Glide line

The symbols for plane groups, the Hermann-Mauguin symbol, have been the standard in crystallography. The first place indicates the type of lattice, p indicates primitive, and c indicates centered. The second place indicates the axial symmetry, which has only 5 possible vales, 1-, 2-, 3-, 4-, and 6-fold. For the rest, the letter m indicates a symmetry under a mirror reflection, and the letter g indicates a symmetry with respect to a glide line, that is, one-half of the unit vector translation followed by a mirror reflection. For example, the plane group pAmm means that the surface has fourfold symmetry as well as mirror reflection symmetries through both x and y axes. 

With class 6 we encounter a new type of translational symmetry, the glide reflection. This is a combination of reflection and translation by one half the unit translation. The glide line is represented by a broken (dashed) line, whereas the mirror is represented by a solid line. 

Continuing with the rectangular lattices, we can add a set of glide lines, thereby obtaining the symmetry pg. We can, further, introduce mutually perpendicular sets of glide lines, whereby the symmetry pgg is obtained. Note that a set of twofold axes arises between the glide lines. 

The diagrams depict the outline of a unit cell (light solid lines) and show within it all the symmetry elements that occur. The symbols used have all been introduced already, namely, those for the various axes perpendicular to the plane, the reflection lines (heavy solid lines), and glide lines (heavy broken lines). The set of diagrams is shown in Figure 11.8. We shall now discuss several of them. 

For pi we have only the outline of the cell since there are no symmetry elements. For p2 the twofold axes are shown. For pm we see the parallel reflection lines at the top and bottom edges and through the middle of the cell, while in pg we see a similar display of the glide lines. 

The diagrams for the remaining symmetries show clearly all of the symmetry elements in these more elaborate cases. For instance, in pAm we see the twofold axes that were required by the presence of the fourfold axes of p4 as well as the glide lines that arise automatically when the reflection lines are introduced. In p4g we see there are actually two networks of glide lines. 

In symmetry pgg, we see that the network of perpendicular glide lines gives rise to a set of twofold axes that lie in the interstices of the network, not at its intersections. To see how this happens, and that it is general, we may refer to Figure 11.9, which shows how an asymmetric object at an initial general... 

Figure 11.9. A diagram showing how an entire set of objects is generated from an initial one (No. 1) at a general position (jc, y) by the combined action of glide lines and the lattice translations.

We can now easily see that 1 and 4" are related also by twofold rotation about an axis at the origin, while 1 and 4 are related by an axis at a - A, b = 0, and 1 and 4 " by an axis at a = 0, b = i, and so forth. Thus, we see that a net of glide lines coupled with the basic translational symmetry generates an entire set of twofold axes, one in the center of each rectangle bounded by the glide lines. 

Example /. The net of vertical and horizontal reflection lines is rather obvious, as are the twofold axes. The fourfold axes may be slightly less obvious. Once they are found, however, it becomes clear that we need to turn the pattern 45° in order to put it into the standard orientation for one of the square symmetries, / 4, p4m, or p4g. Since we have seen the net of reflection lines we know it must be either p4m or / 4g, and when we note that the reflection lines pass between, not through, the fourfold axes we conclude that it is p4g. The presence of the two different nets of glide lines, only one net passing through the fourfold axes, is not obvious. The reader should convince himself that they are there. The second diagram in column C shows one example of each type of glide, g, takes brick AB to A B while g takes brick... 

Example 2. This example shows that the symmetry of the pattern can be less than that of the object used to create it. The blocks may be perfectly square, but the pattern does not have fourfold symmetry. Again, a net of reflection lines is perhaps most obvious and, of course, there are twofold axes at their intersections. If this were all, we would have pmm symmetry. However, it can be seen that the array is centered, and closer inspection will show that there are horizontal and vertical glide lines and an additional set of twofold axes, making this an example of cmm symmetry. 

Example 3. This is another example of p4g symmetry, as indicated by the diagrams in columns B and C. The pattern looks very different from that in Example 1, but its symmetry is exactly the same. The diagram in column C shows only the axes. Reflection lines and glide lines are also present. 

Glide-Reflection. This operation is referred to a symmetry element called a glide plane. We have already employed a glide line (its 2D equivalent) in developing the 2D space groups. 

On Pd(l 10) a c2 x 2 structure is formed at 0 — 0.5. The CO molecules are presumably located in fourfold sites as shown by Fig. 8a. Further adsorption leads to the formation of a 4 x 2 structure at 0 = 0.75 where the adsorbed particles form incoherent chains along the troughs of this plane (Fig. 8b). Saturation is characterized by a 2 x 1 structure in which some characteristic spots are missing (52), which was interpreted (99) as arising from the existence of glide lines within the overlayer arrangement. The result-... 

Both short and full symbols are used for the space groups. In the latter, both symmetry axes and symmetry planes for each symmetry direction are explicitly designated whereas in the former symmetry axes are suppressed. For example, the short symbol lA/mmm designates a body-centered tetragonal space lattice with three perpendicular mirror planes. One of these mirror planes is also perpendicular to the rotation axis, which is denoted by the slash between the 4 and the first m, while the other two mirror planes are parallel with, or contain, the rotation axis. The full symbol for this space group is lA/m Ijm Ijm 42/n 2i/n 2i/c, which reveals the additional presence of screw axes and a diagonal glide line. 

Figure 21. The Cu 100 -p(2x2)-p2gg monolayer alloy (a) top view illustrating the two orthogonal mirror planes (m) and glide lines (g) with the structure consisting of a double layer c(2x2) CuPd alloy with the p(2x2) periodicity introduced by clock rotation of the outermost CuPd monolayer (b) side view illustrating the major geometric parameters (c) motion of top layer Pd and Cu atoms within the clock rotation [160].

Figure 3.8 The plane groups pm and cm (a) the plane lattice op (b) the pattern formed by adding the motif of point group m to the lattice in (a), representative of plane group pm (b) the plane lattice oc (d) the pattern formed by adding the motif of plane group m to the lattice in (c), representative of plane group cm. Mirror lines are heavy, and glide lines in (d) are heavy dashed...

Each two-dimensional plane group is given a symbol that summarises the symmetry properties of the pattern. The symbols have a similar meaning to those of the point groups. The first letter gives the lattice type, primitive ip) or centred (c). A rotation axis, if present, is represented by a number, 1, (monad), 2, (diad), 3, (triad), 4, (tetrad) and 6, (hexad), and this is given second place in the symbol. Mirrors (m) or glide lines (g)... 

Figure 3.9 The glide operation (a) reflection across a mirror line (b) translation parallel to the mirror plane by a vector t, which is constrained to be equal to T/2, where T is the lattice repeat vector parallel to the glide line. The lattice unit cell is shaded...

The location of the symmetry elements within the unit cells of the plane groups is illustrated in Figure 3.10. Heavy lines represent mirror lines and heavy dashed lines represent glide lines. The unit cell has a light outline. 

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