PHYSICS of Materials (PHYS132) Revision Session
N.B. Full course of lectures should be used in preparation for the exam 7 topics (13 lectures) • Introduction: definitions, structure (2) • inter-atomic forces (2) • Thermal properties: States of matter, latent heat, thermal expansion (3) • Mechanical properties: elasticity (1) • Magnetic properties (2) • Electrical properties: band theory, semi-conductors (2) • Optical properties: colour (1) 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
Atoms, ions and molecules Macroscopic matter is made up of assemblies of atoms, ions and molecules ATOMS The smallest particle of an ELEMENT consists of NUCLEUS (Z Protons + N Neutrons) surrounded by Z Electrons. 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 Atoms (or combinations of atoms) which have lost or gained one or more electrons. Charged e.g. Cu++, H+, OH–
1. Introduction to Materials > 1.1 Definitions 2
1 A mole is the amount of a substance that contains as many elementary entities (atoms, molecules, etc.) as there are atoms in 0.012kg of carbon-12 (12C)
Or equivalently: A mole is the amount of a substance that contains 6.022x1023 atoms/molecules.
23 -1 => AVOGADRO CONSTANT NA=6.022x10 mol
1. Introduction to Materials > 1.1 Definitions 3
Molar mass: the mass of a mole of a substance. Symbol: M, unit: kg/mol
Relative atomic (molecular) mass: The ratio of the mass of one atom (molecule) of an element (compound) to 1/12 12 of the mass of one atom of C. Symbol: Ar (or Mr for molecules). Unit ≡ number of atomic mass units (a.m.u. or u) 1 0.012 0.001 Absolute mass 1 u = × = =1.66×10-27 kg of 1 unit: 12 6.023×1023 6.023×1023
Molar mass versus relative −3 −3 atomic (molecular) mass M = Ar ×10 kg/mol (or Mr ×10 kg/mol) M Molar volume: V0 = where ρ is the density of a substance ρ
Volume occupied by one molecule = Vo /NA
1. Introduction to Materials > 1.1 Definitions 4
2 Close-packed structure Consider atoms (as spheres) arranged in a plane. Most closely “packed” if they form a 2D hexagonal lattice.
Atoms in the second layer locate themselves in the hollow sites between 3 adjacent atoms.
1. Introduction to materials > 1.2 Structure: packing and order 5
Stacking sequence cont...
The second layer can be located in two ways (B or C) If the second layer was placed with all the atoms at hollow sites B, the third layer can be placed at sites A (ABABAB stacking sequence, hexagonal close-packed structure) or sites C (ABCABC stacking sequence, face-centred cubic structure) 12 neighbours are touching any particular sphere in these close-packed structures. 1. Introduction to materials > 1.2 Structure: packing and order 6
3 Density of a crystalline solid
•a0 = lattice constant 3 • volume of cell (a0) •Density ρ = (mass of unit cell) / (volume of unit cell) = (effective number of atoms in unit cell) x (mass of atom) / (volume of unit cell)
Neff ⋅M NA ρ = 3 a 0
M = molar mass
Neff = effective number of atoms in unit cell 23 -1 NA= 6.022x10 mol –Avogadro's number
1. Introduction to materials > 1.2 Structure: packing and order 7
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 = √2 a/4 a Total Volume of spheres = [8x(1/8)+6x(1/2)] × [(4/3) π (√2 a/4 )3] = (√2 π a3)/6 effective number of atoms in unit cell Packing fraction = {(√2 π a3 )/6} / a3 = √2 π /6 = 0.74
1. Introduction to materials > 1.2 Structure: packing and order 8
4 Attractive Interatomic Forces • 4 main types: ionic, covalent, metallic and van der Waals
IONIC (e.g. Na+Cl–) In ionic materials there is a complete transfer of electrons between atoms => +ve and –ve ions => Coulomb forces (F ~ 1/r2): attractive or repulsive ∴ attractive forces dominate COVALENT (e.g. diamond) In covalent materials atoms share electrons. +ve nuclei are attracted to the electrons between them and hence to each other => attractive forces between atoms METALLIC (e.g. Na) Metals consist of a lattice of +ve ions in a gas of free electrons. attraction between ions and electrons (not unlike covalent bonds but electron are shared between many atoms) =>attractive force
2. Interatomic Force Models > 2.1 Origins and potentials 9
Attractive force cont.
VAN DER WAALS (e.g. Ar, N2, HCl ) VdW forces arise from an attraction between the dipole moment (either permanent or instantaneous) of one atom (molecule) and that (either permanent or induced) of another 1 theory => F ∝ r 7 occur between all types of atoms or molecules, including neutral* atoms or molecules.
