materials science - university of isfahanengold.ui.ac.ir/~m.heidarirarani/courses/ch03-me.pdf ·...
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Chapter 3 -
In the Name of God
1
Materials Science
DrDrDrDr Mohammad Mohammad Mohammad Mohammad HeidariHeidariHeidariHeidari----RaraniRaraniRaraniRarani
Chapter 3 - 2
ISSUES TO ADDRESS...
• How do atoms assemble into solid structures?
• How does the density of a material depend onits structure?
• When do material properties vary with thesample (i.e., part) orientation?
Chapter 3: Structures of Metals & Ceramics
• How do the crystal structures of ceramic materials differ from those for metals?
Chapter 3 - 3
• Non dense, random packing
• Dense, ordered packing
Dense, ordered packed structures tend to havelower energies.
Energy and PackingEnergy
r
typical neighborbond length
typical neighborbond energy
Energy
r
typical neighborbond length
typical neighborbond energy
Chapter 3 - 4
• atoms pack in periodic, 3D arraysCrystalline materials...
-metals-many ceramics-some polymers
• atoms have no periodic packingNoncrystalline materials...
-complex structures-rapid cooling
crystalline SiO2
noncrystalline SiO2"Amorphous" = NoncrystallineAdapted from Fig. 3.40(b),Callister & Rethwisch 3e.
Adapted from Fig. 3.40(a),Callister & Rethwisch 3e.
Materials and Packing
Si Oxygen
• typical of:
• occurs for:
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Chapter 3 - 5
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
Chapter 3 - 6
• Tend to be densely packed.
• Reasons for dense packing:- Typically, only one element is present, so all atomicradii are the same.
- Metallic bonding is not directional.- Nearest neighbor distances tend to be small inorder to lower bond energy.
- Electron cloud shields cores from each other
• Have the simplest crystal structures.
We will examine three such structures...
Metallic Crystal Structures
Chapter 3 - 7
• Rare due to low packing density (only Po has this structure)• Close-packed directions are cube edges.
• Coordination # = 6(# nearest neighbors)
(Courtesy P.M. Anderson)
Simple Cubic Structure (SC)
Chapter 3 - 8
• APF for a simple cubic structure = 0.52
APF = a3
4
3π (0.5a) 31
atomsunit cell
atomvolume
unit cellvolume
Atomic Packing Factor (APF)
APF = Volume of atoms in unit cell*
Volume of unit cell
*assume hard spheres
Adapted from Fig. 3.42,Callister & Rethwisch 3e.
close-packed directions
a
R=0.5a
contains 8 x 1/8 = 1 atom/unit cell
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Chapter 3 - 9
• Coordination # = 8
• Atoms touch each other along cube diagonals.--Note: All atoms are identical; the center atom is shaded
differently only for ease of viewing.
Body Centered Cubic Structure (BCC)
ex: Cr, W, Fe (α), Tantalum, Molybdenum
2 atoms/unit cell: 1 center + 8 corners x 1/8
Chapter 3 - 10
Atomic Packing Factor: BCC
a
APF =
4
3π ( 3a/4)32
atoms
unit cell atomvolume
a3unit cell
volume
length = 4R =Close-packed directions:
3 a
• APF for a body-centered cubic structure = 0.68
aR
a2
a3
Chapter 3 - 11
• Coordination # = 12
• Atoms touch each other along face diagonals.--Note: All atoms are identical; the face-centered atoms are shaded
differently only for ease of viewing.
Face Centered Cubic Structure (FCC)
ex: Al, Cu, Au, Pb, Ni, Pt, Ag
4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8
Chapter 3 - 12
• APF for a face-centered cubic structure = 0.74Atomic Packing Factor: FCC
maximum achievable APF
APF =
4
3π ( 2a/4)34
atoms
unit cell atomvolume
a3unit cell
volume
Close-packed directions: length = 4R = 2 a
Unit cell contains:6 x 1/2 + 8 x1/8
= 4 atoms/unit cella
2 a
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Chapter 3 - 13
A sites
B B
B
BB
B B
C sites
C C
CA
B
B sites
• ABCABC... Stacking Sequence• 2D Projection
• FCC Unit Cell
FCC Stacking Sequence
B B
B
BB
B B
B sitesC C
CA
C C
CA
AB
C
Chapter 3 - 14
• Coordination # = 12
• ABAB... Stacking Sequence
• APF = 0.74
• 3D Projection • 2D Projection
Hexagonal Close-Packed Structure (HCP)
6 atoms/unit cell
ex: Cd, Mg, Ti, Zn
• c/a = 1.633
c
a
A sites
B sites
A sites Bottom layer
Middle layer
Top layer
Chapter 3 - 15
Theoretical Density, ρ
where n = number of atoms/unit cellA = atomic weight VC = Volume of unit cell = a3 for cubicNA = Avogadro’s number
= 6.022 x 1023 atoms/mol
Density = ρ =
VCNA
n Aρ =
CellUnitofVolumeTotalCellUnitinAtomsofMass
Chapter 3 - 16
• Ex: Cr (BCC) A = 52.00 g/molR = 0.125 nmn = 2 atoms/unit cell
ρtheoretical
a = 4R/ 3 = 0.2887 nm
ρactual
aR
ρ = a3
52.002
atoms
unit cell molg
unit cell
volume atoms
mol
6.022x 1023
Theoretical Density, ρ
= 7.18 g/cm3
= 7.19 g/cm3
Adapted from Fig. 3.2(a), Callister & Rethwisch 3e.
