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24
1.AnIntroductiontotheCrystalStructuresofOxidePerovskites
lyingaboveandbelowthefirstlayer.Iftheoctahedradirectlyabovei.e.
alongc
protateinthesamesense,wewritethistiltsystemasa
0a0c+,the
a
0a0termssignifyingnotiltingaboutthea
pandb
paxes.Alternativelythe
octahedramayalternatetheirsenseofrotationalongthec
p-axisandthis
wewriteasa
0a0c-.Bothtiltsystemscreateatetragonalcrystalstructure,
buta
0a0c+producesapseudocubicunitcellofa
pxa
px2c
p,whilea
0a0c-creates
2a
px2a
px2c
p.
Itisthenasimplemattertoapplythisideaabouttheothertwoaxesto
builduptwoandthree-tiltsystems.Figure1.9showsanexampleofathree-
tiltsystemlabeleda
+b+c+,meaningthattherearein-phasetiltsabouteach
axisofdiferentmagnitudes.Interestinglythetiltsystema
+a+a+inwhich
thethreetiltsareofequalmagnitudeisanother,butrare,cubicperovskite
structure.
a+b+c+
Figure1.9.Anexampleofathree-tiltsystem.
Originally,thereweresuggestedtobe23diferenttiltsystems,butthe
numberwaslaterreducedto15bygrouptheoryarguments[20].Table1.1
showsacompletelisting.
Therearemanyexamplesofperovskiteswithtiltedoctahedra,some-
timescombinedwithatomicdisplacements.Thus,forexample,SrTiO
3has
notiltsatroomtemperaturebutonloweringtoaround105Ktiltsappear
withthetiltsystema
0a0c-withspacegroupI4/mcm.InPZT,ontheZr-
richsideoftheMPBtheR3caveragestructurehasthetiltsystema
-a-a-,
thethreetiltsbeingequalinmagnitude.Intheantiferroelectriccompound
NaNbO
3therearesevendiferentphaseswiththeroom-temperaturephase
consistingofamixtureofdiferenttiltsystemsplusantiparallelNadis-
placements.