week02. chapter 3 -The strucure of crystalline solids.pdf

Introduction to Material Science The Structure of Crystalline Solids Prof. em. Dr. Alex Dommann Artorg Freiburgstrasse 3 3010 Bern [email protected] Introduction to Material Science Callister Chapter 3 1 1 Chapter 3: The Structure of Crystalline Solids ISSUES TO EXPLORE... • What is the di...

Introduction to Material Science The Structure of Crystalline Solids Prof. em. Dr. Alex Dommann Artorg Freiburgstrasse 3 3010 Bern [email protected] Introduction to Material Science Callister Chapter 3 1 1 Chapter 3: The Structure of Crystalline Solids ISSUES TO EXPLORE... • What is the difference in atomic arrangement between crystalline and noncrystalline solids? • What are the crystal structures of metals? • What are the characteristics of crystal structures? • How are crystallographic points, directions, and planes specified? • What characteristics of a material’s atomic structure determine its density? Callister Chapter 3 Introduction to Material Science 2 2 1 Energy and Packing Energy • Non dense, random packing typical neighbor bond length typical neighbor bond energy r Energy • Dense, ordered packing typical neighbor bond length r typical neighbor bond energy Ordered structures tend to be nearer the minimum in energy and are more stable. Introduction to Material Science Callister Chapter 3 3 3 Materials and Atomic Arrangements Crystalline materials... • atoms arranged in periodic, 3D arrays • typical of: -metals -many ceramics -some polymers crystalline SiO2 Si Noncrystalline materials... Adapted from Fig. 3.24(a), Callister & Rethwisch 10e. Oxygen • atoms have no periodic arrangement • occurs for: -complex structures -rapid cooling noncrystalline SiO2 "Amorphous" = Noncrystalline Callister Chapter 3 Introduction to Material Science Adapted from Fig. 3.24(b), Callister & Rethwisch 10e. 4 4 2 What is Crystallography ? SiO2 natural crystals synthetic crystals ADN …the science of crystals and crystalline materials. Introduction to Material Science Callister Chapter 3 5 5 From Crystals… a) …to 3D-Structures a) Minerals b Coordination compounds a b) c a b Macromolecules b) a Organic compounds b c Callister Chapter 3 a Introduction to Material Science 6 6 3 Protein Crystallization John Kendrew shared the 1962 Nobel Prize in chemistry with Max Perutz for determining the first atomic structures of proteins using X-ray crystallography 1965 Callister Chapter 3 Introduction to Material Science 7 7 Crystallography Structure of Materials Crystalline Solids, Definition of a Crystal Lattices Crystallographic Description Symmetry Crystal Systems Indexing of Crystal Directions Indexing of Crystal Planes ("Miller Indices") Simple Crystal Structures Callister Chapter 3 bcc, fcc, hcp Sphalerite, Wurtzite, Diamond NaCl Introduction to Material Science 8 8 4 Structure of Materials The regularity of the external form lead to the belief of a regular repetition of idential bluiding blocks from: C. Kittel "Introduction to Solid State Physics, Wiley 1976 R.J. Haüy, Paris 1784 Introduction to Material Science Callister Chapter 3 9 9 Definition of a Crystal •Characteristics: ‒ solid ‒ homogeneous ‒ anisotropic ‒ flat faces (sometimes) ‒ typical angles between these faces ‒ three-dimensional periodic array of atoms An ideal crystal is constructed by an infinite repetition in space of identical structural units. Callister Chapter 3 Introduction to Material Science 10 10 5 The Crystal Structures • Single crystals comprise an infinite array of ions, atoms, or molecules, known as a crystal lattice. • The lattice energy is dependent on the nature and degree of interactions between adjacent species. • The ions, molecules, or atoms tend to pack in an arrangement that minimizes the total free energy of the crystal lattice. • The overall shape or form of a crystal is known as the morphology. • A crystal is comprised of an infinite 3-D lattice of repeating units, of which the smallest building block is known as the asymmetric unit. Callister Chapter 3 Introduction to Material Science 11 11 The unit