Neutron Powder Diffraction: a powerful technique for studying structural and magnetic phase transitions

Transcription

1 Neutron Powder Diffraction: a powerful technique for studying structural and magnetic phase transitions A-site ordered perovskite YBaMn 2 O 6 : atomic versus larger scales charge ordering Juan Rodríguez-Carvajal Laboratoire Léon Brillouin (CEA-CNRS), Saclay, France

2 Content of the talk Generalities about the properties and use of neutrons for condensed matter research Neutron powder diffraction. Data treatment. Introduction to the programs of the FullProf Suite Study of the phase transitions in YBaMn 2 O 6. Charge/orbital/spin ordering.

3 Neutrons as constituents of matter Chadwick 1932

4 Properties of neutrons

5 Neutron production: nuclear reactors The reactor geometry is optimized to produce the maximum number of neutrons. The fission of a 235 U nucleus produce in average 2.5 neutrons. 1.5 are used to keep the chain reaction, only 1 is used for making neutron beams. Main research reactors in Europe: France, ILL (57 MW) in Grenoble Orpheé (14 MW) in Saclay Germany FRJ-2 (23 MW) in Julich (closed) BER-2 (10 MW) in Berlin FRM-II (24 MW) in Munich (new)

6 Neutron production: spallation sources H + accelerated to 1 GeV Targets of W, Pb, Hg, U 20 to 25 neutrons by H + Pulses of 50 Hz Flux of ns -1 (ISIS) Pulse length of µs Relatively small average flux Switzerland, SINQ quasi-continuous ( 10 MW) United Kingdom, ISIS (1.5 MW) SNS, Oak Ridge (USA) in construction

7 Neutron reactors: ILL

8 Neutron reactors: ILL

9 Particle-wave properties kinetic energy (E) velocity (v) temperature (T). E= m n v 2 /2= k B T = p 2 /2m n =(ħk) 2 /2m n =(h/λ) 2 /2m n momentum (p) ħ=h/2π p= m n v = ħ k wavevector (k) k= 2π/λ = m n v/ħ wavelength (λ)

10 Particle-wave properties (moderators) E = m n v 2 /2 = k B T = (ħk) 2 /2m n ; k = 2π/λ = m n v/ħ Neutrons moderated by heavy water at 300K (Thermal neutrons) Also cold moderating source, liquid deuterium at 25K (Cold neutrons) And hot moderating source, graphite at ~2000K

11 Particle-wave properties (Energy-Temperature-Wavelength) E = m n v 2 /2 = k B T = (ħk) 2 /2m n ; k = 2π/λ = m n v/ħ Energy (mev) Temp (K) Wavelength (Å) Cold Thermal Hot Cold Sources intensity [a.u.] COLD THERMAL HOT velocity [km/s]

12 Interaction neutron-nucleus Φ= number of incident neutrons /cm 2 / second k φ dω r ds θ k Direction θ, φ z σ= total number of neutrons scattered/second/φ Fermi s golden rule gives the neutron-scattering Cross-section number of neutrons of a given energy scattered per second in a given solid angle q Target (the effective area presented by a nucleus to an incident neutron) 2 d σ dω de' = k' k m 2πh 2 2 λ, σ p λ p σ λ', σ ' r k' σ' λ' V r kσλ 2 δ ( hω + E E ) λ λ'

13 Interaction neutron-nucleus Weak interaction with matter aids interpretation of scattering data The range of nuclear force (~ 1fm) is much less than neutron wavelength so that scattering is point-like k Plane wave e ik x Plane wave e ik r k Sample V(r) r k Detector 2θ Spherical wave (b/r)e ik r k Q Fermi Pseudo potential of a nucleus in r j V j = 2πh2 m b jδ(r r j ) Potential with only one parameter

14 Neutron scattering lengths

15 X-rays and neutron scattering lengths Neutron scattering is a nuclear interaction X-Rays H D N Mn Fe Neutrons H D N Mn Fe negative negative

16 Elastic Scattering: Diffraction For elastic scattering: 3 dσ N(2 π) = dω v τ coh 0 2 δ( Q ur r τ) FN ( Q ur Where v ) 0 is the unit cell volume and τ a reciprocal lattice vector The coherent elastic scattering takes place only for r r r τ = k k' = Q= 2ksin( θ) r r r r k k' = Q= τ Bragg Law λ = 2d sin θ

