Earth Materials. John O'Brien

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Earth Materials - John  O'Brien


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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).

Schematic illustration of 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.

Schematic illustration of (a) Mirror plane (m) with the translation vector (t), contrasted with a glide plane (g) with the translation vector (t/2) combined with mirror reflection.

      4.2.2 Compound symmetry operations

      Screw rotation (na) 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).

Image described by caption.

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

      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

      4.3.2 Plane lattices and unit meshes

Schematic illustration of the 10 plane point groups defined by rotational and reflection symmetry.

      Source:


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