# Chapter 3: The Structure of Crystalline Solids

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Chapter 3: The Structure of Crystalline Solids
ISSUES TO ADDRESS... • How do atoms assemble (arrange) into solid structures? (focus on metals) • How does the density of a material depend on its structure? • When do material properties vary with the sample (i.e., part) orientation? Structure Levels: Subatomic  now: Atomic

Energy and Packing • Non dense, random packing
typical neighbor bond length bond energy • Dense, ordered packing Energy r typical neighbor bond length bond energy Dense, ordered packed structures tend to have lower energies.

Materials and Packing Si Oxygen 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

Crystal Systems Unit cell: smallest repetitive volume which contains the complete lattice pattern of a crystal. Fig. 3.4, Callister 7e. There are 6 lattice parameters Their different combinations lead to: 7 crystal systems (shapes) 14 crystal lattices In metals: Cubic and Hexagonal a, b, and c are the lattice constants

Table 3.2 Lattice Parameter Relationships and Figures Showing Unit Cell Geometries for the Seven Crystal Systems See other shapes

Metallic Crystal Structures
How can we stack metal atoms to minimize empty space? 2-dimensions vs. Now stack these 2-D layers to make 3-D structures

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. We will examine three such structures...

Parameters in Studying Crystal Structures
Number and Position of Atoms in unit cell Coordination No. No. of touching-neighbor atoms Cube edge (length) Atomic Packing Factor (APF): Fraction of volume occupied by atoms Fraction of empty space = 1 – APF APF = Total volume of atoms in the unit cell Volume of the unit cell

Simple Cubic Structure (SC)
• Rare due to low packing density (only Po* has this structure) • Close-packed directions are cube edges. • Coordination # = 6 (# nearest neighbors) * Po: Polonium

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 close-packed directions a R=0.5a contains 8 x 1/8 = 1 atom/unit cell atom volume atoms unit cell 4 3 p (0.5a) 1 APF = 3 a unit cell volume

Body Centered Cubic Structure (BCC)
Atoms touch each other along cube diagonals. examples: Cr, Fe(), W, Ta, Mo* Coordination # = 8 No of Atoms per unit cell = 1 center + 8 corners x 1/8 =2 atoms --Note: All atoms are identical; the center atom is shaded differently only for ease of viewing. * Cr: Chromium, W: Tungsten, Ta: Tantalum, Mo: Molybdenum

Atomic Packing Factor: BCC
• APF for a body-centered cubic structure = 0.68 a R a 3 a a 2 length = 4R = Close-packed directions: 3 a APF = 4 3 p ( a/4 ) 2 No. atoms unit cell atom volume a

Face Centered Cubic Structure (FCC)
Atoms touch each other along face diagonals. ex: Al, Cu, Au, Pb, Ni, Pt, Ag Coordination # = 12 No of atoms per unit cell = 6 face x 1/2 + 8 corners x 1/8 = 4 atoms --Note: All atoms are identical; the center atom is shaded differently only for ease of viewing.

Atomic Packing Factor: FCC
• APF for a face-centered cubic structure = 0.74 a 2 a maximum achievable APF Close-packed directions: length = 4R = 2 a APF = 4 3 p ( 2 a/4 ) atoms unit cell atom volume a

FCC Stacking Sequence • ABCABC... Stacking Sequence • 2D Projection
A sites B C sites A B sites C A C A A B C • FCC Unit Cell

Hexagonal Close-Packed Structure (HCP)
• ABAB... Stacking Sequence • 3D Projection • 2D Projection c a A sites B sites Bottom layer Middle layer Top layer • Coordination # = 12 No. of atoms/unit cell = 6 • APF = 0.74 ex: Cd, Mg, Ti, Zn • c/a = 1.633 Cd: Cadmium, Mg: Magnesium, Ti: Titanium, Zn: Zinc

Polymorphism Polymorphous Material: BCC FCC 1538ºC 1394ºC 912ºC -Fe
Same Material with two or more distinct crystal structures Crystal Structure is a Property: It may vary with temperature Allotropy /polymorphism Examples Titanium: , -Ti Carbon: diamond, graphite BCC FCC 1538ºC 1394ºC 912ºC -Fe -Fe -Fe liquid iron system Heating

Theoretical Density, r Density =  = n A  = VC NA
Cell Unit of Volume Total in Atoms Mass Density =  = VC NA n A  = where n = number of atoms/unit cell A = atomic weight VC = Volume of unit cell = a3 for cubic NA = Avogadro’s number = x 1023 atoms/mol

Theoretical Density, r  = a R a 3 52.00 2 atoms unit cell mol g
Example: For Cr Known: A = g/mol and R = nm Crystal Structure: BCC For BCC: n = 2 and a = 4R/ 3 = nm Calculate:  = a 3 52.00 2 atoms unit cell mol g volume 6.023 x 1023 theoretical ractual = 7.18 g/cm3 = 7.19 g/cm3 Compare

