MILLER INDICES q PLANES q DIRECTIONS From the
- Slides: 32
MILLER INDICES q PLANES q 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 = r 1 a + r 2 b + r 3 c a, b, c → fundamental translation vectors
Miller Indices for directions (4, 3) (0, 0) 5 a + 3 b b a Miller indices → [53]
[001] [011] [101] [010] [111] [1 10] [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> → 3 x 2=6 <110> → 6 x 2 = 12 <111> → 4 x 2=8 Symbol Alternate symbol [] <> [[ ]] → Particular direction → Family of directions
Miller Indices for planes (0, 0, 1) (0, 3, 0) (2, 0, 0) q Find intercepts along axes → 2 3 1 q Take reciprocal → 1/2 1/3 1 q Convert to smallest integers in the same ratio → 3 2 6 q 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 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 (1 D) Diagonal of cell face (2 D) Body diagonal (3 D) 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 Direction Plane Point [] [uvw] <> <uvw> () (hkl) {} {hkl} . . . xyz. : : : xyz: → Particular direction → Family of directions → Particular plane (( )) → Family of planes [[ ]] → Particular point → Family of point [[ ]] A family is also referred to as a symmetrical set
q Unknown direction → [uvw] q Unknown plane → (hkl) q Double digit indices should be separated by commas → (12, 22, 3) q In cubic crystals [hkl] (hkl)
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
(100) Number of members in a cubic lattice 6 (110) (111) (210) (211) (221) (310) (311) (320) (321) 12 8 24 24 24 48 Index dhkl The (110) plane bisects the face diagonal 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 hk 0 4 h 0 l 2 hk 0 24* h 0. l 12* h 0 l 8 h 0 l 4 0 k 0 2 hh 0 12 hk. 0 12* hk 0 8* 0 kl 4 hhh 8 hh. 0 6 hh 0 4 h 00 2 h 00 6 h 00 4 0 k 0 2 * Altered in crystals with lower symmetry (of the same crystal class) 00. l 2 00 l 2
Hexagonal crystals → Miller-Bravais Indices a 3 Intercepts → 1 1 - ½ Plane → (1 1 2 0) (h k i l) i = (h + k) a 2 a 1 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 a 3 a 2 a 1 Intercepts → 1 -1 Intercepts → 1 -1 Miller → (1 1 0 ) Miller → (0 1 0) Miller-Bravais → (1 1 0 0 ) Miller-Bravais → (0 1 1 0)
Examples to show the utility of the 4 index notation a 3 a 2 Intercepts → 1 -2 -2 Plane → (2 1 1 0 ) a 1 Intercepts → 1 1 - ½ Plane → (1 1 2 0)
Intercepts → 1 1 - ½ 1 Plane → (1 1 2 1) Intercepts → 1 1 1 Plane → (1 0 1 1)
Directions are projected onto the basis vectors to determine the components a 1 a 2 a 3 Projections a/2 −a Normalized wrt LP 1/2 − 1 Factorization 1 1 − 2 Indices [1 1 2 0]
Transformation between 3 -index [UVW] and 4 -index [uvtw] notations
§ 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:
Directions Planes q Cubic system: (hkl) [hkl] q Tetragonal system: only special planes are to the direction with same indices: [100] (100), [010] (010), [001] (001), [110] (110) ([101] not (101)) q Orthorhombic system: [100] (100), [010] (010), [001] (001) q 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) q Monoclinic system: [010] (010) q Other than these a general [hkl] is NOT (hkl)
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