材料科学基础辅导与习题-上交课件 06ch2-2

§2-2参考书卢光熙等，金属学教程，上海科学技术出版社，1985

胡赓祥等，金属学，上海科学技术出版社，1980

[美]约翰.D.费豪文，物理冶金学基础，上海科学技术出版社，1980

石德珂等，材料科学基础，机械工业出版社，1999潘金生等，材料科学基础，清华大学出版社，1995

1§2-2元素的晶体结构1.周期表中的元素分类2.典型金属的晶体结构3.晶体中原子的堆垛方式4.晶体结构中的间隙5.金属原子半径6.亚金属的晶体结构21.周期表中的元素分类ⅠBⅡBⅢAⅣAⅤA

BCN

Al

SiP

CuZnGaO

GeAs

Ag

CdInA6SnA4A5

Sb

Au

HgRTTlA2A3PbA1

BiA7

Zn、Cd虽为A3结构，但其c/a较大;Tl和Pb的结构和第一类相同，但原子的离子化不完全，原子间距也比典型金属大;Hg和Sn的结构比较复杂;而Ga则具有复杂的正交结构。

31.1第一类为真正的金属包括ⅠB族及其以左元素,另外还有ⅢA族的Al元素。它们绝大多数都具有高对称性的简单结构，其典型结构为:面心立方结构(代号为A1)；体心立方结构(A2)；密排六方结构(A3)。1.2第二类为八个金属元素1.3第三类多数为非金属元素如硅、锗、锑、铋等，这类元素多数具有复杂的晶体结构,每个原子具有(8-N)个近邻原子，N为该原子的族数。4/periodic/periodic.html52.典型金属的晶体结构

金属晶体中的结合键是金属键，由于金属键没有方向性和饱和性，使大多数金属晶体都具有排列紧密、对称性高的简单晶体结构，即A1(FCC)、A2(BCC)、

A3(HCP)晶体结构。62.1FaceCenteredCubic(FCC)Atomsarearrangedatthecornersandcenterofeachcubefaceofthecell.Atomsareassumedtotouchalongfacediagonals72.1FaceCenteredCubic(FCC)Thelatticeparameter,a,isrelatedtotheradiusoftheatominthecellthrough:Coordinationnumber:thenumberofnearestneighborstoanyatom.ForFCCsystems,thecoordinationnumberis12.82.1FaceCenteredCubic(FCC)AtomicPackingFactor:theratioofatomicspherevolumetounitcellvolume,assumingahardspheremodel.FCCsystemshaveanAPFof0.74,themaximumpackingforasysteminwhichallsphereshaveequaldiameter.9

2.1FaceCenteredCubic(FCC)

1、Numberofatomsinunitcell

2、Theradiusoftheatom

3、Coordinationnumberandefficiencyofspacefilling

10ThecoordinationnumberforFCCsystems11DensityCalculations

Inthefcccelltheatomstouchalongthefacediagonals,butnotalongthecelledge:Lengthfacediagonal=a(2)1/2=4r

Usethisinformationtocalculatethedensityofan

fccmetal.12Examplecalculation.

Alhasaccparrangementofatoms.

TheradiusofAl=1.423Å(=143.2pm).

CalculatethelatticeparameteroftheunitcellandthedensityofsolidAl(atomicweight=26.98).Solution:

BecauseAlisccpwehaveanfccunitcell.

