Determination of the magnetic structure from powder neutron diffraction

Transcription

1 Determination of the magnetic structure from powder neutron diffraction Vladimir Pomjakushin Laboratory for Neutron Scattering, ETHZ and PSI A Hands-on Workshop on X-rays, Synchrotron Radiation and Neutron Diffraction Techniques June 8-, 008, Paul Scherrer Institut, Villigen, Switzerland Lecture notes:

2 Literature on (magnetic) neutron scattering Neutron scattering (general) S.W. Lovesey, Theory of Neutron Scattering from Condensed Matter, Oxford Univ. Press, 987. Volume for magnetic scattering. Definitive formal treatment G.L. Squires, Intro. to the Theory of Thermal Neutron Scattering, C.U.P., 978, Republished by Dover, 996. Simpler version of Lovesey. All you need to know about magnetic neutron diffraction. Symmetry, representation analysis Yu. A. Izyumov, V. E. Naish and R. P. Ozerov, Neutron diffraction of magnetic materials, New York [etc.]: Consultants Bureau, 99.

3 Overview of Lecture Principles of magnetic neutron scattering/ diffraction Types of magnetic structures Description of all possible magnetic structures. k- vector formalism for classifying the magnetic modes Real example of magnetic structure determination 3

4 Magnetic neutron scattering on an atom n ˆσ ˆσ µ n =γµ n k R e ŝ µ e = µ B ŝ ri [ µe R ] Magnetic field from an electron H(R) = rot R 3 + transl.part neutron-electron dipole interaction V (R) = γµ n ˆσH(R) averaging over neutron coordinates k V (R) k = γr e ˆσ q [q [ŝ ie iqr i q]] q = k k } ˆQ magnetic interaction operator ˆQ 4

5 Magnetic neutron scattering on an atom. The size magnetic scattering amplitude = γr e ˆQ neutron magnetic moment in μn -.9 classical electron radius γr e = cm= 5.4 fm( S) fm=fermi=0-3 cm, e mc x-ray scattering length: Zr e Comparison of neutron scattering lengths (fm) magnetic Mn 3+ (S=): -0.8, Cu + (S=½): -.65 nuclear Mn : -3.7, Cu: 7.7 5

6 magnetic scattering intensity can be larger than the nuclear one magnetic nuclear 6

7 Magnetic neutron scattering on an atom. q-dependence magnetic scattering amplitude = γr e ˆQ, ˆQ = ŝ i e iqr i = S Fourier image of the spin density in atom or magnetic form-factor drρ s (r)e iqr = Sf(q) q [q ˆQ q] i nuclear 0.8 f(q) Ti + Ni q, Å - q =4π sin θ λ 7

8 Magnetic neutron scattering on an atom 3. geometry magnetic scattering amplitude = γr e ˆQ Q = q Q q q = q/q =[ q S q ]f(q) ϕ S k Q = S sin(ϕ) Q q = k k k 8

9 Elastic scattering intensity Neutron scattering cross-section (for unpolarized neutron beam) dσ dω Q 9

10 Elastic scattering on a lattice of spins incoherent I S ˆ = S(S + ) coherent Bragg scattering I S F HKL incoherent magnetic scattering I S(S + )f (q) q, Å 0

11 Non-polarized neutron diffraction magnetic nuclear I ++ Q σ n + F σ n average over neutron polarization I (Q σ n )(Q σ n )+FF + σ n (F Q + F Q ) σn I Q + F 0 no magnetic/nuclear interference Magnetic and nuclear scattering are completely independent and can be treated as two independent phases in the Rietveld refinement

12 Interference between nuclear and magnetic scattering General note: When the magnetic unit cell is larger than the nuclear one (propagation vector k 0) the interference between nuclear and magnetic scattering is absent in any (un)polarized neutron diffraction experiment. Reason: Magnetic Bragg peaks appear at different positions in reciprocal space

13 Only amplitudes can be determined Spin/atom magnetic moment S = S 0 cos(πkz + φ),k = 3 4 Amplitude I S 0 + S 0 F cos(φ) φ =7π/8 φ = π/ The phase Φ is not accessible and the magnetic moments on the atoms cannot be determined. 3

