![9
Review :
Polymophism --
Difference between theoretical and actual density of
materials—
Interstitial –
Crystal structure vs Crystal system
How many crystal system, structure
APF
Coordination number
Close packed direction, plane
Slip plane
Crystallographic Directions: Cubic cell
A cubic cell with usual coordinates
Example directions = vectors with
lattice constants (side of the cube) =1
Direction of Red arrow:
Temporary new origin O’ (if needed, translate
the vector in parallel so that it starts at origin or at a
corner and contained within unit cell)
Direction vector –1, 1, 0.
Clear fraction : -1, 1, 0.
Format :
O’
0]11[
Example direction: Green Arrow.
Temporary origin:
Direction vector:
Clear Fraction:
Format:
O”
½, 1, -1
1, 2, -2
O”
]221[
½
HCP Directions
a1, a2, a3, z ! 4 directions/coordinates
Same Method/procedure as cubic:
1. Vector in units of coordinates a1, a2, a3, z
2. Clear fraction
3. Format
Example: Red arrow direction.
Vector: 1 -½ -1½ -1 ?
Clear Fraction: 2 -1 -3 -2 (multiplying by 2)
Formatted: [ ?? ]](https://image.slidesharecdn.com/ch03-crystalstructures-fall2016-sent-161102134222/85/Ch03-crystal-structures-fall2016-sent-9-320.jpg)
![10
Linear Density ( of atomic Packing)
ex: linear density of Al in [110] direction
Al - Lattice Constant a = 0.405 nm
Linear Density of Atoms ≡ LD =
Unit length of direction vector
Number of atoms
# atoms
length
1
3.5 nm
a2
2
LD
−
==
a
[110]
Directions having same linear densities are considered
EQUIVALENT DIRECTIONS.
Al crystal- BCC, HCP, FCC, SC ?
How about along [100] direction?
Crystal Planes
Using 3 Miller Indices:
Procedure-
1.If plane goes thru origin,
choose another parallel
plane
2.length of the intercepts
in terms of a, b, c.
3.Calc. Reciprocal
4.Change the numbers to
smallest integer
5. Indices are (hkl)
Ex: Fig (b)-
2. 1 1 ∞
3. 1 1 0
4. (1 1 0)
Ex: A plane in a cubic cell
z
x
y
a b
c
•
•
•
4. Miller Indices (634)
example
1. Intercepts 1/2 1 3/4
a b c
2. Reciprocals 1/½ 1/1 1/¾
2 1 4/3
3. Reduction 6 3 4
(001)(010),
Define: Family of Planes {hkl} ! having same planar density
(100), (010),(001),Ex: {100} = (100),](https://image.slidesharecdn.com/ch03-crystalstructures-fall2016-sent-161102134222/85/Ch03-crystal-structures-fall2016-sent-10-320.jpg)
![12
Planar Density of (111) Iron
1 atom in plane/ unit surface cell
33
3 2
2
R
3
16
R
3
42
a3ah2area =
===
atoms in plane
atoms above plane
atoms below plane
ah
2
3
=
a2
1
= =
nm2
atoms
7.0
m2
atoms
0.70 x 1019
3 2
R
3
16
Planar Density =
atoms
2D repeat unit
area
2D repeat unit
Family of Planes
Family – (most) have common characteristics – same height,
same hair, eye color etc.
Planes in a family, similarly must have same planar densities.
In a given crystal unit cell, all planes having same density
belong to the same family.
E.g. in a BCC cell a family of planes is {100} = (100), (010),
(001), (-100), (101), (0-10), (110), (00-1), (111). 3 are wrong
For a different shaped crystal, this may not be true.
Rule of Thumb for Cubic Cell only:
" Planes in the family have same indices, +ve or –ve, irrespective of
the sequence, are all equivalent. E.g. {123} includes planes (123),
(312), (-231) etc.
REVIEW
Family of planes – all planes have same ____?
If a crystalline direction goes thru origin (of the cell
coordinates)- what should be done?
If a crystal plane goes through the origin, then? _____?
APF =?
COORD. # = ?
