Crystallography of MaterialsShyue Ping OngDepartment of NanoEngineeringUniversity of California, San Diego

Readings¡Chapter 3 of Structure of Materials

NANO 106 - Crystallography of Materials by Shyue Ping Ong - Lecture 1

What is a crystal?¡A crystal is a time-invariant, 3D arrangement of

atoms or molecules on a lattice.

NANO 106 - Crystallography of Materials by Shyue Ping Ong - Lecture 1

Perovskite SrTiO3

The “motif”

repeated on each point in the cubic lattice below…

Nets and Lattices¡A lattice (3D) or net (2D) is an abstract concept of

an infinite array of discrete points generated by a set of translation operations.

NANO 106 - Crystallography of Materials by Shyue Ping Ong - Lecture 1

The motif

Repeated infinitely in all directions

Concept of Symmetry¡You have just encountered

your first symmetry concept – translational symmetry.

¡A symmetry operation is a permutation of atoms such that the molecule or crystal is indistinguishable before and after the operation.

¡ It also means that all lattice points have exactly the same “environment”.

NANO 106 - Crystallography of Materials by Shyue Ping Ong - Lecture 1

Identify the nets in the following patterns

NANO 106 - Crystallography of Materials by Shyue Ping Ong - Lecture 1

Identify the nets in the following patterns

NANO 106 - Crystallography of Materials by Shyue Ping Ong - Lecture 1

Basis and translation vectors

¡ How are points in a lattice related to one another (e.g. how do we get from point A to point B)?

NANO 106 - Crystallography of Materials by Shyue Ping Ong - Lecture 1

A

B

b

a

Let us define basis vectors a and b.

All points in the lattice can therefore be reached by integer linear combinations of the basis vectors, i.e.,

By inspection, we can see that

If we choose A to be the arbitrary origin with coordinates (0, 0), all other lattice points can be represented as (u, v). For example, B = (2, 1). t is then known as the translation vector.

t

t = ua+ vb, u,v ∈ Z

B - A= 2a+ b

Basis and translation vectors

¡ For 3D lattices, there are three basis vectors instead of two. Notationally,

¡ And each lattice point/node can be represented by coordinates (u, v, w)

NANO 106 - Crystallography of Materials by Shyue Ping Ong - Lecture 1

t = ua+ vb+wc, u,v,w ∈ Z

Net and Lattice parameters

NANO 106 - Crystallography of Materials by Shyue Ping Ong - Lecture 1

a = Length of ab = Length of bc = Length of cα =Angle between b and cβ =Angle between a and cγ =Angle between a and b a,b,c,α,β,γ{ }

a = Length of ab = Length of bγ =Angle between a and b

a,b,γ{ }

Lattice math example¡A lattice is given by the following vectors in

Cartesian space:

¡ Calculate the lattice parameters a, b, c, α, β, γ.¡ If a lattice node is given by coordinates (3, 2, 1) in

crystal coordinates, what are its coordinates in Cartesian coordinates?

NANO 106 - Crystallography of Materials by Shyue Ping Ong - Lecture 1

a =100

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&&&, b=

−1/ 23 / 20

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&&&&, c =

003

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Blackboard

Deriving the 2D Crystal Systems¡ What values can take? Or phrased in another way,

what special values of would result in additional symmetry?

NANO 106 - Crystallography of Materials by Shyue Ping Ong - Lecture 1

a,b,γ{ }a,b,γ{ }

Consider arbitrary

a,b,γ{ }What are the symmetry elements in this net?

The oblique net (and all 2D nets) has two-fold rotational symmetry.

Higher symmetry nets

NANO 106 - Crystallography of Materials by Shyue Ping Ong - Lecture 1

a ≠ b,γ = 90° a = b,γ = 90°

Rectangular net Square net

a = b,γ =120°

Hexagonal net

Adding new nodes¡Can we get a new distinct net by adding more

lattice points to existing nets?

NANO 106 - Crystallography of Materials by Shyue Ping Ong - Lecture 1

New centered rectangular net, all nodes have the same environment.

No new net!If we reorient the lattice by 45 deg, we see that what we have is simply a square net with shorter vectors.

Not a net at all. All nodes do not have the same environment.

Rectangular

Five 2D nets

NANO 106 - Crystallography of Materials by Shyue Ping Ong - Lecture 1

p: primitivec: centered(lower case for 2D)

International symbols for 2D nets

Oblique Square HexagonalFour 2D crystal systems

m: monoclinico: orthorhombict: tetragonalh: hexagonal

Unit Cells

NANO 106 - Crystallography of Materials by Shyue Ping Ong - Lecture 1

¡ A unit cell is simply a geometric unit that can be stacked infinitely to reproduce the entire lattice.¡ Primitive unit cells – only 1 lattice point in the cell¡ Non-primitive cells – more than 1 lattice point in the cell

• Which of these cells are primitive?

• How many lattice points are there in each cell?

• Which cell(s) reflect the full symmetry of the net?

The Wigner-Seitz Cell¡ The Wigner-Seitz (WS) cell around a lattice point is

defined as the locus of points in space that are closer to that lattice point than to any of the other lattice points.

NANO 106 - Crystallography of Materials by Shyue Ping Ong - Lecture 1

The WS cell• is a unit cell, i.e., tiles

space to reproduce the lattice;

• can have more than 4 sides in 2D and more than 6 sides in 3D;

• Preserves symmetry of net/lattice.