MATERIALS SCIENCE Data Book This booklet is for use in the written examination

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1 Natural Sciences Tripos Part IA MATERIALS SCIENCE Data Book This booklet is for use in the written examination IA

2 DB1 IA Materials Science - Data Book DB1 Contents Crystal Structures, Planes and Directions 4 Diffraction 6 Polymers 7 Optics of Anisotropic Materials 7 Dielectrics 7 Ferroelectrics, Pyroelectrics & Piezoelectrics 8 Magnetism 8 Diffusion & Electrochemistry 9 Phase Diagrams 10 Mechanical Behaviour of Solids 14 Selected Property Data 16 Materials Selection Maps 17 Periodic Table of the Elements 23

3 DB2 IA Materials Science - Data Book DB2 Nomenclature a unit-cell edge length, along x axis (m) A cross-sectional area (m 2 ) b Burgers vector (m) unit-cell edge length, along y axis (m) B magnetic flux density (T) c crack length (m) unit-cell edge length, along z axis (m) speed of light (m s -1 ) (= in free space) C capacitance (F C V 1 ) composition (wt% or at%) concentration (atoms m -3 ) specific heat capacity (J mol 1 K 1 ) d diameter or linear dimension (m) piezoelectric coefficient (C N 1 ) D diffusivity or diffusion coefficient (m 2 s 1 ) displacement field (C m 2 ) director vector in a liquid crystal (m) e electronic charge (C) (= ) E electric field (V m -1 ) electrochemical cell potential (V) Young s modulus (Pa N m 2 ) f atomic scattering factor/atomic form factor (-) fibre volume fraction (-) fraction of atoms (-) F Faraday s constant (C mol 1 ) (= 96,500) force (N) force per unit length (N m 1 ) structure factor (-) g acceleration due to gravity (m s 2 ) (= 9.81) G Gibbs free energy (J mol 1 ) strain energy release rate (J m 2 ) shear modulus (Pa N m 2 ) h Miller index (x axis intercept = a/h) (-) H enthalpy (J mol 1 ) applied magnetic field (A m -1 ) I current (A C s 1 ) nucleation frequency (m -3 s 1 ) relative intensity of (diffracted) X-rays (-) second moment of area (m 4 ) j current density (A m 2 ) J atomic flux (m 2 s 1 ) k Boltzmann constant (J K 1 molecule 1 ) (= ) Miller index (y axis intercept = b/k) (-) K stress intensity factor (Pa m 1/2 ) l Miller index (z axis intercept = c/l)) (-) polymer segment length (m) L distance or length (m) latent heat for a phase change (J mol -1 ) m mass (kg) multiplicity of a set of lattice planes (-) magnetic moment (A m 2 ) Symbols M bending moment (N m) magnetization (A m -1 ) n integer number (-) number per unit volume (m 3 ) refractive index (-) N A Avogadro s number (mol 1 ) (= ) N number (-) p probability density (m 3 ) pyroelectric coefficient (C m 2 K 1 ) P polarisation (dipole moment / unit vol.)(c m -2 ) pressure (Pa N m 2 ) probability (-) q charge (C) heat energy (J mol 1 ) Q activation energy (J mol 1 ) total charge (C) chemical reaction quotient (-) order parameter (-) r radial distance or radius (m) R electrical resistance () gas constant (J K 1 mol 1 ) (= 8.314) ionic radius (m) radius of curvature (m) s aspect ratio (-) S entropy (J K 1 mol 1 ) t thickness (m) time (s) T temperature (K or C) measured radial distribution function (-) U internal energy (J mol 1 ) strain energy per unit volume (J m 3 ) x-axis component of UVW vector ( a) (-) v velocity (m s -1 ) V volume (m 3 ) electrical potential (V) y-axis component of UVW vector ( b) (-) w mechanical work energy (J mol 1 ) width (m) W energy (J) number of configurations (-) z-axis component of UVW vector ( c) (-) x distance in axial direction (m) X composition (mole fraction) (-) y distance in transverse direction (m) z distance in transverse direction (m) charge number (-) Z coordination number (-)

