Electric quadrupole

The quadrupole is the next electric moment. A molecule has a non-zero electric quadrupole moment when there is a non-spherically symmetrical distribution of charge. A quadrupole can be considered to arise from four charges that sum to zero which are arranged so that they do not lead to a net dipole. Three such arrangements are shown in Figure 2.8. Whereas the dipole moment has components in the x, y and z directions, the quadrupole has nine components from all pairwise combinations of x and y and is represented by a 3 x 3 matrix as follows ... 

Nuclide Natural abundance, % Spin I Sensitivity at constant field relative to NMR frequency for a 1-kG field, Mffz Magnetic moment J-T-1 Electric quadrupole moment Q, 10 m ... 

All of the heteroatoms possess at least one naturally occurring isotope with a magnetic moment (Table 15). The nuclei and also possess an electric quadrupole moment... 

Isotope Natural abundance (%) Nuclear spin Electric quadrupole moment NMR frequency fora 23.5 kO field (MHz) Relative sensitivity... 

Nuclear electric quadrupole QCC (quadrupole coupling constant), (asymmetry parameter) Line-shape analysis, nutation NMR Coordination symmetry... 

Most authors prefer to work with a quantity 0e called the electric quadrupole moment. It is usually defined as... 

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. 

The multipole expansion gives exactly that expression. If the charge distribution shown has an overall charge Q, an electric dipole pe, an electric quadrupole 0g, and so on, then we write... 

Just like the electric quadrupole moment, the electric field gradient matrix can be written in diagonal form for a suitable choice of coordinate axes. 

Molecular electric quadrupole moments are more elusive animals, and they are not particularly easy to determine experimentally. Prior to 1970, the only direct routes to these quantities were from the Kerr and Cotton-Mouton effects. They can now be obtained from microwave Zeeman spectroscopy, to fair accuracy. It is fair to say that direct calculation offers a faster and more reliable route to this property than experiment. 

We often say that an electron is a spin-1/2 particle. Many nuclei also have a corresponding internal angular momentum which we refer to as nuclear spin, and we use the symbol I to represent the vector. The nuclear spin quantum number I is not restricted to the value of 1/2 it can have both integral and halfintegral values depending on the particular isotope of a particular element. All nuclei for which 7 1 also posses a nuclear quadrupole moment. It is usually given the symbol Qn and it is related to the nuclear charge density Pn(t) in much the same way as the electric quadrupole discussed earlier ... 

Thus the formation of tilted analogues of the smectic A phases, i.e. monolayer Cl and bilayer C2, is possible for mesogens with relatively large electric quadrupoles. In the case of strongly sterically asymmetric molecules (e.g., zigzag shaped or dumbell shaped molecules, Fig. 3b) these quadrupolar interactions may be steric in origin. From this point of view observation of molecular tilt in the molecular dynamics simulations for a one-layer film of DOBAMBC in the absence of electrostatic interactions is not so surprising [106]. 

If the nucleus feels both a magnetic field and an electric field gradient, and the electric quadrupole interaction is small, then the excited levels shift further and make the sextet asymmetrical, as observed in the spectrum of Fe203. 

Equation (4.15) would be extremely onerous to evaluate by explicit treatment of the nucleons as a many-particle system. However, in Mossbauer spectroscopy, we are dealing with eigenstates of the nucleus that are characterized by the total angular momentum with quantum number 7. Fortunately, the electric quadrupole interaction can be readily expressed in terms of this momentum 7, which is called the nuclear spin other properties of the nucleus need not to be considered. This is possible because the transformational properties of the quadrupole moment, which is an irreducible 2nd rank tensor, make it possible to use Clebsch-Gordon coefficients and the Wigner-Eckart theorem to replace the awkward operators 3x,xy—(5,yr (in spatial coordinates) by angular momentum operators of the total... 

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