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MILLER INDICES  PLANES  DIRECTIONS From the law of rational indices developed by French Physicist and mineralogist Abbé René Just Haüy and popularized.

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Presentation on theme: "MILLER INDICES  PLANES  DIRECTIONS From the law of rational indices developed by French Physicist and mineralogist Abbé René Just Haüy and popularized."— Presentation transcript:

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 = r 1 a + r 2 b + r 3 c a, b, c → fundamental translation vectors

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

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

5 Family of directions IndexNumber in the family for cubic lattice →3 x 2 = 6 →6 x 2 = 12 →4 x 2 = 8 Symbol Alternate symbol [ ]→Particular direction [[ ]]→Family of directions

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

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

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

9 Entity being divided (Dimension containing the entity) Directionnumber of parts Cell edge (1D)a[100]h b[010]k c[001]l Diagonal of cell face (2D)(100) [011](k + l) (010)[101](l + h) (001)[110](h + k) Body diagonal (3D)[111](h + k + l) 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 Symbol Alternate symbols Direction [ ][uvw]→Particular direction [[ ]]→Family of directions Plane ( )(hkl)→Particular plane { }{hkl}(( ))→Family of planes Point..xyz.[[ ]]→Particular point : :xyz:→Family of point 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 Condition(hkl) will pass through h evenmidpoint of a (k + l) even face centre (001) midpoint of face diagonal (001) (h + k + l) even body centre midpoint of body diagonal

16 Index Number of members in a cubic lattice d hkl (100)6 (110)12 The (110) plane bisects the face diagonal (111)8 The (111) plane trisects the body diagonal (210)24 (211)24 (221)24 (310)24 (311)24 (320)24 (321)48

17 Cubic hklhhlhk0hh0hhhh00 48 * 2424 * 1286 Hexagonal hk.lhh.lh0.lhk.0hh.0h0.000.l 24 * 12 * 662 Tetragonal hklhhlh0lhk0hh0h0000l 16 * 888*8* 442 Orthorhombic hklhk0h0l0klh000k000l Monoclinic hklh0l0k0 422 Triclinic hkl 2 * Altered in crystals with lower symmetry (of the same crystal class) Multiplicity factor

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20 Hexagonal crystals → Miller-Bravais Indices (h k i l) i =  (h + k) a1a1 a2a2 a3a3 Intercepts → ½  Plane → (1 1  2 0) The use of the 4 index notation is to bring out the equivalence between crystallographically equivalent planes and directions

21 a1a1 a2a2 a3a3 Intercepts →  1 -1  Miller → (0 1 0) Miller-Bravais → (0 1  1 0) Intercepts → 1 -1   Miller → (1  1 0 ) Miller-Bravais → (1  ) Examples to show the utility of the 4 index notation

22 a1a1 a2a2 a3a3 Intercepts → ½  Plane → (1 1  2 0) Intercepts →  Plane → (2  1  1 0 ) Examples to show the utility of the 4 index notation

23 Intercepts → ½ 1 Plane → (1 1  2 1) Intercepts → 1   1 1 Plane → (1 0  1 1)

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25 Directions are projected onto the basis vectors to determine the components Directions a1a1 a2a2 a3a3 Projectionsa/2 −a Normalized wrt LP1/2 −1 Factorization11−2 Indices [1 1  2 0]

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

28  Directions in the hexagonal system can be expressed in many ways  3-indices: By the three vector components along a 1, a 2 and c: r UVW = Ua 1 + Va 2 + Wc  In the three index notation equivalent directions may not seem equivalent  4-indices:

29 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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