Thecubic(Fd-3m)structureofpyrochlore (CaNa)Nb2O6F[A2B2O7 orA2B2O6Oʹ]TheAsiteoftenhaslone-paircations (Pb2+orBi3+).Polarmaterialsinthisstructuretypearerarehowever.

Frustrationandice

TheAatomnetworkofconnectedA4 tetrahedra inA2B2O7 isfrustratedwithrespecttocertainkindsofmagneticordering.

SimilaritieswiththecrystalstructureoficeIh:thenotionofspin ice.

Bramwell,Gingras,Science294(2001)1495.

Themorefamiliarspinice

Frustrationandice

ice spinice

Well-known frustrationofspinsoncornersoftriangles.

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Ice,spinels,pyrochlores andspin-ice

ATheoryofWaterandIonicSolution, withParticularReferencetoHydrogenandHydroxylIons,J.D.BernalandR.H.Fowler, J.Chem.Phys. 1 (1933)515-548.

Ice-Ih:a =7.82Å;c =7.36ÅP63cmProtonordering notproved

Atom # OX SITE x y z SOF

O 1 -2 6 c 0.3333 0 0.0625 1. O 2 -2 6 c 0.6667 0 0.9375 1. H 1 +1 6 c 0.3333 0 0.174 1. H 2 +1 6 c 0.438 0 0.026 1. H 3 +1 12 d 0.772 0.105 0.975 1.

ViewsoftheorderedBernal-Fowlerstructure.Hydrogenspositioned through guesswork.

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Ice,spinels,pyrochlores andspin-ice

Actualdisordered structureof Ice-Ih:P63/mmc hexagonaldiamond lattice.

Ice-Ih:a =4.511(3)Å;c =7.346(3)ÅP63/mmcO 1/3 2/3 0.06226(8)H1 1/3 2/3 0.178(3) [Occ.=0.5]H2 0.439(3)0.878(3)0.020(3) [Occ.=0.5]Gotoetal. J.Chem.Phys.93 (1990)1412.

TheBernal-Fowlericerules:1) Eachwatermoleculeisorientedsuchthatitstwohydrogenatoms

aredirectedapproximatelytowardtwoofthefour surroundingoxygenatoms(arrangedalmostinatetrahedron).

2) OnlyonehydrogenatomispresentoneachO-Olinkage.3) Eachoxygenatomhastwonearestneighboring hydrogenatoms

suchthatthewatermoleculestructureispreserved.

5

Ice,spinels,pyrochlores andspin-ice

LinusPaulingandresidualentropy:TheStructureandEntropyofIceandofOtherCrystalswithSomeRandomnessofAtomicArrangement, L.Pauling, J.Am.Chem.Soc.57 (1935)2680-2684.Alsoseehardcopyhandout.

degenerateconfigurationsofhydrogen inice

Theresidualentropyofice,extrapolatedto0KisS =Rln(3/2)

ProvedbyGiaque.

Alsosee:Residualentropyofsquareice,E.H.Lieb,Phys.Rev.162 (1967)162.

http://link.aps.org/abstract/PR/v162/p162

6

Ice,spinels,pyrochlores andspin-ice

Howtoorder thehydrogens inice:addOH—

H2Odoped with0.1moldm-3 ofKOH

PhasetransitioninKOH-dopedhexagonalice,Y.Tajimaetal.Nature299(1982)810.

Phasetransitionnear80Ktoanorderedstructurewithdecreasedresidualentropy.

7

Ice,spinels,pyrochlores andspin-ice

Orderinghydrogens though pressure:Themanyphasesofice.

Ice-IIhasallH(D)atomslocatedat80K.Thestructureisrhombohedral.

Thestructureofanewphaseofice,C.Lobban, J.L.Finney,andW.F.Kuhs,Nature391(1998)268

Thepreciseanalogywithice:Bernal-Fowler(1933)Icerules

§ Oxygens inice-Ih formawurtzite (tetrahedral)lattice,withanO-O distanceof2.76ŧ The0.95ÅOHbondofH2O isretainedinice-Ih§ EachoxygenmusthavetwoHat0.95Åandtwoat1.81Å,but whichtwo?

Allaboutice

Pauling(1935):Ice-Ih hasresidualentropy16waysofarranging

HaroundO.

Allaboutice

CalculatedbyPauling:0.80cal/K/molMeasuredbyGiauque:0.82cal/K/mol

Pauling(1935):Ice-Ih hasresidualentropy16waysofarranging

HaroundO.Only6obeytheicerules:

S=kBlnW andW=6(1/2)(1/2) =3/2

Allaboutice

Ramirez,Hayashi,Cava,Siddharthan,Shastry,Nature294(2001)1495.

Dy2Ti2O7

Heatcapacitysignaturesinspinice

Theincompleteorderingofspinsatlowtemperaturesinspin-iceresultsincharacteristicheatcapacitysignatures.

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Ice,spinels,pyrochlores andspin-ice

Thespinel structure:MgAl2O4Fd-3m(diamond) a~8.5Å

A 1/8 1/8 1/8B 1/2 1/2 ½O 0.264 0.264 0.264**inMgAl2O4AaretetrahedralwithOandBareoctahedral

“Startingwithanarrayofoxygensinccp,weinsertAlincertanoctahedralintersticesandMgincertaintetrahedralinterstices,theselectionofn intersticesbeingmadeinsuchawaythattherepeatdistancealongeachaxisisdoublewhatitwouldbefortheidealclosepacking...”

Megaw

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Ice,spinels,pyrochlores andspin-ice

Magnetism inspinels:FerrimagnetismInthespinelstructure,unlikeperovskiteandpyrochlore, bothAandBionscanbemagnetic(1st rowtransitionmetals).Theycouldwitheachotherantiferromagnetically,butthereisanetmomentbecausetheydonotcancelone-another.

Magnetiteorlodestone, fromwhichthetermmagnetism derives,isactuallyaferrimagneticspinel.

FromLouisNéel’s1970Nobellecture(nobel.se)

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Ice,spinels,pyrochlores andspin-ice

Spinelmagnetismandice:

ThespinelBsitesformanetworkofcorner-connectedtetrahedra.Antiferromagnetism isfrustrated.

OrderingandAntiferromagnetism inFerrites,P.W.Anderson,Phys.Rev. 102(1956)1008.

“Theoctahedralsitesinthespinel structureformoneoftheanomalous latticesinwhichitispossibletoachieveessentiallyperfectshort-rangeorderwhilemaintainingafiniteentropy.Insuchalatticenearest-neighbor forcesalonecanneverleadtolong-rangeorder,whilecalculationsindicatethateventhelong rangeCoulomb forcesareonly5%effectiveincreatinglong-rangeorder.Thisisshowntohavemanypossibleconsequencesboth forantiferromagnetism in"normal" ferritesandforordering in"inverse"ferrites.”

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Ice,spinels,pyrochlores andspin-ice

Gettingridoffrustration:Structuraldistortions inZnCr2O4 andZnV2O4:

ElongateFd-3mà I-41/amd

frustratedordered

CrystallographicandmagneticstructureofZnV2O4:Structuralphasetransitionduetospin-driven Jahn-Tellerdistortions,M.Reehuis,A.Krimmel,N.Büttgen,A.LoidlandA.Prokofiev, Eur.Phys.J.B35,311-316(2003).

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Ice,spinels,pyrochlores andspin-ice

S.T.BramwellandM.J.P.Gingras,Science 294 (2001)1495.

Spin iceinpyrochlores:Dy2Ti2O7