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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 by William Hallowes Miller

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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

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

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

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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

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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

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

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(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

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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:

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The (111) planes:

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

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

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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

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

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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

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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

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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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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

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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

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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

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Intercepts → ½ 1 Plane → (1 1 2 1) Intercepts → 1 1 1 Plane → (1 0 1 1)

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

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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:

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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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