17 Form Factor

18 Magnetic Scattering k I =2π/λu I k F =2π/λ u F Q= k F - k I Dipolar interaction (µ n, m): vector scattering amplitude 1 a ( Q) ( Q) M = reγ f m 2 Q m Q ( ) Q 2

19 Magnetic Scattering - Magnetic interactions are long range and non-central Nuclear and magnetic scattering have similar magnitudes Magnetic scattering involves a form factor: Fourier Transform of unpaired electron spatial distribution Magnetic scattering depends only on the component of m perpendicular to Q Q=Q e m Only the perpendicular component of m to Q contributes to scattering m

20 Content of the talk Generalities about the properties and use of neutrons for condensed matter research Neutron powder diffraction. Data treatment. Introduction to the programs of the FullProf Suite Study of the phase transitions in YBaMn 2 O 6. Charge/orbital/spin ordering.

21 Diffractometers: Powder high flux D1B(ILL) or G41(LLB)

22 Experimental powder pattern A powder diffraction pattern can be recorded in numerical form for a discrete set of scattering angles, times of flight or energies. We will refer to this scattering variable as : T. The experimental powder diffraction pattern is usually given as three arrays : { T, y, σ } = 1,2,... i i i i n The profile can be modelled using the calculated counts: y ci at the ith step by summing the contribution from neighbouring Bragg reflections plus the background.

23 Powder diffraction profile: scattering variable T: 2θ, TOF, Energy Bragg position T h y i zero y i -y ci Position i : T i

24 The calculated profile of powder diffraction patterns y = I Ω( T T ) + b h h ci i i {h} I ( ) = I β h h I Ω=Ω( x, β ) hi P Contains structural information: atom positions, magnetic moments, etc Contains micro-structural information: instr. resolution, defects, crystallite size b i i ( ) = b β B Background: noise, incoherent scattering diffuse scattering,...

25 The calculated profile of powder diffraction patterns y = I Ω( T T ) + b h h ci i i {h} The symbol {h} means that the sum is extended only to those reflections contributing to the channel i. This should be taken into account (resolution function of the diffractometer and sample broadening) before doing the actual calculation of the profile intensity. This is the reason why some Rietveld programs are run in two steps

26 Several phases (φ = 1,n φ ) contributing to the diffraction pattern y = s I Ω( T T ) + b ci φ φ, h i φ, h i φ { φh} Several phases (φ = 1,n φ ) contributing to several (p=1,n p ) diffraction patterns = Ω + p p p p ( ) p ci φ φ, h i φ, h i φ { φh} y s I T T b

27 I h y = I Ω( T T ) + b h h = ci i i {h} { 2 } S LpOACF Integrated intensities are proportional to the square of the structure factor F. The factors are: Scale Factor (S), Lorentz-polarization (Lp), preferred orientation (O), absorption (A), other corrections (C)... h

28 The Structure Factor contains the structural parameters (isotropic case) n { } ( ) = ( ) 2π { } F h O f h T exp i h S t r s j= 1 r j j j j j j j j s = ( x, y, z ) ( j = 1, 2,... n) T j sin θ = exp( B ) j λ 2 2

29 Structural Parameters (simplest case) r j = j j j ( x, y, z ) O j B j Atom positions (up to 3n parameters) Occupation factors (up to n-1 parameters) Isotropic displacement (temperature) factors (up to n parameters)

30 Structural Parameters (complex cases) As in the simplest case plus additional (or alternative) parameters: Anisotropic temperature (displacement) factors Anharmonic temperature factors Special form-factors (Symmetry adapted spherical harmonics ), TLS for rigid molecules, etc. Magnetic moments, coefficients of Fourier components of magnetic moments, basis functions, etc.

31 The Structure Factor in complex cases n { } ( ) = ( ) ( ) 2π { } F h O f h T g h exp i h S t r s h j= 1 j j j j s j s h h T = k = S k s = 1, 2,... N l l ( ) s s G s ( ) h g j s Complex form factor of object j Anisotropic DPs Anharmonic DPs

32 The approximation of the peak shape profile function and microstructural effects Precise refinements can be done with confidence only if the intrinsic and instrumental peak shapes are properly approximated. At present The approximation of the intrinsic profile is mostly based in the Voigt (or pseudo-voigt) function The approximation of the instrumental profile is also based in the Voigt function for constant wavelength instruments For TOF the instrumental+intrinsic profile is approximated by the convolution of a Voigt function with back-to-back exponentials or with the Ikeda- Carpenter function.

