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.
.