1. A SEMINAR ON
CRYSTAL STRUCTURE
PRESENTED
BY
K. GANAPATHI RAO
(13031D6003)
Presence of
Mr. V V Sai sir

2. CONTENT
 INTRODUCTION.
 TRANSLATION VECTOR.
 BASIS & UNIT CELL.
 BRAVAIS & SPACE LATTICES.
 FUNDAMENTAL QUANTITIES.
 MILLER INDICES.
 INTER-PLANAR SPACING.

3. Materials
Solids
Liquids
Gasses

4. Solids
Crystalline
Single
Poly
Amorphous

5. TRANSLATION VECTOR

6. LATTICE PARAMETERS & UNIT
CELL

7. Bravais Lattice
System
Possible Variations
Axial Distances
(edge lengths)
Axial Angles
Examples
Cubic
Primitive, Bodya=b=c
centred, Face-centred
α = β = γ = 90°
NaCl, Zinc Blende, Cu
Tetragonal
Primitive, Bodycentred
a=b≠c
α = β = γ = 90°
White
tin, SnO2,TiO2, CaSO4
Orthorhombic
Primitive, Bodycentred, Facea≠b≠c
centred, Base-centred
α = β = γ = 90°
Rhombic
sulphur,KNO3, BaSO4
Rhombohedral
Primitive
a=b=c
α = β = γ ≠ 90°
Calcite (CaCO3,Cinnaba
r (HgS)
Hexagonal
Primitive
a=b≠c
α = β = 90°, γ =
120°
Graphite, ZnO,CdS
Monoclinic
Primitive, Basecentred
a≠b≠c
α = γ = 90°, β ≠ 90°
Monoclinic sulphur,
Na2SO4.10H2O
Triclinic
Primitive
a≠b≠c
α ≠ β ≠ γ ≠ 90°
K2Cr2O7,
CuSO4.5H2O,H3BO3

8. 1
Cubic
P

Cube
I

F

C
I
P
a b c
Lattice point
90
F

9. P
2
Tetragonal
Square Prism (general height)
I

F
C

I
P
a b c
90

10. P
3
Orthorhombic Rectangular Prism (general height)
I
F
C




One convention
a b c
I
P
a b c
90
F
C

11. P
4
Trigonal
Parallelepiped (Equilateral, Equiangular)
I

Rhombohedral
a b c
90
Note the position of the origin
and of ‘a’, ‘b’ & ‘c’
F
C

12. P
5
Hexagonal

120 Rhombic Prism
a b c
90 ,
A single unit cell (marked in blue)
along with a 3-unit cells forming a
hexagonal prism
120
I
F
C

13. P
6
Monoclinic

Parallogramic Prism
One convention
a b c
a b c
90
Note the position of
‘a’, ‘b’ & ‘c’
I
F
C


14. P
7
Triclinic

Parallelepiped (general)
a b c
I
F
C

15. FUNDAMENTAL QUANTITIES
• NEAREST NEIGHBOUR DISTANCE (2R).
• ATOMIC RADIUS (R).
• COORDINATION NUMBER (N).
• ATOMIC PACKING FACTOR.

16. 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)
(Courtesy P.M. Anderson)

17. ATOMIC PACKING FACTOR
(APF):SC
APF =
Volume of atoms in unit cell*
Volume of unit cell
*assume hard spheres
• APF for a simple cubic structure = 0.52
atoms
unit cell
a
R=0.5a
APF =
1
4
3
a3
close-packed directions
contains 8 x 1/8 =
1 atom/unit cell
Adapted from Fig. 3.24,
Callister & Rethwisch 8e.
volume
atom
(0.5a) 3
volume
unit cell

18. 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
(Courtesy P.M. Anderson)
Adapted from Fig. 3.2,
Callister & Rethwisch 8e.
2 atoms/unit cell: 1 center + 8 corners x 1/8