Densities of Material Classes
In general Graphite/ r metals r ceramics r polymers Metals/ Composites/ > > Ceramics/ Polymers Alloys fibers Semicond 30 Why? B ased on data in Table B1, Callister Metals have... • close-packing (metallic bonding) • often large atomic masses 2 Magnesium Aluminum Steels Titanium Cu,Ni Tin, Zinc Silver, Mo Tantalum Gold, W Platinum *GFRE, CFRE, & AFRE are Glass, Carbon, & Aramid Fiber-Reinforced Epoxy composites (values based on 60% volume fraction of aligned fibers 10 in an epoxy matrix). Ceramics have... • less dense packing • often lighter elements G raphite Silicon Glass - soda Concrete Si nitride Diamond Al oxide Zirconia 5 3 4 (g/cm ) 3 Wood AFRE * CFRE GFRE* Glass fibers Carbon fibers A ramid fibers H DPE, PS PP, LDPE PC PTFE PET PVC Silicone Polymers have... • low packing density (often amorphous) • lighter elements (C,H,O) r 2 Water 1 Composites have... • intermediate values 0.5 0.4 0.3

Structure-Properties
Materials-Structures Amorphous: Random atomic orientation Crystalline: Single Crystal Structure: One Crystal Perfectly repeated unit cells. Polycrystalline Many crystals Properties Anisotropy: depend on direction Isotropy: does not depend on direction

Single Crystals • Some engineering applications require single crystals: --diamond single crystals for abrasives --turbine blades • Properties of crystalline materials often related to crystal structure. Ex: Quartz fractures more easily along some crystal planes than others.

Properties for Single and Polycrystals
E (diagonal) = 273 GPa E (edge) = 125 GPa Single Crystals -Properties vary with direction: anisotropic. -Example: the modulus of elasticity (E) in BCC iron: Polycrystals - Properties may/may not vary with direction. If grains are randomly oriented: isotropic. (Epoly iron = 210 GPa) -If grains are textured, anisotropic. 200 mm

Polycrystals • Most engineering materials are polycrystals.
Anisotropic 1 mm Isotropic • Each "grain" is a single crystal (composed of many unit cells). • If grains are randomly oriented, overall component properties are not directional. • Grain sizes typ. range from 1 nm to 2 cm (i.e., from a few to millions of atomic layers).

Point Coordinates z c y a b x z 2c y b b 111
000 111 Point coordinates for unit cell center are a/2, b/2, c/ ½ ½ ½ Point coordinates for unit cell corner are 111 Translation: integer multiple of lattice constants  identical position in another unit cell z 2c y b b

Crystallographic Directions
Vectors or lines between two points Described by indices Algorithm (for obtaining indices) 1. Vector repositioned (if necessary) to pass through origin. 2. Read off projections in terms of unit cell dimensions a, b, and c 3. Adjust to smallest integer values (multiply by a factor) 4. Enclose in square brackets, no commas [uvw] (note: If the value is –ve put the – sign over the number) Instead of 1 and 2: you can 1. Determine coordinates of head (H) and tail (T) 2. Subtract: head – tail Then as before (steps 3 and 4) z x y Ex 1: 1, 0, ½ 2, 0, 1 [ 201 ] Lecture 2 ended here Ex 2: -1, 1, 1 (overbar represents a negative index) [ 111 ] => families of directions <uvw>

Linear Density 3.5 nm a 2 LD = Linear Density of Atoms  LD = [110]
Number of atoms Unit length of direction vector ex: linear density of Al in [110] direction  a = nm a [110] # atoms length 1 3.5 nm a 2 LD - =

HCP Crystallographic Directions
Algorithm - a3 a1 a2 z 1. Vector repositioned (if necessary) to pass through origin. 2. Read off projections in terms of unit cell dimensions a1, a2, a3, or c 3. Adjust to smallest integer values 4. Enclose in square brackets, no commas [uvtw] dashed red lines indicate projections onto a1 and a2 axes a1 a2 a3 -a3 2 a 1 [ 1120 ] ex: ½, ½, -1, =>

HCP Crystallographic Directions
Hexagonal Crystals 4 parameter Miller-Bravais lattice coordinates are related to the direction indices (i.e., u'v'w') as follows. Fig. 3.8(a), Callister 7e. - a3 a1 a2 z = ' w t v u ) ( + - 2 3 1 ] uvtw [

Crystallographic Planes

Crystallographic Planes
Miller Indices: Reciprocals of the (three) axial intercepts for a plane, Cleared of fractions & common multiples. All parallel planes have same Miller indices. Algorithm (for obtaining Miller Indices) If the plane passes through the origin, choose another origin 1. Read off intercepts of plane with axes (in terms of a, b, c) 2. Take reciprocals of intercepts 3. Reduce to smallest integer values 4. Enclose in parentheses ( ), no commas i.e., (hkl)

