1. Solid State Physics (1) Phys3710
Crystal Structure 3
Lecture 3
Department of Physics
Dr Mazen Alshaaer
Second semester 2013/2014
Ref.: Prof. Charles W. Myles, Department of Physics, Texas Tech University

2. Wigner-Seitz Method to Construct a Primitive Cell
A simple, geometric method to construct a Primitive Cell is called the Wigner-Seitz Method. The procedure is:
Choose a starting lattice point.
Draw lines to connect that point to its nearest neighbors.
At the mid-point & normal to these lines, draw new lines.
The volume enclosed is called
a Wigner-Seitz cell.
Illustration for the 2 dimensional parallelogram lattice. 2

3. 3 Dimensional Wigner-Seitz Cells
Face Centered Cubic Wigner-Seitz Cell
Body Centered Cubic Wigner-Seitz Cell 3

4. Basic definitions – Lattice sites
Define basic terms and give examples of each:
Points (atomic positions)
Vectors (defines a particular direction - plane normal)
Miller Indices (defines a particular plane)
relation to diffraction
3-index for cubic and 4-index notation for HCP

5. Lattice Sites in a Cubic Unit Cell 1) To define a point within a unit cell….
The standard notation is shown in the figure. It is understood that all distances are in units of the cubic lattice constant a, which is the length of a cube edge for the material of interest.
Figure Intensity as a function of viewing angle θ (or position on the screen) for (a) two slits, (b) six slits. For a diffraction grating, the number of slits is very large (≈104) and the peaks are narrower still.

6. 2) Directions in a Crystal: Standard Notation
See Figure Choose an origin, O. This choice is arbitrary, because every lattice point has identical symmetry. Then, consider the lattice vector joining O to any point in space, say point T in the figure. As we’ve seen, this vector can be written
T = n1a1 + n2a2 + n3a3
[111] direction
In order to distinguish a Lattice Direction from a Lattice Point, (n1n2n3), the 3 integers are enclosed in square brackets [ ...] instead of parentheses (...), which are reserved to indicate a Lattice Point. In direction [n1n2n3], n1n2n3 are the smallest integers possible for the relative ratios.
Figure Intensity as a function of viewing angle θ (or position on the screen) for (a) two slits, (b) six slits. For a diffraction grating, the number of slits is very large (≈104) and the peaks are narrower still.

7. Directions in a Crystal
Procedure:
Any line (or vector direction) is specified by 2 points.
The first point is, typically, at the origin (000).
Determine length of vector projection in each of 3 axes in units (or fractions) of a, b, and c.
X (a), Y(b), Z(c)
Multiply or divide by a common factor to reduce the lengths to the smallest integer values, u v w.
Enclose in square brackets: [u v w]: [110] direction. a b c
DIRECTIONS will help define PLANES (Miller Indices or plane normal).
5. Designate negative numbers by a bar
Pronounced “bar 1”, “bar 1”, “zero” direction.
6. “Family” of [110] directions is designated as <110>.

8. Examples X = 1 , Y = ½ , Z = 0 X = ½ , Y = ½ , Z = 1 [1 ½ 0] [2 1 0]
210
X = 1 , Y = ½ , Z = 0
[1 ½ 0] [2 1 0]
Figure Intensity as a function of viewing angle θ (or position on the screen) for (a) two slits, (b) six slits. For a diffraction grating, the number of slits is very large (≈104) and the peaks are narrower still.
X = ½ , Y = ½ , Z = 1
[½ ½ 1] [1 1 2]

9. Negative Directions R = n1a1 + n2a2 + n3a3
When we write the direction [n1n2n3] depending on the origin, negative directions are written as
R = n1a1 + n2a2 + n3a3
To specify the direction, the smallest possible integers must be used.
(origin) O
- Y direction
X direction
- X direction
Z direction
- Z direction
Y direction 9 9

10. Examples of Crystal Directions
X = 1 , Y = 0 , Z = [1 0 0]
X = -1 , Y = -1 , Z = [110] 10 10

11. Examples A vector can be moved to the origin. X =-1 , Y = 1 , Z = -1/6
[ /6] [6 6 1] 11

12. These are called lattice planes.
Crystal Planes
Within a crystal lattice it is possible to identify sets of equally spaced parallel planes.
These are called lattice planes.
In the figure, the density of lattice points on each plane of a set is the same & all lattice points are contained on each set of planes. b a b a
The set of
planes for a
2D lattice. 12

13. Miller Indices
Miller Indices are a symbolic vector representation for the orientation of an atomic plane in a crystal lattice & are defined as the reciprocals of the fractional intercepts which the plane makes with the crystallographic axes.
To find the Miller indices of a plane, take the following steps:
Determine the intercepts of the plane along each of the three crystallographic directions.
Take the reciprocals of the intercepts.
If fractions result, multiply each by the denominator of the smallest fraction. 13

14. Example 1 Axis 1 ∞ 1/1 1/ ∞ Miller İndices (100) X Y Z
(1,0,0)
Axis X Y Z
Intercept points 1 ∞
Reciprocals
1/1
1/ ∞
Smallest Ratio
Miller İndices (100) 14

