Engineering 45
Solid
Crystallography
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Engineering-45: Materials of Engineering
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
Crystal Navigation
 As Discussed
Earlier A Unit Cell is
completely
Described by Six
Parameters
• Lattice Dimensions:
a, b ,c
• Lattice (InterAxial)
Angles:
,  ,
Engineering-45: Materials of Engineering
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 Navigation within a
Crystal is Performed
in Fractional Units of
the Lattice
Dimensions a, b, c
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
Point COORDINATES
 Cartesian CoOrds
(x,y,z) within a Xtal
are written in
Standard Paren &
Comma notation,
but in terms of
Lattice Fractions.
 Example
• Given TriClinic unit
Cell at Right
Engineering-45: Materials of Engineering
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 Sketch the Location
of the Point with
Xtal CoOrds of:
(1/2, 2/5, 3/4)
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
Point CoOrdinate Example
 From The CoOrd
Spec, Convert
measurement to
Lattice Constant
Fractions
• x → 0.5a
• y → 0.4b
• z → 0.75c
 To Locate Point
Mark-Off Dists on
the Axes
Engineering-45: Materials of Engineering
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 Located Point
(1/2, 2/5, 3/4)
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
Crystallographic DIRECTIONS
 Convention to specify crystallographic directions: 3
indices, [uvw] - reduced projections along x,y,z axes

Procedure to
Determine Directions
1. vector through origin, or
translated if parallelism
is maintained
2. length of vectorPROJECTION on each
axes is determined in
terms of unit cell
dimensions (a, b, c);
negative index in
x
opposite direction
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3. reduce indices to smallest
INTEGER values
4. enclose indices in
brackets w/o commas
z
z
[122]
[111]
y
y
[010]
[110]
_
[001]
x
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
Example  Xtal Directions
 Write the Xtal
Direction, [uvw] for
the vector Shown
Below
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 Step-1: Translate
Vector to The Origin
in Two SubSteps
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
Example  Xtal Directions
 After −x Translation,  Step-2: Project
Make −z Translation
Correctly Positioned
Vector onto Axes
Engineering-45: Materials of Engineering
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
Example  Xtal Directions
 Step-3: Convert
Fractional Values to
Integers using LCD
for 1/2 & 1/3 → 1/6
• x: (−a/2)•(6/a) = −3
• y: a•(6/a) = 6
• z: (−2a/3)•(6/a) = −4
 Step-4: Reduce to
Standard Notation:
 3 6  4  364
Engineering-45: Materials of Engineering
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
Crystallographic PLANES
 Planes within Crystals Are Designated
by the MILLER Indices
 The indices are simply the
RECIPROCALS of the Axes
Intersection Points of the Plane, with
All numbers INTEGERS
• e.g.: A Plane Intersects the Axes at (x,y,z)
of (−4/5,3,1/2) Then The Miller indices:

 1 1 1    5 1 2    5 3 1 4 2 12 

  


   15 4 24

  4 5 3 1 2   4 3 1   4 3 3 4 1 12 
Engineering-45: Materials of Engineering
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt

Miller Indices  Step by Step
 MILLER INDICES specify crystallographic planes: (hkl)

Procedure to
Determine Indices
1. If plane passes through
origin, move the origin
(use parallel plane)
2. Write the INTERCEPT
for each axis in terms of
lattice parameters
(relative to origin)
3. RECIPROCALS are
taken: plane parallel to
axis is zero (no
intercept → 1/ = 0)
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4. Reduce indices by
common factor for
smallest integers
5. Enclose indices in
Parens w/o commas
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
Example  Miller Indices
 Find The Miller Indices for the Cubic-Xtal
Plane Shown Below
Engineering-45: Materials of Engineering
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
The Miller Indices Example
 In Tabular Form
Step
Operation
x
y
1
2
3
4
5
Intercepts
3a/4
3/4
4/3
4
3a
3
1/3
1
(4 1 0)
Intercepts in Lattice Dim Multiples
Reciprocals
Reduction to Integers
Enclosure
Engineering-45: Materials of Engineering
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
z
a

0
0
More Miller Indices Examples
 Consider the (001) Plane
z
y
x
y
z

1
0
1
(none needed)
(001)
 Some Others
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Intercepts
Reciprocals
Reductions
Enclosure
x

0
2 3 6
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
FAMILIES of DIRECTIONS
 Crystallographically EQUIVALENT DIRECTIONS →
< V-brackets > notation
• e.g., in a cubic system,
100  100  010  010  001  001 
Also :

123  312  123
Family of <111> directions: SAME Atomic
ARRANGEMENTS along those directions
Engineering-45: Materials of Engineering
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100
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
FAMILIES of PLANES
 Crystallographically EQUIVALENT PLANES →
{Curly Braces} notation
• e.g., in a cubic system,
110  110  101  101  011  010  {110}

