1. Crystal structures Unit-I Hari Prasad Assistant Professor
MVJCE-Bangalore
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2. 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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3. 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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4. Common materials: with various ‘viewpoints’
Graphite
Glass: amorphous
Ceramics
Crystal
Metals
Polymers

5. Common materials: examples
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
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

6. MATERIALS SCIENCE & ENGINEERING
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.
MATERIALS SCIENCE & ENGINEERING
Science of Metallurgy
PHYSICAL
MECHANICAL
ELECTRO-
CHEMICAL
TECHNOLOGICAL
Extractive
Casting
Metal Forming
Welding
Powder Metallurgy
Machining
Structure
Physical Properties
Deformation
Behaviour
Thermodynamics
Chemistry
Corrosion

7. Crystal = Lattice + Motif
Definition 1
Crystal = Lattice + Motif
Motif or Basis: typically an atom or a group of atoms associated with each lattice point
Lattice  the underlying periodicity of the crystal
Basis  Entity associated with each lattice points
Lattice  how to repeat
Motif  what to repeat
Lattice
Crystal
Translationally periodic arrangement of points
Translationally periodic arrangement of motifs

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9. An array of points such that every point has identical surroundings
Space Lattice
A lattice is also called a Space Lattice
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) or
Translationally periodic arrangement of points in space is called a lattice

10. 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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11. Materials and Packing Crystalline materials...
• atoms pack in periodic, 3D arrays
• typical of:
-metals
-many ceramics
-some polymers
crystalline SiO2 (Quartz) Si
Oxygen
Non-crystalline materials...
noncrystalline SiO2 (Glass)
• atoms have no periodic packing
• occurs for:
-complex structures
-rapid cooling
"Amorphous" = Noncrystalline
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12. Crystal Systems
Unit cell: smallest repetitive volume which contains the complete lattice pattern of a crystal.
7 crystal systems
14 crystal lattices
a, b, and c are the lattice constants
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13. A single crystal can have many unit cells.
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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14. Crystal systems Cubic Three equal axes, mutually perpendicular
a=b=c ===90˚
Tetragonal
Three perpendicular axes, only two equal
a=b≠c ===90˚
Hexagonal
Three equal coplanar axes at 120˚ and a fourth unequal axis perpendicular to their plane
a=b≠c == 90˚ =120˚
Rhombohedral
Three equal axes, not at right angles
a=b=c ==≠90˚
Orthorhombic
Three unequal axes, all perpendicular
a≠b≠c ===90˚
Monoclinic
Three unequal axes, one of which is perpendicular to the other two
a≠b≠c ==90˚≠ 
Triclinic
Three unequal axes, no two of which are perpendicular
a≠b≠c ≠ ≠≠90˚
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15. Some engineering applications require single crystals:
--diamond single
crystals for abrasives
--turbine blades
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16. 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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17. Simple Cubic Structure (SC)
• Rare due to low packing density (only Po has this structure)
• Close-packed directions are cube edges.
• Coordination # = 6
(# nearest neighbors)
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21. Body Centered Cubic Structure (BCC)
• Atoms touch each other along cube diagonals.
--Note: All atoms are identical; the center atom is shaded
differently only for ease of viewing.
ex: Cr, W, Fe (), Tantalum, Molybdenum
• Coordination # = 8
2 atoms/unit cell: 1 center + 8 corners x 1/8
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24. Atomic Packing Factor: BCC
• APF for a body-centered cubic structure = 0.68 a R a 3 a a 2
length = 4R =
Close-packed directions:
3 a
APF = 4 3 p (
a/4 ) 2
atoms
unit cell
atom
volume a
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25. Face Centered Cubic Structure (FCC)
• Atoms touch each other along face diagonals.
--Note: All atoms are identical; the face-centered atoms are shaded
differently only for ease of viewing.
ex: Al, Cu, Au, Pb, Ni, Pt, Ag
• Coordination # = 12
4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8
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27. Atomic Packing Factor: FCC
• APF for a face-centered cubic structure = 0.74 a
2 a
maximum achievable APF
Close-packed directions:
length = 4R =
2 a
Unit cell contains:
6 x 1/2 + 8 x 1/8 =
4 atoms/unit cell
APF = 4 3 p ( 2
a/4 )
atoms
unit cell
atom
volume a
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28. FCC Stacking Sequence • ABCABC... Stacking Sequence • 2D Projection
A sites B C
sites A B
sites C A C A A B C
• FCC Unit Cell

29. 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 + + = C A B
FCC A A B B C C

