Measuring Spin-Waves Bella Lake Helmholtz Zentrum Berlin, Germany Berlin Technical University, Germany B.Oliver Lake Pieper,; Oxford, Kolloquium, Sept 2019 28. Mai 2010 Outline • Conventional Magnets • long-range magnetic order, spin-wave excitations • Inelastic Magnetic Neutron Scattering Cross-Section • Measuring spin-waves • Triple-axis and time-of-flight spectrometers • Example B. Lake; Oxford, Sept 2019 Techniques for measuring excitations B. Lake; Oxford, Sept 2019 3 Conventional Magnets B. Lake; Oxford, Sept 2019 4 Conventional Magnetism - Exchange Interactions Heisenberg interactions HJ= ∑ nm, SS n. m nm, J < 0 ferromagnetic 3D magnet J > 0 antiferromagnetic 1D magnet |J1|=|J2|, J3=J4=0 |J1|=|J2|=|J3|=|J4| J1 J2 J1 J2 e.g. RbMnF3 e.g. KCuF3 J3 J3 J3 J3 J3 J1 J2 J1 J2 1D alternating 2D magnet magnet J J J J J ≠ |J1|=|J2|=|J3|, J4=0 3 3 3 3 3 |J1| |J2|, J3=J4=0 e.g. La2CuO4 J1 J2 J1 J2 e.g. CuGeO3 and and CFTD CuWO4 Anisotropic interactions B. Lake; Oxford, Sept 2019 Conventional Magnetism - Ordered Ground State Exchange interactions between magnetic ions often lead to long-range order in the ground state. ferromagnet antiferromagnet spin glass spiral magnet helical magnet B. Lake; Oxford, Sept 2019 Conventional Magnet - Long-range magnetic order Real Space T >TN T <TN • Long-range magnetic order on cooling as thermal fluctuations weaken J1 J1 J1 J1 J0 J0 Reciprocal Space (0,2,0.5) (0,2,1.5) (0,2,0) b* ) 40 2 T = 50 K • Magnetic Bragg peaks c* N 30 (0,1,0.5) (0,1,1.5) appear below the (0,1,0) 20 Nuclear Bragg 10 transition temperatures (~<S> Intensity peaks 0 and grow as a function of (0,0,0.5) (0,0,1.5) Magnetic Bragg peaks 0 20 40 60 80 (0,0,0) (0,0,1) (0,0,2) temperature Temperature (K) B. Lake; Oxford, Sept 2019 Magnetic Excitations – Spin-Waves Real Space Collective motion of spins about an ordered ground state Reciprocal Space Well-defined dispersion in energy and wavevector B. Lake; Oxford, Sept 2019 Spin-Wave Theory The Hamiltonian assuming Heisenberg exchange interactions J(b) J(2a) J(a) HJ=−−∑ (l l ') SSl . l' l, l' l‘ b l a Assumption of fully aligned ground state Excitations are fluctuations about this ground state Aim to diagonalize the Hamiltonian, find eigenstates and eigenvalues. The Hamiltonian is put through a series of transformations 1.Ladder operators S+, S-, Sz 2.Holstein-Primakoff operators, acting on spin deviations 3.Fourier transform of Holstein-Primakoff operators B. Lake4.Bogliubov; Oxford, Sept transformation 2019 for antiferromagnets and complex magnets Spin-Wave Theory http://spinw.org/ S. Toth and B. Lake, By Sandor Toth J. Phys. Condens. Matter 27, 166002 (2014) • Spin-waves are characterised by quantum spin number S=1. • Spin-waves have a well-defined energy as a function of wavevector • For several magetic ions per unit cell it is necessary to define several sublattices • The number of spin-wave branches equals the number, n, of magnetic ions, 1 acoustic branch and (n-1) optic branches. • Spin-wave theory can also describe helical structures, in which case a rotating coordinate frame can be used. • Single-ion and exchange anisotropies can also be included. • Spin-wave models are used to extract value of the exchange interactions B. Lake; Oxford, Sept 2019 Spin-Wave Theory Triangular lattice antiferromagnet with easy plane anisotropy z 2 HJ= ∑∑nm, SS n. m+ D n( S n) nm, n B. Lake; Oxford, Sept 2019 11 Inelastic Magnetic Neutron Scattering Cross-Section B. Lake; Oxford, Sept 2019 12 Basic Properties of the Neutron • The neutron has spin angular momentum Sn=1/2 • And magnetic moment µn= γµN; γ=-1.913; µN=eћ/mp; -1 m • Momentum is p=mnv, and is p=ћk (k units Å ) vk= ; kv= n mn • Its de Broglie wavelength λ (=2π/k) (units Å) 22ππ λ = = k mvn • kinetic energy E (meV where 1eV=1.6x10-19) 22 1 2 k E= mvn = 22mn Values of ν, λ, k and E are all related B. Lake; Oxford, Sept 2019 Scattered Neutrons – Differential Neutron Cross-section Neutrons scattered at kf Ef angle 2θ from direct beam Incident ki, Ei 2θ neutrons sample • Neutrons are scattered by the sample, the scattered pattern is a function of 2θ characteristic of the sample. • During scattering the neutron energy is either unchanged or it gains or loses energy to the sample. o The atom can recoil during the collision with the neutron in which case the neutron loses energy and the sample gains energy (eg a spin-wave). o Alternatively if the spins are already moving e.g. a spin-wave, it gives this energy to the neutron, the neutron gains energy and the sample loses energy. Elastic neutron scattering is when the neutron energy is unchanged. Ei=Ef Inelastic scattering is when the neutron gains or loses energy, Ei ≠Ef B. Lake; Oxford, Sept 2019 Scattering triangles – Elastic Scattering • The total energy and momentum are conserved. The total energy lost by the neutron (ћω) equals the energy gained by the sample. 