4 DB3 IA Materials Science - Data Book DB3 thermal expansion coefficient (K 1 ) unit-cell inter-axial angle (radians or ) unit-cell inter-axial angle (rad. or ) shear strain (-) surface energy (J m 2 ) unit cell inter-axial angle (rad. or ) deflection (m) phase difference (radians or ) normal strain (-) permittivity (F m 1 ) ( in free space) viscosity (Pa s) diffraction angle (rad. or ) inter-polyhedral angle (M-O-M) (rad. or ) angle between light/surface normal (rad. or ) curvature (m 1 ) dielectric constant (relative permittivity) (-) angle (stress axis slip direction) (rad. or ) extension ratio (new length / orig. length) (-) wavelength (m) line tension or energy / unit length (N J m 1 ) electrical dipole moment (C m) magnetic permeability (H m -1 ) (410-7 in free space) Poisson ratio (-) density (kg m 3 ) dislocation density (m 2 ) resistivity ( m) conductivity (S m 1 1 m 1 ) normal stress (Pa N m 2 ) beam stiffness or flexural rigidity (N m 2 ) shear stress (Pa N m 2 ) angle (stress axis slip plane normal) (rad. or ) phase angle for X-ray scattering (rad. or ) phase lag between stress and strain (rad. or ) magnetic susceptibility (-) interaction parameter (J mol 1 ) number of configurations (-) 0 far field / base level / free space / vacuum etc 1 first / in direction 1 etc a atomic A activation / Avogadro c coercive / critical / configurational / crosslinks C coordination / Curie d drift / drag e equilibrium eff effective E eutectic / eutectoid hkl for hkl planes i interfacial f fibre / fictive g glass transition / growth m melting max maximum mix mixing MM mechanical mixture Subscripts n nucleation r in radial direction / remanent / relaxation p plastic P at constant pressure s saturation / solution ss solid solution T at constant temperature v per unit volume V at constant volume Y yield x in x direction X cation y in y direction z in z direction Z anion * fracture or failure in hoop direction remote / at infinity / infinite frequency (hkl) {hkl} [UVW] UVW a UVW n Bracket Conventions lattice plane lattice planes of type (hkl) lattice direction lattice directions of type [UVW] cubic lattice vector

5 DB4 IA Materials Science - Data Book DB4 Crystal Structures, Planes and Directions Miller Indices Lattice Types and Lattice Point Locations Lattice Type Lattice Point Coordinates Lattice Type Lattice Point Coordinates P 0,0,0 A 0,0,0; 0, 1 / 2, 1 / 2 I 0,0,0; 1 /2, 1 / 2, 1 / 2 B 0,0,0; 1 /2, 0, 1 / 2 F 0,0,0; 1 /2, 1 / 2, 0; 1 /2, 0, 1 / 2 ; 0, 1 / 2, 1 / 2 C 0,0,0; 1 /2, 1 / 2, 0 Structure Types Name Formula System Lattice Motif hcp - hexagonal P 0,0,0; 2 /3, 1 / 3, 1 / 2 bcc - cubic I 0,0,0 ccp - cubic F 0,0,0 diamond C cubic F C: 0,0,0; 1 / 4, 1 / 4, 1 / 4 caesium chloride CsCl cubic P Cl: 0,0,0; Cs: 1 / 2, 1 / 2, 1 / 2 sodium chloride NaCl cubic F Cl: 0,0,0; Na: 0,0, 1 / 2 zinc blende ZnS cubic F S: 0,0,0; Zn: 1 / 4, 1 / 4, 1 / 4 wurtzite ZnS hexagonal P S: 0,0,0; 2 / 3, 1 / 3, 1 / 2 Zn: 0,0, 1 / 2 +u; 2 /3, 1 / 3, u (u 1 / 8 ) nickel arsenide NiAs hexagonal P As: 0,0,0; 2 /3, 1 / 3, 1 / 2 Ni: 1 / 3, 2 / 3, 1 / 4 ; 1 /3, 2 / 3, 3 / 4 fluorite CaF 2 cubic F Ca: 0,0,0; F: ±( 1 / 4, 1 / 4, 1 / 4 ) rutile TiO 2 tetragonal P Ti: 0,0,0; 1 / 2, 1 / 2, 1 / 2 O: ± (u,u,0); ±( 1 / 2 +u, 1 / 2 u, 1 / 2 ) (u 0.30 ) perovskite CaTiO 3 cubic P Ti: 0,0,0; Ca: 1 / 2, 1 / 2, 1 / 2 O: 1 / 2,0,0; 0, 1 / 2,0; 0,0, 1 / 2

