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1、材料科学基础FundamentalofMaterials Lessonfive 2 5Indicesofcrystalplanesanddirections What scrystalplanesanddirections Theatomicplanesanddirectionspassingthroughthecrystalarecalled crystal planesanddirectionsrespectively 1 Stepstodeterminatetheplaneindices Establishasetofcoordinateaxes Findtheinterceptsoft
2、heplanestobeindexedona bandcaxes x y z Planeindices Takethereciprocalsoftheintercepts1 x 1 y 1 z Clearfractionsbutdonotreducetolowestintegers Enclosetheminparentheses hkl Example 1 2 1 2 32 1 3 2 423 Planeindicesreferredtothreeaxesa bandcarealsocalledMillerIndices SeveralimportantaspectsoftheMilleri
3、ndicesforplanesshouldbenoted Planesandtheirnegativesareidentical Therefore Planesandtheirmultiplesarenotidentical Incubicsystems adirectionthathasthesameindicesasaplaneisperpendiculartothatplane 2 Theimportantplanesincubiccrystals 110 112 111 001 3 Afamilyofplanesconsistsofequivalentplanessofarasthe
4、atomarrangementisconcerned Total 6 Total 4 Total 12 Total 4 3 24 DirectionIndices 1 DerivationforthecrystallographicdirectionAsthefirstabove settheoriginonthedirectiontobeindexed Findthecoordinatesofanotherpointonthedirectioninquestions Reducetothreesmallestintegers u v w Encloseinsquarebrackets uvw
5、 2 Theimportantdirectionincubiccrystals crystalaxes facediagonal bodydiagonal apicestooppositeface centers3 Familyofdirectionsconsistsofcrystallographicallyequivalentdirections denotede g 2 6Hexagonalaxesforhexagonalcrystals Whychoosefour axissystem Fourindiceshasbeendevisedforhexagonalunitcellsbeca
6、useoftheuniquesymmetryofthesystem Planeindices hkil Itcanbeproved i h k Importantplanes Directionindices uvtw Tomaketheindicesunique anadditionalconditionisimposed Lett u v Importantdirections TransformationofindicesTransformationof3to4indices orviceversa Supposewehaveavector whose3indices uvw and4i
7、ndices uvtw Wehave Since or Forexample 1 Quickwayforindexingthedirectionsincubiccrystals Thevalueofadirectiondependsonitsfeaturewhilethesignondirection ExamplesandDiscussions 2 Thecoordinateorigincanbesetarbitrarily forexampleonapices body center face centersetc butneveronplaneinquestions otherwiset
8、heinterceptswouldbe0 0 0 3 Thecoordinatesystemcanbetransferredarbitrarily butrotationisforbidden 4 Theatomicarrangementandplanardensityoftheimportantdirectionincubiccrystal 5 Theatomicarrangementandlineardensityoftheimportantdirectionincubiccrystal Exercise Calculatetheplanardensityandplanarpackingfractionforthe 010 and 020 planescubicpolonium whichhasalatticeparameterof0 334nm Solution However noatomsarecenteredonthe 020 planes Therefore theplanardensityandtheplanarpackingfractionarebothzero Thankyou 5
