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Home , Amount of substance

PHYSICS of Materials (PHYS132)

Dr Sergey Burdin (Oliver Lodge, Room 314, [email protected]) (based on earlier lectures by Prof. R. McGrath and Dr Joost Vossebeld)

7 topics (13 lectures) • Introduction: definitions, structure (2) • inter-atomic forces (2) • Thermal properties: States of , latent heat, thermal expansion (3) • Mechanical properties: elasticity (1) • Magnetic properties (2) • Electrical properties: band theory, semi-conductors (2) • Optical properties: colour (1)

3 question sheets: 10% of final mark for satisfactory submission. Fairly difficult but good practice for exam.

6 weeks × 3 hours – 1 = 17 hours (13 lectures + 3 problem sessions + 1)

1. Introduction to Materials > 1.1 Definitions 1 PHYSICS of Materials PHYS132

Books: (recommended, not required)

Fundamentals of Physics (Haliday, Resnick, Walker): Electrical and magnetic properties of matter

Properties of Matter (Flowers and Mendoza): Structure, interatomic potentials, thermal properties, mechanical properties

Gases, Liquids and Solids (Tabor): Structure, interatomic potentials, thermal properties

Properties of Materials (White): Optical properties

1. Introduction to Materials > 1.1 Definitions 2 PHYSICS of Materials PHYS132 General approach: Start with: MICROSCOPIC properties of matter: Structure of materials. Forces between , , . (Atomic Physics & )

To arrive at: MACROSCOPIC properties of matter: , elasticity, latent heat, conductivity, etc. (Material science, Engineering, Applied physics)

1. Introduction to Materials > 1.1 Definitions 3 Section 1. Introduction to materials Topic 1.1 Definitions

Atoms, Molecules, Ions.

Mole, Molar ,

Few example calculations

§1. Introduction to Materials > 1.1 Definitions 4 Atoms, ions and molecules Macroscopic matter is made up of assemblies of atoms, ions and molecules He ATOMS n n The smallest particle of an ELEMENT consists of p p NUCLEUS (Z Protons + N Neutrons) surrounded by Z . Electrically neutral. e.g. Cu, H, Ne. MOLECULES The smallest particle of a compound consists of a combination of atoms with some rearrangement of

the electrons. Electrically neutral e.g. CuCl2, H2 etc. IONS He + Atoms (or combinations of atoms) which have n lost or gained one or more electrons charged pnp e.g. Cu++, H+, OH–

1. Introduction to Materials > 1.1 Definitions 5 : The standard (SI) unit of AMOUNT OF SUBSTANCE (Symbol mol)

Definition 1 mol ≡ the amount of substance that contains as many elementary entities (atoms, molecules, etc.) as there are atoms in 0.012kg of carbon-12 (12C)

There are 6.023x1023 atoms in 0.012 kg of 12C.

23 -1 => NA=6.023x10 mol (≡ number of atoms/molecules in a mole)

Mole: the amount of a substance which contains 6.023x1023 elementary entities. The elementary entities may be atoms, molecules, ions, etc., and must be specified.

§1. Introduction to Materials > 1.1 Definitions 6 Examples • The elementary entities must be specified – 1 mol of (molecules) contains 2 mol of

hydrogen atoms (hydrogen : H2 (2 atoms)) • But 1 mol of argon contains 1 mol of argon atoms (argon gas is monatomic: Ar) • Utility of moles – in chemistry

•2H2 + O2 Æ 2H2O ≡ “2 mol of hydrogen plus 1 mol of yields 2 mol of water”

1. Introduction to Materials > 1.1 Definitions 7 : the mass of a mole of a substance. Symbol: M, unit: kg.

Relative atomic (molecular) mass: The ratio of the mass of one (molecule) of an element (compound) to 1/12 of the mass of one atom of 12C Unit ≡ number of atomic mass units (a.m.u. or u) 1 u =

Symbol: Ar (or Mr for molecules)

For a compound Mr = From above definitions M =

(i.e. Ar or Mr gives you the number of per mole!)

Ar =1 u ⇒ M = 1 g 1. Introduction to Materials > 1.1 Definitions 8 Examples

What is the mass of one atom of 12C?