6 DB5 IA Materials Science - Data Book DB5 Crystal Structures and Lattice Parameters of some Common Metals (at Room Temperature) ccp structures hcp structures bcc structures metal a (Å) metal a (Å) c (Å) metal a (Å) Ag 4.09 Cd Cr 2.88 Al 4.05 Co Fe () 2.87 Au 4.08 Mg K 5.33 Cu 3.61 Zn Na 4.29 Ni 3.52 Crystal Systems, Lattices and Symmetry Elements Crystal System Defining Symmetry (rotation or inversion) Conventional Unit Cell Cubic 4 triads a = b = c = = = 90 Hexagonal 1 hexad a = b c = = 90, = 120 Trigonal 1 triad a = b c = = 90, = 120 Tetragonal 1 tetrad a = b c = = = 90 Orthorhombic 3 diads a b c = = = 90 Monoclinic 1 diad a b c = = 90, 90 Triclinic - a b c Conventional Lattice Types P, I, F P P (R) P, I P, C, I, F P, C P Weiss Zone Law If the direction (zone axis) [UVW] lies in the plane (hkl), then: hu kv lw 0 Angle between Pairs of Directions or Plane Normals The angle between [U 1 V 1 W 1 ] and [U 2 V 2 W 2 ] (or between the normals to (h 1 k 1 l 1 ) and (h 2 k 2 l 2 )), in a cubic system, is given by cos U U VV WW U V W U V W cos h h k k l l h k l h k l

7 DB6 IA Materials Science - Data Book DB6 Diffraction The Bragg Equation 2 sin d hkl Spacings between Lattice Planes, for Crystals with Orthogonal Axes (Cubic, Tetragonal and Orthorhombic) h k l dhkl a b c Structure Factor F f cos2 hx ky lz isin 2 hx ky lz hkl n n n n n n n n Relative Intensity of Diffraction Maxima for Randomly Oriented Crystallites 2 2 * hkl hkl hkl hkl hkl n cos2 n n n n sin2 n n n n n I m F F m f hx ky lz f hx ky lz Systematic Absences Lattice Type Absent reflections Lattice Type Absent reflections P None A (k + l) odd I (h + k + l) odd B (h + l) odd F h, k, l mixed odd and even C (h + k) odd X-ray Wavelengths (in Å) Element K l K 2 Mean K K absorption edge Mo Cu Co Fe

8 DB7 IA Materials Science - Data Book DB7 Polymers Mean square of the end-to-end distance exhibited by a polymer chain, in terms of the number of Kuhn segments, and their length R nl 2 2 n Optics of Anisotropic Materials Speed of light in a medium of refractive index n c 0 c n Optical path difference, for a specimen of thickness t, made of a material of birefringence ( n 1 n 2 ) o.p.d. t n n 1 2 Phase difference between slow and fast rays emerging from a birefringent material 2 t n n 1 2 Dielectrics Capacitance in terms of Charge and Potential C Q V Capacitance of Two Parallel Plates, without and with a Dielectric Material between them C A L A L 0 C 0 Charge Density (Polarisation) on the Surface of a Dielectric Material P 0E 1 A L Dielectric Constants (Relative Permittivities) at room temperature Material Material Air ~1 Polyvinylidene 8-13 fluoride (PVDF) Polystyrene 2-3 Inorganic glasses 6-20 Alumina Distilled water 80 Barium titanate up to 7000

9 DB8 IA Materials Science - Data Book DB8 Ferroelectrics, Pyroelectrics & Piezoelectrics ΔP pδ T P d Pyroelectric (p) and Piezoelectric (d) Coefficients at room temperature Material Formula p (C m 2 K 1 ) d (pc N 1 ) Barium titanate BaTiO Lead zirconate titanate (PZT) Pb (Zr x Ti 1-x )O to to 200 Zinc oxide ZnO Polyvinylidene fluoride (CH 2 CF 2 ) n 40 to Curie Temperatures (T C ) and Spontaneous Polarisations (P s ) of Ferroelectrics Material Formula T C ( C) P s (C cm -2 ) at {T} C Perovskites Barium titanate BaTiO {23} Potassium niobate KNbO {250} Lead titanate PbTiO >50 {23} Salts Potassium dihydrogen phosphate KH 2 PO {177} Potassium nitrate KNO to {121} Rochelle salt NaKC 4 H 4 O 6.4H 2 O {5} Polymers PVDF (CH 2 CF 2 ) n ~200 6 {20} Magnetism Magnetisation (defined as magnetic moment m per unit volume), in an applied field M H Resulting magnetic flux density B H H M 0

