晶體對(duì)稱中的平移群和3D對(duì)稱性

SymmetryTranslations(Lattices)Apropertyattheatomiclevel,notofcrystalshapesSymmetrictranslationsinvolverepeatdistancesTheoriginisarbitrary1-Dtranslations=arowSymmetryTranslations(Lattices)Apropertyattheatomiclevel,notofcrystalshapesSymmetrictranslationsinvolverepeatdistancesTheoriginisarbitrary1-Dtranslations=arowaaistherepeatvectorSymmetryTranslations(Lattices)2-Dtranslations=anetSymmetryTranslations(Lattices)2-Dtranslations=anetUnitcellUnitCell:thebasicrepeatunitthat,bytranslationonly,generatestheentirepatternHowdifferfrommotif??SymmetryTranslations(Lattices)2-Dtranslations=anetabPickanypointEverypointthatisexactlynrepeatsfromthatpointisanequipointtotheoriginalTranslationsExercise:Escherprint1.Whatisthemotif?2.Pickanypointandlabelitwithabigdarkdot3.Labelallequipointsthesame4.Outlinetheunitcellbasedonyourequipoints5.Whatistheunitcellcontent(Z)??Z=thenumberofmotifsperunitcellIsZalwaysaninteger?TranslationsWhichunitcelliscorrect??Conventions:1.Celledgesshould,wheneverpossible,coincidewithsymmetryaxesorreflectionplanes2.Ifpossible,edgesshouldrelatetoeachotherbylattice’ssymmetry.3.Thesmallestpossiblecell(thereducedcell)whichfulfills1and2shouldbechosen.TranslationsThelatticeandpointgroupsymmetryinterrelate,becausebotharepropertiesoftheoverallsymmetrypatternTranslationsThelatticeandpointgroupsymmetryinterrelate,becausebotharepropertiesoftheoverallsymmetrypatternGoodunitcellchoice.Why?WhatisZ?Arethereothersymmetryelements?TranslationsThelatticeandpointgroupsymmetryinterrelate,becausebotharepropertiesoftheoverallsymmetrypatternThisiswhy5-foldand>6-foldrotationalsymmetrywon’tworkincrystalsTranslationsThereisanew2-DsymmetryoperationwhenweconsidertranslationsTheGlidePlane:AcombinedreflectionandtranslationStep1:reflect(atemporaryposition)Step2:translaterepeatTranslationsThereare5unique2-Dplanelattices.Thereare5unique2-Dplanelattices.Therearealso172-DPlaneGroupsthatcombinetranslationswithcompatiblesymmetryoperations.ThebottomrowareexamplesofplaneGroupsthatcorrespondtoeachlatticetypeCombiningtranslationsandpointgroupsPlaneGroupSymmetryp211PlaneGroupSymmetryTridymite:OrthorhombicCcell3-DTranslationsandLatticesDifferentwaystocombine3non-parallel,non-coplanaraxesReallydealswithtranslationscompatiblewith323-Dpointgroups(orcrystalclasses)32PointGroupsfallinto6categories3-DTranslationsandLatticesDifferentwaystocombine3non-parallel,non-coplanaraxesReallydealswithtranslationscompatiblewith323-Dpointgroups(orcrystalclasses)32PointGroupsfallinto6categories+c+a+bgbaAxialconvention:“right-handrule”Definitionofthelatticeparametersandtheiruseincubic,orthorhombic,andhexagonalcrystalsystems.(c)2003Brooks/ColePublishing/ThomsonLearning?七個(gè)晶系abcPMonoclinica=g=90o

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1cabcI=CabPTriclinica1b1ga

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1cccaPOrthorhombica=b=g=90oa

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1cCFIba1cPTetragonala=b=g=90oa1=a2

1cIa2a1a3PIsometrica=b=g=90oa1=a2

=a3a2FI3-DTranslationsandLatticesTriclinic:Nosymmetryconstraints.NoreasontochooseCwhencanchoosesimplerPDosobyconvention,sothatallmineralogistsdothesameOrthorhombic:WhyCandnotAorB?IfhaveAorB,simplyrenametheaxesuntilC+c+a+bgbaAxialconvention:“right-handrule”3-DSymmetryCrystalAxes3-DSymmetry3-DSymmetry3-DSymmetry3-DSymmetry3-DSpaceGroupsAsinthe172-DPlaneGroups,the3-Dpointgroupsymmetriescanbecombinedwithtranslationstocreatethe2303-DSpaceGroupsAlsoasin2-D