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Chapter 3

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Chapter 3

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  1. Chapter 3 The structure of crystalline solids Dr. Mohammad Abuhaiba, PE

  2. 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 non-crystalline materials having the same composition. Dr. Mohammad Abuhaiba, PE

  3. 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. Unit Cells • Atoms arrange themselves into an ordered, 3d pattern called a crystal. • Unit cell: small repeating volume within a crystal • Each cell has all geometric features found in the total crystal. Dr. Mohammad Abuhaiba, PE

  4. Metallic crystal structures • The atomic bonding is metallic and thus non-directional in nature • Lattice Parameter: describe size and shape of unit cell, include: • dimensions of the sides of the unit cell • angles between the sides (F3.4). Dr. Mohammad Abuhaiba, PE

  5. Metallic crystal structuresFace Centered Cubic (FCC) • Table 3.1: atomic radii and crystal structures of 16 metals • FCC: AL. Cu, Pb, Ag, Ni • 4 atoms per unit cell • Lattice parameter, aFCC • Coordination Number (CN): # of atoms touching a particular atom, or # of nearest neighbors for that particular atom • For FCC, CN=12 • APF for FCC = 0.74 • Metals typically have relatively large APF to max the shielding provided by the free electron cloud. • Example problem 3.1 • Example problem 3.2 Dr. Mohammad Abuhaiba, PE

  6. Metallic crystal structuresBody Centered Cubic (BCC) • Figure 3.2 • 2 atoms/unit cell • Lattice parameter, aBCC • CN = 8 • APF = 0.68 Dr. Mohammad Abuhaiba, PE

  7. Metallic crystal structuresHexagonal Closed-Packed Structure (HCP) • F3.3 • Lattice parameter • c/a = 1.633 • APF = 0.74 • No of atoms = 6 Dr. Mohammad Abuhaiba, PE

  8. 3.5 Density computation • Example problem 3.3 Dr. Mohammad Abuhaiba, PE

  9. 3.6 Polymorphism and Allotropy • Polymorphism: Materials that can have more than one crystal structure. • When found in pure elements the 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: zirconia (ZrO2): • At 25oC is monoclinic • At 1170oC, monoclinic zirconia transforms into a tetragonal structure • At 2370oC, it transforms into a cubic form • Ex: Fe has BCC at RT which changes to FCC at 912C Dr. Mohammad Abuhaiba, PE

  10. 3.7 Crystal systems • 7 possible systems (T3.2) and (F3.4). Dr. Mohammad Abuhaiba, PE

  11. 3.8 Crystallographic Point Coordinates • F3.5 • Example Problem 3.4 • Example Problem 3.5 Dr. Mohammad Abuhaiba, PE

  12. 3.9 Crystallographic Directions • Metals deform in directions along which atoms are in closest contact. • Many properties are directional. • Miller indices are used to define directions. Procedure of finding Miller indices: • Using RH coordinate system, find coordinates of 2 points that lie along the direction. • Subtract tail from head. • Clear fractions. • Enclose the No’s in [ 634 ], -ve sign (bar above no.) • A direction and its –ve are not identical. • A direction and its multiple are identical. • Example Problem 3.6 • Example Problem 3.7 Dr. Mohammad Abuhaiba, PE

  13. 3.9 Crystallographic DirectionsFamilies of directions • Identical directions: any directional property will be identical in these directions. • Ex: all parallel directions possess the same indices. Sketch rays in the direction that pass through locations: 0,0,0, and 0,1,0 and ½,1,1 • Directions in cubic crystals having the same indices without regard to order or sign are equivalent Dr. Mohammad Abuhaiba, PE

  14. 3.9 Crystallographic DirectionsMiller-Bravais indices for hexagonal unit cells • Directions in HCP: 3-axis or 4-axis system. • We determine No. of lattice parameters we must move in each direction to get from tail to head of direction, while for consistency still making sure that u + v = -t. • Conversion from 3 axis to 4 axis: • u = 1/3(2u` - v`) • v = 1/3(2v` - u`) • t = -(u + v) • w = w` • u`, v`, and w` are the indices in the 3 axis system • After conversion, clear fraction or reduce to lowest integer for the values of u, v, t, and w. • Example Problem 3.8 • Example Problem 3.9 Dr. Mohammad Abuhaiba, PE