Cell unit cell → small repeat entity in a crystal structures for most crystal structures are parallelepipeds or prisms having three sets of parallel faces ‒ is chosen to represents the symmetry of the crystal structure ‒ is the basic structural unit or building block of the crystal structure ‒ defines the crystal structure by virtue of its geometry and the atom positions within Callister Chapter 3 Introduction to Material Science 12 12 6 Metallic Crystal Structures: Atomic Packing Dense atomic packing for crystal structures of metals. • Reasons for dense packing: - Bonds between metal atoms are nondirectional. - Nearest neighbor distances tend to be small in order to lower potential energy. - High degree of shielding (of ion cores) provided by free electron cloud. • Crystal structures for metals simpler than structures for ceramics and polymers. We will examine three such structures for metals... Introduction to Material Science Callister Chapter 3 13 13 Simple Cubic (SC) Crystal Structure • Centers of atoms located at the eight corners of a cube • Rare due to low packing density (only Po has this structure) • Close-packed directions are cube edges. • Coordination # = 6 (# nearest neighbors) ex: Po a Adapted from Fig. 3.3, Callister & Rethwisch 10e. 1 atom per unit cell a = 2R Callister Chapter 3 Introduction to Material Science 14 14 7 Definitions Coordination Number Coordination Number = number of nearest-neighbor or touching atoms Atomic Packing Factor (APF) Volume of atoms in unit cell* APF = Volume of unit cell *assume hard spheres Introduction to Material Science Callister Chapter 3 15 15 Atomic Packing Factor (APF) for Simple Cubic volume atoms atom 4 a unit cell π (0.5a) 1 3 3 R = 0.5a = 0.52 APF = a 3 volume unit cell close-packed directions Unit cell contains 1 atom = 8 x 1/8 = 1 atom/unit cell Callister Chapter 3 Introduction to Material Science 16 16 8 Body-Centered Cubic Structure (BCC) • Atoms located at 8 cube corners with a single atom at cube center. --Note: All atoms in the animation are identical; the center atom is shaded differently for ease of viewing. • Coordination # = 8 ex: Cr, W, Fe (), Ta, Mo Adapted from Fig. 3.2, Callister & Rethwisch 10e. 2 atoms/unit cell: 1 center + 8 corners x 1/8 Introduction to Material Science Callister Chapter 3 17 17 Atomic Packing Factor: BCC • APF for the body-centered cubic structure = 0.68 4R = 3a a 2a For close-packed directions R R= a 3 a/4 volume atoms 4 π ( 3 a / 4) 2 unit cell 3 atom 3 APF = volume Callister Chapter 3 unit cell a 3 Introduction to Material Science 18 18 9 Face-Centered Cubic Structure (FCC) • Atoms located at 8 cube corners and at the centers of the 6 faces. --Note: All atoms in the animation are identical; the face-centered atoms are shaded differently for ease of viewing. ex: Al, Cu, Au, Pb, Ni, Pt, Ag • Coordination # = 12 Adapted from Fig. 3.1, Callister & Rethwisch 10e. 4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8 Introduction to Material Science Callister Chapter 3 19 19 Atomic Packing Factor: FCC • APF for the face-centered cubic structure = 0.74 maximum achievable APF For close-packed directions: 4R = 2 a 2a æ ö çi.e., R = 2a ÷ ç 4 ÷ø è Unit cell contains: 6 x 1/2 + 8 x 1/8 = a 4 atoms/unit cell volume atoms 4 4 unit cell atom π ( 2 a/4)3 3 = 0.74 APF = a 3 volume unit cell Callister Chapter 3 Introduction to Material Science 20 20 10 FCC Plane Stacking Sequence • ABCABC... Stacking Sequence–Close-Packed Planes of Atoms • 2D Projection B B C A B A sites B sites B B C C B B C sites • Stacking Sequence A B C Close-Packed Plane Referenced to an FCC Unit Cell. Introduction to Material Science Callister Chapter 3 21 21 Hexagonal Close-Packed Structure (HCP) • ABAB... Stacking Sequence–Close-Packed Planes of Atoms • 3D Projection c • 2D Projection A sites Top layer B sites Middle layer A sites Bottom layer a • Coordination # = 12 • APF = 0.74 6 atoms/unit cell ex: Cd, Mg, Ti, Zn • Ideal c /a = 1.633 Callister Chapter 3 Introduction to Material Science 22 22 11 Recap Types of unit cells Hexagonal ClosePacked Structure (HCP) Introduction to Material Science Callister Chapter 3 23 23 Theoretical Density for Metals, ρ Density = ρ = ρ = where Callister Chapter 3 Mass of Atoms in Unit Cell Total Volume of Unit Cell = nA VC NA 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 Introduction to Material Science 24 24 12 Theoretical Density Computation for Chromium • Cr has BCC crystal structure A = 52.00 g/mol R = 0.125 nm n = 2 atoms/unit cell a = 4R/ 3 = 0.2887 nm R a VC = a3 = 2.406 x 10-23 cm3 atoms g unit cell n 2 A ρ= mol 52.00 = VC = 7.19 g/cm3 2.406 x 10-23 NA 6.022 x 1023 atoms ρactual volume unit cell mol = 7.15 g/cm3 Introduction to Material Science Callister Chapter 3 25 25 Densities Comparison for Four Material Types In general ρ metals > ρ ceramics > ρ polymers Metals have... • low packing density (often amorphous) • lighter elements (C,H,O) Composites have... • moderate to low densities Callister Chapter 3 ρ Polymers have... (g/cm3 ) • close-packing (metallic bonding) • often large atomic masses • often lighter elements Graphite/ Ceramics/ Semicond Polymers Composites/ fibers 30 Why? Ceramics have... Metals/ Alloys 20 Platinum Gold, W Tantalum 10 Silver, Mo Cu,Ni Steels Tin, Zinc 5 4 3 2 Titanium Aluminum Magnesium 1 0.5 0.4 0.3 Introduction to Material Science Based on data in Table B1, Callister *GFRE, CFRE, & AFRE are Glass, Carbon, & Aramid Fiber-Reinforced Epoxy composites (values based on 60% volume fraction of aligned fibers in an epoxy matrix). Zirconia Al oxide Diamond Si nitride Glass -soda Concrete Silicon Graphite PTFE Silicone PVC PET PS PE Glass fibers GFRE* Carbon fibers CFRE* Aramid fibers AFRE* Wood 26 26 13 Single Crystals When the periodic arrangement of atoms (crystal structure) extends without interruption throughout the entire specimen. -- diamond single crystals for abrasives -- single crystal for turbine blade (Courtesy Martin Deakins, GE Superabrasives, Worthington, OH. Used with permission.) (Courtesy P.M. Anderson) -- Quartz single crystal Fig. 8.35(c), Callister & Rethwisch 10e. (courtesy of Pratt and Whitney) Introduction to Material Science Callister Chapter 3 27 27 Polycrystalline Materials Courtesy of Paul E. Danielson, Teledyne Wah Chang Albany • Most engineering materials are composed of many small, single crystals (i.e., are polycrystalline). large grain 1 mm • Nb-Hf-W plate with an electron beam weld. • Each "grain" is a single crystal. • Grain sizes typically range from 1 nm to 2 cm small grain (i.e., from a few to millions of atomic layers). Callister Chapter 3 Introduction to Material Science 28 28 14 Anisotropy • Anisotropy — Property value depends on crystallographic direction of measurement. - Observed in single crystals. E (diagonal) = 273 GPa - Example: modulus of elasticity (E) in BCC iron E(edge) ≠ E(diagonal) E (edge) = 125 GPa Unit cell of BCC iron Introduction to Material Science Callister Chapter 3 29 29 Isotropy • Polycrystals 200 μm - Properties may/may not vary with direction. - If grains randomly oriented: properties isotropic. (Epoly iron = 210 GPa) - If grains textured (e.g., deformed grains have preferential crystallographic orientation): properties anisotropic. . [Fig. 4.15(b) is courtesy of L.C. Smith and C. Brady, the National Bureau of Standards, Washington, DC (now the National Institute of Standards and Technology, Gaithersburg, MD).] Callister Chapter 3 Introduction to Material Science 30 30 15 Polymorphism/Allotropy • Two or more distinct crystal structures for the same material (allotropy/polymorphism) Titanium: α or β forms Iron system Temperature T Carbon: diamond, graphite,…, graphene liquid 1538°C δ -Fe BCC 1394°C γ -Fe FCC 912°C α -Fe BCC Callister Chapter 3 Introduction to Material Science 31 31 Crystal systems The unit cell geometry is completely defined in terms of 6 parameters: ‒ the 3 edge lengths a, b, and c, and ‒ the 3 interaxial angles 𝛼, 𝛽, and 𝛾. Callister Chapter 3 Introduction to Material Science lattice parameters of a crystal structure 32 32 16 Point Coordinates A point coordinate is a lattice position in a unit cell Determined as