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) (Alternative to Shubnikov symmetry, which are TABULATED crystallographic magnetic space groups : this is however restricted to commensurate structures) Pitfalls

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 core « Classical description» « Quantum description» m = gJ J (J=L+S 4f-rare earths) m = gS S (3d-transition metals)

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

What’s a magnetic structure? Magnet: crystal contaning magnetic atoms kT < Jij, 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 )

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

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 core kF=2/ uF ri Dipolar interaction term with one electron core Total Si si => vector scattering amplitude for one atom

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

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

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

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

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

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

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

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 core kF=2/ uF ri Dipolar interaction term with one electron Total Si si core => vector scattering amplitude for one atom

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

Magnetic structure factors and interaction vectors Q=Q e M(Q) Only the perpendicular component of M to Q=2h contributes to scattering 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

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

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

Propagation vectors and Structure factor in crystal cell Reciprocal space Crystal+AF mag structure b2 Q b1 b1mag R(101) a3 (1) (2) Nuclear reflections a1mag 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)

Propagation vectors and Structure factor in crystal cell Reciprocal space Crystal+AF mag structure Q= H + k k=(½00) « Propagation vector » H b2 Q b1 b1mag R(101) a3 (1) (2) Nuclear reflections a1mag 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

Propagation vectors and Structure factor in crystal cell Reciprocal space Crystal+AF mag structure k=(½00) b2 Q H b1 b1mag R(101) a3 (1) (2) Nuclear reflections a1mag 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

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

Propagation vectors and Structure factor in crystal cell : case 1 Reciprocal space Magnetic structure k=(½00) -k b2 Q H 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 b1 b1mag

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

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

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

Pitfalls : k=(000) Reciprocal space Magnetic structure 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

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

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

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

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

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

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