1. Solid State Physics (1) Phys3710
Crystal structure 2
Lecture 2
Department of Physics
Dr Mazen Alshaaer
Second semester 2013/2014
Ref.: Prof. Charles W. Myles, Department of Physics, Texas Tech University

2. Crystal structures
Crystal: atoms are arranged so that their positions are periodic in all
three dimensions
Atoms are bound to one another →
well defined equilibrium separations;
many identical atoms →
minimum energy requires every
identical atom to be in
identical environment → 3D periodicity
Ideal crystal: perfect periodicity
Real crystals are never perfect:
•surface
•impurities and defects
•thermal motion of atoms (lattice vibrations)

5. Basic definitions
The periodic array of points is called crystal lattice.
For every lattice point there is a group of atoms (or
single atom) called basis of the lattice
Don't confuse with a1, a2, a3 - basis vectors
parallelogram formed by the basis vectors –
unit cell
if a unit cell contains only one lattice point, it is
called a primitive cell (minimum volume)
Bravais lattices – all lattice points
are equivalent

6. Crystal Structure

7. Bravais Lattice (1)
A fundamental concept in the description of crystalline solids is that of a
“Bravais lattice”. A Bravais lattice is an infinite arrangement of points (or
atoms) in space that has the following property:
The lattice looks exactly the same when viewed from any lattice point
A 1D Bravais lattice: b
A 2D Bravais lattice: c b

8. Bravais Lattice (2) d c b

9. Bravais Lattice (3) A Bravais lattice has the following property:
The position vector of all points (or atoms) in the lattice can be written as
follows (Translational Lattice Vectors) :
Example (1D):
Example (2D):

10. Bravais Lattice (4)

11. Bravais Lattice (5)

12. Non-Bravais Lattice
Not only the arrangement but also the orientation must appear exactly the same from every point in a bravais lattice.
Honeycomb
The red side has a neighbour to its immediate left, the blue one instead has a neighbour to its right.
Red (and blue) sides are equivalent and have the same appearance
Red and blue sides are not equivalent. Same appearance can be obtained rotating blue side 180º.

13. Five Bravais Lattices in 2D
General or Oblique
Rectangular
Centered rectangular
Square
Hexagonal

14. aren’t necessarily a mutually orthogonal set!
The Primitive Lattice Vectors a1,a2,a3
aren’t necessarily a mutually orthogonal set!
Usually
Usually, they are neither mutually perpendicular nor all the same length!
For examples, see Fig. 3b (3 dimensions):

15. 2-Dimensional Unit Cells
Unit Cell  The smallest component of the crystal (group of atoms, ions or molecules), which, when stacked together with pure translational repetition, reproduces the whole crystal. S
2D-Crystal a b S S
Unit Cell S S S S S S S S S S S S

16. The choice of unit cell is not unique!
Unit Cell  The smallest component of the crystal (group of atoms, ions or molecules), which, when stacked together with pure translational repetition, reproduces the whole crystal.
The choice of unit cell is not unique!
2D-Crystal S S S

17. 2-Dimensional Unit Cells
Artificial Example: “NaCl”
Lattice points are points with identical environments.

18. 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.

19. 2-Dimensional Unit Cells: “NaCl”
These are also unit cells it doesn’t matter if the origin is at Na or Cl !

20. 2-Dimensional Unit Cells: “NaCl”
These are also unit cells the origin does not have to be on an atom!

21. 2-Dimensional Unit Cells: “NaCl”
These are NOT unit cells - empty space is not allowed!

22. 2-Dimensional Unit Cells: “NaCl”
In 2 dimensions, these are unit cells – in 3 dimensions, they would not be.

23. 2-Dimensional Unit Cells Why can't the blue triangle be a unit cell?

24. 3-Dimensional
Unit Cells

25. 3-Dimensional Unit Cells 3 Common Unit Cells with Cubic Symmetry
Simple Cubic Body Centered Cubic Face Centered Cubic
(SC) (BCC) (FCC)

26. Conventional & Primitive Unit Cells
Simple Cubic (sc)
Conventional Cell = Primitive cell
Body Centered Cubic (bcc)
Conventional Cell ≠ Primitive cell

27. Bravais Lattices in 3D
There are 14 different Bravais lattices in 3D that are classified into 7 different crystal systems (only the unit cells are shown below)

28. Simple Cubic (SC) Structure
Simple Cubic Lattice:
Unit Cell:
It is very cumbersome to draw entire lattices in 3D so some small portion of the lattice, having full symmetry of the lattice, is usually drawn. This small portion when repeated can generate the whole lattice and is called the “unit cell” and it could be larger than the primitive cell

29. Face Centered Cubic (FCC) Structure

30. Conventional & Primitive Unit Cells Face Centered Cubic Lattice
(Shaded)
Primitive Lattice Vectors
a1 = (½)a(0,1,0)
a2 = (½)a(1,0,1)
a3 = (½)a(1,1,0)
Lattice
Constant
Conventional Unit Cell (Full Cube)

31. Elements That Form Solids with the FCC Structure

32. Body Centered Cubic (BCC) Structure

33. Conventional & Primitive Unit Cells Body Centered Cubic Lattice
Primitive Lattice Vectors
a1 = (½)a(1,1,-1)
a2 = (½)a(-1,1,1)
a3 = (½)a(1,-1,1)
Note that the ai’s are
NOT mutually
orthogonal!
Primitive Unit Cell
Lattice
Constant
Conventional Unit Cell (Full Cube)

34. Elements That Form Solids with the BCC Structure
Note: This was the end of lecture 1

35. Conventional & Primitive Unit Cells Cubic Lattices
Simple Cubic (SC)
Primitive Cell = Conventional Cell
Fractional coordinates of lattice points: 000, 100, 010, 001, 110,101, 011, 111
Body Centered Cubic (BCC)
Primitive Cell  Conventional Cell
Fractional coordinates of lattice points in conventional cell: ,100, 010, 001, 110,101, , ½ ½ ½

36. Conventional & Primitive Unit Cells Cubic Lattices
Face Centered Cubic (FCC)
Primitive Cell  Conventional Cell
Fractional coordinates of lattice points in conventional cell: ,100, 010, 001, 110,101, 011,111, ½ ½ 0, ½ 0 ½, 0 ½ ½ ½1 ½ , 1 ½ ½ , ½ ½ 1

37. Simple Hexagonal Bravais Lattice

38. Conventional & Primitive Unit Cells Hexagonal Bravais Lattice
Points of Primitive Cell
120o
Hexagonal Bravais Lattice
Primitive Cell = Conventional Cell
Fractional coordinates of lattice points in conventional cell: , 010, 110, 101, , 000, 001

39. Hexagonal Close Packed (HCP) Structure: (A Simple Hexagonal Bravais Lattice with a 2 Atom Basis)
The HCP lattice is not a Bravais lattice, because the orientation of the environment of a point varies from layer to layer along the c-axis.

40. 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.

41. 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

42. Primitive Unit Cells A 2 Dimensional Example!
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