Review – Cryst. Direction! [ 1 2 1],
Miller indices (h k l) ! ( 2
1 2),
family of planes (cubic){ 1 1 0} includes
(1 0 1), (1 1 0), (1 0 0), (0 0 1), (0 1 1)
(0 1 0), (1 0 1), any plane not in the
family?
Direction indices=[ ]
1/3](https://image.slidesharecdn.com/ch03-crystalstructures-fall2016-sent-161102134222/85/Ch03-crystal-structures-fall2016-sent-12-320.jpg)
![14
19
• Single Crystals- bulk is just
one crystal
-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.
E (diagonal) = 273 GPa
E (edge) = 125 GPa
200 µm
Data from Table 3.3,
Callister 6e.
(Source of data is
R.W. Hertzberg,
Deformation and
Fracture Mechanics of
Engineering Materials,
3rd ed., John Wiley
and Sons, 1989.)
Adapted from Fig.
4.12(b), Callister 6e.
(Fig. 4.12(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].)
SINGLE VS POLYCRYSTALS-
Their effects
What is –ANISOTROPY ?
Basically It is the dependence of materials properties
on the (crystallographic) direction.
-e.g. strength of wood depends on the fiber
orientation
ISOTOPIC MATERIALS- whose properties are
INDEPENDENT of direction of measurement..
Polycrystalline Material – (generally) ISOTROPIC
Which of these are ISOTROPIC?: wood, steel, fabric,
polymer/plastic, fiberglass fishing rod, rubber, ceramics
X Ray Diffraction (XRD)
Did you know x-ray is the main technology to
determine/estimate crystal structure?
Why? – x-ray wavelengths ~ lattice constants
a, b, c etc.
How long are wavelengths of Xray compared
to visible light? Microwave? FM ?
Let’s look at the comparative chart next !](https://image.slidesharecdn.com/ch03-crystalstructures-fall2016-sent-161102134222/85/Ch03-crystal-structures-fall2016-sent-14-320.jpg)
Ch03 crystal structures-fall2016-sent
- 1. 1 Ch 3: CRYSTAL STRUCTURES & PROPERTIES (adapted from Callister) Mir M. Atiqullah, Ph.D. Focus for this chapter: • How do atoms assemble into solid structures? (e.g. metals)- even molecules of compounds • How does the density of a material depend on its structure? • When do material properties vary with the test sample (i.e., part) orientation/direction? 1 Ch3: Crystal Structures 2 • Non dense, random packing • Dense, regular packing Dense, regular-packed structures tend to have lower energy. = stronger bonding! Energy r typical neighbor bond length typical neighbor bond energy Energy r typical neighbor bond length typical neighbor bond energy ENERGY AND PACKING- (motivation for)
- 2. 2 • atoms pack in periodic, 3D arrays • typical of: 3 Crystalline materials... -metals -many ceramics -some polymers (partial) • atoms have no periodic packing • occurs for: Noncrystalline materials... -complex structures -by rapid solidification Si Oxygen crystalline SiO2 noncrystalline SiO2"Amorphous" = Noncrystalline Adapted from Fig. 3.18(b), Callister 6e. Adapted from Fig. 3.18(a), Callister 6e. What is a crystalline Material ? 4 • tend to be densely packed. • have several reasons for dense packing: -Typically, only one element is present, so all atomic radii are the same. -Metallic bonding is non-directional. -Nearest neighbor distances tend to be small in order to lower bond energy. • have the simplest crystal structures. We will look closely at four (4) such structures... METALLIC CRYSTALS A Unit Crystal Cell Crystal Lattice & Lattice Constants • Corners represent atomic sites/positions. • a,b,c,α,β,γ are lattice constant that define the lattice structure Unit cell: Smallest packing pattern of the atoms that repeats… Crystal lattice- a 3 D arrangement of points representing atomic positions
- 3. 3 Unit cell geometries Unit cell= smallest repetitive Pattern/stacking These are 7 Basic types 14 variations possible Here are the 14 varieties What is the difference between crystal structure and crystal system? Answer: A crystal structure is described by both the geometry of, and atomic arrangements within, the unit cell, whereas a crystal system is described only in terms of the unit cell geometry. For example, face-centered cubic and body-centered cubic are crystal structures that belong to the cubic crystal system.
