1. Crystallographic Axes
GEOL 3055 Morphological and optical crystallography JH Schellekens
Crystallographic Axes
Klein (2002) p
5 Crystallographic axes

2. Which Crystal System? Orthorhombic Tetragonal Hexagonal
a=b=c  =  =  = 90
a=b=c  =  =  = 90
a1 = a2 = a3 (120), c perpendicular
Orthorhombic
Tetragonal
Hexagonal

3. Crystallographic axes
When we are describing crystals, it is convenient to use a reference system of three axes, comparable to the axes of analytical geometry.
These imaginary axes are called the ‘crystallographic axes’
These axes are fixed by symmetry &
Coincide with symmetry axes
Parallel to intersections of major crystal faces

4. Crystallographic axes
Ideally crystallographic axes should be parallel to the edges of the unit cell, and lengths proportional to the cell dimensions
REMEMBER
All crystals except hexagonal referred to by 3 axes: a, b and c
Convention:
a is angle between b and c
b is angle between a and c
g is angle between a and b

5. Crystallographic axes
Hexagonal
isometric

6. Crystallographic axes
Axial Ratios
All the crystal systems, except isometric & tetragonal, have crystallographic axes differing in length
The steps on the crystallographic axes, because they are dependent on the unit cell, are different in size

7. Crystallographic axes
For instance orthorhombic sulfur
a= 10.47A, b=12.87A, c=24.49A
We can write a, b and c as ratios of b
a/b : b/b : c/b
10.47/ : : /12.87
: 1 :
We are only interested in the proportional differences, the axial ratios

8. Crystallographic axes
Crystal faces are defined by indicating their intercepts on the crystallographic axes
Face AB is parallel to the c-axis and intercepts a and b
Parameters of this face are
1a:1b: c
It intercepts 1 length of the a axis, one length of the b-axis and is parallel to the c-axis 8
Fig.5.28

9. Crystallographic axes
Crystal faces are defined by indicating their intercepts on the crystallographic axes
Fig.5.28

10. Face Intercepts… Lattice plane A-B B Y or b axis z or c axis
(vertical) A A A’ A’ B
X or a axis
Plane A-A
Intercepts: 1a, ∞b, ∞c
Intersects x axis at one unit (1), is parallel to the y axis ( ∞ ) and the z axis (∞ )
Plane A-B, intersects 1a and 1b, but is parallel to c or ∞c
Parameters: 1a, 1b, ∞c

11. Unit face
If there are several faces of a crystal intersecting all three axes, the largest face at the positive end of the crystallographic axis is taken as the unit face.
Consider this example:
Unit face
(the face with the clear shade)

12. Steps to determine Miller Indices and the Miller-Bravais Indices
1. The first thing that must be ascertained are the fractional intercepts that the plane/face makes with the crystallographic axes. In other words, how far along the unit cell lengths does the plane intersect the axis.
e.g: 1a, ∞b, ∞c and 1a, 1b, ∞c and 1a, 2b, 4c
2. Omit a, b, c and commas
e.g: 1 ∞ ∞ and 1 1 ∞ and 1 2 4
3. Take the reciprocal of the fractional intercept of each unit length for each axis.
e.g.: 1/1 1/∞ 1/∞ and 1/1 1/1 1/∞ and 1/1 ½ ¼

13. Steps to determine Miller Indices and the Miller-Bravais Indices
4. Finally the fractions are cleared (using a common denominator).
so: 1/1 1/∞ 1/∞ and 1/1 1/1 1/∞ and 1/1 ½ ¼
Becomes and and 4 2 1
5. Enclose the integers in parentheses
So: (100) and (110) and (421)
These designate that specific crystallographic plane within the lattice. Since the unit cell repeats in space, the notation actually represents a ‘family of planes’, all with the same orientation.

14. Steps to determine Miller Indices and the Miller-Bravais Indices
(100) and (110) and (421) are called the Miller indices
In the hexagonal system there are 3 horizontal axes and one vertical. The indices are called the Miller-Bravais indices

15. Summary “When intercepts are assigned to the faces of
a crystal, without knowledge of its cell dimensions,
one face that cuts all three axes is arbitrarily assigned
the units 1a,1b,1c”

16. Summary
(hkl)

17. The previous notation is called the Miller Indices and ONLY applies for the Triclinic, Monoclinic, Orthorhombic and Isometric systems…

18. Two very important points about intercepts of faces:
The intercepts or parameters are relative values, and do not indicate any actual cutting lengths.  
Since they are relative, a face can be moved parallel to itself without changing its relative intercepts or parameters.

19. Miller indices
Try & work out how the shaded face (in each case) intersects the axes
(111)
(001)
(110)

20. How is it for hexagonal and trigonal systems?
Recall both systems have 4 crystallographic axes.
In this case, the notation for the intersection of faces is called: Miller -Bravais Indices (hkil)
(1010)
One,zero,bar one, zero
h + k + I (1+0-1)= 0