112 2 1 22 2 • Energy conservation gives Eif−= E mv i − mv f =( kif − k ) =ω 22 2m • Momentum conservation gives ћQ= ћ(ki-kf). where ћQ is the sample momentum • Q is known as the scattering vector Q = ki-kf • For elastic scattering the modulus of the wavevectors are equal |ki | = |kf| (although they point in different directions) • The angle 2θ is known as the scattering angle Elastic scattering |ki|=|kf|=k scattered neutrons kf Incident Wavevector kf neutrons Q transfer 2θ |Q|/2 2θ θ ki ki ki Q Q / 2 4πθ sin sinθθ= ⇒=Qk2 sin = B. Lake; Oxford,k Sept 2019 λ Scattering triangles – Inelastic scattering • Conservation of energy and momentum 112 2 1 22 2 Eif−= E mv i − mv f =( kif − k ) =ω Qkk=if − 22 2m • For elastic scattering the modulus of the wavevectors are not equal |ki | ≠ |kf| • Inelastic Scattering triangles Neutron gains energy Neutron loses energy An excitation is destroyed An excitation is created Q kf Q kf 2θ 2θ ki ki B. Lake; Oxford, Sept 2019 Differential Neutron Scattering Cross-Section d 2σ = number of neutrons scattered per second dΩ dE into solid angle dΩ and dE / Φ dΩ dE 2 dm2σ k 2 =f ppk sρ Vk s ρδ E−+ E ω 2 ∑∑λiis f f f ii i ( ρρi f ) dΩ dE ki 2π ρρ,,ss ii f f Energy Probability of Matrix element for conservation 2 being in moving from initial 22 E=ω =( kkif − ) the initial state to final state 2m V - the magnetic interaction between neutron and electrons The electrons in an atom possess spin and orbital angular momentum, both of which give rise to an effective magnetic field. The neutrons interact with this field because they possess a spin moment The interaction between a neutron at point R away from an electron with momentum l and spin s is ˆˆ −µ γµ2 µ sRjj××1 lR jj V =−µ .B = 0 NB σ.curl + magnetic n π ∑ 22 4 j RR 2π Vbnuclear = ∑ jδ (rr− j ) m j B. Lake; Oxford, Sept 2019 The Magnetic Cross-section Inelastic cross section for spin only scattering by ions 2 2 d σ (γ r ) k 2 =0 f −−δωˆˆ αβ ' F(QQ) exp 2W∑( αβ, QQ α β) S ( , ) dΩ dE2π ki αβ, Dynamical structure factor ∞ Sαβ (Q,ωω) = expiQr . - r Sα(0) S β ( t) exp( i t) dt ∑∑ ( ( ij)) ∫ rij r rrij −∞ F(Q) Magnetic form factor which reduces intensity with increasing wavevector exp<-2W> Debye-Waller factor which reduces intensity with increasing temperature ˆˆ polarisation factor which ensures only components of spin (δαβ, − QQ α β) perpendicular to Q are observed Sαβ0 St rrij ( ) ( ) is the spin-spin correlation function which describes how two B. Lake; Oxford, Sept spins 2019 separated in distance and time a related Distinguishing Phonons and Magnons with Neutrons Wavevector-dependence • Phonon excitations have high intensity at large |Q| and when Q is parallel to the mode of vibration • Magnetic excitations have high intensity at low |Q| and when Q is perpendicular to the magnetic moment direction Temperature dependence • Phonon excitations become stronger as temperature increases • Magnetic excitations become weaker as temperature increases B. Lake; Oxford, Sept 2019 Measuring Spin-Waves B. Lake; Oxford, Sept 2019 20 Instruments for Measuring Inelastic Scattering Inelastic neutron scattering -both the initial and final neutron energy must be known Triple-axis spectrometer The initial and final neutron energies can be selected or measured using monochromator and analyser crystals where the wavelength of the neutrons is determined by the scattering angle. Time-of-flight Spectrometer. The initial and final energies are selected or measured using the time it takes the neutron to travel through spectrometer to the detector from this the velocity and hence kinetic energy are deduced. B. Lake; Oxford, Sept 2019 The Triple Axis Spectrometer - Layout kf 2θ ki Q B. Lake; Oxford, Sept 2019 The Triple Axis Spectrometer – V2/FLEX, HZB monochromator sample table detector analyser B. Lake; Oxford, Sept 2019 The Triple Axis Spectrometer – Monochromator Analyser The monochromator is a crystalline material and selects Vertically focusing a single wavelength from the white neutron beam of the monochromator reactor/spallation source by Bragg scattering where the scattering angle is chosen to select λ. The analyser measures the final neutron energy Selected 2d sinθ= nλ neutrons n=1,2,3…. from graphite, Copper, Germanium, blades can Incident white Diffracted neutron beam Diffracted be focused neutron beam Neutrons to scattered by the Neutrons to sample sample the detector Horizontally focusing Ki Kf analyser θ θ θ d spacing d spacing Transmitted Transmitted B.