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Introduction of Crystal Structure

Text Books:

1. Essentials of Crystallography: by D. Mckie and C. Mckie

2. Geometry of Crystals: by H. K. D. H. Bhadeshia

http://www.msm.cam.ac.uk/phase-trans/2001/crystal.html


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Top 2 reasons to study the structure of crystals1) The properties of materials are strongly

dependent on their internal structure.

2) For a material of given chemicalcomposition, the internal structure is notconstant, but can vary greatly depending

ona. how the material was manufactured (exactly

what processing conditions were involved);

b. under what conditions (temperature, pressure,exposure to radiation, etc.) the material isplaced into service.


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Outlines1. Crystal lattices

2. Stereographic projection3. Crystal symmetry: Point group

4. Internal Structure: Space Group

5. Vector and matrix methods for geometry of crystals

(Application of reciprocal lattice on geometry of crystal)

6. Orientation relationship

7. Diffraction of X-rays

8. Diffraction of electrons


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Chapter 1

Crystal Lattices


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Crystallinity Crystals:

The atoms arrange themselves into an ordered,

repeating,3-dimensional pattern. Such structuresare called crystals.

Unit cells:The unit cell of a crystal structure is the smallestgroup of atoms possessing the symmetry of thecrystal which repeated in all directions, will developthe crystal lattice.

Lattice constants:The lengths of unit cell edges and angles betweencrystallographic axes are referred as latticeconstants, or lattice parameters.


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Crystal lattice

1. All lattice points have the same

environment in the same orientation and

are indistinguishable from one another.

2. Any lattice point is related to any other by a

simple lattice translation.


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A plane in space satisfies the equation

where a, b and c are the unit length of the x, y and z

axes respectively.

intercept on x axis

intercept on y axis

intercept on z axis

the reciprocals of the intercepts: , and

and the indices of the plane ( )

1=++ zc

yb

kxa

h l

h

ax=

k

by =

lcz=

lkh

a

h

b

k

c

l


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Those planes that mutually intersect along a

common direction form the planes of a zone, and

the line of intersection is called the zone axis.

zone axis [ ]planes of zone { }

for cubic, tetragonal, and orthorhombic systems

and all other crystal systems.

0=++ wkvhu l

wvulkh


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1.intersection line of two planes

( ) ( )

2.If the plane ( ) lie in the zones [ ] and

[ ] the plane is given by the solution of

equation:

111 lkh 111 lkh

222 lkh 222 lkh

0111

=++ wvkuh l

0222 =++ wvkuh l

111 lkh 222 lkh

lkh 111 wvu

222 wvu

0111 =++ wvkuh l

0222 =++ wvkuh

l


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3.If two planes ( ) and ( ) lie in the zone

axis[ ],then any other plane lying in this same

zone axis can be expressed as

( ),

where p and q are positive or negative integers.

Addition RuleThe planes of a zone are all parallel to the zone

axis, and their normals from any point must be

coplanar.

111 lkh 222 lkh

wvu

212121 ll qpqkpkqhph +++

0)()()(212121

=+++++ wqpvqkpkuqhph ll


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The condition for a vector [ ] lying in a plane ( ).

If

( and are scalars),

then will lie in plane ( )

Zone equation

All crystal systems satisfy the zone equation.

wvu lkh

][ wvucwbvau =++=

ACAB +=

lkh

bk

ah

AB 11

+=

cah

ACl

11+=

cwbvaucb

k

a

h

++=+++=

l

)(

1

0

)(

=++

=

=

+=

wvkuh

w

vk

uh

l

l


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Diffusivity of Carbon in BCC and

FCCCarbon atom has a greatly higher

diffusivity in ferrite than in austenite, whilethe maximum solubility of carbon in ferrite is

much lower than in austenite.

r/R=0.22r/R=0.410.74FCC

r/R=0.29r/R=0.160.68BCC

tetrahedral intersticeoctahedral intersticeA.P.F


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The locations of all atoms in the unit

cell

in BCC

0,0,0

in FCC

0,0,0

in CsCl

Cs+

Cl- 0,0,0

21

21

21 ,,

0,21,

21

21,0,

21

21,

21,0

2

1

2

1

2

1

,,


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cell

in Diamond

Coordinates for the corresponding atoms in the

Motif, (0,0,0) , (, , )

0,2

1,

2

1

2

1,0,

2

1

2

1,

2

1,0

41,

41,

41

41,

43,

43

43,

41,

43

43,

43,

41

0,0,0


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cell

in Diamond

Coordinates for the corresponding atoms in the

Motif, (0,0,0) , (, , )

