Atomic bonding in solids

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2 Nonmetls AVEE>13eV Metls AVEE<11eV Semimetls 11<AVEE<13eV 5 6 Atomic bonding in solids Gs ions will gglomrte RESULT: SOLID Things re mde of toms little prticles tht move round, ttrcting ech other when they re little distnce prt, but repelling upon being squeezed into one nother. In tht one sentence... there is n enormous mount of informtion bout the world. Richrd P. Feynmn 7 8

3 Energetics of ionic bonding Properties of Ionic Crystls Solid t room temperture High melting nd boiling points Trnsprent to visible light Electricl insultors Hrd nd brittle 9 10 Covlent Bonding Metllic Bonding 11 12

4 13 LECTURE 7,8,9 14 The Big Picture 9/11/ /11/ /11/2009 Electronic Structure Crystl structure of solids Crystl Systems Bonding Bohr tom Bohr-Sommerfeld Quntum numbers Aufbu principle Multielectron toms Periodic tble ptterns Octet stbility Stte of ggregtion Primry: Ionic Covlent Metllic Gs Liquid Solid Clssifiction of Solids: 1. Bonding type 2. Atomic rrngement The surfce of n crystlline Si solr cell

5 Atomic Arrngement SOLID: Smth. which is dimensionlly stble, i.e., hs volume of its own clssifictions of solids by tomic rrngement Energy nd Pcking Non dense, rndom pcking typicl neighbor bond energy Energy typicl neighbor bond length r ordered disordered tomic rrngement regulr rndom* order long-rnge short-rnge nme crystlline morphous crystl glss 17 Dense, ordered pcking typicl neighbor bond energy Energy typicl neighbor bond length Dense, ordered pcked structures tend to hve lower energies. r Mterils nd Pcking Crystlline mterils... toms pck in periodic, 3D rrys typicl of: -metls -mny cermics -some polymers Noncrystlline mterils... toms hve no periodic pcking occurs for: -complex structures -rpid cooling "Amorphous" = Noncrystlline crystlline SiO2 Adpted from Fig. 3.23(), Cllister & Rethwisch 8e. Si Oxygen noncrystlline SiO2 Adpted from Fig. 3.23(b), Cllister & Rethwisch 8e. Metllic Crystl Structures How cn we stck metl toms to minimize empty spce? 2-dimensions vs. Now stck these 2-D lyers to mke 3-D structures

6 Robert Hooke Cnnonblls Crystl must owe its regulr shpe to the pcking of sphericl prticles Niels Steensen ~ 1670 observed tht qurtz crystls hd the sme ngles between corresponding fces regrdless of their size SIMPLE QUESTION: If I see something hs mcroscopic shpe very regulr nd cubic, cn I infer from tht if I divide, divide, divide, divide, divide, if I get down to tomic dimensions, will there be some cubic repet unit? Christin Huygens Studying clcite crystls mde drwings of tomic pcking nd bulk shpe

7 BERYL Erly Crystllogrphy Be 3 Al 2 (SiO 3 ) 6 René-Just Hüy (1781): clevge of clcite Common shpe to ll shrds: rhombohedrl How to model this mthemticlly? Wht is the mximum number of distinguishble shpes tht will fill three spce? Mthemticlly proved tht there re only 7 distinct spce-filling volume elements The Seven Crystl Systems Does it work with Pentgon? BASIC UNIT Specifiction of unit cell prmeters 27 28

8 August Brvis How mny different wys cn I put toms into these seven crystl systems, nd get distinguishble point environments? When I strt putting toms in the cube, I hve three distinguishble rrngements. SC BCC FCC And, he proved mthemticlly tht there re 14 distinct wys to rrnge points in spce Crystl Structure =Brvis Lttice + Bsis (Atomic rrngement in 3D) (Point environment) (Atom grouping t ech lttice point) Brvis Lttice FCC Bsis Atom Crystl Structure tom Au, Al FCC molecule CH 4 FCC (Fullerene Xtl) ion pir NCl Rockslt tom pir C, Si, Ge Dimond cubic 31 32

