Advanced Certificate in Powder Diffraction on the Web Chapter 5 School of Crystallography, Birkbeck College, University of London

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Aims & Objectives

Why do we need to know about symmetry? Molecules often interact symmetrically, producing crystals which intrinsically possess symmetry. The technique of powder diffraction exploits diffraction of the tiny crystallites in a powder in a variety of ways, some of which require a detailed knowledge of symmetry to refine or determine a crystal structure. Even a subject such as indexing requires some knowledge of symmetry. So there is no escaping from it.

Text books often deal with this subject in a way far removed from practical use. It is also a difficult subject. There is no escaping that. This section describes the 3-D symmetry elements with a minimum amount of mathematics, and it does not strive to be exhaustive. Concepts are important here, and it may take a while for the concepts to sink in. This is quite normal, so do not worry if you cannot pick up all the concepts on the first read through of the material. Please do not hesitate to ask questions, either via the the e-mail list, or privately.

Specific objectives are to provide an understanding of the following 3-D symmetry elements:

• Rotational Symmetry

  The first of the 3-D symmetry elements to be discussed in detail is the rotational axis, which uses the highly-symmetric benzene molecule as an illustration. The concept of symmetry operators which relate the coordinates of symmetry equivalent atoms within a crystal structure is then discussed.

• Rotary-Inversion Symmetry

  You will learn here about the concepts of inversion symmetry, mirror symmetry, and rotary-inversion symmetry in general. Note that the high-order rotary-inversion axes are considered reference material (as indicated by the open book icon). This doesn't mean that you should ignore them, but that we don't expect you to develop such a comprehensive knowledge of them during this course!

• Translational Symmetry

  Translational symmetry is now brought in to deal with the extra features resulting from the lattice. This conceptually relates to diffraction from lattices. Note that concepts are important here and mostly not the details in the tables: in the crystal system table and Bravais lattice table only the triclinic, monoclinic and orthorhombic crystal systems will be studied in depth later in the course material. However, you should be aware of the existence of all of the high-symmetry crystal systems, and, in particular, the properties of the tetragonal and cubic lattices.

• Screw Symmetry

  Although a gross simplification again, the combination of rotational and translational symmetry results in screw symmetry. Note that the concepts are important here and not the detail. This means that the helical screw axes table is present for reference only!

• Glide Symmetry

  You need to understand the concepts here: you can let the details of the notation sink in later when studying the section on space group symmetry.

• Symmetry Symbols

  The majority of this table is for reference only! During the course we will be using a very small selection of these symbols as shown below. However, you should be aware of the existence of the others.

We hope that this approach will be a major aid in your understanding of the subject.

Assignments

We shall shortly be setting some assignments on the material covered so far. Further instructions concerning these will follow. The assignments are quite separate from questions posed in the course material itself. The latter, indicated by the light bulb icon, are designed to stimulate your thoughts on the subject.

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