fractional multiples of a, b, and c unit cell edge lengths Example: Unit cell upper corner z 1. Lattice position is a, b, c 111 c a, b, c 2. Divide by unit cell edge lengths (a, b, and c) and remove commas y a b x 3. Point coordinates for unit cell corner are Callister Chapter 3 111 Introduction to Material Science 33 33 Point Coordinates A point coordinate is a lattice position in a unit cell Determined as fractional multiples of a, b, and c unit cell edge lengths Callister Chapter 3 Introduction to Material Science 34 34 17 Crystallographic Directions Algorithm – determine direction indices Example Problem I z pt. 2 head pt. 1: tail x ex: pt. 1 x1 = 0, y1 = 0, z1 = 0 pt. 2 x2 = a, y2 = 0, z2 = c/2 y 1. Determine coordinates of vector tail, pt. 1: x1, y1, & z1 ; and vector head, pt. 2: x2, y2, & z2. 2. Tail point coordinates subtracted from head point coordinates. 3. Normalize coordinate differences in terms of lattice parameters a, b, and c: x 2 - x1 a 4. Reduce to smallest integer values 5. Enclose indices in square brackets, no commas => 1, 0, 1/2 a-0 0-0 c 2-0 a b c Callister Chapter 3 y 2 - y1 z2 - z1 b c [uvw] => 2, 0, 1 => [ 201 ] Introduction to Material Science 35 35 Common Crystallographic Directions Adapted from Fig. 3.7, Callister & Rethwisch 10e. Callister Chapter 3 Introduction to Material Science 36 36 18 Crystallographic Directions Callister Chapter 3 Introduction to Material Science 37 37 Crystallographic Planes Algorithm for determining the Miller Indices of a plane 1. If plane passes through selected origin, establish a new origin in another unit cell 2. Read off values of intercepts of plane (designated A, B, C) with x, y, and z axes in terms of a, b, c 3. Take reciprocals of intercepts 4. Normalize reciprocals of intercepts by multiplying by lattice parameters a, b, and c 5. Reduce to smallest integer values 6. Enclose resulting Miller Indices in parentheses, no commas i.e., (hkl) Callister Chapter 3 Introduction to Material Science 38 38 19 Crystallographic Planes Example Problem I z x y z 2. Relocate origin – not needed a b ∞c Intercepts 3. Reciprocals 1/a 1/b 1/∞c 4. Normalize b/b 1 1 c/∞c 0 0 1. 5. Reduction a/a 1 1 6. Miller Indices (110) c y b a x Introduction to Material Science Callister Chapter 3 39 39 Crystallographic Planes Example Problem II x z z 2. Relocate origin – not needed a/2 ∞b Intercepts 3. Reciprocals 2/a 1/∞b 1/∞c 4. Normalize 5. Reduction 2a/a 2 2 b/∞b 0 0 c/∞c 0 0 6. Miller Indices (200) 1. Callister Chapter 3 y c ∞c Introduction to Material Science y a b x 40 40 20 Crystallographic Planes Example Problem III x y 2. Relocate origin – not needed a/2 b Intercepts 3. Reciprocals 4. Normalize 1. z z c 3c/4 2/a 1/b 4/3c 5. 2a/a 2 Reduction (x3) 6 b/b 1 3 4c/3c 4/3 4 6. Miller Indices a · · · y b x (634) Family of planes – all planes that are crystallographically equivalent (have the same atomic packing) – indicated by indices in braces Ex: {100} = (100), (010), (001), (100), (010), (001) Introduction to Material Science Callister Chapter 3 41 41 Crystallographic Planes Miller Indices (100) Callister Chapter 3 Miller Indices (010) Introduction to Material Science Miller Indices (110) 42 42 21 Crystallographic Planes Miller Indices (101) Miller Indices (111) Introduction to Material Science Callister Chapter 3 Miller Indices (210) 43 43 Common Crystallographic Planes Adapted from Fig. 3.11, Callister & Rethwisch 9e. Callister Chapter 3 Introduction to Material Science 44 44 22 Crystallographic Planes (HCP) • For hexagonal unit cells a similar procedure is used • Determine the intercepts with the a1, a2, and z axes, then determine the Miller-Bravais Indices h, k, i, and l z example a1 c a2 2. Relocate origin – not needed a ∞a Intercepts 3. Reciprocals 4. Normalize 5. Reduction 6. Determine index i = -(h + k) 7. Miller-Bravais Indices 1. 