Electric quadrupole moments

Factoring out D and substituting the Legendre polynomials into the equation where 

If we set up the integrals over the entire volume of the charged object (nucleus), we get 

The first term in the square bracket in this equation is the electric monopole moment, which is equal to the nuclear charge, Ze. The second term in the square bracket is the electric dipole moment while the third term in the square bracket is the electric quadmpole moment. For a quantum mechanical system in a well-defined quantum state, the charge density p is an even function, and because the dipole moment involves the product of an even and an odd function, the corresponding integral is identically zero. Therefore, there should be no electric dipole moment or any other odd electric moment for nuclei. For spherical nuclei, the charge density p does not depend on 0, and thus the quadmpole moment Q is given by 

The quadmpole moment will differ from zero only if the nucleus is not spherical. Thus, the quadmpole moment is a measure of the nonsphericity or shape of the nucleus. We can further elaborate on this by making a simple model (Fig. 2.15) for nonspherical nuclei. We shall assume such nuclei are 

Since we also know the square of the mean radius R of the spheroid is given by 

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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... 

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. 

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. 

Nuclear electric quadrupole moment of ground (g) and excited (e) state (in... 

Both the ground state and the 67.4 keV nuclear excited state of possess a nonzero electric quadrupole moment. If placed in an inhomogeneous electric field (electric field gradient, EFG 0) the Ni nucleus undergoes electric quadmpole interaction with the EFG at the nucleus, as a result of which the 67.4 keV level will split into three substates /, wi) = 5/2, 5/2), 5/2, 3/2), and 5/2, 1/2) and the ground level will split into two substates 3/2, 3/2) and 3/2, 1/2). 

Mossbauer measurements with determination of the electric quadrupole moments have been reported in [253, 254,259]. Wagner et al. [254] measured the quadrupole hyperfine interaction in OSO2 and OSP2 of the Mossbauer isotopes The ratios of the quadrupole moments of the 4 = 72 states in the even osmium isotopes and of the 4 = 5/2 (69.6 keV) and 4 = 3/2 states in Os were deduced very accurately. In Table 7.8, the experimental results [254] are given, from which the following ratios can be calculated ... 

NMR-active nuclei with spin > Vi (these include, as we mentioned previously, deuterium) have an electric quadrupole moment and are thus referred to as quadrupolar nuclei. 

It is worth mentioning that >NH protons may often appear somewhat broader than their -OH counterparts, for another reason >NH protons have another relaxation mechanism available to them (quadrupole relaxation) because the 14N nucleus has an electric quadrupole moment. This extra relaxation capability can lead to a shorter relaxation time for >NH protons, and since the natural linewidth of a peak is inversely proportional to the relaxation time of the proton(s) giving rise to it, a shorter relaxation time will give rise to a broader peak. 

Bleaney, B., Bowers, K.D., and Pryce, M.H.L. 1955. Paramagnetic resonance in dilute copper salts III. Theory, and evaluation of the nuclear electric quadrupole moments of 63Cu and 65Cu. Proceedings of the Royal Society of London A 228 166-174. 

This spurious contribution could be attributed to the presence of boron nuclei in the metallized wafer. Boron is present in nature with two stable isotopes (10B, nB), one having nuclear spin 3/2 and abundance of 80.3%, and the other nuclear spin 3 and abundance of 19.7%. The nuclei have non-zero electric quadrupole moments. 

Fig. 1 (a) Schematic representation of the spherical and non-spherical charge distribution in a nucleus. The value of electric quadrupole moment Q for the quadrupolar nucleus depends on the isotope under consideration, (b) The quadrupolar interaction arises from the interaction of Q with surrounding electric field gradient (EFG)... 

Extraframework cations are needed in anionic zeolites for charge balance, and for several zeolite topologies their locations are well investigated [281, 282]. Different cations have been investigated by solid state NMR in the past with different NMR properties and different project targets. We restrict this section to a tutorial example on sodium cation motion in sodalite and cancrinite structures [283-285], 23Na has a nuclear electric quadrupole moment, and quadrupolar interaction is useful to investigate jump processes, especially when they are well defined. 

For I > 1/2, nuclei also possess an electric quadrupole moment, Q. Quadrupolar nuclei exchange energy with electric fields in the rest of the molecule in which they are located causing profound effects on NMR spectra. Table 3.3 lists properties for some nuclei found in bioinorganic systems. 

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