Report

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 Vector r passing from the origin to a lattice point: r = r1 a + r2 b + r3 c a, b, c → fundamental translation vectors Miller Indices for directions (4,3) 5a + 3b (0,0) b Miller indices → [53] a [001] [011] [101] [010] [111] [110] [100] [110] • Coordinates of the final point coordinates of the initial point • Reduce to smallest integer values Family of directions Index Number in the family for cubic lattice <100> → 3x2=6 <110> → 6 x 2 = 12 <111> → 4x2=8 Symbol Alternate symbol [] <> [[ ]] → Particular direction → Family of directions 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) 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) (111) Family of {111} planes within the cubic unit cell d111 a / 3 a 3 / 3 The (111) plane trisects the body diagonal (111) Plane cutting the cube into two polyhedra with equal volumes Points about (hkl) planes For a set of translationally equivalent lattice planes will divide: Entity being divided (Dimension containing the entity) Cell edge (1D) Diagonal of cell face (2D) Body diagonal (3D) Direction number of parts a [100] h b [010] k c [001] l (100) [011] (k + l) (010) [101] (l + h) (001) [110] (h + k) [111] (h + k + l) The (111) planes: The portion of the central (111) plane as intersected by the various unit cells Tetrahedron inscribed inside a cube with bounding planes belonging to the {111} family 8 planes of {111} family forming a regular octahedron Summary of notations Alternate symbols Symbol [] [uvw] <> <uvw> () (hkl) {} {hkl} .. .xyz. :: :xyz: → Particular direction → Family of directions → Particular plane (( )) → Family of planes [[ ]] → Particular point → Family of point Direction [[ ]] Plane Point A family is also referred to as a symmetrical set Unknown direction → [uvw] Unknown plane → (hkl) Double digit indices should be separated by commas → (12,22,3) In cubic crystals [hkl] (hkl) d hkl a h k l 2 2 2 Condition (hkl) will pass through h even midpoint of a (k + l) even (h + k + l) even face centre (001) midpoint of face diagonal (001) body centre midpoint of body diagonal Index (100) (110) (111) (210) (211) (221) (310) (311) (320) (321) Number of members in a cubic lattice 6 12 8 24 24 24 24 24 24 48 dhkl d100 a d110 a / 2 a 2 / 2 The (110) plane bisects the face diagonal d111 a / 3 a 3 / 3 The (111) plane trisects the body diagonal Multiplicity factor Cubic Hexagonal Tetragonal Orthorhombic Monoclinic Triclinic hkl 48* hk.l 24* hkl 16* hkl 8 hkl 4 hkl 2 hhl 24 hh.l 12* hhl 8 hk0 4 h0l 2 hk0 24* h0.l 12* h0l 8 h0l 4 0k0 2 hh0 12 hk.0 12* hk0 8* 0kl 4 hhh 8 hh.0 6 hh0 4 h00 2 h00 6 h0.0 6 h00 4 0k0 2 * Altered in crystals with lower symmetry (of the same crystal class) 00.l 2 00l 2 00l 2 Hexagonal crystals → Miller-Bravais Indices a3 Intercepts → 1 1 - ½ Plane → (1 12 0) (h k i l) i = (h + k) a2 a1 The use of the 4 index notation is to bring out the equivalence between crystallographically equivalent planes and directions Examples to show the utility of the 4 index notation a3 a2 a1 Intercepts → 1 -1 Intercepts → 1 -1 Miller → (1 1 0 ) Miller → (0 1 0) Miller-Bravais → (1 1 0 0 ) Miller-Bravais → (0 11 0) Examples to show the utility of the 4 index notation a3 a2 Intercepts → 1 -2 -2 Plane → (2 11 0 ) a1 Intercepts → 1 1 - ½ Plane → (1 12 0) Intercepts → 1 1 - ½ 1 Plane → (1 12 1) Intercepts → 1 1 1 Plane → (1 01 1) Directions Directions are projected onto the basis vectors to determine the components [1120] a1 a2 a3 Projections a/2 a/2 −a Normalized wrt LP 1/2 1/2 −1 Factorization 1 1 −2 Indices [1 1 2 0] Transformation between 3-index [UVW] and 4-index [uvtw] notations U u t V v t 1 u (2U V ) 3 W w 1 v (2V U ) t (u v) 3 w W 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: 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)