1. STRUCTURE OF CRYSTALLINE SOLIDS
UNIT I
STRUCTURE OF CRYSTALLINE SOLIDS
Material science
Chapter 1

2. Crystalline material: periodic array Single crystal:
Types of Solids
Crystalline material: periodic array
Single crystal:
periodic array over the entire extent of the material
Polycrystalline material: many small crystals or grains
Amorphous: lacks a systematic atomic arrangement
Crystalline
Amorphous
SiO2

3. Single Crystals and Polycrystalline Materials
Single crystal: periodic array over entire material
Polycrystalline material: many small crystals (grains) with varying orientations.
Atomic mismatch where grains meet (grain boundaries)

4. Polycrystalline Materials

5. Non-Crystalline (Amorphous) Solids
Amorphous solids: no long-range order
Nearly random orientations of atoms
(Random orientation of nano-crystals can be
amorphous or polycrystalline)
Schematic Diagram of Amorphous SiO2

6. Ex: 3D crystalline structure
Unit Cell
The unit cell is building block for crystal.
Repetition of unit cell generates entire crystal.
Ex: 2D honeycomb represented by translation of unit cell
Ex: 3D crystalline structure

7. Unit Cells for Metals
HCP Structure
FCC Structure

8. Two Dimensional Lattices

9. 1- CUBIC CRYSTAL SYSTEMS 3 Common Unit Cells with Cubic Symmetry
Simple Cubic Body Centered Cubic Face Centered Cubic
(SC) (BCC) (FCC)

10. 2- HEXAGONAL CRYSTAL SYSTEMS
In a Hexagonal Crystal System, three equal coplanar axes intersect at an angle of 60°, and another axis is perpendicular to the others and of a different length.
The atoms are all the same.

11. 3 - TRICLINIC & 4 – MONOCLINIC CRYSTAL SYSTEMS
Triclinic crystals have the least symmetry of any crystal systems. The three axes are each different lengths & none are perpendicular to each other. These materials are the most difficult to recognize.
Monoclinic
(Base Centered) a = g = 90o, ß ¹ 90o
a ¹ b ¹ c
Triclinic (Simple) a ¹ ß ¹ g ¹ 90O
a ¹ b ¹ c
Monoclinic (Simple) a = g = 90o, ß ¹ 90o
a ¹ b ¹c

12. 5 - ORTHORHOMBIC CRYSTAL SYSTEM
Orthorhombic (Simple) a = ß = g = 90o
a ¹ b ¹ c
Orthorhombic (Body Centered) a = ß = g = 90o
a ¹ b ¹ c
Orthorhombic
(Base Centered) a = ß = g = 90o
a ¹ b ¹ c
Orthorhombic
(Face Centered) a = ß = g = 90o
a ¹ b ¹ c

13. 6 – TETRAGONAL CRYSTAL SYSTEM
(Body Centered) a = ß = g = 90o
a = b ¹ c
Tetragonal (P) a = ß = g = 90o
a = b ¹ c 13

14. 7 - RHOMBOHEDRAL (R) OR TRIGONAL CRYSTAL SYSTEM
Rhombohedral (R) or Trigonal (S) a = b = c, a = ß = g ¹ 90o 14

15. Metallic Crystal Structures
How can we stack metal atoms to minimize empty space?
2-dimensions
vs.
Now stack these 2-D layers to make 3-D structures

16. Simple Cubic and BCC Structure
• Rare due to low packing density (only Po has this structure)
• Close-packed directions are cube edges.

17. Close-packed Structures (FCC and HCP)
FCC and HCP: APF =0.74 (maximum possible value)
FCC and HCP may be generated by the stacking of close-packed planes
Difference is in the stacking sequence
HCP: ABABAB FCC: ABCABCABC…

18. HCP: Stacking Sequence ABABAB...
Third plane placed directly above first plane of atoms

19. FCC: Stacking Sequence ABCABCABC...
Third plane placed above “holes” of first plane not covered by second plane

20. FCC FCC + + = CLOSE PACKING A B C
Note: Atoms are coloured differently but are the same

21. HCP HCP + + = Shown displaced for clarity B A A
Unit cell of HCP (Rhombic prism)
Note: Atoms are coloured differently but are the same

22. Coordination Number(CN) for SC:

23. FACE CENTERED CUBIC STRUCTURE
COORDINATION NUMBER
In its own plane it touches four face centered atoms.
The face centered atoms are its nearest neighbors.
In a plane, which lies just above this corner atom, it
has four more face centered atoms as nearest
neighbors.
Similarly, in a plane, which lies just below this corner
atom it has yet four more face centered atoms as its
nearest neighbors.
PH0101,UNIT 4,LECTURE 5

24. (or Atomic Packing Fraction)
Atomic Packing Factor
(or Atomic Packing Fraction)
For a Bravais Lattice,
The Atomic Packing Factor (APF) 
the volume of the atoms within the unit cell divided by the volume of the unit cell.
When calculating the APF, the volume of the atoms in the unit cell is calculated AS IF each atom was a hard sphere, centered on the lattice point & large enough to just touch the nearest-neighbor sphere.
Of course, from Quantum Mechanics, we know that this is very unrealistic for any atom!!

