VẬT LÝ CHẤT RẮN 011 lattice, bases and unit cells

Primitive cells, Wigner Seitz cells, and 2D lattices Chapter One Crystal Structures (Lattices, bases, and unit cells) Main Concepts ! To describe crystal structures, we need to understand 2 concepts 1[.]

Chapter One: Crystal Structures (Lattices, bases, and unit cells)

Main Concepts

! To describe crystal structures, we need to

understand 2 concepts:

1. Lattice (what is the underlying symmetry? Cubic?

Hexagonal? Something else?)

2. Basis (what are the contents of the unit cell?)

! The lattice is defined by 3 fundamental translation

vectors a1, a2, a3 such that the atomic

arrangements look the same in every respect when viewed from the point r as when viewed from r'

r' = r + u1 a1 + u2 a2 + u3 a3 Where u1, u2, and u3 are integers

An example of a lattice: the rectangular lattice in 2D

! Define translation vectors a1 and a2 such that any point can be reached by adding u1 a1 +

u2 a2 (where u1 and u2 are integers)

! By “point” we mean any position where the lattice looks the same (ie If the lattice was infinite, you couldn’t tell if you moved or not)

A 2D lattice: rectangular lattice in 2D

a1

a2

1 a2

2 a1

}

r' = r + 1 a1 + 2 a2 Where u1 = 1 and u2 = 2

One can reach any point

in space by adding an integer number of

translation vectors a1 and a2

Crystal structure = lattice + basis

a1

a2

A lattice

A basis

A crystal structure

We need to identify the symmetry (lattice vectors) and the lattice contents (basis) to fully describe a structure.

(1) How do we choose lattice vectors? (This will lead us to thinking about unit cells)

The Unit Cell

! The unit cell is defined by the translational vectors a1, a2 and a3

! This is the basic building block of the crystal structure (it fills space)

! The choice of origin is arbitrary

! The choice of unit cells is arbitrary as well!

! How do we choose unit cells?

How do we pick lattice vectors a 1 , a 2

! The choice of the origin for unit cells is arbitrary!

! (See overhead example for 2D square lattice)

! The choice of the unit cell is arbitrary as well!

! (See overhead example for 2D square lattice)

! Note: I can pick vectors a1 and a2 such that the square lattice looks like the rectangular lattice

! The only requisite is that the lattice must look the same when you translate by a crystal translational vector T:

T = u1 a1 + u2 a2 + u3 a3 Where u1, u2, and u3 are integers

Answer: Choose primitive unit cells

! A primitive unit cell is made

of primitive translation

vectors a1, a2, and a3 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 is defined by

the parallelpiped a1, a2 and

a3 The volume of a primitive

unit cell can be found by

V = | a1 • a2 x a3 |

a1

a2

a3

Cubic cell: Volume = a 3

(homework: show this using the eqn!) (vector products)

Primitive unit cells can have angles between vectors that are not 90°

! Example: Monoclinic unit

cells (eg Like the

monoclinic crystal of

gypsum shown in last class)

! The equation still works, but

a3 is displaced from the

z-axis by an angle β

! Homework: Prove that the

formula still applies and find

the volume (hint: define a

vector which is a2 x a3, and

use vector identities)

a1

a2

a3 z-axis

β

Important points

1 There is only one lattice point/primitive cell

2 There can be different choice for a1, a2

and a3, but the volumes of these cells are all the same.