- 4. 4 5 • Rare due to poor packing (only Po has this structure) • Close-packed directions are cube edges. • Coordination # = 6 (# of nearest neighbors) (Courtesy P.M. Anderson) SIMPLE CUBIC STRUCTURE (SC) 6 APF = Volume of atoms in unit cell* Volume of unit cell *assume hard spheres • APF for a simple cubic structure = 0.52 APF = a3 4 3 π (0.5a)31 atoms unit cell atom volume unit cell volume close-packed directions a R=0.5a contains 8 x 1/8 = 1 atom/unit cell Adapted from Fig. 3.19, Callister 6e. ATOMIC PACKING FACTOR-Simple Cubic • Coordination # = 8 7 Adapted from Fig. 3.2, Callister 6e. (Courtesy P.M. Anderson) • Close packed directions are cube diagonals. --Note: All atoms are identical; the center atom is shaded differently only for ease of viewing. BODY CENTERED CUBIC Crystal (BCC) Ex. Manganese
- 5. 5 a R 8 • APF for a body-centered cubic structure = 0.68 Close-packed directions: length = 4R = 3 a Unit cell contains: 1 + 8 x 1/8 = 2 atoms/unit cell Adapted from Fig. 3.2, Callister 6e. ATOMIC PACKING FACTOR: BCC APF = a3 4 3 π ( 3a/4)32 atoms unit cell atom volume unit cell volume 9 • Coordination # = 12 Adapted from Fig. 3.1(a), Callister 6e. (Courtesy P.M. Anderson) • Close packed directions are face diagonals. --Note: All atoms are identical; the face-centered atoms are shaded differently only for ease of viewing. FACE CENTERED CUBIC STRUCTURE (FCC) APF = a3 4 3 π ( 2a/4)34 atoms unit cell atom volume unit cell volume Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell a 10 • For a face-centered cubic structure APF = 0.74 Close-packed directions: length = 4R = 2 a Adapted from Fig. 3.1(a), Callister 6e. ATOMIC PACKING FACTOR: FCC
- 6. 6 11 • ABCABC... Stacking Sequence • 2D Projection A sites B sites C sites B B B BB B B C C C A A • FCC Unit Cell A B C FCC STACKING SEQUENCE 12 • Coordination # = 12 • ABAB... Stacking Sequence • APF = 0.74 • 3D Projection A sites B sites A sites Bottom layer Middle layer Top layer Adapted from Fig. 3.3, Callister 6e. HEXAGONAL CLOSE-PACKED STRUCTURE (HCP) Atoms per unit HCP cell = 3 + 12* 1/6 + 2* ½ = 6 APF = 0.74 ( same as FCC) Challenge- Show this. 13 • Compounds: Often have similar close-packed structures. • Close-packed directions --along cube edges. • Structure of NaCl Which color is Na ? (Courtesy P.M. Anderson) (Courtesy P.M. Anderson) STRUCTURE OF COMPOUNDS: NaCl Size of compound mol affects the cell size
- 7. 7 Some Crystal molecules diamond Graphite – Hexagonal SiO4-group Structural unit of Quartz 14 ρ = n A VcNA # atoms/unit cell Atomic weight (g/mol) Volume/unit cell (cm3/unit cell) Avogadro's number (6.023 x 1023 atoms/mol) THEORETICAL DENSITY, ρ Balancing units: ... vol g atoms mole vol cell mole g cell atoms mole atoms cell vol mole g cell atoms =⋅⋅⋅= ⋅ ⋅ =ρ Given Crystal structure, atomic wt.,density!VC!R( using ?) Example: Density of Copper • crystal structure = FCC: 4 atoms/unit cell • atomic weight = 63.55 g/mol (1 amu = 1 g/mol) • atomic radius R = 0.128 nm (1 nm = 10 cm) Vc = a3 ; For FCC, a = 4R/ 2 ; Vc = 4.75 x 10-23cm3 Result: theoretical ρCu = 8.89 g/cm3 Data from Table inside front cover of Callister (see next slide): Compare to actual: ρCu = 8.94 g/cm3 Why the difference? Existence of Isotopes- some atoms with more neutrons in nucleus.
