1.3 crystal faces and miller indices

P. 306 – olivine information

Chemistry

Crystallography

Physical Properties

Optical Properties

Crystal Faces Common crystal faces relate simply

to surfaces of unit cell Often parallel to the faces of the unit

cell Isometric minerals often are cubes Hexagonal minerals often are hexagons

Other faces are often simple diagonals – at uniform angles – to the unit cell faces

These relationships were discovered in 18th century and codified into laws: Steno’s law Law of Bravais Law of Huay

Steno’s Law

Angle between equivalent faces on a crystal of some minerals are always the same

Can understand why Faces relate to unit cell, crystallographic

axes, and angular relationships between faces and axes

Strictly controlled by symmetry of the crystal system and class of that mineral

Law of Bravais

Common crystal faces are parallel to lattice planes that have high lattice node densities

Fig. 2-21

All faces parallel unit cell – high density of lattice nodes T has

intermediate density of lattice nodes – fairly common and pronounced face on mineral

Q has low density, rare face

Monoclinic crystal

Faces A, B, and C intersect only one axis – principal faces

Face T intersects two axes a and c, but at same unit lengths

Face Q intersects A and C at ratio 2:1

a b

c t

Law of Haüy Crystal faces intersect axes at simple

integers of unit cell distances on the crystallographic axes

Lengths can be absolute or relative: Absolute distance - lengths have units

(typically Å) and are not integers Unit cell distances - typically small

integers, e.g., 1 to 3, occasionally higher 1 unit length is the absolute length of

crystallographic axis Allows a naming system to describe

planes in the mineral (faces, cleavage, atomic planes etc.)

Miller Indices

Miller Indices

Shorthand notation for where the faces intercept the crystallographic axes

Miller Index Set of three integers (hkl) Inversely proportional to where face or

crystallographic plane (e.g. cleavage) intercepts axes

General form is (hkl) where h represents the a intercept k represents the b intercept l represents the c intercept

h, k, and l are ALWAYS integers

Fig. 2-22

Imagine you extend face t until it intercepts crystallographic axes

Unit cell: each side is one “unit” length

How many unit lengths out along the crystallographic axes?

Consider face t:

Fig. 2-22For face t:

Axial intercepts in terms of unit cell lengths:a = 12b = 12c = 6

Face t, without the rest of the form

Fig. 2-22

Imagine the face is fit within the unit cell so that the maximum intercept is 1 unit length;

The intercepts for a:b:c would be 1:1:1/2

Miller indices are the inverse of the intercepts

Inverting give (112) – note that the higher the number the closer to the origin

Face t is the (112) face

What about faces that parallel axes? For example, intercepts a:b:c could be

1:1:∞ With algorithm, miller index would

be: (hkl) = (1/1 1/1 1/∞) = (110)

If necessary you need to clear fractions E.g. intercepts for a:b:c = 1:2:∞ Invert: 1/1 1/2 1/∞ Clear fractions: 2(1 ½ 0) = (210)

Some intercepts can be negative – they intercept negative axes

E.g. a:b:c = 1:-1:½ Here (hkl) = 1/1:1/-1:1/½ = (112)

Fig. 2-23

It is very easy to visualize the location of a simple face given miller index, or to derive a miller index from simple faces

a

b

c

-b

-c

Hexagonal Miller index There need to be 4 intercepts (hkil)

h = a1

k = a2

i = a3 l = c

Two a axes have to have opposite sign of other axis so that h + k + i = 0

Possible to report the index two ways: (hkil) (hkl)

Klein and Hurlbut Fig. 2-

33

(100)(1010)

(110) (111)

(1120) (1121)

Assigning Miller indices Prominent (and common) faces have

small integers for Miller Indices Faces that cut only one axis

(100), (010), (001) etc Faces that cut two axes

(110), (101), (011) etc Faces that cut three axes

(111) Called unit face