Any set of planes is characterized by 1 their orientation in the crystal hkl Miller indices 2 their d spacing d hkl distance between the planes h k l correspond to the number of segments in which the ID: 1025596
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3. MILLER PLANESAtoms form periodically arranged planesAny set of planes is characterized by: (1) their orientation in the crystal (hkl) – Miller indices (2) their d-spacing (dhkl) – distance between the planesh, k, l correspond to the number of segments in which the a, b, c axes, respectively, are cut by the set of planesOn average, the higher (hkl),the closer is the interplanar distance, dhkl2-D Examples3
4. Crystallographic PlanesWe want to examine the atomic packing of crystallographic planesIron foil can be used as a catalyst. The atomic packing of the exposed planes is important. Draw (100) and (111) crystallographic planes for Fe.b) Calculate the planar density for each of these planes.
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12. Considering 2D lattice- Crystal planes xyab1b-1a1b2b1b∞1a3aThere are many ways connecting lattice points
13. Crystallographic PlanesMiller Indices: Reciprocals of the (three) axial intercepts for a plane, cleared of fractions & common multiples. All parallel planes have same Miller indices.Algorithm 1. Read off intercepts of plane with axes in terms of a, b, c2. Take reciprocals of intercepts3. Reduce to smallest integer values4. Enclose in parentheses, no commas i.e., (hkl)
14. Connecting points 1a and 1b denoted as (1a, 1b) or (1, 1) 3a and 2b (3a, 2b) or (3, 2) -1a and 1b (-1a, 1b) or (-1, 1) ∞a and 1b (+∞a, 2b) or (+∞, 2)(1,1)(3,2)(-1,1)(+∞, 2)
15. d. d-spacing dhkl(110)(010)(210)d110d010
16. Extending to 3D lattice lines connecting points extend to z-direction (c) forming plane 1a-1b designate (1a, 1b, ∞c) or (1, 1, ∞)3a-2b designate (3a, 2b, ∞c) or (3, 2, ∞)-1a-1b designate (-1a, 1b, ∞c) or (-1, 1, ∞)+∞a-1b designate (+∞a, 1b, ∞c) or (∞, 1, ∞)It is not convenient to represent plane coordinates as ∞, a reciprocal representation called “Miller indices, (h k l)” is rather used.(1, 1, ∞) (1/1, 1/1, 1/∞) ≡ (1 1 0)(3, 2, ∞) (1/3, 1/2, 1/∞) ≡ (2 3 0)(-1, 1, ∞) (-1/1, 1/1, 1/∞) ≡ (1 1 0)(∞, 1, ∞) (1/∞, 1/1, 1/∞) ≡ (0 1 0)
17. Crystallographic Planeszxyabc4. Miller Indices (110)examplea b czxyabc4. Miller Indices (100)1. Intercepts1 1 2. Reciprocals1/1 1/1 1/1 1 03. Reduction1 1 01. Intercepts1/2 2. Reciprocals1/½ 1/ 1/2 0 03. Reduction2 0 0examplea b c
18. Crystallographic Planeszxyabc4. Miller Indices (634)example1. Intercepts1/2 1 3/4a b c2. Reciprocals1/½ 1/1 1/¾2 1 4/33. Reduction6 3 4(001)(010),Family of Planes {hkl}(100),(010),(001),Ex: {100} = (100),
19. Crystallographic PlanesAdapted from Fig. 3.9, Callister 7e.
20. HCP Crystallographic Directions1. Vector repositioned (if necessary) to pass through origin.2. Read off projections in terms of unit cell dimensions a1, a2, a3, or c3. Adjust to smallest integer values4. Enclose in square brackets, no commas [uvtw][ 1120 ]ex: ½, ½, -1, 0 =>Adapted from Fig. 3.8(a), Callister 7e.dashed red lines indicate projections onto a1 and a2 axesa1a2a3-a32a22a1-a3a1a2zAlgorithm
21. HCP Crystallographic DirectionsHexagonal Crystals4 parameter Miller-Bravais lattice coordinates are related to the direction indices (i.e., u'v'w') as follows.==='wwtvu)vu(+-)'u'v2(31-)'v'u2(31-=]uvtw[]'w'v'u[®Fig. 3.8(a), Callister 7e.-a3a1a2z
22. Crystallographic Planes (HCP)In hexagonal unit cells the same idea is used examplea1 a2 a3 c4. Miller-Bravais Indices(1011)1. Intercepts1 -112. Reciprocals1 1/1 0 -1-1113. Reduction1 0-11a2a3a1zAdapted from Fig. 3.8(a), Callister 7e.
