Miller Indices & Steriographic Projection

The Miller indices can be determined from the steriographic projection by measuring the angles relative to known crystallographic directions and applying the law-of-cosines.

(Figure 2-39 Cullity)

For ,, and to represent the angles between the normal of a plane and the a1, a2, and a3 axes respectively, then:

lcd

kb

d

ha

d cos,cos,cos

Where a, b, and c are the unit cell dimensions, and a/h, b/k, and c/l are the plane intercepts with the axes. The inner planar spacing, d, is equal to the distance between the origin and the plane (along a direction normal to the plane).

Vector Operations

Dot product: cosabba

Cross product: sinabba

dbcbdacadcba

abba

abba

ba

a

b

Volume: cba V

Reciprocal Lattice

Unit cell: a1, a2, a3

Reciprocal lattice unit cell: b1, b2, b3 defined by:

213

132

321

1

1

1

aab

aab

aab

V

V

V

a1

a2

b3

A

B CP

O

001

213

11

cell ofheight OACB ramparallelog of area

OACB ramparallelog of area

dOP

V

aab

a3

Reciprocal Lattice

Like the real-space lattice, the reciprocal space lattice also has a translation vector, Hhkl:

321 bbbH lkhhkl

Where the length of Hhkl is equal to the reciprocal of the spacing of the (hkl) planes

hklhklhkl d

H1

H

Next overhead and (Figures A1-4, and A1-5 Cullity)

Consider planes of a zone (i.e..: 2D reciprocal lattice).

Zone Axis

Zon e Axis

Top V iew

Planes could be translated so as not tointersect at a common point.

Reciprocal Lattice

023132313 abababab

11

33

OPOP

ab. if,0

, if,1

nm

nmnm

ba

(Zone Axis) 321 bbbH lkhhkl hklH

Zone axis = ua1 + va2 + wa3

0

0321321

lwkvhu

lkhwvu bbbaaa

hklkh 12

321aa

bbbABH

hllkh 13

321aa

bbbACH

A

B

C (hkl)

H

l

a3

k

a2

h

a1

dhkl

N

O