solid state physics

1. Lec. (3)
Crystallography and Structure
1

2. Overview:
 Crystal Structure – matter assumes a
periodic shape
Non-Crystalline or Amorphous “structures”
exhibit no long range periodic shapes
Xtal Systems – not structures but potentials
FCC, BCC and HCP – common Xtal
Structures for metals
 Point, Direction and Planer ID’ing in Xtals
 X-Ray Diffraction and Xtal Structure
2

3. Energy and Packing
• Non dense, random packing
• Dense, ordered packing
Energy
Dense, ordered packed structures tend to have
lower energies & thus are more stable.
r
typical neighbor
bond length
typical neighbor
bond energy
Energy
r
typical neighbor
bond length
typical neighbor
bond energy
3

4. CRYSTAL STRUCTURES
Means: periodic arrangement of atoms/ions
over large atomic distances
Leads to structure displaying
long-range order that is
Measurable and Quantifiable
All metals, many ceramics, and some polymers
exhibit this “High Bond Energy” and a More
Closely Packed Structure
4

5. Amorphous Materials
Are materials lacking long range order
These less densely packed lower bond energy
“structures” can be found in Metals are observed in
Ceramic GLASS and many “plastics”
5

6. Crystal Systems – Some
Definitional information
Unit cell: smallest repetitive volume which
contains the complete lattice pattern of a crystal.
7 crystal systems of varying symmetry
are known
These systems are built by changing the
lattice parameters:
a, b, and c are the edge lengths
, , and  are interaxial angles
Fig. 3.4, Callister 7e.
6

7. General Unit Cell Discussion
• For any lattice, the unit cell &, thus,
the entire lattice, is UNIQUELY
determined by 6 constants (figure):
a, b, c, α, β and γ
which depend on lattice geometry.
• As we’ll see, we sometimes want to
calculate the number of atoms in a
unit cell. To do this, imagine stacking
hard spheres centered at each lattice point
& just touching each neighboring sphere.
Then, for the cubic lattices, only 1/8 of
each lattice point in a unit cell assigned
to that cell. In the cubic lattice in the
figure, each unit cell is associated
with (8)  (1/8) = 1 lattice point.
7

8. Primitive Unit Cells & Primitive Lattice Vectors
• In general, a Primitive Unit
Cell is determined by the
parallelepiped formed by the
Primitive Vectors a1 ,a2, & a3
such that there is no cell of
smaller volume that can be used
as a building block for the
crystal structure.
• As we’ve discussed, a Primitive
Unit Cell can be repeated to fill
space by periodic repetition of it
through the translation vectors
T = n1a1 + n2a2 + n3a3.
• The Primitive Unit
Cell volume can be
found by vector
manipulation:
V = a1(a2  a3)
• For the cubic unit cell in
the figure, V = a3
8

9. Primitive Unit Cells
• Note that, by definition, the Primitive Unit Cell must
contain ONLY ONE lattice point.
• There can be different choices for the Primitive Lattice
Vectors, but the Primitive Cell volume must be independent
of that choice.
A 2 Dimensional
Example!
P = Primitive Unit Cell
NP = Non-Primitive Unit Cell
9

10. 2-Dimensional Unit Cells
Artificial Example: “NaCl”
Lattice points are points with identical environments.
10

11. 2-Dimensional Unit Cells: “NaCl”
The choice of origin is arbitrary - lattice points need not be
atoms - but the unit cell size must always be the same.
11

12. 2-Dimensional Unit Cells: “NaCl”
These are also unit cells -
it doesn’t matter if the origin is at Na or Cl !
12

13. 2-Dimensional Unit Cells: “NaCl”
These are also unit cells -
the origin does not have to be on an atom!
13

14. 2-Dimensional Unit Cells: “NaCl”
These are NOT unit cells - empty space is not allowed!
14

15. 2-Dimensional Unit Cells: “NaCl”
In 2 dimensions, these are unit cells –
in 3 dimensions, they would not be.
15

16. Crystal Systems
Crystal structures are divided into groups according to unit cell geometry (symmetry).
16

17. Metallic Crystal Structures
• Tend to be densely packed.
• Reasons for dense packing:
- Typically, only one element is present, so all atomic
radii are the same.
- Metallic bonding is not directional.
- Nearest neighbor distances tend to be small in
order to lower bond energy.
- Electron cloud shields cores from each other
• Have the simplest crystal structures.
We will examine three such structures (those of
engineering importance) called: FCC, BCC and
HCP – with a nod to Simple Cubic
17

