第二章晶體結構14版

無機材料科學導論AnIntroductiontoScienceforInorganicMaterials(I)南京工業(yè)大學材料科學與工程學院郭露村推薦閱讀文獻

SuggestedReadingM.A.Omar,ElementarySolidStatePhysics,1sted.,Addison-WesleyPublishingCompany,1975C.Kittle,IntroductiontoSolidStatePhysics,4thed.JohnWiley&Sons,Inc.,NewYork,1971,(5thed.or6thed.evenbetter,ifavailable.)G.M.Barrow,PhysicChemistry,5thed.,McGraw-Hill,Inc.,1988L.H.VanVlack,ElementsofMaterialsScienceandEngineering,5thed.,Addison-WesleyPublishingCompany,1985(6thed.,1994)W.S.Kingery,IntroductiontoCeramics,2nded.,JohnWiley&Sons,Inc.,NewYork,1976.W.D.金格瑞等著[美].清華譯.《陶瓷導論》.

北京：建筑工業(yè)出版社，1982L.H.范.弗萊克著[美].夏宗寧等譯.《材料科學與工程基礎》

（第四版）.機械工業(yè)出版社.1984馮端等主編，《材料科學導論》化學工業(yè)出版社.2002.5關振鐸等編.《無機材料物理性能》清華大學出版社，1992顧宜主編.《材料科學與工程基礎》

化學工業(yè)出版社.2002吳剛主編.《材料結構表征及應用》

化學工業(yè)出版社.2002李言榮，惲正中編.《材料物理學概論》清華大學出版社，2001張立德等著.《納米材料和納米結構》.

科學出版社，2001第二章：晶體結構ChapterIICrystalStructures2.1.基本概念Basicconcepts【晶體】crystal（結晶態(tài),crystallinestate）【多晶體】polycrystallinestate【無定形非晶體】amorphous

material【準晶體】quasicrystal【液晶】liquidcrystal【超晶格】superlattice【超材料】supermaterial2.1.1

凝聚態(tài)物質(zhì)的基本形態(tài)

PrincipalStatesofcondensedmatter

自然態(tài)非自然態(tài)【晶體】crystalAquartzcrystal

Ahalitecrystal

Anaquamarinecrystal單晶體外貌內(nèi)部原子排列完全周期性【多晶體】polycrystalline

陶瓷ceramics陶瓷顯微結玻璃陶瓷Li2O-Al2O3-SiO2

偏光顯微鏡ZrO2-Y2O3

陶瓷，TEM（Bright-field）96%Al2O3陶瓷Reflectedlightmicroscopy

冶煉碳化硅【無定形非晶體】amorphous

material

玻璃體、高分子、非晶態(tài)金屬Structureofglass:shortrangecrystallineSiO2玻璃（浮法玻璃）Transmittedpolarisedlightmicroscopy玻璃：玻璃體高分子非晶態(tài)金屬非晶態(tài)物質(zhì)結構SEMofsinglegrainsofquasicrystals:(a)anAluminum-Copper-Ironalloywhichcrystallizesintheshapeofadodecahedron.【準晶體】quasicrystal特點:具有長程取向序,而無平移對稱

(無排列周期性)【液晶】liquidcrystalinasolidcrystalstateinaliquidstateinaliquidcrystalstateTemperatureT1T2液晶相溫區(qū)Withinlayers,themoleculescanslidearoundeachother！液晶顯微結構PhotoofthesmecticCphase

(usingpolarizingmicroscope)PhotoofthesmecticCphase

(usingpolarizingmicroscope)Cross-polarisedlightmicroscopy

SEM【晶格與格點】crystallatticeandlatticesites

從具體的晶體抽象出來的幾何立體格子稱晶格，簡稱格子,其中所有的原子位置(atomicsites)均被格點（latticesitesorlatticepoints）取代。2.1.2.晶體結構基本要素及構成

