Quasicrystals

AlCuLiQCRhombictriacontrahedralgrainTypicaldecagonalQCdiffractionpattern(TEM)1Quasicrystals

Diffractionpatternfor8-foldQCDiffractionpatternfor12-foldQC2Quasicrystals

PrincipaltypesofQCs:icosahedraldecagonal3Quasicrystals

PrincipaltypesofQCs:icosahedraldecagonalmetastable(rapid solidifcation)stable(conventional solidification)4Quasicrystals

PrincipaltypesofQCs:icosahedraldecagonalmetastable(rapid solidifcation)stable(conventional solidification)QCsusuallyhavecompositionsclosetocrystallinephases-the"crystallineapproximants"5Quasicrystals

Whilepentagons(108°angles)cannottiletofill2=Dspace,tworhombsw/72°&36°anglescan-ifmatchingrulesarefollowedN.B.-seedefinitive&comprehensivebookontilingbyGrünbaumandShepherd6Quasicrystals

Whilepentagons(108°angles)cannottiletofill2=Dspace,tworhombsw/72°&36°anglescan-ifmatchingrulesarefollowed7Quasicrystals

FouriertransformofthisPenrosetilinggivesapatternwhichexhibits5(10)-foldsymmetry–verysimilartodiffractionpatternsforicosahedralQCs8Quasicrystals

9Quasicrystals

10Quasicrystals

Diffractionpattern(inreciprocalspace)oficosahedralQCcanbeindexedw/6sixintegers-axesalong6icosahedrondirectionsqi(referredtoCartesianqx,qy,qz)1t11Quasicrystals

Diffractionpattern(inreciprocalspace)oficosahedralQCcanbeindexedw/6sixintegers-axesalong6icosahedrondirectionsqi(referredtoCartesianqx,qy,qz)q1

=(1t0)q2

=(t01)q3

=(t01)q4

=(01t)q5

=(1t0)q6

=(0t1)1t12Quasicrystals

t=

(1+5)2

=1.618…Diffractionpattern(inreciprocalspace)oficosahedralQCcanbeindexedw/6sixintegers-axesalong6icosahedrondirectionsqi(referredtoCartesianqx,qy,qz)

q1

=(1t0)q2

=(t01)q3

=(t01)q4

=(01t)q5

=(1t0)q6

=(0t1)1t13Quasicrystals

Diffractionpattern(inreciprocalspace)oficosahedralQCcanbeindexedw/6sixintegers-axesalong6icosahedrondirectionsqi(referredtoCartesianqx,qy,qz) q1

=(1t0)q2

=(t01)q3

=(t01)q4

=(01t)q5

=(1t0)q6

=(0t1)Thus,icosahedralQCisperiodicin6D14Quasicrystals

Alsoconsider:toperiodicallytilein2-D–needthreetranslationvectorsif5-fold,reasonablecellispentagon–needadditionaldimensiontofillspace(tile)–moretranslationvectors15Quasicrystals

Diffractionpattern(inreciprocalspace)oficosahedralQCcanbeindexedw/6sixintegers-axesalong6icosahedrondirectionsqi(referredtoCartesianqx,qy,qz)q1

=(1t0)q2

=(t01)q3

=(01t)q4

=(1t0)q5

=(t01)q6

=(01t)Thus,icosahedralQCisperiodicin6DButnotin3DTounderstandthis,considerperiodic2Dcrystal:16Quasicrystals

Tounderstandthis,considerperiodic2Dcrystal:The2Dcrystalisnotinourobservableworld-whatISseenisthecutalongEButcutalongEmayormaynotpassthroughlatticenodesCutshownhasslope1/t-doesnotpassthroughlatticenodesexceptorigin17Quasicrystals

Tounderstandthis,considerperiodic2Dcrystal:Butcanobservebothrealstructureanddiffractionpatternforthis1DquasiperiodiccrystalMustbesomekindofstructureintheextendedspace(the2nddimension)-shownhereaslinesthroughthe2Dlatticenodes18Quasicrystals

Tounderstandthis,considerperiodic2Dcrystal:Mustbesomekindofstructureintheextendedspace(the2nddimension)-shownhereaslinesthroughthe2DlatticenodesSomeofthelinesintersect"therealworld"cutE,therebyallowingobservationoftherealquasiperiodicstructure19Quasicrystals

Tounderstandthis,considerperiodic2Dcrystal:Noteshort&longsegmentsinrealrealworldcut-form"Fibonaccisequence":slsllslsllsllslsllslsllsllslsllsl…….. ifs=1,l=t

20Quasicrystals

Tounderstandthis,considerperiodic2Dcrystal:Thinkof2spaces-"parallel"(real)&"perp"(extended)21Quasicrystals

Considerincommensuratecrystals:Needadditionaldimensiontocompletelydescribestructure22Quasicrystals

Considerincommensuratecrystals:Similartoquasiperiodiccase23Quasicrystals

Thereare16spacegroupsforthe6-Dpointgroup532w/P,I,F6-dcubiclatticesThe6-Dstructure&theparallel&perpendicularsubspacesareallinvariantundertheoperationsof53224Quasicrystals

Thereare16spacegroupsforthe6-Dpointgroup532w/P,I,F6-dcubiclatticesThe6-Dstructure&theparallel&perpendicularsubspacesareallinvariantundertheoperationsof532Tovisualize6-Dstructure,mustmake2-Dcutswhichnecessarilymustshowbothparallel&perpspaces25Quasicrystals

Thereare16spacegroupsforthe6-Dpointgroup532w/P,I,F6-dcubiclatticesThe6-D