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Chapter 3 - 17
• Bonding:-- Can be ionic and/or covalent in character.-- % ionic character increases with difference in
electronegativity of atoms.
Adapted from Fig. 2.7, Callister & Rethwisch 3e. (Fig. 2.7 is adapted from Linus Pauling, The Nature of the Chemical Bond, 3rd edition, Copyright 1939 and 1940, 3rd edition. Copyright 1960 byCornell University.
• Degree of ionic character may be large or small:
Atomic Bonding in Ceramics
SiC: small
CaF2: large
Chapter 3 - 18
Ceramic Crystal Structures
Oxide structures– oxygen anions larger than metal cations– close packed oxygen in a lattice (usually FCC)– cations fit into interstitial sites among oxygen ions
Chapter 3 -c03tf02
Chapter 3 - 20
Factors that Determine Crystal Structure1. Relative sizes of ions – Formation of stable structures:
--maximize the # of oppositely charged ion neighbors.
Adapted from Fig. 3.4, Callister & Rethwisch 3e.
- -
- -+
unstable
- -
- -+
stable
- -
- -+
stable2. Maintenance of
Charge Neutrality :--Net charge in ceramic
should be zero.--Reflected in chemical
formula:
CaF2: Ca2+cation
F-
F-
anions+
AmXp
m, p values to achieve charge neutrality
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Chapter 3 - 21
• Coordination # increases with
Coordination # and Ionic Radii
Adapted from Table 3.3, Callister & Rethwisch 3e.
2
rcationranion
Coord #
< 0.155
0.155 - 0.225
0.225 - 0.414
0.414 - 0.732
0.732 - 1.0
3
4
6
8
linear
triangular
tetrahedral
octahedral
cubic
Adapted from Fig. 3.5, Callister & Rethwisch 3e.
Adapted from Fig. 3.6, Callister & Rethwisch 3e.
Adapted from Fig. 3.7, Callister & Rethwisch 3e.
ZnS (zinc blende)
NaCl(sodium chloride)
CsCl(cesium chloride)
rcationranion
To form a stable structure, how many anions cansurround around a cation?
Chapter 3 - 22
Computation of Minimum Cation-Anion Radius Ratio for a Coordination No. of 3
• Determine minimum rcation/ranion for an octahedral site (C.N. = 3)
155.023
231
anion
cation =−=r
r
Chapter 3 - 23
Computation of Minimum Cation-Anion Radius Ratio
• Determine minimum rcation/ranion for an octahedral site (C.N. = 6)
a = 2ranion
2ranion + 2rcation = 2 2ranion
ranion + rcation = 2ranion rcation = ( 2 −1)ranion
arr 222 cationanion =+
414.012anion
cation =−=rr
Chapter 3 - 24
Same concepts can be applied to ionic solids in general. Example: NaCl (rock salt) structure
rNa = 0.102 nm
∴ cations (Na+) prefer octahedral sites
rCl = 0.181 nm
564.0
181.0
102.0
anion
cation
=
=r
r
based on this ratio,-- coord # = 6 because
0.414 < 0.550 < 0.732
AX Crystal StructuresAX–Type Crystal Structures include NaCl, CsCl, and zinc blende
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Chapter 3 - 25
• On the basis of ionic radii, what crystal structurewould you predict for FeO?
• Answer:
5500
14000770
anion
cation
.
.
.rr
=
=
based on this ratio,-- coord # = 6 because
0.414 < 0.550 < 0.732
-- crystal structure is NaCl
Data from Table 3.4, Callister & Rethwisch 3e.