9 BCC Crystl Hrd sphere Unit cell Reduced-sphere Unit cell Chrcteristics of Cubic Lttices SC BCC FCC Unit Cell Volume Lttice Points per cell Unit cell: smllest repetitive volume which contins the complete lttice pttern of crystl. Repetition of the unit cell genertes the entire crystl. By simple trnsltion, it defines lttice. Nerest Neighbor Distnce Number of Nerest Neighbors Atomic Pcking Fctor Volume of toms in unit cell* APF = Volume of unit cell / / *ssume hrd spheres

10 Lst dy Three Types of Solids ccording to tomic rrngement Crystl Systems Unit cell: smllest repetitive volume which contins the complete lttice pttern of crystl. 7 crystl systems 14 crystl lttices Lttice: The periodic rrngement of toms in Xtl., b, nd c re the lttice constnts 37 Fig. 3.4, Cllister & Rethwisch 8e The Importnce of the Unit Cell One cn nlyze the Xtl s whole by investigting representtive volume. Ex: from unit cell we cn Find the distnces between nerest toms for clcultions of the forces holding the lttice together Look t the frction of the unit cell volume filled by toms nd relte the density of solid to the tomic rrngement The properties of the periodic Xtl lttice determine the llowed energies of electrons tht prticipte in the conduction process

11 Simple Cubic Structure (SC) Rre due to low pcking density Close-pcked directions re cube edges. Click once on imge to strt nimtion (Courtesy P.M. Anderson) Coordintion # = 6 (# nerest neighbors) Atomic Pcking Fctor (APF) APF = Volume of toms in unit cell* Volume of unit cell *ssume hrd spheres APF for simple cubic structure = 0.52 close-pcked directions contins 8 x 1/8 = 1 tom/unit cell Adpted from Fig. 3.24, Cllister & Rethwisch 8e. toms unit cell R=0.5 APF = 1 volume 4 3 π (0.5) 3 tom 3 volume unit cell Body Centered Cubic Structure (BCC) Atoms touch ech other long cube digonls. --Note: All toms re identicl; the center tom is shded differently only for ese of viewing. ex: Cr, W, Fe (α), Tntlum, Molybdenum Coordintion # =? 8 Atomic Pcking Fctor: BCC APF for body-centered cubic structure = Click once on imge to strt nimtion (Courtesy P.M. Anderson) Adpted from Fig. 3.2, Cllister & Rethwisch 8e. 2 toms/unit cell: 1 center + 8 corners x 1/ Adpted from Fig. 3.2(), Cllister & Rethwisch 8e. R toms unit cell APF = π (3/4)3 3 Close-pcked directions: length = 4R = 3 unit cell volume tom volume 44 44

12 Fce Centered Cubic Structure (FCC) Atoms touch ech other long fce digonls. --Note: All toms re identicl; the fce-centered toms re shded differently only for ese of viewing. Click once on imge to strt nimtion (Courtesy P.M. Anderson) ex: Al, Cu, Au, Pb, Ni, Pt, Ag Coordintion # = 12 Adpted from Fig. 3.1, Cllister & Rethwisch 8e. 4 toms/unit cell: 6 fce x 1/2 + 8 corners x 1/ Atomic Pcking Fctor: FCC APF for fce-centered cubic structure = 0.74 mximum chievble APF 2 Adpted from Fig. 3.1(), Cllister & Rethwisch 8e. toms unit cell APF = Close-pcked directions: length = 4R = 2 Unit cell contins: 6 x 1/2 + 8 x 1/8 = 4 toms/unit cell π ( 2/4)3 3 volume tom volume unit cell Chrcteristics of Cubic Lttices Unit Cell Volume SC BCC 3 3 Lttice Points per cell 1 2 Nerest Neighbor Distnce 3/2 Number of Nerest Neighbors 6 8 Atomic Pcking Fctor Volume of toms in unit cell* APF = Volume of unit cell *ssume hrd spheres FCC 3 4 2/ Theoreticl Density, ρ where Density = ρ = ρ = Mss of Atoms in Unit Cell Totl Volume of Unit Cell n A V C N A n = number of toms/unit cell A = tomic weight (g/mol) V C = Volume of unit cell = 3 for cubic N A = Avogdro s number = x toms/mol

