ENERGY AND PACKING. Outline: MATERIALS AND PACKING. Crystal Structure
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1 EERGY AD PACKIG Outline: Crstlline versus morphous strutures Crstl struture - Unit ell - Coordintion numer - Atomi pking ftor Crstl sstems on dense, rndom pking Dense, regulr pking tpil neighor ond energ tpil neighor ond energ Energ Energ tpil neighor ond length tpil neighor ond length r r Dense, regulr-pked strutures tend to hve lower energ. Dr. M. Medrj Meh. Eng. Dept. - Conordi Universit MECH221 leture 4/1 Dr. M. Medrj Meh. Eng. Dept. - Conordi Universit MECH221 leture 4/2 MATERIALS AD PACKIG Crstl Struture Crstlline mterils... toms pk in periodi, 3D rrs tpil of: -metls -mn ermis -some polmers rstlline SiO2 Adpted from Fig. 3.23(), Cllister 8e. Motivtion: Mn of the properties of mterils (espeill mehnil) re determined the rrngement of the onstituent toms. This rrngement is lled the mteril s rstl struture. onrstlline mterils... toms hve no periodi pking ours for: -omple strutures -rpid ooling... = onrstlline nonrstlline SiO2 Adpted from Fig. 3.18(), Cllister 6e. Atoms n e rrnged either in regulr, periodi rr (i.e., long-rnge order) or ompletel disordered (morphous). Dr. M. Medrj Meh. Eng. Dept. - Conordi Universit Meh 221 leture 4/3 An importnt distintion Atomi struture reltes to the numer of protons nd neutrons in the nuleus of n tom, s well s the numer nd proilit distriutions of the onstituent eletrons. On the other hnd, rstl struture pertins to the rrngement of toms in the rstlline solid mteril. Dr. M. Medrj Meh. Eng. Dept. - Conordi Universit MECH221 leture 4/4
2 Crstl Struture Crstl Struture We need w to speif rstllogrphi diretions nd plnes. ow ple n tom t eh orner Let s strt with the hrd sphere model (in whih nerest neighor toms touh eh other) This represents the hrd sphere model of.. rstl sstem To illustrte the onept of rstl struture nd lttie sstems, we first identif oordinte sstem (,, ): We n t speif diretions or plnes without knowing wht the referene sstem is. Dr. M. Medrj Meh. Eng. Dept. - Conordi Universit MECH221 leture 4/5 Atoms touh long the rstl es These diretions re referred to s losepked in the simple ui sstem not mn emples of simple ui sstems in nture, eept for Po The ove digrm represents simple ui.. Dr. M. Medrj Meh. Eng. Dept. - Conordi Universit MECH221 leture 4/6 Wht is Unit Cell? A unit ell is the smllest entit tht ehiits the hemil nd. properties of the mteril. Unit ells re the most elementr rrngement of toms whih n generte the entire rstl upon pplition of suitle trnsltion, rottion, mirror, or inversion opertions. Definition: the length of eh unit ell is is lled.. In ui sstems, ll three orthogonl lttie prmeters re equl Lttie prmeters re tpill on the order of few Angstroms (or few tenths of nnometer) Dr. M. Medrj Meh. Eng. Dept. - Conordi Universit MECH221 leture 4/7 How mn toms does the simple ui unit ell ontin? You should e le to onvine ourself tht simple ui struture ontins tom/unit ell. (Rememer, prt of eh tom is shred nother unit ell!) f i 2 Simple Cui Unit Cell The numer of toms/unit ell is n importnt quntit nd determines mn phsil properties. In generl, the numer of toms/unit ell,, is given 8 Where: i = # interior toms, f = # fe toms, = # orner toms Dr. M. Medrj Meh. Eng. Dept. - Conordi Universit Meh 221 leture 4/8
