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Atkins & de Paula: Atkins’ Physical Chemistry 9e Chapter 19: Materials 2: Solids Chapter 19: Materials 2: Solids Crystallography 19.1 Lattices and unit cells space lattice, the pattern formed by points representing the locations of structural motifs (atoms, molecules, or groups of atoms, molecules, or ions). unit cell, an imaginary parellelepiped that contains one unit of a translationally repeating pattern. unit cell Chapter 19: Materials 2: Solids crystal system, a classification based on the rotational symmetry elements of a unit cell. essential symmetry, the elements a unit cell must possess to belong to a particular crystal system. cubic system monoclinic system triclinic system Trigonal Chapter 19: Materials 2: Solids Bravais lattice, the 14 distinct space lattices in three dimensions. primitive unit cell (P), formed by joining neighbouring lattice points by straight lines. body-centred unit cell (I), with lattice points at the corners and at the centre. face-centred unit cell (F), with lattice points at the corners and on each face. side-centred unit cell (A,B,C), with lattice points at the corners and on two opposite faces. Chapter 19: Materials 2: Solids 19.2 The identification of lattice planes Miller indices (hkl), indices that distinguish planes in a lattice. How to define Miller Indices 5. Negative directions are denoted with a bar on top of the number, e.g. 100 Chapter 19: Materials 2: Solids (110) (110) (230) (111) (110) (010) Chapter 19: Materials 2: Solids Some Examples Chapter 19: Materials 2: Solids Common crystallographic terms (hkl); parenthesis designate a crystal face or a family of planes throughout a crystal lattice. [uvw]; square brackets designate a direction in the lattice from the origin to a point. Used to collectively include all the faces of a crystals whose intersects (i.e., edges) parallel each other. These are referred to as crystallographic zones and they represent a direction in the crystal lattice. {hkl}; "squiggly" brackets or braces designate a set of faces that are equivalent by the symmetry of the crystal. [111] {111} Chapter 19: Materials 2: Solids separation of planes (d-spacing) 1 d hk2 0 h2 k 2 a2 or d hk 0 1 h2 k 2 l 2 2 d hkl a2 1 h2 k 2 l 2 2 2 2 2 d hkl a b c a (h 2 k 2 )1/ 2 or d hkl a (h 2 k 2 l 2 )1/ 2 Chapter 19: Materials 2: Solids 19.3 The investigation of structure 19.3(a) X-ray diffraction diffraction, interference caused by an object in the path of waves. diffraction pattern, the pattern of varying intensity that results from diffraction. Bremsstrahlung, X–radiation generated by the deceleration of electrons. K–radiation, X–radiation emitted when an electron falls into a K shell. Chapter 19: Materials 2: Solids four-circle diffractometer, a device used in X–ray crystallography. Chapter 19: Materials 2: Solids 19.3(b) Bragg’s law reflection, an intense beam arising from constructive interference. glancing angle, θ, the angle of incidence of a beam of radiation. Bragg’s law, λ = 2d sin θ. AB+BC = 2d sin θ nλ = 2d sin θ n = 1; first-order reflection 19.3(c) Scattering factors, f, a measure of the ability of an atom to diffract radiation sin kr 2 4 r dr, k sin 0 kr (r ) : electrondensitydistribution f 4 (r ) f; equal to the total # of e in the atom at θ=0 (Justification 19.1) Chapter 19: Materials 2: Solids 19.3(d) The electron density Structure factor, overall amplitude of a wave diffracted by the {hkl} planes. A ax a/h B A a 2d sin 2 sin 2ax sin hx; differencein path h hx phasedifference( ) 2 2hx A f Aeit f B ei (t ) I A ( f Ae it f B e i (t ) )( f Aeit f B ei (t ) ) f A2 f B2 2 f A f B cos 2 A f A f B e i Fhkl f j eihkl ( j ) where hkl ( j ) 2 (hxj ky j lz j ) j Example 19.2 Fhkl 4( f f ) for even h, k , l Fhkl 4( f f ) for odd h, k , l Fhkl 0 for other h, k , l Chapter 19: Materials 2: Solids systematic absences, reflections with values of h + k + l that are absent from the powder diffraction pattern. * I hkl Fhkl Fhkl ( f A f B e ihkl )( f A f B eihkl ) I hkl f A2 f B2 f A f B (eihkl e ihkl ) f A2 f B2 2 f A f B coshkl cubicI ; ( x, y, z ) ( 12 , 12 , 12 ) hkl (h k l ) no reflection from odd (h k l ) of cubicP Au Nanooctahedron Chapter 19: Materials 2: Solids Fourier synthesis, the construction of the electron density distribution from structure factors . 