Example Problem: Predicting the Crystal Structure of FeO
Ionic radius (nm)
0.053
0.077
0.0690.100
0.140
0.181
0.133
Cation
Anion
Al3+
Fe2+
Fe3+
Ca2+
O2-
Cl-
F-Chapter 3 - 26
MgO and FeO
O2- rO = 0.140 nm
Mg2+ rMg = 0.072 nm
rMg/rO = 0.514
∴ cations prefer octahedral sites
So each Mg2+ (or Fe2+) has 6 neighbor oxygen atoms
Adapted from Fig. 3.5, Callister & Rethwisch 3e.
MgO and FeO also have the NaCl structure
Chapter 3 - 27
939.0181.0170.0
Cl
Cs ==−
+
r
r
Cesium Chloride structure:
∴ Since 0.732 < 0.939 < 1.0, cubic sites preferred
So each Cs+ has 8 neighbor Cl-
Chapter 3 - 28
AX2 Crystal Structures
• Calcium Fluorite (CaF2)• Cations in cubic sites
• UO2, ThO2, ZrO2, CeO2
• Antifluorite structure –positions of cations and anions reversed
Fluorite structure
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Chapter 3 - 29
ABX3 Crystal Structures
• Perovskite structure
Ex: complex oxide
BaTiO3
Chapter 3 - 30
Density Computations for Ceramics
A
AC )(NV
AAn
C
Σ+Σ′=ρ
Number of formula units/unit cell
Volume of unit cell
Avogadro’s number
= sum of atomic weights of all anions in formula unit ΣAA
ΣAC = sum of atomic weights of all cations in formula unit
Chapter 3 - 31
Exp: Density of NaclrNa+ = 0.102 nm
rCl- = 0.181 nm
ANa = 22.99 g/mol
ACl = 35.45 g/mol
Chapter 3 - 32
Densities of Material Classesρmetals > ρceramics > ρpolymers
Why?
Data from Table B.1, Callister & Rethwisch, 3e.
ρ (g/
cm
)3
Graphite/ Ceramics/ Semicond
Metals/ Alloys
Composites/ fibersPolymers
1
2
20
30Based on data in Table B1, Callister
*GFRE, CFRE, & AFRE are Glass,Carbon, & Aramid Fiber-ReinforcedEpoxy composites (values based on60% volume fraction of aligned fibers
in an epoxy matrix).10
345
0.30.40.5
Magnesium
Aluminum
Steels
Titanium
Cu,Ni
Tin, Zinc
Silver, Mo
TantalumGold, WPlatinum
GraphiteSilicon
Glass -sodaConcrete
Si nitrideDiamondAl oxide
Zirconia
HDPE, PSPP, LDPE
PC
PTFE
PETPVCSilicone
Wood
AFRE*
CFRE*
GFRE*
Glass fibers
Carbon fibers
Aramid fibers
Metals have...• close-packing
(metallic bonding)• often large atomic masses
Ceramics have...• less dense packing• often lighter elements
Polymers have...• low packing density
(often amorphous)• lighter elements (C,H,O)
Composites have...• intermediate values
In general
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Chapter 3 - 33
Silicate CeramicsMost common elements on earth are Si & O
• SiO2 (silica) polymorphic forms are quartz, crystobalite, & tridymite
• The strong Si-O bonds lead to a high melting temperature (1710ºC) for this material
Si4+
O2-
Adapted from Figs. 3.10-11, Callister & Rethwisch 3e crystobalite
Chapter 3 - 34
Bonding of adjacent SiO44- accomplished by the
sharing of common corners, edges, or faces
Silicates
Mg2SiO4 Ca2MgSi2O7
Adapted from Fig. 3.12, Callister & Rethwisch 3e.
Presence of cations such as Ca2+, Mg2+, & Al3+
1. maintain charge neutrality, and2. ionically bond SiO4
4- to one another
Chapter 3 - 35
• Quartz is crystallineSiO2:
• Basic Unit: Glass is noncrystalline (amorphous)• Fused silica is SiO2 to which no
impurities have been added• Other common glasses contain
impurity ions such as Na+, Ca2+, Al3+, and B3+
(soda glass)
Adapted from Fig. 3.41, Callister & Rethwisch 3e.
Glass Structure
Si04 tetrahedron4-
Si4+
O2-
Si4+
Na+
O2-
Chapter 3 - 36
Layered Silicates• Layered silicates (e.g., clays, mica, talc)
– SiO4 tetrahedra connected together to form 2-D plane
• A net negative charge is associated with each (Si2O5)2- unit
• Negative charge balanced by adjacent plane rich in positively charged cations
Adapted from Fig. 3.13, Callister & Rethwisch 3e.