18. Crystal Structure of Metals – of
engineering interest
18

19. Simple Cubic Structure (SC)
• Rare due to low packing density (only Po – Polonium --
has this structure)
• Close-packed directions are cube edges.
• Coordination No. = 6
(# nearest neighbors)
for each atom as seen
(Courtesy P.M. Anderson)
19

20. Atomic Packing Factor (APF)
• APF for a simple cubic structure = 0.52
APF =
4
3
volume
1 p (0.5a) 3
a3
atoms
unit cell
atom
volume
unit cell
APF =
Volume of atoms in unit cell*
Volume of unit cell
*assume hard spheres
close-packed directions
Adapted from Fig. 3.23,
Callister 7e.
a
R=0.5a
contains (8 x 1/8) =
1 atom/unit cell Here: a = Rat×2
Where Rat is the ‘handbook’ atomic radius 20

21. Body Centered Cubic Structure (BCC)
• Atoms touch each other along cube diagonals within a
unit cell.
--Note: All atoms are identical; the center atom is shaded
differently only for ease of viewing.
• Coordination # = 8
Adapted from Fig. 3.2,
Callister 7e.
(Courtesy P.M. Anderson)
ex: Cr, W, Fe (), Tantalum, Molybdenum
2 atoms/unit cell: (1 center) + (8 corners x 1/8)
21

22. Atomic Packing Factor: BCC
a
2 a
3 a
Close-packed directions:
length = 4R =
atoms
unit cell atom
APF =
4
3
2 p ( 3a/4)3
volume
a3
volume
unit cell
3 a
Adapted from
Fig. 3.2(a), Callister 7e.
• APF for a body-centered cubic structure = 0.68
22

23. 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.
• Coordination # = 12
Adapted from Fig. 3.1, Callister 7e.
(Courtesy P.M. Anderson)
ex: Al, Cu, Au, Pb, Ni, Pt, Ag
4 atoms/unit cell: (6 face x ½) + (8 corners x 1/8)
23

24. Atomic Packing Factor: FCC
• APF for a face-centered cubic structure = 0.74
The maximum achievable APF!
Close-packed directions:
length = 4R = 2 a
Unit cell contains:
6 x 1/2 + 8 x 1/8
= 4 atoms/unit cell
atoms
unit cell atom
APF =
4
3
4 p ( 2a/4)3
volume
a3
volume
unit cell
Adapted from
Fig. 3.1(a),
Callister 7e.
(a = 22*R)
24

25. Comparison of the 3 Cubic Lattice Systems
Unit Cell Contents
Atom Position Shared Between: Each atom counts:
corner 8 cells 1/8
face center 2 cells 1/2
body center 1 cell 1
edge center 2 cells 1/2
Lattice Type Atoms per Cell
P (Primitive) 1 [= 8  1/8]
I (Body Centered) 2 [= (8  1/8) + (1  1)]
F (Face Centered) 4 [= (8  1/8) + (6  1/2)]
C (Side Centered) 2 [= (8  1/8) + (2  1/2)]
25
Counting the number of atoms within the unit cell

26. 2- HEXAGONAL CRYSTAL SYSTEMS
In a Hexagonal Crystal System, three equal coplanar axes
intersect at an angle of 60°, and another axis is
perpendicular to the others and of a different length.
The atoms are all the same.
26

27. Hexagonal Close-Packed Structure (HCP)
ex: Cd, Mg, Ti, Zn
• ABAB... Stacking Sequence
• 3D Projection • 2D Projection
• Coordination # = 12
• APF = 0.74
Adapted from Fig. 3.3(a),
Callister 7e.
Top layer
Middle layer
6 atoms/unit cell
• c/a = 1.633 (ideal)
c
a
A sites
B sites
A sites
Bottom layer
27

28. Hexagonal Close Packed (HCP) Lattice
Crystal Structure 28
Bravais Lattice :
Hexagonal Lattice
He, Be, Mg, Hf, Re
(Group II elements)
ABABAB Type of Stacking
a = b
Angle between a & b = 120°
c = 1.633a,
basis:
(0,0,0) (2/3a ,1/3a,1/2c)
28

29. We find that both FCC & HCP are highest density packing
schemes (APF =0.74) – this illustration shows their
differences as the closest packed planes are “built-up”
29