25. a- The Simple Cubic (SC) Lattice
The SC Lattice has one lattice point in its unit cell, so it’s unit cell is a primitive cell.
In the unit cell shown on the left, the atoms at the corners are cut because only a portion (in this case 1/8) “belongs” to that cell. The rest of the atom “belongs” to neighboring cells.
The Coordinatination Number of the SC Lattice = 6. a b c

26. Number of atoms in SC Figure shows how a single atom in corner of Unit Cell of SC is being shared by other Unit cells

27. Total Number of atom in SC
Number of atoms in each corner = 1/8
Total number of corners = 8
Total number of atoms in corner = 1/8*8=1
No other atoms in any part of unit cell

28. Simple Cubic (SC) Lattice
Atomic Packing Factor

29. b- The Body Centered Cubic
(BCC) Lattice
The BCC Lattice has two lattice points per unit cell so the BCC unit cell is a non-primitive cell.
Every BCC lattice point has 8 nearest- neighbors. So (in the hard sphere model) each atom is in contact with its neighbors only along the body-diagonal directions.
Many metals (Fe,Li,Na..etc), including the alkalis and several transition elements have the BCC structure. b c a 29

30. BCC Structure

31. BCC Structure

32. Total Number of atoms in BCC
Number of atoms in each corner = 1/8
Total number of corners = 8
Total number of atoms in corner = 1/8*8=1
Atoms Inside the Unit cell = 1
Total of atoms = 1+1 = 2

33. Body Centered Cubic (BCC) Lattice
Atomic Packing Factor 2
(0.433a) 33

34. Elements with the BCC Structure
Note: This was the end of lecture 1

35. c- The Face Centered Cubic
(FCC) Lattice
In the FCC Lattice there are atoms at the corners of the unit cell and at the center of each face.
The FCC unit cell has 4 atoms so it is a non-primitive cell.
Every FCC Lattice point has 12 nearest-neighbors.
Many common metals (Cu,Ni,Pb..etc) crystallize in the FCC structure.

36. FCC Structure

37. FCC Structure

38. Total Number of Atoms in FCC
In Coners, Each corner is having 1/8 atom
Total number of corners = 8
Total number of atoms in corner = 1/8*8=1
The face of hexagonal is having 1/2 of atom
Total number of face = 6
Total number of atoms in face = 1/2*6= 3
Total = 1+3= 4

39. Face Centered Cubic (FCC) Lattice
Atomic Packing Factor
FCC
0.74 4
(0.353a)
Crystal Structure 39

40. Elements That Have the FCC Structure

41. Hexagonal Close Packed
(HCP) Lattice
This is another structure that is common, particularly in metals. In addition to the two layers of atoms which form the base and the upper face of the hexagon, there is also an intervening layer of atoms arranged such that each of these atoms rest over a depression between three atoms in the base.

42. Hexagonal Close Packed
(HCP) Lattice
The HCP lattice is not a Bravais lattice, because orientation of the environment of a point varies from layer to layer along the c-axis.

43. Hexagonal Close Packed
(HCP) Lattice

44. Figure shows how a single atom in corner of Unit Cell of HCP is being shared by other Unit cells

45. Total Number of Atoms in HCP
In Coners, Each corner is having 1/6 atom
Total number of corners = 12
Total number of atoms in corner = 1/6*12=2
The face of hexagonal is having 1/2 of atom
Total number of face = 2
Total number of atoms in face = 1/2*2= 1
Atoms Inside the Unit cell = 3
Total = = 6

46. HEXAGONAL CLOSED PACKED STRUCTURE
PH0101,UNIT 4,LECTURE 5

47. ATOMIC PACKING FACTOR (APF)
HEXAGONAL CLOSED PACKED STRUCTURE
ATOMIC PACKING FACTOR (APF)
APF =
v = 6  4/3 r3
Substitute r = ,
v = 6  4/3 
v = a3
PH0101,UNIT 4,LECTURE 5

48. HEXAGONAL CLOSED PACKED STRUCTURE
PH0101,UNIT 4,LECTURE 5

49. HEXAGONAL CLOSED PACKED STRUCTURE
AB = AC = BO = ‘a’. CX = where c  height of the
hcp unit cell.
Area of the base = 6  area of the triangle – ABO
= 6  1/2  AB  OO
Area of the base = 6  1/2  a  OO
In triangle OBO
PH0101,UNIT 4,LECTURE 5

