9/16/ :30 PM. Chapter 3. The structure of crystalline solids. Mohammad Suliman Abuhaiba, Ph.D., PE

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1 Chapter 3 The structure of crystalline solids 1 Mohammad Suliman Abuhaiba, Ph.D., PE

2 2 Home Work Assignments HW 1 2, 7, 12, 17, 22, 29, 34, 39, 44, 48, 53, 58, 63 Due Sunday 17/9/2015

3 3 Why study the structure of crystalline solids? Properties of some materials are directly related to their crystal structure. Significant property differences exist between crystalline and noncrystalline materials having the same composition.

4 4 Crystal structures Fundamental concepts Crystalline materials: Atoms or ions form a regular repetitive, grid-like pattern, in 3D A lattice: collection of points called lattice points, arranged in a periodic pattern so that the surroundings of each point in the lattice are identical

5 5 Crystal structures Unit Cells Atoms arrange themselves into an ordered, 3d pattern called a crystal Unit cell (UC): smallest repeating volume within a crystal Each cell has all geometric features found in the total crystal

6 6 Metallic crystal structures Lattice Parameter: size & shape of UC, includes: 1. dimensions of sides of UC 2. angles between sides

7 7 Metallic crystal structures Face Centered Cubic (FCC) FCC: AL. Cu, Pb, Ag, Ni 4 atoms per unit cell Lattice parameter, a FCC Coordination Number (CN): # of atoms touching a particular atom, or # of nearest neighbors for that particular atom

8 8 Metallic crystal structures Face Centered Cubic (FCC) CN =12 APF = 0.74 Metals typically have relatively large APF to max shielding provided by the free electron cloud

9 9 Metallic crystal structures Face Centered Cubic (FCC)

10 10 Metallic crystal structures Example problem 3.1 Calculate 1. lattice parameter in terms of atomic radius R 2. volume of FCC UC in terms of R 3. atomic packing factor

11 11 Metallic crystal structures Body Centered Cubic (BCC) 2 atoms/uc Lattice parameter, a BCC CN = 8 APF = 0.68

12 12 Metallic Crystal Structures Hexagonal Closed-Packed Structure (HCP) c/a = No of atoms = 6 APF = 0.74

13 13 Density computation Density (No. of atoms/cell)(atomic mass) (volume of unit cell)(n ) (atoms)(g/mole) (cm )(atoms/mole) g cm 3 3 A

14 14 Density computation Example problem 3.2 Copper has an atomic radius of nm, an FCC crystal structure, and an atomic weight of 63.5 g/mol. Compute its theoretical density and compare the answer with its measured density.

15 15 Polymorphism and Allotropy Polymorphism: Materials that can have more than one crystal structure When found in pure elements condition is termed Allotropy A volume change may accompany transformation during heating or cooling. This volume change may cause brittle ceramic materials to crack and fail. Ex: Fe has BCC at RT which changes to FCC at 912 C

16 16 Crystal systems 7 possible systems (T3.2) and (F3.4).

17 17 Crystallographic Point Coordinates Point

18 18 Crystallographic Point Coordinates Example Problem 3.3 For the unit cell shown, locate the point having coordinates

19 19 Crystallographic Point Coordinates Example Problem 3.4 Specify point coordinates for all atom positions for a BCC UC.

20 20 Crystallographic Directions Metals deform in directions along which atoms are in closest contact. Many properties are directional. Miller indices are used to define directions.

21 21 Crystallographic Directions Procedure of finding Miller indices: RH coordinate system Find coordinates of 2 points along direction Subtract tail from head Clear fractions Enclose No s in [634] A direction and its ve are not identical A direction and its multiple are identical

22 22 Example Problem 3.5 Determine the indices for the direction shown in the figure.

23 23 Example Problem 3.6 Draw a direction within a cubic UC

24 24 Crystallographic Directions Families of directions Identical directions: any directional property will be identical in these directions Directions in cubic crystals having same indices without regard to order or sign are equivalent

25 25 Crystallographic Directions Miller-Bravais indices for hexagonal unit cells Directions in HCP: 3-axis or 4-axis system Move in each direction to get from tail to head of direction, while for consistency still making sure that u + v = -t

26 26 Crystallographic Directions Miller-Bravais indices for hexagonal unit cells Conversion from 3 axis to 4 axis: u = 1/3(2u` - v`) v = 1/3(2v` - u`) t = -(u + v) w = w` u`, v`, w`: indices in 3 axis system Clear fraction or reduce to lowest integer for values of u, v, t, and w.