30. Comments on Close Packing
A A
A
A
Crystal Structure 30
Close pack
C C C
A A
B
A A
A A A
B B
A A
A
A A A
A A
A A A
B B
B
B
B B
B
B
C C C
C C
C
C
Sequence ABABAB..
-hexagonal close pack
Sequence ABCABCAB..
-face centered cubic close pack
B
A A
A
A A
B
B B
Sequence AAAA…
- simple cubic
Sequence ABAB…
- body centered cubic
30

31. Theoretical Density, r
Density = r =
Mass of Atoms inUnit Cell
Total Volume of Unit Cell
n A
VCNA
r =
where n = number of atoms/unit cell
A = atomic weight
VC = Volume of unit cell = a3 for cubic
NA = Avogadro’s number
= 6.023 x 1023 atoms/mol
31

32. Theoretical Density, r
• Ex: Cr (BCC)
A = 52.00 g/mol
R = 0.125 nm
n = 2
 a = 4R/3 = 0.2887 nm
a
R
r =
a3
2 52.00
atoms
unit cell
g
mol
volume atoms
unit cell
mol
6.023 x 1023
rtheoretical
ractual
= 7.18 g/cm3
= 7.19 g/cm3
32

33. Locations in Lattices: Point Coordinates
Point coordinates for unit cell
center are
a/2, b/2, c/2 ½½½
Point coordinates for unit cell
(body diagonal) corner are
111
Translation: integer multiple of
lattice constants  identical
position in another unit cell
z
x
y
c
000
111
a b
y
z

2c



b
b
33

34. Crystallographic Directions
1. Vector is repositioned (if necessary) to
pass through the Unit Cell origin.
2. Read off line projections (to principal axes
of U.C.) in terms of unit cell dimensions a, b,
and c
3. Adjust to smallest integer values
4. Enclose in square brackets, no commas
[uvw]
z
y
ex: 1, 0, ½ => 2, 0, 1 => [201 ]
-1, 1, 1
families of directions <uvw>
x
Algorithm
where ‘overbar’ represents a
negative index
=> [111 ]
34

35. What is this Direction ?????
Projections:
Projections in terms of a,b
and c:
Reduction:
Enclosure [brackets]
x y z
a/2 b 0c
1/2 1 0
1 2 0
[120]
35

36. Linear Density – considers equivalance and is
important in Slip
Number of atoms
ex: linear density of Al in [110]
direction
a = 0.405 nm
 Linear Density of Atoms  LD =
a
[110]
Unit length of direction vector
# atoms
length
LD - = =
1 3.5 nm
2
2a
# atoms CENTERED on the direction of interest!
Length is of the direction of interest within the Unit Cell 36

37. Defining Crystallographic Planes
 Miller Indices: Reciprocals of the (three) axial
intercepts for a plane, cleared of fractions &
common multiples. All parallel planes have
same Miller indices.
 Algorithm (in cubic lattices this is direct)
1. Read off intercepts of plane with axes in
terms of a, b, c
2. Take reciprocals of intercepts
3. Reduce to smallest integer values
4. Enclose in parentheses, no
commas i.e., (hkl)  families {hkl}
37

38. Crystallographic Planes
 We want to examine the atomic packing of
crystallographic planes – those with the
same packing are equivalent and part of
families
 Iron foil can be used as a catalyst. The
atomic packing of the exposed planes is
important.
a) Draw (100) and (111) crystallographic planes
for Fe.
b) Calculate the planar density for each of these
planes.
38

39. Planar Density of (100) Iron
Solution: At T < 912C iron has the BCC structure.
(100)
2D repeat unit
4 3
Radius of iron R = 0.1241 nm
R
3
a =
1
Planar Density = =
a2
atoms
2D repeat unit
=
atoms
nm2
12.1
atoms
m2
= 1.2 x 1019
1
2
R
area 4 3
3
2D repeat unit
Atoms: wholly contained and centered in/on plane within U.C., area of plane in U.3C9.

40. Planar Density of (111) Iron
Solution (cont): (111) plane 1/2 atom centered on plane/ unit cell
atoms in plane
atoms above plane
atoms below plane
3
h a
2
=
2 a
Area 2D Unit: ½ hb = ½*[(3/2)a][(2)a]=1/2(3)a2=8R2/(3)
3*1/6
atoms
= =
nm2
7.0
atoms
m2
atoms
2D repeat unit
Planar Density = 0.70 x 1019
area
2D repeat unit
8
R
2 3
40