1、Inelastic Neutron Scattering in Condensed Matter:Phonons,CEF Excitations,Magnons刘本琼刘本琼 11.Basic principles2.Instruments introduction3.Phonons4.Crystal field excitations5.Spin waves:MagnonsOutline2Basic principlesFig.1 Schematic representation of the scattering experiment,and relations among the init
2、ial and final wavevectors ki and kf and the scattering vector Q.2 is the scattering angle,and d is the collection solid angle of the detector.3)(kkQ)(2222kkmBasic principlesInelastic neutron scatteringisanexperimentaltechniquecommonlyusedincondensedmatterresearchtostudyatomicandmolecularmotionaswell
3、asmagneticandcrystalfieldexcitations.4Basic principles5Cold triple-axis spectrometer6monoanadetAn incident beam with wavevector k is selected by the monochromator crystal(1st axis).The monochromatic beam is scattered from the sample(2nd axis).The intensity of the scattered beam with wavevector k is
4、reflected by the analyzer crystal(3rd axis)onto the neutron detector.CTASCMRR:KUNPENG7SamplestageAnalyzerDetectorTOF spectrometers8In neutron scattering experiments,neutron energy can be determined by measuring their flight time t over a distance L of a few meters.The flight time of the neutrons wit
5、h wavevector k and k are t0=L/v,and t=L/v,andDepending on whether k and k is measured by TOF,the method is called direct TOF or inverted TOF,respectively.1.Basic principles2.Instruments introduction3.Phonons4.Crystal field excitations5.Spin waves:MagnonsOutline910Phonons/lattice vibrationsTheatomsin
6、asolidareinconstantmotionandgiverisetolatticevibrations.11Phonons/lattice vibrationsPhonons/lattice vibrations12Fig.2Phonondispersioncurvesforaone-dimensionallineofatomswith(a)asinglemass,(b)twodifferentmassesmandM.Fig.3PhonondispersioncurvesforGealongcertainhighsymmetryaxesintheBrillouinzone.13Long
7、itudinal/transversal phononsFig.4 The momentum transfer of the neutrons Q points always into the direction of the real displacements ui.(a)longitudinal and(b)transversal oscillations.The cross section of phonon excitations contains the scalar product Qu with the polarization of the wave u.Thus,an os
8、cillation is only excited for Q with a component in the polarization direction.14How to measure phonons?In standard experiments,the scans are done at constant Q or constant energy transfer E.Most of the Brillouin zone is normally measured with constant-Q(Fig.(a),while for very stiff dispersion modes
9、,in the vicinity of the Brillouin zone center,constant-E is chosen(Fig.(b).15How to measure phonons?16Example:phonons in GeFig.5 Phonon dispersion curves for Ge along certain high symmetry axes in the Brillouin zone.The characters express the symmetry types and degeneracies of the lattice modes.Grou
10、p theory allows us to identify the high symmetry points where degeneracies occur,which modes stick together,which modes cross,and which modes show anti-crossings.17Examples:NaClFig.6(a)ThespacegroupofNaClis#225.Thefccunitcellcontains4primitiveunitcellswith4Naand4Claotms.(b)Therhombohedralprimitivece
11、llofthefcclatticewhichcontainsoneNaatomandoneClatom.18Examples:NaClThegroupofthewavevectoratk=0fortheNaClstructureisOh.ThepointgroupoperationsforOhis19Theoretical calculations of phononsB.Liu,et al.,Acta Phys.Sin.62,176104(2013)20Books and web sites21Phonons in CeAuAl3 B.Liu,et al.,Phys.Rev.B 98,174
12、306(2018)Fig.7(1)Crystal structure of CeAuAl3.(b)Brillouin zone of the body-centered tetragonal lattice of CeAuAl3 with ca.Fig.8PhonondispersionforCeAuAl3alonghighsymmetrylines-M-S0-.22Imaginary mode 2023Imaginary mode 20B.Liu,et al.,Phys.Chem.Chem.Phys.17,4089(2015)X.Wang,et al.,Phys.Chem.Chem.Phys
13、.16,26974(2014)Fig.9PhonondispersionforU2MoalongsymmetrylinesinthebodycenteredtetragonalBZ.Fig.10ThestructuresofU2Mo.TheredandgreencirclesareMoandUatoms,respectively.24Acoustic modes and macroscopic elasticityFig.11 Atomic displacements associated with a long-wavelength longitudinal acoustic mode pr
14、opagating along 100 in a cubic crystal.Consider a longitudinal acoustic mode in a cubic crystal with wave vector along 100.The magnitude of the wave vector is small but non-zero.Each(100)plane of atoms is displaced in the x direction by a constant amount relative to its neighbouring planes.Therefore
