Space Group Patterns - M.W.Meier.pdf - KRYSTALOGRAFIA - morticore

Space Group Patterns

W. M. Meier

This electronic edition may be freely copied and

redistributed for educational or research purposes

only.

It may not be sold for profit nor incorporated in any product sold for profit

without the express pernfission of The Executive Secretary, International

l,;nion of Crystalk~graphy, 2 Abbey Square, Chester CIII 211U, UK

Crystallography

Published for the

Intemati.onal Union of Crystallography

University College Cardiff Press

Cardiff, Wales

Published by the University College Cardiff Press for the

International Union of Crystallography with the

financial assistance of Unesco Contract No. SC/RP 250.271

This pamphlet is one of a series prepared by the

Commission on Crystallographic Teaching of the

International Union of Crystallography, under the

General Editorship of Professor C. A. Taylor.

Copies of this pamphlet and other pamphlets in

the series may be ordered direct from the

University College Cardiff Press,

P.O. Box 78, Cardiff

CFI IXL, U.K.

ISBN 0 906449 18 9

Printed by J. W. Arrowsmith Ltd., Bristol

Series Preface

The long-term aim of the Commission on Crystallographic Teaching in

establishing this pamphlet programme is to produce a large collection of

short statements each dealing with a specific topic at a specific level. The

emphasis is on a particular teaching approach and there may well, in time,

be pamphlets giving alternative teaching approaches to the same topic. It

is not the function of the Commission to decide on the 'best' approach but

to make all available so that teachers can make their own selection. Similarly,

in due course, we hope that the same topics will be covered at more than

one level.

The first set of ten pamphlets, published in 1981, and this second set of

nine represent a sample of the various levels and approaches and it is hoped

that they will stimulate many more people to contribute to this scheme. It

does not take very long to write a short pamphlet, but its value to someone

teaching a topic for the first time can be very great.

Each pamphlet is prefaced by a statement of aims, level, necessary

background, etc.

C. A. Taylor

Editor for the Commission

The financial assistance of UNESCO, ICSU and of the International Union of Crystal-

lography in publishing the pamphlets is gratefullyacknowledged.

Teaching Aims

For teaching students how to recognize a space group and how to tackle

related problems arising in practice (such as different settings, subgroup

relations).

Level

Should be useful in crystallography courses at an elementary or inter-

mediate level, particularly when the emphasis is on practical rather than

mathematical aspects.

Background

Some knowledge of the elements of crystallography is assumed. The topic

is best preceded by an exercise on plane groups using readily available

periodic patterns (such as suitable gift wrapping paper).

Practical Resources

The work sheets have been specially designed for easy duplication (Xerox)

which is permitted for personal use only. (Actual reprinting ofthe worksheets

is subject to copyright restrictions.)

Time Required for Teaching

An appropriate selection of space groups will take up one or two practical

sessions depending on tho level of the course.

Space Group Patterns

W. M. Meier

ETH Zurich, Switzerland

Introduction

Two-dimensional periodic patterns provide a simple means for teaching

the principles of crystallographic symmetry. In particular they help to

develop some desirable skill in recognizing space groups at an elementary

level. Crystal structure models are generally less useful for demonstrating

space group symmetry. This account shows how 2-dimensional patterns can

be extended to include 3-dimensional space group symmetry. Space group

patterns, as presented in this brief exposition, can furnish a suitable basis

for a non-mathematical introduction to the symmetry of crystal structures

in which the emphasis is on practical rather than theoretical aspects.

A knowledge of elementary symmetry concepts, as outlined in Pamphlet

No. 13 of this series or an introductory text, is assumed.

Two-Dimensional Periodic Patterns

Plane groups can be illustrated using a great variety of 2-dimensional

periodic patterns. Unexcelled classics are the widely known prints by

M. C. Escher' which have found their way into many texts on crystallo-

graphic symmetry. Other examples are provided by decoration and gift

wrapping paper, various fabrics and highly artistic designs including Arabic

geometrical patterns.2

A standard introductory exercise in crystallographic symmetry is the

determination of the plane group of such patterns, e.g. that of an ordinary

brick wall (c2mm). The best approach to such a problem is to begin with

the point symmetry (extended by glide lines if necessary) which will define

the system of axes to be adopted. Thus, the oblique system (comprising

plane groups p 1 and p2) is by no means characterized by a ~ b and y ~ 90°,

a frequent misconception in 'defining' crystal systems. A sheet of stamps,

for example, has symmetry p l in general? and therefore belongs to the

oblique system. The rectangular form of the stamps is not required by

symmetry, it simply happens to be convenient. Another example of this sort

is presented in Fig. 1.

Some knowledge of and practice in recognizing plane groups is not

necessarily a prerequisite but a good basis from which to start an introduc-

tion to space group symmetry as described in the following sections. The

pattern in Plate (i), if considered to represent a 2-dimensional structure,

Space Group Patterns - M.W.Meier.pdf

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