  15. 3.10 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. Dr. Mohammad Abuhaiba, PE

  16. 3.10 Crystallographic planesCalculation of planes • Identify points at which plane intercepts x,y,z coordinates. If plane passes in origin of coordinates; system must be moved. • Take reciprocals of intercepts • Clear fractions but do not reduce to lowest integer ( ). 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 • Example Problem 3.10 • Example Problem 3.11 • Example Problem 3.12 Dr. Mohammad Abuhaiba, PE

  17. 3.10 Crystallographic planesAtomic arrangements and Families of planes {} • 2 or more planes may belong to the same family of planes (all have the same planner density) • In cubic system only, planes having the same indices irrespective of order and sign are equivalent. • ex: in simple cubic • Ex: {111} is a family of planes that has 8 planes. Dr. Mohammad Abuhaiba, PE

  18. 3.11 LINEAR AND PLANAR DENSITIES • Repeat distance: distance between lattice points along the direction • Repeating dist between equiv sites differs from direction to direction. • Ex: in the [111] of a BCC metal, lattice site is repeated every 2R. • Ex: repeating dist in [110] for a BCC is , but for FCC. • Linear density:No of atoms per unit length along the direction. • Equivalent directions have identical LDs • In general, LD equals to the reciprocal of the repeat distance • Example: find linear density along [110] for FCC. Dr. Mohammad Abuhaiba, PE

  19. 3.11 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. • Ex: how many atoms per mm2 are there on the (100) and (111) planes of lead (FCC) • LD and 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 the greatest LD. P Dr. Mohammad Abuhaiba, PE

  20. 3.12 Crystallographic planesClosed Packed Planes and directions • close packed planes are (0001) and (0002) named basal planes. • An HCP unit cell is built up by stacking together CPPs in a …ABABAB… stacking sequence. • Figure 3.15 • Atoms in plane B (0002) fit into the valleys between atoms on plane A (0001). • The center atom in a basal plane is touched by 6 other atoms in the same plane, 3 atoms in a lower plane, and 3 atoms in an upper plane. Thus, CN of 12. Dr. Mohammad Abuhaiba, PE

  21. 3.12 Crystallographic planesClosed 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 and • atoms in plane C fit over valleys in both A and B. • the 4th plane fits directly over atoms in A. • Therefore, a stacking sequence …ABCABCABC… is produced using the (111) plane. • CN is 12. • Figure 3.15 • Unlike HCP unit cell, there are 4 sets of nonparallel CPPs: (111), (111`), (11`1), and (1`11) in FCC. Dr. Mohammad Abuhaiba, PE

  22. 3.13 SINGLE CRYSTALS Properties of single crystal materials depend upon chemical composition and specific directions within the crystal. Dr. Mohammad Abuhaiba, PE

  23. 3.14 POLYCRYSTALLINE MATERIALS • Many properties of polycrystalline materials depend upon the physical and chemical char of both grains and grain boundaries. • Figure 3.18 Dr. Mohammad Abuhaiba, PE

  24. 3.15 ANISOTROPY • A material is anisotropic if its properties depend on the crystallographic direction along which the property is measured. • If properties are identical in all directions, the material is isentropic. • Most polycrystalline materials will exhibit isotropic properties. • Table 3.3 Dr. Mohammad Abuhaiba, PE

  25. 3.16 X-RAY DIFFRACTION: DETERMINATION OF CRYSTAL STRUCTURES • X-rays:electromagnetic radiation with wavelengths between 0.1Å and 100Å, typically similar to the inter-atomic distances in a crystal. • X-ray diffraction is an important tool used to: • Identify phases by comparison with data from known structures • Quantify changes in cell parameters, orientation, crystallite size and other structural parameters • Determine (crystallographic) structure (i.e. cell parameters, space group and atomic coordinates) of novel or unknown crystalline materials. Dr. Mohammad Abuhaiba, PE