(*when, although it is a weak force it is important because other attractive forces are absent)
2. Interatomic Force Models > 2.1 Origins and potentials 10
5 Repulsive Force • Has both ELECTROSTATIC and QUANTUM-MECHANICAL origins. As atoms get close together their electron clouds overlap.
(i) +ve nuclear charges no longer screened => Coulomb repulsion (ii) Electrons near to each-other cannot have the same quantum number (Pauli Exclusion Principle) ∴ cannot have the same energy and position ∴ some electrons have to change (increase) their energy => repulsion − r i) and ii) combined => F ∝ e a 1 or more convenientlyF ∝ where n = 10 – 13 r n
2. Interatomic Force Models > 2.1 Origins and potentials 11
Interatomic forces: Representation • Consider force F between an atom (or molecule) at the origin and another a distance r away. – require F → 0 as r →∞, ∴ F ∝ 1/rp
• Sign convention: repulsive force + ve ; attractive force – ve repulsion Total force F(r)=FR(r)+FA (r) m m (typically 13) > n F const a n: 2 for ionic, > for VdW FR = + m = A repulsive force FR=+const=+A( a)m r m r r r F dominates at short distance (r
At r = a, F(a)=0, I.e.when | FR |=| FA | attractive force FA=+const.Õ=-A(a)n attraction rn r a ≡ equilibrium separation n const a FA = − = B r n r 2. Interatomic Force Models > 2.1 Origins and potentials 12
6 Interatomic potential energies
• Instead of F(r) it is often more convenient to use V(r), the potential energy of the two atoms (molecules). •V(r) is the work done (on the system) in bringing one atom from ∞ to r (where V(∞)≡0), r dV(r) ∴ V(r) =− F(r)dr or F(r) =− ∫ dr ∞
• N.B. System changes to minimize V.
m m−1 a a • If F(r) = const then V (r) = const' r r
2. Interatomic Force Models > 2.1 Origins and potentials 13
The Mie potential • In 1907 Mie proposed a very simple form of the inter-atomic potential A B V(r) = − r m rn
V(r) • A,B,m,n are constants. very steep first term is repulsive second attractive. equilibrium m > n separation
• At equilibrium, less steep dV = 0 ∴ F = 0 equilibrium potential energy dr
2. Interatomic Force Models > 2.1 Origins and potentials 14
7 The Lennard-Jones 6-12 Potential (van der Waals solid) • A good approximation to the potential energy. 12 6 a a V (r) = ε 0 − 2 0 r r
12 6 (equivalent to Mie potential with m=12, n=6, A=εa0 , B=2εa0 ) • At r = ∞ V(r) = 0 V(r)
At r = a0 , (equilibrium separation):
V(r) = -ε, F = -dV/dr = 0
• The L-J 6-12 potential describes the interaction between two isolated atoms (molecules)
2. Interatomic Force Models > 2.1 Origins and potentials 15
L-J potential: in solids or liquids • The forces described by L-J 6-12 are very short range.
e.g. V(a0) = -ε, V(2a0) ~ -ε/30
∴ to a good approximation, only NEAREST NEIGHBOUR interactions need to be considered.
=> Concept of Coordination number n, the number of NEAREST NEIGHBOURS of any particular molecule • SOLIDS n has a definite value (dependent on structure) e.g. for close-packed structure (hcp or fcc) n = 12, for bcc n = 8 etc
• LIQUIDS n has an average value, typically 1 – 2 smaller than in solids
2. Interatomic Force Models > 2.1 Origins and potentials 16
8 Interatomic potential for ionic crystals ∴ Electrostatic potential energy per pair of ions in an ionic crystal equals p.e. of an isolated pair of neighbouring ions multiplied by the Madelung constant α. • Including the p.e. due to the short range repulsive forces the p.e. per pair of ions in the crystal as a function of nearest neighbour separation r A αe2 V(r) = + m − r 4πε 0r
α is the MADELUNG CONSTANT which depends on the particular arrangement of ions. (typicaly 1-2)
for 1D line α = 1.38; for 3D NaCl structure α = 1.75
2. Interatomic Force Models > 2.2 Interatomic potential for ionic crystal 17
Topic 3.1 Thermodynamic aspects of stability
• Solid: - definite volume and shape • Gas: - volume and shape dependent on the container • Liquid:- definite volume, shape dependent on the container This behaviour relates to: COMPRESSIBILITY (C) : response to an attempt to change the volume VISCOSITY (V) and RIGIDITY (R): response* to an attempt to change the shape (*N.B. the measurement time-scale is important) • Solid: - low C, infinite V, high R • Gas: - high C, low V, zero R • Liquid:- low C, intermediate V, low R These properties relate to the PACKING and ORDER of the atoms:- • Solid: close packing, long range order • Gas: dilute packing, no order • Liquid: close packing, short range order
3. Thermal Properties > 3.1 Thermodynamics: states and phases 18
9 States of Matter • The State of a substance depends on the values of P, V and T (pressure, volume and thermodynamic (i.e. Kelvin) temperature). • For a particular amount (e.g. 1 mole) of substance, in a particular state, these quantities are linked by an EQUATION of STATE (e.g. PV = RT for ideal gas) • Possible values of P,V and T form a surface in PVT space. Different regions correspond to different STATES • Boundaries between these regions correspond to transitions between STATES
• The links between PVT and STATE are usually displayed on P-T or P-V PHASE DIAGRAMS.
