1. PRESENTATION
ON
SPACE LATTICE AND
CRYSTAL STRUCTURE
Submitted By
Manmeet Singh
13209015
ME Mechanical
Submitted To
Dr. Sanjeev Kumar
Mechanical Deptt.

2. MATERIALS SOLIDS
1. Crystalline
a. Single crystalline
b. Poly crystalline
“High Bond Energy” and a More Closely Packed Structure
2. Non Crystalline(Amorphous)
These have less densely packed lower bond energy
“structures”

3. • Non dense, random packing
• Dense, ordered packing
Dense, ordered packed structures tend to have
lower energies & thus are more stable.
ENERGY AND PACKING
Energy
r
typical neighbor
bond length
typical neighbor
bond energy
Energy
r
typical neighbor
bond length
typical neighbor
bond energy

4. CRYSTALS GEOMETRY
 Space lattice
 Crystal structure
 Crystal directions and planes

5. SPACE LATTICE
 Arrangement of atoms taken as point periodically
repeating in infinite array in 3 dimensional space
such that every point has identical surrounding.
 Infinite
 Identical surrounding
 In a space lattice we can have more than one kind of
cell, shape of cell, size of cell…..

6. Meaning of
identical
surrounding

7. TO DEFINE MOVEMENT
 2d – two vectors which are non collinear
 3d – three vectors which are non coplanar

8. PRIMITIVE CELL
SMALLEST CELL WITH LATTICE POINTS AT EIGHT
CORNERS HAS EFFECTIVELY ONLY ONE ATOM IN THE
VOLUME OF CELL
PARAMETERS OF PRIMITIVE CELL
 3D lattices can be generated with three basis vectors
 6 lattice parameters
 3 distances (a, b, c)
 3 angles (, , )

9. PARAMETERS OF PRIMITIVE CELL
a,b,c adjacent
sides of cell
α,β,γ interfacial
angles
Points to note----
1. Knowing actual values of (a,b,c) and (α,β,γ) ---form and size of cell.
2. Knowing (α,β,γ) but only rations of (a,b,c)---we can only know
shape of cell not size.

10. UNIT CELL
NOT NECESSARY TO BE PRIMITIVE, CAN BE BIGGER THAN
PRIMITIVE AS LONG AS IT SHOWS ALL POSSIBLE MAXIMUM
SYMMETRIES
It is characterized by:
 Type of atom and their radii,
 Cell dimensions,
 Number of atoms per unit cell,
 Coordination number (CN)–closest neighbors to an atom
 Atomic packing factor, APF
Atomic packing factor (APF) or packing fraction is
the fraction of volume in a crystal structure that is
occupied by atoms.

11. CHOOSING UNIT CELL---
BRAVAIS SPACE LATTICES
 Smallest size
 Maximum possible symmetry
 Symmetry
1. Translational symmetry (inherent in definition
of space lattice is that identical surrounding)
2. Rotation symmetry

15. So there are only 14 bravais space lattices
which belong to 7 crystal classes.
Depending on minimum size and maximum
symmetry.

16. CRYSTAL STRUCTURE
 Crystal– considered as consisting of tiny blocks which are
repeated in 3D pattern.
 Tiny block--- UNIT CELL
Unit cell---arrangement of small group of atoms . It is that
volume of solid from which entire crystal can be constructed by
repeated translation in 3D.
 Lattice point --- each atom in cell is replaced by point.
 Space lattice(infinite lattice array) --- arrangement of
lattice in 3D.every point has identical surrounding.
 Lattice spacing--- distance between atom points.

17. DEFINING CRYSTAL STRUCTURE
A crystal structure is a unique arrangement
of atoms or molecules in a crystalline solid. A crystal
structure describes a highly ordered structure, occurring
due to the intrinsic nature of molecules to form symmetric
patterns.
 Inter atomic spacing
 Number of atoms and their kind
 Orientation in space
 MOTIF –kind of atoms associated with each lattice point
 in polymeric and protein crystals there can be more than
10,000 atoms in a motive

19. ARRANGEMENT OF LATTICE POINTS IN
SPACE LATTICE
1. Primitive simple cubic
8 points or atoms at
each corner
2. 8 corner atom and one
body centered

20. 3. 8 atoms at each
corner and 6 atoms at
center of each face.
4. 8 atoms at each
corner and 2 atoms at
centers of opposite
face.

