MILLER INDICES q PLANES q DIRECTIONS From the

  • Slides: 32
Download presentation
MILLER INDICES q PLANES q DIRECTIONS From the law of rational indices developed by

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

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

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

[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

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

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

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

(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:

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 (111) planes:

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

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

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> ()

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

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

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)

(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

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

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

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

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

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

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

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

§ Directions in the hexagonal system can be expressed in many ways § 3

§ 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

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)