Neutron scattering for magnetism Virginie Simonet virginie.simonet@neel.cnrs.fr Institut Néel, CNRS & Université Grenoble Alpes, Grenoble, France Fédération Française de Diffusion Neutronique The neutron as a probe of condensed matter Neutron-matter interaction processes Diffraction by a crystal: nuclear and magnetic structures Inelastic neutron scattering: magnetic excitations Use of Polarized neutrons Techniques for studying magnetic nano-objects Complementary muon spectroscopy technique Conclusion 1 ESM 2019, Brno
The neutron as a probe of condensed matter 2 ESM 2019, Brno
The neutron as a probe of condensed matter Subatomic particle discovered in 1932 by Chadwik First neutron scattering experiment in 1946 by Shull P ROPERTIES : m N = 1.675 10 -27 kg, s = 1/2, τ = 888 s E = ~ 2 k 2 λ = 2 π/k • Neutron: particle/plane wave with and 2 m N • Wavelength of the order of few Å (thermal neutrons) ≈ interatomic distances Interference diffraction condition 3 ESM 2019, Brno
The neutron as a probe of condensed matter Subatomic particle discovered in 1932 by Chadwik First neutron scattering experiment in 1946 by Shull P ROPERTIES : m N = 1.675 10 -27 kg, s = 1/2, τ = 888 s E = ~ 2 k 2 λ = 2 π/k • Neutron: particle/plane wave with and 2 m N • Wavelength of the order of few Å ≈ interatomic distances diffraction condition • Energies of thermal neutrons ≈ 25 meV ≈ energy of excitations in condensed matter • Neutral: probe volume, nuclear interaction with nuclei 4 ESM 2019, Brno
The neutron as a probe of condensed matter P ROPERTIES : • carries a spin ½ : sensitive to the magnetism of unpaired electrons (spin and orbit) Probe magnetic structures and dynamics ~ µ n = − γµ N ~ σ Possibility to polarize the neutron beam Scattering length ∝ Z X rays neutrons • Better than X rays for light or neighbor elements or isotopes (ex. H, D): complementary • Neutron needs big samples! 5 ESM 2019, Brno
The neutron as a probe of condensed matter ≠ TYPES OF NEUTRON SOURCES FOR RESEARCH : ILL • Neutron reactor (continuous flux) ex. Institut Laue Langevin in Grenoble ESS • Spallation sources (neutron pulses) ex. ISIS UK or ESS future European spallation source (Lund) Images ILL and ESS websites • Compact source projects (neutron pulses) Fission Spallation… 235 U U, W, Hg … Be, Li 6 ESM 2019, Brno
The neutron as a probe of condensed matter ≠ TYPES OF NEUTRON SOURCES FOR RESEARCH : Image ESS website 7 ESM 2019, Brno
The neutron as a probe of condensed matter V ARIOUS ENVIRONMENTS : Temperature: 30 mK-2000 K High magnetic steady fields up to 26 T Pulsed fields up to 40 T Pressure (gas, Paris-Edinburgh, clamp cells) up to 100 kbar Electric field CRYOPAD zero field chamber for polarization analysis Cryopad D23@ILL 15 T magnet 8 ESM 2019, Brno
The neutron as a probe of condensed matter U SE OF NEUTRON SCATTERING FOR MAGNETIC STUDIES : • Most powerful tool to determine complex magnetic structures (non-collinear, spirals, sine waves modulated, incommensurate, skyrmion lattice) • Complex phase diagrams (T, P, H, E) under extreme conditions • Magnetic excitations and hybrid excitations • In situ, in operando measurements • Magnetic domains probe • Short-range magnetism (ex. spin liquid/glass/ice) • Magnetic nano-structures/mesoscopic magnetism • Chirality determination Example: • Materials with hydrogen orthorhombic RMnO 3 Goto et al. PRL (2005) 9 ESM 2019, Brno
Neutron-matter interaction processes 10 ESM 2019, Brno
Neutron-matter interaction processes S CATTERING PROCESS : INTERFERENCE PHENOMENA Q = ~ ~ k i − ~ Momentum transfer=scattering vector k f Born approximation 2 θ sample ~ k f ~ k i plane waves plane waves Detector d Ω source 11 ESM 2019, Brno
Neutron-matter interaction processes S CATTERING PROCESS : INTERFERENCE PHENOMENA Q = ~ ~ k i − ~ Scattering vector k f Energy transfer: ~ ω = E i − E f = ~ 2 2 m ( k 2 i − k 2 f ) 2 θ sample Elastic scattering: ~ ω = 0 ~ k f | k i | = | k f | E i = E f Q = 2 sin θ/λ ~ k i plane waves plane waves Detector d Ω source 12 ESM 2019, Brno
Neutron-matter interaction processes S CATTERING PROCESS : INTERFERENCE PHENOMENA Q = ~ ~ k i − ~ Scattering vector k f Energy transfer: ~ ω = E i − E f = ~ 2 2 m ( k 2 i − k 2 f ) 2 θ sample Inelastic scattering: ~ ω 6 = 0 ~ k f E i < E f | k i | < | k f | ~ k i plane waves plane waves Detector d Ω source 13 ESM 2019, Brno
