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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
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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
http://var/www/apps/conversion/tmp/scratch_6/lattice.ppthttp://var/www/apps/conversion/tmp/scratch_6/Motifs.ppthttp://var/www/apps/conversion/tmp/scratch_6/Motifs.ppthttp://var/www/apps/conversion/tmp/scratch_6/Motifs.ppthttp://var/www/apps/conversion/tmp/scratch_6/Motifs.ppthttp://var/www/apps/conversion/tmp/scratch_6/lattice.ppt
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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
http://var/www/apps/conversion/tmp/scratch_6/lattice.ppthttp://var/www/apps/conversion/tmp/scratch_6/space.ppthttp://var/www/apps/conversion/tmp/scratch_6/space.ppthttp://var/www/apps/conversion/tmp/scratch_6/lattice.ppt
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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
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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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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
Hari Prasad
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
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8
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FCC-coordination number
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4+4+4=12
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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
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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 / ! = "#$%%& nm
ρactual
aR
ρ 5a&
K/)FF/
ato!s
nit #ell !ol
g
nit #ell
+ol!e ato!s
!ol
=)F/& , F/&
T*eoreti#al 'ensit(% ρ
5 G)D g#!&
5 G) g#!&
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Pol(!or"*is!• T;o or !ore distin#t #r(stal str#tres
for t*e sa!e !aterial0allotro"("ol(!or"*is!3
titani!
% β-Ti
#ar8on
dia!ond% gra"*ite
'CC
FCC
'CC
()!%*C
(!+*C
+($*C
δ"Fe
γ"Fe
"Fe
liuid
iron s-stem
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Miller indices
Hari Prasad
Miller indices: defined as the reciprocals of theintercepts made by the plane on thethree axes.
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Procedure for finding Miller indices
Hari Prasad
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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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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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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Hari Prasad
Intercepts : ½a ,a , ∞Fractional intercepts : ½ , 1 , ∞Miller Indices : (210)
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Hari Prasad
Z
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Hari Prasad
(101)
Z
Y
X
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Hari Prasad
(122)
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Hari Prasad
(211)
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Hari Prasad
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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'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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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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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.swfRecommended