33 The peak shape function of powder diffraction patterns contains the Profile Parameters Ω ( x, β ) =Ω( T T, β ) hi P i h P Ω ( x) = g( x) f ( x) = instrumental intrinsic profile + Ω ( xdx ) = 1 In most cases the observed peak shape is approximated by a linear combination of Voigt (or pseudo-voigt) functions Ω( x) L( x) G( x) = V( x)

34 The Voigt function V( x) = Lx ( ) Gx ( ) = Lx ( ugudu ) ( ) + V x V x H H V x β β ( ) = (,, ) = (,, ) L G L G The pseudo-voigt function ( ) = ( ) + (1 ) ( ) pv x ηl x η G x pv ( x) = pv ( x, η, H )

35 The Rietveld Method The Rietveld Method consist of refining a crystal (and/or magnetic) structure by minimising the weighted squared difference between the observed and the calculated pattern against the parameter vector: β χ 2 n = w y y i= 1 { ( β) } 2 i i ci wi = 1 σ 2 i 2 σ i : is the variance of the "observation" y i

36 Least squares: Gauss-Newton (1) Minimum necessary condition: A Taylor expansion of around allows the application of an iterative process. The shifts to be applied to the parameters at each cycle for improving χ 2 are obtained by solving a linear system of equations (normal equations) A kl = 2 χ β = 0 yic( β ) β0 w A δ β 0 b yic ( β0) yic ( β0) β β i i k l yic( β0) b = w( y y ) β = k i i ic i k

37 Least squares: Gauss-Newton (2) The shifts of the parameters obtained by solving the normal equations are added to the starting parameters giving rise to a new set β1 = β0 + δ β The new parameters are considered as the starting ones in the next cycle and the process is repeated until a convergence criterion is satisfied. The variances of the adjusted parameters are calculated by the expression: σ β χ χ ( k) = ( A ) kk ν 2 ν = 0 2 χ N-P+C

38 Neutron and synchrotron powder diffraction Neutrons Constant scattering length. Contrast. Low absorption: easy sample environment Magnetic structures High precision in structure refinement Moderate resolution Synchrotron X-rays Extremely high resolution Subtle distortions Indexing and Structure determination Anomalous scattering Texture effects

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42 Directory structure of the FullProf Suite

43 WinPLOTR: program to access the whole FullProf Suite

44 Configuration of WinPLOTR File: winplotr.set

45 Indexing demo with WinPLOTR New facility: DICVOL04 Two other indexing programs: TREOR, ITO

46 Other features of WinPLOTR Two other indexing programs: TREOR, ITO Access to other programs: EdPCR, Fp_Studio, DICVOL04, BasIreps

47 A program for analysis of diffraction patterns: FullProf A program for : Simulation of powder diffraction patterns Pattern decomposition integrated intensities Structure refinement Powder and single crystal data Crystal and magnetic structures Multiple data sets: simultaneous treatment of several powder diffraction patterns (CW X-rays & neutrons, Energy dispersive X-rays, TOF neutron diffraction) Combined treatment of single crystal and powder data Crystal and magnetic Structure determination capabilities: simulated annealing on integrated intensity data

48 How works FullProf Minimal input: Input control file (extension.pcr ): PCR-file Model, crystallographic/magnetic information PCR file DAT file(s) FullProf Output files, Plot diffr. patterns Eventually, experimental data

49 The PCR file: steep learning curve DAT file(s) PCR file Format depending on the instrument, usually simple Many variables and options Complex to handle Hint: copy an existing (working) PCR-file and modify it for the user case, or... USE the new GUI: EdPCR

50 A new GUI for FullProf: EdPCR GUI using Winteracter:

51 Content of the talk Generalities about the properties and use of neutrons for condensed matter research Neutron powder diffraction. Data treatment. Introduction to the programs of the FullProf Suite Study of the phase transitions in YBaMn 2 O 6. Charge/orbital/spin ordering.

