Translational symmetry elements

The 230 three-dimensional space groups are combinations of rotational and translational symmetry elements. A symmetry operation S transforms a vector r into r ... 

TABLE 2.3 Systematic absences due to translational symmetry elements... 

So far, we have seen that if we measure the Bragg angle of the reflections and successfully index them, then we get information on the size of the unit cell and, if it possesses any translational symmetry elements, also on the symmetry. In addition, we have seen that the intensity of each reflection is different and this too can be measured. In early photographic work, the relative intensities of the spots on the film were assessed by eye with reference to a standard, and later a scanning microdensitometer was used. In modern diffractometers, the beam is intercepted by a detector, either a charge coupled device (CCD) plate or a scintillation counter, and the intensity of each reflection is recorded electronically. 

The 32 crystallographic point groups result from combinations of symmetry based on a fixed point. These symmetry elements can be combined with the two translational symmetry elements the screw... 

Deduction of lattice centering and translational symmetry elements from systemic absences... 

Systematic absences (or extinctions) in the X-ray diffraction pattern of a single crystal are caused by the presence of lattice centering and translational symmetry elements, namely screw axes and glide planes. Such extinctions are extremely useful in deducing the space group of an unknown crystal. 

Symmetry operations which involve shifts, can apply only to regularly repeating infinite patterns, like crystal structmes. A repeated application of such an operation brings the structme not to the original position, but to a different one, separated from the original by an integer number of lattice translations. There are two types of such ( translational ) symmetry elements (see Table 2), besides primitive lattice translations a, b, c. 

Figure 2.34. Illustration of translational symmetry elements. Shown is (a) screw axis and (b) glide plane.

The symmetry of the crystal is indicated in its diffraction pattern. Systematic absences in the diffraction pattern show that there are translational symmetry elements relating components in the unit cell. The translational component of the symmetry elements causes selective and predictable destructive interference to occur when the specific translation in the arrangement of atoms are simple fractions of the normal lattice... 

The ideal crystal is a rigid, three-dimensional array of molecules extending infinitely in all directions. This is the model used to evaluate the symmetry of a group of real atoms. The infinite extent of this array allows us to add new symmetry operations to our list of point group symmetry elements (Section 6.1). Previously, we counted only operations that leave the center of mass unchanged. However, the center of mass is not defined for an infinite number of atoms, so we can ignore that constraint now by adding translational symmetry elements to the list. 

Space group 230 Combines the point group symmetry with the translational symmetry elements Normally needs to be determined in order to solve the crystal structure from a diffraction pattern... 

Translational symmetry elements are symmetry operators that shift the entire function a fixed amount along a specified axis. 

Space groups are groups that combine certain point group operations, translational symmetry elements, and their products. 

Some of the systematic absences due to the presence of translational symmetry elements may be summarized as follows ... 

---