Presentation on theme: "THE STRUCTURE OF CRYSTALLINE SOLIDS"— Presentation transcript:
1 THE STRUCTURE OF CRYSTALLINE SOLIDS CHAPTER 3THE STRUCTUREOFCRYSTALLINE SOLIDS
2 Atoms in an amorphous solid are arranged randomly- No Order 3.2 FUNDAMENTAL CONCEPTSSOLIDSAMORPHOUSCRYSTALLINEAtoms in a crystalline solid are arranged in a repetitive three dimensional pattern Long Range OrderAtoms in an amorphous solid are arranged randomly- No OrderAll metals are crystalline solidsMany ceramics are crystalline solidsSome polymers are crystalline solids
3 LATTICE Lattice -- points arranged in a pattern that LATTICE Lattice -- points arranged in a pattern that repeats itself in three dimensions The points in a crystal lattice coincides with atom centers
8 3.4 METALLIC CRYSTALS • Tend to be densely packed. • Have several reasons for dense packing:-Typically, only one element is present, so all atomicradii are the same.-Metallic bonding is not directional.-Nearest neighbor distances tend to be small inorder to have lower bonding energy.• Have the simplest crystal structures.Let us look at three such structures...4
14 SIMPLE CUBIC STRUCTURE (SC) Only Pd has SC structure
15 Number of atoms per unit cell BCC 1/8 corner atom x 8 corners + 1 body center atom=2 atoms/ucFCC 1/8 corner atom x 8 corners + ½ face atom x 6 faces=4 atoms/ucHCP 3 inside atoms + ½ basal atoms x 2 bases + 1/ corner atoms x 12 corners=6 atoms/uc
16 Relationship between atomic radius and edge lengths For FCC:a = 2R√2For BCC:a = 4R /√3For HCPa = 2Rc/a = (for ideal case)Note: c/a ratio could be less or more than the ideal value of 1.633
21 ATOMIC PACKING FACTOR• APF for a simple cubic structure = 0.526
22 ATOMIC PACKING FACTOR: BCC • APF for a body-centered cubic structure = 0.68a = 4R /√38
23 FACE CENTERED CUBIC STRUCTURE (FCC) • Close packed directions are face diagonals.--Note: All atoms are identical; the face-centered atoms are shadeddifferently only for ease of viewing.• Coordination # = 12
24 ATOMIC PACKING FACTOR: FCC • APF for a face-centered cubic structure = 0.74a = 2R√2
25 3.5 Density ComputationsDensity of a material can be determined theoretically from the knowledge of its crystal structure (from its Unit cell information)Density= mass/VolumeMass is the mass of the unit cell and volume is the unit cell volume.mass = ( number of atoms/unit cell) “n” x mass/atommass/atom = atomic weight “A”/Avogadro’s Number “NA”Volume = Volume of the unit cell “Vc”
27 Example problem on Density Computation Problem: Compute the density of CopperGiven: Atomic radius of Cu = nm (1.28 x 10-8 cm)Atomic Weight of Cu = 63.5 g/molCrystal structure of Cu is FCCSolution: = n A / Vc NAn = 4Vc= a3 = (2R√2)3 = 16 R3 √2NA = x 1023 atoms/mol= 4 x 63.5 g/mol / 16 √2(1.28 x 10-8 cm)3 x x 1023 atoms/molAns = 8.98 g/cm3Experimentally determined value of density of Cu = 8.94 g/cm3
28 3.6 Polymorphism and Allotropy Polymorphism The phenomenon in some metals, as well as nonmetals, having more than one crystal structures.When found in elemental solids, the condition is often called allotropy.Examples:Graphite is the stable polymorph at ambient conditions, whereas diamond is formed at extremely high pressures.Pure iron is BCC crystal structure at room temperature, which changes to FCC iron at 912oC.
29 POLYMORPHISM AND ALLOTROPY BCC (From room temperature to 912 oC)FeFCC (at Temperature above 912 oC)912 oCFe (BCC) Fe (FCC)
30 3.7 Crystal SystemsSince there are many different possible crystal structures, it is sometimes convenient to divide them into groups according to unit cell configurations and/or atomic arrangements.One such scheme is based on the unit cell geometry, i.e. the shape of the appropriate unit cell parallelepiped without regard to the atomic positions in the cell.Within this framework, an x, y, and z coordinate system is established with its origin at one of the unit cell corners; each x, y, and z-axes coincides with one of the three parallelepiped edges that extend from this corner, as illustrated in Figure.
31 The Lattice Parameters a, b, c, , , are called the latticeParameters.
32 Seven different possible combinations of edge lengths and angles give seven crystal systems. Shown in Table 3.2Cubic system has the greatest degree of symmetry.Triclinic system has the least symmetry.
