bruce mayer, pe licensed electrical & mechanical engineer bmayer@chabotcollege

[email protected] • ENGR-45_Lec-04_Crystallography.ppt1

Bruce Mayer, PE Engineering-45: Materials of Engineering

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Engineering 45

CrystallograpCrystallographyhy



Crystal NavigationCrystal Navigation As Discussed

Earlier A Unit Cell is completely Described by Six Parameters• Lattice Dimensions:

a, b ,c

• Lattice (InterAxial) Angles: , ,

Navigation within a Crystal is Performed in Units of the Lattice Dimensions a, b, c



Point COORDINATESPoint 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

Sketch the Location of the Point with Xtal CoOrds of:(1/2, 2/5, 3/4)



Point Coordinate ExamplePoint 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

Located Point (1/2, 2/5, 3/4)



Crystallographic DIRECTIONSCrystallographic 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 vector- PROJECTION on each axes is determined in terms of unit cell dimensions (a, b, c); negative index in opposite direction

3. reduce indices to smallest INTEGER values

4. enclose indices in brackets w/o commas

[111]

[110]

[010]

x

z

y

[001]_

x

z

y



Example Example Xtal Directions Xtal Directions Write the Xtal

Direction, [uvw] for the vector Shown Below

Step-1: Translate Vector to The Origin in Two SubSteps



Example Example Xtal Directions Xtal Directions After −x Translation,

Make −z Translation Step-2: Project

Correctly Positioned Vector onto Axes



Example Example Xtal Directions 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: 463463



Crystallographic PLANESCrystallographic 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:

244151

2

3

1

4

5

21

1

3

1

54

1



Miller Indices – Step by StepMiller 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)

4. Reduce indices by common factor for smallest integers

5. Enclose indices in Parens w/o commas



Example Example Miller Indices Miller Indices Find The Miller Indices for the Cubic-Xtal

Plane Shown Below



The Miller Indices ExampleThe Miller Indices Example In Tabular Form

Step Operation x y z1 Intercepts 3a/4 3a a

2 Intercepts in Lattice Dim Multiples 3/4 3

3 Reciprocals 4/3 1/3 04 Reduction to Integers 4 1 05 Enclosure (4 1 0)



More Miller Indices ExamplesMore Miller Indices Examples Consider the (001) Plane

x y zInterceptsReciprocalsReductionsEnclosure

x

z

y

10 0 1

(001)

Some Others

632

(none needed)



FAMILIES of DIRECTIONSFAMILIES of DIRECTIONS Crystallographically EQUIVALENT DIRECTIONS →

< V-brackets > notation• e.g., in a cubic system,

Family of <111> directions: SAME Atomic ARRANGEMENTS along those directions

123213321 :Also

100100001010010001100



FAMILIES of PLANESFAMILIES of PLANES Crystallographically EQUIVALENT PLANES →

{Curly Braces} notation• e.g., in a cubic system,

Family of {110} planes: SAME ATOMIC ARRANGEMENTS within all those planes

}110{010011101101101110



Hexagonal StructuresHexagonal Structures Consider the Hex

Structure at Right with 3-Axis CoOrds

Plane-B

Plane-A

Plane-C

The Miller Indices• Plane-A → (100)

• Plane-B → (010)

• Plane-C → (110)

BUT• Planes A, B, & C are Crystallographically IDENTICAL

– The Hex Structure has 6-Fold Symmetry

• Direction [100] is NOT normal to (100) Plane



4-Axis, 4-Index System4-Axis, 4-Index System To Clear Up this

Confusion add an Axis in the BASAL, or base, Plane

Plane-B

Plane-A

Plane-C

The Miller Indices now take the form of (hkil)• Plane-A →

• Plane-B →

• Plane-C → 0110

0101

0011



4-Axis Directions4-Axis Directions Find Direction

Notation for the a1 axis-directed unit vector

Noting the Right-Angle Projections find

0112

Operation a1 a2 a3 zProjections 1•a1 -a2/2 -a3/2

Projections in Lattice Multiples 1 -1/2 -1/2

Mult by LCF to Clear Fracs 2 -1 -1 0Enclosure



More 4-Axis DirectionsMore 4-Axis Directions

0112

0121

2011

0011 0211



4-Axis Miller-Bravais Indices4-Axis Miller-Bravais Indices Construct Miller-Bravais (Plane) Index-Sets by

the Intercept Method

0010 :Enclosure

0,0,0,1 :sReciprocal

1,,,:Intercepts Plane

0211 :Enclosure

1,1,-2,0 :sReciprocal

,21,1,1:Intercepts

Plane



4-Axis Miller-Bravais Indices4-Axis Miller-Bravais Indices Construct More Miller-Bravais Indices by the