3 Simple Cui Unit Cell Bod Centered Cui Unit Cell volume of the unit ell = 3 where is the lttie prmeter oordintion # = 6 for simple ui strutures C is the numer of nerest-neighor toms Coordintion numer is importnt in determining the struture of rstlline mterils. Lrge toms tend to hve lrge C, smll toms usull hve smll C it s esier to surround ig tom with lots of toms thn smller one. Dr. M. Medrj Meh. Eng. Dept. - Conordi Universit MECH221 leture 4/9 ow, suppose we dd nother tom t the enter of the ue We no longer hve simple ui struture ut insted,.... (BCC) struture Emples of BCC sstems: Cr, W, Mo, T, Fe (Fe stle elow 912C) otie tht in the BCC struture, toms touh long the od.. These re the lose-pked diretions in the odentered ui struture. BCC unit ells hve C = Dr. M. Medrj Meh. Eng. Dept. - Conordi Universit Meh 221 leture 4/10 Atomi Pking Ftor Atomi Pking Ftor (APF or APE) = totl sphere volume totl ell volume totl sphere volume is just the volume per tomi sphere multiplied the numer of toms in the unit ell Emple: Clulte the APF for BCC unit ell: Fe Centered Cui Unit Cell ow suppose we ple equivlent toms t the orners of the unit ell, AD in the enter of eh fe: This is (FCC) rstl struture O Emples of FCC metls: Cu, i, Au, Ag, Fe (Fe stle ove 912C) Close-pked diretions in FCC metls re long fe digonls Dr. M. Medrj Meh. Eng. Dept. - Conordi Universit MECH221 leture 4/11 Dr. M. Medrj Meh. Eng. Dept. - Conordi Universit MECH221 leture 4/12
4 Fe Centered Cui Unit Cell Hegonl Unit Cell There re other ws in whih toms n e rrnged to form unit ells: For emple This represents n HCP (....) struture Q: How mn toms per unit ell in the FCC struture? rell, i f 2 8 FCC unit ells hve C = 12 nd n APF = 0.75 mimum pking effiien for monosied spheres Dr. M. Medrj Meh. Eng. Dept. - Conordi Universit Meh 221 leture 4/13 Emples of HCP sstems (Mg, Co, Ti, Zn, Zr, RE) The HCP unit ell onsists of 6 toms forming the orners of hegon in the sl plnes + 1 tom in the enter. In ddition, there re 3 interior toms midw etween sl plnes long the -is. # toms/unit ell = (1/6)*12 (orner toms) + (1/2)*2 (enter fe toms) + 3 (interior toms) =. Coordintion # = Dr. M. Medrj Meh. Eng. Dept. - Conordi Universit Meh 221 leture 4/14 Generl onvention for unit ell is nd ngle nottion: Crstl Sstems B onvention origin - 0,0,0 In totl, there re 7 distint nd unique rstl sstems: ui represents onl one of the 7 Cui == Tetrgonl = Orthorhomi Dr. M. Medrj Meh. Eng. Dept. - Conordi Universit Meh 221 leture 4/15 Dr. M. Medrj Meh. Eng. Dept. - Conordi Universit MECH221 leture 4/16
5 Crstl Sstems Crstl Sstems Rhomohedrl == ue Hegonl = Squished tetrgonl Monolini = orthorhomi (in one diretion) Trilini orthorhomi (in two diretions) ote tht these 7 rstl sstems do not ount for ll the possile lttie tpes for emple, the ui sstem ontins SC, FCC, nd BCC s susets Dr. M. Medrj Meh. Eng. Dept. - Conordi Universit MECH221 leture 4/17 Dr. M. Medrj Meh. Eng. Dept. - Conordi Universit MECH221 leture 4/18 Crstl Sstems There re 14 unique lttie tpes from this frmework of 7 rstl sstems (lled Brvis ltties): Crstl sstem Cui Hegonl Tetrgonl Orthorhomi Rhomohedrl Monolin Trilini Tpes of possile lttie rrngements SC, BCC, FCC HCP Simple, od-entered Simple, se-entered, BC, FC Simple Simple, se-entered Simple et time: Crstllogrphi Diretions nd Plnes We will minl e onerned with ui nd hegonl sstems in this lss. But ou need to relie tht mn other tpes of smmetries eist! Dr. M. Medrj Meh. Eng. Dept. - Conordi Universit MECH221 leture 4/19 Dr. M. Medrj Meh. Eng. Dept. - Conordi Universit MECH221 leture 4/20
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