13 Theoreticl Density, ρ Adpted from Fig. 3.2(), Cllister & Rethwisch 8e. volume unit cell toms unit cell ρ = R x Ex: Cr (BCC) A = g/mol R = nm n = 2 toms/unit cell = 4R/ 3 = nm g mol ρ theoreticl ρ ctul toms mol = 7.18 g/cm 3 = 7.19 g/cm Point Coordintes x c z z 000 b b 111 b 2c y y Point coordintes for unit cell center re /2, b/2, c/2 ½ ½ ½ Point coordintes for unit cell corner re Crystllogrphic Directions x z y Algorithm 1. Vector repositioned (if necessry) to pss through origin. 2. Red off projections in terms of unit cell dimensions, b, nd c 3. Adjust to smllest integer vlues 4. Enclose in squre brckets, no comms [uvw] Liner Density Liner Density of Atoms LD = [110] Number of toms Unit length of direction vector ex: liner density of Al in [110] direction = nm ex: 1, 0, ½ => 2, 0, 1 => [ 201 ] -1, 1, 1 => [ 111 ] where overbr represents negtive index fmilies of directions <uvw> Adpted from Fig. 3.1(), Cllister & Rethwisch 8e. # toms LD = length 2 = nm

14 Crystllogrphic Plnes Crystllogrphic Plnes Miller Indices: Reciprocls of the (three) xil intercepts for plne, clered of frctions & common multiples. All prllel plnes hve sme Miller indices. Algorithm 1. Red off intercepts of plne with xes in terms of, b, c 2. Tke reciprocls of intercepts 3. Reduce to smllest integer vlues 4. Enclose in prentheses, no comms i.e., (hkl) Adpted from Fig. 3.10, Cllister & Rethwisch 8e Crystllogrphic Plnes exmple b c 1. Intercepts Reciprocls 1/1 1/1 1/ Reduction Miller Indices (110) exmple b c 1. Intercepts 1/2 2. Reciprocls 1/½ 1/ 1/ Reduction Miller Indices (100) 1. Red off intercepts of plne with xes in terms of, b, c 2. Tke reciprocls of intercepts 3. Reduce to smllest integer vlues 4. Enclose in prentheses, no comms i.e., (hkl) x x c c z z b b y y Crystllogrphic Plnes exmple b c 1. Intercepts 1/2 1 3/4 2. Reciprocls 1/½ 1/1 1/¾ 2 1 4/3 3. Reduction Miller Indices (634) x Fmily of Plnes {hkl} z c b Ex: {100} = (100), (010), (001), (100), (010), (001) y 56 56

15 Summry The Importnce of the Unit Cell Unit cell: smllest repetitive volume which contins the complete lttice pttern of crystl. One cn nlyze the Xtl s whole by investigting representtive volume. Ex: from unit cell we cn Find the distnces between nerest toms for clcultions of the forces holding the lttice together Look t the frction of the unit cell volume filled by toms nd relte the density of solid to the tomic rrngement The properties of the periodic Xtl lttice determine the llowed energies of electrons tht prticipte in the conduction process Atomic Pcking Fctor (APF) APF for simple cubic structure = 0.52 APF = Volume of toms in unit cell* Volume of unit cell *ssume hrd spheres close-pcked directions contins 8 x 1/8 = 1 tom/unit cell Adpted from Fig. 3.24, Cllister & Rethwisch 8e. toms unit cell R=0.5 APF = 1 volume 4 3 π (0.5) 3 tom 3 volume unit cell THEORETICAL DENSITY, ρ # toms/unit cell Atomic weight (g/mol) Volume/unit cell (cm 3 /unit cell) Exmple: Copper ρ= n A V c N A crystl structure = FCC: 4 toms/unit cell tomic weight = g/mol (1 mu = 1 g/mol) tomic rdius R = nm (1 nm = 10 cm) Avogdro's number (6.023 x toms/mol) -7 Vc = 3 ; For FCC, = 4R/ 2 ; Vc = 4.75 x cm 3 Result: theoreticl ρcu = 8.89 g/cm