15. Example 2 Axis 1 ∞ 1/1 1/ 1 1/ ∞ Miller İndices (110) X Y Z
(1,0,0)
(0,1,0)
Axis X Y Z
Intercept points 1 ∞
Reciprocals
1/1
1/ 1
1/ ∞
Smallest Ratio
Miller İndices (110) 15

16. Example 3 Axis 1 1/1 1/ 1 Miller İndices (111) X Y Z Intercept points
(1,0,0)
(0,1,0)
(0,0,1)
Axis X Y Z
Intercept points 1
Reciprocals
1/1
1/ 1
Smallest Ratio
Miller İndices (111) 16

17. Example 4 Axis 1/2 1 ∞ 1/(½) 1/ 1 1/ ∞ 2 Miller İndices (210) X Y Z
(1/2, 0, 0)
(0,1,0)
Axis X Y Z
Intercept points
1/2 1 ∞
Reciprocals
1/(½)
1/ 1
1/ ∞
Smallest Ratio 2
Miller İndices (210) 17

18. Example 5 Axis 1 ∞ ½ 2 Miller İndices (102) a b c 1/1 1/ ∞ 1/(½)
Intercept points 1 ∞
Reciprocals
1/1
1/ ∞
1/(½)
Smallest Ratio 2
Miller İndices (102) 18

19. Example 6 Axis -1 ∞ ½ 2 Miller İndices (102) a b c 1/-1 1/ ∞ 1/(½)
Intercept points -1 ∞
Reciprocals
1/-1
1/ ∞
1/(½)
Smallest Ratio 2
Miller İndices (102) 19

20. Examples of Miller Indices
Plane intercepts axes at 3 2
[2,3,3]
Reciprocal numbers are:
Miller Indices of the plane: (2,3,3)
Indices of the direction: [2,3,3]
(200)
(111)
(100)
(110)
(100) 20

21. Crystal Structure 21

22. Example 7 22

23. Indices of a Family of Planes
Given any plane in a lattice, there is a infinite set of parallel lattice planes (or family of planes) that are equally spaced from each other.
One of the planes in any family always passes through the origin.
The Miller indices (hkl) usually refer to the plane that is nearest to the origin without passing through it.
You must always shift the origin or move the plane parallel, otherwise a Miller index integer is 1/0, i.e.,∞!
Sometimes (hkl) will be used to refer to any other plane in the family, or to the family taken together.
Importantly, the Miller indices (hkl) is the same vector as the plane normal!

24. Indices of a Family of Planes
Sometimes. when the unit cell has rotational symmetry, several nonparallel planes may be equivalent by virtue of this symmetry, in which case it is convenient to lump all these planes in the same Miller Indices, but with curly brackets.
Thus indices {h,k,l} represent all the planes equivalent to the
plane (hkl) through rotational symmetry. 24

25. Classification of Crystal Structures
Crystallographers showed a long time ago that, in 3 Dimensions, there are
14 BRAVAIS LATTICES + 7 CRYSTAL SYSTEMS
This results in the fact that, in 3 dimensions, there are only 7 different shapes of unit cell which can be stacked together to completely fill all space without overlapping. This gives the 7 crystal systems, in which all crystal structures can be classified. These are:
The Cubic Crystal System (SC, BCC, FCC)
The Hexagonal Crystal System (S)
The Triclinic Crystal System (S)
The Monoclinic Crystal System (S, Base-C)
The Orthorhombic Crystal System (S, Base-C, BC, FC)
The Tetragonal Crystal System (S, BC)
The Trigonal (or Rhombohedral) Crystal System (S) 25 25

26. 26 26

27. Simple Cubic (SC) coordination number = 6
For a Bravais Lattice,
The Coordinatıon Number  The number of lattice points closest to a given point (the number of nearest-neighbors of each point).
Because of lattice periodicity, all points have the same number of nearest neighbors or coordination number. (That is, the coordination number is intrinsic to the lattice.)
Examples
Simple Cubic (SC) coordination number = 6
Body-Centered Cubic coordination number = Face-Centered Cubic coordination number = 12
Crystal Structure 27

28. Atomic Packing Factor (or Atomic Packing Fraction)
For a Bravais Lattice,
The Atomic Packing Factor (APF)  the volume of the atoms within the unit cell divided by the volume of the unit cell.
When calculating the APF, the volume of the atoms in the unit cell is calculated AS IF each atom was a hard sphere, centered on the lattice point & large enough to just touch the nearest-neighbor sphere.
Of course, from Quantum Mechanics, we know that this is very unrealistic for any atom!!