Cellcontents:4atoms/cell

[8atcorners(each1/8),6infaces(each1/2)]Latticeparameter:

atomsincontactalongfacediagonal,therefore4rAl=a(2)1/2,a=4(1.432Å)/(2)1/2=4.050Å.Density(=rAl)=Mass/Volume=Massperunitcell/Volumeperunitcell

g/cm3

Massofunitcell=mass4Alatoms=(26.98)(g/mol)(1mol/6.022x1023atoms)(4atoms/unitcell)=1.792x10-22g/unitcellVolumeunitcell=a3=(4.05x10-8cm)3=66.43x10-24cm3/unitcellTherefore

rAl

={1.792x10-22g/unitcell}/{66.43x10-24cm3/unitcell}=2.698g/cm3

132.2BodyCenteredCubic

Atomsarearrangedatthecornersofthecubewithanotheratomatthecubecenter.142.2BodyCenteredCubicSinceatomsareassumedtotouchalongthecubediagonalinBCC,thelatticeparameterisrelatedtoatomicradiusthrough:152.2BodyCenteredCubicCoordinationnumberforBCCis8.Eachcenteratomissurroundedbytheeightcorneratoms.ThelowercoordinationnumberalsoresultsinaslightlylowerAPFforBCCstructures.BCChasanAPFof0.68,ratherthan0.74inFCC16

2.2BodyCenteredCubic

1、Numberofatomsinunitcell

2、Theradiusoftheatom

3、Coordinationnumberandefficiencyofspacefilling

172.3HexagonalClosePackedCellofanHCPlatticeisvisualizedasatopandbottomplaneof7atoms,formingaregularhexagonaroundacentralatom.Inbetweentheseplanesisahalf-hexagonof3atoms.HCPisaverycommontypeofstructureforelementalmetals.

ExamplesincludeBe,Mg,Ti,Zr,etc.182.3HexagonalClosePackedTherearetwolatticeparametersinHCP,aandc,representingthebasalandheightparametersrespectively.Intheidealcase,thec/aratiois1.633,however,deviationsdooccur.Thecoordinationnumberoftheatomsinthisstructureis12.

Theyhave6nearestneighborsinthesameclosepackedlayer,3inthelayeraboveand3inthelayerbelow.Thisisoneofthemostefficientmethodsofpackingspheres(theotherthatisequallyefficientiscubicclosepacking,seebelow).

Inbothcasesthespheresfill74%oftheavailablespace.19Hexagonalclose-packedcrystals:theaxialratio

Theidealaxialratio(c/a)forahexagonalclose-packedcrystalstructurecanbecalculatedbyconsideringnon-interactinghardspherespackedinanh.c.p.lattice.Ifthesphereradiusisr,thenthelatticeparametersa(=b)andccanbewrittenintermsofr:2021Thesetworelationshipscanbesolvedfortheidealaxialratioc/a:2r

=

a

={(a/2cos30°)2+(c/2)2}½a2

=

a2/3+c2/44=4/3+c2/a2c/a

=1.633ManymaterialshavethehexagonalPcrystalsystem,buttheaxialratioisrarelyideal.Cadmium,forexample,hasanaxialratioofc/a=1.886.Thisnon-idealstructurehasimplicationsforthebehaviourofthematerial,forexampleinslip.22