- 8. 8 15 Element Aluminum Argon Barium Beryllium Boron Bromine Cadmium Calcium Carbon Cesium Chlorine Chromium Cobalt Copper Flourine Gallium Germanium Gold Helium Hydrogen Symbol Al Ar Ba Be B Br Cd Ca C Cs Cl Cr Co Cu F Ga Ge Au He H At. Weight (amu) 26.98 39.95 137.33 9.012 10.81 79.90 112.41 40.08 12.011 132.91 35.45 52.00 58.93 63.55 19.00 69.72 72.59 196.97 4.003 1.008 Atomic radius (nm) 0.143 ------ 0.217 0.114 ------ ------ 0.149 0.197 0.071 0.265 ------ 0.125 0.125 0.128 ------ 0.122 0.122 0.144 ------ ------ Density (g/cm3) 2.71 ------ 3.5 1.85 2.34 ------ 8.65 1.55 2.25 1.87 ------ 7.19 8.9 8.94 ------ 5.90 5.32 19.32 ------ ------ Crystal Structure FCC ------ BCC HCP Rhomb ------ HCP FCC Hex BCC ------ BCC HCP FCC ------ Ortho. Dia. cubic FCC ------ ------ Adapted from Table, "Charac- teristics of Selected Elements", inside front cover, Callister 6e. Characteristics of Selected Elements at 20C ρmetals ρceramics ρpolymers 16 ρ(g/cm3) Graphite/ Ceramics/ Semicond Metals/ Alloys Composites/ fibers Polymers 1 2 20 30 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).10 3 4 5 0.3 0.4 0.5 Magnesium Aluminum Steels Titanium Cu,Ni Tin, Zinc Silver, Mo Tantalum Gold, W Platinum Graphite Silicon Glass-soda Concrete Si nitride Diamond Al oxide Zirconia HDPE, PS PP, LDPE PC PTFE PET PVC Silicone Wood AFRE* CFRE* GFRE* Glass fibers Carbon fibers Aramid fibers Why? Metals have... • close-packing (metallic bonding) • large atomic mass Ceramics have... • less dense packing (covalent bonding) • often lighter elements Polymers have... • poor packing (often amorphous) • lighter elements (C,H,O) Composites have... • intermediate values Data from Table B1, Callister 6e. DENSITIES OF MATERIAL CLASSES Polymorphism or Allotropy Some metals/nonmetals may exist in more than one crystal structure depending on both temperature and pressure. Expl: Carbon ! graphite, diamond, Bucky Ball C60 nanotube O2 O3 BCC iron ! high temp. ! FCC iron This condition of multiple crystal structures is also known as ALLOTROPY. See p-48 on Tin-Disease " Tin (Sn) at room Temp BCC. At T<13.2 C, it transforms into a structure similar to diamond ( complex grid structure) with lower density and larger volume. May cause disintegration/crumbling ... Read the story of Russian disaster in 1850. See the full story in Text
- 9. 9 Review : Polymophism -- Difference between theoretical and actual density of materials— Interstitial – Crystal structure vs Crystal system How many crystal system, structure APF Coordination number Close packed direction, plane Slip plane Crystallographic Directions: Cubic cell A cubic cell with usual coordinates Example directions = vectors with lattice constants (side of the cube) =1 Direction of Red arrow: Temporary new origin O’ (if needed, translate the vector in parallel so that it starts at origin or at a corner and contained within unit cell) Direction vector –1, 1, 0. Clear fraction : -1, 1, 0. Format : O’ 0]11[ Example direction: Green Arrow. Temporary origin: Direction vector: Clear Fraction: Format: O” ½, 1, -1 1, 2, -2 O” ]221[ ½ HCP Directions a1, a2, a3, z ! 4 directions/coordinates Same Method/procedure as cubic: 1. Vector in units of coordinates a1, a2, a3, z 2. Clear fraction 3. Format Example: Red arrow direction. Vector: 1 -½ -1½ -1 ? Clear Fraction: 2 -1 -3 -2 (multiplying by 2) Formatted: [ ?? ]