Relative atomic mass:

Ar =

Μolar mass: M =

Ηence the mass of one atom: 12 u =

1. Introduction to Materials > 1.1 Definitions 9 Example: What is the occupied by one molecule of liquid water?

Molecule of water: H2O [Ar(H) ≈ 1u, Ar(O) ≈ 16 u] relative : Mr= ∴ Molar mass: M =

-3] [density of water: ρ = 1000 kg m V = Volume of one mole: 0

∴Vo =

Volume occupied by one molecule =

∴ each molecule occupies a cube of side ≡approximate separation between molecules.

1. Introduction to Materials > 1.1 Definitions 10 1.2: Structure: packing and order

1. Introduction to materials > 1.2 Structure: packing and order 11 PACKING AND ORDER IN SOLIDS AND LIQUIDS • Solids: atoms are essentially touching and in an arrangement that is constant with . In a CRYSTALLINE solid, atoms are in a long-range 3D arrangement called a “LATTICE”

• Liquids: atoms (molecules) are slightly further apart. There is some arrangement of atoms around any atom, but ……………………… ………………………………… …………………………………. • AMORPHOUS solids are similar to liquids but denser and constant or frozen in time. 1. Introduction to materials > 1.2 Structure: packing and order 12 Close packing: stacking sequence Consider atoms (as spheres) arranged in a plane. Most closely “packed” if they form a 2D hexagonal lattice.

If you now add a layer of atoms, where would they fit?

Atoms locate themselves in the hollow sites between 3 adjacent atoms.

1. Introduction to materials > 1.2 Structure: packing and order 13 Stacking sequence cont...

The second layer can be located in two ways (B or C) (Sofar the choices B or C are equivalent.) The crucial decision is where to place the third layer of atoms. If the second layer was placed with all the atoms at hollow sites B, the third layer can be placed at ………………… (i.e. directly over the atoms in layer A or not).

1. Introduction to materials > 1.2 Structure: packing and order 14 Stacking sequence cont.

• The Stacking sequence is the resulting sequence of labels identifying how the layers were laid down on top of each other. Two possible stacking sequences are:

1. Introduction to materials > 1.2 Structure: packing and order 15 Stacking sequences cont.

Hexagonal closed packed (hcp) A

(stacking sequence ABABAB) B

A

As the top hexagonal layer lies directly over the bottom layer, they are labelled with the same letter.

1. Introduction to materials > 1.2 Structure: packing and order 16 Stacking sequences cont. Face centred cubic (fcc) (stacking sequence ABCABC) Note ………………………………. …………………………………….. ……………………………………..

Diagram on the right shows that the fourth layer (A) coincides with the first on. Hence the stacking sequence is ABCABC. It can be seen that the corner atom occupies site A.

1. Introduction to materials > 1.2 Structure: packing and order 17 Some simple structures:

fcc hcp: hexagonal face centred cubic close packed

bcc body centred cubic

1. Introduction to materials > 1.2 Structure: packing and order 18 Simple crystal structures cont.

Two overlapping fcc sub-lattices!

sc simple cubic

1. Introduction to materials > 1.2 Structure: packing and order 19 Density of a crystalline solid

• Simple cubic structure (assume atoms are touching)

•a0 = lattice constant = … = …

3 • volume of cell (a0) number of atoms in cell = …

• Density ρ = mass of atom/ volume of cell

ρ =

(M = molar mass)

1. Introduction to materials > 1.2 Structure: packing and order 20 Packing fraction ≡ Volume of (touching) atoms accommodated by a unit cell divided by the volume of the cell itself is the packing fraction. Example: fcc Conventional unit cell seems to contain 14 atoms: However atoms are shared with neighbouring cells! • 8 “corner”-atoms, shared between 8 cells • 6 “face”-atoms shared between 2 cells Take: Volume cell = a3 Atoms touch along the side-diagonal of the cell, hence Radii of atomic spheres =

Total Volume of spheres = = =

Packing fraction =

1. Introduction to materials > 1.2 Structure: packing and order 21 Packing fraction Example: bcc Conventional unit cell seems to contain 9 atoms: • 1 central atom, not shared. • 8 corner atoms, shared between 8 cells.

Atoms touch along the body diagonal, which defines their radii; therefore:

Radii of atomic spheres =

Total Volume of spheres = = =

Packing fraction =

1. Introduction to materials > 1.2 Structure: packing and order 22