Crystal double groups

Sm, Eu and Gd can be concentrated by crystallization through double magnesium nitrates, followed by crystallization of bismuth magnesium nitrates [30]. Sm and Eu are then removed by a proceedure based on valence change, and Gd is recovered by bromate crystallizations. Yttrium group earths may be conveniently separated by bromate crystallization. 

N-(2-acetamido-4-nitropheryl)pyrrolidene (PAN) is dimorphic. The thermodynamically stable form is non-polar (Fig.2), (14). The crystal space group is P21 with two independent molecules per asymmetric unit which are related by a pseudo center of inversion. This crystal does not display measurable frequency doubling generation. 

A partial character table of the point group D3h is given in Table 8.4. Find the missing characters of the vector and spinor representations of the double group D3h. Determine whether El transitions E-/ E3/2 and E./2 —> E5/2 are allowed in a weak crystal field of D3h symmetry. State the polarization of allowed transitions. 

Describe the splitting of the multiplet 4D under the conditions specified in (i)-(iv) of Problem 8.3, except that the crystal field is of 0 l symmetry. [Hint Since a crystal field does not affect the parity of a state, it is sufficient to work with the double group O.]... 

Nye, J. F. (1957) Physical Properties of Crystals. Oxford Oxford University Press. Onodera, Y., and Okazaki, M. (1966) Tables of basis functions for double groups. J. Phys. Soc. Japan 21, 2400-8. 

Classification of Crystal-field Terms and Multiplets Double Groups... 

While the crystal field potential causes the symmetry lowering (on passing from Rs to its subgroup G), the spin-orbit interaction causes a passage from the group G to its double group G. ... 

The labeling of the crystal-field multiplets according to the irreducible representations of a double group can be performed as follows. 

The crystal-field multiplets are classified according to the irreducible representations of the respective double-group where both, the Bethe and the Mulliken (in parentheses) notations are written. DsH means the spin-Hamiltonian D-value accounting for all excitations AmIi - the lowest energy levels difference using the model-Hamiltonian in the first iteration Affl - the... 

Figure 5. Calculated energy levels for the PtClf ion in D h and Td symmetry. In the center, the energy levels are shown for D h coordination with the inclusion of a spin-orbit coupling parameter = 3000 cm1. The states are designated with the T notation for the D/ double group, which applies with the inclusion of spin-orbit coupling. For D4h dx2 y2 — dxy = 26,100 cm 1 dx2 y2 — dxz>yz = 30,200 cm 1 dx2 y2 — dz2 = 38,300 cm 1, F2 = 1000 cm 1, F = 65 cm 1. For the Td coordination, At = 14,000 cm 1, F2 = 1000 cm 1, and F = 65 cm 1. Observed bands for the crystal are shown at the right side...

The 0/, point group character table and the O (and O double group) multiplication table are given in the Appendix. The irreps are labelled Ti, T2... herein [72], corresponding to Ai, A2. The subscript g or u is usually not included in the labelling of 4fN crystal field states since it is even (odd) for even (odd) N, respectively [73]. 

In this description, the Russell-Saunders coupling scheme is assumed, whereby spin-orbit coupling acts as a perturbation to crystal field effects. It is necessary to use a new notation to describe the splitting of crystal field levels by spin-orbit coupling. This involves double group notation. Some points to note in connection with this are as follows 659). 

The comparison of the calculated spectra of the free ions and the ones in the crystal is not straightforward. Indeed, in the crystal, the presence of the first coordination shell increases the number of electrons and basis functions in the calculations, resulting in a blow-up of the Cl expansion, mainly due to the generated doubly-excited configurations. One should bare in mind that this increase is about six time as fast in double group symmetries as in the non-relativistic symmetry. In a non effective Hamiltonian method, the only way to keep the size of the DGCI matrix to an affordable size of few million configurations, is to cut down the number of correlated electrons. This may essentially deteriorate the quality of electron correlation as the contributions of the spin-orbit interaction... 

For cubic groups one has to work with irreducible representations. The crystal quantum numbers are not useful to classify the states, because of the occurrence of triplet and quartet states. Quartet states are found in the double groups OJ, O and Tj. 

The full-rotational group compatibility tables show how a free-ion J level is broken up into crystal-field levels when the ion is placed in a crystalline environment with a distinct point symmetry. The irreducible representations (irreps) are labelled according to the notations of Koster et al. (1963). The tables are given up to J = 8 for even-electron systems and up to J = 17/2 for odd-electron systems. The double groups are marked by an asterisk. Although higher J values may occur for divalent lanthanide ions, they are of less importance for the study of the energy levels in the ultraviolet, visible and near-infrared parts of the spectra. 

In order for dipole—dipole and dipole-iaduced dipole iateractioas to be effective, the molecule must coataia polar groups and/or be highly polarizable. Ease of electronic distortion is favored by the presence of aromatic groups and double or triple bonds. These groups frequently are found ia the molecular stmcture of Hquid crystal compouads. The most common nematogenic and smectogenic molecules are of the type shown ia Table 2. 

The conditions for surfactants to be useful to form Hquid crystals exist when the cross-sectional areas of the polar group and the hydrocarbon chain are similar. This means that double-chain surfactants are eminently suited, and lecithin (qv) is a natural choice. Combiaations of a monochain ionic surfactant with a long-chain carboxyHc acid or alcohol yield lamellar Hquid crystals at low concentrations, but suffer the disadvantage of the alcohol being too soluble ia the oil phase. A combination of long-chain carboxyHc acid plus an amine of equal chain length suffers less from this problem because of extensive ionisa tion of both amphiphiles. 

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