1/a 1/∞a c 1/c a/a a/∞a 1 0 h=1 k=0 a2 c/c 1 l=1 a3 a1 i = -(1 + 0) = -1 Introduction to Material Science Callister Chapter 3 45 45 Linear Density of Atoms (LD) LD = number of atoms centered on direction vector length of direction vector ex: linear density of Al in [110] direction There are 2 half atoms and 1 full atom = 2 atoms centered on vector # atoms a a = 0.405 nm Callister Chapter 3 LD = 2 2a 2 = = 3.5 nm-1 2 (0.405 nm) length Introduction to Material Science 46 46 23 Planar Density of Atoms (PD) PD = 2D repeat unit number of atoms centered on a plane area of plane ex: planar density of (100) plane of BCC Fe There are 4 quarter atoms = 1 atom centered on plane 4 R 3 a= 4 4 (0.1241 nm) = 0.287 nm R = 3 3 a= # atoms Radius of iron, R = 0.1241 nm PD = 1 1 atom = a2 (0.287 nm)2 = 12.1 atoms nm2 area Introduction to Material Science Callister Chapter 3 47 47 X-ray Analytics X-ray tool & methods optimisation for materials XRD HRXRD SAXS XPCI XRAYS Low to high Energies Callister Chapter 3 Absorption Imaging: 2D, 3D In-situ studies: - Temperature - Humidity - Mechanical stresses Introduction to Material Science 48 48 24 X-Ray Diffraction • To diffract light, the diffraction grating spacing must be comparable to the light wavelength. • X-rays are diffracted by planes of atoms. • Interplanar spacing is the distance between parallel planes of atoms. Callister Chapter 3 Introduction to Material Science 49 49 Interference of waves Δλ = nλ n = 0, 1, 2, ... When a number of waves of the same wavelength propagating in the same direction interfere with each other under continuous phase shift, only the coherent among them will be amplified. In total, the rest will almost completely cancel each other out. Callister Chapter 3 Introduction to Material Science 50 50 25 Bragg Equation Incident angle Bragg Equation: SQ = d sin  reflected angle QT = d sin  n  = 2d sin  This extra distance must be an integral (n) multiple of the wavelength () for the phases of the two beams to be the same. Introduction to Material Science Callister Chapter 3 51 51 Bragg‘s law X-ray diffraction / scattering on different planes of the crystal lattice  d constructive interference disruptive interference Constructive interference of the secondary beam: • Extra distance at the 2nd plane of the crystal lattice: 2·d·sin() • Extra distance is multiple n of wavelength  → Callister Chapter 3 n ·  = 2·d·sin() Bragg‘s law Introduction to Material Science 52 52 26 Information obtained from diffraction patterns: STOE Powder Diffraction System 28-Mar-2010 100.0 1) Peak positions: - refer to geometrical parameters: (unit cell, space group) - influenced by stresses Diffraction pattern Relative Intensity (%) 80.0 60.0 40.0 20.0 0.0 20.0 30.0 40.0 50.0 60.0 70.0 2Theta 2) Peak intensities: - refer to electron density distribution within the unit cell - influenced by preferred orientation / texture in thin films and manufactured materials (rolling processes,…) 3) Peak shape: - refers to the quality (defects) in single crystal materials (broadness) - refers to the crystallite size in polycrystalline materials (broadness) - strain (asymmetry) - (influenced by diffractometer setup) Introduction to Material Science Callister Chapter 3 53 53 X-Rays to Determine Crystal Structure • Crystallographic planes diffract incoming X-rays extra distance travelled by wave “2” reflections must be in phase for a detectable signal, n= 2d sinθ  λ θ θ = 2 (d sin ) Measurement of diffraction angle, θ  c, allows computation of interplanar spacing, d. d spacing between planes X-ray intensity (measured By detector) d= nλ 2 sin θc θ θc Callister Chapter 3 Introduction to Material Science 54 54 27 X-Ray Diffraction Pattern z z Intensity (relative) c a z c b (110) y plane x a c b y a x x b y (211) plane (200) plane Diffraction angle 2θ Diffraction pattern for polycrystalline α-iron (BCC) Introduction to Material Science Callister Chapter 3 55 55 X-Ray diffraction of a crystal X-Ray diffraction of a single crystal detector X-rays The diffraction takes place at different planes of the lattice crystal (200) planes of atoms in NaCl (220) planes of atoms in NaCl Diffraction pattern of a crystal Callister Chapter 3 Introduction to Material Science 56 56 28 X-ray diffraction of a polycristalline sample or of powder In a polycristalline sample or