1 (r ) Fhkl e 2i ( hx kylz) V hkl phase problem, the ambiguity in phase of structure factors obtained from intensities. structure refinement, the adjustment of structural parameters to give the best fit between the observed intensities and those calculated from the model of the structure deduced from the diffraction pattern. Neutron and electron diffraction Chapter 19: Materials 2: Solids 19.5 Metallic solids 19.5(a) Close packing close-packed, a layer of spheres with maximum utilization of space. polytype, structures that are identical in two dimensions but differ in the third dimension. hexagonally close-packed (hcp), the sequence of layers ABABAB.... cubic close-packed (ccp), the sequence of layers ABCABC.... hcp ccp Chapter 19: Materials 2: Solids coordination number, the number of nearest neighbours. packing fraction, the fraction of space occupied by hard spheres. 19.5(b) Less closely packed structures; bcc (cubic I) & cubic P Chapter 19: Materials 2: Solids CCP Primitive Cubic Coordination Number 6 52% Packing Fraction Close-Packed (CCP or HCP) Coordination Number 12 74% Packing Fraction Body-Centered Cubic Coordination Number 8 68% Packing Fraction HCP Chapter 19: Materials 2: Solids 19.6 Ionic solids 19.6(a) Structure (n+,n–)–coordination, the number of nearest neighbours of opposite charge; n+ is the coordination number of the cation and n– that of the anion. . caesium-chloride structure, an ion of one charge occupies the centre of a cubic unit cell with eight counter ions at its corners: (8,8)–coordination rock-salt structure, of two interpenetrating slightly expanded fcc arrays, one of cations and the other of anions: (6,6)–coordination radius ratio, γ = rsmaller/rlarger. radius-ratio rule, a rule suggesting which type of structure is likely based on the radius ratio: γ < 0.414 (zinc blende); 0.414 < γ < 0.732 (rock salt); γ > 0.732 (caesium chloride). Chapter 19: Materials 2: Solids 19.6(b) Energies lattice energy, the difference in potential energy of ions packed together in a solid and widely separated as a gas. lattice enthalpy, ΔHL, the change in molar enthalpy for MX(s) Mz+(g) + Xz–(g). 1 z 2e 2 z 2e 2 z 2e 2 z 2e 2 E p (cation) 4 0 d 2d 3d 4d z 2e 2 1 1 1 z 2e 2 1 ln 2 4 0 d 2 3 4 4 0 d z 2e 2 E p (cation) 2 ln 2 4 0 d 1 z 2 N Ae 2 E p N A E p (cation) E p (anion) 2 ln 2 2 4 0 d For 3D array; E p A z A z B N Ae 2 4 0 d ( A : Made lu n gcon stan)t Chapter 19: Materials 2: Solids Born–Mayer equation, for the total potential energy of an ionic crystal. Born–Haber cycle, a closed path of transformations starting and ending at the same point, one step of which is the formation of the solid compound from a gas of widely separated ions. * d / d repulsivecontribution E p N AC e Minim untotal E , E p ,min * z A z B N Ae 2 4 0 d d* (1 ) A d Chapter 19: Materials 2: Solids 19.7 Molecular solids and covalent networks covalent network solid, a solid in which covalent bonds in a definite spatial orientation link the atoms in a network extending through the crystal. molecular solid, a solid consisting of discrete molecules held together by van der Waals interactions. Graphite Diamond Ice (molecular solid) CNT Chapter 19: Materials 2: Solids Impact on Nanotechnology; CNTs Strong & light: 100 times stronger than steel but 1/6 as heavy. High electrical & thermal conductivities: far better than Cu. [CVD growth] [MWCNTs] [SWCNTs] metallic semiconducting [Nanotube field-effect transistor] [Electrical properties of CNTs] [Mechanical properties of CNTs] Chapter 19: Materials 2: Solids I19.1 X-ray crystallography of biological macromolecules DNA Proteins TLR3 Chapter 19: Materials 2: Solids THE PROPERTIES OF SOLIDS 19.8 Mechanical properties stress, the applied force divided by the area to which it is applied. strain, the distortion of a sample resulting from an applied stress. rheology, the study of the relation between stress and strain. uniaxial stress, stress applied in one direction. hydrostatic stress, a stress applied simultaneously in all directions. pure shear, a stress that tends to push opposite faces of the sample in opposite directions. uniaxial stress shear stress hydrostatic stress Chapter 19: Materials 2: Solids elastic deformation, a deformation that disappears when the stress is removed. plastic strain, a strain from which recovery does not occur when the stress is removed. Young’s modulus, E = (normal stress)/(normal strain). bulk modulus, K = pressure/(fractional change in volume). shear modulus, G = (shear stress)/(shear strain). Poisson’s ratio, vP = (transverse strain)/(normal strain). Chapter 19: Materials 2: Solids 19.9 Electrical properties metallic conductor, a substance with an electrical conductivity that decreases as the temperature is raised. semiconductor, a substance with an electrical conductivity that increases as the temperature is raised. insulator, a semiconductor with a very low electrical conductivity. superconductor, a solid that conducts