15、 the displacement ux of each plane is proportional to its position x.This corresponds to a uniform compressional strain of the crystal,e11=ux/x,which locally makes a cubic unit cell tetragonal.25Acoustic mode frequencies and the elastic constant tensorHow to calculate the slopes of the acoustic phon
16、on dispersion curves in the long-wavelength limit,where the acoustic modes give rise to strain distortions?Using the standard Newton equation of motion,one can obtain:It is common practice to use the Voigt notation,in which pairs of indices are replaced by single indices:26Example:cubic system27Exam
17、ple:cubic systemWe can comment on the stability of the crystal.If C44 is negative,the crystal is unstable against the shear given by one of the transverse acoustic modes with wave vectors in the a*-b*plane.If C11 basis,the CEF Hamiltonian HCEF can be given as37The INS spectra38The bound stateP.ermk,
18、A.Schneidewind,B.Liu,et al.,PNAS,doi/10.1073/pnas.1819664116Fig.15Keycharacteristicsoftheneutron-scatteringexcitationspectraofsingle-crystalCeAuAl3observedinreciprocalspacealong to M to S to at T=5 K.1.Basic principles2.Instruments introduction3.Phonons4.Crystal field excitations5.Spin waves:Magnons
19、Outline39Magnetic orderingMagnetic order normally exists below a critical temperature(Tc or TN)at which a phase transition takes place.For the spin ordered ground state,one can imagine different configurations,e.g.,all spins parallel(ferromagnetism),or antiparallel(antiferromagnetism),(anti-)ferrima
20、gnetic ordering or more complex like helix-form.4041Magnetic ordering42Magnetic ordering43Magnetic structure of MnWO4 Magnetic structure of MnWO4 at 3K(left),13 K(middle),and 10K(right).44Spin Waves in Ferromagnets45Spin Waves in FerromagnetsSpin Waves in Antiferromagnets46Finding the eigenenergies4
21、7Hwhere q and q are Bose operators.If such a transformation exist,the diagonal elements in the matrix will be the eigenergies of the system and therefore non-negative.Calculating the spin wave intensities48Thedifferentialscatteringcross-sectionformagneticscatteringisgivenas:49Calculating the spin wa
22、ve intensitiesdoesnotcontributetotheinelasticscatteringcross-sectioniforisequaltothepreferredspindirection(thez-direction).The spin-canting case:MnWO450Fig.17Themagneticsublattice(4a2b2c)ofAF1ofMnWO4.OnlymagneticMn2+sites(MnainpinkandMnbincyanblue)areplottedtodemonstratetwodifferentspin-cantingtextu
23、resclearlyseenwiththerespectivecantingangles.Fig.16Thenuclearstructureunitofhuebnerite(MnWO4)containstwoindependentMnsites,denotedMnaandMnb.B.Liu,et al.,J.Phys.:Condens.Matter 30,295401(2018).S.Park,B.Liu,et al.,J.Phys.:Condens.Matter 30,135802(2018).51The spin-canting case:MnWO422,1()()2i lilaicii
24、liiHJDD SSa Sc SB.Liu,et al.,J.Phys.:Condens.Matter 30,295401(2018).52The spin-canting case:MnWO453The spin-canting case:MnWO454The spin-canting case:MnWO455The spin-canting case:MnWO4InordertodiagonalizetheHamiltonianHq,itisneededtointroduceatransformationmatrixT,anda=T,so that where is a diagonal
25、matrix and its diagonal elements are eigenvalues of the system.The transformation matrix T is a 1616 matrix with columns that are eigenvectors to I1HqT=T,and it also must respect the Bose commutation rules.Once the correct transformation matrix T is obtained,the differential scattering cross section
26、s for magnetic scattering are calculated as follows,56The spin-canting case:MnWO4Fig.18SpinwavedispersionalongH,0.5,2Hdirectionthroughthemagneticpeak(0.25,0.5,0.5).Theredpointsareexperimentaldata.Fig.19SpinwavespectrumalongH,0.5,2Hdirectionthroughthemagneticpeak(0.25,0.5,0.5),withGaussianfunctioncon
27、voluted.ThecolorcodedenotestheINSintensity.57The spin-canting case:MnWO41.Self-developedprogram(Fortran,Matlab,)2.Density-FunctionalTheory3.SpinW58How to obtain the exchange-coupling constants?59DFT calculationsFig.20 Ordered spin arrangements in each/bc layer of Mn2+ions in the FM,A1,A2,A3,A4,A5,A6,A7,A8,A9 states of MnWO4.The up-spin and down-spin Mn2+sites are represented by filled and unfilled circles,respectively.C.Tian,et al.,Phys.Rev.B 80,104426(2009)60DFT calculationsThanksforyourattention!61The end