14 powder diffraction, + and - + thanks to strong dependence of Q on S and q the powder experiment is in most cases sufficient + experimentally simple, full q-range is detected, no missed incommensurate satellites, better resolution + no difficult corrections, e.g. extinction, crystal shape - peak overlapping/multiplicity in powder data puts a restriction on the determination of spin direction - small spin components (~0 - μb) are difficult to detect - multi-k structures are difficult cases 4

15 Powder neutron diffractometers ILL, FR LLB, FR ISIS, UK FRM-II, DE D0, DB, DA G4, G4 GEM, HRPD, PEARL SPODI FLNP/Dubna, RU HRFD, DN, DN SINQ/PSI, CH DMC, HRPT, POLDI European Portal for Neutron Scattering 5

16 Powder ND at SINQ/PSI HRPT - High Resolution Powder DMC - cold neutron Diffractometer for Thermal Neutrons at SINQ f (q) Ni+ 0.4 Ti powder diffractometer

17 Examples of magnetic structures. Magnetic moment is a real quantity! 0th cell Amplitude is complex k=[0,0] FM Propagation vector k S(r j )= Re(S 0 e πir jk )= (S 0e +πir jk + c.c.) S 0 = S x e iφ x + S y e iφ y + S z e iφ z 0th cell k=[0,0] AFM S 0 = S x + S y S 0 = S x + S y S 0 = S 0

18 Examples of magnetic structures. Amplitude is complex k=[/,/] AFM Propagation vector k 0 Magnetic moment is a real quantity! S(r j )= (S 0e +πir jk + S 0e πir jk ) S 0 = S x e iφ x + S y e iφ y + S z e iφ z k=[0,0,kz] modulated (in)commensurate S 0 = S y S 0 = S x + S y e iπ helix = Sx + is y cycloidal spiral SDW S 0 = S x + is y + S z e iφ z

19 Example of complex magnetic structure Antiferromagnetic three sub-lattice ordering in Tb 4Au5 P6/m k-vector=[/3, /3, 0] Zeroth cell contains 4 spins of Tb 3+. Conventional magnetic unit cell contains 6 spins of Tb 3+.

20 Analysis of magnetic neutron diffraction: computer programs and tutorials/notes INDEXING, K-VECTOR: programs distributed with FullProf Suite [] SYMMETRY: BasIreps[], SARAh[], MODY[3] SOLUTION: FullProf [] (simulated annealing) REFINEMENT: FullProf, GSAS [4] Visualization: FPStudio [] REFERENCES.Juan Rodríguez-Carvajal (ILL) et al, S. Wills (UCL) magnetic_structures.html 3.Wieslawa Sikora et al, 4.Bob Von Dreele (ANL) et al, software/gsas.html 0

21 Description of magnetic structures Magnetic symmetry 65 3D magnetic Shubnikov (Sh) space groups. Derived from 30 space groups G and an additional element: spin inversion operator R. Sh groups contain additional antielements g =(g R), g G (except ) e.g. Pnnm R= time reversal changes S to -S

22 Description of magnetic structures Magnetic symmetry 65 3D magnetic Shubnikov (Sh) space groups. Derived from 30 space groups G and an additional element: spin inversion operator R. Sh groups contain additional antielements g =(g R), g G (except ) e.g. Pnnm y y my=yī my For example: CrCl space group: Pnnm Possible Sh groups derived from the parent space group are: Pnnm Pn nm, Pnnm, Pn n m, Pnn m, Pn n m No one describes CrCl magnetic structure Cr-atoms in (a)-position Cr-spins are antiparallel in 0th cell k=[0 / /] S = [v r] Disadvantages: Sh group is not necessarily made from the parent G. Thus, it is not an ultimate practical tool for obtaining all allowed spin configurations Do not describe modulated structures. No rotations on non-crystallographic angle - no helix. Linear orthogonal transformations preserve the spin size - no SDW One can still find less symmetric Sh group Magnetic symbol {Pnnm; (a) Sh 7 =PsĪ; S=(uvw), S=(-u-v-w)}

