frustration and ice - ucsb mrsecseshadri/2016_286g/frustration-ice.pdf · frustration and ice. the...
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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.
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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
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Ice,spinels,pyrochlores andspin-ice
Howtoorder thehydrogens inice:addOH—
H2Odoped with0.1moldm-3 ofKOH
PhasetransitioninKOH-dopedhexagonalice,Y.Tajimaetal.Nature299(1982)810.
Phasetransitionnear80Ktoanorderedstructurewithdecreasedresidualentropy.
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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