Symmetry space group examples

In this example the two complexes have high internal symmetry and this symmetry allows a high-symmetry space group to be adopted. Complexes of lower symmetry necessarily crystallize in a space group of lower symmetry even though the underlying lattice may still be the same. 

It should be noted that, whereas ferroelectrics are necessarily piezoelectrics, the converse need not apply. The necessary condition for a crystal to be piezoelectric is that it must lack a centre of inversion symmetry. Of the 32 point groups, 20 qualify for piezoelectricity on this criterion, but for ferroelectric behaviour a further criterion is required (the possession of a single non-equivalent direction) and only 10 space groups meet this additional requirement. An example of a crystal that is piezoelectric but not ferroelectric is quartz, and ind this is a particularly important example since the use of quartz for oscillator stabilization has permitted the development of extremely accurate clocks (I in 10 ) and has also made possible the whole of modern radio and television broadcasting including mobile radio communications with aircraft and ground vehicles. 

Symmetry axes can only have the multiplicities 1,2,3,4 or 6 when translational symmetry is present in three dimensions. If, for example, fivefold axes were present in one direction, the unit cell would have to be a pentagonal prism space cannot be filled, free of voids, with prisms of this kind. Due to the restriction to certain multiplicities, symmetry operations can only be combined in a finite number of ways in the presence of three-dimensional translational symmetry. The 230 possibilities are called space-group types (often, not quite correctly, called the 230 space groups). 

A suitable way to represent group-subgroup relations is by means of family trees which show the relations from space groups to their maximal subgroups by arrows pointing downwards. In the middle of each arrow the kind of the relation and the index of the symmetry reduction are labeled, for example ... 

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. 

The occurrence of twinned crystals is a widespread phenomenon. They may consist of individuals that can be depicted macroscopically as in the case of the dovetail twins of gypsum, where the two components are mirror-inverted (Fig. 18.8). There may also be numerous alternating components which sometimes cause a streaky appearance of the crystals (polysynthetic twin). One of the twin components is converted to the other by some symmetry operation (twinning operation), for example by a reflection in the case of the dovetail twins. Another example is the Dauphine twins of quartz which are intercon-verted by a twofold rotation axis (Fig. 18.8). Threefold or fourfold axes can also occur as symmetry elements between the components the domains then have three or four orientations. The twinning operation is not a symmetry operation of the space group of the structure, but it must be compatible with the given structural facts. 

In the preceding paragraphs examples of a number of so-called superstructures have been considered. Generally, it has been observed that a derivative structure has fewer symmetry operations than the reference structure it has either a larger cell or a lower symmetry (or both) than the reference structure. Typically the passage from the reference structure to the derivative structure (superstructure) may be related to the fact that a set of equipoints of a certain structure (the reference one) has to be subdivided into two (or more) subsets in order to obtain the description of the other structure. The structure of the Cu type (cF4 type), for instance, corresponds to 4 Cu atoms in the unit cell, placed in 0, 0, 0 14, 14, 0 14, 0, 14 0, 14, 14, whereas in the cP4-AuCu3 type structure the same atomic sites are subdivided, in another space group, into two sets with an ordered distribution of the two atomic species (1 Au atom in 0, 0, 0 and 3 Cu atoms in 14, 14, 0 14, 0,14 0,14,14). 

A more complex example may be represented by TaSe2 its modification called 2H-TaSe2 is hexagonal (space group P6 mmc, with two formula units in the unit cell). This layered compound shows a displacive 2D modulation (defined by two vectors) its symmetry may be therefore described in terms of a supergroup in a 5D superspace (Janner and Janssen 1980). A general point is therefore denoted by the 5 parameters x,y,z, t, u, and a position vector by the five components xa + yb + zc + td + ue of the superspace, with a, b, c basis in the position space and d and e in the internal subspace . 

To the extent that a crystal is a perfectly ordered structure, the specificity of a reaction therein is determined by the crystallographic symmetry. A crystal is built up by repeated translations, in three dimensions, of the contents of the unit cell. However, the space group usually contains elements additional to the pure translations, such as a center of inversion, rotation axis, and mirror plane. These elements can interrelate molecules within the unit cell. The smallest structural unit that can develop the whole crystal on repeated applications of all operations of the space group is called the asymmetric unit. This unit can consist of a fraction of a molecule, sometimes fractions of two or more molecules, a single whole molecule, or more than one molecule. If, for example, a molecule lies on a crystallographic center of inversion, the asymmetric unit will contain half... 

An example of the second effect is provided by mono-sec-butyl phthalamide (17a) (56). In the crystal the two enantiomers of this molecule are miscible in all proportions. The racemate crystallizes in space group PI (two general positions in the unit cell) with four molecules per unit cell. Thus there are two molecules in the asymmetric unit. The sec-butyl moieties adopt the anti conformation (the two geometries are shown schematically in 17b) and exhibit conformational disorder to different extents at the two symmetry-independent sites. 

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