2.3HexagonalClosePacked

1、Numberofatomsinunitcell

2、Theradiusoftheatom

3、Coordinationnumberandefficiencyofspacefilling23

ThespacelatticeofHCP

把A、B两个阵点作为一个阵点看待，就可看出密排六方晶体点阵实质上就是一个复式简单六方空间点阵。

242.4多晶型性

在周期表中,大约有40多种元素具有两种或两种以上的晶体结构，即具有同素异晶性，或称多晶型性。它们在不同的温度或压力范围内具有不同的晶体构，故当条件变化时，会由一种结构转变为另一种结构称为多晶型性转变或同素异构转变。25PolymorphismandAllotropyPolymorphismisaphysicalphenomenonwhereamaterialmayhavemorethanonecrystalstructure.Amaterialthatshowspolymorphismexistsinmorethanonetypeofspacelatticeinthesolidstate.Ifthechangeinstructureisreversible,thenthepolymorphicchangeisknownasallotropy.Theprevailingcrystalstructuredependsonboththetemperatureandtheexternalpressure.Onefamiliarexampleisfoundincarbon:graphiteisthestablepolymorphatambientconditions,whereasdiamondisformedatextremelyhighpressures.26Thebestknownexampleforallotropyisiron.Whenironcrystallizesat2800oFitisB.C.C.(d-iron),at2554oFthestructurechangestoF.C.C.(g-ironoraustenite),andat1670oFitagainbecomesB.C.C.(a-ironorferrite).Figure1.Coolingcurveforpureiron.(Allotropicbehaviorofpureiron)2728a-iron(alpha)Theothernamefora-ironisferrite.Thiscrystalhasbodycenteredcubicstructure.TheunitcellandthemicrographofthecrystalareshowninFigures(2)and(3).Figure2.Alphairon(B.C.C)unitcellFigure3.Ferritecrystals29g-iron(Gamma)Theothernameforg-ironisaustenite.Thiscrystalhasfacecenteredcubic(F.C.C)structure.TheunitcellandthemicrographofthecrystalareshowninFigures(4)and(5).Figure4.FacecenteredcubiccrystalunitcellFigure5.Austenitecrystals303.晶体中原子的堆垛方式晶体结构配位数致密度是否密排间隙大小Fcc120.74是小Bcc80.68否大Hcp120.74是小313.晶体中原子的堆垛方式3.1晶体中原子的二维排列方式3.2晶体中密排面原子排列方式3.3空隙位置和密排面的堆积方式3.4面心立方结构中原子的堆垛方式3.5密排六方结构中原子的堆垛方式32ClosepackedlayersofatomsIfwetreattheatomsasspheresandconsideralltheatomsinthesolidtobeofequalsize(asisthecaseforelementalmetals),themostefficientformofpackingistheclosepackedlayer.

Thisisillustratedbelowwhereitisclearthatclose-packingofspheresismoreefficientthan,forexample,squarepacking.

333.1晶体中原子的二维排列方式Belowontheleftisasquarepackedarraycomparedtothemoredenselypackedclosepackedarray.

Withinthesquarepackedlayerthecoordination#ofeachatomis4,

intheclosepackedlayeritis6.

343.2晶体中密排面原子排列方式353.3空隙位置和密排面的堆积方式Tobuildour3-dimensionalmetalstructureswenowneedtostacktheclosepackedlayersontopofeachother.

Thereareseveralwaysofdoingthis.

Themostefficientspacesavingwayistohavethespheresinonelayerfitintothe"holes"ofthelayerbelow.

36ClosePackedStructures

EventhoughFCCandHCPareclosepackedstructures,theyarequitedifferentinthemannerofstackingtheirclosepackedplanes.ClosepackedstackinginHCPtakesplacealongthecdirection(the(0001)plane).FCCclosepackedplanesarealongthe(111).Firstplaneisvisualizedasanatomsurroundedby6nearestneighborsinbothHCPandFCC.37ClosePackedStructuresThesecondplaneinbothHCPandFCCissituatedinthe“holes”abovethefirstplaneofatoms.TwopossibleplacementsforthethirdplaneofatomsThirdplaneisplaceddirectlyabovethefirstplaneofatomsABAstacking--HCPstructureThirdplaneisplacedabovethe“holes”ofthefirstplanenotcoveredbythesecondplaneABCstacking--FCCstructure38ClosePackedStructures39SimilaritiesandDifferenceBetweenthe

FCCandHCPStructureThefacecenteredcubicandhexagonalclosepackedstructuresbothhaveapackingfactorof0.74,consistofcloselypackedplanesofatoms,andhaveacoordinationnumberof12.Thedifferencebetweenthefccandhcpisthestackingsequence.Thehcplayerscycleamongthetwoequivalentshiftedpositionswhereasthefcclayerscyclebetweenthreepositions.Ascanbeseenintheimage,thehcpstructurecontainsonlytwotypesofplaneswithanalternatingABABarrangement.Noticehowtheatomsofthethirdplaneareinexactlythesamepositionastheatomsinthefirstplane.However,thefccstructurecontainsthreetypesofplaneswithaABCABCarrangement.NoticehowtheatomsinrowsAandCarenolongeraligned.Rememberthatcubiclatticestructuresallowslippagetooccurmoreeasilythannon-cubiclattices,sohcpmetalsarenotasductileasthefccmetals.40

