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Miller Indices & Steriographic Projection in Crystallography

Learn how to determine Miller indices from steriographic projections by measuring angles relative to crystallographic directions and applying the law of cosines. Understand the relationship between plane normals and crystallographic axes to calculate inner planar spacings and reciprocal lattice unit cells. Explore vector operations like dot and cross products, and grasp the concept of reciprocal space lattice and zone axis planes.

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Miller Indices & Steriographic Projection in Crystallography

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  1. 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).

  2. Vector Operations Dot product: Cross product: b a a Volume:

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

  4. 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)

  5. Zone Axis Planes could be translated so as not to intersect at a common point.

  6. C (hkl) dhkl H N B O f A Reciprocal Lattice (Zone Axis) Zone axis = ua1 + va2 + wa3

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