16 Chrcteristics of Selected Elements t 20 C Element Aluminum Argon Brium Beryllium Boron Bromine Cdmium Clcium Crbon Cesium Chlorine Chromium Coblt Copper Flourine Gllium Germnium Gold Helium Hydrogen Symbol Al Ar B Be B Br Cd C C Cs Cl Cr Co Cu F G Ge Au He H At. Weight (mu) Density (g/cm 3 ) Crystl Structure FCC BCC HCP Rhomb HCP FCC Hex BCC BCC HCP FCC Ortho. Di. cubic FCC Atomic rdius (nm) Adpted from Tble, "Chrcteristics of Selected Elements", inside front cover, Cllister 5 th e Crystllogrphic Nottion POSITION: x,y,z, coordintes, sepd by comms, no enclosure O: 0,0,0 A: 0,1,1 B: 1,0, ½ DIRECTION: move coordinte xes so tht line psses through origin define vector from O to point on the line choose smllest set of integers no comms, enclose in brckets, cler frctions OB 1 0 ½ cler frctions [201] [011] minus denoted by mcron AO cn denote entire fmily of directions by crts < > e.g., ll body digonls: <111> = [111], [111], [111], [111], etc. ll cube edges: <001> ll fce digonls: <011> 62 PLANE: MILLER INDICES 1 Eqution of plne in spce: where, b, c re intercepts of the plne with the x, y, z xes, respectively Crystllogrphic Directions HOW TO DETERMINE CRYST. DIRECTIONS 1. Vector repositioned (if necessry) to pss through origin. 2. Red projections in terms of unit cell dimensions, b, nd c 3. Adjust to smllest integer vlues 4. Enclose in squre brckets, no comms [uvw] Let so tht hx + ky + lz = 1 z LINEAR DENSITY Frction of line length intersected by the toms long direction. no comms, enclose in prentheses (hkl) cn denote entire fmily of plnes by brces { } e.g., ll fces of unit cell: {001} = (001), (001), (100), (010), etc. y Ex: LD long [110], [100] for FCC? cool property: (hkl) [hkl] x 1 Willim Miller, British minerlogist,

17 Crystllogrphic Plnes ex b c 1. Intercepts Reciprocls 1/1 1/1 1/ Reduction Miller Indices ex b c 1. Intercepts 1/2 2. Reciprocls 1/½ 1/ 1/ Reduction Miller Indices (110) (100) HOW TO DETERMINE MILLER INDICES (hkl) 1. Red intercepts of plne with xes:, b, c 2. Tke reciprocls of intercepts 3. Reduce to smllest integer vlues 4. Enclose in prentheses, no comms i.e., (hkl) x x c c z z b b 65 y y Liner Density Liner Density of Atoms LD = Adpted from Fig. 3.1(), Cllister & Rethwisch 8e. [110] Number of toms Unit length of direction vector ex: liner density of Al in [110] direction = nm # toms length LD = 2 = nm Plnr Density of (100) Iron Solution: At T < 912ºC iron hs the BCC structure. (100) 2D repet unit 4 3 = R 3 Plnr Atomic Density Frction of totl plne re occupied by toms Ex: Clculte the plnr density of the (110) plne for FCC nd BCC Adpted from Fig. 3.2(c), Cllister & Rethwisch 8e. toms 2D repet unit 1 Plnr Density = 2 re 2D repet unit = R 3 Rdius of iron R = nm = toms toms 12.1 = 1.2 x nm 2 m

18 Exmples 3.13 Rhodium hs n tomic rdius of nm nd density of g/cm 3. Determine whether it hs n FCC or BCC crystl structure (tomic weight is g/mol) List the point coordintes for ll toms tht re ssocited with the FCC unit cell Determine the Miller indices for the plnes shown in the following unit cell: 71 72

19 3.45 Sketch the tomic pcking of () the (100) plne for the BCC crystl structure, nd (b) the (201) plne for the FCC crystl structure 3.53 () Derive liner density expressions for BCC [110] nd [111] directions in terms of the tomic rdius R. (b) Compute nd compre liner density vlues for these sme two directions for tungsten. (r=0.137nm)

20 POLYMORPHISM & ALLOTROPY Some mterils my exist in more thn one crystl structure, this is clled polymorphism. If the mteril is n elementl solid, it is clled llotropy. An exmple of llotropy is crbon, which cn exist s dimond, grphite, nd morphous crbon. Grphite Dimond Nnotubes 77 78