Quadrupole moment, definition

This definition of the quadrupole is obviously dependent upon the orientation of the chargi distribution within the coordinate frame. Transformation of the axes can lead to alternativi definitions that may be more informative. Thus the quadrupole moment is commonl defined as follows ... 

At the molecular level, electric quadrupoles can lead to useful structural information. Thus, whilst the absence of a permanent electric dipole in CO2 simply means that the molecule is linear, the fact that the electric quadrupole moment is negative shows that our simple chemical intuition of 0 C" 0 is correct. The definition of quadrupole moment is only independent of the coordinate origin when the charges sum to zero and when the electric dipole moment is zero. 

Remarkably, only one nuclear constant, Q, is needed in (4.17) to describe the quadrupole moment of the nucleus, whereas the full quadrupole tensor Q has five independent invariants. The simplification is possible because the nucleus has a definite angular momentum (7) which, in classical terms, imposes cylindrical symmetry of the charge distribution. Choosing x, = z as symmetry axis, the off-diagonal elements Qij are zero and the energy change caused by nuclear... 

The moments defined by Eq. (7.1) are referred to as the unabridged moments. For moments with / > 2, an alternative, traceless definition is often used. In the traceless definition, the quadrupole moment is given by... 

The operators for the potential, the electric field, and the electric field gradient have the same symmetry, respectively, as those for the atomic charge, the dipole moment, and the quadrupole moment discussed in chapter 7. In analogy with the moments, only the spherical components on the density give a central contribution to the electrostatic potential, while the dipolar components are the sole central contributors to the electric field, and only quadrupolar components contribute to the electric field gradient in its traceless definition. 

For q and n the definitions are unambiguous. For higher multipoles it is often convenient to define the quantities in different ways the quadrupole moment is often defined as... 

While the Overhauser shift is due to delocalised electrons, the ENDOR experiments require more localised electrons. Both groups were able to observe ENDOR on the effective mass donor resonance but not on the deep donor signal. An illustrative ENDOR spectrum is shown in FIGURE 4, in which the 69 71Ga ENDOR lines are observed. The 69Ga line is clearly split by quadrupole interactions just as in the Oveihauser shift measurements, while the quadrupolar interaction is much less resolved in the 71 Ga line, due to its smaller quadrupole moment. In the three samples investigated by the two groups, the linewidths were all too broad to resolve any hyperfine interaction and so no definitive identification of the residual donor was possible. 

This equation may be simplified by making use of the equivalence of the nuclei (/, =I2 = In), by neglecting matrix elements off-diagonal in IT, and by noting the definition of the nuclear quadrupole moment Qn, which is given by,... 

We complete the calculation by making use of the definition of the nuclear quadrupole moment of nucleus k ... 

Nuclear quadrupolar interaction arises from the coupling between the nuclear quadrupole moment Q and the EFG at the nuclear position. The EFG varies in space and is described by a traceless second-rank tensor. The EFG tensor is diagonal and its three principal components are VXXr Vyy and Rzz with the definition of VZZ > Vyy > VX < Such a principal-axis system for the EFG tensor is defined with the direction of the external magnetic field, as illustrated in Figure 3(A). It is convenient to express such quadrupolar interactions by using the following two parameters ... 

But the general definition of a quadrupole moment is the magnitude of each charge of quadrupole times the square of the distance of the charge from the origin. Thus, Zlqd is the y component of the quadmpole moment for the particular coordinate system that has been chosen. Similarly, thexcomponentpj of the quadrupole... 

In analogy to the definition of electric dipole moment, electric multipole moments are also defined. In particular, the quadrupole moment Q and the octupole moment U are defined as ... 

There are several different definitions for the second order electric moments. In order to be consistent with the above, the quadrupole moment is defined in a similar way ... 

The Nuclear quadrupole definition used here is consistent with Eq. (II. 1). Normally nuclear quadrupole moments are listed as 2Q/ where Q is given by Eq. (II.l). 

The apparent increase in the kij values for the aromatic hydrocarbon and CC14 binaries is most probably caused by the effect of the quadrupole moment on the properties of C02. That is, the strong quadrupole moment of C02 raises the values of all the ky s for its nonpolar binaries. (The same is also true of the acetylene binaries.) However, the s have a more complex behavior in the case of naphthalene-C02, where a definite dependence of ktj on temperature was found (13) ... 

The definition of the moments becomes unambiguous by specifying the orientation of the nucleus. By convention, in the definition of the electric multipole moments the nuclear angular momentum is assumed to be aligned along the z axis. The intrinsic electric quadrupole moment Qo is then defined as... 

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