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Chapter 3 - 37
Polymorphism
• Two or more distinct crystal structures for the same material (allotropy/polymorphism)
titaniumα, β-Ti
carbondiamond, graphite
BCC
FCC
BCC
1538ºC
1394ºC
912ºC
δδδδ-Fe
γγγγ-Fe
αααα-Fe
liquid
iron system
Chapter 3 - 38
Polymorphic Forms of CarbonDiamond– tetrahedral bonding of
carbon• hardest material known• very high thermal
conductivity
– large single crystals –gem stones
– small crystals – used to grind/cut other materials
– diamond thin films• hard surface coatings –
used for cutting tools, medical devices, etc.
Adapted from Fig. 3.16, Callister & Rethwisch 3e.
Chapter 3 - 39
Polymorphic Forms of Carbon (cont)
Graphite– layered structure – parallel hexagonal arrays of
carbon atoms
– weak van der Waal’s forces between layers– planes slide easily over one another -- good
lubricant
Adapted from Fig. 3.17, Callister & Rethwisch 3e.
Chapter 3 - 40
Polymorphic Forms of Carbon (cont)Fullerenes and Nanotubes
• Fullerenes – spherical cluster of 60 carbon atoms, C60
– Like a soccer ball • Carbon nanotubes – sheet of graphite rolled into a tube
– Ends capped with fullerene hemispheres
Adapted from Figs. 3.18 & 3.19, Callister & Rethwisch 3e.
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Chapter 3 - 41
Fig. 3.20, Callister & Rethwisch 3e.
Crystal Systems
7 crystal systems
14 crystal lattices
Unit cell: smallest repetitive volume which contains the complete lattice pattern of a crystal.
a, b, and c are the lattice constants
Chapter 3 - 42
Chapter 3 - 43
Point Coordinates
Chapter 3 - 44
Example:
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Chapter 3 - 45
Example: For the unit cell shown, locate the point having coordinates ¼ 1 ½.
Chapter 3 - 46
Crystallographic Directions
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 values4. Enclose in square brackets, no commas
[uvw]
ex: 1, 0, ½ => 2, 0, 1 => [ 201]
-1, 1, 1
families of directions <uvw>
z
x
Algorithm
where overbar represents a negative index
[ 111]=>
y
Chapter 3 - 47
Example:
Chapter 3 - 48
Example: Draw a [1 1 0] direction within a cubic unit cell.
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Chapter 3 - 49
Example:
Chapter 3 - 50
ex: linear density of Al in [110] direction
a = 0.405 nm
Linear Density
• Linear Density of Atoms ≡ LD =
a
[110]
Unit length of direction vector
Number of atoms
# atoms
length
13.5 nma2
2LD −==
Chapter 3 - 51
HCP Crystallographic Directions
1. Vector repositioned (if necessary) to pass through origin.
2. Read off projections in terms of unitcell dimensions a1, a2, a3, or c
3. Adjust to smallest integer values4. Enclose in square brackets, no commas
[uvtw]
[ 1120]ex: ½, ½, -1, 0 =>
Adapted from Fig. 3.24(a), Callister & Rethwisch 3e.
dashed red lines indicate projections onto a1 and a2 axes a1
a2
a3
-a3
2
a2
2a1
-a3
a1
a2
zAlgorithm
Chapter 3 - 52
HCP Crystallographic Directions• Hexagonal Crystals
– 4 parameter Miller-Bravais lattice coordinates are related to the direction indices (i.e., u'v'w') as follows.
==
=
'wwt
v
u
)vu( +-
)'u'v2(3
1-
)'v'u2(3
1-=
]uvtw[]'w'v'u[ →
Fig. 3.24(a), Callister & Rethwisch 3e.
-a3
a1
a2
z
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Chapter 3 - 53
Example:
Chapter 3 - 54
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.
• Algorithm1. Read off intercepts of plane with axes in
terms of a, b, c2. Take reciprocals of intercepts3. Reduce to smallest integer values4. Enclose in parentheses, no
commas i.e., (hkl)
Chapter 3 - 55
Crystallographic Planes
Adapted from Fig. 3.25, Callister & Rethwisch 3e.
Chapter 3 - 56
Crystallographic Planesz
x
ya b
c
4. Miller Indices (110)
example a b cz
x
ya b
c
4. Miller Indices (100)
1. Intercepts 1 1 ∞2. Reciprocals 1/1 1/1 1/∞
1 1 03. Reduction 1 1 0
1. Intercepts 1/2 ∞ ∞2. Reciprocals 1/½ 1/∞ 1/∞
2 0 03. Reduction 2 0 0
example a b c
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Chapter 3 - 57
Crystallographic Planesz
x
ya b
c•
••
4. Miller Indices (634)
example1. Intercepts 1/2 1 3/4
a b c
2. Reciprocals 1/½ 1/1 1/¾2 1 4/3
3. Reduction 6 3 4
(001)(010),
Family of Planes {hkl}
(100), (010),(001),Ex: {100} = (100),
Chapter 3 - 58
Example: Determine the Miller indices for the plane shown.
