晶體結(jié)構(gòu)點(diǎn)陣對(duì)稱性

1、晶體對(duì)稱性任山中山大學(xué)，理工學(xué)院納米樓305（南校區(qū)）Symmetry and point groups of crystal晶體的對(duì)稱和點(diǎn)群Sept. 2012Crystal symmetry, SYSU,S Ren2點(diǎn)陣結(jié)構(gòu)層次Sept. 2012Crystal symmetry, SYSU,S Ren3¨ 晶體結(jié)構(gòu) = lattice + motif¨ 7 個(gè)晶系¨ 14種Bravie 陣點(diǎn)¨ 32 個(gè)點(diǎn)群¨ 230 個(gè)空間群2.2.3 Symmetry (宏觀對(duì)稱）¨Acharacteristic of geometrical

2、shapes,equationsandotherobjects;is symmetricanobjectwithrespecttoagivenoperationifthisoperation,whenappliedtotheit.object,doesnotappeartochangeSept. 2012Crystal symmetry, SYSU,S Ren4Symmetricaloperation對(duì)稱操作¨The 5 main symmetrical operations：Rotation (轉(zhuǎn)動(dòng)）Reflection （反映-鏡面） Inversion (反演） Rotatio

3、n+inversion (轉(zhuǎn)動(dòng)反演）translation （）Any crystal has at least one of these symmetrical operationsSept. 2012Crystal symmetry, SYSU,S Ren5Symmetricalelement對(duì)稱元素與操作Sept. 2012Crystal symmetry, SYSU,S Ren6nRotation(旋轉(zhuǎn)對(duì)稱)符號(hào)：Sept. 2012Crystal symmetry, SYSU,S Ren76、3RotationSept. 2012Crystal symmetry, SYSU,S Re

4、n85、7Rotation自然界不具有五次及七次以上的對(duì)稱是否有例外？ 準(zhǔn)晶Sept. 2012Crystal symmetry, SYSU,S Ren9Reflection鏡像（反映）-對(duì)稱面mirrorSept. 2012Crystal symmetry, SYSU,S Ren10Inversion演射對(duì)稱點(diǎn)Sept. 2012Crystal symmetry, SYSU,S Ren11旋轉(zhuǎn)+反演軸¨ 若晶體繞某一軸回轉(zhuǎn)一定角度（360°/n），再以軸上的一個(gè)中心點(diǎn)作反演之后能得到復(fù)原時(shí)，此軸稱為旋轉(zhuǎn)反演軸。旋轉(zhuǎn)反演軸的對(duì)稱操作是一根直線旋轉(zhuǎn)和對(duì)此直線上一點(diǎn)反演。Sep

5、t. 2012Crystal symmetry, SYSU,S Ren12旋轉(zhuǎn)反演軸Sept. 2012Crystal symmetry, SYSU,S Ren13旋轉(zhuǎn)反演軸的表示Sept. 2012Crystal symmetry, SYSU,S Ren14旋轉(zhuǎn)+反演軸¨ Rotation +Inversion =鏡面1 (= i),2 (= m),3,4,6Sept. 2012Crystal symmetry, SYSU,S Ren15反映及演射 = chiral 手性結(jié)構(gòu)Sept. 2012Crystal symmetry, SYSU,S Ren16TranslationSep

6、t. 2012Crystal symmetry, SYSU,S Ren17基本對(duì)稱操作規(guī)律Sept. 2012Crystal symmetry, SYSU,S Ren188種基本宏觀對(duì)稱操作Sept. 2012Crystal symmetry, SYSU,S Ren19The many thousands of lattices classified into 7 crystal systemsSystemMinimum symmetryTriclinic Monoclinic Orthorhombic Tetragonal Cubic Hexagonal Trigonal1 or 1 2 o

7、r 2 three 2s or 2s 4 or 4 four 3s or 3s 6 or 63 or 3Sept. 2012Crystal symmetry, SYSU,S Ren2032種點(diǎn)群晶體的宏觀外形可以只有一種對(duì)稱元素，也可¨以有若干對(duì)稱元素同時(shí)同組合就可以組成形形，由上面八種對(duì)稱元素的不晶體的各種宏觀對(duì)稱性。但是晶體除了對(duì)稱性以外，還必須具有周期性這樣一個(gè)特點(diǎn)。因此這些對(duì)稱元素的組合不能是任意的，必須遵¨循對(duì)稱元素的組合規(guī)律，使對(duì)稱元間相互制約而又相互協(xié)調(diào)，利用數(shù)學(xué)可能有的組合數(shù)為32種，即32種點(diǎn)群?？梢詫?dǎo)出這八個(gè)宏觀對(duì)稱元素了晶體32種宏觀對(duì)稱類型之所以稱其