3. Thermal Properties > 3.1 Thermodynamics: states and phases 19
P-T Phase Diagram • The range of variables where the solid, α β γ liquid and gas phases exist are shown as areas. P • Lines are boundaries and represent P conditions where phase transitions take place SOLID SOLID • TP: Triple Point : Solid, liquid, gas coexist • CP: Critical Point : highest temperature at CP which liquid can exist CP
Changes at constant P: isobars LIQUID TP Changes at constant T: isotherms TP Examples: 3 isotherm line α, β and γ GAS α GAS -> SOLID (below triple point T temperature, sublimation) Brown: sublimation curve β GAS -> LIQUID -> SOLID Blue: melting/fusion curve Red: vaporization/condensation curve γ GAS -> SOLID (above critical (also vapour pressure) temperature, no liquid phase)
3. Thermal Properties > 3.1 Thermodynamics: states and phases 20
10 P-V Phase Diagram T information can be included by ISOTHERMS
ISOTHERMS: α: Critical point curve (fixed T)
compression of gas (ideal gas: P ∝ 1/V) S+G ⇒ solidification (g+s) ⇒ compression of solid β: GAS
compression of gas S+L ⇒ liquefaction/condensation (g+l) L α L+G ⇒ compression of liquid Triple point line ⇒ solidification/fusion (l+s) (fixed T)
⇒ compression of solid SOLID β γ: SOLID+GAS compression of gas γ ⇒ solidification/sublimation (s+g) ⇒ compression of solid
Notice in all (phase transition) regions of mixed phase, P constant along isotherm. 3. Thermal Properties > 3.1 Thermodynamics: states and phases 21
What determines the state of a system? (Microscopically) The balance between the interatomic (intermolecular) potential energy* and the kinetic energy+ (thermal energy) of the atoms (molecules) *depends on separation (i.e. P or V) + depends on temperature (T) • interatomic p.e. dominant (low T or high P) => SOLID • thermal energy dominant (high T or low P) => GAS • both important (intermediate T & P) => LIQUID
3. Thermal Properties > 3.1 Thermodynamics: states and phases 22
11 Critical temperature (For the L-J 6-12 van der Waals potential)
12 6 a a V (r) = ε 0 − 2 0 r r • -ε is the potential energy between two molecules at their equilibrium separation ∴ +ε is the BINDING ENERGY of the pair
≡ energy needed to separate them (to ∞)
–23 • THERMAL ENERGY ≡ kBT (Boltzmann’s constant: kB = 1.38×10 J/deg) when THERMAL ENERGY > BINDING ENERGY molecules do not stay together ∴ substance does not liquefy irrespective of pressure applied
kBTC ≅ ε ∴ at CRITICAL TEMPERATURE ε ∴TC ≅ kB 3 Thermal Properties > 3.2 Critical temperature and latent heat 23
Latent Heat (van der Waals solid/liquid)
• The MOLAR LATENT HEAT OF SUBLIMATION (or
VAPORISATION) is the energy required to change one mole (NA molecules) of a substance from solid (or liquid) to a gas.
