Crystalstructures 141008215641 Conversion Gate02

8/21/2019 Crystalstructures 141008215641 Conversion Gate02

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CRYSTAL STRUCTURES

UNIT-I

Hari Prasad

Hari PrasadAssistantProfessor

MVJCE-Bangalore


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Learning objectives

• After the chapter is completed, you will be able toanswer:

• Difference between crystalline and noncrystalline

structures• Different crystal systems and crystal structures

• Atomic packing factors of different cubic crystalsystems

• Difference between unit cell and primitive cell

• Difference between single crystals and polycrystals

Hari Prasad


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What is space lattice?

• Space lattice is the distribution of points in3D in such a way that every point hasidentical surroundings, i.e., it is an infinitearray of points in three dimensions inwhich every point has surroundingsidentical to every other point in the array.

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Common materials: with various ‘viewpoints’

Glass: amorphous

Ceramics

Crystal

Graphite

PolymersMetals


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Metals and alloys Cu, Ni, Fe, NiAl (intermetallic compound), Brass (Cu-Zn alloys)

Ceramics (usually oides, nitrides, car!ides) Alumina (Al"#$), Zirconia (Zr "#$)

Polymers (thermoplasts, thermosets) (%lastomers) Polythene, Poly&inyl chloride, Polypropylene

Common materials: examples

Based on %lectrical Conduction

Conductors Cu, Al, NiAl

'emiconductors Ge, 'i, GaAs

nsulators Alumina, Polythene

Based on *uctility

*uctile Metals, Alloys

Brittle Ceramics, nor+anic Glasses, Ge, 'i

* some special polymers could be conducting


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MA%A.' 'C%NC% / %NGN%%NG

P01'CA. M%C0ANCA. %.%C#-

C0%MCA.

%C0N#.#GCA.

• %tracti&e• Castin+• Metal Formin+• 2eldin+• Po3der Metallur+y

• Machinin+

• 'tructure• Physical

Properties

Science of Metallurgy

• *e4ormation

Beha&iour

• hermodynamics• Chemistry• Corrosion

he !road scienti4ic and technolo+ical se+ments o4 Materials 'cience are sho3n

in the dia+ram !elo35

o +ain a comprehensi&e understandin+ o4 materials science, all these aspects

ha&e to !e studied5


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Lattice the underlying periodicity of the crystal

Basis Entityassociated with each lattice points

Lattice how to repeat

Motif what to repeat

Crystal =Lattice +

MotifMotif orBasis:typically an atom or a group of atoms associated with each lattice point

Definition 1

Translationally periodicarrangement ofmotifs

Crystal

Translationally periodicarrangement ofpoints

Lattice

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Hari Prasad


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An array of points such that every point has

identical surroundings

InEuclidean space ⇒

infinite array

We can have 1D, 2D or 3D arrays (lattices)

Space Lattice

Translationally periodic arrangement of points in space is called a lattice

or

A lattice is also called a Space Lattice

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Unit cell:A unit cell is the sub-division of the space

lattice that still retains the overall characteristics ofthe space lattice.

Primitive cell:the smallest possible unit cell of a lattice,having lattice points at each of its eight vertices only.

A primitive cell is a minimum volume cellcorresponding to a single lattice point of a structurewith translational symmetry in 2 dimensions, 3dimensions, or other dimensions.

A lattice can be characterized by the geometry of its primitive cell.

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ato!s "a#$ in "eriodi#% &' arra(s

r(stalline !aterials)))

-!etals-!an( #era!i#s-so!e "ol(!ers

ato!s *a+e no "eriodi# "a#$ing

on-#r(stalline !aterials)))

-#o!"le, str#tres-ra"id #ooling

#r(stalline Si./ 01art

!or"*os4 5 Non#r(stalline

Materials andPa#$ing

Si .,(gen

t("i#al of6

o##rs for6

non#r(stalline Si./ 07las

Hari Prasad


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Cr(stal S(ste!s

7 crystal systems

14 crystal lattices

Unit #ell6 smallest repetitive volumewhich contains the complete latticepattern of a crystal.

a, b, and c are t*e latti#e #onstants

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Hari Prasad

The Unite Cell is the smallest group of atom showingthe characteristic lattice structure of a particular metal.It is the building block of a single crystal.A single crystal can have many unit cells.


