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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. => AVOGADRO CONSTANT NA=6.022x1023 mol-1 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 of the mass of one atom of 12C. Symbol: Ar (or Mr for molecules). Unit ≡ number of atomic mass units (a.m.u. or u) Absolute mass of 1 unit: 1u = 1 0.012 0.001 × = = 1.66 ×10-27 kg 23 23 12 6.023×10 6.023×10 Molar mass versus relative atomic (molecular) mass Molar volume: V0 = ( ) M = Ar ×10−3 kg/mol or Mr ×10−3 kg/mol M 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 • • • ρ a0 = lattice constant volume of cell (a0)3 Density = (mass of unit cell) / (volume of unit cell) = (effective number of atoms in unit cell) x (mass of atom) / (volume of unit cell) ρ= N eff ⋅ M N A a 30 M = molar mass Neff = effective number of atoms in unit cell NA= 6.022x1023 mol-1 –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 theory => F ∝ 1 r7 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 i) and ii) combined => F∝e or more conveniently F∝ −ra 1 n r where n = 10 – 13 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 Total force F(r)=FR(r)+FA (r) m (typically 13) > n n: 2 for ionic, > for VdW FR dominates at short distance (r<a) FA dominates at longer distances (r>a) At r = a, F(a)=0, I.e.when | FR |=| FA | a ≡ equilibrium separation 2. Interatomic Force Models > 2.1 Origins and potentials repulsion m const a FR = + m = A m repulsive force r FR=+const=+A r( ar) F rm a 0 (a)n attractive force F A=+const.Õ=-A attraction r rn const a FA = − n = B r r n 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 ∫ ∴ V(r) = − F(r)dr F(r) = − or ∞ dV(r ) dr • N.B. System changes to minimize V. • If a F (r ) = const r m then a V (r ) = const ' r m −1 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 V(r) = • A,B,m,n are constants. first term is repulsive second attractive. m>n • At equilibrium, dV =0 dr ∴ F =0 2. Interatomic Force Models > 2.1 Origins and potentials A B m − n r r V(r) very steep equilibrium separation less steep equilibrium potential energy 14 7 The Lennard-Jones 6-12 Potential (van der Waals solid) • A good approximation to the potential energy. a0 12 a0 6 V (r ) = ε − 2 r r (equivalent to Mie potential with m=12, n=6, A=εa012, B=2εa06) V(r) • At r = ∞ V(r) = 0 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 αe 2 V (r ) = + m − r 4πε 0 r 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, liquid and gas phases exist are shown as areas. Lines are boundaries and represent conditions where phase transitions take place TP: Triple Point : Solid, liquid, gas coexist CP: Critical Point : highest temperature at which liquid can exist Changes at constant P: isobars Changes at constant T: isotherms Examples: 3 isotherm line α, β and γ α GAS -> SOLID (below triple point temperature, sublimation) β GAS -> LIQUID -> SOLID γ GAS -> SOLID (above critical temperature, no liquid phase) 3. Thermal Properties > 3.1 Thermodynamics: states and phases α γ β P P SOLID SOLID CP CP TP TP LIQUID LIQUID GAS GAS T Brown: sublimation curve Blue: melting/fusion curve Red: vaporization/condensation curve (also vapour pressure) 20 10 P-V Phase Diagram T information can be included by ISOTHERMS GAS S+L S+G Critical point curve (fixed T) α L L+G Triple point line (fixed T) SOLID ISOTHERMS: α: compression of gas (ideal gas: P ∝ 1/V) ⇒ solidification (g+s) ⇒ compression of solid β: compression of gas ⇒ liquefaction/condensation (g+l) ⇒ compression of liquid ⇒ solidification/fusion (l+s) ⇒ compression of solid γ: compression of gas ⇒ solidification/sublimation (s+g) ⇒ compression of solid β SOLID+GAS γ 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) • thermal energy dominant (high T or low P) • both important (intermediate T & P) 3. Thermal Properties > 3.1 Thermodynamics: states and phases => SOLID => GAS => LIQUID 22 11 Critical temperature (For the L-J 6-12 van der Waals potential) a0 12 a0 6 V (r ) = ε − 2 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 ∞) • THERMAL ENERGY ≡ kBT (Boltzmann’s constant: kB = 1.38×10–23 J/deg) when THERMAL ENERGY > BINDING ENERGY molecules do not stay together ∴ substance does not liquefy irrespective of pressure applied ∴ at CRITICAL TEMPERATURE k BTC ≅ ε ∴ TC ≅ 3 Thermal Properties > 3.2 Critical temperature and latent heat ε kB 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 MOLAR LATENT HEAT L0 = 1 nN Aε 2 ∴ 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) Coefficient of linear expansion Coefficient of volume expansion ∆L = α ∆T L β = 3α ∆V = β ∆T V MICROSCOPICALLY (QUALITATIVE) Thermal expansion arises from increasing atomic vibration through increasing temperature in combination with the fact that the V(r) curve is not symmetric ∴ increasing temperature ⇒ increasing mean separation 3. Thermal properties > 3.3 Thermal expansion +ε V(r) 0 a0 r Increasing T -ε T=0 ∴ anharmonicity ⇒ 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): F A F/A E= ∆L / L ∆x L G= F/A ∆x / L Bulk Modulus (K) V dV ∆P K = −V ∆V Connection with interatomic energy: 4 Mechanical Properties > 4.1 Elasticity ∴ K r = a0 d 2E d 2 E dr =V ≅V dV 2 dr 2 dV N.B. assumptions: 1) low temperatures 2) small changes from equilibrium 2 27 • External field B0 causes magnetisation of material r r r B = B0 + µ0 M • Resulting magnetic field in the material • 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) Curie law: M = C B0 T (C = Curie constant ) Complete alignment (“saturation”) ⇒ magnetic moment of sample: µmax = Nµatom where N is the number of atoms in sample 1 M/M max 0.5 Magnetization curve for paramagnetic materials Curie law saturation deviation from Curie law Curie law holds 1 3 2 B0 / T (T/K) 4 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 magnetization direction BM x→ • x =BM without B0 I.e. PERMANENT MAGNETISM or MAGNETIC MEMORY B0 • 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 R=ρ L 1 L = A σ A • METAL INSULATOR SEMICONDUCTOR (ρ = resistivity, σ = conductivity ) ρ ≈ 2 x 10-8 Ωm ρ ≈ 1 x 1016 Ωm ρ ≈ 3 x 103 Ωm e.g. Copper e.g. Diamond e.g. Silicon • Also the variation with temperature is very distinctive: α= 1 dρ ρ dT Copper : α ≈ 4 x10 −3 K −1 , Silicon : α ≈ − 7 x10 −2 K −1 • 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 wavefunctions 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 distribution of energy levels (arising from a single atomic energy level) is called an energy band. • Energy Example: electron levels in Na solid 3s: 1 electron 2p: 6 electrons It is the occupation of the outermost band with electrons that determines the electrical properties. 6. Electronic Properties > 6.1 Band theory isolated atom 2s: 2 electrons 1s: 2 electrons a0 Separation 32 16 Occupation Probability The probability of occupation of the states in a band is given by a function p(E). This is called the Fermi function. ( E − E F ) / kT p( E ) = 1 +1 e T = 0K P(E) 1 T > 0K P(E) 1 0.5 0 EF 0 E The highest occupied energy at T = 0K is called the Fermi level EF. EF E At T > 0K the Fermi function ‘smears’ out. (At the Fermi level p(E) = ½) 6. Electronic Properties > 6.1 Band theory 33 Metals Energy Metal EF partially full 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 empty Energy conduction band EF valence band Egap >> kT Example diamond: Egap = 5.5eV ⇒ p(EC ) = 1 ( EC − EF ) / kT e +1 ≈ e−( EC −EF )/ kT =e full − Egap / 2kT ≈ 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: Egap = 1.1eV Energy semi conductor empty conduction band EF – + – + – + – + valence band 6. Electronic Properties > 6.1 Band theory Egap ~ eV full ⇒ p(EC ) = 1 ( EC − EF ) / kT e +1 −( EC − EF ) / kT ≈e =e − Egap / 2kT ≈ 6 ×10−10 at T = 300 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. • Gold does not reflect much high-energy visible light (blue and violet) because of an absence of energy levels in this region. Since gold predominantly reflects at the low-energy (longwavelength) end of the visible range, it appears yellow. silver Reflection 100% 300 gold λ (nm) 700 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