be less efficient in certain steps. We shall not make that transformation and instead will use traced Cartesian moments. It is still appropriate, however, to refer to quadrupoles or octupoles rather than to second or third moments since for interaction energies there is no difference. Logan has pointed out the convenience and utility of a Cartesian form of the multipole polarizabilities [6], and in most cases, that is how the properties are expressed here. 

Figure 2. Effect of the quadrupole interaction on the Zeeman energy levels of an 1 = 1 nucleus with axial symmetry. I is the total spin, m its component, and 6 is the angle between the applied magnetic field and the principal (z) axis of the quadrupole splitting tensor.

ENDOR measurements of nuclear quadrupole coupling tensors of I > Vi nuclei A slightly lengthier analysis is required in the presence of nuclear quadrupole interactions, (nqi), from I > V2 nuclei. When the hyperfine and nuclear Zeeman interactions dominate over that caused by the nqi a simple method involves separate measurements of the parameters, G+ and P+ and/or G and P for ms = Vi of a free radical, as illustrated with the energy diagram in Fig. 3.11 for / = 1. 

The tensor component of the nuclear quadrupole interaction corresponds to an electric field gradient (EFG) that reflects the valence shell configuration of the atomic nucleus. The classical interaction energy is (Davies, 1967)... 

Weak quadrupole perturbation of magnetic levels In this case the quadrupole interaction operator (18.51) must be projected onto the coordinate system associated with the magnetic (Zeeman) hamiltonian (18.1). Since the former is actually a tensor operator, the projection introduces a more complicated angular dependence than in the converse case considered in section 1.3.2.2. The energy levels become, on the basis of first-order perturbation theory ... 

As outlined in section 1.3.2., the nuclear quadrupole interaction is characterized by two parameters, the coupling constant eQVz expressed in suitable energy units, and an asymmetry parameter tj indicating the degree of departure of the electric field gradient (efg) tensor Fy from axial symmetry. Information relevant to the lanthanide-containing metal, alloys, or compound is contained in these two parameters, since from the standpoint of the physics or chemistry of solids the quadrupole moment Q is taken to be a known property of the nuclear state. 

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