50. HEXAGONAL CLOSED PACKED STRUCTURE
cos30º =
 OO = a cos 30º = a
Now, substituting the value of OO,
Area of the base = 6   a  a =
V = Area of the base × height
PH0101,UNIT 4,LECTURE 5

51. HEXAGONAL CLOSED PACKED STRUCTURE
V =  c
APF = =
 APF =
PH0101,UNIT 4,LECTURE 5

52. CALCULATION OF c/a RATIO
In the triangle ABA,
Cos 30º =
AA = AB cos 30º = a
But AX = AA =
i.e. AX =
PH0101,UNIT 4,LECTURE 5

53. CALCULATION OF c/a RATIO
In the triangle AXC,
AC2 = AX2 + CX2
Substituting the values of AC, AX and CX,
a2 =
PH0101,UNIT 4,LECTURE 5

54. CALCULATION OF c/a RATIO
Now substituting the value of to calculate APF of an hcp unit cell,
PH0101,UNIT 4,LECTURE 5

55. HEXAGONAL CLOSED PACKED STRUCTURE
APF = =
 APF =
Packing Fraction =74%
PH0101,UNIT 4,LECTURE 5

56. Imperfections in Solids
The properties of some materials are profoundly influenced by the presence of imperfections.
It is important to have knowledge about the types of imperfections that exist and the roles they play in affecting the behavior of materials.
If volume imperfection is 0.01%, while calculating properties like density, imperfection wont affect.
But impurity of Al is only few parts per million in silicon semiconductor destroy the rectifying and transisting action.

57. Atom Purity and Crystal Perfection
If we assume a perfect crystal structure containing pure elements, then anything that deviated from this concept or intruded in this uniform homogeneity would be an imperfection.
There are no perfect crystals.
Many material properties are improved by the presence of imperfections and deliberately modified (alloying and doping).

58. Types of Imperfections
• Vacancy atoms
• Interstitial atoms
• Substitutional atoms
Point defects
1-2 atoms
Zero dimensional
• Dislocations
Line defects
1-dimensional
• Grain Boundaries
Area defects
2-dimensional

59. Point Defects in Metals By own atoms
• Vacancies:
-vacant atomic sites in a structure.
Vacancy
distortion
of planes
• Self-Interstitials:
-"extra" atoms positioned between atomic sites.
self-interstitial
distortion
of planes

60. Solid Solution Point Defects in Metals By different atoms
The addition of impurity atoms to a metal results in the formation of a solid solution.
The solvent represents the element that is present in the greatest amount (the host atoms). For example, in Lab 8 (MSE 227) Precipitation Hardening of Aluminum, aluminum is the solvent and copper is the solute (present in minor concentration ).
Solid solutions form when the solute atoms (Cu) are added to the solvent (Al), assuming the crystal structure is maintained and no new structures are formed.

61. Solid Solution - continued
A solid solution is a homogenous composition throughout.
The impurity atoms (Cu) are randomly and uniformly dispersed within the solid.
The impurity defects in the solid solution are either substitutional or interstitial.

62. Compressive Stress Fields
Relative size
Interstitial
Compressive Stress Fields
Impurity
Substitutional
Compressive stress fields
SUBSTITUTIONAL IMPURITY  Foreign atom replacing the parent atom in the crystal  E.g. Cu sitting in the lattice site of FCC-Ni
INTERSTITIAL IMPURITY  Foreign atom sitting in the void of a crystal  E.g. C sitting in the octahedral void in HT FCC-Fe
Tensile Stress Fields

63. Ionic Crystals
Overall electrical neutrality has to be maintained
Frenkel defect
Cation (being smaller get displaced to interstitial voids
E.g. AgI, CaF2

64. Schottky defect
Pair of anion and cation vacancies
E.g. Alkali halides

65. Defects in Ceramic Structures
• Frenkel Defect
--a cation is out of place.
• Shottky Defect
--a paired set of cation and anion vacancies.
Shottky
Defect:
Frenkel
Defect
from W.G. Moffatt, G.W. Pearsall, and J. Wulff, The Structure and Properties of Materials, Vol. 1, Structure, John Wiley and Sons, Inc., p. 78.
• Equilibrium concentration of defects

66. Line Defects Are called Dislocations: Schematic of Zinc (HCP): And:
• slip between crystal planes result when dislocations move,
• this motion produces permanent (plastic) deformation.
Schematic of Zinc (HCP):
• before deformation
• after tensile elongation
slip steps which are the physical evidence of large numbers of dislocations slipping along the close packed plane {0001}
Adapted from Fig. 7.8, Callister 7e.

