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8/3/2019 Crystal Structre
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AUTONOMOU
CRYSTALLOGRAPHY
&
CRYSTAL STRUCTURES
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CRYSTALLINE NATURE OF MATTER:
Elements and their chemical compounds are found inthree states, i.e. Solids, liquids and Gases.
Some solids are brittle, some are ductile, some are strong,some are weak, some are good conductors of heat andelectricity.
Solids can be broadly classified as Crystalline or Non-Crystalline solids
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If the atoms or molecules in a solid are arrangedin some regular fashion then it is known asCrystalline solids .
If the atoms or molecules in a solid are arrangedin irregular fashion ,then it is known as Non-Crystalline solids.
Non- Crystalline is also called as Amorphoussolids
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All the bond in Crystalline solids are equallystrong due to their symmetry .When a crystallinesolids is heated all the bonds are ruptured at aunique temperature but amorphous solids are notequally strong .When it is heated, the weakestbonds are ruptured first while stronger ones laterat high temperature .
Crystalline solids have a very sharp melting pointbut in amorphous solids does not have the sharpmelting point
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Space lattice
A crystal is a 3-Dimensional body.
Crystals are made up of a regular and periodic
3-D patterns of atoms in space is called theCrystal Structure
The Crystal Structure may be described in termsof an idealized geometrical concept called aSpace lattice.
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Definition of Space lattice
A space lattice is defined as an infinite
array of points in three dimensions inwhich every point has surroundings
identical to that of every other point in
the array.
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An array of points such that every point has identical
surroundings
In Euclidean 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
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a
b
2D Lattices
Two basis vectors generate the lattice
There are three lattice parameters which describe this lattice
One angle:
Two distances: a, b
= 90in the current example
ba
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Lattice lines
Lattice points
Three dimensional lattice
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Basis and Unit Cell:
Basis : A set of one or more atoms aboutthe lattice point is calledBasis .orA group
of atoms or molecule identical incomposition is called the basis
basis
Lattice point
The crystal structure is formation when a basis of atoms is attached identically to
each lattice point.
Lattice+basis = crystal structure
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Basis and Unit Cell:
Space Lattice + Basis = Crystal Structure
=
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UNIT CELL
UNIT CELL:A unit cell is the smallest geometric figurethe repetition which gives actual crystal structure
Or
The unit cell may also be defined as the fundamentalelementary pattern of minimum number of atoms,
molecules or group of molecules which represent fully
all the characteristics of crystal.
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Square Lattice
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X X X X X X
X X X X X X
X X X X X X
X X X X X X
X X X X X X
X X X X X X
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Lattice Parameters of an unit cellZ
X
Y
a
b
c
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Lattice Parameters of an unit cell
CRYSTALLOGRAPHIC AXES:
The lines drawn parallel to the lines of
intersection of any three faces of the unitcell which do not lie in the same plane are
called crystallographic axes.
PRIMITIVES:
a, b and c are the dimensions of an unit cell and are
known as Primitives.
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Lattice Parameters of an unit cell
INTERFACIAL ANGLES:-
The angles between three crystallographic axesare known as Interfacial angles , and .
The primitives a,b and c and Interfacial angles , and .are the basic lattice parameters .
The unit cell formed by the primitives a,b and c is
called primitive cell A primitive cell will have only one lattice point
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Crystal Systems:
on the basis of the lengths and direction of theaxis of the symmetry may be classified as Seven
Crystal Systems.
Cubic system:
* Lattice parameters
* All three sides are equal,
* All three angles , a b c
e.g :Nacl, diamond etc..
90
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Crystal Systems
Tetragonal System:
Lattice Parameters
* two sides equal
* all three right angles equal
a b c
90
e.g: NiSo4 etc..
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Crystal Systems
Orthorhombic system:
Lattice parameters
* All three sides different
* All three right angles equal
a b c
90
Eg; KNO3
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Monoclinic System :
Lattice parameters
* All three sides
different
Two right angles ,
third arbitrary,
a b c
90
E.g; FeSo4
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Crystal Systems
Triclinic System:
Lattice parameters
*All three sides
different
* All three angles
different
a b c
Eg. CuSO4
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Crystal Systems
Trigonal(rhombohedral):
Lattice parameters
*All three sides equal
All three angles ,
of arbitrary value ,
a b c
90
E.g CaSO4,Quartz,etc.
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Some times an alternate hexagonal cell
is used instead of the Trigonal Cell
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Crystal Systems
Hexagonal System:
Lattice parameters
* Two sides equal,
third arbitrary
* Two right angles ,
third angle
a b c
90 , 120
a b c
E.g SiO2 ,etc..
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Crystal Systems
* Cubic a = b=c = = =90
* Tetragonal a = b c = = =90
* Orthorhombic a b c = = =90
* Monoclinic a b c = = 90
* Triclinic a b c 90
* Trigonal a = b = c = = 90
* Hexagonal a=b=c
=
=90,=120
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Simple Cubic or (Primitive) Or Cubic P
type Lattice:
There is one lattice point at each of eight
corners of the unit cell. There is no latticepoint inside the unit cell.