10 DB9 IA Materials Science - Data Book DB9 Diffusion & Electrochemistry Atomic and Ionic Diffusion Arrhenius expression for the diffusion coefficient (diffusivity) D D exp Q 0 RT Fick s first law expressed as a current density from an ionic flux down a concentration gradient j qd n x Current density down a potential gradient (Ohm s law) j dv dx E Relationship between conductivity and diffusivity (Nernst-Einstein equation) 2 nq D kt Electrochemical cell potential (open-circuit voltage) across an oxygen ion conductor being used to measure the difference in partial pressures of oxygen gas E RT ln po2 I zf p O II where p O 2 I is the product, and O2 II 2 p is the reactant ( z = 4)

11 DB10 IA Materials Science - Data Book DB10 Phase Diagrams The Pb - Sn phase diagram The Al - Si phase diagram

12 DB11 IA Materials Science - Data Book DB11 The Cu - Ni phase diagram The Al - Cu phase diagram

13 DB12 IA Materials Science - Data Book DB12 The Al - Zn phase diagram The Cu - Zn phase diagram

14 DB13 IA Materials Science - Data Book DB13 The Fe - C phase diagram

15 DB14 IA Materials Science - Data Book DB14 Mechanical Behaviour of Solids Stress-Strain Relationships for Elastic Deformation of Isotropic Materials E G where is the normal stress, is the normal strain, E is the Young s modulus, is the shear stress, is the shear strain and G is the shear modulus. Stored Elastic Strain Energy Elastic strain energy per unit volume is given by W 1 2 Beam Bending 2 E (normal strain), W G (shear strain) The bending moment M, is related to the radius of curvature R, the curvature the second moment of area I, and the beam stiffness (=EI), via the relationships EI Σ M Σ R R The axial stress at a distance y from the neutral axis is given by Ey Ey R The second moments of area for a circular section, diameter d, and for a rectangular section, height h and width w, are given by I circle d wh, Irect Single Crystal Slip by Dislocation Glide The resolved shear stress on a slip plane, in a slip direction, during single crystal loading, is given by cos cos (cos cos is known as the Schmid factor) Slip systems ccp metals: <110> {111}; bcc metals: <111> {110} OILS rule This rule applies only to cubic crystals which slip on <110> {111} or <111> {110}. To determine the operative slip system (the one with the highest Schmid factor) in a specimen under uniaxial tension or compression: i) Ignoring the signs, identify the highest (H), intermediate (I) and lowest (L) valued indices of the loading axis [UVW]. ii) The <110> slip direction or {110} slip plane (whichever is appropriate) is the one with zero in the position of the I index and the signs of the other two indices preserved. iii) The {111} slip plane or <111> slip direction (whichever is appropriate) is the one with the L index Sign reversed and the signs of the other two indices preserved.

16 DB15 IA Materials Science - Data Book DB15 Dislocation Stresses and Energies The force per unit length on a dislocation line, due to an applied shear stress oriented parallel to the Burgers vector, is given by F b The Orowan bowing stress for by-passing obstacles is given by Gb L The stored elastic strain energy per unit length (for a screw dislocation), is given by Gb r Gb ln core energy 4 r0 2 Fracture Mechanics 2 2 For a thin plate containing a central through-thickness crack of length 2c, the stress at which brittle fracture will occur is given by EG c * c Fracture toughness (critical value of stress intensity factor) is related to Young s modulus and critical strain energy release rate by K c EG c

