Engineering Science 1021A/B Lecture 10: Atomic and Ionic Arrangements: Lattice, Basis, Unit Cells, and Crystal Structures
17 May 2018
School
Department
Professor
Lattice & Unit Cells
Lattice Point:
Repeating point that has identical surroundings at
every position in the lattice
•
Unit Cell (crystal structure):
The smallest repeating subdivision of the lattice
•
Building black for the lattice
•
Lattice Parameters:
Describes size & shape of the unit cell
•
Lengths of 3 sides and the angles between them
•
Coordination Number:
Number of atoms touching a single atom in the
lattice
Number of nearest neighbours
○
•
Closed-Packed Direction:
Directions in the unit cell along which atoms are in
continuous contact
•
Closed-Packed Plane:
Plane of atoms with perfect hexagonal packing
•
metals deform along planes of atoms that are most
tightly packed together
•
Atomic Packing Factor:
Fraction of the unit cell occupied by atoms•
!"# $%&'()*
%+,-'./011
•
Theoretical Density
Theoretical density can be calculated, if the crystal structure of a material is known•
2 $ 3!
4
+/56
3 $ # of atoms per unit cell
○
! $ atomic weight
○
4
+/ $volume of unit cell
○
56$Avogadro's number
○
•
Unit Cell Structures
Body-Centered Cubic (BCC)
Atoms in
unit cell 7 8 9: 7
9$ ;
Lattice
Parameters < $ = $ > $ ?@
AB
C
○
•
D $ E $ F $ GHI
•
Closed-
packed
Planes
No
Closed-
packed
Directions
Yes, 4 body diagonals
Coordination
Number
8
Atomic
Working
Factor $%&'()*
%+,-'./011 $;: ?J@K
B
L?@
A.B
CM.K $ HNO9
Face-Centered Cubic (FCC)
Atoms in
unit cell O: 7
;8 9: 7
9$ ?
Lattice
Parameters
< $ = $ > $ ;A;
Cr
○
•
D $ E $ F $ GHI
•
Closed-
packed
Planes
Yes, 4
Closed-
packed
Directions
Yes, 6 face diagonals
Coordination
Number
72
Atomic
Working
Factor $%&'()*
%+,-'./011 $?: ?J@K
B
L;A;
C@M.K $ HNP?
Hexagonal Close-Packed (HCP)
Atoms in
unit cell 7 8 ?: 7
7; 8 ?: 7
O$ ;
Lattice
Parameters
< $ = Q >
•
D $ E $ GHI
F $ 7;HI
•
•
Closed-
packed
Planes
Yes, 7
Closed-
packed
Directions
Yes, B
Coordination
Number
7;
Atomic
Working
Factor
$%&'()*
%+,-'./011 $ HNP?
BCC FCC HCP
Atoms in unit cell 7 8 9: 7
9$ ; O: 7
;8 9: 7
9$ ? 7 8 ?: 7
7; 8 ?: 7
O$ ;
Lattice Parameters < $ = $ > $ ?@
AB
C
•
D $ E $ F $ GHI
•
< $ = $ > $ ;A;
Cr
•
D $ E $ F $ GHI
•
< $ = Q >
•
D $ E $ GHI
F $ 7;HI
•
•
Closed-packed Planes 0 4 7
Closed-packed Directions 4 body diagonals 6 face diagonals B
Coordination Number 8 7272
Atomic Working Factor
$;: ?J@K
B
L?@
A.B
CM.K $ HNO9 $?: ?J@K
B
L;A;
C@M.K $ HNP?
$ HNP?
Lattice, Basis, Unit Cells, and Crystal Structures
Lattice & Unit Cells
Lattice Point:
Repeating point that has identical surroundings at
every position in the lattice
•
Unit Cell (crystal structure):
The smallest repeating subdivision of the lattice•
Building black for the lattice•
Lattice Parameters:
Describes size & shape of the unit cell•
Lengths of 3 sides and the angles between them•
Coordination Number:
Number of atoms touching a single atom in the
lattice
Number of nearest neighbours
○
•
Closed-Packed Direction:
Directions in the unit cell along which atoms are in
continuous contact
•
Closed-Packed Plane:
Plane of atoms with perfect hexagonal packing•
metals deform along planes of atoms that are most
tightly packed together
•
Atomic Packing Factor:
Fraction of the unit cell occupied by atoms•
!"# $%&'()*
%+,-'./011
•
Theoretical Density
Theoretical density can be calculated, if the crystal structure of a material is known•
2 $ 3!
4
+/56
3 $ # of atoms per unit cell
○
! $ atomic weight
○
4
+/ $volume of unit cell
○
56$Avogadro's number
○
•
Unit Cell Structures
Body-Centered Cubic (BCC)
Atoms in
unit cell 7 8 9: 7
9$ ;
Lattice
Parameters < $ = $ > $ ?@
AB
C
○
•
D $ E $ F $ GHI
•
Closed-
packed
Planes
No
Closed-
packed
Directions
Yes, 4 body diagonals
Coordination
Number
8
Atomic
Working
Factor $%&'()*
%+,-'./011 $;: ?J@K
B
L?@
A.B
CM.K $ HNO9
Face-Centered Cubic (FCC)
Atoms in
unit cell O: 7
;8 9: 7
9$ ?