Crystallographic Planes
z x y a b c Example 1 a b c Intercepts Reciprocals 1/ / / Integers Miller Indices (110) Example 2 a b c z x y a b c Intercepts 1/   Reciprocals 1/½ 1/ 1/ Integers Miller Indices (100)

Crystallographic Planes
z x y a b c Example 3 a b c Intercepts 1/ /4 Reciprocals 1/½ 1/ /¾ /3 Integers Miller Indices (634) (001) (010), Family of Planes {hkl} (100), (001), Ex: {100} = (100),

Crystallographic Planes (HCP)
In hexagonal unit cells the same idea is used a2 a3 a1 z example a a a c Intercepts -1 1 Reciprocals / -1 1 Integers -1 1 Miller-Bravais Indices (1011)

No. of atoms per area of a plane
Planar Density Concept The atomic packing of crystallographic planes No. of atoms per area of a plane Importance Iron foil can be used as a catalyst. The atomic packing of the exposed planes is important. Example Draw (100) and (111) crystallographic planes for Fe (BCC). b) Calculate the planar density for each of these planes.

Planar Density of (100) Iron
Solution:  At T < 912C iron has the BCC structure. 2D repeat unit R 3 4 a = (100) Radius of iron R = nm = Planar Density = a 2 1 atoms 2D repeat unit nm2 12.1 m2 = 1.2 x 1019 R 3 4 area

Planar Density of (111) Iron
Solution (cont):  (111) plane 1 atom in plane/ unit surface cell 2 a atoms in plane atoms above plane atoms below plane 2D repeat unit 3 h = a 2 3 2 R 16 4 a ah area = ÷ ø ö ç è æ 1 = nm2 atoms 7.0 m2 0.70 x 1019 3 2 R 16 Planar Density = 2D repeat unit area

Determination of Crystal Structure
Wave: has an amplitude and wave length (l) Diffraction: phase relationship between two (or more) waves that have been scattered. Scattering: In-phase Constructive Out-of-phase Destructive Detecting scattered waves after encountering an object: Identify positions of atoms مثل الرؤية البصرية Diffraction occurs when a wave encounters a series of regularly spaced atoms – IF: The obstacles are capable of scattering the wave Spacing between them nearly = wave length Wave length depend on the source (i.e. material of the anode).

X-Ray Diffraction Diffraction gratings must have spacings comparable to the wavelength of diffracted radiation. Can’t resolve spacings  

Constructive versus Destructive Diffraction

Constructive X-Ray Diffraction

X-Ray Experiment

X-Rays to Determine Crystal Structure
• Incoming X-rays diffract from crystal planes. detector “1” incoming X-rays reflections must be in phase for “2” “1” a detectable signal outgoing X-rays l “2” q q extra distance traveled by wave 2 d Interplaner spacing spacing between planes In-phase Constructive diffraction occurs when the second wave travels a distance = n l From Geometry extra traveled distance = 2 d sin q Then X-ray intensity q Constructive d = n l 2 sin q c q c

Calculations d = n l 2 sin q c d = a for plane (h k l) (h2+k2+l2)0.5
Measurement of critical angle, qc, allows computation of planar spacing, d. d = n l 2 sin q c Bragg’s Law, n: order of reflection d = a for plane (h k l) (h2+k2+l2)0.5 Atoms in other planes cause extra scattering For validity of Bragg’s law For BCC h+k+l = even For FCC h, k, l all even or odd Sample calculations Given the plane indices and Crystal structure, get a (from C.S. and R) and d (from indices) Then with wavelength get angle or wavelength

Solved Problem BCC For (310) Brag’s Law d = a for plane (h k l)
3.58 Determine the expected diffraction angle for the first-order reflection from the (310) set of planes for BCC chromium when monochromatic radiation of wavelength nm is used. BCC d = a for plane (h k l) (h2+k2+l2)0.5 For (310) Brag’s Law

X-Ray Diffraction Pattern
z x y a b c z x y a b c z x y a b c (110) (211) Intensity (relative) (200) Diffraction angle 2q Diffraction pattern for polycrystalline a-iron (BCC)

SUMMARY • Atoms may assemble into crystalline or amorphous 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 predict the density of a material, provided we know the atomic weight, atomic radius, and crystal geometry (e.g., FCC, BCC, HCP). • Crystallographic points, directions and planes are specified in terms of indexing schemes. Crystallographic directions and planes are related to atomic linear densities and planar densities.

SUMMARY • Materials can be single crystals or polycrystalline.
Material properties generally vary with single crystal orientation (i.e., they are anisotropic), but are generally non-directional (i.e., they are isotropic) in polycrystals with randomly oriented grains. • 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.