hari prasad assistant professor mvjce-bangalore. learning objectives after the chapter is completed,...
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Hari Prasad
CRYSTAL STRUCTURES
UNIT-I
Hari PrasadAssistant Professor
MVJCE-Bangalore
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Hari Prasad
Learning objectives
• After the chapter is completed, you will be able to answer:
• Difference between crystalline and noncrystalline structures
• Different crystal systems and crystal structures
• Atomic packing factors of different cubic crystal systems
• Difference between unit cell and primitive cell• Difference between single crystals and poly
crystals
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Hari Prasad
What is space lattice?
• Space lattice is the distribution of points in 3D in such a way that every point has identical surroundings, i.e., it is an infinite array of points in three dimensions in which every point has surroundings identical 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 oxides, nitrides, carbides) Alumina (Al2O3), Zirconia (Zr2O3)
Polymers (thermoplasts, thermosets) (Elastomers) Polythene, Polyvinyl chloride, Polypropylene
Common materials: examples
Based on Electrical Conduction Conductors Cu, Al, NiAl Semiconductors Ge, Si, GaAs Insulators Alumina, Polythene*
Based on Ductility Ductile Metals, Alloys Brittle Ceramics, Inorganic Glasses, Ge, Si
* some special polymers could be conducting
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MATERIALS SCIENCE & ENGINEERING
PHYSICAL MECHANICAL ELECTRO-CHEMICAL
TECHNOLOGICAL
• Extractive• Casting• Metal Forming• Welding• Powder Metallurgy• Machining
• Structure• Physical Properties
Science of Metallurgy
• Deformation Behaviour
• Thermodynamics• Chemistry• Corrosion
The broad scientific and technological segments of Materials Science are shown in the diagram below.
To gain a comprehensive understanding of materials science, all these aspects have to be studied.
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Lattice the underlying periodicity of the crystal
Basis Entity associated with each lattice points
Lattice how to repeatMotif what to repeat
Crystal = Lattice + Motif
Motif or Basis: typically an atom or a group of atoms associated with each lattice point
Definition 1
Translationally periodic arrangement
of motifs
CrystalTranslationally periodic
arrangement of points
Lattice
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Hari Prasad
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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
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 of the 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 cell corresponding to a single lattice point of a structure with translational symmetry in 2 dimensions, 3 dimensions, or other dimensions. A lattice can be characterized by the geometry of its primitive cell.
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• atoms pack in periodic, 3D arraysCrystalline materials...
-metals-many ceramics-some polymers
• atoms have no periodic packingNon-crystalline materials...
-complex structures-rapid cooling
crystalline SiO2 (Quartz)
"Amorphous" = Noncrystalline
Materials and Packing
Si Oxygen
• typical of:
• occurs for:
noncrystalline SiO2 (Glass)
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Crystal Systems
7 crystal systems
14 crystal lattices
Unit cell: smallest repetitive volume which contains the complete lattice pattern of a crystal.
a, b, and c are the lattice constants
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The Unite Cell is the smallest group of atom showing the 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 ===90˚
Tetragonal Three perpendicular axes, only two equala=b≠c ===90˚
Hexagonal Three equal coplanar axes at 120˚ and a fourth unequal axis perpendicular to their planea=b≠c == 90˚ =120˚
Rhombohedral Three equal axes, not at right anglesa=b=c ==≠90˚
Orthorhombic Three unequal axes, all perpendiculara≠b≠c ===90˚
Monoclinic Three unequal axes, one of which is perpendicular to the other twoa≠b≠c ==90˚≠
Triclinic Three unequal axes, no two of which are perpendiculara≠b≠c ≠ ≠≠90˚
Hari Prasad
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Some engineering applications require single crystals:--diamond single crystals for abrasives
--turbine blades
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What is coordination number?• The coordination number of a central atom in a
crystal is the number of its nearest neighbours.What is lattice parameter?• The lattice constant, or lattice parameter, refers
to the physical dimension of unit cells in a crystal lattice.
• Lattices in three dimensions generally have three lattice constants, referred to as a, b, and c.
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• Rare due to low packing density (only Po has this structure)• Close-packed directions are cube edges.
• Coordination # = 6 (# nearest neighbors)
Simple Cubic Structure (SC)
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• Coordination # = 8
• Atoms touch each other along cube diagonals.--Note: All atoms are identical; the center atom is shaded differently only for ease of viewing.
Body Centered Cubic Structure (BCC)
ex: Cr, W, Fe (), Tantalum, Molybdenum
2 atoms/unit cell: 1 center + 8 corners x 1/8
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Atomic Packing Factor: BCC
a
APF =
4
3p ( 3 a/4 ) 32
atoms
unit cell atom
volume
a 3
unit cell
volume
length = 4R =
Close-packed directions:
3 a
• APF for a body-centered cubic structure = 0.68
aR
a 2
a 3
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• Coordination # = 12
• Atoms touch each other along face diagonals.--Note: All atoms are identical; the face-centered atoms are shaded differently only for ease of viewing.
