Nuclear Spin-Lattice Relaxation in One-Dimensional Heisenber

Nuclear spin-lattice relaxation in one-dimensional Heisenberg ferrimagnets is studied by means of a modified spin-wave theory. We consider the second-order process, where a nuclear spin flip induces virtual spin waves which are then scattered thermally via

NuclearSpin-LatticeRelaxationinOne-DimensionalHeisenbergFerrimagnets:

Three-MagnonversusRamanProcesses

HiromitsuHoriandShojiYamamoto

arXiv:cond-mat/0404243v2 [cond-mat.stat-mech] 16 May 2004

DivisionofPhysics,HokkaidoUniversity,Sapporo060-0810,Japan

(Received4March2004)

Nuclearspin-latticerelaxationinone-dimensionalHeisenbergferrimagnetsisstudiedbymeansofamodi edspin-wavetheory.Weconsiderthesecond-orderprocess,whereanuclearspin ipinducesvirtualspinwaveswhicharethenscatteredthermallyviathefour-magnonexchangeinteraction,aswellasthe rst-orderprocess,whereanuclearspindirectlyinteractswithspinwavesviathehy-per neinteraction.Wepointoutapossibilityofthethree-magnonrelaxationprocesspredominatingovertheRamanoneandsuggestmodelexperiments.PACSnumbers:75.10.Jm,75.50.Gg,76.50.+g

Designofmolecule-basedferromagnetsisachallengingtopicinmaterialsscience[1]andnumerousquasi-one-dimensionalferrimagnetshavebeensynthesizedinthiscontext.Bimetallicchaincompounds[2,3]areearlyex-amples,oneofwhichhasindeedaccomplishedthethree-dimensionalferromagneticorder.Anotherapproach[4]consistsofbringingintointeractionmetalionsandstableorganicradicals.Genuineorganicferrimagnets[5,6]werealsosynthesized.Thetetramericbond-alternatingchaincompoundCu(C5H4NCl)2(N3)2[7]andthetrimericin-tertwiningdouble-chainmaterialCa3Cu3(PO4)4[8]aredistinctferrimagnetsoftopologicalorigin.

Theoreticalunderstandingofone-dimensionalferri-magnetshasalsosigni cantlybeendeveloped[9–14],butverylittleisknownabouttheirdynamicpropertiesyet[15,16].Althoughtheone-dimensionalspindynamicsisamoderntopicofgreatinterest[17],systematiccalcula-tionofthenuclearmagneticrelaxationisstillabsent.Inthelatesixties,PincusandBeeman[18]formulatedthenuclearspin-latticerelaxationrateforHeisengergferro-magnetsandantiferromagnetsbymeansofthespin-wavetheoryandpointedoutasigni cantcontributionofthethree-magnonscatteringtotherelaxationrate.However,theirtheoryisnote ectiveinonedimensionbutisvalidattheonsetofthethree-dimensionallong-rangeorder.Insuchcircumstances,modifyingtheconventionalspin-wavetheory,wemakeour rstattemptatsystematicallydescribingtheone-dimensionalnuclearspindynamicsonthebasisofthespin-wavepicture.Wepredictthatthethree-magnonrelaxationprocessshouldpredominateovertheRamanoneathightemperaturesandweak eldsinone-dimensionalHeisenbergferrimagnets.

Firstofallourscheme[12,19]ofmodifyingthespin-wavetheoryisdistinctfromtheoriginalideaproposedbyTakahashi[20]andHirschetal.[21].Intheoriginalwayofsuppressingthepergenceofthesublatticemag-netizations,aLagrangemultiplieris rstintroducedandthenthee ectiveHamiltonianisdiagonalizedsubjecttozerostaggeredmagnetization.Thethus-obtainedenergyspectrumdependsontemperatureandfailstodescribetheSchottkypeakofthespeci cheat[22].Inordertoobtainbetterthermodynamics,we rstdiagonalizethe

1

Hamiltonian,keepingthedispersionrelationsfreefromtemperature,andthenintroduceaLagrangemultipliersoastominimizethefreeenergy.Thisschemeworkswellforferrimagnets,asdemonstratedinFig.1.Boththeap-proacheswellreproducethemagneticsusceptibilty,butournewshcemeismuchbetterthantheoriginaloneatdescribingthespeci cheat.Thenonvanishingspeci cheatathightemperatures,thatis,theendlesslyincreas-ingenergywithtemperature,isafatalweakpointoftheoriginalscheme.

WeconsiderHeisenbergferrimagneticchainsofalter-natingspinsSands,asdescribedbytheHamiltonianH=

N

n=1

z z

JSn·sn 1+sn gµBHSn+sn.

(1)

BBC / NkB

kBT/J

kBT/J

FIG.1.Modi edspin-wavecalculationsofthespeci cheatandthemagneticsusceptibilityasfunctionsoftemperatureforthespin-(52)ferrimagneticHeisenbergchain.Theorig-inal(Takahashi)andournew(Yamamoto)schemesarecom-paredwithnumerical ndings(Exact)[10].

(a)

(b)

(c)

(d)

k2

FIG.2.Illustrationoftheelementarynuclearspin-latticerelaxationprocesses.Solidarrowsdesignateferromagneticorantiferromagneticspinwavesinducinganuclearspin ip(×).Brokenarrowsdenotethefour-magnonexchangeinteraction.(a)Thedirectprocess.(b)The rst-orderRamanprocess.(c)The rst-orderthree-magnonprocess.(d)Thesecond-orderthree-magnonprocess,whereq= k4≡k3 k2 k1.

Nuclear spin-lattice relaxation in one-dimensional Heisenberg ferrimagnets is studied by means of a modified spin-wave theory. We consider the second-order process, where a nuclear spin flip induces virtual spin waves which are then scattered thermally via

IntroducingtheFourier-transformedHolstein-Primako magnonoperatorsakandbkforeachsublattice,weexpandtheHamiltonianwithrespectto1/SasH=H2+H1+H0+O(Si),whereweassumethatO(S)=O(s)andHicontainstheO(Si)terms.TheO(S 1)contribu-tionsareneglectedinthefollowing.Thehyper neinter-actionisassumedtobeofthedipolartypeasHhf=gµB¯hγNI+

N 1×

n=1

Sscos(k/2)/(S+s),

createtheferromagneticandantiferromagneticspinwavesofmomentumk,respectively,whoseenergiesare

σ

givenbyωk=ωk+σ(S s) σgµBHwithωk≡

2

[(S s)+4Sssin2(k/2)]1/2.Wecalculatethethermal

distributionfunctions, α ¯ ¯+kαk ≡nkand βkβk ≡nk,byenforcingthezerostaggered-magnetizationconstraint[12].Ifweconsideruptothesecond-orderperturbationwithrespecttoV,theprobabilityofanuclearspinbeingscatteredfromthestateofIz=mtothatofIz=m+1isexpressedas

2 2π

W= i δ(Ei Ef),

Ei Em

(4)

whereiandfdesignatetheinitialand nalstatesofthe

electronic-nuclearspinsystemdescribedbytheunper-turbedHamiltonian.Thenuclearspin-latticerelaxationX(k1,k2)=

timeisthengivenbyT1=(I m)(I+m+1)/2W.Equa-tion(4)containsvariouselementaryrelaxationprocesses,whichareillustratedinFig.2.Duetoth …… 此处隐藏：10545字，全部文档内容请下载后查看。喜欢就下载吧 ……