Presentation on Crystal
structures
“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.”
“Structure of anything is defined as the framework of its body.”
CRYSTAL
STRUCTURES
Lattice The underlying periodicity of the crystalBasis Entity associated with each lattice points
Lattice how to repeat
Motif what to repeat
Crystal = Lattice+BaseMotif or basis: Typically an atom or a group of atoms associated with each lattice point.
Translationally periodic arrangement of motifs.
Translationally periodic arrangement of points.
Lattice
Crystal
Crystal = Lattice (Where to repeat) +
Motif (What to repeat)
=
+
a
a
2
a
Lattice
Motif
Note: all parts of the motif do not sit on the lattice
point
Crystal
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: +
A strict 1D crystal = 1D lattice + 1D motif The only kind of 1D motif is a line segment.
Lattice
Motif
Crystal
=
+
An unit cell is a representative unit of the structure (finite part of a infinite structure) . Which when repeated gives the whole structure.
1-D Crystal
2D crystal = 2D lattice + 2D motif
Lattice
a
b
+Motif
2-D Crystal
Crystal
=
3D crystal = 3D lattice + 3D motifs
3-D Crystal
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
THE 7 CRYSTAL SYSTEM
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
Cubic Crystal
• Cubic unit cell is 3D repeat unit • Rare (only Po has this structure)
• Coordination No. = 6 (# nearest neighbors)
SIMPLE CUBIC STRUCTURE
APF = Volume of atoms in unit cell*
Volume of unit cell
*assume hard spheres
• APF for a simple cubic structure = 0.52
APF = a3
4
3(0.5a)31
atoms
unit cellatom
volume
unit cellvolume
ATOMIC PACKING FACTOR
contains 8 x 1/8 = 1 atom/unit cell
Adapted from Fig. 3.19, Callister 6e.
Lattice constant
close-packed directions
a
R=0.5a
BODY CENTERED CUBIC STRUCTURE
• Coordination No. = 8 (# nearest neighbors)
aR
• APF for a body-centered cubic structure = 3/8 = 0.68
Close-packed directions: length = 4R
= 3 a
Unit cell contains: 1 + 8 x 1/8 = 2 atoms/unit cell
Adapted fromFig. 3.2,Callister 6e.
ATOMIC PACKING FACTOR: BCC
APF = a3
4
3( 3a/4)32
atoms
unit cell atomvolume
unit cell
volume
Atoms are arranged at the corners and center of each cube face of the cell.◦ Atoms are assumed to touch along face diagonals
FACE CENTERED CUBIC STRUCTURE
• Coordination No. = 12 (# nearest neighbors)
APF = a3
4
3( 2a/4)34
atoms
unit cell atomvolume
unit cell
volume
Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell
a
• APF for a body-centered cubic structure = /(32) = 0.74
Close-packed directions: length = 4R
= 2 a
ATOMIC PACKING FACTOR: FCC
• ABCABC... Stacking Sequence
• FCC Unit CellA
BC
FCC STACKING SEQUENCE
A sites
B sites
C sitesB B
B
BB
B BC C
CA
A
• 2D Projection
HEXAGONAL CLOSE-PACKED STRUCTURE (HCP)
Ideally, c/a = 1.633 for close packingHowever, in most metals, c/a ratio deviates from this value
• Coordination NO.= 12
• ABAB... Stacking Sequence
• APF = 0.74, for ideal c/a ratio of 1.633
• 3D Projection • 2D Projection
A sites
B sites
A sites Bottom layer
Middle layer
Top layer
Close Packed Structures
Close packed crystals
A plane
B plane
C plane
A plane
…ABCABCABC… packing[Face Centered Cubic (FCC)]
…ABABAB… packing[Hexagonal Close Packing (HCP)]
Examples of elements with Cubic Crystal Structure
Po
n = 1n = 2 n = 4
Fe Cu
BCC FCC/CCPSC
C (diamond)
n = 8 DC
a = b c = = = 90º
Simple Tetragonal Body Centred Tetragonal -BCT
Elements with Tetragonal structure → In, Sn
Tetragonal Crystal
Example of an element with Body Centred Tetragonal Crystal Structure
BCT
a b c = = = 90º
Simple Orthorhombic Body Centred Orthorhombic Face Centred Orthorhombic End Centred Orthorhombic
Elements with Orthorhombic structure → Br, Cl, Ga
Orthorhombic Crystal
Element with Orthorhombic Crystal Structure
a = b c = = 90º =120º
Elements with Hexagonal structure → Be, Cd, Co, Ti, Zn
Hexagonal Crystal
Simple Hexagonal
Element with Hexagonal Crystal Structure
a = b = c = = 90º
Elements with Trigonal structure → As, B, Bi, Hg
Trigonal/Rhombohedral Crystal
Rhombohedral (simple)
Element with Simple Trigonal Crystal Structure
a b c = = 90º
Elements with Monoclinic structure → P, Pu, Po
Monoclinic Crystal
Simple Monoclinic End Centred (base centered)
Monoclinic
a b c
• Simple Triclinic
Triclinic Crystal
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)
14 Bravais Lattices divided into 7 Crystal Systems
P Primitive
I Body Centred
F Face Centred
C A/B/C- Centred
A Symmetry based concept ‘Translation’ based concept
+
Face Centred Cubic (FCC) Lattice Two Carbon atom Motif(0,0,0) & (¼, ¼, ¼)
=
Diamond Cubic Crystal