Basicelementsandbuildupofcystals【fcc面心立方&bcc體心立方結構】face-centeredcubic&body-centeredcubic)Interstitialsitesinfcc體心立方bcc面心立方fcc2typesofhole–TetrahedralorOctahedral(larger)六方緊密堆積

hcpstructure

【緊密堆積結構】

hcp

＆fcc（face-centeredcubic,面心立方）

立方緊密堆積

fcc

structure

（hexagonally-closepacked）ABCABC..packingABABA..packingNote:Thebccisnotaclose-packedstructure!Close-packedstructures【基】又稱：基元

basis

晶體中最小的周期性重復的單元的原子（離子）或原子團稱為基。例如：NaCl,蛋白質(zhì)晶體的基中含一萬個原子,CH4分子晶體CH4分子晶體中作為基元的CH4fcclattice+CH4Crystal=Lattice+Basis【基矢量、格矢量】basisvectors＆latticevectorsa1a2a1,a2

為該２元晶格的基矢（量）Rn為格矢（量）Rn晶格中任意格點可用格矢Rn（a1,a2,a3

）表示

Rn=n1a1+n2a2+n3a3

a1,a2,a3

為基矢【原胞】primitivecell

由基矢圍成的六面體為原胞（primitivecell，或primitiveunitcell）其具有以下特征：通過平移用其可以“無縫隙地”填滿整個晶格。原胞只含一個格點(基），即只是八頂角有格點。最小體積。原胞選擇的方式可以不是唯一的，但體積是一定的。Bravais點陣不一定是原胞【

W-S單胞】Wigner-Seitzcell（維格納－賽茨晶胞）【單胞】又稱單位晶胞、晶胞（unitcell）為了更加清楚地表示晶體地對稱關系，常?？赡苓x擇更大范圍的六面體作為單位來表示晶胞。通常稱其為單胞。注意：單位晶胞的體積大于或等于原胞。單位胞是原胞體積的整數(shù)倍（2n）。非原胞的單胞可以用原胞來表示，但幾何未必重疊。通常的“晶胞常數(shù)”是指單包。

單胞原胞面心立方的單位胞、原胞及W-S胞體心立方：單胞、原胞、W-S胞面心立方：單胞、原胞、W-S胞2.2Bavais格子及晶系

TheBravaislattices＆

crystalsystems晶體的微觀對稱【Bravais

格子與非Bravais

格子】

Bravaislattice&non-Bravaislattice）Bravais格子:所有的格點（latticepoints）是等價的。它是最基本的格子(fundamentaltypeoflattice)三維晶體可能存在的最多格子形式只有14種非Bravais

格子:看成是Bravais格子+

基元而成。也可以看成是多個Bravais格子套裝而成。亞晶格

sublattice如：ZrO2中的氧離子單獨組成的“晶格”稱為“氧離子亞晶格”【14種Bravais格子】

Bravaislattices在14種格子中，按含格點的差異分４種類:P,I,F,

CSimple(primitive):P,Body-centered:I(innenzentrierte),Face-centered:F,Base-centered:CBravaisLatticeCrystalSystemUnitCellaxesangles1Simple三斜Triclinica≠b≠cα≠β≠γ≠90o2Simple單斜Monoclinica≠b≠c3Base-centered4Simple斜方Orthorhombica≠b≠cα=β=γ≠90o

5Base-centered6Body-centered7Face-centered8Simple四方Tetragonala=b≠cα=β=γ=90o9Body-centered10(Simple)三角Trigonal11(Simple)六方Hexagonalα=β=90o，γ=120o12Simple立方Cubica=b=cα=β=γ=90o13Body-centered14Face-centered【七大晶系】The7crystalsystems14種Bravaislattices抽去含格點的不同(P,I,F,C)即得到７大晶系對稱性TUnitCellandreciprocalcellofHexagonalHexagonalcellsHexagonalrepresentation(4axes)Rhombicrepresentation(3axes)2.3.點群與空間群

pointgroupandspacegroup

【對稱操作要素】【點群與點群對稱】【空間群】【對稱操作與要素】

SymmetryOperationsandElements

ASymmetryoperationisanoperationthatcanbeperformedeitherphysicallyorimaginativelythatresultsinnochangeintheappearanceofanobject，including

