Structure of Crystalline Solids PDF

The Structure of Crystalline Solids Dr. Madan Lal Chandravanshi Assistant Professor Mechanical Engineering Department, IIT(ISM) Dhanbad 1  What is solid state physics? Explains the properties of solid materials. Exp...

The Structure of Crystalline Solids Dr. Madan Lal Chandravanshi Assistant Professor Mechanical Engineering Department, IIT(ISM) Dhanbad 1  What is solid state physics? Explains the properties of solid materials. Explains the properties of a collection of atomic nuclei and electrons interacting with electrostatic forces. Formulates fundamental laws that govern the behavior of solids.  CLASSIFICATION OF SOLIDS SOLID MATERIALS CRYSTALLINE POLYCRYSTALLINE AMORPHOUS (Non-crystalline) Single Crystal Crystal Structure 3  Crystal Structures Groups Lattice Unit cell 4  Crystalline Solids Crystalline materials are solids with an atomic structure based on a regular repeated pattern. The majority of all solids are crystalline. More progress has been made in understanding the behavior of crystalline solids than that of non- crystalline materials since the calculation are easier in crystalline materials. Understanding the electrical properties of solids is right at the heart of modern society and technology.  Energy and Packing Non dense, random packing Energy typical neighbor bond length typical neighbor r bond energy Dense, ordered packing Energy typical neighbor bond length typical neighbor r bond energy a. Dense, ordered packed structures tend to have lower energies in molecules (not bond energy). 6 Electrochemical energy storage materials 7  Materials and Packing Crystalline materials... atoms pack in periodic, 3D arrays typical of: -metals -many ceramics -some polymers crystalline SiO2 Si Oxygen Noncrystalline materials... atoms have no periodic packing occurs for: -complex structures -rapid cooling "Amorphous" = Noncrystalline noncrystalline SiO2 Plastics, glass, rubber, metallic glass, polymers, gel, fused silica, tar, thin layer lubricants, and wax are examples of amorphous solids 8  SINGLE CRYSTALS Single crystals have a periodic atomic structure across its whole volume. At long range length scales, each atom is related to every other equivalent atom in the structure by translational or rotational symmetry Single Pyrite Crystal Amorphous Solid Single Crystals 9  POLYCRYSTALLINE SOLIDS Polycrystalline materials are made up of an aggregate of many small single crystals (also called crystallites or grains). Polycrystalline materials have a high degree of order over many atomic or molecular dimensions. Grains (domains) are separated by grain boundaries. The atomic order can vary from one domain to the next. The grains are usually 100 nm - 100 microns in diameter. Polycrystals with grains less than 10 nm in diameter are Nano crystalline Polycrystalline Pyrite form (Grain) 10  AMORPHOUS SOLIDS Amorphous (Non-crystalline) Solids are made up of randomly orientated atoms , ions, or molecules that do not form defined patterns or lattice structures. These materials have order only within a few atomic or molecular dimensions and do not have any long-range order, Example: amorphous materials include amorphous silicon, plastics which can be used in solar cells and thin film transistors. 11  CRYSTAL LATTICE What is a crystal lattice? In crystallography, only the geometrical properties of the crystal are of our interest, therefore one replaces each atom by a geometrical point located at the equilibrium position of that atom. Platinum Platinum surface (scanning tunneling microscope) Crystal lattice and structure of Platinum 12 Three common Unit Cells in 3D 13 Unit cell 14 Definition of Reciprocal Lattice 15 Types of Lattice Crystal Structure 16  Metallic Crystal Structures Tend to be densely packed. Reasons for dense packing: - Typically, only one element is present, so all atomic radii are the same. - Metallic bonding is not directional. - Nearest neighbor distances tend to be small in order to lower bond energy. - Electron cloud shields cores from each other Have the simplest crystal structures. 