27 27 Crystallographic Directions Example Problem 3.8 a) Convert [111] direction into four-index system for hexagonal crystals. b) Draw [111] direction within a HCP cell that utilizes a three-axis (a 1, a 2, z) coordinate system.

28 28 Example Problem 3.9 Determine the directional indices (fourindex system) for the direction shown.

29 29 Crystallographic planes Crystal contains planes of atoms that influence properties and behavior of a material. Metals deform along planes of atoms that are most tightly packed together. The surface energy of different faces of a crystal depends upon the particular crystallographic planes.

30 30 Crystallographic planes Calculation of planes Identify points at which plane intercepts x,y,z coordinates. If plane passes in origin of coordinates; system must be shifted. Take reciprocals of intercepts Clear fractions but do not reduce to lowest integer.

31 31 Crystallographic planes Calculation of planes Notes: Planes and their ve are identical Planes and their multiple are not identical For cubic crystals, planes and directions having the same indices are perpendicular to one another

32 32 Crystallographic planes Example Problem 3.10 Determine Miller indices for the plane shown in the figure.

33 33 Crystallographic planes Example Problem 3.11

34 34 Crystallographic planes Example Problem 3.12 Determine Miller Bravais indices for the plane shown in the hexagonal unit cell.

35 35 Crystallographic planes Atomic arrangements and Families of planes {} 2 or more planes may belong to same family of planes (same planner density) In cubic system only, planes having same indices irrespective of order & sign are equivalent. Ex: {111} is a family of planes that has 8 planes.

36 36 Linear and Planar Densities Repeat distance: distance between lattice points along the direction Repeating distance between equivalent sites differs from direction to direction. Ex: in the [111] of a BCC metal, lattice site is repeated every 2R. Ex: repeating distance in [110] for a BCC is, but for FCC.

37 37 Linear and Planar Densities Linear density: No of atoms per unit length along the direction. Equivalent directions have identical LDs In general, LD = 1 / repeat distance Example: find linear density along [110] for FCC. LD = number of atoms centered on direction vector length of direction vector

38 38 Linear and Planar Densities Planar packing fraction: fraction of area of the area of a plane actually covered by atoms In cubic systems, a direction that has the same indices as a plane is perpendicular to that plane. PD = number of atoms centered on a plane area of plane

39 39 Linear and Planar Densities PD = number of atoms centered on a plane area of plane

40 40 Linear and Planar Densities Ex: How many atoms per mm 2 are there on the (100) and (111) planes of lead (FCC) LD & PD are important considerations relative to the process of slip. Slip is the mechanism by which metals plastically deform Slip occurs on the most closely packed planes along directions having greatest LD

41 41 Crystallographic planes Closed Packed Planes and directions Close packed planes are (0001) & (0002) named basal planes. An HCP unit cell is built up by stacking together CPPs in a ABABAB stacking sequence.

42 42 Crystallographic planes Closed Packed Planes and directions Atoms in plane B (0002) fit into valleys between atoms on plane A (0001) Center atom in a basal plane is touched by: 6 atoms in same plane 3 atoms in a lower plane 3 atoms in upper plane CN =12

43 43 Crystallographic planes Closed Packed Planes and directions In FCC, CPPs are of the form {111} When parallel (111) planes are stacked: atoms in plane B fit over valleys in plane A atoms in plane C fit over valleys in both A & B 4 th plane fits directly over atoms in A A stacking sequence ABCABCABC is produced using (111) plane CN = 12

44 44 Crystallographic planes Closed Packed Planes and directions

45 45 Single Crystals Properties of single crystal materials depend upon chemical composition & specific directions within crystal Mohammad Suliman Abuhaiba, Ph.D., PE

46 46 Polycrystalline Materials Many properties of polycrystalline materials depend upon the physical and chemical char of both grains and grain boundaries.

