Presentation on crystal structures and packing factor

2. CRYSTAL
“A crystal is a solid in which atoms are arranged in
some regular repetition pattern in all directions.”
“Aggregation of molecules with a definite internal
structure and the external form of a solid enclosed
by symmetrically arranged plane faces.”
STRUCTURES
“Structure of anything is defined as the framework of
its body.”

3. Crystal = Lattice+Base
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 Crystal
Translationally periodic Translationally periodic
arrangement of points. arrangement of motifs.

4. Crystal = Lattice (Where to repeat)
+
Motif (What to repeat)
Crystal
a
=
Lattice
a
+
Motif Note: all parts of the motif do not sit on the lattice
a point
2

5. Let us construct the crystal considered before starting with an
infinite array of points spaced a/2 apart
Put arrow marks pointing up and down alternately on the points:
What we get is a crystal of lattice parameter „a‟ and not „a/2‟!
And the
motif
is: +

6. A strict 1D crystal = 1D lattice + 1D motif
 The only kind of 1D motif is a line segment.
 An unit cell is a representative unit of the structure
(finite part of a infinite structure) .
 Which when repeated gives the whole structure.
Lattice
+
Motif
=
Crystal

7.  2D crystal = 2D lattice + 2D motif
Lattice
+ 
Motif

b

a

8. Crystal
      
      
      
=       
      
      
      
      

9.  3D crystal = 3D lattice + 3D motifs
CRYSTAL OR SPACE LATTICE
 It is defined as an array of points in 3 dimensions
in which every point has surroundings identical to
every other point in array.
According to BRAVAIS there are 14 possible types
of space lattice in 7 basic crystal system

11. a = b= c
= = = 90º
• Simple Cubic (P) - SC
• Body Centred Cubic (I) – BCC
• Face Centred Cubic (F) - FCC
 Elements with Cubic structure →
SC: F, O
BCC: Cr, Fe, Nb, K, W
FCC: Al, Ar, Pb, Ni, Ge

12. SIMPLE CUBIC STRUCTURE
• Cubic unit cell is 3D repeat unit
• Rare (only Po has this structure)
• Coordination No. = 6
(# nearest neighbors)

13. a
R=0.5a
close-packed directions
contains 8 x 1/8 =
1 atom/unit cell
Adapted from Fig. 3.19,
Callister 6e.
0.52
Lattice constant
• APF for a simple cubic structure =

14. BODY CENTERED CUBIC STRUCTURE
• Coordination No. = 8
(# nearest neighbors)

15. Unit cell c ontains:
1 + 8 x 1/8
= 2 atoms/unit cell
R
Adapted from a
Fig. 3.2,
Callister 6e.
• APF for a body-centered cubic structure = 3/8 = 0.68

16. FACE CENTERED CUBIC STRUCTURE
 Atoms are arranged at the corners and center
of each cube face of the cell.
◦ Atoms are assumed to touch along face diagonals

17. • Coordination No. = 12
(# nearest neighbors)

18. Unit cell c ontains:
6 x 1/2 + 8 x 1/8
= 4 atoms/unit cell
a
• APF for a body-centered cubic structure = /(3 2) = 0.74

19. • FCC Unit Cell
• ABCABC... Stacking Sequence
• 2D Projection
A
B B
C
A
A sites B B B
C C
B sites B B
C sites

20. Ideally, c/a = 1.633 for close packing
However, in most metals, c/a ratio deviates from this valu

21. • ABAB... Stacking Sequence
• 3D Projection • 2D Projection
A sites
B sites
A sites
• Coordination NO.= 12
• APF = 0.74, for ideal c/a ratio of 1.633

23. Close packed crystals
A plane
B plane
C plane
A plane
…ABCABCABC… packing …ABABAB… packing
[Face Centered Cubic (FCC)] [Hexagonal Close Packing (HCP)]

24. Examples of elements with Cubic Crystal Structure
Fe Cu
Po
n = 2 BCC n = 4 FCC/CCP
n = 1 SC
n = 8 DC
C (diamond)

25. a=b c
= = = 90º
 Simple Tetragonal
 Body Centred Tetragonal -BCT
 Elements with Tetragonal structure → In, Sn

26. Example of an element with Body Centred Tetragonal Crystal Structure
B
C
T

27. a b c
= = = 90º
 Simple Orthorhombic
 Body Centred Orthorhombic
 Face Centred Orthorhombic
 End Centred Orthorhombic
Elements with Orthorhombic structure → Br, Cl, Ga

28. Element with Orthorhombic Crystal Structure

29. a=b c
= = 90º
=120º
Simple Hexagonal
 Elements with Hexagonal structure → Be, Cd, Co, Ti, Zn

30. Element with Hexagonal Crystal Structure

31. a=b=c
= = 90º
Rhombohedral (simple)
Elements with Trigonal structure → As, B, Bi, Hg

32. Element with Simple Trigonal Crystal Structure

33. a b c
= = 90º
 Simple Monoclinic
 End Centred (base centered) Monoclinic
Elements with Monoclinic structure → P, Pu, Po

34. a b c
• Simple Triclinic

35. 14 Bravais Lattices divided into 7 Crystal Systems
A Symmetry based concept „Translation‟ based concept
Crystal System Shape of UC Bravais Lattices
P I F C
1 Cubic Cube   
2 Tetragonal Square Prism (general height)  
3 Orthorhombic Rectangular Prism (general height)    
4 Hexagonal 120 Rhombic Prism 
5 Trigonal Parallopiped (Equilateral, Equiangular) 
6 Monoclinic Parallogramic Prism  
7 Triclinic Parallopiped (general) 
P Primitive
I Body Centred
F Face Centred
C A/B/C- Centred

36. Face Centred Cubic (FCC) Lattice Two Carbon atom Motif
(0,0,0) & (¼, ¼, ¼)
+
=
Diamond Cubic Crystal