21. 4. Hexagonal with 12
corner atoms and 2
center atoms

22. 7 CRYSTAL CLASSES
 FIRST CRYSTAL CLASS
 CUBIC CRYSTAL CLASS
Symmetry --- THREE -4 fold
1. Simple cubic
2. Body centered cubic
3. Face centered cubic
a = b= c
α = β = γ = 90°
Only one parameter need to be
defined
No end center as they do not show
maximum symmetry

23. • Rare due to low packing density (only Po – Polonium
-- has this structure)
• Coordination No. =
6
(# nearest neighbors)
for each atom as seen
SIMPLE CUBIC STRUCTURE (SC)

24. • APF for a simple cubic structure = 0.52
APF =
a3
4
3
p (0.5a) 31
atoms
unit cell
atom
volume
unit cell
volume
ATOMIC PACKING FACTOR (APF)
APF =
Volume of atoms in unit cell*
Volume of unit cell
*assume hard spheres
close-packed directions
a
R=0.5a
contains (8 x 1/8) =
1 atom/unit cell Here: a = Rat*2
Where Rat’ atomic radius

25. • Coordination # = 8
• Atoms touch each other along cube diagonals within
a unit cell.
--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)

26. ATOMIC PACKING FACTOR: BCC
a
APF =
4
3
p ( 3a/4)32
atoms
unit cell atom
volume
a3
unit cell
volume
length = 4R =
Close-packed directions:
3 a
• APF for a body-centered cubic structure = 0.68
a
R
a2
a3

27. • 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 ½) + (8 corners x 1/8)

28. • APF for a face-centered cubic structure = 0.74
ATOMIC PACKING FACTOR: FCC
The maximum achievable APF!
APF =
4
3
p ( 2a/4)34
atoms
unit cell atom
volume
a3
unit cell
volume
Close-packed directions:
length = 4R = 2 a
Unit cell contains:
6 x1/2 + 8 x1/8
= 4 atoms/unit cella
2 a
(a = 22*R)

29. SECOND CRYSTAL CLASS
TETRAGONAL CRYSTAL CLASS
 Symmetry– One 4 fold
4…Simple tetragonal
5…Body centered tetragonal
a = b not equal to c
α = β = γ = 90°
 Two parameters to define a and c.
 One might suppose stretching
face-centered cubic would result in
face-centered tetragonal, but face-
centered tetragonal is equivalent to
body-centered tetragonal, BCT (with a
smaller lattice spacing). BCT is
considered more fundamental, so that
is the standard terminology

30. THIRD CRYSTAL CLASS
ORTHORHOMBIC CRYSTAL CLASS
 Symmetry–Three 2 fold
6…Simple
7…Body centered
8…Face centered
9…End centered
a≠ b ≠ c
α = β = γ = 90°
 Three parameters to define a
and c.

31. 4. HEXAGONAL CLOSE-PACKED STRUCTURE
(HCP)
 Symmetry–One 6 fold
10…Simple Hexagonal
a = b ≠ c
α = β = 90°
γ = 120°
 Two parameters to define a and c.

32. • Coordination # = 12
• ABAB... Stacking Sequence
• APF = 0.74
• 3D Projection • 2D Projection
4. HEXAGONAL CLOSE-PACKED STRUCTURE
(HCP)4
6 atoms/unit cell
ex: Cd, Mg, Ti, Zn
c
a
A sites
B sites
A sites
Bottom layer
Middle layer
Top layer

34. FIFTH CRYSTAL CLASS
Rhombohedral crystal class
11…Simple rhombohedral
Symmetry– One 3 fold (120°)
a = b = c
α = β = γ ≠ 90°
 Two parameters to define a and angle α.
SIXTH CRYSTAL CLASS
Monoclinic crystal class
Symmetry– One 2 fold
12… Simple monoclinic
13…End centered Monoclinic (A and B not C)
a ≠ b ≠ c
α = β = 90° ≠ γ
 Four parameters to define a ,b ,c and angle γ.