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Crystal systemsCubic Three equal axes, mutually perpendicular

a=b=c α=β=γ =90˚

Tetragnal Three perpendicular axes, nly t! equal

a=b"c α=β=γ =90˚

#exagnal Three equal cplanar axes at 1$0˚ and a %urth unequalaxis perpendicular t their plane

a=b"c α=β= 90˚ γ =1$0˚

&hmbhedral Three equal axes, nt at right angles

a=b=c α=β=γ "90˚

'rthrhmbic Three unequal axes, all perpendicular

a"b"c α=β=γ =90˚

(nclinic Three unequal axes, ne % !hich is perpendicular t thether t!

a"b"c α=γ =90˚" β

Triclinic Three unequal axes, n t! % !hich are perpendicular

a"b"c α" β"γ "90˚Hari Prasad


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Hari Prasad

Some engineering applications require

single crystals:-dia!ond single #r(stals for a8rasi+es --tr8ine 8lades


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9*at is #oordination n!8er:

• The coordination number of a centralatom in a crystal is the number of itsnearest neighbours.

9*at is latti#e "ara!eter:

• The lattice constant , or lattice parameter , refers to the physicaldimension of unit cells in a crystal lattice.

• Lattices in three dimensions generallyhave three lattice constants, referred toas a, b, and c.

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re de to lo; "a#$ing densit( 0onl( Po *as t*is str#tre3

se-"a#$ed dire#tions are #8e edges)

Coordination


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! Coordination " # $

to!s to#* ea#* ot*er along #8e diagonals)Note6 All ato!s are identi#al> t*e #enter ato! is s*aded di?erentl( onl( for ease of +ie;ing)

Bod( Centered C8i# Str#tre 0BCC3

e,6 Cr% 9% @e 0 3% Tantal!% Mol(8den!

/ ato!snit #ell6 #enter D #orners , D

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%tomic Pac&ing 'actor: CC

a

AP@ 5

&

π 0 &a 3&/

ato!s

nit #ell ato!

+ol!e

a&

nit #ell

+ol!e

lengt* 5 R 5

Close-"a#$ed dire#tions6

& a

P@ for a 8od(-#entered #8i# str#tre 5 F)=D

aR

a2

a$

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Coordination < 5 /

to!s to#* ea#* ot*er along fa#e diagonals)

ote6 All ato!s are identi#al> t*e fa#e-#entered ato!s are s*adi?erentl( onl( for ease of +ie;ing)

@a#e Centered C8i# Str#tre 0@CC3

e,6 Al% C% A% P8% Ni% Pt% Ag

ato!snit #ell6 = fa#e , / D #orners , DHari Prasad


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Hari Prasad


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P@ for a fa#e-#entered #8i# str#tre 5 F)G

Ato!i# Pa#$ing @a#tor6 @CC

maximum achievable APF

AP@ 5

&π 0 /a3&

ato!s

nit #ell ato!+ol!e

a&

nit #ell

+ol!e

Close-"a#$ed dire#tions6

lengt* 5 R 5 / a

Unit #ell #ontains6 = , / D , D

5 ato!snit #ella

$ a

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A sites

B B

B

BB

B B

Csites

C C

CA

B

Bsites

ABCABC))) Sta#$ing Seen#e

/' Proe#tion

@CC Unit Cell

@CC Sta#$ing Seen#e

B B

B

BB

B B

Bsites

C C

CA

C C

CA

A

BC


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) *

+ +

CC

=

Putting atoms in the B position in the II layer and in C positions in the III layer we get

a stacking sequence → ABC ABC ABC…. The CCP (CC! crystal

)

*C

)

*C

C


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Coordination < 5 /

ABAB))) Sta#$ing Seen#e

AP@ 5 F)G

&' Proe#tion /' Proe#tion

He,agonal Close-Pa#$ed Str#tre 0HCP3

6 atoms/unit cell

ex: Cd, Mg, Ti, Zn

ca 5 )=&&

c

a

A sites

Bsites

A sites Botto! la(er

Middle la(er

To"la(er

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f


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APF for HCP

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c

a

A sites

B sites

A sites

C=1.633a

Number of atoms in HCP unit cell=(12*1/6)+(2*1/2)+3=6atoms

Vol.of HCP unit cell=area of the hexagonal face X height of the hexagonal

Area of the hexagonal face=area of each triangle X6

a

ha

Area of triangle = Area of hexagon =

Volume of HCP=

APF=6

a=2r

APF =0.74


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SC-coordination number

Hari Prasad

6


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•Coordination # = 6 (# nearest neighbors)


