Chapter 8 The structures and propertiesof metals and alloys8.1 Metallic bond and general properties of metals8.1.1 The "free-electron"model of metalA bulk of metal = valence electrons in free-motions& cationic cores;i)neglectingtheinteractionsbetweenvalenceelectronsandatomiccores;ili) neglecting the interactions between free electrons.A metal solid can be regarded as cationic cores floatinginaseaoffreeelectrons
Chapter 8 The structures and properties of metals and alloys 8.1 Metallic bond and general properties of metals 8.1.1 The “free-electron” model of metal i) A bulk of metal = valence electrons in free-motions & cationic cores; ii) neglecting the interactions between valence electrons and atomic cores; iii) neglecting the interactions between free electrons. A metal solid can be regarded as cationic cores floating in a sea of free electrons
Like an electron confined within a zero-potential 3D box, theSchrodinger equation for a“free electron"ish?H=↑+V---Hp=Ep;(ie.,V=0)8元mThus the behavior of free electrons can be described by a plane wavefunction: (r)=exp(ikr)E= h(2m)withV-the volume of metal,k-wave vectorThe highest occupied energy level is the Fermi level ErE, =h2kz /(2m)This model works well for such metals as Na, K, Rb etc, e.g., for NaE;(calc.)= 5.04*10-19 J 0r 3.15eV vs. Expt. value ~ 3.2 eVMore accurate model: by using pseudo-potentials for cationic coresand by taking into account of the electrostatic interactions betweenfree-electrons and a 3D array of cationic cores
Like an electron confined within a zero-potential 3D box, the Schrödinger equation for a “free electron” is exp( ) 1 ( ) ikr V r k ) ˆ ˆ ˆ ˆ ; ˆ (i.e., 0 8 2 2 2 V m h H E H T V E h k /( m) with k 2 2 2 /(2 ) 2 2 EF h kF m More accurate model: by using pseudo-potentials for cationic cores and by taking into account of the electrostatic interactions between free-electrons and a 3D array of cationic cores. Thus the behavior of free electrons can be described by a plane wave function: The highest occupied energy level is the Fermi level EF : V-the volume of metal, k-wave vector. This model works well for such metals as Na, K, Rb etc, e.g., for Na EF (calc.)= 5.04*10-19 J or 3.15eV vs. Expt. value ~ 3.2 eV
8.1.2 The Band Theory of SolidsConsidering the electrons moving in a periodic potential field of themetal atoms,the Schrodinger equation ish?H=T+V(One-particle equation)Hp=Ep;+V8元mMetal:Insulator:Semiconductor:Empty bandThermallyEConductionbandexcitedelectronsBandgap"BandgapE,≥5eVE.<3eVUEmptystates:energyValenceband"holes"Filled bandPartiallyfilledbands--conductionbands!
8.1.2 The Band Theory of Solids Considering the electrons moving in a periodic potential field of the metal atoms, the Schrödinger equation is (One-particle equation) Band gap Partially filled bands - conduction bands! Eg≥5eV Eg＜3eV Empty band Filled band E V 8 2 2 2 m h H E H T V ˆ ˆ ˆ ; ˆ
8.2 Close-packing of spheres and the structure of pure metalsPacking of metal atoms →> Crystal of metal8.2.1packing ofidentical spheresTypeA1 orABCABC1.Cubicclosepacking(ccp):[111]Layer-A Layer-B Layer-CEach unit cell hasLayered packing of4 spheres (atoms)!Nc=12hexagonal 2D latticesAlso being face-centered cubic (fcc)!
8.2 Close-packing of spheres and the structure of pure metals Packing of metal atoms Crystal of metal Also being face-centered cubic (fcc)! NC=12 [111] Each unit cell has 4 spheres (atoms)! 8.2.1 packing of identical spheres 1. Cubic close packing (ccp)： hexagonal 2D lattice Layered packing of s Type A1 or ABCABC Layer-A Layer-B Layer-C
2.Hexagonal closepacking (hcp) ABAB or TypeA3(Al and A3: The two most common close-packed structures)Lattice:hpEach unit cell has twoNc=12spheres (atoms)hcphexagonal close packing
2. Hexagonal close packing (hcp) ABAB or Type A3 (A1 and A3: The two most common close-packed structures) hcp hexagonal close packing NC=12 Each unit cell has two spheres (atoms). A B Lattice: hp