35. SEVENTH CRYSTAL CLASS
Triclinic crystal class
14… Simple triclinic
a ≠ b ≠ c
α ≠ β ≠ γ ≠ 90°
 Six parameters to define a ,b ,c and anglesα,β ,γ.
 HIGHLY UNSYMMETRIC CRYSTAL

38. 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
 =
CellUnitofVolumeTotal
CellUnitinAtomsofMass

39. CRYSTALLOGRAPHIC DIRECTIONS, AND
PLANES
 Now that we know how atoms arrange themselves to
form crystals, we need a way to identify directions and
planes of atoms
Why?
 Deformation under loading (slip) occurs on certain
crystalline planes and in certain crystallographic
directions. Before we can predict how materials fail,
we need to know what modes of failure are more likely
to occur.
 Other properties of materials (electrical conductivity,
thermal conductivity, elastic modulus) can vary in a
crystal with orientation

40. MILLER INDICES
USED TO SPECIFY DIRECTIONS AND PLANES.
Vectors and atomic planes in a crystal
lattice can be described by a three-
value Miller index notation (hkl).
The h, k, and l directional indices are
separated by 90°, and are thus
orthogonal.
 Choice of origin arbitrary
 Axes and their sense--once chosen
not changed and choice is arbitrary.
 Unit distance need not to be unity,
it is distance between two lattice
points in a particular direction in a
particular crystal.

41. CRYSTAL DIRECTIONS
A crystallographic direction is defined as a line between two
points, or a vector.

42. CRYSTAL DIRECTIONS
 Smallest integer reduction.
 No separator used
 Separator can be used when directions came with
more than one digit number, example [1,14,7]
 Bar is used for negative directions.
 Characteristics of crystal in different directions is
different that is why it is needed to give them
different name.

43. Negative Directions

44. HCP CRYSTALLOGRAPHIC
DIRECTIONS
1. Vector repositioned (if necessary) to
pass
through origin.
2. Read off projections in terms of unit
cell dimensions a1, a2, a3, or c
3. Adjust to smallest integer values
4. Enclose in square brackets, no commas
[uvtw]
[1120]ex: ½, ½, -1, 0 =>
dashed red lines indicate
projections onto a1 and a2 axes a1
a2
a3
-a3
2
a2
2
a1
-a3
a1
a2
z

45. FAMILY OF DIRECTIONS
Directions which look physically
identical not parallel, if it is parallel it
is same direction
o<011>
Permuting it will give 12 different
arrangements
[110]
[1-10]
[-110]
[011]
[101]
[0-11]
[01-1]
[-101]
[10-1]
[-1-10]
[0-1-1]
[-10-1]

46. ex: linear density of Al in [110]
direction
a = 0.405 nm
LINEAR DENSITY – CONSIDERS EQUIVALENCE AND IS
IMPORTANT IN SLIP
 Linear Density of Atoms  LD =
a
[110]
Unit length of direction vector
Number of atoms
# atoms
length
1
3.5 nm
a2
2
LD
-
==

47. DEFINING CRYSTALLOGRAPHIC
PLANES
 Miller Indices: Reciprocals of the (three)
axial intercepts for a plane, cleared of
fractions & common multiples. All parallel
planes have same Miller indices.
 Algorithm (in cubic lattices this is direct)
1. Read off intercepts of plane with axes in
terms of a, b, c
2. Take reciprocals of intercepts
3. Reduce to smallest integer values
4. Enclose in parentheses, no
commas i.e., (hkl)  families {hkl}

48. CRYSTAL PLANES

49. CRYSTALLOGRAPHIC PLANES (HCP)
 In hexagonal unit cells the same idea is
used
example a1 a2 a3 c
4. Miller-Bravais Indices (1011)
1. Intercepts 1  -1 1
2. Reciprocals 1 1/
1 0
-1
-1
1
1
3. Reduction 1 0 -1 1
a2
a3
a1
z

50. FAMILY OF PLANE
 All planes which are physical identical is that
arrangement of atoms is same
 They are also not parallel if parallel this is same
plane.

51. PLANAR DENSITY OF (100) IRON
Solution: At T < 912C iron has the BCC structure.
(100)
Radius of iron R = 0.1241 nm
R
3
34
a =
2D repeat unit
=Planar Density =
a2
1
atoms
2D repeat unit
=
nm2
atoms
12.1
m2
atoms
= 1.2 x 1019
1
2
R
3
34area
2D repeat unit

52. COMPARISON OF FCC, HCP, AND
BCC CRYSTAL STRUCTURES
 Both FCC and HCP structures are close packed
APF = 0.74.
 The closed packed planes are the {111} family for
FCC and the (0001) plane for HCP.
Stacking sequence is ABCABCABC in FCC and
ABABAB in HCP.
 BCC is not close packed, APF = 0.68. Most
densely packed planes are the {110} family.