23 Description of magnetic structures Magnetic symmetry 65 3D magnetic Shubnikov (Sh) space groups. Derived from 30 space groups G and an additional element: spin inversion operator R. Sh groups contain additional antielements g =(g R), g G (except ) e.g. Pnnm y y Representation analysis A universal technique of finding all possible symmetry adapted spin configurations for the given space group G and the propagation vector k. my=yī my S = [v r] Disadvantages: Sh group is not necessarily made from the parent G. Thus, it is not an ultimate practical tool for obtaining all allowed spin configurations Do not describe modulated structures. No rotations on non-crystallographic angle - no helix. Linear orthogonal transformations preserve the spin size - no SDW

24 Case study. Antiferromagnetic order in orthorhombic TmMnO3 4

25 Step Experiment. q-range/resolution. 5

26 Patterns,.9Å HRPT and 4.5Å DMC TmMnO 3 50K TmMnO 3, K 0 4 Neutron counts HRPT λ= Neutron counts HRPT, λ= Θ, deg Θ, (deg).0 TmMnO 3, 35K 3.0 TmMnO 3, K 0 4 Neutron counts DMC, λ= Neutron counts DMC, λ= Θ, (deg) Θ, (deg) 6

27 cf. resolution/q-range HRPT.9Å magnetic contribution DMC range at 4.5Å 7

28 Cf. resolution/q-range excellent resolution DMC, 4.5Å 8

29 Step Finding the propagation vector of magnetic structure (k-vector). Le Bail profile matching fit. 9

30 T-dependence of Bragg peak positions 44.0 int0p50v_p int0p50v_p int0p50onevo_p 43.5 theta (deg) Only one peak for <=3K T (K)

31 Refining the propagation k-vector from profile matching fit In the example we determine incommensurate structure k x (r.l.u.) k=[kx,0,0] T (K) 3

32 Step 3 Symmetry analysis. Classifying possible magnetic structures 3

33 Classifying possible magnetic structures k-vector group Group G: Pnma, no.6: 8 symmetry operators Little group Gk, k=[0.45,0,0]=[q,0,0] Little group of propagation vector Gk contains only the elements of G that do not change k rotation+ translation E x m y m z

34 Classifying possible magnetic structures Magnetic representation group element rotation+ translation E Mn-position position number g g g3 g x element g changes atomic position: a b b a c d d c element g is represented by 4x4 matrix 00 m y Permutation representation m y a b c d 000 a 000 b000 b 000 c = b0 d b a d c b = e πi(ka p) e 0.9πi in addition, element g sometimes moves the atom outside of the zerocell. We have to return the atom back with -ap: -ap a b (000) b a (-00) c d (000) d c (-00) 0 S(r j )=S 0 e πir jk 34

35 Classifying possible magnetic structures Magnetic representation group element rotation+ translation E Mn-position position number 4x4 matrices (P) element g is represented by 3x3 matrix g g g3 g x m y a b c d Permutation representation Axial vector (spin) representation For instance: rotational part of element g: R(g) changes atomic spin direction: R(g) det(r) 000 b b0 00 S x 0 0 S y = 00 S z S x S y S z 00 m y b0 000 b000 35

36 Classifying possible magnetic structures Magnetic representation group element rotation+ translation E g g g3 g x m y m y Mn-position position number a b c d Permutation representation 4x4 matrices (P) 3x3 matrices (A) R(g) det(r) Axial vector (spin) representation

37 Classifying possible magnetic structures Magnetic representation group element Mn-position position number spin 4x4 matrices (P) 3x3 matrices (A) R(g) det(r) direct (tensor) product P A x matrices g g g3 g4 a b c d S S S3 S4 Permutation representation Axial vector (spin) representation e.g. for group element g Magnetic representation 000 b b = b b b b Vector spaces a b c d S x S y S z 37

38 Classifying possible magnetic structures Reducing magnetic representation group element rotation+ translation E g g g3 g x m y m y E.g. for the element g M=P A Matrix of magnetic representation acts on dimensional vector b b b b Magnetic representation is reducible! 4x3= spin components s x s y s z s x s y s z s x3 s y3 s z3 s x4 s y4 s z4 38