Intercept Method

0110 :Enclosure

1,0,-1,0 :sReciprical

,1,,1:Intercepts

1110 :Enclosure

1101 :sReciprical

1,1,,1:Intercepts

,,,

Plane

Plane



3axis↔4axis Translation3axis↔4axis Translation The 3axis Indices

''' wvu

• Where n LCD/GCF needed to produce integers-only

Example [100]

The 4axis Version

uvtw

Conversion Eqns

'

''23

''23

nww

vut

uvnv

vunu

003

101

110233

201233

w

t

v

u

Thus with n = 3

0112100



4axis Indices CheckSum4axis Indices CheckSum

Given 4axis indices• Directions → [uvtw]

• Planes → (hkil)

Then due to Reln between a1, a2, a3

1110 1211 1112

0110

0or ikhkhi

0or tvuvut



Linear & Areal Atom DensitiesLinear & 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 PLANE• PD is also called The Areal Density

In General, LD and PD are different for Different• Crystallographic Directions

• Crystallographic Planes



Silicon CrystallographySilicon Crystallography Structure = DIAMOND; not ClosePacked

a (pm) b (pm) c (pm) 543.1 543.1 543.1 90 90 90

Lattice Constants InterAxial 's



LD & PD for SiliconLD & PD for Silicon

Si

30cos222

1

2

1aabhA



LD and PD For SiliconLD and PD For Silicon

For 100 Silicon• LD on Unit Cell EDGE

matpm

atom

a

atomLD /10841.1

1.543

15.02 9

For {111} Silicon• PD on (111) Plane

– Use the (111) Unit Cell Plane

2182 /10830.730cos1.5432

30cos1.54321.54325.0

2

21

5.031667.03

matompmatPD

pmpm

atom

hb

atomatomPD

eightase

HIGHER 15.5% is

/10781.6

111

218)100(

PD

matomPD



X-Ray Diffraction → Xtal Struct.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

Constructive Scattering Destructive Scattering

Amplitude100% Added

Amplitude100% Subtracted



XRD QuantifiedXRD 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



XRD Constructive InterferenceXRD Constructive Interference The Path Length

Difference is Line Segment SQT

1

2

1’ 2’

Waves 1 & 2 will be IN-Phase if the Distance SQT is an INTEGRAL Number of X-ray WaveLengths• Quantitatively

Now by Constructive Criteria Requirement

sinhkldSTSQ

nddSTSQ hklhkl sinsin

Thus the Bragg Law

sin2 hkldn



XRD CharateristicsXRD Charateristics The InterPlanar

Spacing, d, as a Function of Lattice Parameters (abc) & Miller Indices (hkl)

By Geometry for Orthorhombic Xtals

2

2

2

2

2

2

2

1

c

l

b

k

a

h

dhkl

For Cubic Xtals a = b = c, sod

222

2

222

2

2

2

2

2

2

2

1

lkh

ad

a

lkh

a

l

a

k

a

h

d

hkl

hkl



XRD ImplementationXRD Implementation X-Ray Diffractometer

Schematic• T X-ray Transmitter

• S Sample/Specimen

• C Collector/Detector

Typical SPECTRUM• Spectrum Intensity/Amplitude vs. Indep-Index

Pb



XRD Example XRD Example Nb Nb Given Niobium, Nb

with• Structure = BCC

X-ray = 1.659 Å

• (211) Plane Diffraction Angle, 2 = 75.99°

FIND• ratom

• d211

Find InterPlanar Spacing by Bragg’s Law

BCC Niobium

Å348.1

299.75sin2

Å659.11

sin2

case in thisor sin2

211

211

d

nd

dn hkl



Nb XRD contNb XRD cont To Determine ratom

need The Cubic Lattice Parameter, a • Use the Plane-

Spacing Equation For the BCC Geometry

by Pythagorus

Å4298.14

Å302.33

4

3

4 2222

Nb

atom

atom

r

soa

r

aaar

ÅÅa

da

Solkh

ad

Nb

Nb

hkl

302.36348.1

112 222211

222

aR



PolyCrystals → GrainsPolyCrystals → 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 typ. range from 1 nm to 20 mm• (i.e., from a few to millions of atomic layers).



19

• 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.

E (diagonal) = 273 GPa

E (edge) = 125 GPa

Single vs PolyCrystalsSingle vs PolyCrystals

200 m



WhiteBoard WorkWhiteBoard Work

Problem 3.47• Given Three Plane-Views, Determine Xtal

Structure ccgmacro /91.18Also:



All Done for TodayAll Done for Today

xTal Planesin

Simple CubicUnit Cell

bruce mayer, pe licensed electrical & mechanical engineer bmayer@chabotcollege

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