Earth Materials. John O'Brien

of the simple symmetry operations to visualize. Inversion involves the repetition of motifs by inversion through a point called a center of inversion (i). Inversion occurs when every component of a pattern is repeated by equidistant projection through a common point or center of inversion. The two “letters” in Figure 4.5 illustrates this enantiomorphic symmetry operation and shows the center through which inversion occurs. In some symbolic notations centers of inversion are symbolized by (c) rather than (i).

Figure 4.5 Inversion through a center of symmetry (i) illustrated by the letter “m” repeated by inversion through a center (inversion point).

      One test for the existence of a center of symmetry is that all the components of a pattern are repeated along straight lines that pass through a common center and are repeated at equal distances from that center. If this is not the case, the pattern does not possess a center of symmetry.

Figure 4.6 (a) Mirror plane (m) with the translation vector (t), contrasted with (b) a glide plane (g) with the translation vector (t/2) combined with mirror reflection.

      4.2.2 Compound symmetry operations

Three other symmetry operations exist. However, unlike those discussed so far, they are compound symmetry operations that combine two simple symmetry operations. Glide reflection (g) is a symmetry operation that combines translation (t) parallel to a mirror plane (m) with reflection across the mirror plane to produce a glide plane (Figure 4.6).

Rotoinversion (n^–) is an operation that combines rotation about an axis with inversion through a center to produce an axis of rotoinversion (Figure 4.7). Figure 4.7b illustrates an axis of fourfold rotoinversion ( ) in which the motif is repeated after 90° rotation followed by inversion through a center so that it is repeated four times by rotoinversion during a 360° rotation. Axes of twofold rotoinversion ( ) are unique symmetry operations, whereas axes of threefold rotoinversion ( ) are equivalent to a threefold axis of rotation and a center of inversion (3i), and axes of sixfold rotoinversion ( ) are equivalent to a threefold axis of rotation perpendicular to a mirror plane (3/m).

      Screw rotation (n_a) is a symmetry operation that combines translation parallel to an axis with rotation about the axis. This is similar to what occurs when a screw is inserted into a wall (Figure 4.7c). Much more detailed treatments of the various types of compound symmetry operations can be found in Klein and Dutrow (2007), Wenk and Bulakh (2016) or Nesse (2016).

Figure 4.7 (a) An axis of fourfold rotation (4). This contrasts with (b) an axis of fourfold rotoinversion () that combines rotation with inversion every 90°, and (c) a fourfold screw axis (4₁) that combines translation with 90° rotations every one‐fourth translation.

      Source: Wenk and Bulakh (2004). © Cambridge University Press.

4.3 TWO‐DIMENSIONAL MOTIFS AND LATTICES (MESHES)

      The symmetry of three‐dimensional crystals can be quite complex. Understanding symmetry in two dimensions provides an excellent basis for understanding the increased complexity that characterizes three‐dimensional symmetries. It also provides a basis for learning to visualize planes of constituents within three‐dimensional crystals. Being able to visualize and reference lattice planes is of the utmost importance in describing cleavage and crystal faces and in the identification of minerals by X‐ray diffraction methods.

      4.3.1 Plane point groups

As discussed earlier, any fundamental unit of two‐dimensional pattern, or motif, can be repeated by various symmetry operations to produce a larger two‐dimensional pattern. All two‐dimensional motifs that are consistent with the generation of long‐range two‐dimensional arrays can be assigned to one of ten plane point groups based on their unique plane point group symmetry (Figure 4.8). Using the symbolic language discussed in the previous section on symmetry, the ten plane point groups are 1, 2, 3, 4, 6, m, 2mm, 3m, 4mm, and 6mm. The numbers refer to axes of rotation that are perpendicular to the plane (or page); the m refers to mirror planes perpendicular to the page. The first m refers to a set of mirror planes that is repeated by the rotational symmetry and the second m to a set of mirror planes that bisects the first set. Note that the total number of mirror planes is the same as the number associated with its rotational axis (e.g., 3m has three mirror planes and 6mm has six mirror planes).

      4.3.2 Plane lattices and unit meshes

As discussed previously, any motif can be represented by a point called a node. Points or nodes can be translated some distance in one direction by a unit translation vector t_a or t₁ to produce a line of nodes or motifs. Nodes can also be translated some distances in two directions t_a and t_b or t₁ and t₂ to produce a two‐dimensional array of points called a plane mesh or plane net. Simple translation of nodes in two directions produces five basic types of two‐dimensional patterns (Figure 4.9). The smallest units of such meshes, which contain the unit translation vectors and at least one node, are called unit meshes (unit nets). Unit meshes contain all the information necessary to produce the larger two‐dimensional pattern. They contain only translation symmetry information. The five basic types of unit mesh are classified on the basis of (1) the unit translation vector lengths (equal or unequal), the angles between them (90°, 60°, and 120° or none of these) and (2) whether they have nodes only at the corners (primitive = p) or have an additional node in the center (c) of the mesh.

Figure 4.8 The 10 plane point groups defined by rotational and reflection symmetry.

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