23. The distance between each plane can be calculated ФФFor 2 (h k 0) planes
24. Similarly for 2 (h k l) planesIf a, b, c and h, k, l are known , d can be computed
25. Example 7.1 Calculate the separation of (a) the (123) planes and (b) the (246) planes of an orthorhombic cell with a = 0.82 nm, b = 0.94 nm, and c = 0.75 nmFrom (a)
26. (b)
27. 1. Choose a point on the direction as the origin.2. Choose a coordinate system with axes parallel to the unit cell edges.xy3. Find the coordinates of another point on the direction in terms of a, b and c4. Reduce the coordinates to smallest integers. 5. Put in square bracketsMiller Indices of Directions[100]1a+0b+0cz1, 0, 01, 0, 0Miller Indices 2
28. yzMiller indices of a direction represents only the orientation of the line corresponding to the direction and not its position or senseAll parallel directions have the same Miller indices[100]xMiller Indices 3
29. xyzOA1/2, 1/2, 1[1 1 2]OA=1/2 a + 1/2 b + 1 cPQxyzPQ = -1 a -1 b + 1 c-1, -1, 1Miller Indices of Directions (contd.)[ 1 1 1 ]__-ve steps are shown as bar over the number
30. Miller indices of a family of symmetry related directions[100][001][010]= [uvw] and all other directions related to [uvw] by the symmetry of the crystal = [100], [010], [001]= [100], [010]CubicTetragonal[010][100]Miller Indices 4
31. 5. Enclose in parenthesisMiller Indices for planes3. Take reciprocal2. Find intercepts along axes in terms of respective lattice parameters1. Select a crystallographic coordinate system with origin not on the plane4. Convert to smallest integers in the same ratio1 1 11 1 11 1 1(111)xyzO
32. Miller Indices for planes (contd.)origininterceptsreciprocalsMiller IndicesABCDOABCDO1 ∞ ∞1 0 0(1 0 0)OCBEO*1 -1 ∞ 1 -1 0(1 1 0)_PlanexzyO*xzEZero represents that the plane is parallel to the corresponding axis Bar represents a negative intercept
33. Miller indices of a plane specifies only its orientation in space not its positionAll parallel planes have the same Miller IndicesABCDOxzyE(100)(h k l ) (h k l )_ _ _(100) (100)_
34. Miller indices of a family of symmetry related planes= (hkl ) and all other planes related to (hkl ) by the symmetry of the crystal {hkl }All the faces of the cube are equivalent to each other by symmetryFront & back faces: (100)Left and right faces: (010)Top and bottom faces: (001){100} = (100), (010), (001)
35. {100}cubic = (100), (010), (001){100}tetragonal = (100), (010)(001)CubicTetragonalMiller indices of a family of symmetry related planesxzyzxy
36. Some IMPORTANT ResultsCondition for a direction [uvw] to be parallel to a plane or lie in the plane (hkl):h u + k v + l w = 0Weiss zone lawTrue for ALL crystal systemsNot in the textbook
37. CUBIC CRYSTALS[hkl] (hkl)Angle between two directions [h1k1l1] and [h2k2l2]: C[111](111)
38. dhklInterplanar spacing between ‘successive’ (hkl) planes passing through the corners of the unit cellOx(100)BOxzE
39. [uvw] Miller indices of a direction (i.e. a set of parallel directions)(hkl) Miller Indices of a plane (i.e. a set of parallel planes)<uvw> Miller indices of a family of symmetry related directions {hkl} Miller indices of a family of symmetry related planesSummary of Notation convention for Indices
40. 7.1.4 Crystal Planes and Miller Indicesa.Lattice planesIt is possible to describe certain directions and planes with respect to the crystal lattice using a set of three integers referred to as Miller Indices. Miller indices describe the orientation and spacing of a family of planes.Simple Tetragonal Lattices: projections of crystals planes (parallel to the c-axis) on the (001) plane.(110)(010)(210)
41. b.Miller indices (hkl)Miller indices are the reciprocal intercepts of the plane on the unit cell axes.Example: 1/3:1/2:1/1 = 2:3:6 The Miller index is (236)
42. abc(110)abc(111)abc(100)(010)originExamples of Miller indices
43. (0001)abchexagonal, four axis (a1,a2,-(a1+a2),c)(hkil), i=-(h+k) hexagonal, four axis (a1,a2,a3,c)
44. C. Directions in lattice
45. Example: Directions on the (111) plane.
46. Miller indices [hkl] are used to specify a direction in space with respect of the unit cell axes. <hkl> are used to specify a set of symmetry- equivalent directions.[uvw] zone axisMiller indices (hkl) are used to specify the orientation and spacing of a family of planes. {hkl} are used to specify all symmetry- equivalent sets of planesDirection Vector = ua + vb + wc
47. Cubic : 1/d2 = (h2+k2+l2)/a2 or d2 = a2/(h2+k2+l2)Tetragonal: 1/d2 = (h2+k2)/a2 + l2/c2Orthorhombic: 1/d2 = h2/a2+k2/b2 + l2/c2Hexagonal: 1/d2 = (4/3)(h2+hk+k2)/a2 + l2/c2Monoclinic: 1/d2 = [(h/a )2 + (k/b )2sin2 + (l/c )2- (2hl/ac)cos]/sin2Triclinic:The spacing between adjacent planes in a family is referred to as a “d-spacing”