02

1

2

1,,

2

1,0,

2

1

2

1,

2

1,0

43

43

43 ,,

43

41

41 ,,

41

43

41 ,,

41

41

43 ,,

0,0,0


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cell

in NaCl

Cl- 0,0,0

Na+

0,2

1,

2

1

2

1,0,

2

1

2

1,

2

1,0

2

1,0,0

2

1,

2

1,

2

10,0,

2

10,

2

1,0


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cell

in ZnS (Zinc Blende -Type)

S-

Zn+

0,0,0 0,2

1,

2

1

2

1,0,

2

1

2

1,

2

1,0

4

1,

4

1,

4

1

4

1,

4

3,

4

3

4

3,

4

1,

4

3

4

3,

4

3,

4

1


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The Locations of Interstices

in FCC and in BCC


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The Locations of Intersticesin FCC

Octahedral interstices

2

1,

2

1,

2

1

2

1,0,0

0,2

1,0 0,0,

2

1


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The Locations of Intersticesin FCC

Tetrahedral interstices

4

1,

4

1,

4

1

4

1,

4

3,

4

3

43,

41,

43

43,

43,

41

4

3,

4

3,

4

3

4

3,

4

1,

4

1

41,

43,

41

41,

41,

43


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The Locations of Intersticesin BCC

Octahedral interstices

0,2

1,

2

1

2

1,0,

2

1

2

1

,2

1

,0 2

1

,0,0

0,2

1,0 0,0,

2

1


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The Locations of Intersticesin BCC

Tetrahedral interstices

0,4

1,

2

1

4

1,0,

2

10,

2

1,

4

1

2

1,0,4

1

4

1,2

1,0 2

1,4

1,0

0,4

3,

2

1

4

3,0,

2

10,

2

1,

4

3

2

1,0,4

3

4

3,2

1,02

1,4

3,0


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The effect of crystal orientation on the mechanical behaviors

Example 1. Explain the typical face-centered single crystal stress-strain curve in the following figure.

E l 2


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Example 2.

(a) If the tensile stress axis is along [2 1 -1], then how many slip

systems are provided for the start deformation in FCC single

crystal? List out the slip systems.

(b) If the compressive stress axis is along [110], repeat the above

question.

Further Reading:

1. R. E. Reed-Hill and R. Abbaschian, Physical Metallurgy

Principles. PWS-KENT Publishing Company, Boston, 1992, Ch.

1 and Ch. 5.

2. W. C. Leslie, The Physical Metallurgy of Steels, McGraw-Hill.

Auckland, 1981, Ch. 2.


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From: [email protected] (MDavis3206)

Newsgroups: sci.engr.metallurgy

Subject: TWA 800 Failure Analysis

Date: 31 Jul 1996 18:05:40 -0400Organization: America Online, Inc. (1-800-827-6364)

To determine if an explosion occurred on TWA Flight 800,

Metallurgists will look for the presence of mechanical or deformation

twins in aluminum.

Being FCC, aluminum doesn't normally mechanically twin at ambient

Temperatures unless it is deformed at extremely high strain rates over a short

time interval (i.e. a short duration shock wave induced by an explosion).

Impacts with the ground or water most likely involve deformation at relatively

low strain rates where dislocation slip, as opposed to twinning, is the primary

mode of plastic deformation. This may also apply to any steel structures in

the vicinity of the explosion as well.

It should be noted that this is only one aspect of the failure analysis.

Fracture mode, for example, also changes with strain rate (i.e. ducti le to

brittle).


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Lattice and Motif in Three Dimensions

Unit cell a parallelepiped having lattice points atits corners. Stacking unit cells parallel to each

other in 3-D will generate the pattern or structure.

As for meshes, unit cells may be primitive

(having lattice points only at their corners, i.e.,

having a volume such as to contain only one

lattice point), or non-primitive (containing

more than one lattice point).


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If one corner of the unit cell is taken as the origin, the three

edges of the cell meeting at that comer define the vectors a, b

and c, whose lengths a, b and c are the lattice parameters and

whose directions are the crystallographic reference axes x, y

and z. The interaxial angles y : z, z : x and x : y are,

respectively ,and(taken to be 90for cubic crystal ).


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The lattice describes only the periodicity,

and for a full description we need also:Lattice motif the components of a pattern

or structure associated with one lattice point.

That is: LATTICE + MOTIF = STRUCTURE

If the lattice and the motif are each described by

mathematical functions, a more precise meaning has

to be attached to the operation indicated by the " + "

in the above expression. The operation is called

convolution.