53. ATOMIC DENSITIES
WHY DO WE CARE?
Properties, in general, depend on linear and planar
density.
Examples:
 Speed of sound along directions
 Slip (deformation in metals) depends on linear &
planar density. Slip occurs on planes that have the
greatest density of atoms in direction with highest
density (we would say along closest packed directions
on the closest packed planes)

54. Symbol
Alternate
symbols
Direction
[ ] [uvw] → Particular direction
< > <uvw> [[ ]] → Family of directions
Plane
( ) (hkl) → Particular plane
{ } {hkl} (( )) → Family of planes
Point
. . .abc. [[ ]] → Particular point
: : :abc: → Family of point

55. POLYMORPHISM: ALSO IN METALS
 Two or more distinct crystal structures for the
same material (allotropy/polymorphism)
titanium
 (HCP), (BCC)-Ti
BCC
FCC
BCC
1538ºC
1394ºC
912ºC
-Fe
-Fe
-Fe
liquid
iron system:
•Carbon (diamond, graphite,)
•Silica (quartz, tridymite,
cristobalite, etc.)
•Iron (ferrite, austenite)

56.  Anisotropic – Direction dependent properties
Example carbon fibers, whiskers
 Isotropic – Direction independent properties
Each grain may be anisotropic ,but a specimen
composed by grain aggregate behaves isotropically.
The degree of anisotropy increases with decreasing
structural symmetry. So TRICLINICS are highly
anisotropic

57. LOOKING AT THE CERAMIC UNIT CELLS

58. CERAMIC CRYSTAL STRUCTURE
 Broader range of chemical composition than metals with
more complicated structures
 Contains at least 2 and often 3 or more atoms.
 Usually compounds between metallic ions (e.g. Fe,Ni, Al)
called cations and non-metallic ions (e.g.O, N, Cl) called
anions.
How do Cations and Anions arrange themselves in
space???
Structure is determined by two characteristics:
 1. Electrical charge
Crystal (unit cell) must remain electrically neutral .Sum
of cation and anion charges in cell is 0
 2. Relative size of the ions

59. CERAMIC CRYSTAL STRUCTURE
AX-TYPE CRYSTAL STRUCTURES
Some of the common ceramic materials are those in
which there are equal numbers
of cations and anions.
Example– NaCl, MgO,Feo,CsCl,Zinc blende
Am Xp -TYPE CRYSTAL STRUCTURES
Not equal number of charges
Example – Fluorite CaF2, UO2, ThO2
Am Bn Xp -TYPE CRYSTAL STRUCTURES
Barium titanate BaTiO3

60. CESIUM CHLORIDE (CSCL) UNIT CELL SHOWING
(A) ION POSITIONS AND THE TWO IONS PER LATTICE POINT AND

61. SODIUM CHLORIDE (NACL) STRUCTURE SHOWING
(A) ION POSITIONS IN A UNIT CELL,
(B) FULL-SIZE IONS, AND
(C) MANY ADJACENT UNIT CELLS.

62. FLUORITE (CAF2) UNIT CELL SHOWING
(A) ION POSITIONS AND
(B) FULL-SIZE IONS.

65. ARRANGEMENT OF POLYMERIC CHAINS IN THE UNIT CELL
OF POLYETHYLENE. THE DARK SPHERES ARE CARBON ATOMS,
AND THE LIGHT SPHERES ARE HYDROGEN ATOMS. THE UNIT-
CELL DIMENSIONS ARE 0.255 NM × 0.494 NM × 0.741 NM.

66. DENSITIES OF MATERIAL CLASSES
metals > ceramics >polymers
Why?
(g/cm)3
Graphite/
Ceramics/
Semicond
Metals/
Alloys
Composites/
fibers
Polymers
1
2
20
30
*GFRE, CFRE, & AFRE are Glass,
Carbon, & Aramid Fiber-Reinforced
Epoxy composites (values based on
60% volume fraction of aligned fibers
in an epoxy matrix).10
3
4
5
0.3
0.4
0.5
Magnesium
Aluminum
Steels
Titanium
Cu,Ni
Tin, Zinc
Silver, Mo
Tantalum
Gold, W
Platinum
Graphite
Silicon
Glass -soda
Concrete
Si nitride
Diamond
Al oxide
Zirconia
HDPE, PS
PP, LDPE
PC
PTFE
PET
PVC
Silicone
Wood
AFRE*
CFRE*
GFRE*
Glass fibers
Carbon fibers
Aramid fibers
Metals have...
• close-packing
(metallic bonding)
• often large atomic masses
Ceramics have...
• less dense packing
• often lighter elements
Polymers have...
• low packing density
(often amorphous)
• lighter elements (C,H,O)
Composites have...
• intermediate values
In general

67. THANKYOU
ANY QUESTION