Space group symmetries Crystallographic symmetry

Table 17.1 Crystallographic data of the hexagonal and cubic closest-packings of spheres. +F means +(j,0), +(j,0, j), +(0, j, j) (face centering). Values given as 0 or fractional numbers are fixed by space-group symmetry (special positions)...

Electron Diffraction (CBED) and Large-Angle Convergent-Beam Electron Diffraction (LACBED) allow the identification of the crystal system, the Bravais lattice and the point and space groups. These crystallographic features are obtained at microscopic and nanoscopic scales from the observation of symmetry elements present on electron diffraction patterns. 

Compounds 39, 42 and 44 exist as discrete monomeric molecules. The Zn—C bond distances are close to the values of simple diafkylzinc compounds obtained from gas-phase electron diffraction studies and are also close to the values predicted by DFT computational studies . Because the zinc atoms in 39, 42 and 44 are located at special positions in the crystallographic unit ceU (at 0.25, 0.25, 0.5 in 39, and at 0, 0, 0 in both 42 and 44) the C—Zn—C bond angles are by definition 180° as a consequence of space-group symmetry. Compound 43 forms dimers in the solid state via hydrogen bridges between... 

By far the most insidious source of error, and also a common one, in making inferences about molecular symmetry from space group symmetry is the occurrence of disorder. This usually takes the form of what we may call a systematic disorder, as opposed to a random one. A molecule, or a component of the formula unit, will lie on a crystallographic symmetry element that would seem to require more symmetry of it than it is capable of having. It accomodates to this by systematically taking each of two (or more) orientations an equal number of times in different unit cells. 

International Tables for X-Ray Crystallography, Vol. 1, Symmetry Groups, N.F.M. Henry and K. Lonsdale, Eds, International Union of Crystallography, Kynoch Press, Birmingham, 1952. The complete source for space groups and crystallographic information. 

To describe the contents of a unit cell, it is sufficient to specify the coordinates of only one atom in each equivalent set of atoms, since the other atomic positions in the set are readily deduced from space group symmetry. The collection of symmetry-independent atoms in the unit cell is called the asymmetric unit of the crystal structure. In the International Tables, a portion of the unit cell (and hence its contents) is designated as the asymmetric unit. For instance, in space group P2 /c, a quarter of the unit cell within the boundaries 0asymmetric unit. Note that the asymmetric unit may be chosen in different ways in practice, it is preferable to choose independent atoms that are connected to form a complete molecule or a molecular fragment. It is also advisable, whenever possible, to take atoms whose fractional coordinates are positive and lie within or close to the octant 0 < x < 1/2,0 < y < 1/2, and 0 < z < 1 /2. Note also that if a molecule constitutes the asymmetric unit, its component atoms may be related by non-crystallographic symmetry. In other words, the symmetry of the site at which the molecule is located may be a subgroup of the idealized molecular point group. 

Asymmetric unit The smallest part of a crystal structure from which the complete structure can be derived by use of the space-group symmetry operations (including translations). The asymmetric unit may consist of only one molecule or ion, part of a molecule, or of several molecules that are not related by crystallographic symmetry. 

By repeating the same process in combination with the base-centered lattice, C, two new space groups symmetry, Cm and Cc can be obtained. Therefore, the following four monoclinic crystallographic space groups... 

We might note in passing that were the Fourier equation applied to asymmetric units related by space group symmetry in a crystallographic unit cell, the expressions for symmetry equivalent atomic positions assume considerable value. Their application can reduce the number of terms in the summation by the number of symmetry equivalent positions. We need, in practice, to consider only the atoms comprising a single asymmetric unit in the actual calculations. 

Assume, however, that the asymmetric unit is composed of two or more identical copies of a molecule. These may be related by exact rotational symmetry, such as a dyad or triad, that is not coincident with any crystallographic operator and, hence, is not a part of the space group symmetry. The molecules may also be related by some completely general rotation plus translation. The asymmetric unit will give rise, in either case, to a continuous transform that is essentially a superposition of the transforms of the two independent molecules. This is illustrated schematically in Figure 8.9. The two molecular transforms are identical because the two molecular structures are the same, but the two transforms will be rotated relative to... 

A crystallographic space group describes the symmetry of a three-dimensional repeating pattern, such as found in a crystal. Each space group is thus a collection of symmetry operators. There are 230 space groups, which can be derived by... 

Taking into account submultiples of the interplanar spacing diiki due to space group symmetry, the most important (MI) crystallographic forms (hkl) will have the greatest interplanar spacings. Hence, if there are two forms (/ii,ki,/,) and (h2, k2, h) and... 

Having more than one molecule in the asymmetric unit occurs predominantly in space groups of low symmetry like P or P2i. In most of these cases the two (or more) independent molecules are not related by simple symmetry operators such as twofold axes, mirror planes or inversion centres, but are different rotamers of the same molecule. Those cases are not what this chapter is about. This chapter deals with structures where there is in fact non-crystallographic symmetry to be found, relating two or more crystallographically independent molecules. 

When the symmetry elements are introduced into the possible lattices, the symmorphic elements combine to yield a total of 32 so-called point groups, while inclusion of symmorphic and nonsymmorphic elements leads to the 230 crystallographic space groups. Space groups, symmetry elements, and other aspects of crystal symmetry are tabulated in great detail in the International Tables for Crystallography. ... 

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