52 Example: Study of A-site ordered manganite YBaMn 2 O 6 Introduction to the half-doped (Mn 3+ /Mn 4+ ) manganites Phase transitions, crystal and magnetic structures of mixed valence YBaMn 2 O 6

53 Colossal Magnetoresistance Regain of interest in electronic properties of (semi)-conducting magnetic oxides Our team started few years ago to answer some important questions! Charge ordering: In the charge ordered state, is the charge localization at atomic level? What does the conventional ionic states Mn 3+ /Mn 4+ mean? CDW, charge disproportionation, polarons, Orbital ordering: Is that concept really relevant for explaining the electronic/structural transitions in manganites? Is there any difference with respect to a MO 6 distorting structural transition?

54 Electronic transitions and phase separation: Role of the chemistry and the disorder. Homogeneous/inhomogeneous states: what is a phase? Role of the defects and microstrains Magnetic exchange interactions: Are the Goodenough-Kanamori-Anderson rules still applicable? How to explain the canted magnetic structures? Are there other magnetic structures compatible with the observed magnetic powder diffraction patterns? Is the CE structure really collinear?

55 Nature of the structural transition at T CO in half-doped manganites R 1/2 D 1/2 [Mn 3.5+ ]O 3 Our previous work : A new interpretation of the electronic localization in charge ordered manganites The single Mn-site in the HT phase breaks up into two sublattices in the COlow temperature phase, having very similar characteristics concerning the average Mn-O distances. The observed distortions are different of what is expected from the generally accepted picture of a concomitant charge (Mn 3+ -Mn 4+ ) and orbital ordering. Formation of molecular ferromagnetic Mn-O-Mn pairs stabilised by local DE and a suitable structural distortion: Zener polaron. O 2 O 1 e - Mn 2 Mn 1 O 3 O 2 O 1 A. Daoud-Aladine et al., Phys. Rev. Lett. 89, (2002). O 4 q O 4

56 Magnetic Susceptibility: two kinds of paramagnetic states The temperature dependence electrical resistivity, and inverse magnetic susceptibility of the Bi 1/2 Sr 1/2 MnO 3 crystal. The slope of 1/χ in the charge-ordered state for K closely approaches the value for Zener pairs with total spin S=7/2 ( µ exp =7.83 µ B, µ calc =7.94 µ B ). J. Hejtmánek et al., J. Appl. Phys. 93, 7370 (2003)

57 Our previous strategy to study the structural distortions in half-doped manganites: Minimise the size mismatch between atoms in A sites of perovskites AMnO 3 (Pr, Ca) In common manganites the A-site doping induces disorder and local inhomogeneities (fluctuations of chemical composition) Role of the A-site disorder Using Ba and a heavy rare earth, or Y, a new ordered family of half-doped manganites is obtained Y 1/2 Ba 1/2 MnO 3 YBaMn 2 O 6 A first report on this compound was published by: T. Nakajima et al. J. Phys. Chem. Solids 63 (2002) 913

58 Approximate phase diagram of R 1/2 Ba 1/2 MnO 3 Random Potential Effect near the Bicritical Region in Perovskite Manganites as Revealed by Comparison with the Ordered Perovskite Analogs D. Akahoshi, M. Uchida, Y. Tomioka,1 T. Arima, Y. Matsui and Y. Tokura, Physical Review Letters 90, (2003)

59 Phase transitions, crystal and magnetic structures of mixed valence YBaMn 2 O 6

60 The Y 3+ and the Ba 2+ ions may be ordered or disordered depending on the synthesis conditions 773K (O 2 ) YBaMn 2 O K (purified Ar) Y 2 O 3 +BaCO 3 +MnCO 3 YBaMn 2 O K (air) Y 1/2 Ba 1/2 MnO 3

61 YBaMn 2 O 6 : High temperature phases Nakajima et al. J. Phys. Soc. Jpn. 71, (2002) At high temperature, 3 transitions: 2 of them were reported by Nakajima et al. (T 1 & T 2 ) New transition at T 3

62 Phase transitions in YBaMn 2 O 6 : resistivity vs T T K T CO 480 K (?) T K T K T TM 780 K (?)

63 X-ray diffraction versus temperature (112) (200) (004) (212) (114) (220) (204) Region D Region C Region B Region A Radiation: pure CuKα 1 spectral line (λ=1.5406å)

