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Magnetic Neutron Diffraction the basic formulas

Magnetic Neutron Diffraction the basic formulas. A.Daoud-aladine, (ISIS-RAL). What’s a magnetic structure? Why study magnetic order? Example : Manganites The magnetic structure factor Using the k-vector Formalism (as required by the Fullprof input)

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Magnetic Neutron Diffraction the basic formulas

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  1. Magnetic Neutron Diffractionthe basic formulas A.Daoud-aladine, (ISIS-RAL)

  2. What’s a magnetic structure? Why study magnetic order?Example : Manganites The magnetic structure factorUsing thek-vector Formalism(as required by the Fullprof input) (Alternative to Shubnikov symmetry, which are TABULATED crystallographic magnetic space groups : this is however restricted to commensurate structures) Pitfalls

  3. core What’s a magnetic structure? (atomic) magnetic moments (m) arise from quantum effects in atoms/ions with unpaired electrons Ni2+ Intra-atomic electron correlation Hund’s rule(maximum total S) + Spin-orbit + CEF « Classical description» « Quantum description» m = gJJ (J=L+S 4f-rare earths) m = gSS (3d-transition metals)

  4. What’s a magnetic structure? Magnet: crystal containing magnetic atoms kT >> Jij, Jij paramagnetic (disordered) state Temperature (entropy) overcomes magnetic energy: Entropy essentially dominated by local magnetic moment fluctuations

  5. Jij What’s a magnetic structure? Magnet: crystal contaning magnetic atoms kT < Jij, Exemple here: Jij>0 Antiferromagnetic coupling (AF) Magnetic energy overcomes the entropy :  Quasi-static configuration of magnetic moment with small fluctuations that are made cooperative by the magnetic exchange (spin waves excitations )

  6. Why study magnetic order? • Fundamental properties of condensed matter. Exchange • interactions related to the electronic structure. • The first (necessary) step before determining the exchange interactions (generally, with inelastic neutron scattering) • Permanent magnet industry. Chemical substitutions • controlling single ion anisotropy, strength of effective • interactions, canting angles, etc: NdFeB materials, • SmCo5, hexaferrites, spinel ferrites. • Spin electronics, thin films and mutilayers

  7. Dipolar interaction term with one electron core core Total Si si => vector scattering amplitude for one atom Magnetic scattering of a single atom From Introduction to the Theory of Thermal Neutron Scattering, G SQUIRES – Cambridge University Press (1978) Q= kF - kI r R kI=2/uI kF=2/uF ri

  8. Example: RMnO3 manganites LaMnO3 TN=150K Ideal cubic Perovskite TbMnO3 TN=41K O Mn Pnma La/Tb Site A m [010]O [001]C [101]O [010]C n Mn-O-Mn m n a a

  9. Example: RMnO3 manganites LaMnO3 TN=150K AF Commensurate structure T. Goto, et al. Phys. Rev. Lett. 92, 257201 (2004)

  10. LaMnO3 : 50K and 150 K Example: RMnO3 manganites LaMnO3 TN=150K AF Commensurate structure

  11. O Jahn-Teller Distorsion dx2-y2 dxy dz2 dyz dxz Example: RMnO3 manganites LaMnO3 TN=150K Electronic structure of Mn3+ (x=0) Hund JH Crystal field x2 eg 3d4 AF S=3/2 Commensurate structure t2g (e- localized) • scattering amplitude for one atom, tabulated f(Q)

  12. Example: RMnO3 manganites LaMnO3 TN=150K AF Commensurate structure T. Goto, et al. Phys. Rev. Lett. 92, 257201 (2004)

  13. Example: RMnO3 manganites LaMnO3 TN=150K TbMnO3 TN2<T<TN1=41K Sinusoidal T<TN2=23K Cycloid AF Commensurate structure Incommensurate structures

  14. What’s a magnetic structure? Why study magnetic order?Example : Manganites The magnetic structure factor Pitfalls

  15. core core Magnetic scattering of a single atom From Introduction to the Theory of Thermal Neutron Scattering, G SQUIRES – Cambridge University Press (1978) Q= kF - kI r R kI=2/uI kF=2/uF ri Dipolar interaction term with one electron Total Si si => vector scattering amplitude for one atom

  16. Magnetic scattering from a magnet Magnetisation density: M(r)=MS(r)+ML(r) kI=2/uI kF=2/uF n Q= kF - kI • Magnetic interaction vector for a crystal • Magnetic structure factor => vector scattering amplitude for one atom In the magnetic cell

  17. Only the perpendicular component of M to Q=2h contributes to scattering Magnetic structure factorsand interaction vectors Q=Q e M(Q) q q M(Q)xQ M(Q) • Magnetic interaction vector for a crystal • Magnetic structure factor => vector scattering amplitude for one atom In the magnetic cell

  18. Q=(hmagkmaglmag) b1mag a1mag = 2a1 a2mag = a2 a3mag = a3 b1mag = b1/2 Propagation vectors : Structure factor in crystal cell Reciprocal k-space Crystal+AF mag structure b2 k b1 R(101) a3 a1mag Nuclear reflections a2 a1 H=h.b1+k. b2+l. b3

