Directions_Miller_indices

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1、CrystalDirections,WavePropagationandMillerIndicesD.K.Ferry,D.VasileskaandG.KlimeckCrystalDirections;WavePropagationElectronbeamgeneratedpatterninTEM(001)(110)CrystalDirectionsWehavereferredtovariousdirectionsinthecrystalas(100),(110),and(111).Whatdothesemean?Howarethesedirectionsdeterm

2、ined?Consideracube-1andaplane-2/31/2Theplanehasintercepts:x=0.5a,y=0.667a,z=a.CrystalDirectionsWewanttheNORMALtothesurface.Sowetaketheseintercepts(inunitsofa),andinvertthem:123,,12,,1232Thenwetakethelowestcommonsetofintegers:123,,12,,14,3,2232ThesearetheMIL

3、LERINDICESoftheplane.TheNORMALtotheplaneisthe(4,3,2)direction,whichisnormallywrittenjust(432).(Anegativenumberisindicatedbyabaroverthetopofthenumber.)CrystalDirections1(111)11(110)11CrystalDirections(100)1(220)11/21/2CrystalDirectionsElectronbeamgeneratedpatterninTEM(001)(110)Sofar,weh

4、avediscussedtheconceptofcrystaldirections:1(111)11(110)11Wewanttosayafewmorethingsaboutthis.zConsiderthisplane.Theinterceptsare0,0,,ywhichleadsto1,1,0forMillerindices.Whatarethenormalstotheplane?xTheinterceptsare-1,1,,whichleadsto110zThenormaldirectionisxˆyˆ110yx-1-1Theinterce

5、ptsare1,-1,,whichleadsto110Thenormaldirectionisxˆyˆ110Wecaneasilyshifttheplanesbyonelatticevectorinxoryzyx-1-1Thesearetwodifferentnormalstothesameplane.z110y110x-1-1Directionshavesquarebrackets[…]Planeshaveparentheses(…)Nowconsiderthefollowingplane:zy110110x-1-1Thisdirec

6、tionliesintheoriginalplane.Nowconsiderthefollowingplane:zy001x-1-1Thisdirectionliesintheoriginalplane.zyx-1001111-1110110pointsintothepageCrystalDirectionsWewanttheNORMALtothesurface.Sowetaketheseintercepts(inunitsofa),andinvertthem:123Thisinversion,,12,,1createsunit

7、sof1/cm232Thenwetakethelowestcommonsetofintegers:123These“numbers”,,12,,14,3,2232defineanewVECTORinthis“reciprocalspace”ThesearetheMILLERINDICESoftheplane.TheNORMALtotheplaneisthe[4,3,2]direction,whichisnormallywrittenjust[432].(Anegativenumberisindicatedbyabarove