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 r, s, and t to represent the angles between the normal of a plane and the a1, a2, and a3 axes respectively, then: 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: Cross product: b a a Volume:

Reciprocal Lattice Unit cell: a1, a2, a3 Reciprocal lattice unit cell: b1, b2, b3 defined by: b3 P B a3 C a2 O A a1

Reciprocal Lattice Like the real-space lattice, the reciprocal space lattice also has a translation vector, Hhkl: Where the length of Hhkl is equal to the reciprocal of the spacing of the (hkl) planes Consider planes of a zone (i.e..: 2D reciprocal lattice). Next overhead and (Figures A1-4, and A1-5 Cullity)

Zone Axis Planes could be translated so as not to For planes of a zone, all plane normals lie in a common plane which is perpendicular to the zone axis. For example, if the zone axis labeled above is parallel to the coordinate system z -axis, then all plane normals lie in the x-y plane. Planes could be translated so as not to intersect at a common point.

Reciprocal Lattice (Zone Axis) Zone axis = ua1 + va2 + wa3 f A B C (hkl) H dhkl N O H·AB = 0 H·AC = 0 Therefore the plane must be perpendicular to H. dhkl = (a1/h)cosf = (a1/h)·n = (a1/h)·(H/H) = (a1/h)·{(hb1 + kb2 + lb3)/H} dhkl = 1/H Also: Inner planar spacing: (Zone Axis) Zone axis = ua1 + va2 + wa3 Angles between reciprocal lattice vectors.