Site symmetry symbols

Site symmetry symbols, I, 128 Six-coordinate compounds stereochemistry, 1, 49-69 Six-membered rings metal complexes, 2, 79 Skeletal muscle sarcoplasmic reticulum calcium pump, 6, 565 Slags... 

Priming convention. The configuration index is especially useful for bis(tridentate) complexes and for more complicated cases. Bis(tridentate) complexes exist in three di-astereoisomeric forms which serve to illustrate the utility of a priming convention. These isomers are represented below, along with their site symmetry symbols and configuration indexes. For Examples I, 2, and 3, the two ligands are identical and the ligating-atom priority numbers are indicated. 

The difference between oriented site-symmetry groups of different Wyckoff positions is due to different orientations of the elements of the site-symmetry group G, with respect to the lattice. The difference arises when similar symmetry elements (reflections in planes and rotations about twofold axes of symmetry) occur in more than one class of elements of the point group F. Only eleven site groups [C2(2), Cs m), C2h 2./m), C 2 (2mm), CsyiZmm), 2(222), Ds(322), D2d(42m), D3d 32m), D hijnrnm), and >3 (62m)] can have different orientations with respect to the Bra-vais lattice. Oriented site- symmetry symbols show how the symmetry elements at a site are related to the symmetry elements of a space group. The site-symmetry... 

The Wyckoff symbol is a short designation it consists of a numeral followed by a letter, for example 8/. The cipher 8 states the multiplicity, that is, the number of symmetry-equivalent points in the unit cell. The / is an alphabetical label (a, b,c,...) according to the sequence of the listing of the positions a is always the position with the highest site symmetry. 

Every space group listed in the family tree corresponds to a structure. Since the space group symbol itself states only symmetry, and gives no information about the atomic positions, additional information concerning these is necessary for every member of the family tree (Wyckoff symbol, site symmetry, atomic coordinates). The value of information of a tree is rather restricted without these data. In simple cases the data can be included in the family tree in more complicated cases an additional table is convenient. The following examples show how specifications can be made for the site occupations. Because they are more informative, it is advisable to label the space groups with their full Hermann-Mauguin symbols. 

Inspection of Equation 1.23 and consideration of the properties of 3-y and 6-j symbols confirm that only even A--values contribute to crystal field splitting. Further, it indicates that mixing between levels belonging to different / multiplets can only occur if terms with k site symmetry of the lanthanide, in much the same way as discussed above for the Stevens formalism. 

The three symbols of an atomic position represent (1) the multiplicity of the position (ii) a letter that Identifies the position and (Hi) the site symmetry. (See International Tables for Crystallography, Vol. A, Reidel Publishing Co., 1983). 

The symmetry elements that intersect at a Wyckoff position determine its site symmetry. For example, a Wyckoff position that lies on the intersection of two mirror planes has mm2 (C2v) symmetry (Fig. 10.6(a)) while one that lies at the intersection of a mirror plane and a three-fold rotation axis along its normal has 3/m (Csh) symmetry (Fig. 10.6(b)). An atom lying on a general position has no symmetry other than a one-fold axis which is represented by the symbol 1 (Ci). 

Both are body-centered Bravais lattices and for both the site symmetry of the origin is identical with the short space group symbol. The body-center position is of the lowest multiplicity (two-fold) and highest symmetry, and thus is considered as the origin in the lA/mmm space group. However, in the tetragonal lattice, a = b c. Hence, the body center position is not an inversion center. It possesses four-fold rotational symmetry (the axis is parallel to c) with a perpendicular mirror plane and two additional perpendicular mirror planes that contain the rotation axis. 

For noncentrosymmetric space groups, the point with the highest site symmetry and lowest multiplicity is chosen as the origin. This means that either the Ti atom or the Ca atom could be chosen as the origin since they both have the same multiplicity (one-fold) and site symmetry m3m). However, it is important to note that the Ca and Ti atom are not at the same position. The Bravais lattice is primitive, as indicated by the space group symbol. Therefore, if we allow the Ti atom positions to coincide with lattice points, the Ca atoms and O atoms cannot. 

As well as the chemical composition, information about point defects, site symmetry, and site occupancy can be given by using additional symbols. These symbols may also be used to write quasi-chemical equilibria between point defects.6... 

If X = 1/4, the coordinates x, y and 1/2 - x, y are identical. They represent a point on the reflection line. The orbit only includes two objects per unit cell, but these are invariant with respect to reflection. The multiplicity is thus 2, but the site symmetry is m. The two periods (.) in the symbol m. mark the positions in the international symbol of the plane group (Section 2.7.1) which refer to the rotation point and the directions a and b . m. indicates that the reflection line is perpendicular to a. 

Examples of such results are illustrated by fig. 32 which at the same time demonstrates experimentally the good linearity between A i and Ny. All materials for which both A i and are available, whatever their site symmetry, are indicated by circled symbols. This figure illustrates a large scale of scalar crystal-field parameters of known materials the weakest is for NdCh, the strongest are for LaOF and YAG. 

The first two columns of the table contain the labels of the induced reps in the q-basis (these labels number the rows of the table) the international symbols (Roman letters a, b, c and so on) of the Wyckoff positions (sites in direct space) and the Mulliken symbols of the irrep>s of the site-symmetry groups for these Wyckoff positions. For example, d a2u) and d(e ) are the labels of induced reps in q-basis for space... 

The inversion notation has become standard for a number of good reasons. In the Hermann-Mauguin terminology, the center of symmetry is dropped and the inversion axes maintained. In the Schonflies terminology, the center of symmetry is a key element and all the mirror reflections and simple or inversion rotation axes are dropped and replaced by other symbols these are described in Section V, as they are important in relation to site symmetry, group theory, and Raman scattering. 

The elementary unit cell can be quite easily described starting from the four mineral ion sites of the crystal F, Ca f+, Ca(ll)2+ and P04 , where the symbols I and II represent the two different crystallographic sites of the cations, with the application of all the symmetry operations relevant to the space group P63/m. Among the principal symmetry elements, one can cite mirror planes perpendicular to the c-axis (at z = 1/4 and 3/4), which contain most of the ions of the structure (F , Ca +(ll), P04 ), three-fold axes parallel to the c-axis (at x = 1/3, y = 2/3 and x = 2/3, y = 1/3) along which are located the Ca + (I) ions, screw axes 63 at the corners of the unit cell and parallel to the c-axis and screw axes 2i parallel to the c-axis and located at the midpoints of the cell edges and at the centre of the unit cell itself [3]. 

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