19. ATOMIC PACKING FACTOR: BCC
• APF for a body-centered cubic structure = 0.68
3a
a
2a
Adapted from
Fig. 3.2(a), Callister &
Rethwisch 8e.
atoms
R
Close-packed directions:
length = 4R = 3 a
a
4
2
unit cell
3
APF =
( 3a/4) 3
a3
volume
atom
volume
unit cell

20. 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
Adapted from Fig. 3.1, Callister & Rethwisch 8e.
(Courtesy P.M. Anderson)
4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8

21. ATOMIC PACKING FACTOR: FCC
• APF for a face-centered cubic structure = 0.74
maximum achievable APF
Close-packed directions:
length = 4R = 2 a
2a
a
Adapted from
Fig. 3.1(a),
Callister &
Rethwisch 8e.
Unit cell contains:
6 x 1/2 + 8 x 1/8
= 4 atoms/unit cell
atoms
4
4
unit cell
3
APF =
( 2a/4) 3
a3
volume
atom
volume
unit cell

22. MILLER INDICES
• PROCEDURE FOR WRITING DIRECTIONS IN MILLER INDICES
• DETERMINE THE COORDINATES OF THE TWO POINTS IN THE
DIRECTION. (SIMPLIFIED IF ONE OF THE POINTS IS THE ORIGIN).
• SUBTRACT THE COORDINATES OF THE SECOND POINT FROM
THOSE OF THE FIRST.
• CLEAR FRACTIONS TO GIVE LOWEST INTEGER VALUES FOR ALL
COORDINATES

23. MILLER INDICES
• INDICES ARE WRITTEN IN SQUARE BRACKETS WITHOUT
COMMAS (EX: [HKL])
• NEGATIVE VALUES ARE WRITTEN WITH A BAR OVER THE
INTEGER.
[hkl]
• EX: IF H<0 THEN THE DIRECTION IS
•

24. MILLER INDICES
• CRYSTALLOGRAPHIC PLANES
• IDENTIFY THE COORDINATE INTERCEPTS OF THE PLANE
• THE COORDINATES AT WHICH THE PLANE
INTERCEPTS THE X, Y AND Z AXES.
• IF A PLANE IS PARALLEL TO AN AXIS, ITS INTERCEPT IS
TAKEN AS .
• IF A PLANE PASSES THROUGH THE ORIGIN, CHOOSE
AN EQUIVALENT PLANE, OR MOVE THE ORIGIN
• TAKE THE RECIPROCAL OF THE INTERCEPTS

25. Miller Indices for planes
(0,0,1)
z
y
(0,3,0)
x
(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)

26. MILLER INDICES
• CLEAR FRACTIONS DUE TO THE RECIPROCAL,
BUT DO NOT REDUCE TO LOWEST INTEGER
VALUES.
• PLANES ARE WRITTEN IN PARENTHESES, WITH
BARS OVER THE NEGATIVE INDICES. [hkl]
• EX: (HKL) OR IF H<0 THEN IT BECOMES

27. z
z
y
y
x
x
Intercepts → 1
Plane → (100)
Intercepts → 1 1
Plane → (110)
z
y Intercepts → 1 1 1
x
Plane → (111)
(Octahedral plane)

28. INTER-PLANAR SPACING
• FOR ORTHORHOMBIC, TETRAGONAL AND CUBIC UNIT
CELLS (THE AXES ARE ALL MUTUALLY
PERPENDICULAR), THE INTER-PLANAR SPACING IS
GIVEN BY:
h, k, l = Miller indices
a, b, c = unit cell dimensions
• For cube a = b = c than
a
d hkl
h2 k 2 l 2

29. REFRENCES
• APPLIED PHYSICS BY P.K. PALANISAMY
• http://en.wikipedia.org/wiki/crystal_structure
• http://journals.iucr.org/c/
• http://www.scirp.org/journal/csta/
• http://www.asminternational.org/portal/site/www/subje
ctguideitem/?vgnextoid=ad7cdc8cc359d210vgnvcm10
0000621e010arcrd