67. Linear Defects (Dislocations)
Are one-dimensional defects around which atoms are misaligned
Edge dislocation:
extra half-plane of atoms inserted in a crystal structure
b (the berger’s vector) is  (perpendicular) to dislocation line
Screw dislocation:
spiral planar ramp resulting from shear deformation
b is  (parallel) to dislocation line
Burger’s vector, b: is a measure of lattice distortion and is measured as a distance along the close packed directions in the lattice

68. Edge Dislocation Edge Dislocation
Atoms above the dislocation line is under compression
Atoms below the dislocation line is under tension
That can be seen dy the curvature of vertical planes of atoms as they bend around the extra half-plane of atoms
The inverted T represents the extra half plane in top of the crystal.
Edge dislocation occurs due to mechanical and thermal stresses, etc
Dislocation line
Fig. 4.3, Callister 7e.

69. Definition of the Burgers vector, b, relative to an edge dislocation
Definition of the Burgers vector, b, relative to an edge dislocation. (a) In the perfect crystal, an m× n atomic step loop closes at the starting point. (b) In the region of a dislocation, the same loop does not close, and the closure vector (b) represents the magnitude of the structural defect. For the edge dislocation, the Burgers vector is perpendicular to the dislocation line.

70. Dislocation Motion Plastically stretched zinc single crystal.
• Produces plastic deformation,
• Depends on incrementally breaking
bonds.
Plastically
stretched
zinc
single
crystal.
• If dislocations don't move,
deformation doesn't happen!

71. b parallel () to dislocation line
Screw dislocation:
Named for the spiral stacking of crystal planes around the dislocation line; results from shear deformation
b parallel () to dislocation line

72. Screw dislocation. The spiral stacking of crystal planes leads to the Burgers vector being parallel to the dislocation line.

73. B vectors in Edge and screw dislocstions

74. Mixed dislocation. This dislocation has both edge and screw character with a single Burgers vector consistent with the pure edge and pure screw regions.

75. Edge, Screw, and Mixed Dislocations
Adapted from Fig. 5.10, Callister & Rethwisch 3e.

76. Imperfections in Solids
Dislocations are visible in electron micrographs; TEM Titanium alloy, dark lines are dislocations 51,450x
Fig. 5.11, Callister & Rethwisch 3e.

77. Polycrystalline Materials
Grain Boundaries
regions between crystals
transition from lattice of one region to another
The atoms near the boundaries of the 3 grains do not have an equilibrium spacing or arrangement; slightly disordered.
Grains and grain boundaries in a stainless steel sample. low density in grain boundaries

78. Typical optical micrograph of a grain structure, 100×
Typical optical micrograph of a grain structure, 100×. The material is a low-carbon steel. The grain boundaries have been lightly etched with a chemical solution so that they reflect light differently from the polished grains, thereby giving a distinctive contrast.

79. Simple grain-boundary structure
Simple grain-boundary structure. This is termed a tilt boundary because it is formed when two adjacent crystalline grains are tilted relative to each other by a few degrees (θ). if θ is between 10 to 15 deg that is non- crystalline structure. If Orientation difference between two crystals is less than 10 deg. Then it is called tilt boundary

80. Planar Defects in Solids - Twinning
A shear force that causes atomic displacements such that the atoms on one side of a plane (twin boundary) mirror the atoms on the other side. A reflection of atom positions across the twin plane.
Displacement magnitude in the twin region is proportional to the atom’s distance from the twin plane.
Takes place along defined planes and directions depending upon the system.
Ex: BCC twinning occurs on the (112)[111] system

81. Twinning
(c) 2003 Brooks/Cole Publishing / Thomson Learning
Applied stress to a perfect crystal (a) may cause a displacement of the atoms, (b) causing the formation of a twin. Note that the crystal has deformed as a result of twinning.

82. Properties of Twinning
Of the three common crystal structures BCC, FCC and HCP, the HCP structure is the most likely to twin.
FCC structures will not usually twin because slip is more energetically favorable.
Twinning occurs at low temperatures and high rates of shear loading (shock loading) conditions where there are few present slip systems (restricting the possibility of slip)
Small amount of deformation when compared with slip.

83. Stacking faults
For FCC metals an error in ABCABC packing sequence
Ex: ABCAB—ABC
C is missing in the stacking order
For HCP metals an error in ABAB packing sequence
Ex: ABCABAB
Extra layer called C is present in the stacking order.

84. A micrograph of twins within a grain of brass (x250).
(c) 2003 Brooks/Cole Publishing / Thomson Learning
A micrograph of twins within a grain of brass (x250).