17 DB16 IA Materials Science - Data Book DB16 Selected Property Data Material Metals Density (Mg m -3 ) Melting Point T m ( C) Young s Modulus E (GPa) Yield Stress Y (MPa) Tensile Strength u (MPa) Fracture Energy G c (kj m -2 ) Pure aluminium High strength Al alloy (7075) Pure copper 8.9 1, Copper 40%zinc (- brass) Mild steel 7.8 1, High strength steel (TRIP) 7.8 1, Titanium alloy (Ti-6Al-4V) 4.5 1, Nickel superalloy (Hastalloy) 7.9 1, Tungsten , ,000 2, Ceramics Diamond 3.2-1,000 (50,000) Silicon carbide (SiC) (10,000) Alumina (Al 2 O 3 ) 3.9 2, (5,000) Polymers Polyethylene Nylon Polycarbonate Rubber Composites Wood (spruce, loaded // grain) Bone (compact, loaded //axis) Cermet (WC-Co) , Concrete (20) Carbon-fibre crossply NB Density and Young s modulus are well-defined properties and the values given should be reliable. Melting point data are complicated by the fact that some materials melt over a range of temperature and some do not melt properly at all they may sublime or become chemically degraded instead. The data given for yield stress, tensile strength and fracture energy are very approximate. These properties tend to be quite sensitive to microstructure, which in turn depends on processing conditions etc. In some cases, it is very difficult to induce macroscopic yielding: for example, the yield stress values given in brackets would often be impossible to attain during tensile loading, because brittle fracture would normally intervene. For brittle materials (those with low fracture energies), the tensile strengths may be dependent on specimen preparation, the presence of surface flaws etc.

18 DB17 IA Materials Science - Data Book DB17 Materials Selection Maps Granta Design (

19 DB18 IA Materials Science - Data Book DB18 Granta Design (

24 Periodic Table of the Elements 1 1 Atomic number 2 H H Chemical Symbol He Mass number s1 1s1 Outer electron configuration 1s Li Be B C N O F Ne s1 2s2 2s2 2p1 2s2 2p2 2s2 2p3 2s2 2p4 2s2 2p5 2s2 2p Na Mg Al Si P S Cl Ar s1 3s2 3s2 3p1 3s2 3p2 3s2 3p3 3s2 3p4 3s2 3p5 3s2 3p K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr s1 4s2 4s2 3d1 4s2 3d2 4s2 3d3 4s1 3d5 4s2 3d5 4s2 3d6 4s2 3d7 4s2 3d8 4s1 3d10 4s2 3d10 4s2 3d10 4p1 4s2 3d10 4p2 4s2 3d10 4p3 4s2 3d10 4p4 4s2 3d10 4p5 4s2 3d10 4p Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe s2 5s2 5s2 4d1 5s2 4d2 5s1 4d4 5s1 4d5 5s2 4d5 5s1 4d7 5s1 4d8 (5s0) 4d10 5s1 4d10 5s2 4d10 5s2 4d10 5p1 5s2 4d10 5p2 5s2 4d10 5p3 5s2 4d10 5p4 5s2 4d10 5p5 5s2 4d10 5p Cs Ba La Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn s1 6s2 6s2 4f1 6s2 4f14 5d2 6s2 4f14 5d3 6s2 4f14 5d4 6s2 4f14 5d5 6s2 4f14 5d6 6s2 4f14 5d7 6s2 4f14 5d8 6s2 4f14 5d9 6s2 4f14 5d10 6s2 4f14 5d10 6p1 6s2 4f14 5d10 6p2 6s2 4f14 5d10 6p3 6s2 4f14 5d10 6p4 6s2 4f14 5d10 6p5 6s2 4f14 5d10 6p Fr Ra Ac Rf Db Sg Bh Hs Mt Ds Rg Cn Uut Uuq Uup Uuh Uus Uuo s1 7s2 7s2 5f1 7s2 5f14 6d2 7s2 5f14 6d3 7s2 5f14 6d4 7s2 5f14 6d5 7s2 5f14 6d6 7s2 5f14 6d7 7s2 5f14 6d8 7s2 5f14 6d9 7s2 5f14 6d10 7s2 5f14 6d10 7p1 7s2 5f14 6d10 7p2 7s2 5f14 6d10 7p3 7s2 5f14 6d10 7p4 7s2 5f14 6d10 7p Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu s2 4f2 6s2 4f3 6s2 4f4 6s2 4f5 6s2 4f6 6s2 4f7 6s2 4f8 6s2 4f9 6s2 4f10 6s2 4f11 6s2 4f12 6s2 4f13 6s2 4f14 6s2 4f14 5d Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr s2 5f2 7s2 5f3 7s2 5f4 7s2 5f5 7s2 5f6 7s2 5f7 7s2 5f8 7s2 5f9 7s2 5f10 7s2 5f11 7s2 5f12 7s2 5f13 7s2 5f14 7s2 5f14 6d1