47 47 Polycrystalline Materials Figure 3.18: stages in solidification of a polycrystalline material; square grids depict unit cells a) Small crystallite nuclei b) Growth of crystallites c) Upon completion of solidification, grains having irregular shapes have formed d) Grain structure as it would appear under the microscope; dark lines are grain boundaries

48 48 Polycrystalline Materials Figure 3.18

49 49 Anisotropy Anisotropic material: properties depend on the crystallographic direction along which the property is measured Isotropic material: properties are identical in all directions Most polycrystalline materials will exhibit isotropic properties. Table 3.3

50 50 Anisotropy

51 51 X-Ray Diffraction: Determination of Crystal Structures X-rays Electromagnetic radiation Wavelengths between 0.1Å & 100Å Similar to inter-atomic distances in a crystal

52 52 X-Ray Diffraction: Determination of Crystal Structures X-ray diffraction is a tool used to: 1. Identify phases by comparison with data from known structures 2. Quantify changes in cell parameters, orientation, crystallite size and other structural parameters 3. Determine crystallographic structure of novel or unknown crystalline materials.

53 53 X-Rray diffraction: Determination of Crystal Structures - Bragg s law An X-ray incident upon a sample will either: 1. Be transmitted: ray will continue along its original direction 2. Be scattered by electrons of the atoms in the material Interested in peaks formed when scattered X-rays constructively interfere.

54 54 X-Rray diffraction: Determination of Crystal Structures - Bragg s law Constructive Interference

55 55 X-Rray diffraction: Determination of Crystal Structures - Bragg s law Destructive Interference

56 56 X-Rray diffraction: Determination of Crystal Structures - Bragg s law Angle between transmitted & Bragg diffracted beams is always equal to 2θ This angle is readily obtainable in experimental situations Results of X-ray diffraction are frequently given in terms of 2θ

57 57 X-Rray diffraction: Determination of Crystal Structures - Bragg s law

58 58 X-RAY Diffraction: Determination of Crystal Structures - Inter-planar spacing Distance between adjacent parallel planes of atoms with the same Miller indices, d hkl In cubic cells, it s given by d hkl a o h k l

59 59 Example Calculate distance between adjacent (111) planes in gold which has a lattice constant of A.

60 60 X-RAY Diffraction: Determination of Crystal Structures - Powder diffraction A powder is a polycrystalline material There are all possible orientations of the crystals Similar planes in different crystals will scatter in different directions

61 61 X-RAY Diffraction: Determination of Crystal Structures - Powder diffraction In single crystal X-ray diffraction there is only one orientation. For a given wavelength & sample setting relatively few reflections can be measured: possibly zero, one or two As other crystals are added with slightly different orientations, several diffraction spots appear at the same 2θ value and spots start to appear at other values of 2θ.

62 62 X-RAY Diffraction: Determination of Crystal Structures - Powder diffraction Schematic diagram of an x-ray diffracto - meter T = x-ray source S = specimen C = detector O = axis of rotation of specimen & detector

63 63 X-RAY Diffraction: Determination of Crystal Structures - X-ray Diffractometer As the counter moves at constant angular velocity, a recorder automatically plots diffracted beam intensity as a function of 2q.

64 64 X-RAY Diffraction: Determination of Crystal Structures - X-ray Diffractometer Fig 3.22: a diffraction pattern for a powdered specimen High-intensity peaks result when Bragg diffraction condition is satisfied by some set of crystallographic planes. These peaks are plane-indexed in the figure

65 65 X-RAY Diffraction: Determination of Crystal Structures - X-ray Diffractometer Unit cell size & geometry may be resolved from angular positions of diffraction peaks Arrangement of atoms within unit cell is associated with relative intensities of these peaks.

66 66 X-RAY Diffraction: Determination of Crystal Structures Indexing Diffraction Pattern λ = 2 d sin θ d hkl a o h k l

67 67 Example Problem 3.13 For BCC iron, compute a. inter-planar spacing b. diffraction angle for (220) set of planes The lattice parameter for Fe is nm. Also, assume that monochromatic radiation having a wavelength of nm is used, and the order of reflection is 1.

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