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MILLER INDICES

MILLER INDICES. PLANES DIRECTIONS. From the law of rational indices developed by French Physicist and mineralogist Abb é Ren é Just Ha üy and popularized by William Hallowes Miller. Vector r passing from the origin to a lattice point: r = r 1 a + r 2 b + r 3 c

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MILLER INDICES

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  1. MILLER INDICES • PLANES • DIRECTIONS From the law of rational indices developed by French Physicist and mineralogist Abbé René Just Haüy and popularized by William Hallowes Miller

  2. Vector r passing from the origin to a lattice point: r = r1a + r2 b + r3c a, b, c → fundamental translation vectors

  3. Miller Indices for directions (4,3) (0,0) 5a + 3b b a Miller indices → [53]

  4. [001] [011] [101] [010] [110] [110] [100] [111] • Coordinates of the final point  coordinates of the initial point • Reduce to smallest integer values

  5. Family of directions

  6. Miller Indices for planes (0,0,1) (0,3,0) (2,0,0) • Find intercepts along axes → 2 3 1 • Take reciprocal → 1/2 1/3 1 • Convert to smallest integers in the same ratio → 3 2 6 • Enclose in parenthesis → (326)

  7. Intercepts → 1   Plane → (100) Family → {100} → 3 Intercepts → 1 1  Plane → (110) Family → {110} → 6 Intercepts → 1 1 1 Plane → (111) Family → {111} → 8 (Octahedral plane)

  8. (111) Family of {111} planes within the cubic unit cell The (111) plane trisects the body diagonal (111) Plane cutting the cube into two polyhedrawith equal volumes

  9. Points about (hkl) planes For a set of translationally equivalent lattice planes will divide:

  10. The (111) planes:

  11. The portion of the central (111) plane as intersected by the various unit cells

  12. Tetrahedron inscribed inside a cube with bounding planes belonging to the {111} family 8 planes of {111} family forming a regular octahedron

  13. Summary of notations A family is also referred to as a symmetrical set

  14. Unknown direction → [uvw] • Unknown plane → (hkl) • Double digit indices should be separated by commas → (12,22,3) • In cubic crystals [hkl]  (hkl)

  15. Multiplicity factor * Altered in crystals with lower symmetry (of the same crystal class)

  16. a3 a2 a1 Hexagonal crystals → Miller-Bravais Indices Intercepts → 1 1 - ½  Plane → (1 12 0) (h k i l) i = (h + k) The use of the 4 index notation is to bring out the equivalence between crystallographically equivalent planes and directions

  17. Examples to show the utility of the 4 index notation a3 a2 a1 Intercepts →  1 -1  Miller → (0 1 0) Miller-Bravais → (0 11 0) Intercepts → 1 -1  Miller → (1 1 0 ) Miller-Bravais → (1 1 0 0 )

  18. Examples to show the utility of the 4 index notation a3 a2 a1 Intercepts → 1 1 - ½  Plane → (1 12 0) Intercepts → 1 -2 -2  Plane → (2 11 0 )

  19. Intercepts → 1 1 - ½ 1 Plane → (1 12 1) Intercepts → 1   1 1 Plane → (1 01 1)

  20. Directions Directions are projected onto the basis vectors to determine the components

  21. Transformation between 3-index [UVW]and 4-index [uvtw] notations

  22. Directions in the hexagonal system can be expressed in many ways • 3-indices:By the three vector components along a1, a2 and c: rUVW = Ua1 + Va2 + Wc • In the three index notation equivalent directions may not seem equivalent • 4-indices:

  23. Directions  Planes • Cubic system: (hkl)  [hkl] • Tetragonal system: only special planes are  to the direction with same indices:[100]  (100), [010]  (010), [001]  (001), [110]  (110)([101] not  (101)) • Orthorhombic system: [100]  (100), [010]  (010), [001]  (001) • Hexagonal system: [0001]  (0001) (this is for a general c/a ratio; for a Hexagonal crystal with the special c/a ratio = (3/2) the cubic rule is followed) • Monoclinic system: [010]  (010) • Other than these a general [hkl] is NOT  (hkl)

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