41Thetablebelowshowsthestableroomtemperaturecrystalstructuresforseveralelementalmetals

Ananometer(nm)equals10-9meteror10Angstromunits.42MetalCrystalStructureAtomicRadius(nm)AluminumFCC0.1431CadmiumHCP0.1490ChromiumBCC0.1249CobaltHCP0.1253CopperFCC0.1278GoldFCC0.1442Iron(Alpha)BCC0.1241LeadFCC0.1750MagnesiumHCP0.1599MolybdenumBCC0.1363NickelFCC0.1246PlatinumFCC0.1387SilverFCC0.1445TantalumBCC0.1430TitaniumAlphaHCP0.1445TungstenBCC0.1371ZincHCP0.1332433.4CubicClosePacking(CCP)WhilefortheHCPstructurethethirdclosepackedlayerwaspositionedabovethefirst,analternatemethodofstackingistoplacethethirdlayersuchthatitliesinanuniqueposition,inthiswayan"ABCABC..."

closepackedlayersequencecanbecreated,seebelow.

ThismethodofstackingiscallCubicClosePacking(ccp)44453.4CubicClosePacking(CCP)463.5HexagonalclosepackingIfwecallthefirstlayer"A",thenthesecondlayer("B")ispositionedasshownontheleftofthediagrambelow.

Thethirdlayercanthenbeaddedintwoways.

InthefirstwaythethirdlayerfitsintotheholesoftheBlayersuchthattheatomslieabovethoseinlayerA.

ByrepeatingthisarrangementoneobtainsABABAB...stackingorhexagonalclosepacking.

4748Top:Anotherlayercanbeplacedontopinoneoftwoways:overtheupward-pointinggaps(blue)orthedownward-pointinggaps(yellow).LayersBandCshowlayersinthesepositions.

Bottom:Iflayersalternate(left)orarerandomlystacked(right)theoverallstructurehasonlythehexagonalsymmetryofthe

individualsheets.ThealternatingpatterniscalledHexagonalclosepacking.Theoxygenatomsincorundumandhematitehavethispacking.494.晶体结构中的间隙4.1面心立方结构中的间隙4.2体心立方结构中的间隙4.3密排六方结构中的间隙4.4晶体结构中各种间隙汇总504.1面心立方结构中的间隙4.1.1FCC结构中间隙的位置4.1.2FCC结构中间隙的大小4.1.3FCC四面体间隙的位置与个数4.1.4FCC八面体间隙的位置与个数514.1.1FCC结构中间隙的位置524.1.2FCC结构中间隙的大小534.1.3FCC四面体间隙的位置与个数4+4544.1.4FCC八面体间隙的位置与个数1+3554.2体心立方结构中的间隙6×1/2＋12×1/4=66×4×1/2=1256体心立方结构中间隙的位置▲四面体空隙是否为八面体空隙的一部？574.3密排六方结构中的间隙四面体：C轴2

＋

竖直棱6×2×1/3＋中心3×2=12584.4晶体结构中各种间隙汇总晶体结构八面体间隙四面体间隙间隙数rB/rA

间隙数rB/rA

Bcc60.155120.291Fcc40.41480.225Hcp60.414120.225595.Atomicandionicradii根据X射线测定的晶体结构类型和点阵常数大小，可计算出元素的原子半径R。实际上任何元素的R都不是固定不变的，而是受外界条件、原子间结合力、结合键类型、配位数等多种因素的影响而变化。605.1外界温度和压力的影响温度改变，原子热振动及晶体内点阵缺陷平衡浓