64 (110) (102) T=793K T=750K T=730K T=720K T=535K T=530K T=520K T=510K T=500K T=458K T=438K T=398K

65 T=793K T=750K T=730K T=720K T=535K T=530K T=520K T=510K T=500K T=458K T=438K T=398K (200) (004)

66 Phase transitions in YBaMn 2 O 6 as seen by X-ray diffraction d Mn-O (Å) <Mn-O 1a > Mn-O 1b Mn-O 2 Mn-O 1 P4/mmm a ~ 3.90Å c ~ 7.78Å The picture provided by X-rays is wrong! T Paramètres (Å) a/ 2 b/2 2 b a Mn-O 3 c/ beta( ) P2/m a ~ 3.92Å b ~ 3.85Å c ~ 7.74Å β ~90.3 T c/ T(K) Intermediate phase? P2/m?

67 Neutron diffraction versus temperature (D20, high resolution mode at ILL) q q

68 YBaMn 2 O 6 high temperature phase ILL, D20 (λ =1.37 Å) YBaMn 2 O 6 : 815K P 4/nbm a=5.51å c=7.78å (χ 2 =2.99 R B =2.40) Wavy background due to SiO 2 tube A sigle site for Mn cations <Mn-O>=1.955(1)Å Valence Sum=3.49(1)

69 YBaMn 2 O 6 : Intermediate phase ILL, D20 (λ =1.37Å et 1.88 Å) YBaMn 2 O 6 : 570K P2 1 /m (or C2/m) a=7.85å b=7.69å c=7.73å β=90.3 (χ 2 =2.69, R B =3.03) <Mn 2 -O>=1.962(7) Å <Mn 1 -O> =1.953 (7) Å Valence Sum=3.49(7) Valence Sum=3.58(7)

70 YBaMn 2 O 6 :Room temperature phase Nakajima et al. : Monoclinic space group P2 (a p 2 a p 2 2a p ) with two non-equivalent MnO 6 octahedra showing a marked volume difference Conclusion of the authors: This phase is characterised by a CO with a checkerboard pattern in the ab-plane. Williams et al. for TbBaMn 2 O 6 (neutron diffraction pattern) The space group P2 1 /m initially used (a p 2 a p 2 2a p ), does not take into account few superstructure reflections, with a propagation vector q=(0,1/2,0). A good fit to the profiles was obtained by transforming the structure to a b- doubled cell with triclinic P-1 symmetry. Scarce number of observed superstructure peaks : constrained parameter refinements. strain-relieving displacements in the TbO layer orbital ordering in the MnO 2 layers Conclusion of the authors: The obtained structure is characterised by a rocksalt three-dimensional Mn 3+ /Mn 4+ charge ordering.

71 Combined XR-N refinement of the crystal structure of YBaMn 2 O 6 3T2 observed vs. calculated pattern P-1 a ~ 5.55Å α ~90 b ~ 11.0Å β ~90.3 c ~ 7.60Å γ ~90

72 Combined refinement RX - Neutrons (3T2) Model for TbBaMn 2 O 6 of J. Williams and J. P. Attfield, Phys. Rev. B R (2002) (P-1 with constraints) YBaMn 2 O 6 (298K) The superstructure reflections cannot be accounted with the published model

73 Combined refinement RX - Neutrons (3T2) (P-1 without the published constraints) YBaMn 2 O 6 (298K)

74 Structure of YBaMn 2 O 6 obtained from the combined RX-N refinement in P-1 without constraints Mn1a Mn1b Mn2a Mn2b <Mn-O> : 1.940(8) <Mn-O>: 1.957(8) <Mn-O>: 1.979(8) <Mn-O>: 1.970(8) A comparison of the Mn-O distances within each octahedron shows that the four non-equivalent MnO 6 can be associated in pairs New refinement in which we applied constraints for some coordinates and restraints for the Mn-O distances

75 Structure of YBaMn 2 O 6 obtained from the combined RX-N refinement in P-1 with constraints/restraints Mn1a Mn1b Mn2a Mn2b <Mn-O> : 1.947(3) <Mn-O>: 1.949(2) <Mn-O>: 1.975(2) <Mn-O>: 1.974(3) : 17 x10-4 : 17 x10-4 : 20 x10-4 : 20 x10-4 BVS : 3.62(3) BVS: 3.61(2) BVS: 3.45(3) BVS: 3.46(3)

76 First hypothesis: YBaMn 2 O 6 with alternating Y/Ba layers giving rise to a rocksalt charge order Mn 1 =Mn 3.6+ Mn 2 =Mn Small separation with respect to average Mn 3.5+ charge. d z2 orbitals associated with the longest Mn-O bonds.

