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Atoms, Ions and Molecules 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<a) R a FA dominates at longer distances (r>a) 0 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.
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