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BCC-coordination number

Hari Prasad

8


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FCC-coordination number

Hari Prasad

4+4+4=12


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Hari Prasad


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HCP-coordination number

Hari Prasad

3+6+3=12


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T*eoreti#al 'ensit(% ρ

;*ere n 5 n!8er of ato!snit #ell A 5 ato!i# ;eig*t

V C 5 Vol!e of nit #ell 5 a& for

#8i# NA 5 A+ogadros n!8er

5 =)F/& , F/& ato!s!ol

'ensit( 5 ρ 5

V C N

A

n Aρ 5

CellUnitof Vol!eTotal

CellUnitinAto!sof Mass

Hari Prasad


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• E,6 Cr 0BCC3 A 5 K/)FF g!ol

R 5 F)/K n!

n 5 /

ρ

theoretical

a = R / ! = "#$%%& nm

ρactual

aR

ρ 5a&

K/)FF/

ato!s

nit #ell !ol

g

nit #ell

+ol!e ato!s

!ol

=)F/& , F/&

T*eoreti#al 'ensit(% ρ

5 G)D g#!&

5 G) g#!&

Hari Prasad


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Pol(!or"*is!• T;o or !ore distin#t #r(stal str#tres

for t*e sa!e !aterial0allotro"("ol(!or"*is!3

titani!

% β-Ti

#ar8on

dia!ond% gra"*ite

'CC

FCC

'CC

()!%*C

(!+*C

+($*C

δ"Fe

γ"Fe

"Fe

liuid

iron s-stem

Hari Prasad


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Miller indices

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Miller indices: defined as the reciprocals of theintercepts made by the plane on thethree axes.


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Procedure for finding Miller indices

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Determine theintercepts of the plane along the axesX,Y and Z in terms of thelattice constants a, b and c.

Step 1


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Determine the

reciprocals of thesenumbers.

Step 2


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Find the least commondenominator (lcd)and multiply each bythislcd

Step 3


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The result is written inparenthesis.

This is called thèMillerIndices’ of the plane inthe form (h k l).

Step 4


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Find intercepts along axes → 2 3 1

Take reciprocal → 1/2 1/3 1

Convert to smallest integers in the same ratio → 3 2 6

Enclose in parenthesis → (326)

(2,0,0)

(0,3,0)

(0,0,1)

#iller Indices $or planes


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X

Z

Y

Plane ABC has intercepts of2 unitsalong X-axis, 3 units along Y-axis and 2units along Z-axis.

A

C

B


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DETERMINATION OF ‘MILLER INDICES’

Step 1:The intercepts are2, 3 and 2 on the threeaxes.

Step 2:The reciprocals are1/2, 1/3 and 1/2.

Step 3:The least common denominator is ‘6’.Multiplying each reciprocal by lcd,

we get, 3,2 and 3.

Step 4:HenceMiller indices for the plane ABC is (32 3)

IMPORTANTFEATURESOFMILLERINDICES


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Hari Prasad

For the cubic crystal especially, the important features ofMiller indices are, A plane which is parallel to any one of the co-ordinateaxes has an intercept of infinity (∞).

Therefore the Miller index for that axis is zero; i.e. for anintercept at infinity, the corresponding index is zero.

A plane passing through the origin is defined in termsof aparallel plane having non zero intercepts.

Allequally spaced parallel planes have same ‘Millerindices’i.e. The Miller indices do not only define a

particular plane but also a set of parallel planes. Thus the planes whose intercepts are 1, 1,1; 2,2,2; -3,-3,-3etc., are all represented by the same set of Miller indices.

IMPORTANT FEATURES OF MILLER INDICES


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Hari Prasad

Worked Example: Calculate the miller indices for the plane with intercepts 2a, - 3band 4c the along the crystallographic axes.

The intercepts are 2, - 3 and 4

Step 1:The intercepts are 2, -3 and 4 along the 3 axes

Step 2: The reciprocals are

Step 3: The least common denominator is 12.

Multiplying each reciprocal by lcd, we get 6 -4 and 3

Step 4: Hence the Miller indices for the plane is( )6 7 $


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Intercepts → 1 ∞ ∞

Plane → (100

!amily → "100# → $

Intercepts → 1 1 ∞

Plane → (110

!amily → "110# → %

Intercepts → 1 1 1

Plane → (111

!amily → "111# → &

(Octahedral plane)


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Miller Indices : (100)


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Intercepts :a ,a , ∞Fractional intercepts : 1 , 1 , ∞Miller Indices : (110)


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Intercepts :a ,a ,a

Fractional intercepts : 1 , 1 , 1Miller Indices : (111)


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Intercepts : ½a ,a , ∞Fractional intercepts : ½ , 1 , ∞Miller Indices : (210)


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Z


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(101)

Z

Y

X


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Hari Prasad

(122)


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Hari Prasad

(211)


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Crystallographic Directions

The crystallographic directions are fictitious lineslinking nodes (atoms, ions or molecules) of acrystal.