Chapter 3 - 59
Crystallographic Planes (HCP)
• In hexagonal unit cells the same idea is used
example a1 a2 a3 c
4. Miller-Bravais Indices (1011)
1. Intercepts 1 ∞ -1 12. Reciprocals 1 1/∞
1 0 -1-1
11
3. Reduction 1 0 -1 1
a2
a3
a1
z
Adapted from Fig. 3.24(b), Callister & Rethwisch 3e.
Chapter 3 - 60
Crystallographic Planes
• We want to examine the atomic packing of crystallographic planes
• Iron foil can be used as a catalyst. The atomic packing of the exposed planes is important.
a) Draw (100) and (111) crystallographic planes for Fe.
b) Calculate the planar density for each of these planes.
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Chapter 3 - 61
Planar Density of (100) IronSolution: At T < 912°C iron has the BCC structure.
(100)
Radius of iron R = 0.1241 nm
R3
34a =
Adapted from Fig. 3.2(c), Callister & Rethwisch 3e.
2D repeat unit
= Planar Density =a2
1
atoms2D repeat unit
= nm2
atoms12.1
m2
atoms= 1.2 x 1019
12
R3
34area
2D repeat unitChapter 3 - 62
Planar Density of (111) IronSolution (cont): (111) plane 1 atom in plane/ unit surface cell
333 2
2
R3
16R3
42a3ah2area =
===
atoms in plane
atoms above plane
atoms below plane
ah2
3=
a2
1
= = nm2
atoms7.0m2
atoms0.70 x 1019
3 2R3
16Planar Density =
atoms2D repeat unit
area2D repeat unit
Chapter 3 - 63
• Some engineering applications require single crystals:
• Properties of crystalline materials often related to crystal structure.
(Courtesy P.M. Anderson)
-- Ex: Quartz fractures more easily along some crystal planes than others.
-- diamond singlecrystals for abrasives
-- turbine blades
Fig. 9.40(c), Callister & Rethwisch 3e. (Fig. 9.40(c) courtesy of Pratt and Whitney).
(Courtesy Martin Deakins,GE Superabrasives, Worthington, OH. Used with permission.)
Crystals as Building Blocks
Chapter 3 - 64
Polycrystals
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Chapter 3 - 65
• Single Crystals-Properties vary withdirection: anisotropic.
-Example: the modulusof elasticity (E) in BCC iron:
• Polycrystals
-Properties may/may notvary with direction.
-If grains are randomlyoriented: isotropic.(Epoly iron = 210 GPa)
-If grains are textured,anisotropic.
200 µm
Single vs PolycrystalsE (diagonal) = 273 GPa
E (edge) = 125 GPa
Chapter 3 - 66
X-Ray Diffraction
• Diffraction gratings must have spacings comparable to the wavelength of diffracted radiation.
• Can’t resolve spacings < λ• Spacing is the distance between parallel planes of
atoms.
Chapter 3 - 67
X-Rays to Determine Crystal Structure
X-ray intensity (from detector)
θ
θc
d = nλ2 sin θc
Measurement of critical angle, θc, allows computation of planar spacing, d.
• Incoming X-rays diffract from crystal planes.
Adapted from Fig. 3.37, Callister & Rethwisch 3e.
reflections must be in phase for a detectable signal
spacing between planes
d
θλ
θextra distance travelled by wave “2”
Chapter 3 - 68
X-Ray Diffraction Pattern
Adapted from Fig. 3.20, Callister 5e.
(110)
(200)
(211)
z
x
ya b
c
Diffraction angle 2θ
Diffraction pattern for polycrystalline α-iron (BCC)
Inte
nsity
(re
lativ
e)
z
x
ya b
cz
x
ya b
c
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Chapter 3 - 69
• 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).
SUMMARY
• 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.
• 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.
• Ceramic crystal structures are based on:-- maintaining charge neutrality-- cation-anion radii ratios.
• Interatomic bonding in ceramics is ionic and/or covalent.
Chapter 3 - 70
• Some materials can have more than one crystal structure. This is referred to as polymorphism (or allotropy).
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.
• X-ray diffraction is used for crystal structure and interplanar spacing determinations.