Engineering-45: Materials of Engineering
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Family of {110} planes:
SAME ATOMIC
ARRANGEMENTS
within all those planes
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
Hexagonal Structures
 Consider the Hex
Structure at Right
with 3-Axis CoOrds
Plane-C
 The Miller Indices
Plane-B
• Plane-A → (100)
• Plane-B → (010)
• Plane-C → (110)
 BUT
Plane-A
• Planes A, B, & C are Crystallographically IDENTICAL
– The Hex Structure has 6-Fold Symmetry
• Direction [100] is NOT normal to (100) Plane
Engineering-45: Materials of Engineering
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
4-Axis, 4-Index System
 To Clear Up this
Confusion add an
Axis in the BASAL,
or base, Plane
Plane-C
 The Miller Indices
now take the
form of (hkil)
 
 
 
• Plane-A → 1010
• Plane-B → 0110
• Plane-C → 1100
Engineering-45: Materials of Engineering
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Plane-B
Plane-A
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
4-Axis Directions
 Find Direction
Notation for the a1
axis-directed unit
vector
 Noting the RightAngle Projections
find
Operation
a1
a2
a3
Projections
1•a1
1
2
-a2/2
-1/2
-1
-a3/2
-1/2
-1
Projections in Lattice Multiples
Mult by LCF to Clear Fracs
Enclosure
Engineering-45: Materials of Engineering
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2110
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
z


0
More 4-Axis Directions
1120
1100
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1210
2110
1120
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
4-Axis Miller-Bravais Indices
 Construct Miller-Bravais (Plane) Index-Sets
by the Intercept Method
Plane
Intercepts : , , ,1
Intercepts : 1,1,1 2 , 
Reciprocal s : 0,0,0,1
Reciprocals :1,1,2,0
Enclosure : 0001

Enclosure : 1120
Plane
Engineering-45: Materials of Engineering
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt

4-Axis Miller-Bravais Indices
 Construct More Miller-Bravais Indices by the
Intercept Method
Plane
Intercepts : 1, ,1, 
Intercepts : 1, ,1,1
Recipricals :1,0,1,0
Reciprical s :1,0,  1,1
Enclosure : 1010
Enclosure : 1011
 
 
Plane
Engineering-45: Materials of Engineering
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
3axis↔4axis Translation
 The 3axis Indices
u' v' w'
 The 4axis Version
uvtw
 Conversion Eqns
u  n 32u 'v '
v  n 32v 'u '
t  u  v
w  nw'
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• Where n  LCD/GCF
needed to produce
integers-only
 Example [100]
u  3 32 1  0   2
v  3 32  0  1  1
t  1  0  1
w  1 0  0
 Thus with n = 1
100  2110
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
4axis Indices CheckSum
1011
 Given 4axis indices
• Directions → [uvtw]
• Planes → (hkil)
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2111
1121
1010
 Then due to Reln
between a1, a2, a3
 t  u  v or u  v  t  0
 i  h  k or h  k  i  0
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
Linear & Areal Atom Densities
 Linear Density, LD  Number of Atoms
per Unit Length On a Straight LINE
 Planar Density, PD  Number of Atoms
per Unit Area on a Flat PLANE
• PD is also called The Areal Density
 In General, LD and PD are different
for Different
• Crystallographic Directions
• Crystallographic Planes
Engineering-45: Materials of Engineering
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
Silicon Crystallography
 Structure = DIAMOND; not ClosePacked
Lattice Constants
InterAxial 's
a (pm) b (pm) c (pm)   
543.1
543.1
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543.1
90
90
90
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
LD & PD for Silicon
 Si
 
1
1
A  bh  a 2 a 2 cos 30
2
2
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt

LD and PD For Silicon
PD(100)  6.781 1018 atom / m 2
 For 100 Silicon
• LD on Unit Cell EDGE
LD 
PD111 is  15.5% HIGHER
2  0.5atom
1atom

 1.841109
a
543.1 pm
at / m
 For {111} Silicon
• PD on (111) Plane
– Use the (111) Unit Cell Plane
3  0.1667atom  3  0.5atom
2atom
PD 