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Lattice Point per Unit cell:
Simple Cubic lattice :one lattice point at eachof the eight corners of the unit cell. Each latticepoint forming the simple cubic lattice is amember of 8 surrounding cell.
the share of each lattice point to the cubiclattice=1/8.
Number of lattice points in unit cell=(8*1/8)=1
Hence it is Primitive.
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Body Centred Cubic (bcc) I Lattice:
* There is one lattice point at each of the
eight corners and one lattice point at centreof each cell. So there is a lattice point at
the centre of each unit cell
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lattice Points Per Unit Cell:
There are eight lattice points at the eight corners and eachof them is a member of 8 surrounding cells.
Also there in one lattice point at the centre of the body.
** total number of body centre of lattice point per unitcell =1
** total number of corner lattice point attached to unitcell=(8*1/8)=1
** total number of lattice point in unit cell 1+1=2
*** hence its not primitive.
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Face Centred Cubic or Cubic F Lattice:
* There is one lattice Point at each of the
eight corners and one lattice point at thecentres of each of the six faces of the unit
cell.
* Thus there is an extra point point in at thecentre of each face.
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LatticePoints Per Unit Cell:
unit cell consists of 8 corner lattice point ,eachbeing a member of 8 surrounding cells.
*** Six face lattice points,each being a memberof two cells.
** total no. of corner lattice point concerned withunit cell=8*1/8=1.
** total no.of face centred lattice atomsconcerned with unit cell=6*1/2=3
*** total no.of lattice points concerned with unitcell=1+3=4
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Position of lattice
points
Effective number of Lattice points /
cell
1 P 8 Corners = [8 (1/8)] = 1
2 I8 Corners+
1 body centre
= [1 (for corners)] + [1 (BC)] = 2
3 F
8 Corners
+
6 face centres
= [1 (for corners)] + [6 (1/2)] = 4
4
A/
B/
C
8 corners
+
2 centres of opposite
faces
= [1 (for corners)] + [2 (1/2)] = 2
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1 Cubic Cube
P I F C
P
I
F90 a b c
BRAVAIS LATTICES
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P I F C
2 Tetragonal Square Prism (general height)
IP
a b c 90
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P I F C
3Orthorhomb
icRectangular Prism (general height)
PI
F
C
a b c 90
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6 Monoclinic Parallogramic Prism
I F
P I F C
a b c
P C
90 a b c
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P I F C
7 Triclinic Parallelepiped (general)
a b c
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P I F C
5 TrigonalParallelepiped (Equilateral,
Equiangular)
90 a b c
Rhombohedral
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P I F C
4 Hexagonal 120 Rhombic Prism
Note: there is only one type of hexagonal
lattice (the primitive one)
a b c 90 , 120
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BRAVAIS LATTICES
**Cubic Simple p
Body centred I
Face centred F
**Tetragonal simple p
Body centered I
**Orthorhombic simple pBase centred C
Body centred I
Face centred F
**Monoclinic sim ple p
Base centred c
**Triclinic simple p
**Trigonal simple p
**Hexagonal simple p
`
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The distance between the centers of two nearest
neighboring atoms is called nearest neighbordistance.
Atomis radius (r). Half the distnce between the NND
i.e 2r/2=r
NEAREST NEIGHBOUUR DISTANCE
(2r):
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CO-ORDINATION NUMBER:
Co-ordination number is defined as thenumber of equidistance nearest neighbors
that an atom has in a given structure.
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ATOMIC PACKING FACTOR:
Atomic packing factor is the ratio of volumeoccupied by the atoms in an unit cell to the total
volume of the unit cell. It is also called packingfraction.
V
vA.P.F =
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Void space:
Void space in the unit cell is the vacantspace left or unutilized space in unit cell ,and more commonly known as interstitialspace.
Void space = ( 1-APF ) X 100
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SIMPLE CUBIC STRUCTURE-
Effective number of atoms per unit cell
(8 x 1/8) =1
Nearest neighbour distance 2r = a
. Atomic radius r = a / 2
co-ordination number = 6 a
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SIMPLE CUBIC STRUCTURE
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SIMPLE CUBIC STRUCTURE
5. Atomic packingfactor
Vol. of all atoms in the unit cell
Vol.of the unit cell
6.Void space = (1-APF) X 100
= (1-0.52)X 100
=48%
Ex: polonium.
3
3
3
3
41
3
2
41
3
(2 )0.52
(52%)
r
a
wherea r
r
r
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BCC STRUCTURE
Close packed directions are cube diagonals.
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BCC STRUCTURE
Effective number of atoms per unit cell (8 x 1/8) + 1 =2
Atomic radius r = 3a /4
Nearest neighbour distance
2r =3a/2
Co-ordination number = 8
C
Aa
aB
a2
a3
D
D
C
a
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BCC STRUCTURE
5.Atomic packing factor
6.Void space = (1-APF) x 100
= (1-0.68) x 100
= 32%
Ex:Na, lithium andChromium.
