Transcript Slide 1

Part of

MATERIALS SCIENCE &

A Learner’s Guide

ENGINEERING AN INTRODUCTORY E-BOOK

Anandh Subramaniam & Kantesh Balani

Materials Science and Engineering (MSE) Indian Institute of Technology, Kanpur- 208016

Email: [email protected], URL: home.iitk.ac.in/~anandh http://home.iitk.ac.in/~anandh/E-book.htm

The rotations compatible with translational symmetry are

(1, 2, 3, 4, 6)

   In general any n-fold rotational axis is possible, where ‘n’ divides 360  without leaving a reminder. E.g. 1, 2, 3, 4, 5, 8, 12- fold axes are possible.

Apart from 1 (trivial), 2, 3, 4, 6 - fold axes, others are not compatible with translational symmetry.

We now see “why” this is so.

 Disallowed crystallographic symmetries like 5, 8, 12- fold are seen in QUASICRYSTALS .

    Consider two rows of lattice points: R 1 and R 2 A is taken to A’ by the lattice translation vector

t

(modulus is t) Point A is taken to B’ by a rotation axis and similarly A’ is taken to B by a rotation axis If B and B’ have to qualify as lattice points then BB’ must be an integral multiple of AA’, i.e. b = mt (where m is an integer)

b

m t b

mt m

 

Cos

 2

Cos

m

m & (1

m) are integers Say (1

m) is M Cos

 

M

2

Thus Cos

can take only half-integral values.

The permitted values are in the table in the next slide.

M

3

2

1 0 1 2 3 Cos

 

3/2

1

1/2 0 ½ 1 3/2

-

2

/3

/2

/3 0 n = 2

/

 2 3 4 6   1

-

 When we talk about n-fold, we also include the roto-reflection and roto-inversion axes.

.