Crystallographic space groups, list

Between classes a and / there has been found a true crystallographic distinction as will be noted on consideration of the space groups listed in Table 1. The non-phototropic anils occur in lattices which are centro-symmetric, whereas the lattices of the phototropic anils have no centre of symmetry. 

Table 1.17. The 230 crystallographic space groups arranged according to seven crystal systems and 32 crystallographic point groups as they are listed in the International Tables for Crystallography, vol. A. The centrosymmetric groups are in bold, while the non-centrosymmetric groups that do not invert an object are in italic. The remaining are non-...

An example of how each of the 230 three-dimensional crystallographic space groups is listed in the International Tables for Crystallography is shown in Table 1.18. Explanations of what information is found in different fields (the fields are numbered in the first column) follows. 

As briefly mentioned in the previous section, equivalent positions (or sites) that are listed in the field No. 8 in Table 1.18 for each crystallographic space group, represent sets of symmetrically equivalent points found in one unit cell. All equivalent points in one site are obtained from an initial point by applying all symmetry operations that are present in the unit cell. The fractional coordinates (coordinate triplet) of the initial (or independent) point are usually marked as x, y, z. 

In the previous section the 230 crystallographic space groups, with the symbols listed in Table 2.16, have been introduced. Yet, one should be aware that in the symbol of a space group are included the minimum symmetries necessary to deduce the other ones and does not reflect all the symmetries that a space group involves. Here we analytically unfold such reality the present discussion follows (Chiriac-Putz-Chiriac, 2005). 

Table 1-4 lists the point symmetry elements and the corresponding symmetry operations. The notation used by spectroscopists and chemists, and used here, is the so-called Schoenflies system, which deals only with point groups. Crystallographers generally use the Hermann-Mauguin system, which applies to both point and space groups. 

The summation over 5 (n = 2 in the case of the example) indicates the importance of orientation in the unit cell on the magnitude of x Even large values of P in noncentrosymmetric space groups can result in small values of x depending on the details of orientation. Oudar and Zyss (25, 26) have analyzed the orientational dependence of molecules in all the polar crystallographic point groups. The maximum value of x that can be obtained for a properly phase matched interaction for optimal molecular orientation in that space group is listed in Table 6.3. The values are relative to an... 

Table 1.3 List of the simpler three-periodic minimal surfaces (IPMS), together with their crystallographic symmetries. Those surfaces that carve space into two interpenetrating open labyrinths are marked with a tick, a cross denotes IFMS that are self-intersecting. In most cases, two space groups are listed for each IPMS, the first is that of the surface ctssuming both sides are equivalent, the second is the s)munetry displayed by the surface assuming inequivalent sides.

Information that is put into this Database is derived from published reports of crystal structure determinations. The data extracted from the scientific literature in this way include the atomic coordinates, information on the space group, chemical connectivity, and the literature reference to each structure determination. Each compound listed in the Database is identified by a six-letter code (the REFCODE), unique to each crystal structure determination. Duplicate structures and remeasurements of the same crystal structure are identified by an additional two digits after the REFCODE. Scientific journals are scanned regularly by the Database staff for reports of crystal structure determinations, and the data are then entered into this Database. Structural data are also deposited by journals, for example. Chemical Communications, that publish articles, but do not have space for atomic coordinates. All crystallographic data reported in the literature are tested by the Database staff for internal consistency, precision, and chemical reasonableness. In... 

Donnay, J. D. H. (ed.). Crystal Data. Washington American Crystallographic Association (Monograph 5). 2nd ed. 1963. A list of crystalline substances classified in terms of space group and cell dimensions. 

The crystal structures of selenium dithiocyanate 181) and selenium diselenocyanate 6) have been determined by X-ray diffraction. The crystals of these compounds and of sulfur dithiocyanate 94) are isomorphous, and the three structures are accordingly analogous. The space group is Dihlt-Pnma with four molecules per unit cell, of dimensions as listed in Table I. A mirror plane of molecular symmetry is crystallographically... 

The description of the symmetry elements of the space groups is similar to that of the point groups [9-19]. The main difference is that the order in which the symmetry elements of the space groups are listed may be of great importance, except for the triclinic system. The order of the symmetry elements expresses their relative orientation in space with respect to the three crystallographic axes. For the monoclinic system, the unique axis may be the c or the h axis. For the P2 space group, the complete symbol may be P112 or... 

A unit cell reflects the symmetry of the crystal structure. Thus, an atom at a position (x, y, z) in a unit cell may require the presence of atoms at other positions in order to satisfy the symmetry of the structure. For example, a unit cell with a centre of symmetry will, of necessity, require that an atom at (x,y,z) be paired with an atom at (—x, —y, —z). To avoid long repetitive lists of atom positions in complex structures, crystallographic descriptions usually list only the minimum number of atomic positions which, when combined with the symmetry of the structure, given as the space group, generate all the atom positions in the unit cell. Additionally, the Bravais lattice type and the motif are often specified as well as the number of formula units in the unit cell, written as Z. Thus, in the unit cell of rutile, given above, Z = 2. This means that there are... 

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