Crystals are solid structures with repeating patterns of atoms or molecules. They consist of a lattice, which defines points in space that repeat periodically, and a motif or basis made of one or more atoms associated with each lattice point. Different crystal structures are defined by their lattice type (e.g. cubic, hexagonal) and motif. Common 3D crystal structures include simple cubic, body-centered cubic, and face-centered cubic lattices combined with different motifs like single atoms or groups of atoms. The document provides examples of crystal structures for various elements and discusses key properties like coordination number and atomic 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
arrangement of motifs.
Translationally periodic
arrangement of points.

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

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

a

b
+

Motif

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

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

15. a
R
Unit cell c ontains:
1 + 8 x 1/8
= 2 atoms/unit cell
Adapted from
Fig. 3.2,
Callister 6e.
• APF for a body-centered cubic structure = p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 = p/(32) = 0.74

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

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

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
[Face Centered Cubic (FCC)]
…ABABAB… packing
[Hexagonal Close Packing (HCP)]

24. Examples of elements with Cubic Crystal Structure
Po
n = 1
Fe Cu
n = 2 n = 4
SC BCC FCC/CCP
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