Neutron-matter interaction processes S CATTERING PROCESS : INTERFERENCE PHENOMENA Q = ~ ~ k i − ~ Scattering vector k f Energy transfer: ~ ω = E i − E f = ~ 2 2 m ( k 2 i − k 2 f ) 2 θ sample Inelastic scattering: ~ ω 6 = 0 ~ k f | k i | > | k f | E i > E f ~ k i plane waves plane waves Detector d Ω source 14 ESM 2019, Brno
Neutron-matter interaction processes S CATTERING PROCESS : INTERFERENCE PHENOMENA The cross-sections (in barns 10 -24 cm 2 ) = quantities measured during a scattering experiment: Total cross-section : number of neutrons scattered per second /flux of incident neutrons σ dσ Differential cross section : per solid angle element d Ω d 2 σ Partial differential cross section : per energy element d Ω dE 15 ESM 2019, Brno
Neutron-matter interaction processes F ERMI ’ S G OLDEN RULE Partial differential cross section d 2 σ d Ω dE = k f ( m N 2 π ~ 2 ) 2 X X p λ p σ i | ⇥ k f σ f λ f | V | k i σ i λ i ⇤ | 2 δ ( ~ ω + E − E 0 ) k i λ , σ i λ 0 , σ f Energy conservation Initial and final wave vector and spin state of the neutrons Initial and final state of the sample Interaction potential = Sum of nuclear and magnetic scattering 16 ESM 2019, Brno
Neutron-matter interaction processes Interaction potential = Sum of nuclear and magnetic scattering neutron Nuclear interaction potential Magnetic interaction potential � r very short range Longer range (e - cloud) Scatterer j isotropic Anisotropic ~ R j r ) = (2 ⇥ ~ 2 Dipolar interaction of the neutron r − ⇤ X V ( ⇤ ) b i δ ( ⇤ R i ) magnetic moments µ n with magnetic m N field from unpaired e - i µ n . ~ V ( ~ r ) = − ~ B ( ~ r ) b Scattering length depends on isotope and nuclear spin r − ⇥ r − ⇥ [rot( ⇥ µ ei × ( ⇥ R i ) ) − 2 µ B p i × ( ⇥ ⇥ R i ) r ) = µ 0 ⇥ X B ( ⇥ ] r − ⇥ r − ⇥ 4 π ~ R i | 3 R i | 3 | ⇥ | ⇥ i Orbital contribution Spin contribution 17 ESM 2019, Brno
Neutron-matter interaction processes d 2 σ d Ω dE = k f ( m N 2 π ~ 2 ) 2 X X p λ p σ i | ⇥ k f σ f λ f | V | k i σ i λ i ⇤ | 2 δ ( ~ ω + E − E 0 ) k i λ , σ i λ 0 , σ f Some algebra (hyp. no spin polarization) Z + ∞ d 2 σ 1 d Ω dE = k f j (0) A j 0 ( t ) e − i ~ Q ~ R j 0 (0) e i ~ Q ~ X R j ( t ) i e − i!t dt h A ∗ 2 π ~ k i −∞ jj 0 A j ( t ) with the scattering amplitude Scattering experiment FT of interaction potential 18 ESM 2019, Brno
Neutron-matter interaction processes Z + ∞ d 2 σ 1 d Ω dE = k f j 0 (0) A j ( t ) e − i ~ Q ~ R j 0 (0) e i ~ Q ~ X R j ( t ) i e − i!t dt h A ∗ 2 π ~ k i −∞ jj 0 neutron neutron � r � r nucleus j electron i ~ � pf j ( Q ) ~ M j ⊥ ( ~ b j R j R i Q, t ) Magnetic form factor p = 0.2696x10 -12 cm of the free ion X-rays neutrons 19 ESM 2019, Brno
Neutron-matter interaction processes Z + ∞ d 2 σ 1 d Ω dE = k f j 0 (0) A j ( t ) e − i ~ Q ~ R j 0 (0) e i ~ Q ~ X R j ( t ) i e − i!t dt h A ∗ 2 π ~ k i −∞ jj 0 neutron neutron � r � r nucleus j electron i ~ ~ pf j ( Q ) ~ M j ⊥ ( ~ b j R j R j Q, t ) Projection of the ⊥ ~ magnetic moment Q 20 ESM 2019, Brno
Neutron-matter interaction processes Z + ∞ d 2 σ 1 d Ω dE = k f j 0 (0) A j ( t ) e − i ~ Q ~ R j 0 (0) e i ~ Q ~ X R j ( t ) i e − i!t dt h A ∗ 2 π ~ k i −∞ jj 0 pf j ( Q ) ~ M j ⊥ ( ~ b j Q, t ) = Double FT in space and time of the pair correlation function of the ⊥ ~ nuclear density magnetic density Q 21 ESM 2019, Brno
Neutron-matter interaction processes Z + ∞ d 2 σ 1 d Ω dE = k f j 0 (0) A j ( t ) e − i ~ Q ~ R j 0 (0) e i ~ Q ~ X R j ( t ) i e − i!t dt h A ∗ 2 π ~ k i −∞ jj 0 Separation elastic/inelastic: Keeps only the time-independent terms in the cross-section and integrate over energy elastic scattering (resulting from static order) dσ j 0 A j e − i ~ Q ( ~ R j 0 − ~ X R j ) i h A ∗ d Ω = jj 0 22 ESM 2019, Brno
Diffraction by a crystal: nuclear and magnetic structures 23 ESM 2019, Brno
Diffraction by a crystal: nuclear and magnetic structures N UCLEAR DIFFRACTION crystal lattice � a + v n � R n = u n � b + w n � c Crystal = lattice + basis � a ∗ + k � b ∗ + l � c ∗ H = h � reciprocal lattice d σ R j ) > < b j b j 0 e − i � Q ( � R j 0 − � X d Ω = j,j 0 d Ω = (2 ⇥ ) 3 d ⇤ | F N ( ⌅ Q ) | 2 δ ( ⌅ Q − ⌅ X H ) V � H diffraction condition (lattice) Coherent elastic scattering from crystal Bragg peaks at nodes of reciprocal lattice 24 ESM 2019, Brno
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