3
3
3
3
42
3
34
4 32 ( )
3 4
( )
0.68
(68%)
r
a
wherer a
a
a
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FCC Crystal Structure
Effective number of atoms per unit cell (8 x 1/8) + 1/2 X 6 = 4
2.Atomic radius r = a / 22
3.Nearest neighbour distance
2r = a /2
4.Co-ordination number = 12
A B
C
4r
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FCC Crystal Structure
Close packed directions are face diagonals.
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FCC Crystal Structure
How the co-ordination number is 12 in FCCStructure ?
Ans. Centre of each face as one atom. this centreatom touches four corner atoms in its plane, four
body centered atoms in each of the two planes oneither side of its plane. thus the co-ordinationnumber is 12.
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FCC coordination number
4
C
1
2
35
6
7
89
10
11
12
a
a /2
a /2
a
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5.Atomic packing factor
6.Void space = (1-APF) X 100
= (1-0.74) X 100
= 26%
Ex: Copper , Aluminum, silver and Led
3
3
3
3
443
2 2
44 ( )
3 2 2
( )
0.74
(74%)
r
a
a
r
a
a
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Hexagonal (Unit cell)
H l l k d
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Hexagonal close packed
structure: Effective number of atoms per unit cell
2 x (6x 1/6) + 2 x 1/2 + 3 = 6.
2.Atomic radius r = a / 2.
3.Nearest neighbor distance
2r = a
4.Co-ordination number = 12.
CC/2
a
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HCP Co ordination Number
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Calculation of c/a ratio
Let c be the hight of the unit cell and abe its edge, the three body atoms lie in a
horizontal plane at height c/2 from the
bottom layer
a
a
cc
B
A
O
B
Ox
A30
0
C
XA
/
O/
a2
cCX
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A/AB=300
From the figure, Triangle ABA/
Cos
=A
300
A/
AB
A
=
A/
But AX=2/3 A=A/
23a
3
a
=
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8
32
4
2
3
2
3
2
2
22
2
222
222
a
c
actherefore
cbutCX
aa
a
AXACCX
CXAXACIn Triangle AXC,
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Atomic packing factor
Area of the base=6* Area of the triangle ABO
6*1/2*AB*OO/
2/332
a
2/332
a* C
aa
2
3**3
VOL. OF UNIT CELL IS=Area of the base * hight
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7000
2
3
2
3
74
23
8
3
33
2
2.
33
16
2
33
3
4*6
c
a
Since
c
a
arSub
ca
r
ca
r A.P.F=
Eg; Mg,Zn,Cd
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Diamond structure:
Diamond structure:
Diamond is a combination of twointerpenetrating FCC - sub lattices alongthe body diagonal by 1/4thCube edge.
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1 23
4
5
6
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x p
y
z
a/4
a/4
a/2
a
a
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Diamond - APF
Effective number of atoms perunit cell (8 x 1/8) + 1/2 X 6+ 4 = 8.
2.Atomic radius r = 3a / 8.
3.Nearest neighbor distance 2r = 3a / 4.
4.Co-ordination number = 4.
a/4
a/4
a/4
x p
z
y
2r
8
a
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Diamond Structure
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Diamond Structure
* The fraction such as 1/4,1/2,3/4 denote
the height above the base in units of cubeedge.
1/2
1/2
0 1/2 0
00
1/4
0
3/4
1/2
1/43/4
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5. Atomic packing
factor
3
3
3
3
48
3
38
4 38 ( )
3 8
( )
0.34
(34%)
r
a
r a
a
a
6. Void space = (1-APF) x 100
= (1-0.34) x 100
= 66%Ge, Si and Carbon atoms are
possess this structure
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Structure of a zinc sulphide:
ZnS structure is interpenetration of two fcclattices where One of zinc and other is of
sulpher atoms. This structure is similar to
Diamond crystal structure
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Zn
S
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Sodium chloride structure:
Nacl crystal is an ionic crystal.
In Nacl structure Na+ and Cl- are situated side by side
NaCl Structure is consists of two fcc sub lattices.
One of the chlorine ion having its origin at the(o, o, o) point. and other of the sodium ions having itsorigin at (a/2,0,0).
Each ion in a Nacl lattice has six nearest neighbor ions ata distance a/2.i,e its co-ordination number is 6.
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2a
2
a
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2
a
a
a2
a
2
a
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Structure of cesium chloride
CsCl is an ionic compound. Ithas simple cubic structure.
The Unit cell of CsCl as shownIn Fig.
From the figure Cs+ ions aresituated at the corners of thecubic cell and Cl- ions at thebody centre or vice versa
So the crystal may be consideredto be a combination of twosimple cubic Sublattice one is
cs+ ions and other of Cl- ion.
The co-ordination number is 8 Cs+
Cl
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