30. Hexagonal Close-Packed Structure (HCP)
• ABAB... Stacking Sequence
• 3D Projection
• 2D Projection c a
A sites B
sites
Bottom layer
Middle layer
Top
layer
• Coordination # = 12
6 atoms/unit cell
• APF = 0.74
ex: Cd, Mg, Ti, Zn
• c/a = 1.633
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31. APF for HCP c a A sites B 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=2r
Area of triangle = 𝒃𝒉 𝟐 = 𝒂𝒉 𝟐 = 𝟏 𝟐 𝒂. 𝒂 𝟑 𝟐
Area of hexagon = 𝟔. 𝒂𝟐 𝟑 𝟒
Volume of HCP= 𝟔. 𝒂𝟐 𝟑 𝟒 .𝐂=𝟔. 𝒂𝟐 𝟑 𝟒 .𝟏.𝟔𝟑𝟑𝐚
APF= 6∗ 𝟒𝝅𝒓𝟑 𝟑 /( 𝟑 𝟒 ∗𝟔∗𝟏.𝟔𝟑𝟑∗𝐚𝟑) a h a
APF =0.74
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32. SC-coordination number6
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33. • Coordination # = 6
(# nearest neighbors)
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34. BCC-coordination number8
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36. FCC-coordination number
4+4+4=12
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38. HCP-coordination number
3+6+3=12
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39. Theoretical Density, r Cell Unit of Volume Total in Atoms Mass
VC NA
n A
 =
where n = number of atoms/unit cell
A = atomic weight
VC = Volume of unit cell = a3 for cubic
NA = Avogadro’s number
= x 1023 atoms/mol
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40. Theoretical Density, r  = Ex: Cr (BCC) A = 52.00 g/mol R = 0.125 nm
a = 4R/ 3 = nm
 = a 3
52.00 2
atoms
unit cell
mol g
volume
6.023 x 1023
theoretical
= 7.18 g/cm3
ractual
= 7.19 g/cm3
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41. Polymorphism
Two or more distinct crystal structures for the same material (allotropy/polymorphism)     titanium
  , -Ti
carbon
diamond, graphite
BCC
FCC
1538ºC
1394ºC
912ºC
-Fe
-Fe
-Fe
liquid
iron system
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42. Miller indices
Miller indices: defined as the reciprocals of the intercepts made by the plane on the three axes.
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43. 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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44. Determine the reciprocals of these numbers. Step 2
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45. Find the least common denominator (lcd) and multiply each by this lcd
Step 3
Find the least common denominator (lcd) and multiply each by this lcd
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46. The result is written in parenthesis.
Step 4
The result is written in parenthesis.
This is called the `Miller Indices’ of the plane in the form (h k l).
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47. Miller Indices for planes
(0,0,1)
(0,3,0)
(2,0,0)
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)

48. Plane ABC has intercepts of 2 units along X-axis, 3 units along Y-axis and 2 units along Z-axis.
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49. 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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50. IMPORTANT FEATURES OF MILLER INDICES
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.
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51. The intercepts are 2, - 3 and 4
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 and 3
Step 4: Hence the Miller indices for the plane is
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52. Intercepts → 1  
Plane → (100)
Family → {100} → 3
Intercepts → 1 1 
Plane → (110)
Family → {110} → 6
Intercepts → 1 1 1
Plane → (111)
Family → {111} → 8
(Octahedral plane)

53. Miller Indices :   (100)
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54. Fractional intercepts : 1 , 1 , ∞ Miller Indices : (110)
Intercepts :   a , a , ∞
Fractional intercepts :   1 , 1 , ∞
Miller Indices :   (110)
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55. Fractional intercepts : 1 , 1 , 1 Miller Indices : (111)
Intercepts :   a , a , a
Fractional intercepts :   1 , 1 , 1
Miller Indices :   (111)
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56. Fractional intercepts : ½ , 1 , ∞ Miller Indices : (210)
Intercepts :   ½ a , a , ∞
Fractional intercepts :   ½ , 1 , ∞
Miller Indices :   (210)
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58. Z
(101) Y X
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59. (122)
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60. (211)
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61. 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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62. Find the Miller indices of that perpendicular plane.
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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63. 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

64. The above image shows [100], [110], and [111] directions within a
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 over
the appropriate index. For example, the 1
The above image shows [100], [110], and [111] directions within a
unit cell
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65. 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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67. The two directions are [2 1 1] and [1 1 2]
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,
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68. In this case, u1 = 2, v1 = 1, w1 = 1, Type equation here
In this case, u1 = 2, v1 = 1, w1 = 1, Type equation here.u2 = 1, v2 = 1, w2 = 2
(or) cos  = 0.833
 = 35° 3530.
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69. Reference
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