in powders there are not single beams, but cones Why cones? • For every set of planes, there is a small percentage of crystallites that are properly oriented to diffract. • The rotation of these crystallites is still a free parameter, ideally randomly distributed. Sample with a poor particle statistics → random „errors“ Introduction to Material Science Callister Chapter 3 57 57 X-ray Diffraction: Application Identification of crystal structures: Example: SiO2 compound Reference spectra amorphous Comparing with reference spectra, the crystalline part of the compound can be determined and quantified Callister Chapter 3 Introduction to Material Science 58 58 29 X-ray Diffraction Analysis Summarizing some fundamentals: • For electromagnetic radiation to be diffracted the spacing in the grating should be of the same order as the wavelength • In crystals the typical inter atomic spacing ~ 2-3 Å so the suitable radiation is X-rays • Hence, X-rays can be used for the study of crystal structures nλ = 2 d sin θ • Constructive interference only occurs for certain θ’s correlating to a (hkl) plane, specifically when the path difference is equal to n wavelengths. cubic Callister Chapter 3 Face Centred Cubic Body centred Orthorhombic Introduction to Material Science 59 59 X-ray diffractometry The structural properties of materials at different temperatures for different chemical compositions can be investigated by X-Ray Diffraction (XRD) techniques → Comprehensive compositional and structural analysis Callister Chapter 3 Introduction to Material Science 60 60 30 Structural properties addressed by X-ray techniques SAXS / GISAXD XRD / HRXRD Phases Nano-particle Size Distribution Crystallite Size Texture Orientation Mosaicity Domain ordering 3D molecular structures for new materials Layer thickness Stress / Strain Density Defects Reflectivity (Interface) roughness Introduction to Material Science Callister Chapter 3 61 61 X-ray Analytics: Infrastructure BRUKER Linac-CT 3D-CT with 450 kV X-ray source and flat panel detector HRXRD / XRD XRD / HRXRD Tomography SC-XRD SC-XRD Center for X-Ray Analytics Image Analysis BRUKER SAXS XPCI Micro-CT Callister Chapter 3 SAXS / WAXS Introduction to Material Science XPCI 62 62 31 http://www.iupui.edu/~bbml/ Introduction to Material Science Callister Chapter 3 63 63 63 Summary • Atoms may assemble into crystalline (ordered) or amorphous (disordered) structures. • Common metallic crystal structures are FCC, BCC, and HCP. Coordination number and atomic packing factor are the same for both FCC and HCP crystal structures. • We can calculate the theoretical density of a metal, given its crystal structure, atomic weight, and unit cell lattice parameters. • Crystallographic points, directions and planes may be specified in terms of indexing schemes. • Atomic and planar densities are related to crystallographic directions and planes, respectively. Callister Chapter 3 Introduction to Material Science 64 64 32 Summary (continued) • Materials can exist as single crystals or polycrystalline. • For most single crystals, properties vary with crystallographic orientation (i.e., are anisotropic). • For polycrystalline materials having randomly oriented grains, properties are independent of crystallographic orientation (i.e., they are isotropic). • Some materials can have more than one crystal structure. This is referred to as polymorphism (or allotropy). • X-ray diffraction is used for crystal structure and interplanar spacing determinations. Callister Chapter 3 Introduction to Material Science 65 65 CHAPTER 3 THE STRUCTURE OF CRYSTALLINE SOLIDS Fundamental Concepts 3.1 What is the difference between atomic structure and crystal structure? Unit Cells Metallic Crystal Structures 3.2 If the atomic radius of lead is 0.175 nm, calculate the volume of its unit cell in cubic meters. Density Computations 3.5 Strontium (Sr) has an FCC crystal structure, an atomic radius of 0.215 nm and an atomic weight of 87.62 g/mol. Calculate the theoretical density for Sr. Callister Chapter 3 Introduction to Material Science 66 66 33