Rotation,reflection,andinversion

對稱操作完成后晶格保持不變！對稱分類中心對稱（至少一點保持不動）

反演中心條件r→-r晶格保持不變

14種Braivs格子全有inversioncentern度旋轉(zhuǎn)軸鏡面：立方有9鏡面，三斜晶系零鏡面、

n度旋轉(zhuǎn)反演軸非中心對稱螺旋軸(screwaxes)、滑移反映面(gladeplanes)【旋轉(zhuǎn)對稱】RotationalSymmetryn-度旋轉(zhuǎn)軸-n-FoldRotationAxis

2-fold1-fold8-fold5-fold3-fold4-fold6-fold【反映對稱】

MirrorSymmetry

【鏡面】reflectionplanes【反演對稱】

inversion【對稱中心】

CenterofSymmetry

又稱：反演中心(inversioncenter)：

Inthisoperationlinesaredrawnfromallpointsontheobjectthroughapointinthecenteroftheobject,calledasymmetrycenter(symbolizedwiththeletter"i").【n度旋轉(zhuǎn)反演軸

】(n-foldrotationaxis)2-fold3-fold6-fold螺旋軸(screwaxes)、

滑移反映面(gladeplanes)【點群與點群對稱】

pointgroupanditssymmetries

在非Bavais點陣中，每一格點有一原子團簇（cluster）,即基元（basis）,稱為點群。該點群的對稱性稱為點群對稱。

點群對稱特點：有一點始終保持不變。共有32種點群。晶體的宏觀對稱CombinationsofSymmetryOperations

Infact,incrystalsthereare32possiblecombinationsofsymmetryelements.

These32combinationsdefinethe32CrystalClasses.

1

-

4-foldrotationaxis(A4)4-

2-foldrotationaxes(A2),2cuttingthefaces&2cuttingtheedges.5mirrorplanes(m),2cuttingacrossthefaces,2cuttingthroughtheedges,andonecuttinghorizontallythroughthecenter.Notealsothatthereisacenterofsymmetry(i).Thesymmetrycontentofthiscrystalisthus:i,1A4,4A2,5mExternalSymmetryofCrystals,

32CrystalClasses

1Tetragonal41A4Tetragonal-Pyramidal4Tetragonal-disphenoidal4/mi,1A4,1mTetragonal-dipyramidal4221A4,4A2Tetragonal-trapezohedral4mm1A4,4mDitetragonal-pyramidal2m4,2A2,2mTetragonal-scalenohedral4/m2/m2/mi,1A4,4A2,5mDitetragonal-dipyramidal四方晶系所屬7個晶形偏三角面體偏方三八面體【空間群】spacegroup點群對稱+平移對稱（translationsymmetries）

72種空間群72種空間群+螺旋軸和滑移反映面對稱

230種空間群晶體結構與各種微觀及宏觀對稱間的關系

(晶系,Bravais格子,32點群,72及230空間群的關系)7大晶系+格點差異→

14Bravais

格子14Bravais+32點群

→72空間群

72空間群+螺旋軸,滑移反映→230空間群2.4.格點、晶向、晶面及密勒指數(shù)

Crystalpoints,directions,crystalplanes

andMillerIndices【格點指數(shù)】【晶向指數(shù)】Orthorhombicunitcell.(a)Indicesofpoint(b)Indicesofdirections.【晶向指數(shù)】CrystaldirectionsRn=n1a+n2b+n3c[n1n2n3

]

如有共同系數(shù)則約去：[111]注意：不是指特定的通過原點的直線，而是所有同方向的線。<100>則表示等價的方向“晶向族”：[100][][][100][010][001]指數(shù)越高，線密度越小?！久芾罩笖?shù)】MillerIndices

就是晶面指數(shù)（hkl）planeindices方法：(a/xb/yc/z)

注意：不是指特定的面，而是所有平行的面。｛hkl

｝則表示等價的方向“晶面族”：（100）（100）（010）（001）指數(shù)越高，面密度越小，面間距dhkl也越小MillerIndices

(hkl)

