Wyckoff-Positions.ppt

Wyckoff-Positions.ppt

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FromthespacegrouptablesweseethattheatomsarelocatedonthefollowingWyckoffsitesThenumberassociatedwiththeWyckoffsitestellsushowmanyatomsofthattypethereareintheunitcell.Sothereare40atomsintheunitcell,withstoichiometrySr8Al4Ta4O24whichreducestotheempiricalformulaSr2AlTaO6.Sincethenumberofatomsintheunitcellisfourtimesthenumberofatomsintheformulaunit,wesaythatZ=4.OneofthemostusefulpiecesofinformationcontainedintheInternationalTablesaretheWyckoffpositions.TheWyckoffpositionstelluswheretheatomsinacrystalcanbefound.TounderstandhowtheyworkconsiderthemonoclinicspacegroupPm.Thisspacegrouphasonlytwosymmetryelements,bothmirrorplanesperpendiculartotheb-axis.Oneaty=0andoneaty=（halfwayuptheunitcellinthebdirection）.Nowletsplaceanatomintheunitcellatanarbitraryposition,x,y,z.Ifwenowcarryoutthesymmetryoperationassociatedwiththisspacegroupasecondatomwillbegeneratedbythemirrorplaneatx,-y,z.However,ifweweretoplacetheatomononeofthemirrorplanes（itsycoordinatewouldhavetobeeither0or）thenthereflectionoperationwouldnotcreateasecondatom.AlloftheinformationintheproceedingparagraphiscontainedinWyckoffpositionssectionoftheInternationalTables.PmhasthreeWyckoffsitesasshowninthetablebelow:

Themultiplicitytellsushowmanyatomsaregeneratedbysymmetryifweplaceasingleatomatthatposition.Inthiscaseforeveryatomweinsertatanarbitraryposition（x,y,z）intheunitcellasecondatomwillbegeneratedbythemirrorplaneatx,-y,z.ThiscorrespondstotheuppermostWyckoffposition2c.Theletterissimplyalabelandhasnophysicalmeaning.Theyareassignedalphabeticallyfromthebottomup.Thesymmetrytellsuswhatsymmetryelementstheatomresidesupon.TheuppermostWyckoffposition,correspondingtoanatomatanarbitrarypositionneverresidesuponanysymmetryelements.ThisWyckoffpositioniscalledthegeneralposition.Thecoordinatescolumntellsusthecoordinatesofallofthesymmetryrelatedatoms（twointhiscase）.AlloftheremainingWyckoffpositionsarecalledspecialpositions.Theycorrespondtoatomswhichlieupononeofmoresymmetryelements,becauseofthistheyalwayshaveasmallermultiplicitythanthegeneralposition.Furthermore,oneormoreoftheirfractionalcoordinatesmustbefixed.Inthiscasetheyvaluemustbeeither0orortheatomwouldnolongerlieonthemirrorplane.GeneratingaCrystalStructurefromitsCrystallographicDescriptionUsingthespacegroupinformationcontainedintheInternationalTableswecandomanythings.Onepowerfuluseistogenerateanentirecrystalstructurefromabriefdescription.ConsiderthefollowingdescriptionofthecrystalstructureofSr2AlTaO6.SpaceGroup=Fm3ma=7.80AtomicPositionsWyckoff位置（WyckoffPositions）用来表示晶胞中等价原子的对称性的，WyckoffPositions包括多重度（Multiplicity），Wyckoff字母（WyckoffLetter）,位置的对称性（SiteSymmetry）以及该位置的分数坐标。

其中前面所说的”1a2b3c16f.“实际上就是Wyckoff字母,一般它是由a开始往下写：

b,c,d,.一直到最大的字母。

最大的一个字母所代表的那个位置上的多重度最大,因为这些位置上如果有原子，那么就会由对称操作转出最多数目的其他等价原子。

这个最大的一个字母所代表的那个位置代表的是一般性的位置，故也称为”一般性位置“（Generalposition）；其他多重度小的位置用较小序号的字母表示，这些位置都是特殊的几个位置（往往其分数坐标的一个或多个分量是特定值，这往往是对称元素的位置确定的），故也称”特殊位置“（SpecialPosition）。

