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
3. Other types of close-packed structures:ABAC......ABABCBCAC,etc(Eachlayerbelongs to ahexagonal2Dlattice!(bcporbce)A24.Body-centred cubicpackingEach unit cell has twospheres(atoms)(0,0,0),(0.5,0.5,0.5)
3. Other types of close-packed structures: ABAC., ABABCBCAC., etc. (Each layer belongs to a hexagonal 2D lattice!) Each unit cell has two spheres(atoms). 4. Body-centred cubic packing (bcp or bcc) A2 (0,0,0), (0.5,0.5,0.5)
8.2.2 Packing density1)ccp---fcc4R= √2aaRa=4R/V2cell =α3 =(4R/ /2)3 =16/2RThe volume of the unit cellThe total volume of the four spheres in the unit cell= 4 ×(4元R /3) = 16元R3 /3spheresPacking coefficient:/ Vcell = 元 /(3/2) = 74.05%spheresNote: The hexagonal close packing (hcp) of identical spheresgives the same packing density. (74.05%)
Note: The hexagonal close packing (hcp) of identical spheres gives the same packing density. (74.05%) a 4R/ 2 R a 3 3 3 Vcell a (4R/ 2) 16 2R 4 (4 /3) 16 /3 3 3 Vspheres R R Vspheres /Vcell /(3 2) 74.05% 8.2.2 Packing density 1) ccp -fcc The volume of the unit cell : The total volume of the four spheres in the unit cell: Packing coefficient: 4R 2a
hcp structureα=b=2R, =232R2RC= 4V2R/ /3 =cacoSy=(4R//2)3=8/2Rcell2Rsphere = 2(4元R3 /3)= 8元R /3/Vcel = 元 /(3/2) = 74.05%spher
hcp structure 2R 2R c b 2R a a b 2R， 2/3 2 3 3 Vcell ca cos ( 4R / 2 ) 8 2R 2 4 3 8 3 3 3 V R / R / sphere （ ） Vsphere/Vcell /( 3 2 ) 74.05% c 4 2R / 3
2) Body-centred cubic packing (bcp or bcc)Two spheres in a unit cell14R=/3a=a=R64R0元RCelspheres9R64pheresa/3元 =68.02%8 Thus bcp has a lower density than ccpbcpisnot a close-packed structure!
2) Body-centred cubic packing (bcp or bcc) • Thus bcp has a lower density than ccp. • bcp is not a close-packed structure! a R 3 3 3 9 3 64 3 4 Vspheres 2( )R ; Vcell a R 68.02% 8 3 ) 9 3 64 ) /( 3 4 / 2 ( 3 3 Vspheres Vcell R R R a a R 3 4 4 3 Two spheres in a unit cell
8.2.3Intersticesa) octahedral holes in ccp:a=4R//2Holeradius:1a/2 -R = 0.414 RFor a close-packed structure formed from identical spheresof radius R, the octahedral hole size is 0.414R
8.2.3 Interstices For a close-packed structure formed from identical spheres of radius R, the octahedral hole size is 0.414R. a) octahedral holes in ccp a/2 R = 0.414 R Hole radius: a 4R / 2 a