Step1:

Identifytheinterceptsonthex-,y-andz-axes.Intercepts:

a,∞,∞Step2:

Specifytheinterceptsinfractionalco-ordinatesCo-ordinatesareconvertedtofractionalco-ordinatesbydividingbytherespectivecell-dimension-forexample,apoint(x,y,z)inaunitcellofdimensions

axbxc

hasfractionalco-ordinatesof(x/a,y/b,z/c).Inthecaseofacubicunitcelleachco-ordinatewillsimplybedividedbythecubiccellconstant,a.ThisgivesFractionalIntercepts:

a/a,￥/a,￥/a

i.e.

1,￥,￥Step3:

TakethereciprocalsofthefractionalinterceptsThisfinalmanipulationgeneratestheMillerIndiceswhich(byconvention)shouldthenbespecifiedwithoutbeingseparatedbyanycommasorothersymbols.TheMillerIndicesarealsoenclosedwithinstandardbrackets(….)whenoneisspecifyingauniquesurfacesuchasthatbeingconsideredhere.Thereciprocalsof1and￥are1and0respectively,thusyieldingMillerIndices:

(100)陶瓷晶體的結構特點?CrystalStructuresfromanionpackingCeramiccrystalstructurescan

oftenbeconsideredasclose-packedanionstructureswithcationsininterstitialholes?2typesofhole–TetrahedralorOctahedral(larger)?2typesofpacking–FCCorHCP?Examples–NaCl–FCCanionpacking,cationsinoctahedralholes–ZnS–FCCanionpackingwithcationsintetrahedralholes–Al2O3–HCPanionpackingwithAlin2/3octahedralsites【NaCl結構】RocksaltstructureAnionlattice:FCCCationfilling:alloctahedralsitesTherC/rA:0.414~0.732Stoichiometry:MXExample:NaCl,KCl,LiF,MgO,CaO,SrO,NiO,CoO,MnO,PbO,etal.2.5.

典型的晶體結構

Typicalcrystalstructures【CsCl結構】【Diamond與ZnS結構】纖鋅礦結構【鈣鈦礦結構

】CaTiO3結構ABO3BaTiO3TetragonalBaTiO3(below120oC):astructureleadingtoitspiezoelectricity.Diagramshowinghowpiezoelectricmaterialworks【熒石結構

】

Fluoritestructure

(反熒石結構

Antifluoritestructure)FCaZrO【

ZrO2】ZrOZrOZrOZrOZrO2中存在大量的”空位”！OZrElectrolyteMaterialsI:ZrO2(YSZ)

Verylowelectronicconduction

(bandgap:>7eV)VeryhighthermodynamicstabilityEasilydopedwithlowervalencecations(e.g.Ca2+,Y3+,Sc3+)tocreateoxygenvacanciesDopedmaterialishighlyoxideionconductive>0.1Ω-1cm-1at1000oC)ThegeneralformulahastheformAO2,whereAisusuallyabigtetravalentcation,e.g.U,Th,Ce.SinceZr4+istoosmalltosustainthefluoritestructureatlowtemperatures,ithastobepartlysubstitutedwithalargercation,calleddopant.Dopinginvolvesusuallysubstitutinglowervalencecationsintothelattice.Inordertomaintainchargeneutralityoxygenvacancieshavetobeintroduced,whichallowoxygenionmigration.Fluoritestructure

andZrO2(YSZ)OZrFluoritestructure

andZrO2(YSZ)OZrYZirconia(zirconiumdioxide,ZrO2)initspureformhasahighmeltingtemperatureandalowthermalconductivity.Theapplicationsofpurezirconiaarerestrictedbecauseitshowspolymorphism.Itismonoclinicatroomtemperatureandchangestothedensertetragonalphasefromcirca1000