Similarly, the crystallographic planes arefictitious planes linking nodes.

The length of the vector projection on each of the

three axes is determined;these are measured interms of the unit cell dimensions a ,b ,and c.


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Hari Prasad

To find the Miller indices of a direction,Choose a perpendicular plane to that direction.

Find the Miller indices of that perpendicularplane.

The perpendicular plane and the direction havethe same Miller indices value.

Therefore, the Miller indices of theperpendicular plane is written within a squarebracket to represent the Miller indices of thedirection like [ ].


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Summary of notations

SymbolAlternat

e

symbols

Directio

n

[ ] [uvw] → Particular direction

< > [[ ]] → Family of directions

Plane( ) (hkl) → Particular plane

{ } {hkl} (( )) → Family of planes

Point. . .xyz. [[ ]] → Particular point

: : :xyz: → Family of point

*A family is also referred to as a symmetrical set


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Hari Prasad

'or each of the three a*es+ there wille*ist both positive and negativecoordinates.

,hus negative indices are also possible+which are represented by a bar overthe appropriate inde*. 'or e*ample+ the-

The above image shows [100], [110], and [111] directions within a

unit cell


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Hari Prasad

The vector, as drawn, passes through the origin of the coordinate system,and therefore no translation is necessary. Projections of this vector onto thex ,y , andzaxes are, respectively,1/2,b , and 0c , which become 1/2, 1, and 0 interms of the unit cell parameters (i.e., when thea ,b , andcare dropped).Reduction of these numbers to the lowest set of integers is accompanied bymultiplication of each by the factor 2.This yields the integers 1, 2, and 0,which are then enclosed in brackets as [120].


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Worked Example

Find the angle between the directions [2 1 1] and [1 1 2] in acubic crystal.

The two directions are [2 1 1] and [1 1 2]

We know that the angle between the two directions,

8 " 8 " 8 "

" " " " " "

8 8 8 " " "

9 9

u u & & 3 3cos

(u & 3 ) (u & 3 )

+ +θ =

+ + × + +


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Hari Prasad

In this case, u1 = 2, v1 = 1, w1 = 1, u2 = 1, v2 = 1, w2 = 2

(or) cosθ = 0.833

θ = 35° 3530′.

" " " " " "

(" 8) (8 8) (8 ") :cos

6" 8 l 8 8 "

× × × + ×∴ θ = =

+ + × + +


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http://

core.materials.ac.uk/repository/doitpoms/tlp/miller_indices/indexing_a_plane_e

mbed.swf

http://

core.materials.ac.uk/repository/doitpoms/tlp/miller_indices/drawing_lattice_plane

s.swf

http://core.materials.ac.uk/repository/doitpoms/tlp/miller_indices/miller.swf

Reference
http://core.materials.ac.uk/repository/doitpoms/tlp/miller_indices/indexing_a_plane_embed.swfhttp://core.materials.ac.uk/repository/doitpoms/tlp/miller_indices/indexing_a_plane_embed.swfhttp://core.materials.ac.uk/repository/doitpoms/tlp/miller_indices/indexing_a_plane_embed.swfhttp://core.materials.ac.uk/repository/doitpoms/tlp/miller_indices/drawing_lattice_planes.swfhttp://core.materials.ac.uk/repository/doitpoms/tlp/miller_indices/drawing_lattice_planes.swfhttp://core.materials.ac.uk/repository/doitpoms/tlp/miller_indices/drawing_lattice_planes.swfhttp://core.materials.ac.uk/repository/doitpoms/tlp/miller_indices/miller.swfhttp://core.materials.ac.uk/repository/doitpoms/tlp/miller_indices/miller.swfhttp://core.materials.ac.uk/repository/doitpoms/tlp/miller_indices/miller.swfhttp://core.materials.ac.uk/repository/doitpoms/tlp/miller_indices/drawing_lattice_planes.swfhttp://core.materials.ac.uk/repository/doitpoms/tlp/miller_indices/drawing_lattice_planes.swfhttp://core.materials.ac.uk/repository/doitpoms/tlp/miller_indices/drawing_lattice_planes.swfhttp://core.materials.ac.uk/repository/doitpoms/tlp/miller_indices/indexing_a_plane_embed.swfhttp://core.materials.ac.uk/repository/doitpoms/tlp/miller_indices/indexing_a_plane_embed.swfhttp://core.materials.ac.uk/repository/doitpoms/tlp/miller_indices/indexing_a_plane_embed.swf

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