1 2 base  height
0.5  2  543.1 pm  2  543.1 pm  cos 30


PD  2at 543.1 pm cos 30  7.830 1018 atom / m 2
2
Engineering-45: Materials of Engineering
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
X-Ray Diffraction → Xtal Struct.
 As Noted Earlier X-Ray Diffraction (XRD) is
used to determine Lattice Constants
 Concept of XRD → Constructive Wave
Scattering
 Consider a Scattering event on 2-Waves
Amplitude
100% Added
 Constructive Scattering
Engineering-45: Materials of Engineering
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Amplitude
100% Subtracted
 Destructive Scattering
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
XRD Quantified
 X-Rays Have WaveLengths, , That are
Comparable to Atomic Dimensions
• Thus an Atom’s Electrons or Ion-Core Can
Scatter these X-rays per The Diagram Below
Path-Length Difference
Engineering-45: Materials of Engineering
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
XRD Constructive Interference
 The Path Length
Difference is Line
Segment SQT
 Waves 1 & 2 will
be IN-Phase if the
Distance SQT is an
INTEGRAL Number
of X-ray
WaveLengths
• Quantitatively
SQ  ST  d hkl sin 
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1
1’
2’
2
 Now by Constructive
Criteria Requirement
SQ  ST  dhkl sin   dhkl sin   n
 Thus the Bragg Law
n  2d hkl sin 
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
XRD Charateristics
 The InterPlanar
Spacing, d, as a
Function of Lattice
Parameters (abc) &
Miller Indices (hkl)
d
 By Geometry for
OrthoRhombic Xtals
1
h2 k 2 l 2
 2 2 2
2
d hkl a
b
c
 For Cubic Xtals
a = b = c, so
1
h2 k 2 l 2 h2  k 2  l 2
 2 2 2 
2
d hkl a
a
a
a2
 d hkl 
Engineering-45: Materials of Engineering
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a
h2  k 2  l 2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
XRD Implementation
 X-Ray Diffractometer
Schematic
• T  X-ray Transmitter
• S  Sample/Specimen
• C  Collector/Detector
 Typical SPECTRUM
• Spectrum  Intensity/Amplitude vs. Indep-Index
Pb
Engineering-45: Materials of Engineering
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
X-Ray Diffraction Pattern
z
z
c
Intensity (relative)
c
a
z
y (110)
a
b
x
c
b
y
x
a
x
b
y
(211)
(200)
Diffraction angle 2θ
Diffraction pattern for polycrystalline α-iron (BCC)
Engineering-45: Materials of Engineering
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
33
XRD Example  Nb
 Given Niobium, Nb
with
• Structure = BCC
• X-ray = 1.659 Å
• (211) Plane
Diffraction Angle,
2∙θ = 75.99°
• n = 1 (primary diff)
 FIND
• ratom
• d211
Engineering-45: Materials of Engineering
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BCC Niobium
 Find InterPlanar
Spacing by Bragg’s Law
n  2d hkl sin 
or in this case
n
11.659Å
d 211 

2 sin 
2 sin 75.99 2
d 211  1.348Å
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
Nb XRD cont
 To Determine ratom
need The Cubic
Lattice Parameter, a
R
• Use the PlaneSpacing Equation
d hkl 
a
h k l
2
2
2
a Nb  d 211 2 2  12  12
So
a
 For the BCC Geometry
by Pythagorus
4ratom 2  a 2  a 2  a 2
3a
so
4
a Nb  1.348 Å 6  3.302 Å
3  3.302Å
rNb 
 1.4298Å
4
Bruce Mayer, PE
Engineering-45: Materials of Engineering

35

 ratom 
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
PolyCrystals → Grains
 Most engineering materials are POLYcrystals
Nb-Hf-W plate
with an electron
beam weld
1 mm
 Each "grain" is a single crystal.
• If crystals are randomly oriented, then overall
component properties are not directional.
 Crystal sizes typically range from 1 nm to 20 mm
•
(i.e., from a few to millions of atomic layers).
Engineering-45: Materials of Engineering
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
Single vs PolyCrystals
• Single Crystals
-Properties vary with
direction: anisotropic.
-Example: the modulus
of elasticity (E) in BCC iron:
• Polycrystals
-Properties may/mayNot
vary with direction.
-If grains are randomly
oriented: isotropic.
200 mm
(Epoly iron = 210 GPa)
-If grains are textured,
anisotropic.
Engineering-45: Materials of Engineering
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
19
WhiteBoard Work
 Planar-Projection (Similar to P3.48)
• Given Three Plane-Views, Determine Xtal
Structure
Also:  macro  18.91 g / cc
Engineering-45: Materials of Engineering
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Find Aw
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
Engineering-45: Materials of Engineering
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
Engineering-45: Materials of Engineering
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
All Done for Today
xTal Planes
in
Simple Cubic
Unit Cell
Engineering-45: Materials of Engineering
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
Planar Projection
101
101
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
Planar Projection
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
Engineering-45: Materials of Engineering
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt
Engineering-45: Materials of Engineering
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-04_Crystallography.ppt