°C.Thisinvolvesalargechangeinthevolumeandcausesextensivecracking.Hencezirconiahasalowthermalshockresistivity.Theadditionofsomeoxidesresultsinstabilisingthecubicphaseandthecreationofoneoxygenvacancy.Partiallystabilizedzirconia(PSZ)isamixtureofzirconiapolymorphs:acubicandametastabletetragonalZrO2phaseisobtained,sinceaninsufficientamountofstabilizerhasbeenadded.PSZisalsocalledtetragonalzirconiapolycrystal:TZP.PSZisatransformation-toughenedmaterialsincetheinducedmicrocracksandstressfieldsabsorbenergy.PSZisusedforcruciblesbecauseithasalowthermalconductivityandahighmeltingtemperature.Theadditionof16

mol%CaOor16

mol%MgOor8

mol%Y2O3(8YSZ)isenoughtoformfullystabilizedzirconia.Thestructurebecomescubicsolidsolution,whichhasnophasetransformationwhenheatingfromroomtemperatureupto2500

°C.Becauseofitshighoxideionconductivity,YSZisoftenusedforoxygensensoringandsolidoxidefuelcells.FurtherReading

IonicconductionandcrystalstructureIonicconductiondependsonthemobilityoftheionsandthereforeontemperature.Athightemperatures,theconductivitycanreach1

S

cm-1,whichisofthesameorderofmagnitudeasforliquidelectrolytes.Thecrystalhastocontainunoccupiedsitesthatareequivalenttotheoccupiedsitesbylatticeoxygenions.Theenergybarrierformigrationfromanoccupiedsitetoanunoccupiedsitemustbesmall(≤1

eV).Thismightseemunusualsincetherelativesizeoftheoxygenionsisbiganditseemsmorelikelythatthesmallermetalionsmigrateinanelectricfield.Thatiswhythereareonlyafewspecialstructuresthatmakeoxygenionmigrationpossible:

fluoriteandperovskitesoxides.【

Al2O3】AlORutileTiO2【金紅石結構】Rutilestructure2.6.倒格子與布里淵區(qū)

Reciprocallattices&BrillouinZone定義：以晶格a1,a2,a3,

為基矢的R格矢，可定義另一組矢量為：

b1=2π/V（a2×a3

）

b2=2π/V（a1×a3

）

b3=2π/V（a2×a1

）

V=a·（a2×a3

）由矢量b1、

b2、b3構成的

G=n1b1+n2b2+n3

b3

稱R的倒格子，而b1、

b2、b3則稱為倒基矢,而原晶格則稱為正格子（directlattices,reallattices）。由G構成的空間稱為倒空間?！镜垢褡印康垢褡拥奶卣鳎海ńY論）

倒格子的特征：所有晶體結構都存在相應的倒格子，與正格子具有同樣的對稱性必屬同一晶系，但可以是不同的Bravais格子,如bcc與fcc.正格子倒格子V=a1·（a2×a3

）六方格子的倒格子仍舊是六方格子，但方向改變了。正格子倒格子fcc與bcc互為倒格子正格子倒格子【布里淵區(qū)】BrillouinZone定義：W-S胞的倒格子即為第一Brillouin區(qū)（FirstBrillouinZone）。特點：倒空間原胞與W-S胞同樣是格點在中心。bcc與fcc互為倒格子包含所有的基本信息二維WS胞及BrillouinZone【幾何空間】

晶體學（crystallography）【狀態(tài)空間】

就是倒空間又稱k空間，在固體物理學中用于描述電子（electron）、光子(photon)、聲子（phonon）等在晶體中的狀態(tài)和行為【幾何空間與狀態(tài)空間】Whyreciprocallattice？Electronwaveslikeallwavesexperiencediffractioneffectsinperiodicstructureslikecrystals.Thisisbestdescribedinthereciprocallatticeofthecrystalinquestion.Thereareseveralwaystoconstructareciprocallatticefromaspacelattice.

Definitionsofthereciprocallatticeinthreeways:1.

ThereciprocallatticeistheFouriertransformofthespacelattice.

2.Thereciprocallatticewithanelementarycell(EC)asdefinedbythebasevectors

b1,2,3isobtainedformthespacelatticeasdefinedbyitsbasevectorsa1,2,3bytheequations3.Thebasevectorsofthereciprocallatticecanbeconstructedbydrawingve