33 The Lattice Parameters Lattice parameters area, b, c, , , are called the latticeParameters.
35 3.8 Point Coordinates in an Orthogonal Coordinate System Simple Cubic
36 3.9 Crystallographic Directions in Cubic System Determination of the directional indices in cubic system:Four Step Procedure (Text Book Method)Draw a vector representing the direction within the unit cell such that it passes through the origin of the xyz coordinate axes.Determine the projections of the vector on xyz axes.Multiply or divide by common factor to obtain the three smallest integer values.Enclose the three integers in square brackets [ ].e.g. [uvw]u, v, and w are the integers
37 Crystallographic Directions in Cubic System 
39 Head and Tail Procedure for determining Miller Indices for Crystallographic Directions Find the coordinate points of head and tail points.Subtract the coordinate points of the tail from the coordinate points of the head.Remove fractions.Enclose in [ ]
40 Indecies of Crystallographic Directions in Cubic System Direction AHead point – tail point(1, 1, 1/3) – (0,0,2/3)1, 1, -1/3Multiply by 3 to get smallest integers3, 3, -1A = [33Ī]Direction BHead point – tail point(0, 1, 1/2) – (2/3,1,1)-2/3, 0, -1/2Multiply by 6 to get smallest integers_ _ B = C = [???]D = [???]
41 Indices of Crystallographic Directions in Cubic System Direction CHead Point – Tail Point(1, 0, 0) – (1, ½, 1)0, -1/2, -1Multiply by 2 to get the smallest integers C = [0I2]Direction DHead Point – Tail Point(1, 0, 1/2) – (1/2, 1, 0)1/2, -1, 1/2Multiply by 2 to get the smallest integersD = [I2I]B= [???]A = [???]
42 Crystallographic Directions in Cubic System 
46 3.10 MILLER INDICES FOR CRYSTALLOGRAPHIC PLANES Miller Indices for crystallographic planes are the reciprocals of the fractional intercepts (with fractions cleared) which the plane makes with the crystallographic x,y,z axes of the three nonparallel edges of the cubic unit cell.4-Step Procedure:Find the intercepts that the plane makes with the three axes x,y,z. If the plane passes through origin change the origin or draw a parallel plane elsewhere (e.g. in adjacent unit cell)Take the reciprocal of the interceptsRemove fractionsEnclose in ( )
47 Miller Indecies of Planes in Crystallogarphic Planes in Cubic System
48 Drawing Plane of known Miller Indices in a cubic unit cell Draw ( ) plane
49 Miller Indecies of Planes in Crystallogarphic Planes in Cubic System Origin for AOrigin for BOrigin for AA = (IĪ0) B = (I22)A = (2IĪ) B = (02Ī)
50 CRYSTALLOGRAPHIC PLANES AND DIRECTIONS IN HEXAGONAL UNIT CELLS Miller-Bravais indices -- same as Miller indices for cubic crystals except that there are 3 basal plane axes and 1 vertical axis. Basal plane -- close packed plane similar to the (1 1 1) FCC plane. contains 3 axes 120o apart.
51 Direction Indices in HCP Unit Cells – [uvtw] where t=-(u+v) Conversion from 3-index system to 4-index system: Miller Bravais indices are h,k,i,l with i = -(h+k). Basal plane indices ( )
52 Miller-Bravais Indices for crystallographic planes in HCP _ (1211)
53 Miller-Bravais Indices for crystallographic directions and planes in HCP
57 3.11 Linear and Planar Atomic Densities Linear Density “LD”is defined as the number of atoms per unit length whose centers lie on the direction vector of a given crystallographic direction.
58 Linear Density LD for  in BCC. # of atom centered on the direction vector = 1/2 +1/2 = 1Length of direction vector  = 2 aa = 4R/ 3 2a
59 Linear Density LD of  in FCC # of atom centered on the direction vector  = 2 atomsLength of direction vector  = 4RLD = 2 /4RLD = 1/2RLinear density can be defined as reciprocal of the repeat distance ‘r’LD = 1/r
60 Area of the plane Planar Density Planar Density “PD” is defined as the number of atoms per unit area that are centered on a given crystallographic plane.No of atoms centered on the planePD = —————————————Area of the plane
61 Planar Density of (110) plane in FCC # of atoms centered on the plane (110)= 4(1/4) + 2(1/2) = 2 atomsArea of the plane= (4R)(2R 2) = 8R22(111) Plane in FCCa = 2R 24R
62 Closed Packed Crystal Structures FCC and HCP both have:CN = 12 and APF = 0.74APF= 0.74 is the most efficient packing.Both FCC and HCP have Closed Packed PlanesFCC ----(111) plane is the Closed Packed PlaneHCP ----(0001) plane is the Closed Packed PlaneThe atomic staking sequence in the above two structures is different from each other
66 Crystalline and Noncrystalline Materials 3.13 Single Crystals For a crystalline solid, when the periodic and repeated arrangement of atoms is perfect or extends throughout the entirety of the specimen without interruption, the result is a single crystal.All unit cells interlock in the same way and have the same orientation.Single crystals exist in nature, but may also be produced artificially.They are ordinarily difficult to grow, because the environment must be carefully controlled.Example: Electronic microcircuits, which employ single crystals of silicon and other semiconductors.
67 Polycrystalline Materials 3.13 Polycrytalline MaterialsPolycrystalline crystalline solids composed of many small crystals or grains.Various stages in the solidification :Small crystallite nuclei Growth of the crystallites.Obstruction of some grains that are adjacent to one another is also shown.Upon completion of solidification, grains that are adjacent to one another is also shown.Grain structure as it would appear under the microscope.
68 3.15 AnisotropyThe physical properties of single crystals of some substances depend on the crystallographic direction in which the measurements are taken.For example, modulus of elasticity, electrical conductivity, and the index of refraction may have different values in the  and  directions.This directionality of properties is termed anisotropy.Substances in which measured properties are independent of the direction of measurement are isotropic.