- 10. 10 Linear Density ( of atomic Packing) ex: linear density of Al in [110] direction Al - Lattice Constant a = 0.405 nm Linear Density of Atoms ≡ LD = Unit length of direction vector Number of atoms # atoms length 1 3.5 nm a2 2 LD − == a [110] Directions having same linear densities are considered EQUIVALENT DIRECTIONS. Al crystal- BCC, HCP, FCC, SC ? How about along [100] direction? Crystal Planes Using 3 Miller Indices: Procedure- 1.If plane goes thru origin, choose another parallel plane 2.length of the intercepts in terms of a, b, c. 3.Calc. Reciprocal 4.Change the numbers to smallest integer 5. Indices are (hkl) Ex: Fig (b)- 2. 1 1 ∞ 3. 1 1 0 4. (1 1 0) Ex: A plane in a cubic cell z x y a b c • • • 4. Miller Indices (634) example 1. Intercepts 1/2 1 3/4 a b c 2. Reciprocals 1/½ 1/1 1/¾ 2 1 4/3 3. Reduction 6 3 4 (001)(010), Define: Family of Planes {hkl} ! having same planar density (100), (010),(001),Ex: {100} = (100),
- 11. 11 Example: Planes in cubic cell Example 3.9: Miller Indices? Fig. (a) Plane thru origin. 1. Move the plane/origin along y-axis. 2.intercepts: inf., -1, ½ 3. Reciprocal: 0, -1, 2 4. Smallest integers: 0, -1, 2 5. Indices are (hkl): )210( Problem 3.41 Plane A. 1. Move the origin to O’, all intercepts within 1 unit.. 2. Intercepts: ½, -½ , infinity. 3. Reciprocal: 2, -2, 0 4. Integers: 2, -2, 0 5. Indices are (hkl): O’ 0)22( Atomic Arrangements What is the atomic packing in a given crystallographic plane? Packing of atoms often determine the characteristic of that materials in certain application, e.g. catalyst. Iron foil can be used as a catalyst. The atomic packing of the exposed planes is important. " Consider (100) and (111) crystallographic planes for Fe. " What are the planar densities for each of these planes? Planar Density: Example =(100) Iron Solution: At T < 912°C iron has the BCC structure. (100) Radius of iron R = 0.1241 nm R 3 34 a = Adapted from Fig. 3.2(c), Callister 7e. 2D repeat unit =Planar Density = a2 1 atoms 2D repeat unit = nm2 atoms 12.1 m2 atoms = 1.2 x 1019 1 2 R 3 34area 2D repeat unit
- 12. 12 Planar Density of (111) Iron 1 atom in plane/ unit surface cell 33 3 2 2 R 3 16 R 3 42 a3ah2area = === atoms in plane atoms above plane atoms below plane ah 2 3 = a2 1 = = nm2 atoms 7.0 m2 atoms 0.70 x 1019 3 2 R 3 16 Planar Density = atoms 2D repeat unit area 2D repeat unit Family of Planes Family – (most) have common characteristics – same height, same hair, eye color etc. Planes in a family, similarly must have same planar densities. In a given crystal unit cell, all planes having same density belong to the same family. E.g. in a BCC cell a family of planes is {100} = (100), (010), (001), (-100), (101), (0-10), (110), (00-1), (111). 3 are wrong For a different shaped crystal, this may not be true. Rule of Thumb for Cubic Cell only: " Planes in the family have same indices, +ve or –ve, irrespective of the sequence, are all equivalent. E.g. {123} includes planes (123), (312), (-231) etc. REVIEW Family of planes – all planes have same ____? If a crystalline direction goes thru origin (of the cell coordinates)- what should be done? If a crystal plane goes through the origin, then? _____? APF =? COORD. # = ? Review – Cryst. Direction! [ 1 2 1], Miller indices (h k l) ! ( 2 1 2), family of planes (cubic){ 1 1 0} includes (1 0 1), (1 1 0), (1 0 0), (0 0 1), (0 1 1) (0 1 0), (1 0 1), any plane not in the family? Direction indices=[ ] 1/3
- 13. 13 17 • Some engineering applications require single crystals: • Crystal properties reveal features of atomic structure. (Courtesy P.M. Anderson) --Ex: Certain crystal planes in quartz fracture more easily than others. (True for large salt crystals.) --diamond single crystals for abrasives --turbine blades --silicon wafers Fig. 8.30(c), Callister 6e. (Fig. 8.30(c) courtesy of Pratt and Whitney). (Courtesy Martin Deakins, GE Superabrasives, Worthington, OH. Used with permission.) CRYSTALS AS BUILDING BLOCKS Solidification leading to polycrystalline structure Neucleation – forming of first solid Boundary – formation Grain boundary Under microscope 18 • Most engineering materials are polycrystals. • Nb-Hf-W plate with an electron beam weld. • Each "grain" is a single crystal. • If crystals are randomly oriented, overall component properties are not directional. • Crystal sizes typ. range from 1 nm to 2 cm (i.e., from a few to millions of atomic layers). Adapted from Fig. K, color inset pages of Callister 6e. (Fig. K is courtesy of Paul E. Danielson, Teledyne Wah Chang Albany) 1 mm POLYCRYSTALS