  19. Propagation vectors : Structure factor in crystal cell In the magnetic cell Crystal+AF mag structure Q=(hmagkmaglmag) Reciprocal space Atom 1 Real space Atom 2 R(101) a3 (1) (2) a1mag a2 a1 the AF arrangement =>… hmag necessarily odd (type, 2h+1)

  20. Propagation vectors and Structure factor in crystal cell Reciprocal space Crystal+AF mag structure b2 Q b1 R(101) b1mag a3 (1) (2) a1mag Nuclear reflections a2 a1 Magnetic reflections a1mag = 2a1 a2mag = a2 a3mag = a3 Q=hmag.b1mag+k. b2+l. b3 b1mag = b1/2 the AF arrangement =>… hmag necessarily odd (type, 2h+1)

  21. k=(½00) H « Propagation vector » Q= H+ k Propagation vectors and Structure factor in crystal cell Reciprocal space Crystal+AF mag structure b2 Q b1 R(101) b1mag a3 (1) (2) a1mag Nuclear reflections a2 a1 Magnetic reflections a1mag = 2a1 a2mag = a2 a3mag = a3 Q=(2h+1).b1/2+k. b2+l. b3 b1mag = b1/2 Q=h.b1+k. b2+l.b3 + b1/2

  22. Propagation vectors and Structure factor in crystal cell Reciprocal space Crystal+AF mag structure k=(½00) b2 Q H b1 R(101) b1mag a3 (1) (2) a1mag Nuclear reflections a2 a1 a1 Magnetic reflections m[n]i = Si. (-1)n1 Q=(2h+1).b1/2+k. b2+l. b3 Q=h.b1+k. b2+l.b3 + b1/2 because k.Rn=n1/2 Q= H+ k

  23. Propagation vectors and Structure factor in crystal cell Reciprocal space Crystal+AF mag structure k=(½00) b2 Q H b1 R(101) b1mag a3 (1) (2) a1mag a2 a1 m[n]i = Si. (-1)n1 because k.Rn=n1/2

  24. Propagation vectors and Structure factor in crystal cell : case 1 Reciprocal space Magnetic structure k=(½00) -k b2 Q In our example (AF): - k equivalent to –k (2k is a reciprocal lattice vector, or k at the border of the brillouin zone) - Only k sufficient to index all the magnetic reflections, - And Skj must be real H b1 b1mag

  25. Propagation vectors and Structure factor in crystal cell : case 2 Reciprocal space Magnetic structure k b2 -k H In general k NOT equivalent to –k, so a set of {k}=k,-k is needed - Skj are complex vectors b1 - necessary condition for real mnj : Formalism Required to describe Incommensurate structures

  26. Magnetic scattering and structure factors For non-polarised neutrons Nuclear Phase: Magnetic Phase: Scattering vector h=H Arrangement of the moments Atomic positions Structural model Structure Factor/ Intensity

  27. What’s a magnetic structure? Why study magnetic order?Example : Manganites The magnetic structure factor Pitfalls

  28. Pitfalls : k=(000) Reciprocal space Magnetic structure k=(000) b2 When k=(000) (primitive P): - mni=Skj whatever n: this only means that the cell of the the magnetic structure is the same as that of the nuclear structure (not necessarily ferromagnetic, if more than 2 atoms/cell) b1 Nuclear reflections Magnetic reflections h= H

  29. LaMnO3 : 50K and 150 K Pitfalls : k=(000) LaMnO3 TN=150K Orthorhombic Pnma mni=Skj k=(000) Sk1= -Sk2= -Sk3= Sk4, all reals vectors 3 2 4 1 AF Commensurate AF structure

  30. Pitfalls : Centred cells Reciprocal space Magnetic structure (110) (-110) b2 b1p b2p I/F-centred lattice (2D): b1 a2 a2p a1p a1 Nuclear reflections

  31. Pitfalls : Centred cells Reciprocal space Magnetic structure (110) (-110) b2 b1p b2p I/F-centred lattice (2D): b1 a2 a2p a1p k=(000) a1 Nuclear reflections Magnetic reflections h= H

  32. k=(100) kp=(½½0) Magnetic reflections h= H+ k Pitfalls : Centred cells Reciprocal space Magnetic structure (110) (-110) b2 b1p b2p AF order on a centred lattice b1 a2 a2p a1p a1 Nuclear reflections

  33. Pitfalls : Centred cells Reciprocal space Magnetic structure (110) (-110) b2 b1p b2p • I-centred lattice : • - k equivalent to –k • (2k is a reciprocal lattice vector) • This is because {Rn} contains type (½½½)… translations, • - k vectors, which can have integer components, have in fact non-integer components in the primitive cell b1 k=(100) kp=(½½0) Nuclear reflections Magnetic reflections h= H+ k

  34. Magnetic scattering and structure factors For non-polarised neutrons Nuclear Phase: Magnetic Phase: Scattering vector h=H Arrangement of the moments Atomic positions Structural model Structure Factor/ Intensity

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