具体解释，请参见：

http:

/www.chemistry.ohio-state.edu/woodward/ch754/sym_itc.htm对于某个具体的空间群（SGorSpaceGroup），都对应有一套WyckoffPositions，大家可以从http:

/www.cryst.ehu.es/cryst/get_wp.html上查询。

但是对于属于该空间群的某个特定的晶体晶体，并不是可能具有所有可能的WyckoffPositions的（往往只具有某一些）。

如果知道了某个晶体的空间群和相应的WyckoffPositions，我们就可以利用各个WyckoffPositions上的Coordinates关系从非对称原子（其总数一般用Z表示）导出所有的等价原子，于是，整个晶胞中的所有原子我们就可以推导出来了。

详细实例，参见：

http:

/www.chemistry.ohio-state.edu/woodward/ch754/sym_itc.html等效点系（等效位置）在晶体构造中，由一个任意的起始点开始，通过空间群所有对称要素的作用，重复出来一系列规则分布的点（等效点）的总和。

空间群把同一个等效点系中的各等效点联系在一起。

l在晶体构造中，同一个等效点系中的等效点必为等质点；l由于起始点与空间群中对称元素之间的相对位置不同，同一个空间群可以推导出不同的等效点系、或不同数目的等效点；l不同的等效点系的等效点之间，无法用空间群中任何对称操作使之相互重合，但它们具有同一个空间群，对称特点相同。

l特殊等效点系等效点位于空间群的某个或某些对称元素上的等效点系；l一般等效点系等效点位于空间群对称元素外的等效点系。

Usingthefacecenteringgenerators（0,0,0）,（,0）,（,0,）,（0,）togetherwiththecoordinatesofeachWyckoffsitewecangeneratethefractionalcoordinatesofallatomsintheunitcell:

Sr1:

（0.25,0.25,0.25）,2:

（0.75,0.75,0.25）,3:

（0.75,0.25,0.75）,4:

（0.25,0.75,0.75）5:

（0.25,0.25,0.75）,6:

（0.75,0.75,0.75）,7:

（0.75,0.25,0.25）,8:

（0.25,0.75,0.25）Al1:

（0.0,0.0,0.0）,2:

（0.5,0.5,0.0）,3:

（0.5,0.0,0.5）,4:

（0.0,0.5,0.5）Ta1:

（0.5,0.5,0.5）,2:

（0.0,0.0,0.5）,3:

（0.0,0.5,0.0）,4:

（0.5,0.0,0.0）O1:

（0.24,0.0,0.0）,2:

（0.74,0.5,0.0）,3:

（0.74,0.0,0.5）,4:

（0.24,0.5,0.5）5:

（0.76,0.0,0.0）,6:

（0.26,0.5,0.0）,7:

（0.26,0.0,0.5）,8:

（0.76,0.5,0.5）9:

（0.0,0.24,0.0）,10:

（0.5,0.74,0.0）,11:

（0.5,0.24,0.5）,12:

（0.0,0.74,0.5）13:

（0.0,0.76,0.0）,14:

（0.5,0.26,0.0）,15:

（0.5,0.76,0.5）,16:

（0.0,0.26,0.5）17:

（0.0,0.0,0.24）,18:

（0.5,0.5,0.24）,19:

（0.5,0.0,0.74）,20:

（0.0,0.5,0.74）21:

（0.0,0.0,0.76）,22:

（0.5,0.5,0.76）,23:

（0.5,0.0,0.26）,24:

（0.0,0.5,0.26）FromthesefractionalcoordinatesyoucansketchoutthestructureofSr2AlTaO6.WithsomeluckIwillprovidealinktoapictureofthestructurehere,atsomepointinthefuture.Wecanalsoworkoutbonddistancesfromthisinformation.ThefirstAlionisoctahedrallycoordinatedbysixoxygens（1,5,9,13,17,21）andtheAl-OdistanceisWyckoffPositionsofGroup1（P1）WyckoffPositionsofGroup2（P-1）WyckoffPositionsofGroup3（P2）uniqueaxisbuniqueaxiscWyckoffPositionsofGroup4（P21）uniqueaxisbuniqueaxiscWyckoffPositionsofGroup5（C2）uniqueaxisbuniqueaxiscWyckoffPositionsofGroup6（Pm）uniqueaxisbuniqueaxiscWyckoffPositionsofGroup7（Pc）uniqueaxisbuniqueaxiscWyckoffPositionsofGroup8（Cm）uniqueaxisbuniqueaxiscWyckoffPositionsofGroup9（Cc）uniqueaxisbuniqueaxiscWyckoffPositionsofGroup10（P2/m）uniqueaxisbuniqueaxiscWyckoffPositionsofGroup11（P21/m）uniqueaxisbuniqueaxiscWyckoffPositionsofGroup12（C2/m）uniqueaxisbuniqueaxiscWyckoffPositionsofGroup13（P2/c）uniqueaxisbuniqueaxiscWyckoffPositionsofGroup14（P21/c）uniqueaxisbuniqueaxiscWyckoffPositionsofGroup15（C2/c）uniqueaxisbuniqueaxiscWyckoffPositionsofGroup16（P222）WyckoffPositionsofGroup17（P2221）WyckoffPositionsofGroup18（P21212）WyckoffPositionsofGroup19（P212121）WyckoffPositionsofGroup20（C2221）WyckoffPositionsofGroup21（C222）WyckoffPositionsofGroup22（F222）WyckoffPositionsofGroup23（I222）WyckoffPositionsofGroup24（I212121）WyckoffPositionsofGroup25（Pmm2）WyckoffPositionsofGroup26（Pmc21）WyckoffPositionsofGroup27（Pcc2）WyckoffPositionsofGroup28（Pma2）WyckoffPositionsofGroup29（Pca21）WyckoffPositionsofGroup30（Pnc2）WyckoffPositionsofGroup31（Pmn21）WyckoffPositionsofGroup32（Pba2）WyckoffPositionsofGroup33（Pna21）WyckoffPositionsofGroup34（Pnn2）WyckoffPositionsofGroup35（Cmm2）WyckoffPositionsofGroup36（Cmc21）WyckoffPositionsofGroup37（Ccc2）WyckoffPositionsofGroup38（Amm2）WyckoffPositionsofGroup39（Abm2）WyckoffPositionsofGroup40（Ama2）WyckoffPositionsofGroup41（Aba2）WyckoffPositionsofGroup42（Fmm2）WyckoffPositionsofGroup43（Fdd2）WyckoffPositionsofGroup44（Imm2）WyckoffPositionsofGroup45（Iba2）WyckoffPositionsofGroup46（Ima2）WyckoffPositionsofGroup47（Pmmm）WyckoffPositionsofGroup48（Pnnn）originchoice1WyckoffPositionsofGroup48（Pnnn）originchoice2WyckoffPositionsofGroup49（Pccm）WyckoffPositionsofGroup50（Pban）originchoice1WyckoffPositionsofGroup50（Pban）originchoice2WyckoffPositionsofGroup51（Pmma）WyckoffPositionsofGroup52（Pnna）WyckoffPositionsofGroup53（Pmna）WyckoffPositionsofGroup54（Pcca）WyckoffPositionsofGroup55（Pbam）WyckoffPositionsofGroup56（Pccn）WyckoffPositionsofGroup57（Pbcm）WyckoffPositionsofGroup58（Pnnm）WyckoffPositionsofGroup59（Pmmn）originchoice1WyckoffPositionsofGroup59（Pmmn）originchoice2WyckoffPositionsofGroup60（Pbcn）WyckoffPositionsofGroup61（Pbca）WyckoffPositionsofGroup62（Pnma）WyckoffPositionsofGroup63（Cmcm）WyckoffPositionsofGroup64（Cmca）WyckoffPositionsofGroup65（Cmmm）WyckoffPositionsofGroup66（Cccm）WyckoffPositionsofGroup67（Cmma）WyckoffPositionsofGroup68（Ccca）originchoice1WyckoffPositionsofGroup68（Ccca）originchoice2WyckoffPositionsofGroup69（Fmmm）WyckoffPositionsofGroup70（Fddd）originchoice1WyckoffPositionsofGroup70（Fddd）originchoice2WyckoffPositionsofGroup71（Immm）WyckoffPositionsofGroup72（Ibam）WyckoffPositionsofGroup73（Ibca）WyckoffPositionsofGroup74（Imma）WyckoffPositionsofGroup75（P4）WyckoffPositionsofGroup76（P41）WyckoffPositionsofGroup77（P42）WyckoffPositionsofGroup78（P43）WyckoffPositionsofGroup79（I4）WyckoffPositionsofGroup80（I41）WyckoffPositionsofGroup81（P-4）WyckoffPositionsofGroup82（I-4）WyckoffPositionsofGroup83（P4/m）WyckoffPositionsofGroup84（P42/m）WyckoffPositionsofGroup85（P4/n）originchoice1WyckoffPositionsofGroup85（P4/n）originchoice2WyckoffPositionsofGroup86（P42/n）originchoice1WyckoffPositionsofGroup86（P42/n）originchoice2WyckoffPositionsofGroup87（I4/m）WyckoffPositionsofGroup88（I41/a）originchoice1WyckoffPositionsofGroup88（I41/a）originchoice2WyckoffPositionsofGroup89（P422）WyckoffPositionsofGroup90（P4212）WyckoffPositionsofGroup91（P4122）WyckoffPositionsofGroup92（P41212）WyckoffPositionsofGroup93（P4222）WyckoffPositionsofGroup94（P42212）WyckoffPositionsofGroup95（P4322）WyckoffPositionsofGroup96（P43212）WyckoffPositionsofGroup97（I422）WyckoffPositionsofGroup98（I4122）WyckoffPositionsofGroup99（P4mm）WyckoffPositionsofGroup100（P4bm）WyckoffPositionsofGroup101（P42cm）WyckoffPositionsofGroup102（P42nm）WyckoffPositionsofGroup103（P4cc）WyckoffPositionsofGroup104（P4nc）WyckoffPositionsofGroup105（P42mc）WyckoffPositionsofGroup106（P42bc）WyckoffPositionsofGroup107（I4mm）WyckoffPositionsofGroup108（I4cm）WyckoffPositionsofGroup109（I41md）WyckoffPositionsofGroup110（I41cdWyckoffPositionsofGroup111（P-42m）WyckoffPositionsofGroup112（P-42c）WyckoffPositionsofGroup113（P-421m）WyckoffPositionsofGroup114（P-421c）WyckoffPositionsofGroup115（P-4m2）WyckoffPositionsofGroup116（P-4c2）WyckoffPositionsofGroup117（P-4b2）WyckoffPositionsofGroup118（P-4n2）WyckoffPositionsofGroup119（I-4m2）WyckoffPositionsofGroup120（I-4c2）WyckoffPositionsofGroup121（I-42m）WyckoffPositionsofGroup122（I-42d）WyckoffPositionsofGroup123（P4/mmm）WyckoffPositionsofGroup124（P4/mcc）WyckoffPositionsofGroup125（P4/nbm）originchoice1WyckoffPositionsofGroup125（P4/nbm）originchoice2WyckoffPositionsofGroup126（P4/nnc）originchoice1WyckoffPositionsofGroup126（P4/nnc）originchoice2WyckoffPositionsofGroup127（P4/mbm）WyckoffPositionsofGroup128（P4/mnc）WyckoffPositionsofGroup129（P4/nmm）originchoice1WyckoffPositionsofGroup129（P4/nmm）originchoice2WyckoffPositionsofGroup130（P4/ncc）originchoice1WyckoffPositionsofGroup130（P4/ncc）originchoice2WyckoffPositionsofGroup131（P42/mmc）WyckoffPositionsofGroup132（P42/mcm）WyckoffPositionsofGroup133（P42/nbc）originchoice1WyckoffPositionsofGroup133（P42/nbc）