29. 1- CUBIC CRYSTAL SYSTEMS 3 Common Unit Cells with Cubic Symmetry
Simple Cubic Body Centered Cubic Face Centered Cubic
(SC) (BCC) (FCC)

30. 3 Common Unit Cells with Cubic Symmetry

31. a- The Simple Cubic (SC) Lattice
The SC Lattice has one lattice point in its unit cell, so it’s unit cell is a primitive cell.
In the unit cell shown on the left, the atoms at the corners are cut because only a portion (in this case 1/8) “belongs” to that cell. The rest of the atom “belongs” to neighboring cells.
The Coordinatination Number of the SC Lattice = 6. a b c 31

32. Simple Cubic (SC) Lattice Atomic Packing Factor32

33. b- The Body Centered Cubic (BCC) Lattice
The BCC Lattice has two lattice points per unit cell so the BCC unit cell is a non-primitive cell.
Every BCC lattice point has 8 nearest- neighbors. So (in the hard sphere model) each atom is in contact with its neighbors only along the body-diagonal directions.
Many metals (Fe,Li,Na..etc), including the alkalis and several transition elements have the BCC structure. b c a 33

34. BCC Structure

35. Body Centered Cubic (BCC) Lattice Atomic Packing Factor2
(0,433a)
Crystal Structure 35

36. Elements with the BCC Structure
Note: This was the end of lecture 1

37. c- The Face Centered Cubic (FCC) Lattice
In the FCC Lattice there are atoms at the corners of the unit cell and at the center of each face.
The FCC unit cell has 4 atoms so it is a non-primitive cell.
Every FCC Lattice point has 12 nearest-neighbors.
Many common metals (Cu,Ni,Pb..etc) crystallize in the FCC structure.
Crystal Structure 37

38. FCC Structure

39. Face Centered Cubic (FCC) Lattice Atomic Packing Factor
0,74 4
(0,353a)
Crystal Structure 39

40. Elements That Have the FCC Structure

41. FCC & BCC: Conventional Cells With a Basis
Alternatively, the FCC lattice can be viewed in terms of a conventional unit cell with a 4-point basis.
Similarly, the BCC lattice can be viewed in terms of a conventional unit cell with a 2- point basis.

42. Comparison of the 3 Cubic Lattice Systems
Unit Cell Contents
Counting the number of atoms within the unit cell
Atom Position Shared Between: Each atom counts:
corner cells /8
face center cells /2
body center cell
edge center cells /2
Lattice Type Atoms per Cell
P (Primitive) [= 8  1/8]
I (Body Centered) [= (8  1/8) + (1  1)]
F (Face Centered) [= (8  1/8) + (6  1/2)]
C (Side Centered) [= (8  1/8) + (2  1/2)]
Crystal Structure 42

43. 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.
Crystal Structure

44. Simple Hexagonal Bravais Lattice

45. Hexagonal Close Packed (HCP) Lattice

46. Hexagonal Close Packed (HCP) Lattice
This is another structure that is common, particularly in metals. In addition to the two layers of atoms which form the base and the upper face of the hexagon, there is also an intervening layer of atoms arranged such that each of these atoms rest over a depression between three atoms in the base.
Crystal Structure 46

47. Hexagonal Close Packed (HCP) Lattice
The HCP lattice is not a Bravais lattice, because orientation of the environment of a point varies from layer to layer along the c-axis.

48. Hexagonal Close Packed (HCP) Lattice
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)
Crystal Structure 48

50. Comments on Close PackingB A B C
Sequence AAAA…
- simple cubic
Sequence ABABAB..
hexagonal close pack
Sequence ABAB…
- body centered cubic
Sequence ABCABCAB..
-face centered cubic close pack
Crystal Structure 50

51. Hexagonal Close Packing

52. HCP Lattice  Hexagonal Bravais Lattice with a 2 Atom Basis

53. Comments on Close Packing

56. Close-Packed Structures
ABCABC… → fcc
ABABAB… → hcp

57. Crystal Structure 57

58. 3 - TRICLINIC & 4 - MONOCLINIC CRYSTAL SYSTEMS
Triclinic crystals have the least symmetry of any crystal systems.
The three axes are each different lengths & none are perpendicular to each other. These materials are the most difficult to recognize.
Triclinic (Simple) a ¹ ß ¹ g ¹ 90
oa ¹ b ¹ c
Monoclinic (Base Centered) a = g = 90o, ß ¹ 90o
a ¹ b ¹ c,
Monoclinic (Simple) a = g = 90o, ß ¹ 90o
a ¹ b ¹c 58

59. 5 - ORTHORHOMBIC CRYSTAL SYSTEM
Orthorhombic (BC) a = ß = g = 90o
a ¹ b ¹ c
Orthorhombic (Simple) a = ß = g = 90o
a ¹ b ¹ c
Orthorhombic (Base-centred) a = ß = g = 90o
a ¹ b ¹ c
Orthorhombic (FC) a = ß = g = 90o
a ¹ b ¹ c
Crystal Structure 59

60. 6 – TETRAGONAL CRYSTAL SYSTEM
Tetragonal (P) a = ß = g = 90o
a = b ¹ c
Tetragonal (BC) a = ß = g = 90o
a = b ¹ c
Crystal Structure 60

61. 7 - RHOMBOHEDRAL (R) OR TRIGONAL CRYSTAL SYSTEM
Rhombohedral (R) or Trigonal (S) a = b = c, a = ß = g ¹ 90o
Crystal Structure 61