STATE OF SOLIDIFICATION & CRYSTAL STRUCTURE

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1 STATE OF SOLIDIFICATION & CRYSTAL STRUCTURE

2 Chapter Outline Determination of crystal properties or properties of crystalline materials. Crystal Geometry! Crystal Directions! Linear Density of atoms! Crystal Planes! Planar Density of Atoms

3 Crystalline Geometry Determination of crystal properties or properties of crystalline materials is based on crystal geometry. Crystal geometry is described by means of MİLLER İNDECES. 1. Crystal Directions Many properties of crystals change with crystal directions. In a space lattice there are infinite number of parallel planes.

4 Crystalline Geometry 1. Crystal Directions A vector starting from the origin and ending at a corner of the next unit cell is used to describe directions. The components of this vector on each axis are described in terms of lattice constant to obtain the smallest possible whole number of sequence as [hkl].

5 Crystalline Geometry 1. Miller İndeces of the Directions z (x,y,z) Direction İndeces Line 1 has a,0,0 [100] 6 5 x 4 1 a 3 a a Line has a,a,0 [110] Line 3 has a,a,a [111] Line 4 has a,0,a [10] y Line 5 has a,-a,0 [1ī0] Line 6 has 0,-a,a [0ī] Ī

6 Example: Draw the following direction vectors in a cubic unit cells. z [011] From Miller İndeces [hkl], find the Position Coordinates (hkl). a) [010] & [011] b) [11] & [11] x y c) [ī10] & [11ī] & [1ī0] d) [131] & [11] & [130]

7

8 Linear Density of Atoms The spacing of atoms in a crystal lattice changes with respect to direction. The linear density (LD) of atoms affects certain properties. Linear Density [ hkl] # of atoms length of the direction

9 FCC Linear Density of Atoms ex: linear density of Al in [110] direction. a= nm

10 Linear Density of Atoms In a lattice system, equivalent directions form a family which is designated by hkl. For example in a cubic cell, 100 is composed of [100], [010], [001] Cubic edge directions collectively. z x [001] [100] [010] y 111 = [111], [ī 11], [1 ī 1], [11 ī], [ī ī ī] cubic body diagonals. 110 = [110], [1ī 0], [ī 10], [ī ī 0] cubic face diagonals.

11 Linear Density of Atoms For certain calculations, the angle between two directions may be necessary. Cos ( h 1 k h h 1 1 l 1 k 1 k )*( h l. Example: =? Between [110] & [111] directions in the cubic system. 1 l k l ) İs the angle between [h 1 k 1 l 1 ] & [h k l ] Cos 1*11*1 0*1 35, 3 (1 1 0 )*(1 1 1 ) 6

12 Crystal Planes A crystal contains planes of atoms, and these planes influence the properties and behaviour of crystals. Shear may occur between neighboring planes, permitting the crystal to shear and conferring ductility in metals. If the bond between planes is weak, they may apart from each other thus failure occurs. Then it is necessary to identify planes in crystal structures. Such identification is carried out by means of Miller İndeces..

13 Crystal Planes: Miller İndeces Miller Indices (hkl) is a specific crystallographic plane {hkl} is a family of planes with each one ahving the same atomic arrangement Determination of Miller İndeces (MI) Determine values where plane intercepts x,y,z axes Take reciprocals of intercepts. Put the resultants over their lowest common denominators The numerators then give the required indeces, i.e., (hkl)

14 Crystal Planes: Miller Indeces abc Plane Intercepts = 1, 1, Reciprocals = 1/1, 1/1, 1/ L.C.D = 1/1, 1/1, 0/1 MI = (110) abc Plane Intercepts = 1/,, Reciprocals = 1/1/, 1/, 1/ =, 0, 0 L.C.D = /, 0/, 0/ MI = (00) or (100)

15 Crystal Planes: Miller İndeces

16 Crystal Planes: Miller İndeces

17 Family of planes All equivalent planes in a structure are conventionally represented by one set of Miller indices in curly brackets {}. The family of the {100} planes. The family of the {110} planes.

18 Octahedral plane in an FCC or BCC structure Octahedral planes are (111) type planes. A 3-fold symmetry axis is perpendicular to those planes. That axis is of the type < 111 >. The octahedral plane. Note that for the cubic systems that the [hkl] direction is always normal to the (hkl) plane. A crystal plane & its normal directions have the same Miller İndeces. e.g. The normal to the (001) plane is [001] direction.

19 PLANAR DENSITY When slip occurs under stress, it takes place on the planes on which the atoms are most densely packed. Planar Density, hkl) Example: FCC unit cell ( # of atoms Area of plane z δ (100) = 4*1/4+1 = a a 1 a = 4 r δ (100) = 4r x (100) a y

20 Example: [100] of cubic unit cell δ [100] = 1/a a Example: Calculate planar density of the face plane (100) and linear density on the face diagonal [011] of an FCC structure. a 0 (100) [011] (100) [011] a a0 a 0 0

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