17  Simple Cubic Structure (SC) Rare due to low packing density (only Po (Polonium) has this structure) Close-packed directions are cube edges. Coordination # = 6 (# nearest neighbors) (Courtesy P.M. Anderson) 18  Atomic Packing Factor (APF) Volume of atoms in unit cell* APF = Volume of unit cell *assume hard spheres APF for a simple cubic structure = 0.52 volume atoms atom a 4 unit cell 1 p (0.5a) 3 3 R=0.5a APF = a3 volume close-packed directions unit cell contains 8 x 1/8 = 1 atom/unit cell Adapted from Fig. 3.24, Callister & Rethwisch 8e. 19  Body Centered Cubic Structure (BCC) Atoms touch each other along cube diagonals. --Note: All atoms are identical; the center atom is shaded differently only for ease of viewing. ex: Cr, W, Fe (), Tantalum, Molybdenum Coordination # = 8 Adapted from Fig. 3.2, Callister & Rethwisch 8e. (Courtesy P.M. Anderson) 2 atoms/unit cell: 1 center + 8 corners x 1/8 20  Atomic Packing Factor: BCC APF for a body-centered cubic structure = 0.68 3a a 2a Close-packed directions: R length = 4R = 3 a a atoms volume 4 unit cell 2 p ( 3a/4) 3 3 atom APF = volume a3 unit cell 21  Face Centered Cubic Structure (FCC) Atoms touch each other along face diagonals. --Note: All atoms are identical; the face-centered atoms are shaded differently only for ease of viewing. ex: Al, Cu, Au, Pb, Ni, Pt, Ag Coordination # = 12 Adapted from Fig. 3.1, Callister & Rethwisch 8e. 4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8 22  Atomic Packing Factor: FCC APF for a face-centered cubic structure = 0.74 maximum achievable APF Close-packed directions: length = 4R = 2 a 2a Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell a atoms volume 4 unit cell 4 p ( 2a/4) 3 3 atom APF = 3 volume a Adapted from Fig. 3.1(a), Callister & Rethwisch 8e. unit cell 23 Common Crustal Structures 24  FCC Stacking Sequence ABCABC... Stacking Sequence 2D Projection B B C A A sites B B B C C B sites B B C sites A FCC Unit Cell B C 25 Hexagonal Close-Packed Structure (HCP) ABAB... Stacking Sequence 3D Projection 2D Projection A sites Top layer c B sites Middle layer A sites Bottom layer a Adapted from Fig. 3.3(a), Callister & Rethwisch 8e. Coordination # = 12 6 atoms/unit cell APF = 0.74 ex: Cd, Mg, Ti, Zn c/a = 1.633 26 Hexagonal Close-Packed Structure (HCP) 27  Theoretical Density,  Ex: Cr (BCC) A = 52.00 g/mol (Atomic weight) R = 0.125 nm n = 2 atoms/unit cell R a a = 4R/ 3 = 0.2887 nm atoms g unit cell 2 52.00 theoretical = 7.18 g/cm3 mol = actual = 7.19 g/cm3 a3 6.022 x 1023 volume atoms unit cell mol 28 Miller indices 29 Miller indices – Example 30 31  X-Ray Diffraction Pattern z z z c c c y (110) y y a b a b a b Intensity (relative) x x x (211) (200) Diffraction angle 2q Diffraction pattern for polycrystalline -iron (BCC) Adapted from Fig. 3.22, Callister 8e. 32 Crystallographic planes 33 Crystallographic planes 34 Hexagonal Crystals 35 Planes and Directions in Hexagonal Unit Cells – MILLER Bravais Indices 36 Planes and Directions in Hexagonal Unit Cells 37 Planes and Directions in Hexagonal Unit Cells 38 Hexagonal Unit Cell- Example 39 Hexagonal Unit Cell- Examples 40  Hexagonal Crystal The first three indices pertain to projections along the respective a1, a2, and a3 axes in the basal plane the four values of miller bravaias may be obtained from equation: Conversion and Construction of Miller Bravais Directional Indices for a Hexagonal Unit Cell from Miller Indices for hexagonal crystals. 