8、為點(diǎn)群，是因?yàn)槊糠N宏觀對(duì)稱類型中的各個(gè)對(duì)稱元素必須至少相交于一點(diǎn)，此點(diǎn)稱為點(diǎn)群中心。¨Sept. 2012Crystal symmetry, SYSU,S Ren2132點(diǎn)群的四類操作組合¨ 八種宏觀對(duì)稱元素按照一定規(guī)則進(jìn)行組合，可以得到32種點(diǎn)群。這32種點(diǎn)群概括起來(lái)屬于以下四種組合之一。 1對(duì)稱軸的組合，當(dāng)五個(gè)對(duì)稱軸分別與一個(gè)二次軸組合后，可以得到4個(gè)雙面群，2個(gè)等軸旋轉(zhuǎn)群，加上原來(lái)5個(gè)群，共11個(gè)點(diǎn)群；旋轉(zhuǎn) 2反演中心分別加到上述11個(gè)點(diǎn)群上，可以得到另外11個(gè)新的點(diǎn)群； 3對(duì)稱面與第一11個(gè)點(diǎn)群組合（分別以垂直和平行的方式加入鏡面），這樣又可以增加9個(gè)點(diǎn)群； 4四次

9、反演，因?yàn)樗荒芘c其它對(duì)稱元素組新的對(duì)稱群，所以它成為一個(gè)點(diǎn)群。Sept. 2012Crystal symmetry, SYSU,S Ren2232點(diǎn)群的對(duì)稱操作組合Sept. 2012Crystal symmetry, SYSU,S Ren23320種空間群Sept. 2012Crystal symmetry, SYSU,S Ren24空間群¨ 晶體外形的對(duì)稱用點(diǎn)群來(lái)說(shuō)明，而晶體內(nèi)部結(jié)構(gòu)原子、離子、分子類別和排列的對(duì)稱性類別則用空間群來(lái)說(shuō)明。¨ 這種微觀的對(duì)稱性不但了所有宏觀對(duì)稱元素而且又多出三類微觀對(duì)稱元素：、螺旋軸以及滑移面。¨ 導(dǎo)致這種差異的根本在于晶體內(nèi)

10、部結(jié)構(gòu)具有特有的對(duì)稱性。¨ 晶體內(nèi)部結(jié)構(gòu)中所有對(duì)稱操作的組合叫做空間群。Sept. 2012Crystal symmetry, SYSU,S Ren25空間群的數(shù)量Sept. 2012Crystal symmetry, SYSU,S Ren26微觀對(duì)稱元素¨ 宏觀對(duì)稱元素在晶體內(nèi)部結(jié)構(gòu)中仍然。此外，由于介入操作又出現(xiàn)了一些新的對(duì)稱元素，將這些微觀特有的對(duì)稱元素稱為微觀對(duì)稱元素。¨ 微觀對(duì)稱元素主要分為以下三類。軸螺旋軸滑移面Sept. 2012Crystal symmetry, SYSU,S Ren271軸¨ 晶體最根本的特點(diǎn)是其內(nèi)部結(jié)構(gòu)具有空間點(diǎn)陣的特

11、征，即晶體內(nèi)部結(jié)構(gòu)的周期性。若點(diǎn)陣沿著點(diǎn)陣中某一方向上任何兩點(diǎn)的矢量進(jìn)行，點(diǎn)陣必然復(fù)原，由這種操作所組合的對(duì)稱群稱為群，可以用下式來(lái)表達(dá)：¨ Tmnpm a n b p c¨ 其中： m、n、p0、±1、±2Sept. 2012Crystal symmetry, SYSU,S Ren28軸點(diǎn)陣與相應(yīng) 要的性質(zhì)上：群間的對(duì)應(yīng)還體現(xiàn)在下列兩個(gè)重¨連接點(diǎn)陣中任意兩點(diǎn)的矢量必屬于矢量；群T中的一個(gè)¨屬于群T中的任何矢量必定通過(guò)點(diǎn)陣中的二個(gè)節(jié)¨點(diǎn)。所以不難得出：。群矢量之組合仍屬于群晶體的宏觀對(duì)稱性與微觀對(duì)稱性的區(qū)別就在于：宏觀對(duì)稱操