77 4.0Å 3.9Å 4.0Å Second Hypothesis: formation of Zener polarons within the ab-plane Representation showing the possible formation of Mn-Mn pairing Elongation of the Mn-O bonds

78 Third hypothesis: Zener polarons along c <Mn-O-Mn> ~172 <Mn-O-Mn> ~155 <Mn-O-Mn> ~ Å 3.67Å 3.93Å Representation showing the possible formation of Mn-Mn pairing Widest Mn-O-Mn angle in the structure

79 Comparison of low-q range neutron diffraction patterns of half-doped manganites (½,0,½) (½,1,½) (½,2,½) (-½,0,3/2) (½,1,3/2) YBaMn 2 O 6 La 1/2 Ca 1/2 MnO 3 (½,2,1) (3/2,1,1) Y 1/2 Ca 1/2 MnO 3 Pr 1/2 Ca 1/2 MnO 3 Nd 1/2 Ca 1/2 MnO 3 (½,0,1) (½,1,1) (½,3,1) Tb 1/2 Ca 1/2 MnO 3

80 Magnetic scattering of YBaMn 2 O 6 The magnetic propagation vector is different to disordered manganites: k=(½ 0 ½ ) Instead of k=(½ 0 0 )

81 Inverse of the magnetic susceptibility of YBaMn 2 O 6 µ (4) = 13.33µ eff θ= 427Κ µ (4) = 7.97µ eff B θ=335κ B Spin-only effective moment for Z=4 Mn ions: µ Mn 4+ = 7.94µ B µ µ ( Mn 4+, Mn 3+ ) = 8.81 B AF Charge Ordered Paramagnetic Charge ordered phase Paramagnetic Charge disordered Spin-only effective moment for Z=4 Zener Polarons: µ 2Mn ZP= 11.23µ B µ 4Mn ZP= 14.97µ B

82 Constraints: Collinear model: CE with different stacking along c (three degrees of freedom) Alternative non-collinear model: A single amplitude of the magnetic moment for all Mn atoms Two angular variables within the ab-plane µ(mn1a)= µ(mn1b); µ(mn2a)= µ(mn2b) Free rotation of the single axis within the ab-plane

83 YBaMn 2 O 6 Y 1/2 Ca 1/2 MnO 3 a b Ionic ordering picture Mn 3+ Mn 4+ AF? AF Mn 3+ Mn 4+ c d ZP ordering picture 2-Mn ZP 4-Mn ZP

84 Ionic model ZP model Comparison of two simple models (three degrees of freedom)

85 Propagation vector (doubling c) k=(1/2,0,1/2) ( ) Zener polarons involving more than 2 Mn atoms? Susceptibility data supports the formation of ZP involving more than 2 Mn atoms

86 Magnetic Structure of YBaMn 2 O 6

87 Perspectives of research in CO transitions An image of non-atomic localisation, conserving the intermediate valence of Mn-ions, is more appropriate and explain most of the observed experimental facts for formally Mn 3+ /Mn 4+ perovskites. Extension: confinement of de-localised electron in ferromagnetic supraatomic units,. CO in manganites Order of Zener polarons? Implication for orbital ordering pseudo-molecular orbitals Implication for super-exchange theory extension of GKA rules Single crystal x-rays and neutron diffraction to obtain accurate structures Polarized neutron scattering (3D-polarimetry): magnetic structures Electron density studies Electronic structure calculations

88 Access to the LLB neutron beams

89 Access to the LLB neutron beams

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93 The end FullProf Suite and related programs: ftp://ftp.cea.fr/pub/llb/divers/ Set of directories with different programs, documents, tutorials and examples of powder diffraction data analysis Downloading of Software Graphical tutorial run-through of most of this software is located via ( look before you try ):