Lattice
Parameters
< $ = $ > $ ;A;
Cr
○
•
D $ E $ F $ GHI
•
Closed-
packed
Planes
Yes, 4
Closed-
packed
Directions
Yes, 6 face diagonals
Coordination
Number
72
Atomic
Working
Factor $%&'()*
%+,-'./011 $?: ?J@K
B
L;A;
C@M.K $ HNP?
Hexagonal Close-Packed (HCP)
Atoms in
unit cell 7 8 ?: 7
7; 8 ?: 7
O$ ;
Lattice
Parameters
< $ = Q >
•
D $ E $ GHI
F $ 7;HI
•
•
Closed-
packed
Planes
Yes, 7
Closed-
packed
Directions
Yes, B
Coordination
Number
7;
Atomic
Working
Factor
$%&'()*
%+,-'./011 $ HNP?
BCC FCC HCP
Atoms in unit cell 7 8 9: 7
9$ ; O: 7
;8 9: 7
9$ ? 7 8 ?: 7
7; 8 ?: 7
O$ ;
Lattice Parameters < $ = $ > $ ?@
AB
C
•
D $ E $ F $ GHI
•
< $ = $ > $ ;A;
Cr
•
D $ E $ F $ GHI
•
< $ = Q >
•
D $ E $ GHI
F $ 7;HI
•
•
Closed-packed Planes 0 4 7
Closed-packed Directions 4 body diagonals 6 face diagonals B
Coordination Number 8 7272
Atomic Working Factor
$;: ?J@K
B
L?@
A.B
CM.K $ HNO9 $?: ?J@K
B
L;A;
C@M.K $ HNP?
$ HNP?
Lattice, Basis, Unit Cells, and Crystal Structures
Lattice & Unit Cells
Lattice Point:
Repeating point that has identical surroundings at
every position in the lattice
•
Unit Cell (crystal structure):
The smallest repeating subdivision of the lattice•
Building black for the lattice•
Lattice Parameters:
Describes size & shape of the unit cell•
Lengths of 3 sides and the angles between them•
Coordination Number:
Number of atoms touching a single atom in the
lattice
Number of nearest neighbours
○
•
Closed-Packed Direction:
Directions in the unit cell along which atoms are in
continuous contact
•
Closed-Packed Plane:
Plane of atoms with perfect hexagonal packing•
metals deform along planes of atoms that are most
tightly packed together
•
Atomic Packing Factor:
Fraction of the unit cell occupied by atoms
•
!"# $%&'()*
%+,-'./011
•
Theoretical Density
Theoretical density can be calculated, if the crystal structure of a material is known
•
2 $ 3!
4
+/56
3 $
# of atoms per unit cell
○
! $
atomic weight
○
4
+/ $
volume of unit cell
○
56$
Avogadro's number
○
•
Unit Cell Structures
Body-Centered Cubic (BCC)
Atoms in
unit cell
7 8 9: 7
9$ ;
Lattice
Parameters
< $ = $ > $ ?@
A
B
C
○
•
D $ E $ F $ GHI
•
Closed-
packed
Planes
No
Closed-
packed
Directions
Yes, 4 body diagonals
Coordination
Number
8
Atomic
Working
Factor $%&'()*
%+,-'./011 $;: ?J@K
B
L?@
A.B
CM.K $ HNO9
Face-Centered Cubic (FCC)
Atoms in
unit cell O: 7
;8 9: 7
9$ ?
Lattice
Parameters
< $ = $ > $ ;A;
Cr
○
•
D $ E $ F $ GHI
•
Closed-
packed
Planes
Yes, 4
Closed-
packed
Directions
Yes, 6 face diagonals
Coordination
Number
72
Atomic
Working
Factor $%&'()*
%+,-'./011 $?: ?J@K
B
L;A;
C@M.K $ HNP?
Hexagonal Close-Packed (HCP)
Atoms in
unit cell 7 8 ?: 7
7; 8 ?: 7
O$ ;
Lattice
Parameters
< $ = Q >
•
D $ E $ GHI
F $ 7;HI
•
•
Closed-
packed
Planes
Yes, 7
Closed-
packed
Directions
Yes, B
Coordination
Number
7;
Atomic
Working
Factor
$%&'()*
%+,-'./011 $ HNP?
BCC FCC HCP
Atoms in unit cell 7 8 9: 7
9$ ; O: 7
;8 9: 7
9$ ? 7 8 ?: 7
7; 8 ?: 7
O$ ;
Lattice Parameters < $ = $ > $ ?@
AB
C
•
D $ E $ F $ GHI
•
< $ = $ > $ ;A;
Cr
•
D $ E $ F $ GHI
•
< $ = Q >
•
D $ E $ GHI
F $ 7;HI
•
•
Closed-packed Planes 0 4 7
Closed-packed Directions 4 body diagonals 6 face diagonals B
Coordination Number 8 7272
Atomic Working Factor
$;: ?J@K
B
L?@
A.B
CM.K $ HNO9 $?: ?J@K
B
L;A;
C@M.K $ HNP?
$ HNP?
Lattice, Basis, Unit Cells, and Crystal Structures