• In 1 mole there are 1/2 n NA pairs. – factor of 1/2 ensures pairs are not counted twice, – n is coordination number (lecture 2.1). For sublimation use n for solid ; For vaporisation use n for liquid 1 MOLAR LATENT HEAT L = nN ε 0 2 A ∴ using assumptions that kinetic energy is small compared to potential energy (i.e. low temperatures) and only nearest neighbours are considered
For ionic crystal: L0 = N Aε
3 Thermal Properties > 3.2 Critical temperature and latent heat 24
12 Thermal expansion in solids MACROSCOPICALLY (HRW) ∆L = α ∆T Coefficient of linear expansion L β = 3α Coefficient of volume expansion ∆V = β ∆T V MICROSCOPICALLY (QUALITATIVE) +ε Thermal expansion arises from increasing atomic vibration through V(r) increasing temperature a0 in combination with 0 the fact that the V(r) curve is not r symmetric Increasing T ∴ increasing temperature -ε ⇒ T=0 increasing mean separation ∴ anharmonicity ⇒ thermal expansion 3. Thermal properties > 3.3 Thermal expansion 25
4: Mechanical Properties: Topic 4.1: Elasticity • Elasticity is a macroscopic property related to how the potential energy of (≡ forces between) the atoms changes as a result of changes in separation from equilibrium. Elasticity is the change in shape or volume of a body in response to external forces.
• For small changes (~0.1→ 1 %) the behaviour is REVERSIBLE (ELASTIC) and the original shape or volume is restored when the forces are removed.
• For very small changes: Change in shape or volume ∝ force producing it
Elastic modulus = Stress / Strain
Elastic modulus is a constant Strain = fractional change in relevant dimension Stress = (change in) pressure producing it
4 Mechanical Properties > 4.1 Elasticity 26
13 There are 3 main elastic moduli corresponding to different types of deformation: Shear Modulus (G) Young Modulus (E): FA ∆x F / A F / A G = E = L ∆x / L ∆L / L
Bulk Modulus (K)
V ∆P dV K = −V ∆V 2 Connection with d 2 E d 2 E dr interatomic energy: ∴Kr=a = V ≅ V 0 dV 2 dr 2 dV N.B. assumptions: 1) low temperatures
4 Mechanical Properties > 4.1 Elasticity 2) small changes from equilibrium 27
• External field B0 causes magnetisation of material r r r • Resulting magnetic field in the material B = B0 + µ0M • There are 3 main types of magnetic behaviour of materials – Diamagnetism (a very small effect exhibited by all materials) • the ORBITAL motion of all electrons changes in a way that gives
• Î very small magnetic moment opposite to B0, • Î B very slightly less than B0 – Paramagnetism (in atoms/ions of certain elements with unpaired electrons)
• The atom magnetic moments are aligned by external magnetic field B0 • Î small magnetic moment parallel to B0 • Î B slightly greater than B0 – Ferromagnetism (a few elements and their alloys which have non-zero atomic magnetic moments and exchange coupling) • Magnetic domains formed due to exchange coupling are aligned by external magnetic field B0
• Î large magnetic moment parallel to B0 • Î B greater than B0 – (effects can remain when B0 is taken back to 0 !)
5. Magnetic properties > 5.2: Magnetism in matter 28
14 Magnetization M = (magnetic moment of sample)/(volume of sample) (Units of M: (Am2)/m3 = Am-1) B Curie law: M = C 0 ()C = Curie constant T
Complete alignment (“saturation”) ⇒ magnetic moment of sample: µmax = Nµatom where N is the number of atoms in sample
Magnetization curve for paramagnetic materials 1 Curie law saturation M/Mmax 0.5 deviation from Curie law
Curie law holds
1 2 34
B0 / T (T/K)
5. Magnetic properties > 5.2: Magnetism in matter 29
Hysteresis • In FERROMAGNETIC materials the magnetization curve does not
retrace itself for B0 increasing then B0 decreasing. • This phenomenon results from the irreversibility of: – moving domain boundaries – changing domain BM magnetization direction x→
B0 •x =BM without B0 I.e. PERMANENT MAGNETISM or MAGNETIC MEMORY
• The above curve is called a HYSTERESIS LOOP – Broad loop means high reminiscent magnetisation (used for permananent magnets) – Narrow loop means low reminiscent magnetisation (used in transformers) 5. Magnetic properties > 5.2 Magnetism in matter 30
15 Section 6: Electronic properties Topic 6.1.1 The Band Theory of Conduction Conduction of Electricity in Solids • Conduction is measured by resistance R where for a block of area A, length L, the resistance (R = V/I) is given by L 1 L R = ρ = ()ρ = resistivity, σ = conductivity A σ A • METAL e.g. Copper ρ ≈ 2 x 10-8 Ωm INSULATOR e.g. Diamond ρ ≈ 1 x 1016 Ωm SEMICONDUCTOR e.g. Silicon ρ ≈ 3 x 103 Ωm
• Also the variation with temperature is very distinctive: 1 dρ α = Copper : α ≈ 4x10−3 K −1, Silicon : α ≈ − 7x10−2 K −1 ρ dT • Finally the density of charge carriers is very different Copper: n ≈ 9x1028 m-3, Silicon: n ≈ 1x1016 m-3
6. Electronic Properties > 6.1 Band theory 31
Energy Levels in a Crystal • In a solid (typically N ~ 1028 atoms/m3) we have many atoms and hence electron wave- functions brought close together, modifying the energy levels: • The most significant effect is at the least tightly bound energy levels where the overlap will be greatest. • Due to the Pauli Principle each electron must still occupy its own unique state, however these states have almost infinitesimally closely packed energy levels.
