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## Chapter 3

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**Chapter 3**The structure of crystalline solids Dr. Mohammad Abuhaiba, PE**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**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**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**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**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**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**3.5 Density computation**• Example problem 3.3 Dr. Mohammad Abuhaiba, PE**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**3.7 Crystal systems**• 7 possible systems (T3.2) and (F3.4). Dr. Mohammad Abuhaiba, PE**3.8 Crystallographic Point Coordinates**• F3.5 • Example Problem 3.4 • Example Problem 3.5 Dr. Mohammad Abuhaiba, PE**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**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**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**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**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**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**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**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**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**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**3.13 SINGLE CRYSTALS**Properties of single crystal materials depend upon chemical composition and specific directions within the crystal. Dr. Mohammad Abuhaiba, PE**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**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**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**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**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**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**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**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**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**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**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**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**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