Lecture 10 (10/16/2006)
Crystallography
Part 3:
Crystallographic Axes
Numerical Notation of Crystal Faces
and Atomic Planes – Miller Indices
Point Groups of Crystal Systems
Crystallographic Axes
Isometric Crystal System

Axes equal length, 90º

Aligned with A4 (432, 43m, 4/m32/m)
or with A2 (23, 2/m3)
Hexagonal Crystal System


1 long or short axis (c), 90º to 3 axes of equal
length (a1, a2, a3), 120º to each other
c aligned with A6 or A3; a1, a2, a3 are aligned
parallel to A2 or m planes or arbitrary (3, 3, 6, 6)
Tetragonal Crystal System


1 long or short axis (c), two axes of equal length
(a1, a2); all axes 90º to each other
c aligned with A4; a1 and a2 are aligned parallel to
A2 or m planes or arbitrary (4, 4)
Orthorhombic Crystal System


three axes of unequal length (a, b, c); all axes
90º to each other
axes aligned parallel to A2; for mm2, c aligned
parallel with A2, a & b normal to m planes
Monoclinic Crystal System


three axes of unequal length (a, b, c); β ≠ 90º,
α=γ= 90º,
b axis is parallel to A2 (2, 2/m) or normal to
mirror (m); a and c axes normal to b and normal
to crystal faces or edges
Triclinic Crystal System


three axes of unequal length (a, b, c); α ≠ β ≠ γ
≠ 90º
a, b, & c axes normal to crystal faces or edges
Numerical Notation for Crystal
Planes and Faces
•relative values, not absolute distances
•units defined by largest face cutting all three axes – Unit Face
Unit
Face
Crystal Face Intercepts
1c
UNIT
FACE
2/3c
1/3c
1b
or 1a, 1b, 1/3c
or 3a, 3b, 1c
2b
3b
Miller Indices
Invert intercept values and clear fractions
Law of Rational Indices –
common faces have simple
whole numbers for Miller Indices
Negative
Indices
Crystal Face Intercepts
1c
UNIT
FACE
2/3c
1/3c
or 1a, 1b, 1/3c
or 3a, 3b, 1c
Miller Indices
(1/1,1/1,1/1)
(1/2,1/2,3/2)
(1/1,1/1,3/1)
(1/3,1/3,1/1)
1b
=
=
=
=
111
113
113
113
2b
3b
Miller
Indices in
Hexagonal
System
Crystal Zone and Zone Axis
Zone [100]
Zone [001]
Zone [100] axis
Zone [001] axis
Next Lecture
Crystal Symmetry (Continued)
 Crystal Forms
 Twinning
Read: Chapter 5, p. 201-213
Read: Chapter 6, p. 240-251 for Lab
No Lecture Next Week