• The quasi-continuous Example: electron levels in Na distribution of energy isolated atom levels (arising from a solid single atomic energy level) is called an energy band. Energy 3s: 1 electron
• It is the occupation of 2p: 6 electrons the outermost band with electrons that 2s: 2 electrons determines the electrical properties. 1s: 2 electrons
a0 Separation 6. Electronic Properties > 6.1 Band theory 32
16 Occupation Probability The probability of occupation of the states in a band is given by a function p(E). 1 This is called the p(E) = Fermi function. e(E−EF )/ kT +1
T = 0K T > 0K P(E)1 P(E)1
0.5
0 0 EF E EF E The highest occupied energy at T = 0K At T > 0K the Fermi function ‘smears’ out. is called the Fermi level EF. (At the Fermi level p(E) = ½) 6. Electronic Properties > 6.1 Band theory 33
Metals
Metal
E partially F full Energy full
• In metals, the highest occupied energy levels lie in the middle of an energy band which is called the conduction band.
• There are therefore plenty of adjacent empty states almost infinitesimally higher energies for an electron to be raised to by the application of an electrical potential (or by heat) so that the Pauli principle poses no obstacle to conduction.
6. Electronic Properties > 6.1 Band theory 34
17 Insulators • When the levels in one band are exactly filled and the lowest energy level in the next band is at a considerably higher energy, a material is an insulator.
• This is because any externally applied electrical field would have to be very high to excite any electrons into the next available energy levels.
•Also, as the band-gap, Egap is much greater than the typical thermal excitation energy there will be no electrons thermally excited to the (empty) conduction band. (e.g. diamond Egap = 5.5eV) Insulator Example diamond: E = 5.5eV conduction band empty gap 1 ⇒ p(EC ) = ()E −E /kT e C F +1 E >> kT EF gap −()E −E /kT ≈ e C F Energy = e−Egap /2kT valence band full ≈ 6×10−47 at T = 300K
6. Electronic Properties > 6.1 Band theory 35
Semiconductors
• When Egap is near kT, we have a semiconductor.
• Although p(EC) may be small at room temperature, we can still have a significant number of electrons in the conduction band through thermal excitation. • These electrons (and the holes they leave in the top of the valence band!) explain the conduction properties of semiconductors.
• Because there is an exponential dependence of p(E) on temperature, the value of α is large and negative for semiconductors, i.e. better conductivity at higher T. Example silicon: E = 1.1eV semi conductor gap 1 ⇒ p(EC ) = ()E −E /kT conduction band empty e C F +1 −()E −E /kT ≈ e C F EF Egap ~ eV −Egap /2kT Energy +– +– +– +– = e valence band full ≈ 6×10−10 at T = 300
6. Electronic Properties > 6.1 Band theory 36
18 Difference in colour between metals
• The colours of silver and gold are quite distinct from one another.
• This is due to differences in the number of states above the Fermi edge.
• For silver there are plenty of states available to reflect photons of all wavelengths of visible light with high efficiency. silver 100% • Gold does not reflect much high-energy visible light (blue and violet) because of gold an absence of energy levels in this region. Since gold predominantly reflects at the low-energy (long- Reflection wavelength) end of the visible range, it 300 λ (nm) 700 appears yellow.
7. Optical Properties > 7.1 Colour 37
Colours of pure semiconductors • For pure semiconductors the colour depends on the energy gap Eg.
•If Egap< the lowest energy of visible light (λ = 700nm, red, E = 1.7eV) then any wavelength will be absorbed and the colour will be black or metallic (depending on the efficiency of re-radiation):
–Si (Egap= 1.1eV) ⇒ grey metallic
–GaAs(Egap= 1.4eV) ⇒ black
•If Egap > the highest energy of visible light (λ = 400nm, violet, E = 3eV), then no visible light is absorbed and the
material is colorless/transparent (e.g. diamond, Egap= 5.4eV).
•If Egap falls in the range of visible light, the colour is the complementary colour to the absorbed wavelengths.
–HgS(Egap= 2.1eV): λTRANSMITTED> 590nm ⇒ red HgS is a red pigment “Vermilion”.
7. Optical Properties > 7.1 Colour 38
19