Glide planes translation vectors

Atoms and molecules in solids arranged in a lattice can be related by four crystallographic symmetry operations - rotation, inversion, mirror, and translation - that give rise to symmetry elements. Symmetry elements include rotation axis, inversion center, mirror plane, translation vector, improper rotation axis, screw axis, and glide plane. The reader interested in symmetry and solving crystal stmctures from diffraction data is encouraged to refer to other sources (7-... 

Figure 4.1 Schematic dislocation line a simple cubic crystal structure. The line enters the crystal at the center of the left-front face. It emerges at the center of the right-front face. The shortest translation vector of the structure is the Burgers Vector, b. The line bounds the glided area of the glide plane (100) from the unglided area.

Fig. 229. Left When a crystal has a plane of symmetry normal to b, the distances of atoms from this plane are given by maxima along the line OyO of the vector cell. Right When there is a glide plane perpendicular to b, with translation c/2, the distances oi atoms from this plane are given by maxima along the line OyJ of the vector cell.

It turns out that three types of glide plane can be differentiated. In the first type, the translation is in the direction of a principal lattice vector, that is, it is given by one of the vectors a/2, b/2, or c/2. For each of these, the plane must be parallel to the plane defined by the translation direction and one of the other two principal directions. Thus, if we have a glide plane parallel to the plane of a and c, the glide component may be either a/2 or c/2. Planes of this type are called axial glide planes and are symbolized a, b> or c, according to the direction of the glide. 

Accordingly, glide planes are those planes which have the shortest b vectors a/2 <110> for fee, a/2 <111> for bcc, and a/3 <211.0) for hep lattices. Dislocations can split into so-called Shockley partials b = bx +b2, if b2>b +b. Since b and b2 are not translational vectors of the crystal lattice, they induce a stacking fault. The partial dislocation therefore bounds the stacking fault. 

In eq. (1), v is not necessarily a lattice translation t, since w may be either the null vector 0 or the particular non-lattice translation associated with some screw axis or glide plane. If v C a VR, then there are no screw axes or glide planes among the symmetry elements... 

Symmetry plane or symmetry line Graphic symbol Glide vector in units of lattice translation vectors parallel and normal to the projection plane Printed symbol... 

It is noteworthy to point out that two sequential screw-axis or glide-plane operations will yield the original object that has been translated along one of the unit cell vectors. For example, a 63 axis yields an identical orientation of the molecule only after 6 repeated applications - 4.5 unit cells away (i.e., 6 x 3/4 = 4.5). However, since glide planes feature a mirror plane prior to translation, the first operation will... 

The symmetry of the Patterson function is the same as the Laue symmetry of the crystal. The Patterson function for space groups that have symmetry operations with translational components (screw axes and glide planes) has an added property that is very useful for the determination of the coordinates of heavy atoms. Specific peaks, first described b David Harker, are associated with the vectors between atoms related by these symmetry operators. These peaks are found along lines or sections (Figure 8.17). For example, in the space group P2i2i2i there are atoms at... 

Symmetry operations, therefore, can be visualized by means of certain symmetry elements represented by various graphical objects. There are four so-called simple symmetry elements a point to visualize inversion, a line for rotation, a plane for reflection and the already mentioned translation is also a simple symmetry element, which can be visualized as a vector. Simple symmetry elements may be combined with one another producing complex symmetry elements that include roto-inversion axes, screw axes and glide planes. 

Glide planes a, b and c, which after reflecting in the plane translate an object by 1/2 of the length of a, b and c basis vectors, respectively. 

Because of this, for example, glide plane, a, can be perpendicular to either b or c, but it cannot be perpendicular to a. Similarly, glide plane, b, cannot be perpendicular to b, and glide plane, c, cannot be perpendicular to c. Since the translation is always by 1/2 of the corresponding basis vector, these planes produce two symmetrically equivalent objects within one full length of the corresponding basis vector (and within one unit cell), i.e. their order is 2. 

Infinite symmetry elements interact with one another and produce new symmetry elements, just as finite symmetry elements do. Moreover, the presence of the symmetry element with a translational component (screw axis or glide plane) assumes the presence of the full translation vector as seen in Figure 1.28 and Figure 1.29. Unlike finite symmetry, symmetry elements in a continuous space (lattice) do not have to cross in one point, although they may have a common point or a line. For example, two planes can be parallel to one another. In this case, the resulting third symmetry element is a translation vector perpendicular to the planes with translation (t) twice the length of the interplanar distance d) as illustrated in Figure 1.30. 

It is practically obvious that simultaneously or separately acting rotations (either proper or improper) and translations, which portray all finite and infinite symmetry elements, i.e. rotation, roto-inversion and screw axes, glide planes or simple translations can be described using the combined transformations of vectors as defined by Eqs. 1.38 and 1.39. When finite symmetry elements intersect with the origin of coordinates the respective translational part in Eqs. 1.38 and 1.39 is 0, 0, 0 and when the symmetry operation is a simple translation, the corresponding rotational part becomes unity, E, where... 

There are parallels between the two-and three-dimensional cases. Naturally, mirror lines in two dimensions become mirror planes, and glide lines in two dimensions become glide planes. The glide translation vector, t, is constrained to be equal to half of the relevant lattice vector, T, for the same reason that the two-dimensional glide vector is half of a lattice translation (Chapter 3). 

The translation vector associated with a glide plane is ... 

The two types of symmetry transformation considered thus far are the only ones, aside from translations, that occur in a symmorphic space group (composed of rotation and reflection operations). Most molecular crystals, however, belong to nonsymmorphic space groups which contain screw axes and glide planes in addition to pure proper and improper rotations. The space groups C (naphthalene, anthracene) and 75 (a-N2, CO2) are both examples. For a twofold screw axis operation (e.g., axis parallel to ) the rotation is accompanied by a translation composed of half unit cell vectors, (e.g., x + y). Application of such an operation maps one pair of molecules on another pair, neither of them remaining the same ... 

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