12、作至少要求有一點(diǎn)不動(dòng)，而微觀對(duì)稱操作要求¨全部點(diǎn)。因此，宏觀對(duì)稱性無(wú)法反映微觀對(duì)稱性中的部分。，當(dāng)宏觀對(duì)稱元素一旦與結(jié)合起來(lái)即可形成新的微觀對(duì)稱元素。Sept. 2012Crystal symmetry, SYSU,S Ren292螺旋軸螺旋軸是設(shè)想的直線，晶體內(nèi)部的相同部分繞其周期¨轉(zhuǎn)動(dòng)，并且附以軸向得到重復(fù)。螺旋軸是一種復(fù)合的對(duì)稱要素，其輔助幾何要素為：一根假想的直線及與之平行的直線方向。相應(yīng)的對(duì)稱變換為，此直線旋轉(zhuǎn)一定的角度和此直線方向的聯(lián)合。螺旋軸的n只能等于1、2、3、4、6，所包含的變換其距離應(yīng)等于沿螺旋軸方向結(jié)點(diǎn)間距的s/n，s為小于n的自然數(shù)。螺旋軸的國(guó)際符

13、號(hào)為ns。旋轉(zhuǎn)軸根據(jù)其軸次和距離的大小的不同可分為21¨、31、32、41、42、43、61、62、63、64、65共11種螺旋軸。螺旋軸根據(jù)其旋轉(zhuǎn)方向可分為左旋、右旋和中性旋轉(zhuǎn)軸。Sept. 2012Crystal symmetry, SYSU,S Ren30螺旋軸表達(dá)Sept. 2012Crystal symmetry, SYSU,S Ren31螺旋軸各種圖像Sept. 2012Crystal symmetry, SYSU,S Ren323滑移面滑移面是設(shè)想的平面。晶體內(nèi)部的相同部分沿平行于¨該面的直線方向后再反演而會(huì)得到重復(fù)?；泼嬉彩且环N復(fù)合的對(duì)稱要素，其輔助對(duì)稱要

14、素有兩個(gè)： 一個(gè)是假想的平面和平行此平面的某一直線方向。相應(yīng)的對(duì)稱變換為：對(duì)于此平面的反映和沿此直線方¨向的聯(lián)合，其的距離等于該方向行列結(jié)點(diǎn)間距的一半。根據(jù)成分的大小，滑動(dòng)面可歸納為以¨下三種：¨¨¨軸滑移面對(duì)角滑移面石滑移面Sept. 2012Crystal symmetry, SYSU,S Ren33軸滑移面Sept. 2012Crystal symmetry, SYSU,S Ren34對(duì)角滑移面Sept. 2012Crystal symmetry, SYSU,S Ren35石滑移面Sept. 2012Crystal symmetry, S

15、YSU,S Ren36滑移面的類型Sept. 2012Crystal symmetry, SYSU,S Ren37230space groups¨ all230spacegroupsisfromthecombinationofthe32the14onecrystallographic point groupswithBravais lattices ofwhichbelongto7 crystal systems.¨ Thisresultsinaspacegroupbeingacombinationofaunit cellwithsomeformofmotifcenteri

16、ng,alongwiththepointoperationsofreflection,rotationandimproper-rotation.Sept. 2012Crystal symmetry, SYSU,S Ren38230 Space GroupsSept. 2012Crystal symmetry, SYSU,S Ren39點(diǎn)群與空間群的對(duì)比Sept. 2012Crystal symmetry, SYSU,S Ren402.2.7，空間點(diǎn)群的表示Sept. 2012Crystal symmetry, SYSU,S Ren41Patersonnotation¨ Consist

17、sofasetoffour symbols.Thefirstdescribes thecentering or F).oftheBravais lattice (P,C,IThenextthreedescribethemostprominent projectedsymmetryoperationvisible facewhenfromthea,band crespectively.¨ Thesesymbolsare withthethesameasusedinpoint groups,additionofglide above.planesandscrewaxis,desribed

18、Sept. 2012Crystal symmetry, SYSU,S Ren42空間點(diǎn)群表示實(shí)例Sept. 2012Crystal symmetry, SYSU,S Ren43What kind of symmetry does this object have?4 (旋轉(zhuǎn)軸)Sept. 2012Crystal symmetry, SYSU,S Ren44What kind of symmetry does this object have?4m (鏡)Crystal symmetry, SYSU,S RenSept. 201245What kind of symmetry does this

19、 object have?4mSept. 2012Crystal symmetry, SYSU,S Ren46What kind of symmetry does this object have?4mCrystal symmetry, SYSU,S RenSept. 201247What kind of symmetry does this object have?44mm(點(diǎn)群)mCrystal symmetry, SYSU,S RenSept. 201248Another example:Sept. 2012Crystal symmetry, SYSU,S Ren49Another example:6點(diǎn)群：6mmmSept. 2012Crystal symmetry, SYSU,S Ren50And another:Sept. 2012Crystal symmetry, SYSU,S Ren51And another:2點(diǎn)群：2Sept. 2012Crystal symmetry, SYSU,S Ren52Stereographic projections of symmetry groupsRotation + mirr