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Key Crystallographic Formulas and Stacking Modes in Material Science

This document presents essential crystallographic formulas essential for understanding crystal structures and their properties. It covers zone equations for crystal planes, interplanar distances in different crystal systems (cubic, orthorhombic, hexagonal), and stacking modes of crystal structures (simple cubic, HCP, FCC). The relationships between indices, angles, and volumes of unit cells are also discussed, with comparisons between normal and fault stacking orders. This summary is vital for materials science students and professionals engaging in crystallography.

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Key Crystallographic Formulas and Stacking Modes in Material Science

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  1. §2.6 Some Important Crystallographic Formulas＊ Lan Yu Faculty of Material Science and Engineering Kunming University of Science and technology

  2. 1. Zone and zone equation All Planes parellel to common axis [u v w] constitute a (crystal) zone, the [u v w] zone. If (h k l) belongs to zone [u v w], then hu＋kv＋lw＝0 2. If hu＋kv＋lw＝0，then [u v w] lies in (h k l) 3. For cubic crystals, [h k l] ⊥(h k l)

  3. 4. The zone [u v w] contains two planes (h1 k1 l1) and (h2k2l2),then 5. The plane (h k l) belongs to two zones [u1v1w1] and [u2v2w2] if

  4. 6. The distance between adjacent plane ( interplanar distance ) d(hkl)＝f(a,b,c α,β,γ h,k,l) For orthorhombic: For cubic: For hexagonal crystals:

  5. 7. The length of [u v w] For cubic: 8. The angle Ф between (h1k1l1) and (h2k2 l2)

  6. 8. The angle between (h1k1l1) and (h2k2 l2) For cubic: For orthorhombic:

  7. For hexagonal:

  8. 9. The angle between [u1v1w1] and [u2v2w2] For orthorhombic: For cubic:

  9. For hexagonal:

  10. 10. The volume of unit cellsV

  11. §2.7 Stacking Modes of Crystals Ⅰ. A crystal can be considered as the result of stacking the atomic layers, say (h k l ), one over another in a specific sequence. For simple cubic (001) aaaa…… (110) abab…… This sequence is called the stacking order

  12. Ⅱ.Comparison of stacking mode of HCP and FCC 1. HCP HCP stacking order: ABABABAB……

  13. 2. FCC Stacking order of (111)： ABCABCABC……

  14. A B A A C B B C C

  15. Ⅲ. Stacking fault • For HCP: normal order: ABABAB…… fault order: ABCABAB…… • For FCC： normal order: ABCABCABC…… fault order: ABCACABCA…… ABCACBCABC …… ABCABABAB …… ABCACBA ……

  16. cH cR bR aR bH aH Ⅳ.Transformation of hexagonal to rhombohedral indices and vice versa

  17. Thanks !

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