The space group of a crystal

The space group G of a crystal is the set of all symmetry operators that leave the appearance of the crystal pattern unchanged from what it was before the operation. The most general kind of space-group operator (called a Seitz operator) consists of a point operator R (that is, a proper or improper rotation that leaves at least one point invariant) followed by a translation v. For historical reasons the Seitz operator is usually written R v. However, we shall write it as (R ) to simplify the notation for sets of space-group operators. When a space-group operator acts on a position vector r, the vector is transformed into 

In each of the equations (l)-(4) the crystal pattern appears the same after carrying out the operation signified. It follows from eq. (2) that the pattern, and therefore the subset of lattice translations 

When a Seitz operator acts on configuration space all functions defined in that space are transformed, and the rule for carrying out this transformation is the same as that for rotation 

Equation (11) shows that the set of lattice translations T form an Abelian subgroup of G. Moreover, T is an invariant subgroup of G, since 

We now remove the inconvenience of the translation subgroup, and consequently the Bravais lattice, being infinite by supposing that the crystal is a parallelepiped of sides Aja,-where ay, j 1,2,3, are the fundamental translations. The number of lattice points, N1N2N3, is equal to the number of unit cells in the crystal, N. To eliminate surface effects we imagine the crystal to be one of an infinite number of replicas, which together constitute an infinite system. Then 

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Any symmetry in the packing of objects is related (in a reciprocal way) to symmetry in its diffraction pattern, and this symmetry in the diffraction pattern can be used to determine the crystal symmetry (see Tables 4.2 and 4.3). This is of great importance to the X-ray crystallo-grapher because this is the way the space group of a crystal is determined. 

Is there any advantage in choosing the Lane method to determine the space group of a crystal ... 

The space group of a crystal structure can be considered as the set of all the symmetry operations which leave the structure invariant. All the elements (symmetry operations) of this set satisfy the characteristics of a group and their number (order) is infinite. Of course, this definition is only valid for an ideai structure extending to infinity. For practical purpose, however, it can be applied to the finite size of real crystals. Lattice translations, proper or improper rotations with or without screw or gliding components are all examples of symmetry operations. 

Conclusion The space group of a crystal is the collection of symmetry elements (macroscopic and microscopic) which, when considered to be distributed in space according to the Bravais Lattice, provides knowledge of total symmetry present in the crystal amongst the different array of atoms or molecules within it. 

Crystal lattices can be depicted not only by the lattice translation defined in Eq. (7.2), but also by the performance of various point symmetry operations. A symmetry operation is defined as an operation that moves the system into a new configuration that is equivalent to and indistinguishable from the original one. A symmetry element is a point, line, or plane with respect to which a symmetry operation is performed. The complete ensemble of symmetry operations that define the spatial properties of a molecule or its crystal are referred to as its group. In addition to the fundamental symmetry operations associated with molecular species that define the point group of the molecule, there are additional symmetry operations necessary to define the space group of its crystal. These will only be briefly outlined here, but additional information on molecular symmetry [10] and solid-state symmetry [11] is available. 

Perhaps even more interesting is the behavior of 2-benzyl-5-p-bromobenzylidenecyclopentanone (124, with X = 4-Br, Y = Z = H). Although, on reaction, the space group of this crystal remains unaltered, there are appreciable changes in the cell dimensions. Thus, a changes by —3.77%, b by - 5.61%, and c by 6.52%. The reaction normally involves fracture of the crystal, but can be induced to proceed homogeneously by careful control of such reaction conditions as temperature and rate of conversion. 

Molecular or ionic symmetry. If the space-group of a particular crystal has been determined unequivocally, this knowledge may make it possible to draw certain definite conclusions about the symmetry of the molecules or ions of which the crystal is composed—and this without any attempt to discover the positions of individual atoms. 

A truly radical improvement was the development of quartz crystals, used in high-Q resonators. The vibrations of a quartz timing force cantilever can be transformed into a voltage across the faces of the crystal, by the piezoelectric effect The space group of a-quartz is P3221, and a voltage... 

K-M and Kossel patterns are extremely sensitive to crystal orientation, and are therefore particularly useful for tilting a crystal into a precise orientation. However, their most important use is in obtaining three-dimensional crystal symmetry information, including the complete determination of the point group and space group of a crystal. Extensive reviews of... 

The evidence for the existence of screw axis symmetry is manifested in certain subclasses of reflections that are systematically absent. These systematic absences, we will see, fall along axial lines (/tOO, OkO, 00/) in reciprocal space and clearly signal not only whether an axis in real space is a screw axis or a pure rotation axis, but what kind of a screw axis it is, for instance, 4i or 42, 6i or 63. Thus the inherent symmetry of the diffraction pattern (which we call the Laue group), plus the systematic absences, allow us to unambiguously identify (except for a few odd cases) the space group of any crystal. 

FIGURE 9.2 A section from a difference Patterson map calculated between a heavy atom derivative and native diffraction data (known as a difference Patterson map). This map is for a mercury derivative of a crystal of bacterial xylanase. The plane of Patterson density shown here corresponds to all values of u and w for which v =. Because the space group of this crystal is P2, this section of the Patterson map is a Harker section containing peaks denoting vectors between 2t symmetry related heavy atoms. 

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