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Formal description of structures

Formally the description must give:

the dimensions of the unit cell;

the lattice type;

the positions of the atoms in the lattice motif.

The description may be simplif ied by choosing

a unit cell that displays the symmetry of the

structure and has the simplest possible shape

consistent with that symmetry.


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Of the six unit cell shapes in common use three have

orthogonal reference axes:

Cubic

Tetragonal

Orthorhombic .

The others are:

Hexagonal

Monoclinic taken to be obtuse

Triclinic

cba ==

cba =

cba

cba = oo 120,90 ===

cba ,90,90 oo ==

o90 cba


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If we choose these unit cells, some lattice types wil l be non-

primitive. The conventional non-primit ive lattice types are:Lattice points at 0, 0, 0 and

F

I

A

B

C

R

The R lattice occurs only in crystals that possess

one triad axis, that is in the trigonal system.

Note: If there is an atom at vector distance r

from the origin, there are exactly equivalent atoms

at r + T where T is a lattice translation.

.,,0;,0,;0,, 212121212121

.,, 21

2

1

2

1

.,,0 2121

.,0, 2121

.0,, 2121

.,,;,, 323231313132

G hi


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Graphite

Formally graphite is hexagonal, lattice type P,with

and

The lattice motif contains four C atoms with coordinates .

Graphite is a layer structure.

o

A46.2=aoA73.6=c

21

32

31

21

31

32 ,,;,0,0;0,,;0,0,0

P i t ithi it ll


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Points within unit cell

1Every point within a unit cell can be identified in terms of the

coefficients along the three coordinate axes.

2A translation from any selected site within a unit cell by an

integer multiple of lattice constants (a, b, or c) leads to an

identical position in another unit cell.


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3A body-centered lattice has identical sites that are related

by translation.

This gives us identical sites per unit cell.

4A face-centered lattice has identical sites that are related

by translation.

This gives us identical sites per unit cell.


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Seven Crystal Systems and Fourteen Bravais Lattices

The description of crystal structures by means

of unit cell has an important advantage. All

possible structures reduce to a small number of

basic unit cell geometry. This is demonstrated

in two ways.

FIRST, there are only seven, unique unit cellshapes that can be stacked together to fill

three-dimensional space. These are so-called

the seven crystal systems.


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SECOND, we must consider how atoms (viewed as hardspheres) can be stacked together within a given unit cell.To do this in a general way, we begin by consideringlattice points, theoretical points arranged periodically

in three-dimensional space, rather than actual atoms orspheres.

There are so-called fourteen Bravais lattices. Theselattices are skeletons upon which crystal structures are

built by placing atoms or group of atoms on the latticepoints.


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Seven Crystal Systems and Fourteen


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Bravais Lattices

Note:

1. All lattice points have the same environment

in the same orientation and are

indistinguishable from one another.





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The simplest possibility is with one atom

centered on each lattice point; BCC and FCC

metal structures are of this type. However, a

very large number of actual crystal structureshave more than one atom associated with a

given lattice point.



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By placing a motif unit of one or more atoms at

every lattice point, the regular structure of a perfect

crystal is obtained.

Seven crystal systems:

Cubic, Tetragonal,

Orthorhombic, Hexagonal,

Rhombohedral, Monoclinic,

Triclinic.


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Bravais Lattices

Fourteen Bravais lattices:

Triclinic 1 : P.

Monoclinic 2 : P, C.

Orthorhombic 4 : P, C, I, F. Tetragonal 2 : P, I.

Hexagonal 1 : P.

Trigonal 1 : R. Cubic 3: P, I, F.


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Note:

1. All lattice points have the same environment in

the same orientation and are indistinguishablefrom one another.




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1/2 1/2

1/21/2


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lattice + motif = structure

primitive cubic lattice

motif = Cu at 0,0,0

Zn at 1/2, 1/2, 1/2


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1/4

1/4

1/4

1/4

1/43/4

3/4

3/4

3/4

Lattice: face-centred cubic

Motif: C at 0,0,0 C at 1/4,1/4,1/4


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1/4

1/43/4

3/4


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1/4

1/43/4

3/4

Lattice: face-centred cubic

Motif: Zn at 0,0,0 S at 1/4,1/4,1/4


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Graphiteo


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Formally graphite is hexagonal, lattice type P,withand

The lattice motif contains four C atoms with coordinates .

Graphite is a layer structure.

A46.2=

aoA73.6=c

21

32

31

21

31

32

,,;,0,0;0,,;0,0,0

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