- 14. 14 19 • Single Crystals- bulk is just one crystal -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. E (diagonal) = 273 GPa E (edge) = 125 GPa 200 µm Data from Table 3.3, Callister 6e. (Source of data is R.W. Hertzberg, Deformation and Fracture Mechanics of Engineering Materials, 3rd ed., John Wiley and Sons, 1989.) Adapted from Fig. 4.12(b), Callister 6e. (Fig. 4.12(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].) SINGLE VS POLYCRYSTALS- Their effects What is –ANISOTROPY ? Basically It is the dependence of materials properties on the (crystallographic) direction. -e.g. strength of wood depends on the fiber orientation ISOTOPIC MATERIALS- whose properties are INDEPENDENT of direction of measurement.. Polycrystalline Material – (generally) ISOTROPIC Which of these are ISOTROPIC?: wood, steel, fabric, polymer/plastic, fiberglass fishing rod, rubber, ceramics X Ray Diffraction (XRD) Did you know x-ray is the main technology to determine/estimate crystal structure? Why? – x-ray wavelengths ~ lattice constants a, b, c etc. How long are wavelengths of Xray compared to visible light? Microwave? FM ? Let’s look at the comparative chart next !
- 15. 15 Electro magnetic Spectrum Where is the range for cell phones? Constructive and destructive Interference of light waves In phase Out of phase 20 • Incoming X-rays diffract from crystal planes. • Measurement of: Critical angles, θc, for X-rays provide atomic spacing, d. Adapted from Fig. 3.2W, Callister 6e. X-RAYS TO CONFIRM CRYSTAL STRUCTURE reflections must be in phase to detect signal spacing between planes d incom ing X-rays outgoing X-rays detector θ λ θ extra distance travelled by wave “2” “1” “2” “1” “2”
- 16. 16 X-ray Diffraction Calculations-Bragg’s law 222 cells,cubicAlso sin2 law)s(Bragg'sin2 distanceextrathe lkh a d n d dn QTSQn hklhkl hkl ++ = ⋅ = ⋅⋅= += θ λ θλ λ Knowing type of cubic, we can select the PEAK and its angle and the reflecting plane. (Goniometer in a) Xray Diffractometer T – source of x ray C - Detector (goniometer, with angle measurement) S – powder crystalline sample – with possibly many grains reflecting x-ray. 2Ө = angle of diffraction d=nλ/2sinθc x-ray intensity (from detector) θ θc Bragg’s law Demo How it works? Material is prepared as fine powder – that still keeps many crystals. X-ray will be reflected from various dense reflecting planes with relatively higher intensity. Angles (2θ) of all these intense reflections are measured using the goniometer (of the diffractometer).
- 17. 17 Typical Diffraction Pattern Material : Pb (SC, BCC or FCC ?) powder Type h2+k2+l2 Simple Cubic (All but 7) 1, 2, 3, 4, 5, 6, 8 BCC (Even) 2, 4, 6, 8, 10, 12, 14, 16 FCC 3, 4, 8, 11, 12, 16 Crystals of various materials but same structure, will diffract at different angles but in the same sequence of planes. TRUE/FALSE ? • Atoms may assemble into crystalline or amorphous 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). • Material properties generally vary with single crystal orientation (i.e., they are anisotropic), but properties are generally non-directional (i.e., they are isotropic) in polycrystals with randomly oriented grains. 23 SUMMARY