  26. 3.16 X-RAY DIFFRACTION: DETERMINATION OF CRYSTAL STRUCTURES - Bragg′s law An X-ray incident upon a sample will either: • Be transmitted, in which case it will continue along its original direction • Be scattered by electrons of the atoms in the material. We are primarily interested in peaks formed when scattered X-rays constructively interfere. Dr. Mohammad Abuhaiba, PE

  27. 3.16 X-RAY DIFFRACTION: DETERMINATION OF CRYSTAL STRUCTURES - Bragg′s law • When two parallel X-rays from a coherent source scatter from two adjacent planes their path difference must be an integer number of wavelengths for constructive interference to occur. • Path difference = n λ = 2 d sin θ • λ = 2 dhkl sinθhkl Dr. Mohammad Abuhaiba, PE

  28. 3.16 X-RAY DIFFRACTION: DETERMINATION OF CRYSTAL STRUCTURES - Bragg′s law • The angle between the transmitted and Bragg diffracted beams is always equal to 2θ as a consequence of the geometry of the Bragg condition. • This angle is readily obtainable in experimental situations and hence the results of X-ray diffraction are frequently given in terms of 2θ. • The diffracting plane might not be parallel to the surface of the sample in which case the sample must be tilted to fulfill this condition. Dr. Mohammad Abuhaiba, PE

  29. 3.16 X-RAY DIFFRACTION: DETERMINATION OF CRYSTAL STRUCTURES - Inter-planar spacing • Distance between adjacent parallel planes of atoms with the same Miller indices, dhkl. • In cubic cells, it’s given by • Ex:calcdist between adj (111) planes in gold which has a lattice constant of 4.0786A. Dr. Mohammad Abuhaiba, PE

  30. 3.16 X-RAY DIFFRACTION: DETERMINATION OF CRYSTAL STRUCTURES - Powder diffraction • A powder is a polycrystalline material in which there are all possible orientations of the crystals so that similar planes in different crystals will scatter in different directions. Dr. Mohammad Abuhaiba, PE

  31. 3.16 X-RAY DIFFRACTION: DETERMINATION OF CRYSTAL STRUCTURES - Powder diffraction • In single crystal X-ray diffraction there is only one orientation. • This means that for a given wavelength and 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. • Rings consisting of spots and then rings of even intensity are formed. • A powder pattern consists of rings of even intensity from each accessible reflection at the 2θ angle defined by Bragg's Law. Dr. Mohammad Abuhaiba, PE

  32. 3.16 X-RAY DIFFRACTION: DETERMINATION OF CRYSTAL STRUCTURES - X-ray diffractometer • An apparatus used to determine angles at which diffraction occurs for powdered specimens. A specimen S in the form of a flat plate is supported so that rotations about the axis labeled O are possible. Dr. Mohammad Abuhaiba, PE

  33. 3.16 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. • Fig 3.22 shows a diffraction pattern for a powdered specimen. • The high-intensity peaks result when Bragg diffraction condition is satisfied by some set of crystallographic planes. • These peaks are plane-indexed in the figure. • The unit cell size and geometry may be resolved from the angular positions of the diffraction peaks • Arrangement of atoms within the unit cell is associated with the relative intensities of these peaks. Dr. Mohammad Abuhaiba, PE

  34. EXAMPLE PROBLEM 3.13 Inter-planar Spacing and Diffraction Angle Computations For BCC iron, compute • the inter-planar spacing • the diffraction angle for the (220) set of planes. The lattice parameter for Fe is 0.2866 nm. Also, assume that monochromatic radiation having a wavelength of 0.1790 nm is used, and the order of reflection is 1. Dr. Mohammad Abuhaiba, PE

  35. Home Work Assignments • 2, 7, 12, 17, 22, 29, 34, 39, 44, 48, 53, 58, 63 • Due Saturday 1/10/2011 Dr. Mohammad Abuhaiba, PE