Point symmetry cubic

Among all the isomers of fullerenes C32 we choice the molecule with cubic point symmetry group Oh. The molecule cage is formed by 12 hexagonal and 6 square faces. The atoms disposed in vertex of squares take part in forming of intermolecular bonds in dimer (C32)2 and cuban-like cluster (C32)s-... 

Another special case occurs if the nucleus is at a site of cubic point symmetry (actually tetrahedral symmetry or higher will do). If this condition holds, the electric field gradient must by symmetry disappear, and the quadrupole coupling vanishes. Common examples are tetrahedral ions such as SO -, MoO -, TcOj, and simple salts, such as NaCl, in which the ions reside in sites of cubic symmetry. Unfortunately, this situation almost never occurs for the quadrupolar nuclei, such as l70, l4N, and 35C1, that one might wish to study in organic compounds. The only possible exceptions are symmetrically tetrasubstituted ammonium salts. 

In this Appendix, Htickel theory calculations are used to demonstrate that the polynomials of Appendix 1 can be applied as basis functions for the irreducible subspaces of a Hamiltonian invariant under icosahedral point symmetry, while extended Htickel theory calculations on cubium cages of cubic point symmetry are used to demonstrate the same result for the kubic harmonics, since single bond-length regular orbits are not possible in all cases. 

From its sensitivity to short-range structure and low-frequency dynamics, NMR spectroscopy provides some constraints on the nature of the P-cristobalite structure. For cristobalite, only powder NMR techniques can be applied, because large single crystals do not survive the a-P transition intact. It is helpful to consider also AlP04-cristobalite, because it is similar in essential respects to Si02 and the cubic point symmetry of the Al-position in the average structure of the P-phase provides some additional constraints. 

Figure 10. NMR spectra (without MAS) of AIPO4 cristobalite taken just below (473 K) and above (523 K) the a-f transition, with y-axis scaling proportional to absolute intensity. Spectrum of the a-phase contains only the central transition, whereas that for the f-phase contains also the satellite transitions, indicating Cq = 0 (cf Fig. 4) and consistent with cubic point symmetry. For the a-phase, the central transition comprises 9/35 of the total intensity the remainder occurs in the satellite transition powder spectrum that spans a wide frequency range according to Equation 3, with Cq = 0.9 MHz. Origin of the frequency scale is arbitrary. [Redrawn from data of Phillips et al. (1993).]...

Millard et al. 1992 Maekawa et al. 1997), because of the distinct chemical shift ranges for " Al and Al in oxides. In spinel, " Al gives a narrow peak near +70 ppm, whereas the Al gives a broader peak near +10 ppm (Fig. 27). The tetrahedral site in spinel has cubic point symmetry which requires Cq = 0, but crystal defects and the [Mg,Al] disorder results in a small distribution of EFG s at the Al site, so that the peak at 70 ppm contains only the central transition. The earlier studies suffered from low MAS rates, which yield spinning sidebands that overlap the centerbands, and unequal excitation of the two Al NMR signals. 

Within a manifold of angular momentum J composed of f electron wavefunctions the most general operator equivalent potential with cubic point symmetry may be expressed as... 

The interaction of the nuclear quadrupole moment Q with the local electric field gradient (EFG) at the nuclear site causes a splitting of the nuclear energy levels. Table lc illustrates the situation for an Fe57 absorber atom embedded in a crystal with lower than cubic point symmetry for the iron lattice site. Since the nuclear ground state of Fe57 has zero quadrupole moment it remains unsplit. The excited state of Fe57, characterized by Q>0, however, splits into two sublevels. 

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