electricity without resistance. Chapter 19: Materials 2: Solids 19.9(a) The formation of bands nearly-free-electron approximation, a model of a metal in which the valence electrons are assumed to be trapped in a box with a periodic potential. tight-binding approximation, a model of a metal in which the valence electrons are assumed to occupy molecular orbitals delocalized throughout the solid. s- and p-bands, a band formed from overlap of s- and p-orbitals, respectively. band gap, a range of energies to which no orbital corresponds. From Hückel secular determinant k Ek 2 cos k 1,2,..., N N 1 E N E1 4 as N Chapter 19: Materials 2: Solids 19.9(b) The occupation of orbitals Fermi level, the highest occupied molecular orbital in a solid at T = 0. Fermi–Dirac distribution, P = 1/(e(E – μ)/kT + 1); a version of Boltzmann distribution that takes into account the effect of the Pauli principle. Chapter 19: Materials 2: Solids 19.9(c) Insulators and semiconductors valence band, a filled band in a solid. conduction band, an empty band in a solid. intrinsic semiconductor, where semiconduction is a property of the pure material. compound semiconductor, an intrinsic semiconductor being a compound of different elements. extrinsic semiconductor, becomes semiconducting when it is doped with other atoms. dopant, introduced atoms. p- and n-type semiconductivity, conduction by holes and particles, respectively. p–n junction, a junction between p- and n-type semiconductors. p-type n-type Reverse bias Forward bias Chapter 19: Materials 2: Solids 19.10(a) Light absorption by excitions in molecular solids exciton, an electron–hole pair. Frenkel exciton, the electron and hole jump together from molecule to molecule. Wannier exciton, the electron and hole are on different but nearby molecules. exciton bands, the structure of an absorption spectrum due to exciton formation: there are N exciton bands when there are N molecules in each unit cell Davydov splitting, the splitting between exciton bands. Chapter 19: Materials 2: Solids 19.10(b) Light absorption by metals and semiconductors 19.10(c) Nonlinear optical phenomena frequency doubling (or second harmonic generation), the process in which an intense laser beam is converted to radiation with twice its initial frequency as it passes though a suitable material. optical Kerr effect, the change in refractive index of a well chosen medium (Kerr medium) when it is exposed to intense electric fields . Kerr lens, the self-focussing of the laser beam by using the Kerr effect. frequency doubling 1 2 E E 2 : hyperpolarizability 1 2 E 2 E 0 2 cos2 t E 2 (1 cos 2t ) Chapter 19: Materials 2: Solids 19.11 Magnetic properties 19.11(a) Magnetic susceptibility magnetization, the magnetic dipole moment density, M = χH. volume magnetic susceptibility, the proportionality constant χ. molar magnetic susceptibility, χm = χVm. magnetic flux density, B = μ0(H + M) = μ0(1 + χ) H. paramagnetic, a material for which χ is positive. diamagnetic, a material for which χ is negative. magnetizability, ξ, a measure of the extent to which a magnetic dipole moment may be induced in a molecule. Curie law, χm = A + C/T, A = NAμ0ξ and C = NAμ0m2/3k. Gouy balance, a device for determining the magnetic susceptibility of a sample. superconducting quantum interference device (SQUID), a superconducting device for determining the magnetic susceptibility of a sample. Gouy balance Chapter 19: Materials 2: Solids 19.11(b) The permanent magnetic moment ferromagnetism, strong, persistent magnetization arising from the cooperative alignment of spins. antiferromagnetic phase, a phase in which spins are locked into a low–magnetization arrangement. Curie temperature, the temperature of a ferromagnetic transition. Néel temperature, the temperature of an antiferromagnetic transition. Temperature–independent paramagnetism (TIP), orbital paramagnetism. paramagnetic ferromagnetic antiferromagnetic Chapter 19: Materials 2: Solids 19.12 Superconductors superconductor, a substance that conducts electricity without resistance. high-temperature superconductor (HTSC), a substance that is superconducting at relatively high temperatures. Type I superconductor, a superconductor that shows an abrupt loss of superconductivity when exposed to a magnetic field above a critical value; completely diamagnetic below Hc Meissner effect, the exclusion of a magnetic field from a superconductor. Type II superconductor, a superconductor that shows a gradual loss of superconductivity when exposed to a magnetic field. Cooper pair, a pair of electrons that exists as a result of interactions with the lattice. YBa2Cu3O7 Cooper pair