39 Classifying possible magnetic structures Reducing magnetic representation group element g g g3 g4 Mn-position Magnetic representation is reducible to a block-diagonal shape that is a direct sum of irreducible square matrices (dimensions can be from to 6.) τ τ τ 3... = τ τ τ Each of these matrices acts only on a subspace of the spin components. Sτ, Sτ,... are vectors with dimension of matrix S τ S τ S τ3.. each componet of Sτ is a linear combinations of spin components s x s y s z s x s y s z s x3 s y3 s z3 s x4 s y4 s z4 39

40 Landau theory of phase transitions says that only one irreducible representation is needed to describe the structure Why the Landau theory does work for magnetic phase transition is a separate topic. 40

41 Classifying possible magnetic structures basis vectors/functions Sτ, Sτ, Sτ3,... Pnma, k=[0.45,0,0] Mn in (4a)-position Magnetic representation is reduced to four one-dimensional irreps 3τ 3τ 3τ 3 3τ4 g g g 3 g 4 τ a a τ a a τ 3 a a τ 4 a a a = e πik x Mn-position 3 4 S τ3 = +e x a e x e 3x + a e 4x S τ3 = +e y + a e y +e 3y + a e 4y S τ3 = +e z + a e z e 3z a e 4z Assuming that the phase transition goes according to one irreducible representation τ3 the spins of all four atoms are set only by 3 variables instead of! C S τ3 + C S τ3 + C 3 S τ3 4

42 Steps 3-4 in practice Solving/refining the magnetic structure by using one irreducible representation. construct basis functions for single irreducible representation irrep (use BasIreps, SARAh, MODY). plug them in the FULLPROF and try to fit the data. In difficult cases the Monte-Carlo simulated annealing search is required 3. If the fit is bad go to and choose different irrep. If the fit is good it is still better to sort out all irreps. 4

43 Refinement of the data for τ3 S(r) = (C S τ3 + C S τ3 + C 3 S τ3)e πikr + c.c. S τ3 = +e x a e x e 3x + a e 4x S τ3 = +e y + a e y +e 3y + a e.0 4y S τ3 = +e z + a e z e 3z a e 4z 0 4 Neutron counts k=[0.45,0,0] TmMnO 3, 35K DMC, λ=4.5 at T=35K C=.()μB, C=0, C3=0.67() e iφ μb φ can be fixed to any value. Experiment data are insensitive to φ Θ, (deg) 43

44 Visualization of the magnetic a cycloid structure propagating along x-direction structure S(r) =Re [(C S τ3 + C 3 exp(iϕ)s τ3) exp(πikr)] S τ3 = +e x a e x e 3x + a e 4x S τ3 = +e z + a e z e 3z a e 4z Propagation of the spin, e.g. for atom no. S (x) =C cos(kx)e x + C 3 cos(kx + ϕ)e z k=[0.46,0,0]

45 Visualization of the magnetic structure: xz-projection for arbitrary φ: both direction and size of S are changed Propagation of the spin, e.g. for atom no. S (x) =C cos(kx)e x + C 3 cos(kx + ϕ)e z 4 z 4 x k=[0.46,0,0] 45

46 Visualization of the magnetic structure: xz-projection for φ=0: only the size of S are changed Propagation of the spin, e.g. for atom no. S (x) =(C e x + C 3 e z ) cos(kx) 4 z 4 x k=[0.46,0,0] 46

47 literature, programs and tutorials/notes All you need to know about magnetic neutron diffraction. Symmetry, representation analysis Yu. A. Izyumov, V. E. Naish and R. P. Ozerov, Neutron diffraction of magnetic materials, New York [etc.]: Consultants Bureau, 99. COMPUTER PROGRAMS, TUTORIALS.Juan Rodríguez-Carvajal (ILL) et al, S. Wills (UCL) magnetic_structures/magnetic_structures.html 3.Wieslawa Sikora et al, This lecture 47