結(jié)晶學(xué)chap3symmgp1教學(xué)幻燈片

Symmetryintwo-dimension2DunitcellPeriodicityin3-dim.–smallestrepeatedunit

unitcellSymmetryintwo-dimensionSymmetryelementsPeriodicityin3-dimetions---SmallestrepeatedUnit---UnitcellSymmetryelements,symbols,matrixrepresentation：Basicsymmetryelements☆properrotationCn→n;symbolin‘pointgroup→spacegroup’e.g.4zTosuitinsidearepeatedunitinthespace1,2,3,4,5,6foldmatrixrepresentation☆mirrorplanes

v,h,d---m(a,b,c,d,n)☆centerofsymmetry☆translationalongedgesofthecellbyfractionsoftheedgelength☆improperrot.s→☆screwaxisrot.+tr.☆glideplanesm+tralonga,b,c,diagonal,

a,b,c,n,d☆translationalongedgesofthecellbyfractionsoftheedgelength+tralongc

DerivedSymmetrywithinthelatticeUnitcellorcrystallatticeformedby3-non-planarvectorsLimitationofsymmetrybyperiodicitytt

n

ttLatticeCentering---Puretranslationalrot:+translationLatticeCentering–puretranslationalP

I

A

B

C

F

R

PI,A,B,CR4

FLatticecenteringCrystalsystemMin.sym.Max.sym.CellparametersPTriclinic1abc,90P,IMonoclinic2abc,90P,I,F,BOrthorhombic222abc,90P,ITetragonal4abc,90PHexagonal6abc,90,120PRTrigonalRhombohedral333mabc,90,120,Va’b’c’,’’’90,V’=1/3VP,I,FCubic23abc,90UnitcellclassificationsSystemandPointGroupPositioninPoint-groupSymbolStereographicrepresentationPrimarySecondaryTertiaryTriclinicOnlyonesymbolwhichdenotesalldirectionsinthecrystal.MonoclinicThesymbolgivesthenatureoftheuniquediadaxis(rotationand/orinversion).1stsetting:z-axisunique(001)2ndsetting:y-axisunique(010)OrthorhombicDiad(rotationand/orinversion)alongx-axis(100)Diad(rotationand/orinversion)alongy-axis(010)Diad(rotationand/orinversion)alongz-axis(001)TetragonalTetrad(rotationand/orinversion)alongz-axis(001)Diads(rotationand/orinversion)alongx-andy-axes

(100)or(010)Diads(rotationand/orinversion)along[110]and[10]axes(110)(10)TrigonalandhexagonalTriadorhexad(rotationand/orinversion)alongx-axis(001)Diads(rotationand/orinversion)alongx-,y-andu-axes(100)____Diads(rotationand/orinversion)normaltox-,y-,u-axesintheplane(001);(100)……CubicDiadsortetrads(rotationand/orinversion)along(100)axes(100)Triads(rotationand/orinversion)along(111)axes(111)Diads(rotationand/orinversion)along(110)axes(110)xyzxTriclinicyzyzxMonoclinic1stsetting2ndsettingyzxOrthorhombicyzxTetragonalyzxTrigonalandhexagonal

uyzxCubicOtherofPositionsintheSymbolsoftheThree-dimensionalPointGroupsasappliedtoLatticesPolesofdirectionsforprimarypositionSecondarydittoTertiarydittoSymmetryOperationsandSpaceGroupsabc

abc

abc

abcabcabcabcThe14Bravaislatticescabcabcab120

cabcabcab續(xù)上頁(yè)a1b1c1abcorP3m1

Lauesymmetry

uniquepartofsphereTriclinic 1/2Monoclinic 2/m1/4Orthorhombic (mmm)1/8Tetragonal1/8 1/16Hexagonal 1/12 1/24Trigonal 1/12 1/12 1/6 Cubic 1/24

1/48SpaceGroupGroupDefinition1.ai

aj=akwhere

akmustbeanelementinthegroup2.musthaveanidentityelement,I,sothat

ai

I

=ai

3.Theinverseofeveryelementmustalsobeanelementinthegroup

4.associativelaw

(ai

aj)ak

=ai(aj

ak)

32PointGourpsCn:rot.1,2,3,4,6(Sn)

inverserot.Cnh:rot.+m

mCnv

rot.+m

3m

Dn

3rot.222,32(2),422,622,23,432

5531

6

Dnh

rot.+m

+

m

mm24mm6mm

93m(7)(32)(230)(7)(32)(230)CrystalsystempointgroupspacegroupsSpacegroupP21/coriginshiftbasic

symelementsrot.tr.tr.

afteroriginshiftto(000)afteroriginshiftof(0,?,?)

P21/cMonoclinicP121/c1PattersonsymmetryP12/m12/mUNIQUEAXISb,CELLCHOICE1P21/cSpacegroup

Pnc2FigCompletedworksheet=

n

basicsymderivedsymP6mm6mmHexagonalP6mmPattersonsymmetryP6mmP6mmPCFI[det]2[det]2PT

R(Trigonal

RhombohedralCell)[det]3R(0,0,0);(2/3,1/3,1/3);(1/3,2/3,2/3)R(0,0,0);(1/3,2/3,1/3);(2/3,1/3,2/3)[det]3P21/cMonoclinic2/mUNIQUEAXISb,DIFFERENTCELLCHOICE1P121/c1UNIQUEAXISb,CELLCHOICE1P121/n1UNIQUEAXISb,CELLCHOICE2P121/a1UNIQUEAXISb,CELLCHOICE3Inverse

Inverse

transpose

transposedirectspacereciprocalspaceCellTransformationCell1Cell2(a,b,c)(h.k.l)(x,y,z);(a*,b*,c*);(u,v,w)ux,vy,wzwhereu,v,wintegerTransformationbetweena1a2c1c2a,b,ch,k,lreversetransposetransposea*,b*,c*x,y,zu,v,wreversePIPFFITrigonalTSrhombohedralcelltrigonalcellobverse(positive)reverse(negative)Trigonallatticesahex

aR

bRbhex

bR

cRchex

aR

bR

cRahex

bR

cRbhex

cR

aRchex

aR

bR

cRororahex

cR

aRbhex

aR

bRchex

aR

bR

cRAsforthehexagonalcell,intheconventionaltrigonalcellthethreefoldaxisischosenparalleltoc,withab,unrestrictedc,90,and120.CentredcellsareeasilyamenabletotheconventionalPtrigonalcell.BecauseofthepresenceofatreefoldaxissomelatticescanexistwhichmaybedescribedviaaPcellofrhombohedralshape,withunitvectorsaR,bR,cRsuchthataRbRcR,R

R

R,andthethreefoldaxisalongtheaR

bR

cRdirection.Suchlatticesmayalsobedescribedbythreehexagonalcellswithbasisvectorsahex,bhe