41 X-RAY DIFFRACTION: DETERMINATION OF CRYSTAL STRUCTURES Our understanding regarding the atomic and molecular arrangements in solids has resulted from x-ray diffraction investigations; furthermore, x-rays are still very important in developing new materials. Diffraction occurs when a wave encounters a series of regularly spaced obstacles that (1) are capable of scattering the wave, and (2) have spacings that are comparable in magnitude to the wavelength. Furthermore, diffraction is a consequence of specific phase relationships established between two or more waves that have been scattered by the obstacles. https://www.youtube.com/watch?v=xBA09PXPPR4 42 diffraction is a consequence of specific phase relationships established between two or more waves that have been scattered by the obstacles. 43  X-Ray Diffraction and Bragg’s Law X-rays are a form of electromagnetic radiation that have high energies and short wavelengths—wavelengths on the order of the atomic spacings for solids. When a beam of x-rays impinges on a solid material, a portion of this beam will be scattered in all directions by the electrons associated with each atom or ion that lies within the beam’s path. Two parallel planes of atoms A– and B– in Figure 3.20, the same h, k, and l Miller indices are separated by the interplanar spacing dhkl. Now assume that a parallel, monochromatic, and coherent (in-phase) 44  The diffractometer used to determine the angles at which X-ray Diffraction diffraction occurs for powdered specimens Figure 3.20 Diffraction of x-rays by planes of atoms (A–A’ and B–B’). 45 Bragg’s law 46 47 Relative Sizes and Microscopy Examination tools 48  Theoretical Density,  Mass of Atoms in Unit Cell Density =  = Total Volume of Unit Cell nA  = VC NA where n = number of atoms/unit cell A = atomic weight VC = Volume of unit cell = a3 for cubic NA = Avogadro’s number = 6.022 x 1023 atoms/mol 49 Linear and Planer densities 50 Linear Density- Example 51  Densities of Material Classes In general Graphite/ metals > ceramics > polymers Metals/ Composites/ Ceramics/ Polymers Alloys fibers Semicond 30 Based on data in Table B1, Callister 20 Platinum *GFRE, CFRE, & AFRE (Aramid Fiber- Gold, W Metals have... Tantalum Reinforced Epoxy composites) are Glass, close-packing 10 Silver, Mo (metallic bonding) Cu,Ni Steels often large atomic masses Tin, Zinc Zirconia  (g/cm3 ) 5 Ceramics have... 4 Titanium Al oxide less dense packing 3 Diamond Si nitride Aluminum Glass -soda often lighter elements Concrete Silicon PTFE Glass fibers 2 GFRE Polymers have... Magnesium Graphite Silicone Carbon* fiber PVC low packing density PET PC CFRE 1 HDPE, PS Aramid fibers (often amorphous) PP, LDPE lighter elements (C,H,O) AFRE 0.5 PET-polyethylene Terephthalate. Wood Composites have... 0.4 Polycarbonates (PC) intermediate values 0.3 Data from Table B.1, Callister & Rethwisch, 8e. 52  Crystals as Building Blocks Some engineering applications require single crystals: -- diamond single crystals for abrasives -- Ex: Quartz fractures more easily along some crystal planes than others. -- Turbine blades 53  Single vs Polycrystals Single Crystals E (diagonal) = 273 GPa -Properties vary with direction: anisotropic. -Example: the modulus of elasticity (E) in BCC iron: E (edge) = 125 GPa Polycrystals -Properties may/may not 200 mm vary with direction. -If grains are randomly oriented: isotropic. (Epoly iron = 210 GPa) - 54  Polymorphism Two or more distinct crystal structures for the same material (allotropy/polymorphism) titanium Iron system , -Ti liquid 1538ºC carbon BCC -Fe diamond, graphite, Uckyball – C60 1394ºC FCC -Fe 912ºC BCC -Fe 55 Polymer structures 56  Polymer Structures Linear Branched Cross-Linked Network 57 Thermo-sets – Thermo-plasts 58  Ceramic crystal structures Ceramic Crystal structures 59  Ceramic crystal structures Ceramic Crystal structures 60  Ceramic crystal structures Silicates 61 Carbon 62 Thanks For Your Attention 63