Face Centered Cubic Structure (FCC)
ex: Al, Cu, Au, Pb, Ni, Pt, Ag
4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8
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Hari Prasad
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• APF for a face-centered cubic structure = 0.74
Atomic Packing Factor: FCC
maximum achievable APF
APF =
4
3 p ( 2 a/4 )34
atoms
unit cell atom
volume
a3unit cell
volume
Close-packed directions: length = 4R = 2 a
Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cella
2 a
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A sites
B B
B
BB
B B
C sites
C C
CA
B
B sites
• ABCABC... Stacking Sequence• 2D Projection
• FCC Unit Cell
FCC Stacking Sequence
B B
B
BB
B B
B sitesC C
CA
C C
CA
AB
C
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A B
+ +
FCC
=
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 (FCC) crystal
A
BC
A
BC
C
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• Coordination # = 12
• ABAB... Stacking Sequence
• APF = 0.74
• 3D Projection • 2D Projection
Hexagonal Close-Packed Structure (HCP)
6 atoms/unit cell
ex: Cd, Mg, Ti, Zn
• c/a = 1.633
c
a
A sites
B sites
A sites Bottom layer
Middle layer
Top layer
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APF for HCP
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 hexagonalArea 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
6
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• Coordination # = 6 (# nearest neighbors)
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BCC-coordination number
8
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FCC-coordination number
4+4+4=12
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HCP-coordination number
3+6+3=12
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Theoretical Density, r
where n = number of atoms/unit cell A = atomic weight VC = Volume of unit cell = a3 for cubic NA = Avogadro’s number = 6.023 x 1023 atoms/mol
Density = =
VC NA
n A =
Cell Unit of VolumeTotalCell Unit in Atomsof Mass
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• Ex: Cr (BCC) A = 52.00 g/mol R = 0.125 nm n = 2
theoretical
a = 4R/ 3 = 0.2887 nm
ractual
aR
= a 3
52.002
atoms
unit cellmol
g
unit cell
volume atoms
mol
6.023 x 1023
Theoretical Density, r
= 7.18 g/cm3
= 7.19 g/cm3
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Polymorphism • Two or more distinct crystal structures for the same
material (allotropy/polymorphism) titanium , -Ti
carbondiamond, graphite
BCC
FCC
BCC
1538ºC
1394ºC
912ºC
-Fe
-Fe
-Fe
liquid
iron system
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Miller indices
Miller indices: defined as the reciprocals of the intercepts made by the plane on the three axes.
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Procedure for finding Miller indices
Determine the intercepts of the plane along the axes X,Y and Z in terms of the lattice constants a, b and c.
Step 1
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Determine the reciprocals of these numbers.
Step 2
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Find the least common denominator (lcd) and multiply each by this lcd
Step 3
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The result is written in parenthesis.This is called the `Miller Indices’ of the plane in the 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)
Miller Indices for planes
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X
Z
Y
Plane ABC has intercepts of 2 units along X-axis, 3 units along Y-axis and 2 units along Z-axis.
A
C
B
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DETERMINATION OF ‘MILLER INDICES’
Step 1: The intercepts are 2, 3 and 2 on the three axes.
Step 2: The reciprocals are 1/2, 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:Hence Miller indices for the plane ABC is (3 2 3)
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For the cubic crystal especially, the important features of Miller indices are, A plane which is parallel to any one of the co-
ordinate axes has an intercept of infinity (). Therefore the Miller index for that axis is
zero; i.e. for an intercept at infinity, the corresponding index is zero.
A plane passing through the origin is defined in terms of a parallel plane having non zero intercepts.
All equally spaced parallel planes have same ‘Miller indices’ 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,-3 etc., 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, - 3b and 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 4 3
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Intercepts → 1 Plane → (100)Family → {100} → 3
Intercepts → 1 1 Plane → (110)Family → {110} → 6
Intercepts → 1 1 1Plane → (111)Family → {111} → 8(Octahedral plane)
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Miller Indices : (100)
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Hari Prasad
Intercepts : a , a , ∞ Fractional intercepts : 1 , 1 , ∞ Miller Indices : (110)
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Hari Prasad
Intercepts : a , a , a Fractional intercepts : 1 , 1 , 1 Miller Indices : (111)
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Hari Prasad
Intercepts : ½ a , a , ∞ Fractional intercepts : ½ , 1 , ∞ Miller Indices : (210)
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Hari Prasad
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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 lines linking nodes (atoms, ions or molecules) of a crystal.
Similarly, the crystallographic planes are fictitious planes linking nodes.
The length of the vector projection on each of the three axes is determined; these are measured in terms of the unit cell dimensions a, b, and c.
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To find the Miller indices of a direction, Choose a perpendicular plane to that direction.
Find the Miller indices of that perpendicular plane.
The perpendicular plane and the direction have the same Miller indices value.
Therefore, the Miller indices of the perpendicular plane is written within a square bracket to represent the Miller indices of the direction like [ ].
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Summary of notations
Symbol
Alternate
symbols
Direction
[ ] [uvw] →Particular direction
< > <uvw> [[ ]] →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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For each of the three axes, there will exist both positive and negative coordinates.Thus negative indices are also possible, which are represented by a bar overthe appropriate index. For example, the 1
The above image shows [100], [110], and [111] directions within aunit 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 the x, y, and z axes are, respectively,1/2, b, and 0c, which become 1/2, 1, and 0 in terms of the unit cell parameters (i.e., when the a, b, and c are dropped). Reduction of these numbers to the lowest set of integers is accompanied by multiplication 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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Hari Prasad
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Worked Example Find the angle between the directions [2 1 1]
and [1 1 2] in a cubic crystal.
The two directions are [2 1 1] and [1 1 2]
We know that the angle between the two directions,
1 2 1 2 1 2
2 2 2 2 2 21 1 1 2 2 2
½ ½
u u v v w wcos
(u v w ) (u v w )
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Hari Prasad
In this case, u1 = 2, v1 = 1, w1 = 1, u2 = 1, v2 = 1, w2 = 2
(or) cos = 0.833
= 35° 3530.
2 2 2 2 2 2
(2 1) (1 1) (1 2) 5cos
62 1 l 1 1 2
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http://core.materials.ac.uk/repository/doitpoms/tlp/miller_indices/indexing_a_plane_embed.swf
http://core.materials.ac.uk/repository/doitpoms/tlp/miller_indices/drawing_lattice_planes.swf
http://core.materials.ac.uk/repository/doitpoms/tlp/miller_indices/miller.swf
Reference