Introduction Part 2-Solid State Physics-Lecture Slides, Slides of Solid State Physics

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Crystal Structure 51

• or if you don’t start from an atom

Crystal Structure 52

This is NOT a unit cell even though they are all the same - empty space is not allowed!

Crystal Structure 54

Why can't the blue triangle

be a unit cell?

Crystal Structure 55

Unit Cell in 3D

Crystal Structure 57

Three common Unit Cell in 3D

Crystal Structure 58

UNIT CELL

Primitive Conventional & Non-primitive

 Single lattice point per cell  Smallest area in 2D, or Smallest volume in 3D

 More than one lattice point per cell  Integral multibles of the area of primitive cell

Body centered cubic(bcc) Conventional ≠ Primitive cell

Simple cubic(sc) Conventional = Primitive cell

Crystal Structure 60

Primitive and conventional cells of FCC

1

2

3

( ˆ^ ˆ^ ˆ)

a x y z

a x y z

a x y z

Primitive and conventional cells of BCC

Primitive Translation Vectors:

Crystal Structure 63

Body centered cubic (bcc): primitive (rombohedron) conventional cell

a

b c

Fractional coordinates: 000, 100, 101, 110, 110,101, 011, 211, 200

Face centered cubic (fcc): conventional primitive cell

a

b c

Fractional coordinates: 000,100, 010, 001, 110,101, 011,111, ½ ½ 0, ½ 0 ½, 0 ½ ½ ,½1 ½ , 1 ½ ½ , ½ ½ 1

Primitive and conventional cells

Crystal Structure 64

Hexagonal close packed cell (hcp): conventional primitive cell

Fractional coordinates: 100, 010, 110, 101,011, 111,000, 001

points of primitive cell

a

b

c

Primitive and conventional cells-hcp

Crystal Structure 66

 A primitive unit cell is made of primitive translation vectors a 1 ,a 2 , and a 3 such that there is no cell of smaller volume that can be used as a building block for crystal structures.

 A primitive unit cell will fill space by repetition of suitable crystal translation vectors. This defined by the parallelpiped a 1 , a 2 and a 3. The volume of a primitive unit cell can be found by

 V = a 1 .(a 2 x a 3 ) (vector products) Cubic cell volume = a^3

Primitive Unit Cell and vectors

Crystal Structure 67

 The primitive unit cell must have only one lattice point.

 There can be different choices for lattice vectors , but the volumes of these primitive cells are all the same.

P = Primitive Unit Cell NP = Non-Primitive Unit Cell

Primitive Unit Cell

a 1

Crystal Structure 69

Wigner-Seitz Cell - 3D

Crystal Structure 70

Lattice Sites in Cubic Unit Cell